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dl 2 = dx 2 + dy 2 + dz 2 = 

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dx 

i=1,3 

2 i 

x ′ i = ci + 

−1ØÔ××ÖÙÒ×Ý×ØÑÖÓØÙÒÙØÚ 

bijxj 

bijbkj = δik 

j 

j 

| b |= ±Ä×|b|= 

dx ′ 

i = bijdxj 

j 

dV ′ = 

dx 

i=1,3 

′ i = ∂(x′ 1, x ′ 2, x ′ 

3) 

dxi = 1 × 

∂(x1, x2, x3) 

i=1,3 

 

dxi = dV 

i=1,3 

|

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λÔÖÓÙÖØÐÖÓØ 

B |= 1 

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k 

Ai − 

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dxi 

∂xi i 

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= ∂P 

∂x j 

∂x j 

∂x ′ i

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= ÒÜdxi 

dP ∂P 

i ∂x = ′ i 

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· = gji 

∂xi ∂xj v 

2 

v 

v 

v v x 1 

1 

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1 

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a ′ i ∂x 

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P 

µÚÖ 

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(t, r) = (t, x, y, 

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δs ′ 

2 2 2 

= c γ t − v 

c2x ′ 1 = (γ(t − vx/c 2 ), γ(x − vt), 0, 0) 

2 − γ 2 (x − vt) 2 = δs 2

x’ = 

y’ = 

z’ 

= 

γ ( x − v t ) = γ ( x − β c t ) 

y 

z 

t’ γ ( t − v x / c 2 = ) = γ ( t − β x / c ) 

ou β = v / c < 1 et γ = ( 1 − β 2 ) −1/2 0’ 

x’ 

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x 

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⎞ 

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⎜ 

g = ⎜ −1 ⎟ 

⎝ −1 ⎠ 

−1 

, gµν = g µν 

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x 0 ≡ ct, x 1 ≡ x, x 2 ≡ y, x 3 ×ØÒÚÖ× ÊÑÖÕÙÓÒ×ÕÙ×ÐÓÒÙØÐ×ÐÑØÖÕÙ 

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⎛ 

⎞ 

γ −γβ 0 0 

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Λ = ⎜ −γβ γ 0 0 ⎟ 

⎝ 0 0 1 0 ⎠ 

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y 

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Λ α β = gβρg ασ Λ ρ σ 

δτ 

δ t 

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t 

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d d d d d d 

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1×ØÐÐØØÓÒÙØÑÔ×Ç 

− vt1) = 0 

Ø ÔÔÖ×ÕÙ×ÙØÒØÔÖÓÜ×Ò×ÓÑÔØÖÐ×ÓÙÚÖ×ËÆÛ Ö×ÙÐØØÓÒÙØÙÔÖÓÜ×ÙÑÙÜÒÓÒÚØÓÑÔØ ÇÒÔÖÓÓÒÒÐÓÙÔÓÙÖÑÓÒØÖÖÐÓÒØÖØÓÒ×ÐÓÒÙÙÖ× 

= ′×ØÔÐÙ×ÐÓÒÕÙt ÓÒt1 

ÐØØÓÒÙØÑÔ×ÍÒÑ×ÙÖÕÙÒØØØÚÔÒÓÑÒ×ØÐÝÔÖÓÙØ ÐÜÔÖÒ ØÓÒÒÔÝ×ÕÙ×ÔÖØÙÐ××ÔÖÙÚ×ÜÔÖÑÒØÐ×ÖØ×Ð 

ÒØ×Ò×ÙÒÒÒÙ×ØÓÑÑØÖÄ×ÔÓÒ×Ö×× ØÙÖÝÖÓÑÒØÕÙÙÑÙÓÒ ×ÒØÖÒØÒÚÓÐÒÓÒÒÒØ×ÑÙÓÒ×π→µνÄ×ÑÙÓÒ××ÔÐÒØ×ÙÖ ÚÓÖÙÖ× Ò×ØØÜÔÖÒ×ÔÓÒ×Ö××ÓÒØ ÝÒØÓÒÙØÐØÖÑÒØÓÒÙ 

×ÓÖØ××ØÐ×Ø××ÒØÖÒØ×ÐÓÒÐ×Ñµ→eννÄ×ÐØÖÓÒ× 

ÐÒÒÙ Ñ××ÓÒØØØ×ÔÖ×ÓÑÔØÙÖ××ÒØÐÐØÓÒ×ÔÓ×××ÙÖÐÔÓÙÖØÓÙÖ 

ÙÒÔÒØÙØÙÜÓÑÔØÓÑÔØÐÚγτµµ× ÉÐÔÖÓÜ×ÙÑÙÜ×ØÐÒ×ÚÖ ÄÑÔÙÐ×ÓÒ×µ×Ø Ä×ÑÙÓÒ×ÙÖÔÓ×ÓÒØÙÒØÑÔ×Úτµµ×ÒÚÓÐÓÒÓ×ÖÚ Î ÕÙÓÒÒÙÒγÑ 

ÙÑÙÓÒ ÇÒÖÔÖÒÖÐ×Ù××ÓÒØØÜÔÖÒÐÓÖ×ÕÙÓÒÔÖÐÖ×ÔÖÓÔÖØ× 

ÄÒÓØÓÒÚØ×××ØÒÖÐ×ÔÖÐÒØÖÓÙØÓÒÙÕÙÖÚØÙÖ ÉÙÖÚØÙÖÒÖÑÔÙÐ×ÓÒ 

ÔÖØÙÐ×ØÙÖÔÓ× 

τ×ØÐØÑÔ×ÔÖÓÔÖÐÔÖØÙÐ×ØÖÐØÑÔ×Ò×Ð×Ý×ØÑÓÐ 

ÄÅÖÖÌÑÒØ×ÔØÖÚÐÐÖÍÒÚÓÈÒÒ×ÝÐÚÒÈÖ×× 

 

ÈÖØË ÂÐÝØÐÆÙÐÈÝ× ÂÅÖÐÝØÈ××ÓÒÒÙÊÚÆÙÐ 

x ′ 

1 = 0Ñ×Òx1 = 

x ′ 

t ′ 

′ 

γt1 

t0 

= t ′ 

0 

= 0 

1 = γ(t1 − vx1/c 2 ) = γ(t1 − v 2 t1/c 2 ) = t1/γ 

u µ = dxµ 

 

dt 1 dx 

≡ c , 

dτ dτ c dτ

Ð×ÑÒØ×ÔÓÐÖ×ØÐ× ÑÒØÔÓÐÖÒÓÖÑ Ð×ÙÑÙÓÒ×ØØÑÖ×ØÙ×ØÒ×ÐÒØÖÖ×ÑÒØ× ÒÒÙ×ØÓÑÙÓÒ×ÚÙ×ÑØÕÙ××Ù×Ú ÓÑÔØÙÖ×ÔÖØÙÐ×ÚÙÒÓÙÔÙÒ ÚÙÒÓÙÔÐÑÖÚÓÖÙÐ 

 

ÖÔÓ×ÓÒÓØÒØÐÕÙÖÚØÙÖÒÖÑÔÙÐ×ÓÒ ÒÑÙÐØÔÐÒØÔÖÑÐÑ××ÐÔÖØÙÐÒÒ××ÓÒ×Ý×ØÑ 

ÇÒdt 

 

= ÇÒÚÓØÕÙÙ2 

p µ Ú ≡ (γmc, γmv) = (E/c, p) ÓÒp 2×ØÙÒÒÚÖÒØÕÙ×ØØØÒÙÖÐ×ØÐÒÖÑ×× ÐÔÖØÙÐÒ××ÓÒ×Ý×ØÑÙÖÔÓ× ÄÒÖÒØÕÙÙÒÔÖØÙÐ×ØÒÔÖ 

 

dτ 

ÉÑÓÒØÖÖÕÙT = 

dx dt 

= γØdx 

= 

dτ dt dτ = vγÓÒ 2×ØÙÒÒÚÖÒØ 

u 

c 

p 

T = E − mc 2 

mv 2 /2ÕÙÒv≪c 

µ 

≡ c γ, γ v 

c 

2 = c 2 m 2

Ð× ÖÔÔÓÖØÐÐÐ×ÐÓÒÐ×××Ð×ÙØÖ×ÓÙÖ××ÖÔÔÓÖØÒØ×ÐÐ× ÌÙÜÓÑÔØÒÓÒØÓÒÙØÑÔ×ÄÓÙÖ×ÙÔÖÙÖ× 

ÍÒÜÑÔÐÜÔÖ××ÓÒÓÚÖÒØ×ØÐÐÕÙÓÒÓØÒØÒÖÚÒØÐÕÙ ÓÚÖÒÒÐØÖÓÝÒÑÕÙ ØÓÒÓÒØÒÙØdρ ∂µj µ ÓÐÓÙÖÒØÚÖÑÑÓÒÔÙØÖÓÖÑÙÐÖÐ×ÕÙØÓÒ×ÅÜÛÐÐ ÁÐÙØÒÖÐ×ÖÕÙÐÓÚÖÒÐÕÙØÓÒÓÒØÒÙØÜÔÖÑÙÒ× ÓÒÓÚÖÒØÇÒÒØÐØÒ×ÙÖÙÑÔÑ ØÙØÓÒÔÝ×ÕÙÔÖ×ØÓÙØÓ×ÖÚØÙÖÓØÓ×ÖÚÖÙÒÙÑÙÐØÓÒ ÔÖØÓÒÐÒÖÓØ 

= 0 ÓÐÓÒÔÓ×j 

ÇÒÔÙØÒ×ÓÑÒÖÐ×ÙÜÕÅÜÛÐÐÔÓÙÖÚ(E)Ørot(B)Ò 

c jν 

 

ÜÔÖÓÙÚÖÕÙÐÒØ×ÝÑØÖF µν 

dt 

0×ÓÙ×ÐÓÖÑ 

+ div(j) = 

= cρ 

0 

F µν ⎛ 

⎞ 

0 −Ex −Ey −Ez 

⎜ 

= ⎜ Ex 0 −Bz By ⎟ 

⎝ Ey Bz 0 −Bx ⎠ 

Ez −By Bx 0 

∂µF µν = 4π 

= −F νµÑÔÐÕÙ

Ä×ÙÜÕÅÜÛÐÐÓÑÓÒ×ÓÙÐÒØÙØÓÑØÕÙÑÒØ ÙÒÓ× 

ËÐÓÒÓÑÒ×ÔÓØÒØÐ× ÕÙÒÓÙ×ÚÓÒ×ÒØÖÓÙØÐ×ÔÓØÒØÐ×ÚØÙÖ(A)Ø×ÐÖ(V ) 

A µ ÕÙÔÖÑØÖÖÖÐÕÓÒØÖÓÙÚ 

ÇÖÓÒÔÙØØÙÖÙÒÖÒØÓÒÙÔÓØÒØÐÕÙÓÒÔÔÐÐÙÒØÖÒ× 

≡ (V, A) 

ÓÖÑØÓÒÙÐ×ØÔÓ××ÐÖÐØÖÒ×ÓÖÑØÓÒ 

A µ → A µ + ∂ µ ÒÒÔ×Ð×ÖÔØÓÒÔÝ×ÕÙ ÇÒÑÔÓ×ÓÒÐÓÒØÖÒØÙÄÓÖÒØÞ 

f(t, x) 

∂µA µ ÕÙÓÒÒ 

= 0 

∂µ∂ ν = 4π 

c jν 

ÇÒÓØÒØÒ××ÜÔÖ××ÓÒ×ÓÑÔØ×Ò×Ð×ÕÙÐÐ×ÐÓÚÖÒ×Ø ÜÔÐØÈÖÜÑÔÐÐÒÚÖÒÙÔÖÓÙØF µν ÉÕÙÚÙØÔÖÓÙØ 

Fµν×ØÙØÓÑØÕÙ 

ÔÓÒÒØÖÐØÚ×ØÊÉÅÚÖØ×ØÙÖÒÓÙÙÖÇÒÑÖØÖÖÐ× ÄØÖÒ×ØÓÒÐÑÒÕÙÕÙÒØÕÙÒÓÒÖÐØÚ×ØÉÅ×ÓÒÓÖÖ× Ä×ÕÙØÓÒ×ÃÐÒÓÖÓÒØÖ 

ÔÖÒÔ×× ÜÔÖÑÔÖÙÒÓÒØÓÒØØψÕÙ×ØÓÒØÓÒÙÒÖØÒÒÓÑÖ Ò×ÐÑÒÕÙÕÙÒØÕÙÒÓÒÖÐØÚ×ØÙÒØØ×ØÑØÑØÕÙÑÒØ Ä×ÕÓÑÓÒ×ÔÙÚÒØ×ÓØÒÖ∂µF µν ÎÓÖÖØ ÂÂ×ÓÒÐ××ÐÐØÖÓÝÒÑ×ÂÓÒÏÐÝËÓÒ×ÁÒ 

ËØ 

ØE= −grad(V ) − 1 

✷×ØÐÐÑÖØÒ ∂µ∂ 

∂A 

c ∂t 

F µν = ∂ µ A ν − ∂ ν A µ 

∂µ∂ µ A ν − ∂ ν (∂µA µ ) = 4π 

c jν 

µ A ν = 4π 

c jν ÓÙ 

✷A 

µ = ∂ µ ∂µ 

= 0ÓF 

: B = rot(A) 

µν = 1 

2εµναβ FαβÎÓÖ

×ÓÒØÐ×ÓÓÖÓÒÒ×Ð××ÕÙ××Ð×ÔÒÓÙØÓÙØÙØÖÖÐÖØÒØÖÒØ ÔÓÙÖÐÔÖØÙÐ×ØÖÓÙÚÖÒ×ØØØ 

t)ÓÕ Ö×ÐÖØÈÜÓÒÔÙØÖÖÔÓÙÖÐØØÙÒÔÖØÙÐψ(q, s, ÐØÑÔ×| ψ ÌÓÙØÓ×ÖÚÐΩ×ØÖÔÖ×ÒØÔÖÙÒÓÔÖØÙÖÖÑØÕÙ 

ÍÒØØÔÝ×ÕÙ×ØÙÒÚØÙÖÔÖÓÔÖÐÓ×ÖÚÐΩ× ψ 

p 

ÍÒØØÖØÖÖ×ÜÔÖÑÓÑÑÙÒ×ÙÔÖÔÓ×ØÓÒÐÒÖÙÒÒ×ÑÐ ÓÑÔÐØÓÒØÓÒ××ÚØÙÖ×ÔÖÓÔÖ× ÚÐÓÖØÓÓÒÐØ 

Óωn×ØÐÚÐÙÖÔÖÓÔÖÖÐÐÕÙÓÖÖ×ÔÓÒÙÚØÙÖÔÖÓÔÖφn 

Ä×ÔÖÒÑØÑØÕÙÐÓ×ÖÚÐΩ×ØÓÒÒÔÖ 

 

| 2×ØÙÒÕÙÒØØÒÔÓ×ØÚÒØÖÔÖØÓÑÑÐÔÖÓÐØ 

ÒÔÖØÙÐÖÐÑÔÙÐ×ÓÒpÐÔÖØÙÐ×ØÓÒÒÔÖ 

 

Ωψ = 

Ωψ ψ 

pi → ∂ 

i ∂qi 

Ωφn = ωnφn 

= i∇ 

ψ = 

 

anϕn 

ϕn 

ϕm = δnm 

n 

| an | 2ÖÔÖ×ÒØÐÔÖÓÐØØÖÓÙÚÖÐ×Ý×ØÑÒ×ÐØØÒ 

〈Ω〉ψ = 〈ψ|Ω|ψ〉 = 

| an | 

n 

2 ÓÖÖ×ÔÓÒÐÒØÖØÓÒ×ÙÖÐ×ÓÓÖ 

ÒÖ ÄÚÓÐÙØÓÒÙÒ×Ý×ØÑÔÝ×ÕÙ×ØÖÔÖ×ÒØÔÖÐÕÙØÓÒËÖ ÓÒÒ×ØÐ×ÓÑÑØÓÒ×ÙÖÐ×ØØ×ÒØÖÒ× 

ωn Ò×Ð×ÖÐØÓÒ×××Ù×Ð×Ò 

i ∂ψ 

∂t = Hψ Ø××ÚÐÙÖ×ØÚØÙÖ×ÔÖÓÔÖ×ÖÔÖ×ÒØÒØÐ×ØØ××ØØÓÒÒÖ×ÔÓ××Ð×Ù À×ØÐÀÑÐØÓÒÒÙ×Ý×ØÑËÐ×Ý×ØÑ×ØÖÑÀ×ØÒÔÒÒØ 

ÙÒÔÓØÒØÐÖÐÎÐÕÙØÓÒÚÒØ 

 

− 2∇2 

+ V ψ = i 

2m ∂ 

∂t ψ 

 

×Ý×ØÑÙÖÔÓ×ËÐÓÒÓÒ×ÖÙÒÔÖØÙÐÒÖÒØÕÙÔ2ÑÒ×

ÓÒÒÐÓÙ ÓÒÔÓ× 

E → i ∂ 

∂t 

ÇÒÚÑÒØÒÒØ×Ö×ØÖÒÖÙ××ÔÖØÙÐ×ÐÖ×Î ËÐÓÒÑÙÐØÔÐÐÕ ÚÎ ÔÖψ ∗ØÓÒ×ÓÙ×ØÖØ×ÓÒÓÑÔÐÜ ÓÒÙÙÑÙÐØÔÐÔÖψÓÒÓØÒØÐÕÙØÓÒÓÒØÒÙØ 

×ÐÓÒÑÙÐØÔÐρØjÔÖÐÖÐÔÖØÙÐÓÒÜÔÖÑÐÓÒØÒÙØÙ Ü×ØÒÐÔÖØÙÐØ×ÖØÖ×ØÕÙ×ÕÙÐÐØÖÒ×ÔÓÖØÈÖÜÑÔÐ ØØÕÙØÓÒÜÔÖÑÐÓÒ×ÖÚØÓÒÐÔÖÓÐØÓÒÐÔÖÓÔÖØ ÓÙÖÒØÐØÖÕÙ ÄÔÖÑÖØÒØØÚØÖÒ×ÔÓ×ÖØÓÙØÐÒ×ÙÒÓÒØÜØÖÐØÚ×ØÔÖØ ÐÜÔÖ××ÓÒÐÒÖE 2 ÓØÒØ Ø ÓÒ 

ÕÙÔÙØ×ÖÖ×ÓÙ×ÐÓÖÑÙÒÕÙØÓÒÓÒ 

− 

Ç×ÖÚÓÒ×ÕÙ ÐÓÙÐÙØÕÙÐÓÒÙÒÑÙØ×ÒÒ×Ð×ÓÐÙØÓÒ exp(±iEt/) 

ÓÑÔØÐÐ×ÖÙØÐÔÓÙÖÖÖÐØØ×ÒØÔÖØÙÐ× ÇÒÚÖÖÕÙØØ×ØÙØÓÒÒ×ØÔØÓÐÓÕÙÕÙÒÔÔÖÒØÕÙÒÒ 

ÑÔÐÕÙ××ÓÐÙØÓÒ×ÒÖÒØÚ∼ 

ÈÖØÒØ ÐÔÖÓÙÖ××Ù×ÑÒÐÕÙØÓÒÓÒØÒÙØ 

ÄÔÖÓÐÑ×ØÕÙÙ×Ù×ÓÒÓÖÖÐÖÚØÑÔÓÖÐÐÒ×ÐÕ ÐÜÔÖ××ÓÒ 

 

(ψ 

dρ 

+ ∇ · j = 

dt 0Úρ = ψ ∗ ψØj = 

 

2mi [ψ∗ ∇ψ − (∇ψ ∗ )ψ] 

= (Ôc) 2 2ÒÙØÐ×ÒØ + (Ñ2 ) 2 ∂ 2 t ψ = (−2c 2 ∇ 2 + m 2 c 4 

)ψ 

ÕÙØÓÒÃÐÒÓÖÓÒ 

 

mc 

 

2 

✷ + ψ = 0 

 

E = ± (pc) 2 + (mc2 ) 2 

∂ 

∂t (ψ∗∂tψ − ψ∂tψ ∗ ) + ∇(ψ ∗ ∇ψ − ψ∇ψ ∗ ÓÙ ) = 0 , 

∂ µ (ψ ∗ ∂µψ − ψ∂µψ ∗ ) = ∂ µ jµ = 0 

∗ ∂tψ − ψ∂tψ ∗ )

ÒØ×ÐÓÒÓÒ×ÖÙÒ×ÓÐÙØÓÒ Ò×ØÔ×ÒÔÓ×ØÚÕÙÑÔÐ××ÓÖÙÒÒ×ØÔÖÓÐØ 

ÖÕÙÑÒØÔÖÓÐÑÑÒÐÒÓÒÔÖÓÚ×ÓÖÐÕÙØÓÒÃÐÒ ÓÖÓÒ ÓÒÓØÒØÙÒÒ×ØÆ2ÕÙ×ØÒØÚÔÓÙÖÐ×ÓÐÙØÓÒÚ

Ò× Ø Ð×ØÐ×ÖÔÖ×ÒØÒØ×ÑØÖ×ÜØÐ×σ i×ÓÒØ Ð×ÑØÖ×ÈÙÐÇÒÒØÖÓÙØÙ×× 

γ 5 = iγ 0 γ 1 γ 2 γ 3 ÄÓÒØÓÒÓÒÓÒÐÓÖÑÙÒÓÙÐ×ÔÒÙÖ 

= 

ÆÓØÓÒ×ÕÙÐÒ×ØÔ×ÙÒÕÙÖÚØÙÖ ÄÓÙÖÒØÔÓÙÖÐ×ÔÖØÙÐ×Ö 

ÒÔÖÒÒØÐ† ψ×Ø×ØÐÕÙØÓÒÖÓÒØÕÙÐÓÒÓØÒØ 

= ÇÒÒØÐÓÙÖÒØÔÖÆ† 

ψ×ØÔÔÐÐÓÒØψ ØÒÑÙÐØÔÐÒØÔÖγ 0ÇÒØÖÓÙÚÕÙØØÜÔÖ××ÓÒ ×Ø×ØÐÕÙØÓÒÓÒØÒÙØ∂µj µ ÕÙÔÓÙÖÓÑÔÓ×ÒØØÑÔÓÖÐÐÐÕÙÒØØÒÔÓ×ØÚj 0 ÆÓØÓÒ×ÕÙψ † ØÖÖÕÙψψÔÖÓÒØÖ×ØÒÚÖÒØ Ä××ÓÐÙØÓÒ×ÐÕÙØÓÒÖ 

ψÒ×ØÔ×ÙÒ×ÐÖÙÒÒÚÖÒØÄÓÖÒØÞÓÒÔÙØÑÓÒ 

ÔÖ×ÓÐÙØÓÒ× 0Ò××ÓÒÐ 

Ú 

ÓÒ×ÖÓÒ×ØÓÙØÓÖÐ×ØÙØÓÒ×ØØÕÙp = 

 

ψ±(t) 

 

ÐØÖÓÒØÙÒÔÓ×ØÖÓÒ×ÔÒ ÇÒÖÓÒÒØÙØÖÔÖØÒ× ÔÓÒÒÓÖ×ÒÖ×ÒØÚ×ÇÒ××ÓÖ××ÓÐÙØÓÒ×ÙÜÒØÔÖØÙÐ× Ñ2×ØÐÒÖÐÔÖØÙÐÙÖÔÓ×Ð×ÓÐÙØÓÒÐÜÔÓ×ÒØÔÓ×ØÓÖÖ× ÙÜ×ÔÒÙÖ×ÙØÐ×Ð×ÔÓÙÖÖÖÙÒ 

ψ+ 

 

⎛ 

⎜ 

ψ = ⎜ 

⎝ 

ψ1 

 

0 1 

1 0 

ψ2 

ψ3 ⎠ 

t 

ψ4 

∗ (a ) 

j µ (x) = cψ † (x)γ 0 γ µ ψ(x) = cψ(x)γ µ ψ(x)Óψ = ψ † (x)γ 0 

cψ † ψÓÑÑØØÒÙ 

= 

⎞ 

⎟ 

= 0 j µ×ØÙÒÕÙÖÚØÙÖÄÓÖÒØÞ 

= ρc = cψγ0ψ = 

= e +i(mc2 t/) ψ±(0) 

ψ1 

ψ2 

 

et ψ− = 

ψ3 

ψ4

ÒÓÒ××ÓÐÙØÓÒ×ÒÓÖÑÓÒÔÐÒ 0ÇÒ×ØÒØÖ×××ØØ×ÒÖ 

ÕÙÓÒÖØÓÒÓÒÒ× 

ÎÒÓÒ×ÒÐ×ØÙØÓÒÝÒÑÕÙp = 

ψ(x) = ae ixp/ ËÐÓÒÒ×Ö Ø Ò×ÐÕÙØÓÒÖÓÐ×ÑØÖ×γ×ÓÒØÒ×Ò ÓÒÓØÒØÐ××ÓÐÙØÓÒ×ÒÓØØÓÒ×ÐÖ 

u(p) 

u (1) = N 

u (3) = N 

⎛ 

⎜ 

⎝ 

⎛ 

⎜ 

⎝ 

1 

0 

cpz 

E+mc 2 

c(px+ipy) 

E+mc 2 

cpz 

E−mc 2 

c(px+ipy) 

E−mc 2 

1 

0 

ψ(r, t) = ae −i(Et−p·r)/ u(E, p) 

⎞ 

⎟ 

⎠ , u(2) = N 

⎞ 

⎟ 

⎠ , u(4) = N 

⎛ 

⎜ 

⎝ 

⎛ 

⎜ 

⎝ 

0 

1 

c(px−ipy) 

E+mc 2 

−cpz 

E+mc 2 

×E = m2c4 + p2c2 

×E = − m2c4 + p2c2 ÇÒÓ×ÐÒÓÖÑÐ×ØÓÒÔÖØÙÐ×ÔÖÙÒØÚÓÐÙÑÖØ Ú N 

(4)ÙÔÓ×ØÖÓÒ 

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c(px−ipy) 

E−mc 2 

−cpz 

E−mc 2 

0 

1 

⎞ 

⎟ 

⎠ 

⎞ 

⎟ 

⎠ 

= (| E | +mc2 )/c 

u † 

u = 2 | E | /c 

ÊÑÖÕÙÓÒ×ÕÙÙØÖ×ÒÓÖÑÐ×ØÓÒ××ÓÒØÔÓ××Ð×u † (2)×ÓÒØ×ÚØÙÖ×ÔÖÓÔÖ×ÐÓÔÖØÙÖÒÖÚÐÚÐÙÖÔÖÓÔÖ 

u| E |Ñ2 

u (1)Øu 

| E | 

u (3)Øu (4)×ÓÒØ×ÚØÙÖ×ÔÖÓÔÖ×ÚÐÚÐÙÖÔÖÓÔÖ| E | 

(3)Øu

ÔÖØÐ××ÓÐÙØÓÒ×ÔÓÙÖ×ÔÖØÙÐ×ÒØÚÓÑÑ××ÓÐÙØÓÒ×ÔÓÙÖ Ð×ÒØÔÖØÙÐ×ÔÓ×ØÚ ×ÐÓÒÒÐ×ÒEØ×ÑÙÐØÒÑÒØÐÙÔ ÓÒÓÒÖÒØÖ 

ÇÒÓØÒØ 

 

Ä×u×Ø×ÓÒØÐÕÙØÓÒ ÓÖÒÚÒØÓÒÐ××Ö (4)Ð×ÚØÙÖ×ÔÖÓÔÖ× 

ØÐ×vÐÕÙØÓÒ 

ØÐ×ÒÓØØÓÒ×u (2)Ð×+ 

ÊÑÖÕÙÓÒ×ÕÙÔÓÙÖÖÚÒÖÐÓÖÑ ÐÙØÖÐ×Ù×ØØÙØÓÒ 

(γ 

γ 0 ÉÙÐÕÙ×ÕÙ×ØÓÒ××ÔÓ×ÒØØ 

ÓÒÚÓÖÔÒ×ÞÑÖÓÒÒÐÖÔÓÒ××ÙÚÒØØØÒØÖ ÇÒÒÚÙØÔ×ÔÖØÙÐ×ØØ×ÓÖØÍÒÑ××ÒØÚ×ØÐ ÉÙÒØØÚÑÒØÐ××ÓÐÙØÓÒ×ÒÖÒØÚ 

= β 

ÖÓØÓÒ×ÓØ×ÒÖÒØÚÜ×ØÒØÒ×ÐÍÒÚÖ×Ñ×Ð×ØØ× 

• 

ÔÖÔÖØÙÐØÖÓÙÔÙØ×ÓÖÑÖÔÖÔ××ÙÒ×Ô×ÙÓÔÖØÙÐ× ÙÒÓÖÑÑÒØÒØÖÐ×ØØ××ÔÓÒÐ×ÚÓÖÐÔÖÒÔÜÐÙ×ÓÒÈÙÐ ÙÒÓÖÑÑÒØÓÙÔ×ÒØÓÒÓÒ×Ö×ÖÑÓÒ×ÕÙ××ØÖÙÒØ Ñ××ÑØÖqÕÙÓÖÖ×ÔÓÒÒØ(−E, 

ØØÒØÖÔÖØØÓÒÒ×ÔÔÐÕÙÓÒÔ×ÙÜÓ×ÓÒ× 2Ñ2Ù×Ý×ØÑÚÙÒ 

−p), 

 

ËÐÓÒÔÔÓÖØÙÒÖØÒÕÙÒØØÒÖδE > 

u (1)Øu (2)ÖÔÖ×ÒØÒØÐ×−ØÐ×ÚØÙÖ×ÔÖÓÔÖ×v 

v (1) (E, p) = u (4) (−E, −p) = N 

⎛ 

⎜ 

⎝ 

⎛ 

v (2) (E, p) = u (3) ⎜ 

(−E, −p) = N ⎜ 

⎝ 

c(px−ipy) 

E+mc 2 

−cpz 

E+mc 2 

0 

1 

cpz 

E+mc2 c(px+ipy) 

E+mc2 1 

0 

⎞ 

⎟ 

⎠ 

⎞ 

⎟ 

⎠ 

(3)Øu 

(1)Øv 

(γ µ pµ − mc)u = 0 

γ 

µ pµ + mc)v = 0 

i = βαi 

E > 0×ÓÒØÒÓÖÑÐÑÒØØÓÙ×

2 

+m 

E 

particule 

 

−m 

2 

ÓÖÑØÓÒÙÒÔÖÔÖØÙÐØÖÓÙ×ÐÓÒÖ 

trou=antiparticule 

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p)Ñ××mØÖ−qÍÒØÖÓÙÕÙÚÒ×Ø 

(2)ÖÔÖ×ÒØÒØÐ×ÙÒÐØÖÓÒÒ××ØØ× ÁÐÙØÖÑÖÕÙÖÕÙ ØÔÖØÙÐÖØÖ×ÔÖ(E, 

• Ä×ÚØÙÖ×ÔÖÓÔÖ×u (1)Øu 

×ÔÒ±1/2 ÄÖÔÓÒ××ØÒÓÒËÐÓÒÒØÐÓÔÖØÙÖ×ÔÒ 

S = 

ÐÕÙØÓÒÖÑÔÐÕÙØØÓÒ×ÖÚØÓÒ 

 

σ 0 

ÜÖÑÓÒØÖÖÕÙ 

S×ØÙÒÕÙÒØØÓÒ×ÖÚØ 

2 0 σ ÕÙ×ÙÐÐÑÓÑÒØÒÙÐÖØÓØÐJ = L + 

S1 = i 

2 γ2γ 3 

S2 = i 

2 γ3γ 1 ØÔÖØÖÐÕÙ S = − 1 

ÐÓÒÐÖØÓÒÙÑÓÙÚÑÒØÞØÔÖL=0)ÕÙÓÒÓØÒØÐ××ÓØÓÒ ×Ø×ÙÐÑÒØÒ×Ð×ÔÖØÙÐÖÔxÔy 

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ÓÒÔÙØÚÖÖÕÙÐÓÒÒÔ××ÚØÙÖ×ÔÖÓÔÖ×ËzÄÖ×ÓÒÒ×Ø 

S3 = i 

2 γ1γ 2 

2 γ0γ 5 

γ

Ö×ÔÓÒÒØ×Ø |−〉ØØÔÖÓØÓÒ×ÔÔÐÐÐÐØØÐÓÔÖØÙÖÓÖ Ù(1) = |+〉ØÙ(2) = 

λ ≡ S · ˆp = Ä×ØØ×ÔÓ××Ð××ÓÒØλ=±/2ÐØ×ÔÓ×ØÚØÒØÚÕÙÐÓÒÖÔÖ ×ÒØÔÖÙÖ Ó 

 

σ · ˆp 0 

ˆp = p/ | p | 

2 0 σ · ˆp 

+1/2 

ÄÔØØÓÒÒÐÖØÓÒÐÔÖØÙÐÐÖÓ××ÖÔÖ×ÒØÐÔÖÓ ØØ×ÐØÔÓ×ØÚØÒØÚ 

−1/2 

ÕÙÒØÕÙ ØÓÒÙ×ÔÒ×ÙÖØØÖØÓÒ ÄÓÔÖØÙÖÐØÓÑÑÙØÚÐÀÑÐØÓÒÒÀλ×ØÙÒÓÒÒÓÑÖ 

ÓÙÓÒØÓÙØØÔÓÙÖ ÈÓÙÖÐÑÓÒØÖÖÐÙØÖÖÙÒÔÖ×ÖÔØÓÒÕÙÔÖÑØØÔ××Ö ÐØØψ(x)ÖØÔÖÐÓ×ÖÚØÙÖÇÐØØψ ′ (x ′ )ÖØÔÖÇÓψ ′ (x ′ ×Ø×ØÐÕÙØÓÒÖ 

) 

 

µ ∂ 

iγ 

∂x ′ 

− mc ψ 

µ ′ (x ′ ) = 0 

(γ µ p ′ µ − mc)ψ ′ (x ′ ÄØÖÒ×ÓÖÑØÓÒÓØØÖÐÒÖ 

) = 0 

ψ ′ (x ′ ) = ˜ 

•ÄÕÙØÓÒÖÑØÐÐÙÒÐÑØÒÓÒÖÐØÚ×ØÖ×ÓÒÒÐ ÒØÐ×ÓÐÙØÓÒÜ×ØÚÓÖÔÜÖ ÔÖØÖ Ø ÓÒÓØÒØ 

S(ÇÇ′ )ψ(x),Ó˜ 

 

ÉÐÐØ×ØÐÐÙÒÒÚÖÒØÄÓÖÒØÞ 

• ØÓÒÚÖÑÒØÓØÒÙÙÒÓÖÑÙÐØÓÒÓÚÖÒØÁÐÚÖØØÖÐÖÕÙ 

S×ØÙÒÑØÖÜ

Hψ = (cp · α + mc 2 β)ψ = Eψ 

 

0 σ 

c · p + mc 

σ 0 

2 ÓÐÓÙÔÐ 

 

1 0 

ψ = Eψ 

0 −1 

 

cσ · pψ− = E − mc2 )ψ+ 

cσ · pψ+ = E + mc2 Ò×ÐÐÑØÒÓÒÖÐØÚ×Ø×ØÓÑÒÔÖÑ2 )ψ− 

cσ · p 

ψ− = 

E + mc2ψ+ σ · p 

−→ 

2mc ψ+ ÇÒÔÔÐÐψ−ÐÔØØÓÑÔÓ×ÒØÖØÔÖÐØÙÖ ÇÒÑÓÒØÖÕÙ×ÐÓÒÓÒ×ÖÐÔÖØÙÐÒÒØÖØÓÒÚÙÒÑÔÑ Ñ 

ÜØÖÒØÕÙÓÒÒØÖÓÙØÐÓÙÔÐÑÒÑÐ 

p µ → p µ − e 

c Aµ ÓÒØÓÑ×ÙÖÐÕÙØÓÒÈÙÐ 

A ≡ (V, A), 

i ∂ 

2 (p − (e/c)A) 

ϕ = 

− 

∂t 2m 

e 

 

σ · B + eV ϕ 

2mc ÚB = rot(A)ØϕÙÒ×ÔÒÙÖ(ϕ = eimc2 ØÖÑÕÙÖØÐÒØÖØÓÒÒØÖÐÐØÖÓÒØÐÑÔÑÒØÕÙ ÇÒÚÓØÕÙÒ××ÐÑØÒÓÒÖÐØÚ×ØÐÕÙØÓÒÖÓÑÔÖÒÙÒ 

t/ψ+) − e e 

σ · B ≡ −g S · 

2mc 2mc BÚS = 1 

2 σØg 

•ÉÙÒ×ØÐ×ÔÖØÙÐ××ÔÒÒØÖ ×ØÐØÙÖÄÒÐÔÖØÙÐ 

ÕÙÔÓÙÖÐÕÖÓÒÓÒ×ÖÐ×ÒÖ×ÒØÚ×ÓÑÑØÒØ××Ó ÇÒÖÚÒØÐÕÙØÓÒÃÐÒÓÖÓÒØÓÒÝÔÔÐÕÙÐÑÑÓÒÚÒØÓÒ Ø 

= 

Ô×ÙÓ×ÐÖÓÑÑÐÓÒÚÖÖÔÐÙ×ØÖÐÐ×ØÒÚÖÒØÚ×Ú×× ØÖÒ×ÓÖÑØÓÒ×ÄÓÖÒØÞÇÒÔÙØÚÓÖÐÓÒÒØÙØÚÒÖÑÖÕÙÒØ ×ÙÜÒØÔÖØÙÐ×ÄÕÃ×ØÔØÖÖÙÒÔÖØÙÐ×ÐÖÓÙ 

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2 

ÕÙÐÓÔÖØÙÖ✷Ñ22℄×ØÙÒÒÚÖÒØÓÒÔÓÙÖÕÙÐÔÖÓÙØÔÖψÖ×Ø 

Ø

ÖÐÐ× ÇÒ×ÑÔÐ×ÓÙÚÒØÐÖØÙÖ×ÓÖÑÙÐ×ÔÖÐÓÔØÓÒ×ÙÒØ×ÒØÙ Ä×ÙÒØ×ÒØÙÖÐÐ× 

ÄÕÖÚÒØ 

c 

ÕÙ×ÓÙÚÒØÐÓÒÖÔÖ 

ÉÚÖÖÕÙÅÎÑ−1 ÈÖÓÔÖØ××ÑØÖ×γ 

 

ØÖÒÙÐÐØ×ÔÒ 

ÈÓÙÖ×ÒÓÖÑØÓÒ×ÔÐÙ×ÓÑÔÐØ×ÚÓÖÔÖÜÊ ÇÒÓØØÖØÖÔÖØÐÐÑØÑ→Ä×ÙÐÔÖØÙÐÓÒÒÙÑ××ÔÙØ ×ØÐÒÙØÖÒÓ ÔÔÒÜ 

ÄÕÙØÓÒÖÔÓÙÖm = 

 

= h/2π = = 1 

(γ µ pµ − m)ψ = 0 

(/p − m)ψ = 0 avec : /a = γ µ aµ 

{γ µ , γν } 2g µν 

γ5 iγ 0γ1γ 2γ3 {γ µ , γ5 } 

γ µ γµ 

γµγνγ µ −2γ ν 

γµγνγ λγ µ 4g νλ 

γµγ ν γ λ γ σ γ µ−2γ σ γ λ γ ν 

0

Ä×Õ Ø 0ÓÒÒÒØ ÔÓÙÖm = 

γ µ Ø pµu = 0 µ 

pµv = 0 

ÄÕÔÓÙÖuÕÙÓÖÖ×ÔÓÒÐÔÖØÙÐÔÙØ×ÖÖ γ 

γ i piu = −γ 0 p0u = −γ 0 ÓÙÙ×× Ö 

Eu p0 = E 

ÕÙÓÒÒ 

γ 

S · pu = 1 

2 γ5 ÓÙ S · p 1 

Eu u = λu = 

| p | 2 γ5 

u ÇÒÙØÐ×ÐØÕÙÔÓÙÖÙÒÔÖØÙÐÑ××ÒÙÐÐ| p | ÅÙÐØÔÐÓÒ×ÔÖγ 5ØÓÙØÓÒ×Ø×ÓÙ×ØÖÝÓÒ×ÐÖ×ÙÐØØÓØÒÙÐÕÙ ØÓÒÔÖØ λ(1 + γ 5 )u = 1 

2 (1 + γ5 )u 

λ(1 − γ 5 )u = − 1 

2 (1 − γ5 Óλ×ØÓÒÒÔÖ 

)u ÓÒ(1 

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± ×ÓÒØ×ØØ×ÐØpositive ÐÓÒØÓÒ×ØÐÓÖ×vØE= − | p 

ÇÒÑÙÐØÔÐÕÙ ØÔÖ− 1 

2 γ0 γ 5ØÓÒÙØÐ×ÐÕ 

· pu = γ 0 Eu 

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mγ < 3 10 −33ÅÎÔÖÐÚÐÙÖÙÑÔÑÒØÕÙÐØÕÙÇÒÓÒÒØ

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p/vØØÑØÓØÙØÐ× ØÖÑÒØÓÒÐÑ××ÐÔÖØÙÐm = 

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×ÐÑ××ÐÐØÖÓÒ 

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±0.2 

ËÔØÖÐÑ××ØÚÔÖ×ÔÓØÓÒ× 

ÏÀÖ×ÏÖÒÙÑÅËÑØÈÝ×ÊÚ

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ËÔØÖÐÑ××ØÚÔÖ×ÔÓÒ× ÔÖÜÑÔÐÐ×ÒØÖØÓÒπ 0 γ1γ2ÓÒØÖÙ×ÓÒ×ÐÕÙÒØØ 

→ 

m 2 γ1γ2 = (pγ1 + pγ2) 2 Ò×Ð×ÔÖØÙÐÖÔÓØÓÒ×Ñ××ÒÙÐÐÓÒÓØÒØ Ó 

m 2 γ1γ2 = 2E1E2(1 E2ØÐÒÐθ12ÔÖÑØØÒØØÖ Ó − cos θ12) Ä×Ñ×ÙÖ×ÓÑÒ××ÒÖ×E1 ÑÒÖmγ1γ2Ñ××ØÚËÐ×ÙÜÔÓØÓÒ×ÓÒ×Ö××ÓÒØ××Ù×ÙÒπ 0 

p 

cos 

= (E, p) 

θ12 = p 1 · p 2 

| p 1 || p 2 |

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ρf 

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×ØØÓÒÒÖψ =sin(kxx) 

= ÓØLkx 

×Ý×ØÑ×ÙÜÐÔØÓÒ×ÒÖØÓØÐ 

E0 = Ee + Eν 

sin(kzz)ÚÐ×ÓÒØÓÒ×ÙÜÓÖ×Ð 

ρf = dN/dE0 

 

sin(kyy) 

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= −dV/dr1 = −dV/d(r1 − r2) = ... = −F 

F 1 + F 0 

⇕

ÓØÒÙ×ÙÄÈ×ÓÒØÒÚÖÒØ×ÔÖØÖÒ×ÐØÓÒÙÄÈÔÜ×ÙÖÙÒÔÐÒØÒ ÓÖØÙØÓÙÖβÇÖÓÒ×ÇÒÔÖÐÒ××ØÖÒ×ÓÖÑØÓÒØÚ ØÐÓÒÚØØ×ÙÖÙÒØÖÒ×ÐØÓÒxÙ×Ý×ØÑØÙÐ×Ö×ÙÐØØ×ÔÝ×ÕÙ× ÇÒ×ÖØÖÖÚÙÜÑÑ×ÓÒÐÙ×ÓÒ××ÐÓÒÚØÓÒ×ÖÐÓ×ÖÚØÙÖÜ 

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ËÐÓÒÒÚÖ× ψ 

x = f −1 (x ′ ÓÒÔÙØÖÖ 

) 

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)) 

x ′×ØÕÙÐÓÒÕÙÓÒÓÒÔÙØÐ××ÖØÓÑÖÐ×ÔÖÑ××ÙÖx 

ψ ′ (x) = ψ(f −1 

(x)) Ð×ÙÖÙÒÒÓÙÚÐÐÒØÖÔÖØØÓÒÐØÖÒ×ÓÖÑØÓÒψ ′ ÒÒØÔÓÙÖÐÓ×ÖÚØÙÖÇÐÓÒØÓÒÓÒÙ×Ý×ØÑØÚÑÒØØÖÒ×ÓÖÑ ÔÐØÓÙÖÒ ÒÔÖÒÔÐ×ØÖÒ×ÓÖÑØÓÒ×ØÚØÔ××Ú×ÓÒØÕÙÚÐÒØ×Ò×ÙÒ (x)×ØÑÒØ 

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ÐØÖÒ×ÓÖÑØÓÒf ×ÓØÖÓÔÚ×Ú×ÐÓØËÇÓÒ×ØØÖÙÒÒÑÒØÓÒÙÖØÓÒ ÇÇÒÖÒÖÔ×ÓÑÔØÙÒÑÒØ×ÙÖËÈÖÓÒØÖ×Ð×ÔÒ×ØÔ× ÇÒÖØÐØÖÒ×ÓÖÑØÓÒØÚÔÖÙÒÓÔÖØÙÖÍÙÒØÖÕÙÔÒ 

ψ ′ 

ÎÓÖÖÖÒ 

= Uψ 

 

x → x ′ = x + aÕÙÐÓÒ×ÝÑÓÐ×ÔÖ 

x → x ′ = f(x) 

′ (x ′ ) = ψ(x) 

ÄÙÒØÖØÔÖÑØÓÒ×ÖÚÖÐÒÓÖÑψ 

 

Uψ Uψ = ψ ψ U † U = 1 U −1 = U † 

(x ′ )

ÈÖÐØÖÒ×ÓÖÑØÓÒfÐÓ×ÖÚÐÚÒØÐÓ×ÖÚÐÓÒØÐ×ÔÖÒ ÑØÑØÕÙÔÓÙÖÐØØψ ′×ØÒØÕÙÐÐÔÓÙÖÐØØψ 

〈ψ|A|ψ〉 = 〈ψ ′ |A ′ |ψ ′ ÉÑÓÒØÖÖÕÙ 

〉 

ÓÒÓØÒØÍÍ†ØÍÍ Ëf×ØÙÒ×ÝÑØÖÚ×Ú× ÓÒÓÑÑÙØÚÍ 

A 

ÚÒØÔ××ÖÙØÖØÑÒØ×ÖÒØ××ÝÑØÖ×ÖÑÖÕÙÓÒ×ÕÙÓÒÓÒ× 

×ØÖ××ØÒÚÖÒØ×ÓÙ×f 

Ö×ÓÙÚÒØ×ØÖÒ×ÓÖÑØÓÒ×ÒÒØ×ÑÐ×ØÖÒ×ÐØÓÒÖÓØØÓÒ ÕÙ×ÑÔÐÐ×ÐÙÐ×ÇÒÔÙØÒ×ÙØÒØÖÖÔÓÙÖÓØÒÖÐØÖÒ×ÓÖÑØÓÒ ÒÏÛÐÐÓØÓ×ÖÒÓÛÌÓÖ×ØØØÑÓÑÒØÖ××Ò×ØÓÔÐÙ 

 

[U, A] = 0 

Ì×ÛÓÖÐÛ×ÑÙÖÖÓÒÒÓÐÖ×ØÐÐ ÑÓÑÒØÑÒÙØÐÝÓÙØÓÓÙ×ÒØÒ×ÒÔÔÒ××ÙÒÐÝÖÒØÛÓÖÐ ØÛ×ØØÒØÖÛÓÖÐØÖÓÙÐÓÒØÔÖØÓÖÚÖÝØÒ×ØÛ×ÓÖ ÒÔÔÐÙØÒ×ØÓÔÐÙÒÑØÒÝ×ÖÔØÙÖÒÒÑÓÚÑÒØÌØÛ×× 

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×ØÒÚÖÒØÔÖÖÓØØÓÒ×ÐÓ×ÖÚÐ×ØÐÓÙÐÙÖ ËÓÒØ ψÐØØÙ×Ý×ØÑ 

OÙÒÓ×ÖÚÐ××ÓÙÒØØÙÖØÙÒÜÔÖÑÒØØÙÖ 

UÙÒÓÔÖØÓÒÙØÝÔÖÓØØÓÒØÖÒ×ÐØÓÒÔÖØ ØÝÔÕÙÑÒØÐÒÖÙ×Ý×ØÑ×ÓÒÀÑÐØÓÒÒ ØØÐÐÕÙ 

U † ÉÙÐÓÔÖØÓÒU×ÓØ××ÓÙÒ×ÝÑØÖÓÙÒÓÒÐÙØÒÒÑÓÒ×× ÓÒØÓÒ×ÙØÝÔ×ÓØÖÓÔÐ×ÔØ ×ÐÓÒØÖÒ×ÓÖÑÐ×Ý×ØÑ 

U = 1 

ψ ′ = UψØ×ÑÙÐØÒÑÒØÐØØÙÖU(O) = O ′ = UOU †ÐÖ×ÙÐØØ ÒØ ÐÜÔÖÒÖ×ØÒÒ ÓÙÐ×Ñ×ÌÄÓÒÖÌØÑÓØËÓÙÐ 

′ = UAU †

Ñ×ÙÖO ′ÔÓÙÖψ ′ ) =< ψ ′ | O ′ | ψ ′ >=< Uψ | UOU † ÕÙ×ÖØÙ×× 

| Uψ > 

ÍÒØ×ØÒÚÖÒ ÓÒÜÐØØÙÖOØÓÒÓÙÐ×Ý×ØÑ ÓÒÜÐ×Ý×ØÑØÓÒÓÙOÒ×Ò×ÒÚÖ× ×ÝÑØÖÔÙØ×ÖÙÜÓÒ× ÇÒÓØÒØÓÒ< ψ 

ψ 

ψ ÚO ′′ÓÒÒÔÖU −1 Ø×ÓÒØÕÙÚÐÒØ××Ð×Ô×Ø×ÓØÖÓÔØ ÒØ ËÐÓÒÙÒ×ØÙØÓÒ×ÝÑØÖÐÓÖ× 

< 

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ÓÒ×ÖÓÒ×ÙÒÓØÖØÔÖ×ÓÒØÓÒÓÒψ(xÍÒØÖÒ×ÐØÓÒÒ Ò×ÐØÑÔ×ØÔÖÖÓØØÓÒÒ×Ð×Ô ÄØÖÒ×ÐØÓÒÒ×Ð×Ô 

ÒÒØ×ÑÐ ÒÙÒØ×ÒØÙÖÐÐ×p≡−i∇ÇÒÔÙØÓÒÒØÖÓÙÖÐÓÔÖØÙÖØÖÒ×ÐØÓÒ 

ÒØ×ÑÐx → 

ÕÙÓÒÒ 

D(δx) ÍÒØÖÒ×ÐØÓÒÒXÔÙØ×ÓÑÔÓ×ÖÒNÔØØ×Ô×δx = 

 

D(X) 

(ψ † U † UOU † Uψ) = (ψ † O ψ) = (ψ † Oψ) 

Ú(ψ, ϕ) = d3xψ∗ ψØϕÔÔÖØÒÒØÐ×ÔÀÐÖØ×ØØ× 

(x)ϕ(x), ′ | O ′ | ψ ′ >ÓÑÑÔÖÚÙ 

>=< ψ | O | ψ 

→ ψ ′ O → O 

→ ψ O → O ′′ 

(O) ≡ U † OU 

ψ > 

< ψ ′ | O | ψ ′ >= (ψ † U † OUψ) = ψ † (U † OU)ψ =< ψ | O ′′ | ψ > 

< ψ ′ | O | ψ ′ >=< ψ | O ′′ | ψ >=< ψ | O | ψ >=< ψ | O ′ | ψ > 

[U, O] = 0 

ψ ′ | O | ψ ′ >=< ψ | O ′′ | 

δxÐÓÖÖÇδx×ÜÔÖÑÔÖ 

x + 

ψ ′ ≡ ψ(x − δx) ∼ = (1 − δx · ∂ 

∂x )ψ(x) = (1 − iδx · p)ψ(x) 

 

≡ 1 − iδx · p 

X/N 

D(X) ≡ D(δx) N = (1 − iδx · p) N 

(iNδx · p)2 (iNδx · p)3 

≈ 1 − iNδx · p + − + ... 

2! 3! 

≡ exp(−iX · p)

ÕÙØÓÒ× ØÑÖÒÖÐ×ØØ×ÐÙÚÒØØÔÖ×ÐØÖÒ×ÐØÓÒÐ× ÄÒÚÖÒÔÖØÖÒ×ÐØÓÒ×ÒÕÙÐ×ØÑÔÓ××ÐÔÖÐÑ×ÙÖÙ×Ý× ÊÚÒÓÒ×Ù×ÒÒØ×ÑÐD(δx)ÕÙÐÓÒÚÖÖÔÓÙÖÐÐÖÐ× 

ØÖÙØÔÖÐÒÚÖÒÐÒÖÙ×Ý×ØÑÐÓÐÑÒØÔÐδxÄÀ ÑÐØÓÒÒÀÓØ×Ø×ÖH(x) 

ÓÓÒÙØÐ× 

= 

ÇÒÒØÖ 

ÔÔÐÕÙÓÒ×ÐÓÔÖØÙÖD×ÙÖÐØØφ(x) = 

ÓÒØÓÒÓÒψ(x)ÕÙÐÓÒÕÙÓÒ 

Ð××ÒÚÖÒØÐÀÑÐØÓÒÒÚÓÖ ÓÑÑÙØÚÐÀÑÐØÓÒÒÕÙ×ØÙÒÓÒÕÙÚÐÒØÒÕÙÖÕÙ Ò×Ù×ØØÙÒØ Ò×ÓÒ 

ÒÓÒÐÙ×ÓÒÓÒÜÔÖÑÐÒÚÖÒÀÐÓÖ×ÙÒØÖÒ×ÐØÓÒÙ×Ý×ØÑ ÕÙÓÒ×ØØÙÙÒ×ÝÑØÖØÖÒ×ÐØÓÒÐÒÓÙ×ÓÒÙØÐÐÓ 

ÑØÖδÍδÕÙÐÓÒÜÔÖÑÔÖ ÊÔÖÒÓÒ×Ð×ÒÖÐÙÒØÖÒ×ÓÖÑØÓÒÙÒØÖÒÒØ×ÑÐÔÖ ÓÒ×ÖÚØÓÒp 

U(δ) = 1 + iδG + O(δ 2 ×ØÔÔÐÐÒÖØÙÖÐØÖÒ×ÓÖÑØÓÒÇÒÔÙØÖÑÓÒØÖÐØÖÒ× 

δ 

) ÓÖÑØÓÒÒ∆ 

ÐÐÑØ 

U(∆ 

 

 

δ iGU(∆) 

 

H(x − δx) 

H(x)ψ(x) 

0ÔÓÙÖÙÒ 

D(H(x)ψ(x)) = Dφ(x) = φ(x−δx) = H(x−δx)ψ(x−δx) = H(x)ψ(x −δx) = 

H(x)Dψ(x) 

 

D(H(x)ψ(x)) = H(x)Dψ(x)Ó(DH(x) − H(x)D)ψ(x) = 

ÓØÒØÐÐÓÓÒ×ÖÚØÓÒÐÕÙÒØØÑÓÙÚÑÒØ 

[D, H] = 0 

 

[p, H] = 0 

+ δ) = (1 + iδG)U(∆) 

[U(∆ + δ) − U(∆)]/δ = iGU(∆) 

→ 0 :Í(∆) 

∆ =

ØÖÒ×ÓÖÑØÓÒÍÓØØÖÙÒØÖ ÄÒØÖØÓÒÓÒÒ ÐÓÒ×ØÒØÔÙØØÖÓ×ÐÔÓÙÖ×Ø×ÖÐÓÒØÖÒØÍÄ 

U(∆) 

U † U = 1 ⇒ 1 − iδ(G † − G) + O(δ 2 

) = 1 ÐÑÓÒØÖÕÙG † ÚH GÓÑÑÙØÙ×× 

ÐÓÐÓÔÖØÙÖÕÙÒØØÑÓÙÚÑÒØ×ØÔÔÐÐÒÖØÙÖ×ØÖÒ×Ð ÓÒÐÓ×ÖÚÐG×ØÙÒÓÒ×ØÒØÒÓÒØÓÒÙØÑÔ×Ò×ØØØÖÑÒÓ ØÓÒ×Ò×Ð×Ô 

ÙØÖÔÖØ×UÓÑÑÙØÚHHÒÚÖÒØ×ÓÙ×U 

ÄØÖÒ×ÐØÓÒÒ×ÐØÑÔ× 

[G, 

ÑÒØÐÓÖÒÐÐÐØÑÔÓÖÐÐ ØÑØØÓÒ×ÕÙÐ×Ý×ØÑÔÝ×ÕÙÒ ×ÓØÒÖÒÐÔÖ×ÙÒØÖÒ×ÐØÓÒ ÓÒ×ÖÓÒ×ÑÒØÒÒØÙÒÒ 

ÐØØÖÑÒÖÙÒÓÖÒ×ÓÐÙÙ Ò×ÐØÑÔ×ÇÒÒÓÒÔ×ÐÔÓ×× ØÑÔ×ÔÖØÖ×ÖØÖ×ØÕÙ×Ù×Ý× ÖÚÒØÐ×Ý×ØÑÒÔÒÒØÔ×Ü ØÑÄÀÑÐØÓÒÒÓÙÐÄÖÒÒÕÙ 

ÔÖ××ÓÒÐÓÒ×ÖÚØÓÒÐÒÖ ÓÒ×ØÒØÐÒ×ØÖÒÙØÖÕÙÐÜ 

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ØÔÖ×ÐÖÓØØÓÒ×ÓÒØÒ×ÖÒÐ×ÇÒÚÑÓÒØÖÖÕÙÐÒÚÖÒÔÖÖÓØ ÄÒÚÖÒÔÖÖÓØØÓÒ×ÒÕÙÐ×ØØ×ÙÒ×Ý×ØÑÔÖØÙÐ×ÚÒØ ÄÖÓØØÓÒÒ×Ð×Ô 

ÒÖÐ×Ð×Ù×ÙÒÔÖØÙÐÚ×ÔÒ ÍÒÖÓØØÓÒÒÒØ×ÑÐδθÙØÓÙÖÐÜÞ×ØÖÔÖ×ÒØÔÖÐÑØÖÅ ÔÖØÙÐ×Ò××ÔÒÑ××Ò×ÐÔÖÓÙÚÖÓÒÔÙØÖÑÖÕÙÐ×ÓÒÐÙ×ÓÒ××ÓÒØ ØÓÒ×Ø××ÓÐÓÒ×ÖÚØÓÒÙÑÓÑÒØÒÙÐÖÇÒÓÒ×ÖÐ×ÙÒ 

 

⎠ ÊÓØØÓÒ:x↦→Mx M 

 

G×ØÀÖÑØÕÙÙÒÖØÖ×ØÕÙ×Ó×ÖÚÐ× 

= exp(iG∆) + C 

 

= G, 

H] = 0 

⎛ 

⎞ ⎛ 

cosδθ − sin δθ 0 

≡⎝ 

sin δθ cosδθ 0 ⎠ ∼ = ⎝ 

0 0 1 

1 −δθ 0 

δθ 1 0 

0 0 1 

⎞

ÇÒÒØÖÓÙØÐÓÔÖØÙÖÊzÕÙØÖÒ×ÓÖÑÐØØψ 

Rz(δθ)ψ(x) ≡ ψ(M −1 x) = ψ(x + yδθ, −xδθ + y, z) ∼ ×ÒØ×ÐÓÒÐÜÞÐÓÔÖØÙÖÑÓÑÒØÒÙÐÖ ÄÜÔÖ××ÓÒÒØÖÔÖÒØ×ÙÖÒÖØÖÑ×ØÔÖÓÔÓÖØÓÒÒÐÐÐÓÑÔÓ 

= ψ(x) + δθ(y∂x − x∂y)ψ(x) 

L = r × ≡ pÄz i(y∂x − x∂y)ÕÙÓÒÒi 2 1 

= −1 

Rz(δθ)ψ(x) = (1 − iδθLz)ψ(x) ÇÒÔÙØÒÖÐ×ÖÙÒÖÓØØÓÒÙØÓÙÖÐÜn | n |= 

Rnψ(x) = (1 − iδθnL · n)ψ(x) 

H(x)ψ(x)ÓÒÒÓØx ′ = M −1x) ÔÔÐÕÙÓÒ×ØÓÔÖØÙÖÐØØφ(x) = 

Rn(H(x)ψ(x)) = Rnφ(x) = φ(x ′ ) = H(x ′ )ψ(x ′ ÓÓÒØÙ×ÐÓÒØÓÒÒÚÖÒ ) 

ËÐÓÒÓÒ×ÖÐ×ÔÖØÙÐ×Ú×ÔÒÓÒÔÙØ×Ù×ØØÙÖ ÖÐÖÓØØÓÒ×ØÕÙÐÓÒÕÙ 

×ØÒØÐÑÓÑÒØÒØÕÙØÓØÐÕÙ×ØÓÒ×ÖÚ 

JÓÙÐÖÐÒÖØÙÖÐÖÓØØÓÒ ÓψÓÒØÒØÒÔÐÙ×ÐÒÓÖÑØÓÒ×ÙÖÐ×ÔÒÐÔÖØÙÐØJ = 

ÊÑÖÕÙÙÔÓÒÚÖÖÙÒÙØÖ×ÓÖØ×ÝÑØÖØÒÚÖÒ ÙÄÒÚÖÒÙÑÑÔÐÕÙÐÓÒ×ÖÚØÓÒÐÖÑ 

×ÓÒÖÖÒÚÖ×ÑÒØÙØÑÔ× ÆÓÙ×ÐÐÓÒ×ÑÒØÒÒØÓÒ×ÖÖÐ×××ÝÑØÖ××ÖØ×ÔÖØÓÒÙ ÄÔÖØ 

 

ÓÑÑÔÖÚÙ 

[Rn, H] = 0 ÕÙ×ØÕÙÚÐÒØ 

= H(x)ψ(x ′ ) = H(x)Rnψ(x) 

H(x) = H(x ′ 

) 

[L, H] = 0 

Rnψ(x) = (1 − iδθnJ · n)ψ(x) 

[J, H] = 0 

L + S

ØÓÒ ÄØÖÒ×ÓÖÑØÓÒÔÖÔÖØÙÒ×Ý×ØÑÔÖØÙÐ××ØÒÔÖÐÓÔÖ 

ÓÒÙÑÖÓØÐ×ÔÖØÙÐ×ÁÐ×ØÙÒ×ÝÑØÖÔÖÖÔÔÓÖØÐÓÖÒ 

P : xi ↦→ −xi 

ÁÐ×ØÐÖÕÙ 

ÙÖÔÖÄÕÙÖÚØÙÖ(t, x)×ØÖÒ×ÓÖÑÓÒÒP(t, x) (1, −1, ÊÑÖÕÙÞÕÙÒÓÓÖÓÒÒ×ÔÓÐÖ×x=R sin 

P 2 

È×ØÕÙÚÐÒØÙÒÖÜÓÒÒ×ÙÒÑÖÓÖÔÐÒ×ÙÚÙÒÖÓØØÓÒ Ð×ÐÑÒØ×ÙÖÓÙÔÔÖØ×ÓÒØÓÒÈØÈ2Á É×ÓÒÚÒÖÕÙÐÓÒÒÔÙØÔ×ÖÙÖÈÙÒÖÓØØÓÒÑÓÒØÖÖÕÙ ÉÕÙÐ×ØÐÓÑÔÓÖØÑÒØ×ÓÙ×PÐÕÙÒØØÑÓÙÚÑÒØpÙ 

=ÒØØ. 

ÓÒ×Ð×Ý×ØÑÓ×ÖÚ×ØÒÚÖÒØ×ÓÙ×Ð×ÖÓØØÓÒ×ÐØÙ×ÓÒÓÑÔÓÖ ÄÓÔÖØÓÒÈÕÙÚÙØÙÒÖÓØØÓÒÔÖ×ÒÖÐÖÓØØÐÙ p 

ÙÒÚÖ×ÐÐËÙÖØÖÖÐ×ØÐÖÕÙÐ×ÓÖÒ×Ñ×ÚÚÒØ×ÓÒØØÙÙÒ ØÑÒØ×ÓÙ×È×ØÕÙÚÐÒØÐØÙ×ÓÒÓÑÔÓÖØÑÒØ×ÓÙ×ÙÒÖÜÓÒ 

Ò××ÒÐÚÈÖÜÑÔÐÐÖ×ÙÖÐ×ÝÑØÖÔÓÙÖÖØ×ÜÔÐÕÙÖÔÖ ÓÜØÓÙØÓ×ÐÔÓÙÖÖØØÖÙÜÓÒØÓÒ×ÒØÐ×ÔÖ×ÒØ×ÐÓÖ×Ð ÇÒÔÙØ×ÑÒÖ×Ð×ØÒØÓÒÒØÖÐÖÓØØÐÙÙÒÚÐÙÖ 

ÐØÕÙÐ×ÔÖÑÖ×ÓÖÒ×Ñ×××ÓÒØÓÖÑ×ÔÖØÖ×ÑÓÐÙÐ×ÓÖÒØ× ÄÔÖÑÖÓÖÒ×ÑÕÙÐ×ÖØÜØÖÓÖÄÔÖÓÒØÙÖÒÔ×Ù ×ÓÙ×ÐØÙÑÔÑÒØÕÙØÖÖ×ØÖÓÙ×ÓÙ×ÐÒÙÒÐÖÓØØÓÒÐ ÓÜÌÓÙØÓ×ÓÒÔÙØÑÒÖÕÙÒÑÓÝÒÒÐÚÒ×ÐÍÒÚÖ××ØØÓÙØ ØÖÖ ØÑÓÖØÕÙÐÝÙØÒØÚÐ×ØÓÒ×ÕÙ××ÐÙÒØÒ××ÖÖÒØÐ ÇÒÔÙØÑÑÑÒÖÙÒ×ØÙØÓÒØÓØÐÑÒØ×ÝÑØÖÕÙÐÓÖÒ 

ÖÓØ ÑÒÓÙØÒØÙÐÓÙÔÒÓÙ ÙÕÙÚÐ×ØÓÒ×ÕÙ××ÖÖÒØÐ 

ÁÐ×ØÙÒ×ÝÑØÖÜØÒ×ÐÐÑØÐÔÖ×ÓÒÜÔÖÑÒØÐÔÓÙÖ Ð×ÒØÖØÓÒ×ÑØÓÖØÑ×ÕÙ×ØÚÓÐÔÖÐÒØÖØÓÒ×ÐÓÒ ÐÆØÙÖØÙÒÖÒÒØÖÐÖÓØØÐÙ 

z = R cos θÐØÖÒ×ÓÖÑØÓÒ×ØÖÙØÔÖR 

−1, −1) 

′ 

≡ (t, −x) : P ≡ 

θ cosφ y = R sin θ sin φ 

= R, θ ′ = π − θ φ ′ = π + φ 

pÙ×ÔÒS×ÑÔ×EØBÙØÑÔ× ÑÓÑÒØÒÙÐÖÓÖØÐL = r × ÙÒÕÙÖÚØÙÖÐÖÐØÖÕÙQÙÔÖÓÙØ×ÐÖS · 

ÈÓÙÖÙÒ×Ý×ØÑÔÖØÙÐ×ÒÚÖÒØ×ÓÙ×ÐÔÖØÓÒ 

H({x ′ 

i }) ≡ H(P {xi}) = H({xi})

ØÙÖ ÒÓÒ×Ð×ÙÒÔÖØÙÐÒÔÖÐÓÒØÓÒÓÒψÇÒÒØÖÓÙØÐÓÔÖ ÓÒ×ÖÓÒ×ÑÒØÒÒØÐ×ØÙØÓÒÓÐÔÖØ×ØÙÒÓÒÒ×ÝÑØÖÈÖ 

Ð×ÚÐÙÖ×PaÔÓ××Ð××ÓÒØ+1Ø−1ÇÒÔÔÐÐPaÐÔÖØÒØÖÒ×ÕÙ ÔÖÓÖÐÚÐÙÖPaÔÒÐÔÖØÙÐÓÒÒØÖÓÙØ×ÚÐÙÖ×ÒÜ×ÔÖ ÄÓÒØÓÒ ÑÔÐÕÙÕÙ 

ÉÑÓÒØÖÖÕÙÐÓÒØÓÒÓÒÙÒÔÖØÙÐÙÖÔÓ××ØÙÒÚØÙÖÔÖÓÔÖ ÐÔÖØÙÐÖÐÐÖØÖ×ÐÔÖØÙÐÙÖÔÓ× 

Pψ(x, 

ÅÓÒØÖÖÕÙ×ÐÔÖØÙÐÙÒÑÓÑÒØÓÖØÐÒÖØÖ×ÔÖÐÒÓÑÖ 

ÐÒÓÑÐÔÖØÙÐÒÕÙ×ØÓÒPγÈπ Pp 

ÕÙÒØÕÙÄ×ÓÒØÓÒÓÒ×ØÙ××ÙÒÚØÙÖÔÖÓÔÖÈÚÐÚÐÙÖ ÈÚÚÐÙÖÔÖÓÔÖÈa 

ÓÒÔÓÙÖÐ×ÔÖØÙÐ×ØÒ×ÙÒØØÑÓÑÒØÒÙÐÖÓÖØÐÄÓÒ ×Ø 

ÓØÒØ 

ÄÒÖÐ×ØÓÒÔÐÙ×ÙÖ×ÔÖØÙÐ×Òx1 x2 

ÕÙ ÈÓÙÖÙÒ×Ý×ØÑÖØÖ×ÔÖ×ÓÒÑÐØÓÒÒÀÐÒÚÖÒ×ÓÙ×ÈÑÔÐÕÙ 

Pψ(x1, 

ÑÑ×Ð×ÔÖØÙÐ×ÓÒ×ØØÙÒØ×Ý×ØÑÒØÖ××ÒØØ×ØÖÒ×ÓÖÑÒØÒ ÙØÖ×ÔÖØÙÐ×ØØÓÒÐÙ×ÓÒ×ØÚÖÔÓÙÖÐ×ÒØÖØÓÒ×ÓÖØØÑ Ð×Ý×ØÑ×ØÒÚÖÒØÙÓÙÖ×ÙØÑÔ×È×ØÙÒÓÒ×ØÒØÙÑÓÙÚÑÒØ ÄÓÒ×ÖÚØÓÒÐÔÖØ×ÒÕÙÐÔÖØÐÓÒØÓÒÓÒÕÙÖØ 

Ò×ØÔ×Ð×ÔÓÙÖÐÒØÖØÓÒÐ 

[H, 

Ä×ÖÙÑÒØ×ÕÙÓÒÔÔÐÕÙÙÜÒÖØÙÖ××ØÖÒ×ÓÖÑØÓÒ×ÓÒØ ÒÙ×Ò×ÓÒØÔÐÙ×ÚÐÐ×Ò×Ð×ÙÒØÖÒ×ÓÖÑØÓÒ×ÖØÌÓÙ ØÓ××È×ØÐÓÔÖØÙÖÔÖØÐÓØ×Ø×Ö(P) 2 IÓÒÔÖ 

= ÐÙÒØÖØP PP † = IÓÒÙØÕÙP P † ÚÐÙÖ×ÔÖÓÔÖ×Pa×ÓÒ×ÖØÓÒ××ÔÔÐÕÙÒØÙÜ×ÝÑØÖ×ÙØÝÔ ÓÔÖØÙÖÀÖÑØÕÙØÐÔÙØØÖ××ÑÐÙÒÓ×ÖÚÐÓÐ× Ñ×Ô×ÐÒÚÖ×ÓÒÙ ÕÙÐ×ØÙÒ 

 

ØÑÔ×T 

t) = Paψ(−x, t) 

ÔÖÓÔÖPa(−1) LPψnℓm(x, t) = Pa(−1) L 

ψnℓm(x, t) 

Pψ(x1, x2, ..., t) = P1P2...ψ(−x1, −x2, ...t) 

x2t) = P1P2(−1) L 

 

ψ(x1, x2, t) 

P] = 0 

U 2 = IÓÒÙ×ÓÒÖCÔÖØG 

=

×ÓÙ×P ÇÒÐ×ÒÓÑÒØÓÒ××ÙÚÒØ×ÒÓÒØÓÒÙÓÑÔÓÖØÑÒØÐÖÒÙÖ ËÐÖ È×ÙÓ×ÐÖ ÎØÙÖÔÓÐÖ È×ÙÓÚØÙÖÜÐ P(s) 

P(p)−p 

P(v)−v 

P(a) ÍÒÔÖØÙÐ×ÔÒ0×ØØ×ÐÖ××ÔÖØ×ØÔÓ×ØÚ(JÔÖØ= 0 ØÔ×ÙÓ×ÐÖ××ÔÖØ×ØÒØÚ0 −Ø 

Ô×ÙÓÚØÙÖÔ×ÙÓ×ÐÖÚØÙÖÜÐ→−Ô×ÙÓ×ÐÖÚØÙÖÜÐ ÑÑÔÓÙÖÙÒÚØÙÖÓÒ ÚØÙÖ×ÐÖÚØÙÖ →×ÐÖ−ÚØÙÖ 

c×ÐÓÒÓÒÒØÐ×ÑÓÑÒØ×ÒÙÐÖ×ÓÖØÙÜLa + bØℓa ÓØÚÓÖPaPb(−1) L = PaPbPc(−1) ℓ •ËÔÖÓÒØÖÐÔÖØÙÐ×ØØÓÙÓÙÖ×ÔÖÓÙØÒ××ÓØÓÒÚÙÒÔÖ ×ØÙÒ×ÔÖÚÐÔÓÙÖØÖÑÒÖÐÔÖØÙÒÔÖØÙÐ 

ÇÒÐÖÐÒÖÐ PdÔÓÙÖÖØÖØÖÑÒ 

ÕÙÓÒÒPc 

ØÙÐ×ÙÐÐÔÖØÖÐØÚPc · 

s 

a 

+ ) 

cÓP×ØÓÒ×ÖÚÓÒÓØÒØÐÔÖØ cÇÒ 

•Ò×ÐÔÖÓ××Ù×a + b → a + b + 

+ b + 

= (−1) L+ℓ 

ÔÓÙÖÐ×ÖÑÓÒ× ÔÓÙÖÐ×Ó×ÓÒ× Èparticule−Èantiparticule 

Èparticule Èantiparticule É×ψ×ØÙÒÓÙÐ×ÔÒÙÖÖÑÓÒØÖÖÕÙÈψ(r, t) = γ0ψ(−r, t) ÉÙÚÙØÐÓÙÖÒØÖJ µ = ψγ µ ψØÖÒ×ÓÖÑ×ÓÙ×ÐÓÔÖØÓÒÈØ 

A µ = ψγ µ γ5 ÈÖØ×ÐÔØÓÒ× 

ψ 

ÒØÖÑÓÒ ÕÙ×ÓÐÙØÓÒ×ÐÕÙØÓÒÖÇÒØÖÓÙÚÕÙÔÓÙÖÙÒÔÖÖÑÓÒ ÈÖØÙÐØÒØÔÖØÙÐÔÔÖ××ÒØÒ×ÙÒÑÑÓÙÐ×ÔÒÙÖÒØÒØ 

PfP 

 

 

f = −1

Ô×Ø×ÔÖÐÒØÖØÓÒÓÖØÁØÓÑÑÓÒÐÑÒØÓÒÒÐÒØÖØÓÒ ÈÓÙÖØ×ØÖÐÓÒÓØÖÖÓÙÖ×ÐÒØÖØÓÒÑ ÖÐ×ÐÔØÓÒ×Ò×ÓÒØ 

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γγÑÓÒØÖÕÙ×Ø 

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ÈÖØÙÔÓÒ ÓÒÓÒØØÖÙÙÔÓØÓÒÙÒÔÖØÒØÚÄÔÓØÓÒ×ØÙÒÚØÙÖ ÐEÔÖE= −∇V ÈÙ×ÕÙP(E) 

ÓÒ×ÖÓÒ×ØÓÙØÓÖÐÔÖØÙÔÓÒÖÇÒÙØÐ×ÐÖØÓÒ 

= 

 

Ð×ÚÙÒÑÓÑÒØÒÙÐÖÓÖØÐÄ +− 

J P = 1 − 

= ÙÒÔÖØÒØÖÒ×ÕÙ 

ud PÑ×ÓÒ= PqPq(−1) 

L = (−1) L+1 

= +1 

= −1ÓÒÐÔÖÓØÓÒØÐÒÙØÖÓÒÓÒØÙÒÔÖØ+1 

(V, A) A×Ø 

− ∂A µ 

≡ −∂µA ∂t 

−EØP(∂t, ∇) = (∂t, −∇)ÓÒÓØÒØP(V, A) = (V, −A).

π− Ò Ò 

ÂÄ Ü ÓÒ×ÖÚØÓÒÙÂ 

ÐÓÖØËℓ ÄÔÓÒ×ÑØÒÓÖØÙØÓÙÖÙØÓÒÐ×ØÔÖÐ×ÙØÔØÙÖÔÙ× ÕÙÐÔÐÙ×ÖÒÖÓÙÚÖÑÒØÚÐÔÖÓØÓÒÙÙØÓÒÄ 

⇒ 

ÔÖØÐØØÒØÐ×ØÓÒPπPd(−1) ℓ PπPdÄÙØÓÒ×ØÓÖÑÙÒ ×ØÖÚÙÒ 

= ÒÙØÖÓÒØÙÒÔÖÓØÓÒÐ×ÔÖÒÔÐÑÒØÒ×ÐØØ3 S1 ×ÔÒËØÙÒÑÓÑÒØÒÙÐÖÓÖØÐÄ×ÔÖØÚÙØ(+1)(+1)(−1) L +1ÄØØÒØÐÐÖØÓÒ×ØÓÒÖØÖ×ÔÖÙÒÔÖØÐÐÐ ÙÔÓÒ ÐØØÒÐÐÖØÓÒÓÒÔÙØÓÖÑÖ×ÓÑÒ×ÓÒ××ÔÒÒØ×ÝÑ ÎÓÝÓÒ×ÓÑÑÒØØØÖÙÖÐ×ÑÓÑÒØ×ÒÙÐÖ×ÚÐ×ÙÜÒÙØÖÓÒ× 

1ÙØÝÔ|↑↑〉 ØÖÕÙS= 0ÙØÝÔ|↑↓〉Ø×ÝÑØÖÕÙS= 

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Ä×ÙÐÔÓ××ÐØÕÙ×Ø××××ÑÙÐØÒÑÒØÐÓÒ×ÖÚØÓÒJØÐ Ù×Ý×ØÑ×ÙÜÒÙØÖÓÒ×ÓØØÖÒØ×ÝÑØÖÕÙψ(n1, n2) 

×ØØ×ØÕÙÖÑÖ×Ø Si ÒÓÒÐÙ×ÓÒÐÔÖØÐØØÒÐÐÖØÓÒ×ØÓÒÒÔÖPnPn(−1) 1 −1ÄÓÒ×ÖÚØÓÒÐÔÖØÒ×ÐÒØÖØÓÒÓÖØÑÔÓ×ÓÒÙÒÔÖØ 

 

ÒØÖÒ×ÕÙÙÔÓÒÖÐ−1 ÖÔÔÐÐÒÓØØÓÒ×ÔØÖÓ×ÓÔÕÙ2S+1 LJ 

ËËi 

→ 

|0, 0〉 = 1 

√ 1 1 

, 1 

, −1 

2 2 1 2 2 2 

2 

− 

1 

, −1 1 1 

, 2 2 1 2 2 2 

|1, +1〉 = 

1 1 , 1 1 , 2 2 1 2 2 2 

|1, 0〉 = 1 

√ 1 1 , 1, 

−1 

2 2 1 2 2 2 

2 

+ 

1, 

−1 1 1 , 2 2 1 2 2 2 

|1, −1〉 = 

1 

, −1 1 

, −1 

2 

2 

1 

2 

2 

2 

= 1 x = 1ØJ = 1 

= 

= −ψ(n2, n1) 

=

ËiÖÕÙ×ÔÖ ÖÕÙ×ÔÖ 

ÄØÙÐ×ÒØÖØÓÒπ 0 ÔÓÒÒÙØÖ×ØÙ××ÒØÚÄ×ØØ×ÖÙÔÓÒÓÖÑÒØÓÒÙÒØÖÔÐØ 

γγÑÓÒØÖÕÙÐÔÖØÒØÖÒ×ÕÙÙ 

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Ùθ + 

Ü 

θ 

S Sθ + 

L 0 

ÖÑÖ ÓÒ×ÖÚØÓÒÂ 

x = 0, 2, 4, ... x = 1 

x = 1, 3, 5, ... x = 0, 1, 2 

JP = 0−ÔÖØÙÐ×Ô×ÙÓ×ÐÖ× + → π + π 0 

J Sθ + ⇒ Ëθ + ÓÒ×ÖÚØÓÒÙJ ÓÜ=Äππ =Âππ =Ëθ + 

Èππ = P π+ P π0 

2 (−1)Äππ = (−1) (−1)Äππ = (−1)Äππ ÒÓÒØÓÒÐÚÐÙÖËτ + Ò×Ð×ÒØÖØÓÒÙτ + ÁÐØØÓÑÑÓÔÓÙÖÐÒÐÝ×ÓÑÔÓ×ÖÐÑÓÑÒØÒÙÐÖÓÖØÐ ÖØÖ×ÒØÐÑÓÙÚÑÒØÖÐØÙπ −ÔÖÖÔÔÓÖØÙ×Ý×ØÑπ + π +ÄÔÓÒ 

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ÒÝÒØÔ××ÔÒÓÒ| 

 

L − ℓ |≤ Jπππ ≤| L + ℓ | 

ÄÔÖØθ +ÔÓÙÚØÓÒØÖÖØÖ×ÔÖÐ×ÕÙÒJ Pθ ++ − + 

ÒÐÒÙÜØÖÑ×ÄÖØÖ×ÒØÐÑÓÙÚÑÒØÖÐØ×ÙÜπ +Øℓ 

Ø ±−8× ÒÙÒØÑ××ÐØÖÓÒÕÙ ± Ø± ±

ℓ 

τ + → π + π + π− Ä S Sτ + 

L 

J Sτ + ⇒ Ëτ + ÄÔÖØÐØØÒÐ×ØÓÒÒÔÖ 

ÓÒ×ÖÚØÓÒÙJ 

Pπππ = Pπ +Pπ +Pπ −(−1)L (−1) ℓ = (−1) 3 (−1) L+ℓ 

ÓØØÖÔÖ ÒØÕÙ×ÚÒØ×Ø×ÖÐ×ØØ×ØÕÙÓ×Ò×ØÒÐÙÖÓÒØÓÒÓÒ ÓÙØÓÒ×ÕÙÄÒÔÙØÔÖÒÖÕÙ×ÚÐÙÖ×ÔÖ×ÖÐ×ØÙÜÓ×ÓÒ× ÇÒÔÓÙÚØÓÒ××ÒÖÐ×ÚÐÙÖ××ÔÒÔÖØ×ÙÚÒØ×ÐÔÖØÙÐτ + ÒÓÒØÓÒ×ÚÐÙÖ×ℓØÄÚÓÖÙÖ 

π 

+ 

0 0 

L 

+ 

1 0 1 

− 

2 0 2 

− 

0 2 2 

+ + + 

π + 

1 2 1 2 3 

− − − − − 

2 2 0 1 2 3 4 

ÍÒÒÐÝ××ÖØÖ×ØÕÙ×ÒÑØÕÙ×Ð×ÒØÖØÓÒÒ ÒØÓÒℓØÄÔÓÙÖÐ×Ý×ØÑØÖÓ×ÔÓÒ× 

π − 

ÓÖÔ× ÐÔÖØÙÐτ +ØÔÖÐØÞÚØÑÓÒØÖÕÙÐ××ÒØÓÒJ Pτ + = 0− ×ÔÒÔÖØØÔÓÙÖÐÔÖØÙÐθÆÓØÓÒ×ÕÙÐ×ÓÖ×Ù××Ø×ÐÓÒ ØØÒØØÑÒØÚÓÖ×ØØÔÖØÓÒØØÒÓÑÔØÐÚÐÔÖØÓÒ ÔÖÒÐ×ÔÖÑÖ×ØØ×ÜØØÓÒÓÖØÐÙ×Ý×ØÑ×ØÖÓ×ÔÓÒ× ×ÔÖØÙÐ××ØÒØ××ÓØÐ×ØÐÑÑÔÖØÙÐØÐÙÒ×ÔÖÓ××Ù× ×ÒØÖØÓÒÚÓÐÐÔÖØ ÊÀÐØÞÈÐÅ 

×Ö×ÙÐØØ×ÚÒØÓÒÙØÙÜÓÒÐÙ×ÓÒ××ÙÚÒØ××ÓØÐθØÐτ×ÓÒØ 

ÇÒ×ØÙÓÙÖÙÕÙÐ×ØÒÐÑÑÔÖØÙÐÕÙÐÓÒ×Ò×ÓÙ×ÐÒÓÑ 

ÓÒ 

L 

J P 

0 −

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ÄÜÔÖÒÏÙØÐ Æ 

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ÑÓÒØÖÕÙÐÓÒÙÖØÓÒÙ×ØÔÐÙ×ÔÖÓÐÕÙÐÐÖÓØ ÓÙÓÒÈÖÓÒØÖÐÖØÓÒÑ××ÓÒ×−×ØÓÒ×ÖÚÄÜÔÖÒ 

ÓÐ×ÙÜÐÔØÓÒ××ÓÒØÑ×Ó×Ó×ÈÙ×ÕÙÐÒØÒÙØÖÒÓ×ØÜÐÙ×ÚÑÒØ ØÓÒÒÒÔ×ÐÖ×ÙÐØØÐ×Ù××ÓÒ ÂCoØÐ60Æ∗ÙÒ×ÔÒÂNiÓÒ×ÖÓÒ×ÚÓÖÙÖ ÌÄÆÒÈÝ×ÊÚ 

ÄÒØÖÔÖØØÓÒÑÖÓ×ÓÔÕÙÙÔÖÓ××Ù××ØÐ×ÙÚÒØÐ60ÓÙÒ×ÔÒ 

Ð×ÜØÑ ·p ÐÄÖÓØØÓÒ×Ù×ÕÙÒØ×◦ÒØÔ×ÐÚÐÙÖÙÔÖÓÙØJCo 

ËÏÙÑÐÖÊÀÝÛÖÀÓÔÔ×ÊÀÙ×ÓÒÈÝ×ÊÚ 

ÏÙØÐØÙÒØÐ×ÒØÖØÓÒØÏÁÙÓÐØ 

60 Co → 60 Ni ∗ + e − + νe 

ÐÓÔÖØÙÖÔÖØÔÖÓÙØ×ÐÖÓÒÓØÒØ 

P(JCo · p e) = −JCo · p e 

(JCo · p e×ØÙÒÔ×ÙÓ×ÐÖ 

P(< JCo · p e >) = − < JCo · p e > 

×ØÓÒÒÓÒÒÙÐÐØÔÐÙ×ÒØÚÖ×ÙÐØØÔÔÓÖØÐ 

· p e >= 0 

0ÐÔÖØ×ØÓÒÚÓÐÒ×ÐÔÖÓ××Ù××ÒØÖØÓÒ 

< JCo · pe > < 

e

i 

i 

e 

JCo 

ÖÓØÖÐØ ÊÔÖ×ÒØØÓÒ×ÑØÕÙÐÜÔÖÒÏÙØÐ 

ÓÖØÖÙÖÐØ−1ØØÖÑ×ÒÖØÓÒÓÔÔÓ×Ù×ÔÒÙ ÓÒÓÑÔÖÒÕÙÒ×ØØÓÒÙÖØÓÒÐÐØÖÓÒ×ÓØ 

Miroir 

ÜÑÔÐÒ×ÐÍÒÚÖ×ÓÒØÖÓÙÚ ÓÓÒ××ÙÖÖÐÓÒ×ÖÚØÓÒÙÑÓÑÒØÒÙÐÖ ÐÓÔÔÓ××ØÚÖÔÓÙÖÐ×ÒØÒÙØÖÒÓ× ÇÒÔÙØÖÑÖÕÙÖÕÙÐÚÓÐØÓÒÓ×ÖÚÐÔÖØ×ØÑÜÑÐÈÖ 

ÔÖÐÒØÖØÓÒÐÔÖÜÑÔÐÒ×Ð×ÒØÖØÓÒÑÙÓÒ×ÔÓÐÖ×× ÄÚÓÐØÓÒÐÔÖØØÓ×ÖÚÔÖÐ×ÙØÒ×ÙØÖ×ÔÖÓ××Ù×Ö× νÙÖ×Ø0%νÖÓØÖ× 

ÕÙÐ××ÓÒØÒÚ×Ð× Ñ×ÕÙ×ÖÒÖ××ÓÒØ×ØÖÐ× ØÝÔÖÓÒ×ÔÓÐÖ×× 

×ÐÖØÐÈÝ×ÊÚ ÊÄÖÛÒØÐÈÝ×ÊÚ ÍÒÙØÖÔÓÒØÚÙ×ØÕÙÐÜ×ØÙÒÒÓÑÖÐÒÙØÖÒÓ×ÙÖ×ØÖÓØÖ× ÕÙÐ×ÒÓÙÔÐÒØÚÙÙÒÙØÖÔÖØÙÐÓÒ ØËÖÛÓÖØÐÈÝ×ÊÚ 

 

JCo 

e 

i 

i

z 

J z = 5 

J z = 4 

J z = 1 

e 

ÇÒÖÔÖÒÖ×ÙØÔÐÙ×ÒØÐÙÔØÖÓÒ×ÖÐÒØÖØÓÒ ×ÒØÖØÓÒÙ60ÓÓÒÙÖØÓÒÚÓÖ× 

60 

60 

Co 

Ni * 

Ð ÔÐÐ×ÔÖØÙÐ×ÙÒ×Ý×ØÑÔÖÐÙÖ×ÒØÔÖØÙÐ××Ò×ÐØÖÖÐ×ÙØÖ× ÄÓÒÙ×ÓÒÖCÓÔÖØÙÖÕÙÒØÕÙC×ØÐÓÔÖØÓÒÕÙÖÑ ÄÓÒÙ×ÓÒÖ 

Ð×Ö×ØÐ×ÑÓÑÒØ×ÑÒØÕÙ××ÓÒØÒÚÖ××ÓÒÐÐÒÒÖØÕÙ 

p)×ÔÒÐØ ÈÖØØÓÔÖØÓÒØÓÙØ× ÖØÖ×ØÕÙ×ÕÙÖÚØÙÖ(E, 

××ÓÐÒØÖØÓÒÑÒ×ÙØÔ×ÒÑÒØ 

[Hem, C] = 0 

×ØÙ××ÙÒ×ÝÑØÖÐÁÓÒ 

[Hem + HIF, C] = 0 

ÈÖÓÒØÖÐÀÑÐØÓÒÒÐÒØÖØÓÒÐÒ×ØÔ×ÒÚÖÒØ×ÓÙ× 

[HWI, C] = 0 

 

νe

ÇÒÚ×ÐÑØÖÔÓÙÖÐÑÓÑÒØ 

ÁÐÙØ×ØÒÙÖÙÜÐ×××ÔÖØÙÐ× H = Hem 

a 

+ HIF 

ÓÒÔÓÙÖÐ×ÕÙÐÐ×= 

 

Ð×ÔÖØÙÐ×ÕÙ×ÓÒØÖÒØ×ÐÙÖ×ÒØÔÖØÙÐ×ÔÒπ + , K + Ð×ÔÖØÙÐ×ÕÙ×ÓÒØÒØÕÙ×ÐÙÖ×ÒØÔÖØÙÐ×α=γ, π0 ÉØÙÖÐ×ÙÒÙØÖÓÒÒØÖÑ××ØÖÙØÙÖÕÙÖ×ÁÑÔÓÙÖÐ , η 

×Ý×ØÑ×ÃÒÙØÖ× 

ÐÑÒØÖ ÕÙ×ØÒ×ÒÔÖÕÙÔÖÓÙÚÕÙÐÒ×ØÔ×ÙÒÓÒ×ØØÙÒØ ÒÕÙÐÒÙØÖÓÒØÙÒÖÒÙÐÐÐÙÒÑÓÑÒØÑÒØÕÙÒÓÒÒÙÐ 

Ò×ÒÕÙÒØØÑÓÙÚÑÒØ×ÔÒ ψ〉Ð×ÔÒ 

ÚØØÒÓØØÓÒÓÒÔÓÙÖÐ×ÙÜÐ×××ÔÖØÙÐ×××Ù× 

ÆÓÙ×Ò××ÓÒ×ÐÓÒØÓÒÓÒÔÜÙÒÔÖÓØÓÒÔÖ|p, ØÒØÓÒØÒÙ×Ò×ÐÓÒØÓÒψ 

Ò×ÐÔÖÑÖ×ÓÒÒÔ×ØØÔÖÓÔÖÒ×Ð×ÓÒÓÒÙÒØØÔÖÓÔÖ ÚÐÚÐÙÖÔÖÓÔÖCαÄØÙÖÔ×Cα×Ø×ÑÐÖÐÙÕÙÐÓÒ ÖÒÓÒØÖÚÐÓÔÖØÙÖPÔÙ×ÕÙC 2ÓÒÓØÒØCα Ô××ØÖØÖÖÖØÓÙØÓÒÒÓÒÑ×ÙÖÐÓÒÖÒÓÒØÖÐ×ÙÜ ÔÖÓ×ØÙÖÐÔÖØCÒ×Ð××ÔÖØÙÐ×ØÝÔÐØÙÖ 

±1ÇÒÔÔÐÐ 

ÈÓÙÖÙÒ×Ý×ØÑÔÖØÙÐ× 

= 

ÍÒ×Ý×ØÑÓÑÔÓ×ÙÒÔÖØÙÐØ×ÓÒÒØÔÖØÙÐÑÑØÝÔ×Ø 

= ÓÒÚÒØÓÒ×Ca 

ÙÒØØÔÖÓÔÖÖÒÔÖØÕÙÐÓÔÖØÙÖØÙÐÒ×ÙÜ ÔÖØÙÐ× 

 

C|a, 

ÍÒ×Ý×ØÑÔÓÒÒÙÐÓÒ×ÖÖØÔÜÔÖ 

|π + , ψ1; p, ψ2〉 ≡ |π + , ψ1〉|p, ψ2〉 etc... 

C|a, ψ〉 = η|a, ψ〉 Ø 

C|α, 

ψ〉 = Cα|α, ψ〉 

 

±1 

C|a1, ψ1; a2, ψ2; ...; an, ψn; αn+1, ψn+1; ...; αm+n, ψm+n〉 = 

Cαn+1...Cαn+m|a1, ψ1; a2, ψ2; ...; an, ψn; αn+1, ψn+1; ...; αm+n, ψm+n〉 

 

ψ1; a, ψ2〉 = |a, ψ1; a, ψ2〉 = ±|a, ψ1; a, ψ2〉

Ð×Ò±ÔÒÙÖØÖ×ÝÑØÖÕÙÓÙÒØ×ÝÑØÖÕÙÐØØ ×Ý×ØÑÐÓÖ×ÐÒØaÈÖÜÑÔÐÔÓÙÖÐ×Ý×ØÑπ + ÙÔÓÒØÚÙÐÓ×ÖÚØÙÖÐÓÔÖØÓÒÓÒÙ×ÓÒÖÖÚÒØ 

, π− : 

C|π + π− ; L〉 = (−1) L |π + π− 

; L〉 

π π 

r r 

1 π 1 π 

ÌÖÒ×ÓÖÑØÓÒÔÖÙ×Ý×ØÑÙÜÔÓÒ× 

C 

r2 r2 

ÓØÒØÙÒ×Ý×ØÑÕÙÚÐÒØÙÔÖÒØÒ×ÐÕÙÐÐÔÓ×ØÓÒÖÐØÚ× ÒÖÐÔÓ×ØÓÒ×ÙÜÔÖØÙÐ×Ë×ØÙÒ×ÝÑØÖÙ×Ý×ØÑÓÒ 

ÑÔÖ 

r2ÚÒØ−RÕÙ×ÖÔÖÙØ 

ψ(−R)ÓÑÑ ÙÜÔÖØÙÐ×Ò×ÒR 

ÊÔØØÙÚÒØÚÐ×ÙÜ×ÔÒ× 

= 

ÓÒÔÙØÓÖÑÖÐ×ÓÑÒ×ÓÒ××ÔÒ 

r1 − ×ÙÖÐÔÖØ×ÔØÐÐÓÒØÓÒÓÒC(ψ(R)) 

1)×ÙÚÒØ× 

= ×ÙØØØÔÖØ×ÔØÐ×Ø×ÝÑØÖÕÙÔÓÙÖLÔÖÒØ×ÝÑØÖÕÙÔÓÙÖL ÓÒ×ÖÓÒ×Ð×Ù×Ý×ØÑÖÑÓÒÒØÖÑÓÒff ÒØ×ÝÑØÖÕÙ(S = 0)Ø×ÝÑØÖÕÙ(S = 

|0, 0〉 = 1 

√ 1 1 , 1, 

−1 

2 2 1 2 2 2 

2 

− 

1, 

−1 1 1 , 2 2 1 2 2 2 

|1, +1〉 = 

1 1 , 1 1 , 2 2 1 2 2 2 

|1, 0〉 = 1 

√ 1 1 

, 1 

, −1 

2 2 1 2 2 2 

2 

+ 

1 

, −1 1 1 

, 2 2 1 2 2 2 

|1, −1〉 = ËÓÙ×ÐÓÔÖØÓÒÔÖØÐÔÖØ×ÔÒÓÖÐÐÐÓÒØÓÒÓÒÑÒ 

LØÐÔÖØ 

 

1, 

−1 1, 

−1 

2 2 1 2 2 2 ÙÒØÙÖÑÙÐØÔÐØ(−1) S+1ÐÔÖØ×ÔØÐÙÒØÙÖ(−1)

ÒØÖÒ×ÕÙÚÓÖÔØÖ ÖÑÓÒÒØÖÑÓÒ ÙÒØÙÖ−1ÓÒÙØÓØÐÔÓÙÖÐ×Ý×ØÑ 

C 

S+L 

f, f, J, L, S = (−1) 

ÇÒÔÙØ×Ò×ÔÖÖÐÔÖÓÙÖÙØÐ×ÔÓÙÖÐÔÖØÈËÓÙ×ÐÔÓØÒØÐ ÄÔÖØÙÔÓØÓÒ 

 

f, f, J, L, S 

×Ò ÄÑÔEØÐÔÓØÒØÐ×ÐÖVÒÒØ×Ò×ÓÙ×CÖØÓÙØ×Ð× ÓÒ 

 

Ö××ÓÒØÒÚÖ××E= −∇V 

Cγ ÄÔÖØÙπ 0 

ÓÒÖÑÜÔÖÑÒØÐÑÒØÔÖÐ×ÒØÖØÓÒÓ×ÖÚÒÙÜÔÓØÓÒ× Ò×ÐÑÓÐ×ÕÙÖ×ÐÔÓÒÒÙØÖ×ØÙÒØØ1Ë0Ù×Ý×ØÑ1 √2 

uu + ddÇÒÐÙÔÖØÓÒÙÒÔÖØπ π 0 ÁÐ×ØÙÒ×ÒØÖØÓÒÑÕÙÓÒ×ÖÚÓÒÔÙØÓÒÖÖ 

→ γγ 

Cπ0 = CγCγ ÐÙÖ ÔÖÓ××Ù××ØÒØÖÔÖØÐÙÖÑÑÒÒÐØÓÒØÖÒÙÐÖ ËÙÖÐ×ØÖØÔÐÒÙÓÙÔÐÑÐ×ÒØÖØÓÒÒÔÓØÓÒ× 

= 1 

×ÖÒÓÑÔØÐÔÙ××ÒÖØÖ×ÐØÓÒ ØÙÖÙÑÓÒ× ÜÔÖÑÒØÐÑÒØÓÒØÖÓÙÚÕÙÐÚÓÒ ÎÓÖÔÜÃÓØØÖÒÎÏ××ÓÔÓÒÔØ×ÓÈÖØÐÈÝ××ÚÓÐ −8ÕÙ×ØÙÒÓÒ×ÕÙÒÐÓÒ×ÖÚØÓÒÇÒ ÔÓØÓÒ××Ø×ÙÔÔÖÑÔÖÙÒ 

ÍÒÈÖ××ÔÒÓØ 

ÇÜÓÖ 

ÚØÙÖ×ØÖÒ×ÓÖÑÓÑÑ 

C(A(t, x)) = CγA(t, x) 

− ∂A µÓÒØÖÕÙAÒÙ×× 

 

≡ ∂µA ∂t 

= −1. = (−1)L+S = (−1) 0+0 1Ð×Ø = 

0 

ÚÖØ×ÑÒØÖÒ×ÐÔÖÓÔÓÖØÓÒ 

Γ(π 0 → γγγ) 

(Γπ 0 → γγ) ≃Ç(αem) ≈ 1 

137 ≈ 1%

ËÑ×ÒØÖØÓÒÙπ 0 

ÈÖØÙη Äη×ØÙÒÔÖØÙÐÔ×ÙÓ×ÐÖÓÑÑÐπ 0ÇÒÐ×ÑÓ××Ò ØÖØÓÒ 

×ØÑÔ×ØÝÔÕÙÐÒØÖØÓÒÑÄ×ÖÔÔÓÖØ×ÑÖÒÑÒØ×ÓÒØ× 

η 

ÒÚÖÒ×ÓÙ×ÐÒØÖØÓÒÑË×ØÙÒÓÒÒ×ÝÑØÖÐØØÒÐ ÑÐÖ×ÓÒÐ×ØÖÓ×ÔÖÓ××Ù××ÓÒØÒØÙÖÑÄ×ÙÜÔÖÑÖ×ÚÓ× ÓÒÒÒØÙÒη ×Ò×ÑÙØÇÒÙØÐ×ÐÖÒÖÚÓÔÓÙÖÙÒØ×Ø ÄØÑÔ×Ú×ØØÐÈτ = 

ÕÙ×ØÓÒÖÑÜÔÖÑÒØÐÑÒØÙÒÚÙ ÒÓÒ×ÕÙÒÐ××ÔØÖ××ÙÜÔÓÒ×Ö×ÓÚÒØØÖÒØÕÙ× 

−3 

→ γ + γ Br = 0.39 

η → π0 + π0 + π0 Br = 0.32 

η → π0 + π + + π− 

Br = 0.24 

Γ = 6.6 10−22 /1.2 10−3 = 5.5 10−19 π 0 (p) + π + (p1) + π − ÓØØÖÒØÕÙ×ÓÒÓÒÙÙ×ÓÙ× 

(p2) 

π 0 (p) + π − (p1) + π + 

(p2)

ÆÓÒÓÒ×ÖÚØÓÒÒ×ÐÒØÖØÓÒÐ ÒÔÖÐ×Ø×ÖØÑÒØÓ×ÖÚÐ×ÙÜÕÙÐ×ÐÐÓÒÒÐÙÈÖÒÓÒ×ÔÖ ÄÒÓÒÓÒ×ÖÚØÓÒÒ×ÐÒØÖØÓÒÐÔÙØØÖÑ×ÒÚ ÜÑÔÐÐ×ÔÖÓ××Ù××ÒØÖØÓÒÐπ ± ×ÔÖÓÔÖØ×ÙÒÙØÖÒÓØÐÓÒ×ÖÚØÓÒÙÑÓÑÒØÒÙÐÖÐµ +Ø Ðµ −ÓÒØ×ÐØ×Ø×ÔÓÐÖ×ØÓÒ×ÓÔÔÓ××ÁÑÒÓÒ×ÙÒÜÔÖÒ ÖÐ×Ò×ÙÒ×ÙÑÜØπ ±ÚÙÒÔÔÖÐÐÔÐ×ÐØÓÒÒÖ Ð×ÑÙÓÒ×Ò×ÙÒØØÐØÔÓÐÖ×ØÓÒÓÒÒÄØÙÜÓÑÔØ ÐØØÒÐ××ÒØÖØÓÒ×××Ù× ÓØÒÙ×ÖÒÓÒÒÙÐÒ×ÐÙÒ×ÚÓ×ØÒÙÐÒ×ÐÙØÖÄÚÓÐØÓÒ 

ÉÑÓÒØÖÖÕÙ×ÐÓÒØ×ÙÚÖÈÓÒÖ×ØÙÖÐ×ÝÑØÖÙ×Ý×ØÑ ÑÒÓÒ××ØÙØÓÒ×ÒÓÒÕÙÚÐÒØ×ÒÕÙÓÒÖÒÐ×ÔÖØÙÐ× 

ØÈ 

 

ÒÚÖ×ÖÐÙØÑÔ×ÓÒÚÖÖØ××ÒØÐÐÑÒØÐÑÑÔÝ×ÕÙØÐ ÄÒÚÖÒÔÖÖÒÚÖ×ÑÒØÙØÑÔ×ÓÙÐÐÕÙ×ÐÓÒÔÓÙÚØ ÄÖÒÚÖ×ÑÒØÙØÑÔ× 

×ÓØ ν, 

×Ý×ØÑÔÓÙÖÐÕÙÐ 

×ÖØÒ×ÖÒÐÐÓÖÒÐÄÑÒÕÙÆÛØÓÒ×ØÒÚÖÒØÔÖÖÒ 

T 

ÓÒ×ÙÒÔÖØÙÐÔÖÓÙÖØÙÒØÖØÓÖr(t)ØØÑÑØÖØÓÖ×Ö ÙÒ×ÓÐÙØÓÒÔÓÙÖÐ×Ý×ØÑÙÑÓÙÚÑÒØÒÚÖ×r TØÔ××ÔÖÐ×ÑÑ× ÔÓÒØ×ÕÙr(t)Ñ××ØÑÔ×−tr T ÐÐ×ÔÜÐÓÖ×ØÙÒÓÒØÓÒÐÚØ××ÆØÙÖÐÐÑÒØØÓÙØÐÓØ ØÖÓÒ×ÖÙÒÚÙÑÖÓ×ÓÔÕÙÖÓÒ×ØÕÙÙÒ×Ý×ØÑÑÖÓ×ÓÔÕÙ 

ØÙÖÐÙÑÓÙÚÑÒØÒÖØÓÒÐÕÙÒØØÑÓÙÚÑÒØp×Ø ÚÓÐÙÚÖ×ÙÒÓÒØÓÒÔÐÙ×ÓØÕÙ ÑÒÙÒ×Ý×ØÑÝÒÑÕÙÑÒØÕÙÚÐÒØÐÓÖÒÐÔÓÙÖÐÕÙÐØÓÙØÚ ÒÒ−pÐÑÓÑÒØÒÙÐÖJÒ−JÐÓÙÖÒØÑÓÙÙØÖjÒ ËÐÖÒÚÖ×ÑÒØÙØÑÔ××ØÙÒ×ÝÑØÖÙ×Ý×ØÑ×ÓÒÔÔÐØÓÒ 

−jØ ×ÓÒØÒ× ÁÐÙØÓÙØÖÕÙÐÓÖ×ÙÒÒØÖØÓÒÐØØÒØÐØÐØØÒÐ ÆÔÖÐ×ÙØÓÒÙØÐ×ÖÐÒÓØØÓÒA T 

→ µ ± +νµ(νµ)ÒÓÒ×ÕÙÒ 

ÐØØÙÒÙØÖÒÓÙÖÉÙÚÐÒØ 1 λ = − ν, 1 

 

λ = − 2 

2 

ν, λ = −1 2 

ÚÖ×ÑÒØÙØÑÔ×ÖÐÐ×ÜÔÖÑÔÖ×ÕÙØÓÒ×Ù×ÓÒÖÒØ 

: (t, r) ↦→ (−t, r) 

m d2 

dt2r(t) = F (r(t)) 

r(−t)ØØ×ÝÑØÖÒ×ØÔ×Ú 

(t) = 

= T(A)

ÔÐÙ×ÓÑÔÐÕÙØÖØÖÕÙÐÙ××ÝÑØÖ×PØCÖÓÒÒÔ×ÕÙÒ 

TØØÒØÐ→ØØÒÐ ÄÔÖÓÐÑÐ×ÝÑØÖT×ØÓÒÔØÙÐÐÑÒØØÑØÑØÕÙÑÒØ ØTØØÒÐ→ØØÒØÐ 

ØØÓÒ×ÖÚÅØÑØÕÙÑÒØÐ×ÜÔÐÕÙÔÖÐØÕÙÐÓÒÒ×ØÔ× ÙÒÓÔÖØÙÖÐÒÖÖÑØÕÙÍÒ×Ý×ØÑ 

ËÐÓÒ×ÙÔÔÓ×HÒÚÖÒØ×ÓÙ×TØÓÒÒÐ×ÒtÓÒÓØÒØ 

ÓÒ×ØÖÙÖÙÒÓ×ÖÚÐT 

ÈÖÓÒØÖ×ÐÓÒÔÖÒÐÓÒÙÙÓÑÔÐÜ t) ÇÒÓÒ×ØØÕÙÐÓÒØÓÒψ(r, −t)ÒÓØÔ×ÐÑÑÕÙØÓÒÕÙψ(r, ÒØÒÒØÓÑÔØH ∗ ÓÒ H = 

Hψ ÇÒÚÓØÕÙψ(r, t)Øψ ∗ −t)Ó××ÒØÐÑÑÕÙØÓÒÄØØØÖÒ×ÓÖÑ 

ÐÒ×ØÔ××ÙÖÔÖÒÒØÖÓÒÓ×ÖÚÕÙÐ×ØØ×ÒØÐØÒÐÓÚÒØ 

r, ÔÙØÓÒØÖÖØÔÖT(ψ(r, t)) 

×Ò××ÔÒÖØÖ×ÔÖÙÒÀÑÐØÓÒÒHÓØÐÕÙØÓÒËÖÒÖ 

Hψ(r, t) = i ∂ 

∂t ψ(r, t) 

Hψ(r, −t) = −i ∂ 

∂t ψ(r, −t) 

∗ (r, −t) = i ∂ 

∂t ψ∗ 

(r, −t) 

= ψ ∗ ØÖÒ×T×ØÙÒÓÔÖØÙÖÒØÙÒØÖÒÒÓØØÓÒÖØÓÒ 

(r, −t) 

T |ψ〉 = |ψ〉 T 

T 〈ψ| = T 〈ψ| 

T f 

T i = f 

T 

i = i f = f ∗ 

i ÚÐÓÒ×ÕÙÒÕÙ|ψ〉 = a|ψ1〉 + b|ψ2〉 ⇒ T |ψ〉 = a∗T |ψ1〉 + b∗ |ψ2〉 

Ü×ØÖØÔÖ ÒÖÙÒÕÙÒØØÓÒ×ÖÚ ÙØÖÔÖØÓÒÒÔÙØÔ×ÓÒ×ØÖÙÖÙÒÚØÙÖÔÖÓÔÖTØÓÖØÓÖ 

p)×ÔÖÓÔÒØÐÐÓÒÐÜ 

T 

ÄØØÙÒÔÖØÙÐÕÙÖÚØÙÖ(E, ψ(x, 

t) ∝ e i(p·x−Et) ÄÔÔÐØÓÒÙÖÒÚÖ×ÑÒØÙØÑÔ×ÓÒÒ 

Tψ(x, t) ∝ e i(−p·x−Et)

ÕÙ×ØÕÙÚÐÒØÐØØÙÒÔÖØÙÐ×ÔÖÓÔÒØ×ÐÓÒÐÜÜ ÉÔÖÓÙÚÞÕÙ 

ØÔÖÒÐÓ 

ÉÕÙÐÐ×ØÐØÓÒÌ×ÙÖÐÐØÙÒÔÖØÙÐ 

 

−Sz〉ØØÓÔÖØÓÒ×ØÓÒ Sz〉ÒØÖÒ 

ÇÒÔÙØÑÓÒØÖÖÕÙÔÓÙÖÙÒÔÖØÙÐ×ÔÒ ÖØÔÖÐ×ÔÒÙÖ 

ÄÓÔÖØÓÒÌÔÔÐÕÙÙÒÔÖØÙÐÒ×ÐØØ×ÔÒ|S, ÙÒÖÒÚÖ×ÑÒØÙ×ÔÒØÑÒÐØØ|S, 

(r,ØÐØØØÖÒ×ÓÖÑ×ØÓÒÒÔÖ 

Ó 

ÄÓÖ×ÙÒÐÙÐÐÑÒØÑØÖÅ××ÓÙÒØÖÒ×ØÓÒ→ÐÒÚ ÖÒÔÖÖÒÚÖ×ÑÒØÙØÑÔ×ÑÒÐÕÙÚÐÒ 

σy 

×ØÓÒÐØØÚÖ×ÐØØ×ØÒØÕÙÐÐÐØØØÖÒ×ÓÖÑÚÖ× ËÐÖÒÚÖ×ÑÒØÙØÑÔ××ØÙÒ×ÝÑØÖÙ×Ý×ØÑÐÑÔÐØÙØÖÒ 

 

 

T ψ 

rψT 

(t) = 〈ψ|r|ψ〉(−t) ÑÑÓÒÓØÒØ T ψ 

pψT 

(t) = −〈ψ|p|ψ〉(−t) 

 

T ψ 

LψT 

(t) = −〈ψ|L|ψ〉(−t) 

T ψ 

SψT 

ÕÙÚÐÒØÙÒÖÓØØÓÒ◦ÙØÓÙÖÐÜÝ 

(t) = −〈ψ|S|ψ〉(−t) 

 

T |S, Sz〉 ∝ Ry(π)|S, Sz〉 

ψ+ 

ψ− 

 

 

T ψ+ 

ψ− 

 

(r,Ø= −iσy 

ψ+ 

ψ− 

 

0 −i 

= 

i 0 

〈ψB|M|ψA〉 = ψT ÐØØØÖÒ×ÓÖÑ ×ØÕÙÚÐÒØ 

 

 

A MψT B 

A → B B 

 

(r, −Ø 

T → A T

×ÓÒØÖÒØ×ÖÐ×ÔÔ×××ÐÒ×ØÔ×ÐÑÑÔÓÙÖÐ×ÙÜ ÍÒÓÒØ×ØÖÐÒÚÖÒ×ÓÙ×Ì×ØÓÑÔÖÖÐØÙÜÙÒÖØÓÒ ÐÙ×ÓÒÒÚÖ× →Ó=Ò= 

ØØ×ÔÖÙÒ×ÔÓ×ØÒ×Ò×ÐÙ×ÔÒÓÒ ÖØÓÒ× Ò×Ð×ÓÐ×ÔÖÓØÐ××ÓÒØÒÓÒÔÓÐÖ××ØÐ×ÔÖÓÙØ×ÖØÓÒ×ÓÒØ 

→ ÁÐÙØ×ÓÙÐÒÖÕÙÑÑ×| MAB 

ÖÖÒØÐÙÑÐÖØÓÒØÐ×Si×ÓÒØÐ×ÚÐÙÖ×Ù×ÔÒ×ÔÖØÙÐ× ÓpabØpcd×ÓÒØÐ×ÚÐÙÖ×ÐÕÙÒØØÑÓÙÚÑÒØ×ÔÖØÙÐ×Ò×Ð 

ØÐ×ÚÐÝÔÓØ×ÒÚÖÒ×ÓÙ×ÌÌÓÙØÓ×ÐÔÖ×ÓÒÜÔÖÑÒØÐ ×ÙÖÒÖØ×ØÒ×ØÕÙÐÓÖÖÙ ×ØÐÔÖÒÔÙÐÒØÐÐÚÓÖÔØÖ Ä×Ö×ÙÐØØ××Ø×Ø×ØÙ××ÙÖ×ÒØÖØÓÒ×ÑØÓÖØ×ÓÒØÓÑÔ 

ÓÖÑÖ×qi×ØÙ×ÒriÔÖ ÐØÖÕÙÇÒÒØÐ××ÕÙÑÒØÐÑÓÑÒØÔÓÐÖÐØÖÕÙÙÒ×Ý×ØÑ ÍÒÙØÖØ×ØÒÚÖÒ×ÓÙ×ÌÖÔÓ××ÙÖÐÑ×ÙÖÙÑÓÑÒØÔÓÐÖ 

TÐ××ÒÒ×qØrÓÒd×ØÙ××ÒÒËT×ØÙÒ×ÝÑØÖÐ 

ÑÓÑÒØÔÓÐÖdÓØØÖÒÙÐÔÐÙ×Ð×ÙÐÚØÙÖÕÙÖØÖ×ÙÒ ÐØÖÕÙ×ÓÒØÔÖÐÐÐ×ÓÒÒØÙØÚÓÒÖØ ÔÖØÙÐÙÖÔÓ××Ø×ÓÒ×ÔÒÇÒ×ØØÒÓÒÕÙ×ÔÒØÑÓÑÒØÔÓÐÖ 

d×ØÒÚÖÒØÐÓÖ×ÕÙSÒ×Ò 

d = fS ËÓÙ×ÐÓÔÖØÓÒT 

ÑØÓ×Ö×ÓÒÒÑÒØÕÙÍÒ×ÑØÓ×ÓÒ××ØÔÓÐÖ×ÖÐ× ÒÙØÖÓÒ×ÔÖÓÙØ×Ò×ÙÒÖØÙÖÔÖÙ×ÓÒ×ÙÖÙÒÖÒÖÑÒØ× ÄÑÓÑÒØÐØÖÕÙÔÓÐÖÙÒÙØÖÓÒÔÙØØÖÑ×ÙÖÔÖÖÒØ× 

T : d ↦→ −fS. ÄØÙÖÓØÓÒØÖÒÙÐÓd 

ÙÒÖØÓÙÖÒÑÒØ××ÔÒ×ÔÖÔÔÐØÓÒÙÒÊÐÖÕÙÒÕÙØË ×ÙÔÖØÐÐÑÒØÔÓÐÖ×ØÖÚÖ×ÙÒÖÓÒÓ×ÓÒØÔÔÐÕÙ×× ÑÔ×ÐØÖÕÙØÑÒØÕÙÔÖÐÐÐ×ØÙÒÓÖÑ×ÇÒÖØÙÖ 

= 

ÐØÖÒØÚÑÒØÒÔÖ×ÒØÒÐ×ÒÑÔÐØÖÕÙÓÒÖÙÖ ØØÖÕÙÒÔÖÖÔÔÓÖØÐÚÐÙÖÄÖÑÓÖÄÑ×ÙÖ×ØØÙ ÐÑÓÑÒØÐØÖÕÙ×ØÒÓÒÒÙÐÓÒ×ØØÒØÖÓÙÚÖÙÒÐÖÔÐÑÒØ 

 

|=| MBA |Ð×ØÙÜÚÒÑÒØ×ØØÒÙ× 

(2Sa + 1)(2Sb + 1) p 2 dσab 

ab 

dΩ = (2Sc + 1)(2Sd + 1) p 2 dσcd 

cd 

dΩ 

d = 

i 

0 

qiri

ÐÑØ×ÙÔÖÙÖÄÑÓÝÒÒ×ÜÔÖÒ×ØÙÐÐ×Ð×ÔÐÙ×ÔÖ××ÓÒÒ ÐÖ×ÕÙ×ÜÔÖÑÒØÐÄÖ×ÙÐØØÓØÒÙ×ØÜÔÖÑ×ÓÙ×ÐÓÖÑÙÒ 

ÉÖÖÒ×ÐØÐ×ÔÖØÙÐ×Ð×Ö×ÙÐØØ×ÔÓÙÖÐÐØÖÓÒØÐÔÖÓØÓÒ 

< −25ÑÓÒÒÄÚÐÄ 

ÎÓÐØÓÒÈØÐ×Ý×ØÑ×ÃÓÒ×ÒÙØÖ× 

ØÐÈneutron 

ÕÙÐÓÔÖØÓÒÈØØÐÑÒØÙÒÓÒÒ×ÝÑØÖÒ×ÐÒØÖØÓÒÐ ÉÙÒ×ØÐÒ×ÐÒØÖØÓÒÐÂÙ×ÕÙÙÜÒÚÖÓÒ×ÓÒÓÒ×Ö ÕÙÔÔÖ××ØÖÓÒÓÖØÒØÐÚÓÐØÓÒÈØÒØÔÖÙÒÓÒÓÙÖ×ÖÓÒ× È×ØÙÒÖÒÙÖÓÒ×ÖÚÒ×Ð×ÒØÖØÓÒ×ÓÖØØÐØÖÓÑÒØÕÙ 

Ð×ÓÈ×ØÑÒ×ØÑÒØÚÓÐ ØÒ×ÓÑÔÒ×ÔÖÐÚÓÐØÓÒÔÙ×ÐÓÖ×ÓÒÓÙÚÖØ×ÔÖÓ××Ù× Ä×Ý×ØÑ×ÃÓÒ×ÒÙØÖ× ÄÑ×ÓÒK 0Ø×ÓÒÒØÔÖØÙÐK 0ÓÒØÐØÖÒØËK0 1ØËK ÒØÖÑ×ØÖÙØÙÖÒÕÙÖ×ÓÒK 0 ØÓÒÐÄ××ÒØÖØÓÒ× ÐØÖÒØ−1ÄÒÓÑÖÕÙÒØÕÙØÖÒØÒ×ØÔ×ÓÒ×ÖÚÒ×ÐÒØÖ 

dsÆÐÕÙÖ× 

×ÓÒØÓ×ÖÚ×ÐÐ×ÔÙÚÒØØÖÖØ×ÔÖÙÒÖÔÙØÝÔÖÔÖ×ÒØÒ× 

= dsØK = 

ÐÙÖ Ð×ÙÜÔÓÒ×ÒÙÜÒ×ÐÙÖ×ÖÒØ×ØØ×Ö ÚÐ×ÙÜÕÙÖ×ØÐ×ÙÜÒØÕÙÖ××ÓÖØÒØ×ÓÒÔÙØÓÖÑÖ 

0 

K K 0 ( ) 

d (d) 

( ) 

s 

s 

K 0 (K 0 ) → π + π − Ø 

K 

+ 

W (W ) 

0 

0 (K 0 ) → π 0 π 0 

− 

d (d) 

u ( u) 

d ( u ) 

u ( d) 

= 

0 = −1 

π π− + 

ou 

0 0 π π 

ÖÑÑ×ÒØÖØÓÒÙK 0 (K 0 ÐÔÓ××ÐØÙÒÑÒ×ÑØÐÕÙÐÙÖÔÖ×ÒØÐÙÖ 0×ÙÖ 

ÁÐ×ØÙÒ 

) ÄÓ×ÖÚØÓÒ×ÑÓ××ÒØÖØÓÒÓÑÑÙÒ×ÙK 0ØÐK

0 

d u s 

+ 

K W W 

s u d 

ÐÒØÖÚÒØÓÒÐÒØÖØÓÒÐÙÒØØØÖÒØË−1(+1)×ØÖÒ×ÓÖÑ ÖÑÑÒÐÐÓÑÔÖÒÒØÙÒÓÙÐÒÓ×ÓÒÚØÙÖÏÈÖ ×ÔÓÒØÒÑÒØÒÙÒØØØÖÒØË+1(−1)ØÖÚÖ×ÙÒØØÚÖØÙÐÙÜ 

ÖÑÑÓÒÚÖ×ÓÒÙÒK 0ÒÙÒK 0 

ÔÓÒ× K 0 ↔ 2π ↔ K 0 ÒÖ×ÓÒÑÒ×ÑÐÓÒØÓÒÓÒÙÒÓÒÙÒÒ×ØÒØÓÒÒ ÔÙØØÖÖÔÖ×ÒØÓÑÑÙÒ×ÙÔÖÔÓ×ØÓÒ|K 0 K 

〉 = |ds〉Ø 0 

= ÓÒ×ÖÓÒ××ÓÒ×ÙÖÔÓ×ØÐÙÖÒÚÙÒÖØÕÙÐÔÐÙ××Ð× 

ds P |K0 〉 = −|K0 Ø 

 

〉 PK 

0 

 

 

= −K 

0 ÙØÖÔÖØÚÐÓÒÚÒØÓÒη=−1ÓÒ 

C|K0 

 

〉 = −K 

0 Ø 

 

CK 

0 

ÖÒÒÖÙÜÓÒÐÙ×ÓÒ×ÔÝ×ÕÙ×ÄÓÔÖØÓÒÓÒÓÒØØÈÓÒÒ ÊÑÖÕÙÞÕÙÐÓÜη ÒÖØÐ××Ò××ÓÖÑÙÐ×ÕÙ×ÙÚÒØ×Ò× 

CP |K0 

 

〉 = K 0 

 

ØCP K 0 Ò××ÓÒ×Ð×ØØ× 

|K0 1 

 

1 〉 = √ |K 

2 

0 

 

〉 + K 0 

et |K0 2 〉 = 1 

√ |K 

2 

0 

 

〉 − K 0 

 

×ÓÒØÖØÖ××ÔÖÂP−×ØÖÕÙ 

+ 

K 

0 

= −|K0 

〉 

= |K0 

〉

ÔÔÐÕÙÓÒ××ØØ×ÐÓÔÖØÙÖÈ 

CP |K0 1〉 = CP 1 ÑÑ 

√ 

2 

ØÖÓ×ÔÓÒ× ÇÒÓÒÙÜØØ×ÓÖØÓÓÒÙÜÚ×ÚÐÙÖ×ÔÖÓÔÖ×ÈØÖÑÒ× Ò×ÐØØÒÐÐ×ÒØÖØÓÒÒÙÜÔÓÒ×ÓÒÔÙØÒØÖÓÙÖÐ ÓÒ×ÖÓÒ×ÑÒØÒÒØÐ×ÒØÖØÓÒÐÙÓÒÒÙØÖÒÙÜØ 

CP 

0ÔÖ×ÙØ ÑÓÑÒØÓÖØÐÖÐØÄππÄÓÒØÐÔÓÒØÒØ×Ò××ÔÒÄππ = ÐÓÒ×ÖÚØÓÒÙÑÓÑÒØÒÙÐÖÈÓÙÖÐÚÓÒπ 0 Ó CP(π ÈÓÙÖÐÚÓÒπ + Ó CP(π + π − ×ØÙØÓÒÚÓ×ÒÐÐÖØÔÖÐÙÖ ÓÒÓØÒØÖÓÙÖÙÜÑÓ 

π3ÓÒ×ØÒ×ÙÒ 

) = +1 Ò×ÐØØÒÐÐ×ÒØÖØÓÒÒØÖÓ×ÔÓÒ×π1, π2, 

+ ÑÒØ×ÒÙÐÖ×ÓÖØÙÜÄπ1π2Øℓπ3ØÐ×ÕÙÄπ1π2 ÓÒ×ÖÚØÓÒÙÑÓÑÒØÒÙÐÖÈÓÙÖÐÚÓπ 0 Ò×ØÒÖÕÙÖØÕÙÄπ0 Ó CP 

|K 0 1 〉Ø|K 0 2 〉×ÓÒØ×ØØ×ÔÖÓÔÖ×ÈÚÐ×ÚÐÙÖ×ÔÖÓÔÖ× 

 

|K0 

 

〉 + K 0 

= 1 

√ CP |K 

2 

0 

 

〉 + CP K 0 

= 1 

 

K 0 

√ 

2 

 

+ |K0 

〉 = +|K0 1 〉 

Ø−1 

π0ÓÒÔÖØÕÙ π 0 π 0 π 0ÁÐ×Ò×ÙØÕÙ 

CPπ0 1π0 2 

CPπ0 3 

|K 0 2 〉 = −|K0 2 〉 

P(π 0 π 0 ) = (Pπ) 2 · (−1) Lππ = (−1) 2 (−1) 0 = +1 

C(π 0 π 0 ) = (Cπ) 2 = +1 

0 π 0 ) = +1 

π −P(π + π − ) = +1 

C(π + π − ) = (−1) Lππ = +1 

0ÔÖ×ÙØÐ 

ℓπ3 = 

π0π0Ð×ØØ×ØÕÙÓ× 1π0 3×ÓÒØÔÖ×ÒÖ×ÓÒÐ×ÝÑØÖÙ×Ý×ØÑ 

2Øℓ π0 = +1 

= −1 (Cπ0 = +1, P 

3 π0 = (−1)(−1) 

3 ℓπ0 

 

3 = −1) 

(π0π0π0 ) = −1

ÈÓÙÖÐÚÓπ + π−π0ØÖÙÑÒØ×ÝÑØÖÒ×ØÔ×ÚÐÐÈÖÓÒØÖ ÓÒÔÙØÖÑÖÕÙÖÕÙÐÐÒÒÖØÕÙØØ×ÒØÖØÓÒ×ØÔØØÒ 

≃ÅÎÐØØÒÐ×ØÓÒÓÑÒ ÖÖÐÑ×××ÔÖØÙÐ×Éπππ ÔÖÐ×ÓÒ×Äπ + 0 ×ØÙØÓÒÓÑÒÒØÓÒÔÖØÕÙ 

= ÓÒÓ×ÔÖÓÑÑÓØπ3 CP(π + π− Ó ÑÔÓÙÖÐ××ÙÜÔÓÒ× 

ÒÖ×ÙÑ 

) = +1 

CP(π 

π 0Ò×ØØ 

π−Øℓπ CP(π 0 ) = −1 (Cπ0 = +1, Pπ0 = (−1)(−1)ℓπ0 = −1) 

+ π − π 0 

) = −1 

|K0 1 〉 → π0π0 ÓÙπ + π− |K0 2 〉 → π0π0π0 |K0 2 〉 → π+ π−π0ÓÒÒÔÔÖÓÜ |K0 1 〉 π0π0π0 |K0 1 〉 π+ π−π0ÓÒÒÔÔÖÓÜ |K0 2 〉 π0π0 ÓÙπ + π− 

ÙÜØØ×ÓÒ×ÒÙØÖ×ÝÒØ×Ñ×××ÔÖØÕÙÑÒØÒØÕÙ×∼ ÅÎØ×ÙÖ×ÚØÖ×ÖÒØ× ÄØÙÜÔÖÑÒØÐ×ÔÖÓ××Ù××ÒØÖØÓÒÖÚÐÐÜ×ØÒ ÄÙÒÔÔÐKÞÖÓSÓÖØ(K 0 S )ÙÒØÑÔ×Ú(0.8926 ± 0.0012) 10−10 ×Ø××ÒØÖÒÙÜÔÓÒ× 

K 0 S → π+ π − 

K 0 S → π0 π 0 

 

Br = 68.61 ± 0.28% 

Br = 31.39 ± 0.28%

ÄÙØÖÔÔÐKÞÖÓÄÓÒ(K 0 L )ÙÒØÑÔ×Ú(5.7 ××ÒØÖÒØÖÓ×ÔÓÒ×Ò×ÕÙÒÑÓ××ÑÐÔØÓÒÕÙ× 

ÉÔÓÙÖÕÙÓØØÖÒÖÒÒ×ÐØÑÔ×Ú ÄÓÑÔÖ×ÓÒÚÐ×ÖÐ× Ø ×ÙÖÐ××ÓØÓÒ 

Óℓ = 

ÃÞÖÓÇÒÔÖÓÙØÓÒÒÓÖÑÐÑÒØÔÖÒØÖØÓÒÓÖØÕÙÓÒÒÐ× ÁÐÙØ×ÓÙÐÒÖÐÔÖÓÐÑÓÒÔØÙÐÔÓ×ÔÖÐÒØÓÒÐÔÖØÙÐ ØØ×ÔÖÓÔÖ×ÐØÖÒØÃ0ØK 0ÄÔÖØÙÐ××ÒØÖÔÖÒØÖØÓÒ 1Ø 

×ÒØÖØÓÒ ØÖÑÒÓÙÐØØÝÒØÙÒØÑÔ×ÚÒÒ ÒÓÒÑ×ÒÚÒÔÓÙÖÐÔÖÑÖÓ×ÐÜ×ØÒÐÚÓ 

ÐÒÓÒÒÒØ×ØØ×ÙÖÚÒØØ×ÔÖÓÔÖ×ÈÃ0 

ÐÓÖÖ ÁÐ×ØÙÒ×ÒÐØÖ×ØÒÙÔÙ×ÕÙÐÖÔÔÓÖØÑÖÒÑÒØ×Ø 

Ð ÙÒÖÔÔÓÖØÑÖÒÑÒØÙÑÑÓÖÖÖÒÙÖØØÓ×ÖÚØÓÒ ÔÔÓÖØÐÔÖÑÖÔÖÙÚÕÙÈÔÙØØÖÚÓÐÒ×ÙÒÔÖÓ××Ù×ÒØÖØÓÒ 

−3ÄÚÓÒÙÜÔÓÒ×ÒÙØÖ×ØÓ×ÖÚÙÐØÖÙÖÑÒØÚ 

K 

ÐÙ×ÔÓ×ØÐÙÖ Ç×ÖÚØÓÒÐÚÓÐØÓÒÈ 

Ö×ÙÖÙÒÐÈÖÑÐ×ÔÖØÙÐ××ÓÒÖ×ÔÖÓÙØ×Ð×Ö× ÔÓÒ×ÓÒ× ×ÓÒØÐÝ×ÔÖÐÑÔÙÒÑÒØÐ×ÔÓØÓÒ×ÔÖÓÚÒÒØ ÍÒ×ÙÔÖÓØÓÒ× Î×Ø ÄÚÓ×ÒØÖØÓÒÃ0 L 

×ÒØÖØÓÒπ 0 ØÐÖ×ØÔÖÒÔÐÑÒØ×ÓÒ×ÒÙØÖ× ÉÕÙÐÐ×ØÐÖØÓÒÔÓØÓÒ×ÙØÒÖÐÑÒ 

γγ×ÓÒØ×ÓÖ×Ò×ÙÒÖÒÈÑÔ××ÙÖ 

ÂÀÖ×ØÒ×ÓÒØÐÈÝ×ÊÚÄØØ 

→ 

e, µ 

± 0.04) 10−8×Ø Br = 12.38 ± 0.21% 

Br = 21.6 ± 0.8% 

K 0 2 = K 0 

L 

K0 L → π0π + π − 

K0 L → π0π 0 π 0 

K0 L → π± + ℓ ± 

+ νℓ(νℓ) Br = 65.7 ± 0.6% 

K 0 1 = K 0 Ø 

2ÈÖÖØÓÒÓÖÖÐ×ØØÙØÖÐØÐØØÚÙÒÓÒØÒÙÒÕÙÖ× 

S 

Ã0 0 L → π+ π − 

→ π + π −ØÓ×ÖÚÔÓÙÖÐÔÖÑÖÓ×

P 

Aimant 

Cible Pb K 0 

S 

π − 

K 0 

ËÑÙ×ÔÓ×ØÑ×ÙÖÐÚÓÐØÓÒÈÚÐ×ÃÓÒ× ÒÙØÖ× 

L 

ÐÓÐÐÑØÙÖÒØÖÙ×ÔÓ×ØØØÓÒÐÒÖ×ØÔÖØÕÙÑÒØÔÐÙ×ÕÙ 

S×ØØÒÙÖÔÑÒØØÐ×ØÒÑÓ××ØÙ ÄÓÑÔÓ×ÒØÃ0 ÐÓÑÔÓ×ÒØÃ0 L ÉÕÙÐÐ×ØÐÖØÓÒK 0 S×ÙÖÚÚÒØÔÖ××Ñ ÄÖÓÒÓÐ××ÒØÖØÓÒ×K 0 L → π+ π−×ÓÒØÖÖ××ØÒÔÖ ÙÒÒÒØÓÒØÒÒØÐÐÙÑÞÙÜÔÓÙÖÑÒÑ×ÖÐ×ÒØÖØÓÒ×K 0 ÚÐÑÐÙÄ×ÙÜÔÓÒ×Ö××ÓÒØØØ×Ò×ÙÜ×Ö×ÑÖ× ×ÔØÖÓÑØÖÑÒØÕÙ×ØÓÑÔÐØÔÖ××ÒØÐÐØÙÖ×ËØÔÖÙÒ ØÒÐÐ×ÒØÖÐ×ÕÙÐÐ××ØÒØÖÐÙÒÑÒØÒÐÝ×ÓÙÐ 

ÒÖÕÙÖÒØ Ñ×ÙÖ×ØÒÓÒÒÒØÖÐ×ÙÜÖ×Ä××ÓÙÖ×ÖÙØ×ÓÒØÖÙØ× >Ä 

ÕÙÐÑ××ØÚ×ÙÜÔÓÒ××ÓØÓÑÔØÐÚÐÑ××ÙÓÒ ÕÙÐ×ÔÓÒ××ÓÒØÖ×ÓÔÔÓ×× 

ØØÙÖÖÒÓÚÕÙ×ÐØÓÒÒÐ×ÔÖØÙÐ×Ö×ÖÔ×β 

ØÖØÓÖ××ÓØÔÖÓÐÐÒ×Ù ÕÙÐÚÖØÜ×ÒØÖØÓÒÖÓÒ×ØÖÙØÔÖÐÜØÖÔÓÐØÓÒ×ÙÜ 

L××ÒØÖÒÙÜ ÄÜÔÖÒÓÒÙØÙÖ×ÙÐØØÑÔÓÖØÒØÕÙÐÃ0 ÔÓÒ×ÚÙÒÖÔÔÓÖØÑÖÒÑÒØÊÃ0 L → π + π− Ö× →ØÓÙ×Ð×ÑÓ× Ã0 

L 

±·10 −3 ÇÒÔÙØÒÚ×ÖÙÜÑÒ×Ñ× 

π + 

18 m 

E B E 

S 

K 0 

L 

C

ÖÔÖ×ÒØÖÔÕÙÑÒØÔÖÐ× 2×ØÙÒ×Ý×ØÑÕÙÔÙØÚÓÐÖÈÓÑÑ ÐÙÖ 

SØ 1) 

ÙÒÑÒ×ÑÖØÃ0 LÃ0 ÙÒÑÒ×ÑÒÖØÔ××ÒØÔÖÐÑÐÒ×ØØ×Ð×ØØ×Ã0 ØØ×ÖØÖ×ÔÖÙÒÔÖÑØÖǫÔÖÓÖÓÑÔÐÜØØÐÕÙ|ǫ|≪ 

Ã0 LÒ×ÓÒØÔ××ØØ×ÔÙÖ×Ò|K 0 1〉Ø|K 0 2〉Ñ×ÙÒ×ÙÔÖÔÓ×ØÓÒ× 

1) 

π 

π 

K 0 

K L 

0 

0 

L 

K 1 

ÅÒ×Ñ×ÒÚ×Ð×Ð×ÒØÖØÓÒÙK 0 L 

|K0 1 

S 〉 = 

1+ | ǫ | 2 (|K0 1〉 − ǫ|K0 Ø 

2〉) 

|K0 1 

L 〉 = ÄÔÖ×ÒÐÓÑÔÓ×ÒØ|K 0 1〉Ð×ÐÙÖÒ×ÐÑÔÐØÙ ×ÒØÖØÓÒÐØØK 0 LÓÒÒÙÒÔÖÓÐØÓ×ÖÚÖÐÚÓK 0 L → 2π 

|ǫ| 2 ÍÒÒÐÝ×ØÐÐÑÓÒØÖÕÙÐÑÒ×Ñ | 

ÕÙÐÐÐÚÓÐØÓÒÈÄÚÓÐØÓÒÈ×ØÑÜÑÐØÓÒÚØØÖÓÙÚÐ ÑÒ×Ñ 

ÈÒ×ØÔ×ÖÒØÐÒØÖÔÖØØÓÒÔØØØÒ×ØÔ×ÚÒØÇÒ ÓÒØÜØØÓÖÕÙÔÔÖÓÔÖÔÖÑØØÒØÖÖÐÈÖÓÒØÖÐÚÓÐØÓÒ ÄÒØÖÔÖØØÓÒÐÚÓÐØÓÒÈ×ØÙÒÖØÒÓÒÔÐÙ×ÓÑÔÐÜ ×ØÓÑÒÒØÚ| ǫ 

ÖÔÖÒÖ×ÙØÔÐÙ×ØÖ ÄØÙ×ÑÓ××ÑÐÔØÓÒÕÙ××ÒØÖØÓÒÙK 0 ÑÒØÐÓ×ÖÚØÓÒÙÒÚÓÐØÓÒÈËÈØØÓÒ×ÖÚÐ×ÚÓ× 

LÓÒÙØÐ 

K0 L → π−ℓ + νℓØπ + ℓ−νℓ×ÖÒØÕÙÔÖÓÐ×ÜÔÖÑÒØÐÑÒØÓÒÓØÒÙ Γ(π−ℓ + νℓ)−Γ(π + ℓ−νℓ) Γ(π−ℓ + νℓ)+Γ(π + ℓ− Ò××ÔÖÓ××Ù×Ð× 

ÔÓ××Ð×ØÒÙÖÓÒ×ÓÐÙÐÔÖÓÙØÓÒÑØÖØÒØÑØÖ 

10−3ÒÓÒ×ÕÙÒÐÚÓÐØÓÒÈÐ×ØÓÒ 

≃ 3 · νℓ) 

 

1+ | ǫ | 2 (ǫ|K0 1 〉 + |K0 2 〉) 

1 + |ǫ| 2 = |ǫ|2 −3ÔÐÙ×ÙÒÐÓÑÔÓ×ÒØÐÓÖÖ −6××ÓÙ ± 

2) 

π 

π

ÚÑÑÒØ Ö××ÓÖØØØÒÙÑ×Ð×ÔÖÓØÓÒ×Ö×ØÒØ×ÔÖÓØÓÒ××ÙÙÜÕÙÓÒØÒØÖ ÉÙÒÙÒ×ÙÔÖÓØÓÒ×ÔÜÔ××ØÖÚÖ×ÙÒÐÓÑØÖÐÒ ÄÖÒÖØÓÒÙÃ0 

ÑÐÒØØ×ÓÒÜǫ LØÖÚÖ×ÙÒÖÒ 

S 

L×ÓÑÔÓ×Ò SËÐÓÒÒÓÖÐ ÈÖÓÒØÖ×ÐÓÒØÔ××ÖÙÒ×ÙÔÙÖÃ0 ÑØÖÓÒÓ×ÖÚÐ×ÓÖØÐÔÖ×ÒÙÒÓÑÔÓ×ÒØÃ0 

ÚÐ×ÒÙÐÓÒ×ÐÑØÖÔÖ×ÚÓÖÔÖÓÙÖÙÙÒÖØÒØÖØÐ×ÙÜ 

0ÒØÖ××ÒØÓÒÖÒØÒØÖØÓÒÓÖØ 

ÙÒ×ÙÃ0 

ÑÓÑÒØÐØØÙ×Ù×ÖÖØÔÖ Ä×ÓÑÔÓ×ÒØ×K 0ØK ÓÑÔÓ×ÒØ××ÓÒØØØÒÙ×ÔÖÙÒØÙÖØÖ×ÔØÚÑÒØØ< 

2 (a − b)|K0 S 〉 ÕÙÐÓÑÔÓ×ÒØ 

ØÒ×ÐÒØÖØÓÒÓÖØÙØÖÔÖØÐÖØÓÒ 

K 0 + n → K − 

+ p 

Ö×ØÐÓÙÖ× 

|K0 L 〉 = |K0 2〉 = 1 

 

√ |K 

2 

0 

 

〉 − K 0 

|f〉 = 1 

√ a |K 

2 

0 

 

〉 − b K 0 ÍØÐ×ÒØ Ò×ÕÙK 0 

1 

S = √ K 

2 

0 

 

> + K 0ÓÒÔÙØÖÖ×ÓÙ×Ð ÓÖÑ |f〉 = 1 

2 (a + b)|K0 1 

L 〉 + ÇÒÚÓØÕÙ×=ÙÒÓÑÔÓ×ÒØK 0 SÔÔÖØÒ×Ð×ÙÒØÐÑÒØ ÔÙÖÒK 0 ÙÒÔÖØÐÖØÓÒ LÇÒÓ×ÖÚÜÔÖÑÒØÐÑÒØÕÙ< 0×ØÔÐÙ×ÓÖØÑÒØ×ÓÖÕÙÐÓÑÔÓ×ÒØK 0 

K 

K 0 + p → π + + Λ 0 ÒÔ×ÓÒØÖÔÖØÔÓÙÖÐK 0ÒÓÒ×ÕÙÒÐÓÒ×ÖÚØÓÒÐØÖÒ 

×ØÔÐÙ×ÖÕÙÒØÕÙÐÖØÓÒ K 0 + p → K + ÖÐÝÔÐÙ×ÒÙØÖÓÒ×ÕÙÔÖÓØÓÒ×Ò×Ð×ÒÓÝÙÜØÓÑÕÙ×ÒØÖÑ 

+ n

ÄÓ×ÐÐØÓÒØÖÒØ ×ÓÒ×ÒÙØÖ×ØÖÒØËÒ×ÓÒØÔÖÓÙØ×Ò×ÙÒÖØÓÒÓÑÑ 

π− + p → K0 + Λ0 

S = 0 0 1 −1 0ÚÔÔÖØÖÒ×ÐÒØÐÐÓÒK ÙÓÙÖ×ÙØÑÔ×ÙÒÓÑÔÓ×ÒØK 0 

2 2 

α(t) et α(t) 

1 

0.8 

0.6 

0.4 

0.2 

K 0 

0 

− K 

∆mτ 

=0.5 

S 

ÊÔÖ×ÒØØÓÒÐÓ×ÐÐØÓÒØÖÒØÒÓÒØÓÒÙØÑÔ× 

0 

0 2 4 6 8 10 12 t/τ S 

ÔÖÐÑÒ×ÑÔÖÑÑÒØÖØÔÐÙ×ÐÓÑÔÓ×ÒØÓÙÖØÙK 0Ú ×ÔÖØÖÔÐÙ×ÖÔÑÒØÕÙ×ÓÑÔÓ×ÒØÐÓÒÙÓÒ×ÖÓÒ×ÒØÐØØ 

|K0 〉 = 1 

√ (|K 

2 0 L 〉 + |K0 S 〉) 

Ð×ÒØÖØÓÒ×ÔÓÒØÒÖÒÖØÖÑÓÖÖ×ÔÓÒÐ×ÔÖØÓÒ× 

−imxtÓxËÓÖØØÄÓÒÑÙÐØÔÐÔÖÐØØÒÙØÓÒexp(−Γxt/2)Ù Ä×Ý×ØÑÚÓÐÙÒÓÒØÓÒÙØÑÔ×t×ÙÚÒØÐÓ×ÐÐØÓÒexp(−iEt) = 

(exp ÓÒ×Ò×Ð×Ù××ÓÙÒØÑÔ×Úτx = ÐØØÖÓÙÚÖÐ×Ý×ØÑÒ×ÐØØΨ(t)×Ø| Ψ(t) | 2ÕÙ×ØÔÖÓÔÓÖØÓÒÒÐ ÇÒÔÓ×ÈÓÙÖÚÓÖÐ×ÑÒ×ÓÒ×Ù×ÙÐÐ××ÖÒÙÖ×ÐÙØÖÐ×Ù×ØØÙØÓÒ exp Γxt) 

E → E/, m → m/, Γ → Γ/Ó ≃ 6.582122 · 10−22ÅÎ·× 

(Γx) −1ÒØÐÔÖÓ

ÖÚÓÒ× 

Ú 

|Ψ(t)〉 

ÖÚÓÒ×ÐÑÐÒ×ÓÑÔÓ×ÒØ×ØÖÒØËØË−1×ÓÙ×ÐÓÖÑ 

ax(t) 

Ú 

≪ ÙØÑÔ×ØØÐÕÙτS ×Ù××ØÁÒØÖ××ÓÒ×ÒÓÙ×ÔÖÓÒØÖÙÖÑÒ×ÐÖÓÒØÐÓÖÖτS 

ÒÙØÐ×ÒØ Ø×Ø ÐÔÖÓÐØØÖÓÙÚÖÙÒØØØÖÒØË ÙØÑÔ× 

α(t) 

ÙÖ 

−1ØØÔÖÓÐØ×ØÓÒÒÔÖÚÓÖ 

 

| 

 

Ú∆m =| Ò×Ð×ÐØÖÒØS = 

ØÕÙ× ÓÒÒÓÒØÓÒÙØÑÔ×ØØÓ×ÖÚÒÙØÐ×ÒØÐ××ÒØÙÖ×ÖØÖ× ÓÑÔÓÖØÑÒØØÚÖÐÓÖ×ÙÒÔÖÑÖÜÔÖÒØÙÖÓÓ 

 

ÄÜÔÖÒØÖÔØÔÐÙ×ÙÖ×ÖÔÖ××ÔÙ×ÐÓÖ×ÄÚÐÙÖ∆mØÙÐ 

ÚÒÍËÄÔÔÖØÓÒK 

×Ø K 

∆m = (0.5307 ± 0.0015) · 10 10 s −1 ÅÙÐÐÖØÐÈÝ×ÊÚÄØØ 

 

= 1 

√ 2 (aL(t)|K 0 L 〉 + as(t)|K 0 S 〉) 

τLÐÓÑÔÓ×ÒØÓÙÖØ×ÔÖÙØ×ÙÐÐÐÓÒÙ 

=ÄË = exp(−imxt) exp(−Γxt/2) x 

t 

 

|Ψ(t)〉 = α(t)|K0 

 

〉 + α(t) K 0 

= 1 

2 [aS(t) Ø + aL(t)] α(t) = 1 

2 [aS(t) 

− aL(t)] 

α(t) | 2 = 1 

4 

mS − mL | 

| α(t) | 2 = 1 

 

4 

 

e −Γ S t + e −Γ L t + 2 cos(∆mt)e 

t 

−(ΓS+ΓL)· 2 

e −ΓSt + e −ΓLt t 

−(ΓS+ΓL)· 

− 2 cos(∆mt)e 2 

0ÒÓÒØÓÒÐ×ØÒÐ×ÓÙÖK 0 

0 + p → π + + Λ et π 0 + Σ +

ÖÑØÈÌÄÙÖ×ØÙÑÒÓØÈÙÐÒØÓÖÑ ÄÓÔÖØÓÒ×ÝÑØÖÈÌÔÖ×ÒØÙÒÒØÖØÔÖØÙÐÖÙ×ÙØÓ ÄØÓÖÑÈÌ 

ÒÓÒÕÙÒØÓÖÕÙÒØÕÙ×ÑÔ×ÐÒÚÖÒÐÀÑÐØÓÒÒ×ÓÙ×Ð× 

ÕÙÚÓÐÈÚÓÐÙ××ÌÔÖÓÑÔÒ×ØÓÒÔÓÙÖ××ÙÖÖÐÒÚÖÒ×ÓÙ×ÈÌ ØÖÒ×ÓÖÑØÓÒ×ÄÓÖÒØÞÑÔÐÕÙÐÒÚÖÒØÀÑÐØÓÒÒ×ÓÙ×ÐÓÔÖ ÈØÌØÙ××ÔÖÑÒØÒÓÒ×ÕÙÒÙÒ×Ý×ØÑÓÙÙÒÒØÖØÓÒ ØÓÒÓÑÒÈÌÑÑ×ØØÒÚÖÒÒ×ØÔ×ÚÖ×ÓÙ×Ð×ÓÔÖØÓÒ× Ò××ÙÑÓÒ×ÙÒ××ÝÑØÖ×ÓÙÌ×ØÚÓÐÑÑ×Ì×Ø ÙÒÓÒÒ×ÝÑØÖÐÓÖ×È×ØÙ××ÓÒ×ÖÚÓÒØÈ×ÓÒØØÓÙØ×ÙÜ ÓÒ×ÖÚ×ÓÙØÓÙØ×ÙÜÚÓÐ× ÈÌ Ü Ü Ü ÔÔÐØÓÒ× ÁÑ ÙÙÒ 

ÌÓÒ×ÕÙÒ×ÙØÓÖÑÈÌÄÔÖ×ÒÙÜ×ÒÕÙÐ Ü Ü Ü ÙÙÒ 

×ÝÑØÖÒÕÙ×ØÓÒ×ØÚÓÐ 

ÏÁ×ÒØβ 

ÄØÐÙÑÓÒØÖÐ×ÓÑÒ×ÓÒ×ÔÓ××Ð×Ú×ÔÔÐØÓÒ× 

K 

Ó×ÖÚÐÓÖÖ×ÔÓÒÒØÓÒÔ×ÚÐÙÖØÚØÙÖÔÖÓÔÖ×ÈÌÔÓÙÖ ÄÓÔÖØÙÖÕÙÒØÕÙO=ÈÌÐÑÑÓÑÔÓÖØÑÒØÕÙÌÓÒÒÔ× ÓÒ×ÖÓÒ×Ð×ÙÒÔÖØÙÐ×ØÒØ×ÓÒÒØÔÖØÙÐØÝÔ 

ÐÀÑÐØÓÒÒ×ÓÙ×OÑÔÐÕÙÕÙ Öp×ØÒÚÖ××ÓÙ×PØ×ÓÙ×TÐÓÖ×ÕÙJÒ×ØÒÚÖ×ÕÙÔÖTÄÒÚÖÒ 

ØÒÚÖ×ÖÐÕÙÖÚØÙÖ(t, r) 

ÈÓÙÖÐÔÖØÙÐÙÖÔÓ×ÀÖÔÖ×ÒØ×Ñ××ØÓÒÒØÖÕÙÐÒØÔÖØÙÐ 

 

ÐÑÑÑ××ÕÙÐÔÖØÙÐma = 

0 L 

→ 2π 

 

:ÈÌ(t, r) = (−t, −r) 

O|a, p, J...〉 = η〈a, p, −J, ...| 

[H, O] = 0 O −1 ØÔÓÙÖÙÒÑÔÐØÙØÖÒ×ØÓÒÐØØÐØØ 

HO = H 

〈b|H|a〉 = 〈b|O −1 HO|a〉 = 〈Ob|H|Oa〉 = 〈a|H b 

ma

ØØÓÒ×ÕÙÒÔÙØØÖÙØÐ×ÓÑÑØ×ØÐÓÒ×ÖÚØÓÒ×ÓÙ×ÈÌ 

ÈÌÐ×ÑÑ×Ø×ÕÙ×ÙÖÐØØÐÔÖØÙÐ×Ò×ÑÔÐÕÙÖÐÒÚÖÒ ÖÓØØÓÒ◦ÙØÓÙÖ×Ü×ÝÓÙÜËÐÓÒÑØÐÒÚÖÒ×ÓÙ×ÐÖÓØØÓÒ ÒØÔÖØÙÐÚÐÔÖÓØÓÒ−JzÇÒÖØÖÓÙÚÐÓÖÒØØÓÒÓÖÒÐÔÖÙÒ ÈÌØÖÒ×ÓÖÑÐÔÖØÙÐÙÖÔÓ×ÚÐÔÖÓØÓÒÙ×ÔÒJzÒ×ÓÒ 

ÐÒØÔÖØÙÐÓÒØÐÑÑØÑÔ×Ú ×ÓÙ×Ä×ÝÑØÖÈÌ×ØÓÒÔÖÓÑÒÒØ ÚÐ×ÑÑ×ÖÙÑÒØ×ÕÙ××Ù×ÓÒÔÙØÑÓÒØÖÖÕÙÐÔÖØÙÐØ 

ÁÐÒ×ØÑÑÔÓÙÖÐÑÓÑÒØÑÒØÕÙÒ×ÐØÖÑÒØÖØÓÒÑ 

maÁÐ×Ò×ÙØÕÙ 

Γa = 

 

Γa 

×ÙÖÐÑÙÓÒÚÓÖÔØÖ Ä×ÜÔÖÒ×ÙØÝÔ ÓÒØÔÖÑ×Ø×ØÖÐÒÚÖÒ×ÓÙ×ÈÌÚ 

→ 

ÓÒÓØÒÙ 

2)ÙÊÆÔÓÖØÒØ 

Øma 

ÖÒØ×ØÝÔ×ÜÔÖÒ×ÓÒØØØÙ××ÙÖÐÐØÖÓÒÄÜÔÖÒÐÔÐÙ× 

ÙÒØÖ×ÖÒÔÖ×ÓÒÈÖÜÒ×Ð×ÜÔÖÒ×(g 

ÔÖ×ÓÒÒÐÖ×ÙÐØØ×ÙÚÒØ 

− 

(ge + − ge−)/ < ge >= (−0.5 ± 2.1) · 10−12 ÖÒ×ÓÙ×ÈÌÈØÌØÙ×ÚÐ×ÐÔØÓÒ×Ö× ÂÀÐØÐÓÒÒÒØÙÒÓÑÔØÖÒÙÙÒÒ×ÑÐ×Ø×Ø×ÒÚ 

×ÔÖØÙÐ××ÓÒØØÖ×ÚÓ×Ò×ÖÒ×ÐÓÖÖÙÔÓÙÖÒØÓÙÑÓÒ× Ä×ÑÐÐ×ÖÓÒ××ÖØÖ×ÒØÒØÖÙØÖÔÖÐØÕÙÐ×Ñ××× Ä×Ó×ÔÒ 

ØÒÙØÖ×ØÇÒÑÒÕÙØØÖØÖ×ØÕÙ×ØÐÖØÐ×ØÖÙØÙÖ ÔÖÜÐ×Ñ×ÓÒ×Ö×ØÒÙØÖ×ÐÔÖÓØÓÒØÐÒÙØÖÓÒÐ×ÝÔÖÓÒ×Ö× 

ÐÖØÐÐ×ÙÖØØÕÙ×ØÓÒ ÒØÖÒ×ÖÓÒ×ÒÒÑÓÒ×ÐÜÑÒØØÒØ×ÖÒØ×ÑÐÐ×ÑÓÒØÖ ÕÙÐÒ×ØÔ×ÙÒÕÙÑÒØØ×ÑÄÓÖÑÐ×ÑÐ×Ó×ÔÒÔÔÓÖØÙÒ 

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a) 

|×Ô〉|ËÔÒ〉|p〉|p〉ØΨnn = |×Ô〉|ËÔÒ〉|n〉|n〉 

1 

√ (|p〉|n〉 + |n〉|p〉) b) 

2 

|p〉 ↔ 

1 

0 

 

|n〉 ↔ 

 

1 

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2 

0 

1

ÕÙÐ×Ó×ÔÒÙÖØÓÙÖÒÒ×ÙÒ×ÔÑÒÖÖÓÙ×Ó×ÔÒÍÒ ØÙØÐ×ÖÐ×ÑÑ×ÖÐ×ØÓÙØÐ×ÔÓÙÖÐ××Ó×ÔÒÙÖ×ÕÙÔÓÙÖÐ××ÔÒÙÖ× ÄÖÒ×ØÕÙÐÖÓØØÓÒÙÒ×ÔÒÙÖÐÙÒ×Ð×ÔÔÝ×ÕÙÐÓÖ× 

 

ÕÙ×ÒÕÙÐÓØÙÒÔÖÓÐØ|a| 2ØÖÙÒÔÖÓØÓÒ| ÒÙØÖÓÒÄ×ÑØÖ×ÈÙÐÖÒÓÑÑ×τiÔÓÙÖÐÓ×ÓÒÔÖÑØØÒØ ÓÒ×ØÖÙÖÐ×ÓÑÔÓ×ÒØ×ÙÚØÙÖÓÔÖØÙÖ×Ó×ÔÒI 

b 

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×ØÖÒ×ÓÖÑÒØÓÒÓÖÑÑÒØÙÜÖÐ×ÙÖÓÙÔËÍ ×ÒØÓÒÔÝ×ÕÙÓÒ×ÖÓÒ×Ð×ØØ××Ó×ÔÒÙÒ×Ý×ØÑÙÜÒÙ ÓÖÑÒØÙÒÖÔÖ×ÒØØÓÒÓÒÑÒØÐÑÒ×ÓÒ ÄÔÖÓØÓÒØÐÒÙØÖÓÒ 

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2 

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1 

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0 

 

0 

+ b 

1 

| 2ØÖÙÒ 

I1 = 1 

2 τ1 = 1 

 

0 1 

, I2 = 

2 1 0 

1 

2 τ2 = 1 

 

1 −i 

, I3 = 

2 i 0 

1 

2 τ3 = 1 

 

1 0 

2 0 −1 

I 2 |p〉 = I(I + 1) 1 I = 2 , I3 = + 1 

3 

= 2 4 |p〉, I3|p〉 = + 1 

2 |p〉 

I 2 |n〉 = I(I + 1) 1 I = 2 , I3 = −1 3 

= 2 4 |n〉, I3|n〉 = − 1 

2 |n〉 

= I1 ± iI2 

ÉÐÙÐÖI+|p〉, I+|n〉, I−|p〉, I−|n〉 

Q = 1 

2 + I3 

 

ǫijkIkÐÐ× 

1 0 

= 

0 0 

= i 

|I = 1, I3 = +1〉 = |p1p2〉 

|I = 1, I3 = 0〉 = 1 

|I = 1, I3 = −1〉 = |n1n2〉

|I = 0, I3 = 0〉 = 1 Ð×Ó×ÒÙÐØ 

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2 

ÄÓÒØÓÒÓÒÐÓÐ×ÙÜÒÙÐÓÒ×ÓØØÖÒØ×ÝÑØÖÕÙÔÖÒ ÇÒÔÙØÑÒØÒÒØÒÓÒÖÐÔÖÒÔÜÐÙ×ÓÒÈÙÐÒÖÐ× 

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Ψ(x1, S1, I1; x2, S2, I2) = −Ψ(x2, S2, I2; x1, S1, I1) 

ÔÒÙ×Ý×ØÑÔÖÒÖÒØ×ÚÐÙÖ××ÔÖ×ÙÒÙÒØ×ØÙ×ÒØÖØ ØÙÖ××Ó×ÔÒÓÑÑÔÓÙÖÐ×ÚØÙÖ×ÑÓÑÒØÒÙÐÖÈÓÙÖÒÙÐÓÒ×Ð×Ó× ÄÒÖÐ×ØÓÒÙÒÒ×ÑÐÒÙÐÓÒ××ØÔÖÓÑÔÓ×ØÓÒ×Ú ÔÓÙÖÔÖØÒØÖ Ø 

(−1) 

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Q = 1 

2 B × + I3 ØÓÒÐÙÖØØÖÙÖÙÒÒÓÑÖÖÝÓÒÕÙ−1ÁÐÒ×ØÑÑÔÓÙÖÐ Ä×ÖÐ×××Ù××ÔÔÐÕÙÒØÙ×ÒØÔÖÓØÓÒ×ØÒØÒÙØÖÓÒ×ÓÒ 

Ð×ØØ× 0×ÓÒØØÖÙÙÒÙÜÔÓÒ×Ø×ÐÓÒÒØ ØÖÔÐØÓÑÔÓ×π ±Øπ |I = 1, I3 = +1〉 ↔ |π + 〉 

|I = 1, I3 = 0〉 ↔ |π0 〉 

|I = 1, I3 = −1〉 ↔ |π− 

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I+S+L 

= −1 

Ò Ò×Ð×ÙÜÒÙÐÓÒ×ÐÖÐØÖÕÙ×ØÓÒÒÔÖÁ3 

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[HIF, I] = 0

ÐÀÑÐØÓÒÒÐÓÖ×ÐÕÙØÓÒÔÙØØÖØÒÙÐÀÑÐØÓÒÒØÓØÐ ÈÓÙÖ×ÑÔÐÖÐ×Ù××ÓÒ×ÙÔÔÓ×ÓÒ×ÒÐÐ×Ð×ÔÖØ×ÑØÐ 

ÇÒÒÙØÐ×ÖÐ××ÙÚÒØ× 

Á3×ØØ×ÑÑÁØÖÒØ×ÚÐÙÖ×Á3×ÓÒØÒÖ× ×ØØ×ÒÖÒÓÒØ×ÚÐÙÖ×ÒØÖÑÒ×Á2Ø 

[H, I] = 0 

ÒÒÖ Á2ØÁ3×ÓÒØÓÒ×ÖÚ×ÔÒÒØÙÒØÖÒ×ØÓÒ 

ÚÓÐØÓÒØÐ×ØØ×ÑÑÁÒ×ÖÒØÔÐÙ×ÒÖ×ÒÁ3ÇÒ ×ÖØÒ×ÙÒ××ÝÑØÖÔÔÖÓ ÓÒÒÓÒ×ÙÒÔÔÐØÓÒÐÖÐ ÆÓØÓÒ×ÕÙÒÔÖ×ÒÙÒÔÖØÙÖØÓÒÑÙ×Ý×ØÑÓÒÙÖØÙÒ Ù×Ý×ØÑÙÜÒÙÐÓÒ×ØÒ××ÓÒ× 

ÉÑÓÒØÖÖÕÙ×ØÓÑÔØÐÚ 

Ù×Ý×ØÑ 

ÒÓÑÒÒØÚÓÒÓØÒØ 

ÇÒÓÒÙÒÑÓÝÒÕÙÒØÖÐÖÒÒØÖÐØØ×ÒÙÐØÙØÓÒØ 

ÔÓÒÒÙÐÓÒ ÐØØØÖÔÐØ×Ó×ÔÒØØ×ÒÓÒÐ×ÙÒ×Ý×ØÑÙÜÒÙÐÓÒ× ÁÐÐÙ×ØÖÓÒ×ÑÒØÒÒØÐÖÐ ÔÖÐ×ÖÔØÓÒ×ÔÖÓ××Ù×Ù×ÓÒ 

π 

π − + p → π 0 ÒÖ 

+ n 

ÙÒÀÑÐØÓÒÒÔÓÙÖÐÒØÖØÓÒÒÙÐÓÒÒÙÐÓÒ 

Hint = U + V I1 · I2 

I2×ØÐ×Ó×ÔÒØÓØÐ 

I1ØI2××ÒØ×ÙÖÐ×ÒÙÐÓÒ×ÒÚÙÐ×ØI = I1 + ÇÒI 2 = (I1 + I2) 2 = I 2 1 + I 2 2 + 2I1 · I2 = 3 3 

+ 

4 4 + 2I1 · I2 

 

I = 1 → I1 · I2 = +1/4 → Hint = U + V/4 

I = 0 → I1 · I2 = −3/4 → Hint = U − 3V/4 

± + p → π ± + p Ù×ÓÒÐ×ØÕÙ

ÑÑ×ËÓÒÚØÖ×ØØ×ÔÙÖ××Ó×ÔÒÐÑÔÐØÙØÖÒ×ØÓÒ ÔÓÒÒÙÐÓÒÒ×Ð×ØØ×ÒØÐØÒÐÙÒ×ÔÖÓ××Ù××ÓÒØÐ× ÒÓÒ×ÕÙÒÐÓÒ×ÖÚØÓÒÐ×Ó×ÔÒÐ×ÚÐÙÖ×ÁÁ3Ù×Ý×ØÑ ×ÖØÐÓÖÑ〈Ψ(I ′ ÓÁÁ3×ÖÔÔÓÖØÒØÐØØÒØÐØÁÁ3ÐØØÒÐÔÐÙ×ÐÐÑÒØ ÑØÖM 2I×ØÒÚÖÒØ×ÓÙ×ÐÖÓØØÓÒÒ×Ð×Ô×Ó×ÔÒÓÒÒÔÒ ÒØÁ3ÓM2IÁ℄ ÍØÐ×ÒØÙÒØÐÓÒØ×Ð×ÓÖÒÓÒØÖÓÙÚØÚÑÒØ ÓÒÔÙØÓÖÑÖ××Ý×ØÑ×ÖØÖ××ÔÖÙÒ×Ó×ÔÒØÓØÐ ÚÙÒÔÓÒ×Ó×ÔÒÁØÙÒÒÙÐÓÒ×Ó×ÔÒ ÓÙ 

 

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ÇÒ×ØÓÒÑÒÒÖÙÜÐÑÒØ×ÑØÖ××Ó×Ð×Ó×ÔÒØÓØÐ 

 

Ø 

 

 

Ò×Ð×ÐÙ×ÓÒÐ×ØÕÙπ + Ð×ØÓÒØÖÒ×ØÓÒ×ÖÙØ p×ÙÐÐØØ×Ó×ÔÒØÓØÐ ×ØÓÒÖÒ 

ÃÓÒØÒØÐ×ØÙÖ××ÔÔ××ÔÒ Ñ××ÒØÖÔÖØÙÐ×Ö×ÐØÙÖÃ×ØÐÑÑÔÓÙÖÐ×ÖØÓÒ× ËÐÓÒÒÐÐÖÒ ÄÐÑÒØÑØÖÔÓÙÖÐ×Ó×ÔÒØÓØÐÁ×ØÖØÓÒÚÒØÓÒÒÐÐÑÒØM 2I 

 

3 

2 

 

3 

2 

 

3 

2 

 

1 

2 

 

3 

2 

 

1 

2 

, I ′ 3)|M|Ψ(I, I3)〉 = δI ′ IδI ′ 3 I3 M2I 

 

, +3 = |1, 1〉 1 1 + , ≡ |π p〉 

2 2 2 

 

, −3 = |1, −1〉 1 

− , −1 ≡ |π n〉 

2 

2 2 

 

, +1 = √3 

1 

|1, 1〉 2 

 

1 2 

, −1 + 2 2 3 |1, 0〉 

1 

, +1 ≡ √3 

1 

|π 2 2 

+ n〉 + 

 

2 , +1 = 2 3 |1, 1〉 

1, 

−1 − √3 1 |1, 0〉 2 2 

 

1 

2 , +1 ≡ 2 2 

 

, −1 = √3 1 |1, −1〉 2 

 

1 1 2 , + 2 2 3 |1, 0〉 

1, 

−1 ≡ √3 1 |π 2 2 

−p〉 + 

 

1 1 1 , −1 = − , + √3 |1, 0〉 

1 

2 , −1 ≡ − 

Ø M 

2 

1 = 1 

3 |1, −1〉1 2 

, I3 2 

2 

2 

2 

2 

3 |π0 p〉 

3 |π+ n〉 − 1 √ |π 

3 0p〉 

2 

|π + p〉 = 3 

, +3 

2 2 

|π−p〉 = 1 

2 

√ 3, 

−1 − 1, 

−1 

2 2 2 2 

3 3 

|π0 

2 

1 

n〉 = 3 

, −1 + √31 

, −1 

2 2 2 2 

3 

 

M1, I3 

M 2 3 = 

3, 

I3 

M3 2 2 

3 |π0 n〉 

3 |π− p〉 + 1 √ 3 |π 0 n〉 

, I3 

σπ + p→π + p = K | M 3 | 2

ØÈÖÓÒØÖÒ×Ð×ÐÙ×ÓÒÐ×ØÕÙπ − ÓÑÔÓ×ÒØ×Ó×ÔÒØÓØÐ ØÙÒÓÑÔÓ×ÒØ×Ó×ÔÒØÓØÐ pÐÒØÖÚÒØÙÒ ÓÒÓÒ 

2 

2 σπ−p→π − 

1 

p = K 

3 M3 + 2 

3 M1 ÒØÐØÐØØÒÐ×ÓÒØÖÒØ× ÁÐÒ×ØÑÑÒ×Ð×ÐÒÖÓÐ×ÔÖØÙÐ×ÐØØ 

σπ−p→π0n = 〈π0n|M|π− √ 

2 

p〉 = K 

3 M3 √ 

2 

− 

3 M1 ÇÒÒÙØÐ×ÖÔÔÓÖØ×××ØÓÒ×× 

σπ + p→π + p : σπ−p→π −p : σπ−p→π0n :=| M 3 | 2 

1 

: 

3 M3 + 2 

3 M1 

√ 2 

2 

: 

3 M3 √ 

2 

− 

3 M1 

ÄÙÖÑÓÒØÖÐ×ÚÐÙÖ×Ñ×ÙÖ×××ØÓÒ××ÔÓÒÔÖÓØÓÒØ 

2 ÒÚÖÓÒ ÔÓÒÙØÓÒÒÓÒØÓÒÐÒÖËÐÓÒ×ÔÐÙÒÚÙÙÔÖÑÖÔ Ó×ÖÚ ÑÑÑØØÖÒÖ×ØÐÙÐÒ×ÓÙ×ØÖÝÒØÐ Î Ò×ÐÅÓÒØÖÓÙÚ×ÚÐÙÖ××ØÓÒ×× ×ØÓÒØÓØÐπ − Â ÇÒÓÒ×ÖÔÔÓÖØ×Ñ×ÙÖ×ÒÚÖÓÒØØÒÖÐ×ÔÖÓ××Ù× Ø×ÓÒØÓÑÒ×ÔÖÐÓÖÑØÓÒÐØØÖ×ÓÒÒÒØ∆Ñ Á pÒÚÖÓÒÑÐ×ØÓÒÐ×ØÕÙÑ Î ÇÒÔÙØÓÒÑØØÖÒ××ÓÒØÓÒ×ÕÙM 3 ≫ M1 ØÒÐÖÐÓÒØÖÙØÓÒM 1Ò×Ð×ÖÐØÓÒ×ÇÒÓØÒØ ÐÜÔÖÒ ÉØÖØÖÐÑÑÓÒÐ×ÐÙ×ÓÒÒÙÐÓÒÒÙÐÓÒÒÒÐÝ×ÒØ ÐÓÖ××ÖÔÔÓÖØ×ÐÙÐ×ÒÚÖÓÒÒÓÒÓÖÚÐ×Ö×ÙÐØØ× 

Ð×ÖØÓÒ× p + p → π + + d, p + n → π0 + d, n + n → π− ÑÓÒØÖÖÕÙÐÖÔÔÓÖØ××ØÓÒ×××ØÖÓ×ÖØÓÒ×××ÙÖÐ ÓÒ×ÖÚØÓÒÐ×Ó×ÔÒ×Ø 

d + 

ÇÒÒØÔÒ×ÔÖÖÔ×ÙÖÐØÙÐ×ØÖÙØÙÖÒÕÙÖ×× Ä×Ó×ÔÒËÍ ØÐ×ÕÙÖ× 

ÕÙÚÙØÁÔÓÙÖÐÔÖØÙÐρÄÕÙÐÐ××ÒØÖØÓÒ×ÓÖØ××ØÒØÖØ 

ρ + → π + π0 , ρ− → π−π0 , ρ0 → π−π0 , ρ− → π−π + , ρ0 → π0π0 ÖÓÒ×ÔÖ×ÒØÙÔØÖ×ÙÖÐÑÓÐ×ÕÙÖ×

ËØÓÒ××ØÓØÐ×ØÐ×ØÕÙ×ÔÓÒÒÙÐÓÒØÔÓÒÙØÓÒ 

ÒÓÒØÓÒÐÕÙÒØØÑÓÙÚÑÒØÙ×ÙÔÓÒ×ØÐÒÖÒ× 

ÐÑ

ÔÓÒ×ÓÒØÙÒÑ××ØÖ×ÚÓ×ÒÄÖÒÑ××Ò×ØÕÙÕÙÐÕÙ× ÕÙÖ×ÙØÕÙ×ÓÒØÐ×ÓÒ×ØØÙÒØ××ÔÖØÙÐ×Ð×ÔÐÙ×ÓÑÑÙÒ×ÔÒ ÇÒÔÙØÖÖÑÓÒØÖÐÖ×ÓÒØÖÙÓÖÑÐ×Ñ×Ó×ÔÒÙØÕÙÐ× 

Ò×Ð×ØÖÙØÙÖ×ÖÓÒ×ËÓÒÒÐÐØ×ÒØÖØÓÒ×ÑØÐ ÙÔÓÒØÚÙÐÒØÖØÓÒÓÖØ×ÙÜÕÙÖ×ÓÙÒØÙÒÖÐÒØÕÙ 

ÓÖÑÒØÐ×ÐÑÒØ××ÙÒÖÔÖ×ÒØØÓÒÑËÍ ÙÑÜÑÙÑ ÄÒÙÒÕÙÖÙØÙÒÕÙÖÙÒØÐÓÖÖÕÙÐÕÙ×ÔÓÙÖÒØ× 

Ö Ò×ÐÓÖÑÐ×Ñ×Ó×ÔÒÔÔÐÕÙÙÜÕÙÖ×ÓÒÓÒ×ÖÕÙÙ ×Ø 

Ä×ÒÙÐÓÒ×ØÐ×ÔÓÒ××ÓÒØ×ÖÔÖ×ÒØØÓÒ×ÓØÒÙ×ÔÖÓÑÒ×ÓÒ ÖÕÙ×ÙØÈÓÙÖÐ×ÖÓÒ×ÓÖÑ×ÔÖÙØÖ×ÕÙÖ×ÓÑÑÔÖÜÑÔÐ 

Ð×ÓÒ×ÐÙØÓÙØÖ×Ö×ÐÖØ×ÙÔÔÐÑÒØÖ×ÕÙÓÒÙØ 

 

u 

ÙÒÖÔÖ×ÒØØÓÒÖÔÕÙÒØÙØÚÐÐÙ×ØÖÒØÐÓÑÔÓ×ØÓÒÙÜ×Ó×ÔÒ× ÙÜ×Ó×ÔÒ×ÓØÙÜÖÐ×Ù×ÙÐÐ×ØÓÒ××ÔÒ×ÄÙÖ ×ÔÒ×ØØÑØÓÖÔÕÙ×ÖÙØÐÒ×Ð×ÙËÍÒ ÄØÓÒ 

ÄÔÔÐØÓÒÐÖÐØÓÒ ÙÜÕÙÖ×ÙØÓÒÒÙÒÖÐ ×Ø ÖÓÙÖÖ×ÖÓÙÔ×ÑÒ×ÓÒ×ÙÔÖÙÖËÍÒÒ> 

1/3ÕÙÕÙÖ 

ÐØ× ËÐÓÒØÓÒÒ 

ØÖÕÙ1/6 + 

ÕÙÒÙØÐ×ÒØÐÑÒ×ÓÒ×ÖÔÖ×ÒØØÓÒ×ÑÁ ×Ó×ÔÒ−1/2ÇÒÔÙØÓÒ××ÓÖÐÔÖÔÒÙÒ ÓÒÓØÒØÙÒÕÙÖÙÔÐØ ØÙÜÓÙ 

ÚÐÙÖB= ÈÓÙÖÐÔÖÓØÓÒ×Ý×ØÑÙÙÁ3ÚÙØ1/2 + 

= ÐÓÒÙÖØÓÒÙÓÒÒI3 ÓÙÐØI = 

ÇÒÔÙØ××ÓÖÐÙÒ×ÓÙÐØ×Ù×Ý×ØÑÔÒØÐÕÙÖÙÔÐØÙ×Ý×ØÑ ×ÖØÓÖÑÐÐÑÒØ2 ⊗ 

ÅÎÕÙ×ØÔØØÒÖÖÔÜÐÑ××ÙÔÖÓØÓÒÇÒ 

| mu − md | /mp = O(10 −3 ) 

Ð×ÝÑØÖÙ↔ÔÔÖØÐÓÖ×ÕÙÓÒÓÑÔÖÐÑ×××ÒÙÐÓÒ× 

p = uud , mp = 938 MeV ; n = udd , mn = 940 MeV 

ÄÓÒ×ØÙÒÙØÖÜÑÔÐÚÐÕÙÖ×Ò×ÐÖÐ×ÔØØÙÖ 

K + = us mK + = 494 MeV K0 = ds mK 0 = 498 MeV 

= I = 1 

1/2 = 2/3ØÖ×ÔØÚÑÒØ1/6 

1/2 

2 , I3 = + 1 

2 

 

d = I = 1 

2 , I3 = −1 2 

−1/3ÒØØÖÙÒØÐ 

1/2ÔÓÙÖÐÒÙØÖÓÒ 

− 1/2 = 

1/2 − 1/2 = 

2 ⊗ 2 = (3 ⊕ 1) ⊗ 2 = 4 ⊕ 2 ⊕ 2 

△ ++ , △ + , △ 0 , △ −ÓÑÔÓ×××ÕÙÖ×ÙÙÙÙÙÙØÖ×ÔØÚÑÒØ

−1/2 

= 

+1/2 

2 

2 2 

3 1 

4 2 

 

+ 

+ 

ÙÒ×Ó×ÔÒÙÓÑÔÓ×ØÓÒÙÜ×Ó×ÔÒ× ØÙÒ×Ó×ÔÒ ÖÓØÓÑÔÓ×ØÓÒ 

I I 

3 

3 

−1 −1/2 0 1/2 1 

−3/2 −1 −1/2 0 1/2 1 3/2 

Ä×ÖÒØ×ØØ×ÖÙ△ÓÒØ×Ñ×××ØÖ×ÔÖÓ×ÒÚÖÓÒ ÅÎÈÓÙÖ×ØÒÙÖÐÒÙ△ 0ÔÜÐÙØÖÖÜÔÐØÑÒØÐÓÑÔÓ×ØÓÒ ËÍ ØÖÔÐØ×ÔÓÒ×ØÐηÓÑÑÒØÐÔÓ×ØÓÒÙ×ÒÙÐØ ÁËÐÓÒØÓÒÒÙÒÕÙÖØÙÒÒØÕÙÖÓÒÔÙØÓÖÑÖÙÒØÖÔÐØ×Ó×ÔÒ ×ÙÜÖÓÒ×ÕÙ×ÖØÒ×ÐÔØÖ×ÙÖÐÑÓÐ×ÕÙÖ× 

ÓÒ×ÖÓÒ×ÑÒØÒÒØÐ×Ý×ØÑÙÓÒÒ××Ð×Ó×ÔÒÒ×ÔÔÐÕÙ ØÙÒ×ÒÙÐØ×Ó×ÔÒÁ Ò×ÐÒØÙÖÓÒØÖÓÙÚØÚÑÒØÐ 

ÕÙÙÜÓÑÔÓ×ÒØ×ÙØÐÓÑÔÓ×ÒØ××ØÓÒ×ÖÖÔÖØÒÒÐÙÒØ ÐØÖÒØÐÖÐØÓÒ ÚÒØ 

Q = 1 

2 (B + S) × + I3 = 1 

Y×ØÔÔÐÐhypercharge×ØÐÖÝÒØÖÐÖÙÑÙÐØÔÐØ 

Y × + I3 

2 Ä×Ã±ÓÒØBØS= ± ÓÒQ = (0 ± 1)/2 ± 1 ÓÒ×ØØÙÒØ×ÓÒ 

±1ÈÓÙÖÐ×ÕÙÖ× 

= 2 

 

u 

Q = 

d 

1 

 

1 

+ 0 ± 

2 3 1 

2 = 

 

+2/3 

, Q(s) = 

−1/3 

1 

 

1 

− 1 + 0 = − 

2 3 1 

ÒÓÑÔØÐÒ×ÑÐ××ÚÙÖ×ÕÙÖÙËÅ ÇÒÔÙØ×ÐÓÖ×ÒØÔÖÐÓÖÑÕÙÔÖÒÐÓÔÖØÙÖQÕÙÒÓÒÔÖÒ 3 

Q = 1 

2 (B + S + Cha. + Bot. + Top.) × + I3, 

 

avec : 

= 

2 

3 

3 2

ChaÖÑ 

BotÓØØÓÑÒ×× 

TopÌÓÔÒ×× 

ÔÓÙÖÐ×ÔÖØÙÐ×Ò×ÖÒÐ×ÐÙÖÒØÔÖØÙÐÁÐ×ØÓÒÔÖØÙÐ× ÇÒÚÙÕÙÐÓÔÖØÙÖÓÒÙ×ÓÒÖCÒ×ÓÒØÓÒ×ÔÖÓÔÖ×ÕÙ ÙØÖÔÖØÓÒ×ÒØÒØÐÒØÓÒY = ÄÔÖØG 

ÒÙØÖ×ÓÑÑÐπ 0 C|π0 〉 = Cπ0|π0 〉 = +1|π0 −ØÚÚÖ× 

〉 

ÖÓØØÓÒ¦Ò×Ð×Ô×Ó×ÔÒÔÖÜÑÔÐÙØÓÙÖÐÜÝ××Ó 

ÈÖÓÒØÖÙÒπ +×ØØÖÒ×ÓÖÑÒπ 

Á2ÓÒÙÖØÔÙÓ×ÖÐÜÜ××ÓÁ1 

−ÓÒÔÙØØÙÖÙÒ Å×ÐÝÙÒÙØÖÓÒØÖÒ×ÓÖÑÖÙÒπ 

ÐÓÔÖØÓÒÔÓÙÖØÒÖÐ×ÒÁ3ÄÔÔÐØÓÒÊ2(π)ÐØØ 

+Òπ 

0〉 = (−1) I ÉÑÓÒØÖÖ 

|I, 0〉 

×ØÔÔÐÐÔÖØÓÙÙ××Ð×ÓÔÖØ ÇÒÓØÒØÔÓÙÖÐ×ÔÓÒ×ÒÙØÖ× ÇÒÒØÐÓÔÖØÓÒÓÑÒG = 

ÔÓÒ×ÖØÖ×ÓÒÔÖ ÈÓÙÖÐ×ÔÓÒ×Ö×ÓÒÙÒÐÖØÓÜÐÔ×ÕÙÐÓÒÙØÐ× ÓÒÓØÒÖÐÑÑÚÐÙÖÔÖÓÔÖ−1ÕÙÔÓÙÖÐÔÓÒÒÙØÖÄÑÐÐÙ 

 

 

|I1, I3 = 0〉ÓÒÒ 

R2(π)|I, 

B + S 

C|π + 〉 = |π − 〉 

C|π − 〉 = |π + 〉 

R2(π) = exp(iπI2) 

C R2(π) 

G|π 0 〉 = C R2(π)|π 0 〉 = C R2(π)|I = 1, I3 = 0〉 = C(−1) 1 |1, 0〉 

= −C|1, 0〉 = −C|π 0 〉 = −|π 0 〉 

G|π〉 = Gπ|π〉ÓGπ = −1

ÒÙØÖ×ÓÒ ×ØÙÒÒÓÑÖÕÙÒØÕÙÑÙÐØÔÐØÔÓÙÖÙÒ×Ý×ØÑÆÔÓÒ×Ö×ÓÙ 

GN = (−1) N Ò××Ø ÙÒÓÒÒÖÐÐ×Ñ×ÓÒ××Ò×ØÖÒØ×Ò×ÖÑ×Ò×ÓØØÓÑ ×ÓÒØ×ØØ×ÔÖÓÔÖ×ÚÐ×ÚÐÙÖ×ÔÖÓÔÖ× 

ÓC0×ØÐÔÖØCÐÐÑÒØÒÙØÖÙÑÙÐØÔÐØ×Ó×ÔÒ ×ØÓÒ×ÖÚÒ×ÐÒØÖØÓÒÓÖØÔÖÓÒØÖ×ØÚÓÐÒ×Ð×Ò 

ÜÑÔÐ× ÓÖØρ→ππ×ØÔÖÑ×ÖÐÒ×ÑÐÙÜÔÓÒ×ÙÒÔÖØÔÓ×ØÚ ØÖØÓÒ×ÐØÖÓÑÒØÕÙØÐÁÐÐÙ×ØÖÓÒ×ØØÔÖØÓÒÔÖÕÙÐÕÙ× 

ÈÖÓÒØÖÐ×ÒØÖØÓÒÓÖØÙρÒÙÒÒÓÑÖÑÔÖÔÓÒ××ØÒØÖØ 

−1ÓÒρ Ä×ÒØÖØÓÒ 

ÔÖÓÔÖÚÐ×ÚÐÙÖ×ÔÖÓÔÖ× ÆÓÙ×ÚÓÒ×ÚÙÕÙÐÔÖÖÑÓÒÒØÖÑÓÒÙÒÓÒÙ×ÓÒÖ Ä×Ý×ØÑÒÙÐÓÒÒØÒÙÐÓÒÓÖÑÙÒ×ÓÑÙÐØÔÐØÁÁÐ×ØÓÒÙÒØØ 

= ÄÑ×ÓÒρ×ØÖØÖ×ÔÖÁ0 

ÒÓÒ×ÕÙÒÐÓÒ×ÖÚØÓÒÐ×Ý×ØÑNNÒ×ÐØØÑÓÑÒØ ÓÖØÐÄ×ÒÒÐÒÙÒÒÓÑÖÑÔÖÔÓÒ××Ë×ÒÙÐØ×ÔÒØ 

ÔÖ×ÒØÐÒØÖØÒÖ×ÓÒÐÓÒØÖÒØ×ÐØÓÒÕÙÐÒØÖÓÙØÒ× ÒÙÒÒÓÑÖÔÖÔÓÒ××ËØÖÔÐØ×ÔÒ ÖØÒ×ÔÖÓ××Ù×ÓÙÚÖÒ×ÔÖÐÒØÖØÓÒÓÖØ ÇÒÚÓØÕÙÐÒØÖÙÖ×ÐÑØ××ÓÒÑÔÔÔÐØÓÒÐÓÔÖØÙÖ 

×ÖÑÓÒ×ÒØÕÙ×ÓÑÑÐ×ÐØÖÓÒ×ÐØÓÑÄØÓÖÑÕÙ××Ó ÄÔÖÙÖ×ÙÖØÓÖÑ×ØÐÔÖÒÔÜÐÙ×ÓÒÈÙÐÕÙ×ÔÔÐÕÙ ÄØÓÖÑ×ÔÒØ×ØØ×ØÕÙ 

Ù×ÐØÐ×Ñ×××ÒØÖÙÜÔÖØÙÐ×ÒÔÙÚÒØÔ××ÔÖÓÔÖÔÐÙ× ×ÔÒØ×ØØ×ØÕÙØÒÓÒÔÖËÛÒÖÒÐ×ÐÝÐÔÖÒÔ ÚØÕÙcÄÖÐ×ØÐ×ÙÚÒØ 

Ò×ØÒÐ×ÔÖØÙÐ××ÔÒÑÒØÖÖÑÓÒ×ÐÐÖÑÖ Ð×ÔÖØÙÐ××ÔÒÒØÖÐ×Ó×ÓÒ××ÓÒØ×ÓÙÑ×Ð×ØØ×ØÕÙÓ× 

ÁÎÎÓÖÔÖÜÃÓØØÖÎÏ××ÓÔÓÒÔØ×ÓÈÖØÐÈÝ××ÎÓÐÁÁÔÔÒÜ 

(−1) L+S 

G = (−1) I C0 

G = (−1) I+L+S = (−1)(−1) L+S

ÕÙÙÒÒÚÚÒØÒ×ÐÙÒÚÖ×UÒÙÑÖÓØÙÜÔÖØÙÐ×ÚÐ×ØÕÙØØ× ÙÒÔÖÔÖØÙÐ×ÒØÕÙ×ÐÓÖ×ÕÙÓÒÒÐÙÖÖÐÇÒÔÙØÑÒÖ ÄÓÜÐ×ØØ×ØÕÙØÖÑÒÐ×ÝÑØÖÐÓÒØÓÒÓÒÕÙÖØ 

Ä×ÓÒØÓÒ×ÓÒÕÙÖÚÒØÐ×Ý×ØÑ Ò×ÐÑÑÓÒÙÖØÓÒÕÙÐÚØÒ×UØÖÔÖÓÕÙÑÒØÔÓÙÖÐ ØØÕÙÐÙ×ÙÐÔÙØÐÖÐ×ØÕÙØØ×ÍÒÔØØÐÖÖÙÒ 

′Ð×ÙÜÔÖØÙÐ×ÓÒØØÒ×ÐÜØÑÒØ ÙØÖÙÒÚÖ×U ′Ò×U Ò×U ′ØÒ×U×ÓÒØ ÔÖØÙÐ××ÓÒØ×ÖÑÓÒ× ÑÑ×Ò×Ð×ÙÜÔÖØÙÐ××ÓÒØ×Ó×ÓÒ×Ø×ÒÓÔÔÓ××Ð×ÙÜ 

+ΨÔÓÙÖÐ×Ó×ÓÒ× 

ÔÖØ×ÓÖÖ×ÔÓÒÒØÙÜÖ×ÐÖØ×ÒØÖÒ××ÔÒ×Ó×ÔÒÓÙÐÙÖ ÄÓÒØÓÒÓÒÙÒ×Ý×ØÑ×ØÓÑÔÓ××ÔÖØ×ÔÐØ× 

Ψ → Ò1 ←→ 

ÔÓ×ØÓÒÒÓÑÖ×ÕÙÒØÕÙ×ÐÔÖØÙÐ×ÙÖÐÔÖØÙÐØÚØÚÖ× ÄÒÙÜÔÖØÙÐ×Ù×Ý×ØÑ×ÒØÖÒ×ÔÓÖØÖÐ×ÒØØÙÖ× 

... 

χ(×ÔÒ)i t(×Ó×ÔÒ)i 

ÓÙÒØÕÙ×ÇÒÓØÒØÐ×ØØ×Ù×Ý×ØÑ×ÙÜÔÖØÙÐ×ÒÓÖÑÒØ ×Ó×ÓÒ×ÓÙ×ÖÑÓÒ×ÔÖÓÖÐÔÙØ×ÖÓ×ÓÒ×ÖÑÓÒ×ÖÒØ× ÔÔÐÓÒ×ψaÐÓÒØÓÒÓÒÕÙÖÔÖ×ÒØÐØØÐÔÖØÙÐaØψbÐ ÓÒØÓÒÓÒÕÙÖÔÖ×ÒØÐØØÐÔÖØÙÐb×ÔÖØÙÐ×ÔÙÚÒØØÖ 

ÖÒØ××ØÙÖ×ÐÔÝ×ÕÙÈÓÙÖ×ÜÑÔÐ×ÔÔÐØÓÒ×ÒÔÝ×ÕÙ ÙÔÖÒÔÈÙÐÄ×ÓÒ×ÕÙÒ×ØÓÖÑ×ÓÒØÒÓÑÖÙ××Ò×Ð× 

ψbÓÒΨÖÑÓÒ×=0ÇÒ×ØÖÑÒ 

Ð×ØØ×ØÕÙÓ××ÙÖÐÑ××ÓÒÔÓÒ×Ò××ÓÐÐ×ÓÒ×ÙØÒÖ ×ÔÖØÙÐ× ÁÐÙØÙ××ÑÒØÓÒÒÖÐ×ØÒØØÚ×ÖÒØ×ÑØØÖÒÚÒÐ×Ø× 

= ËÐ×ØÙÜÖÑÓÒ×ÒØÕÙ×ψa 

×ÓØÑÓÔÖÐ×ØØ×ØÕÙÓ× 

ÖÑÒØØÓÒÖ×ØÖÒØÓÒ×ØØÒÕÙÐ×ØÖÙØÓÒÒÙÐÖÖÐØÚ ÈÓÙÖ×ÔÓÒ×ÒØÕÙ×ÑÑÖÕÙ×ÓÒØÑ×Ò×ÙÒÖÓÒ× 

2 : 

Ψ → −ΨÔÓÙÖÐ×ÖÑÓÒ× 

 

ψi = ψ(×Ô)i 

Ψ(x1, S1, I1; x2, S2, I2) −→ ±Ψ(x2, S2, I2; x1, S1, I1) 

ΨÓ×ÓÒ×= 1 ΨÖÑÓÒ×= 1 

Ψ(1, 2) −→ ±Ψ(2, 1) 

√ 2 (|ψa〉 1 |ψb〉 2 + (|ψb〉 1 |ψa〉 2 ) 

√ 2 (|ψa〉 1 |ψb〉 2 − (|ψb〉 1 |ψa〉 2 )

q 

q 

ÈÖÓÙØÓÒÖÓÒ×ÔÖÖÑÒØØÓÒÒ×ÐÔÖÓ××Ù×e + e− qÄÖÑÒØØÓÒÙÒÔÖÕÙÖÒØÕÙÖ××ÙÙÒÓÐÐ×ÓÒÐØÖÓÒ ÔÓ×ØÖÓÒÒÖ×Ø×ÖÓÒ×ÚÓÖÙÖ ÙØÓÒ×ÒÐ×αÒØÖÐ×ÔÖ×ÔÓÒ×Ö×ÑÑ×ÒØ×Ò× 

→ 

ÓÔÔÓ××ÈÓÙÖÕÙÐØÓÒÒ×ØÓÒÐ×ØØ×ØÕÙÓ××ÓØØÐ ÇÒ×ÒØÖ××Ð×ØÖ 

q + 

ÐÙÖÓÒØÓÒÓÒ ØÒÓÒÒÙÐÐÓÖ×ÕÙÐ×ÙÜÔÖØÙÐ×ÓÒØÙÒÕÙÒØØÑÓÙÚÑÒØÔÖÓ ÙØÕÙÐ×ÔÖØÙÐ×ÔÖÓÚÒÒÒØÐÑÑ×ÓÙÖÇÒ×ØØÒÓ×ÖÚÖÙÒ ÒÑÓÙÐØÖØÓÒÓÒÔÖ×ÕÙÙÖÔÓ×Ò×ÐÙÖÑÐÖÖÓÙÚÖÑÒØ ÒÔÖØÕÙÓÒÓÒ×ÖÙÒÓÒØÓÒÓÖÖÐØÓÒ 

R(p1, p2) = σ(p ÕÙÓÒÔÖÑØÖ×ÔÖÐÓÒØÓÒ 1, p2) σ(p1)σ(p2) R(Q) = 1 + λ exp(−r 2 Q 2 

) ÓQ 2 = (p1 + p2) 2 − 4m2 π = M2 ÐÑÔÓÖØÒÐÓÖÖÐØÓÒ ÙÐ×ÔÖÓ×rÖÔÖ×ÒØÐ×ÑÒ×ÓÒ×ÐÖÓÒ×ÓÙÖÒÒλÑ×ÙÖ 

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∆W = IbN dΩ , dΩ = sinθ dθ dφ 

dΩ 

∆Ω 

Ódσ(θ) 

××Ð×ÐÔÖØÙÐÙ×ÐÒ×Ø×ÓÙÚÒØÔ×Ð×ÖÐÝÙÒ×ÙÐ 

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1 ÈÓÙÖÐ×ÙÒÙ×ÓÒÐ×ØÕÙ| p 

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d 4 

x(jac)f(q)(jbd) 

Eseuil 

ÐÐÑÒØÑØÖÓÙÑÔÐØÙØÖÒ×ØÓÒM 

dσ = 2π 

|M|2 d×ÔÔ× × 

dσ 

dΩ = 

2 c Stat 

8π (Ea + Eb) 2 

| p2 | 

| p1 | × | M |2 

|ØÓÒÔÙØÒØÖÓÙÖ 

≡ pa = −pb p2 ≡ pc = −pd 1 |=| p2 Ei = Ea + Eb 

ÁÐÖ×ØÒÐÙÖÐÔÓØÓÒÚÖØÙÐÒÖØÖ×ÔÖÐÕÙÖÚØÙÖ 

q µ = (pa − pc) µ = (pd − pb) µ 

ÔÐØÙÕÙÔÖÒÐÓÖÑ 

M = αeacebd 

 

d 4 xj1j2ÙØÖÔÖØÐÒØÖ

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)Ψ = −V (t)Ψ 

ËÐÓÒÖÑÔÐÒ× ÓÒ×ÖÕÙÐÔÖØÔÒÒØÙØÑÔ×ÐÒØÖÐÒ× ΨaØΨcÔÖÐÓÒØÓÒÓÒ Ø×ÐÓÒÒ ÚÒØ 

V Ψa 

Ð×ÔÖØÙÐ××ÓÒØÖØ×ÒÓÖ×ÐÖÓÒÒØÖØÓÒ 

ÒÐÙÖÓÒØÓÒÓÒ×ÖØ 

Ψi = Ni exp(−ipix) i = a, b, c, d 

 

d 4 xΨ ∗ c V Ψa ∝ 

 

 

M = −i d 4 xΨ ∗ 

c 

dt exp{i(Ec − ω − Ea)} = 2πδ(Ec − ω − Ea)

ÒØØÕÙÐÓÒÔÔÐÕÙÐÓÒÚÒØÓÒÝÒÑÒÕÙÒØÐÒØÔÖØÙÐ ÒØÚÚÓÖÙÖ ÝÒØÙÒÒÖÔÓ×ØÚÚÙÒÔÖØÙÐÖÑÓÒØÒØÐØÑÔ×ÝÒØÙÒÒÖ ÊÑÖÕÙÞÕÙÐÓÒÓØÒØÙÒÖ×ÙÐØØÒØÕÙ×ÐÓÒÓÒ×ÖÙÒÔÓÒ = Ea+ω ÇÒÓÒÙÒØÖÒ×ÖØÒÖÙÔÓØÒØÐÚÖ×ÐÔÖØÙÐØÐÕÙEc 

π 

E0 

π− 

ØØ×ÒØÐØÒÐÓÒÔÖÐ×ÓÙÚÒØØØ×ÒØÖØ×ÓÖØÓÒÒ ÙÒÖ×ÙÐØØÜÔÖÑÒØÓÖÖØÑÒØÐÓÒ×ÖÚØÓÒÐÒÖÜÖ ÑÑÓÒÓØÒØÙÖ ÔÓÙÖÐÖØÓÒÔÖ×ÐÖ×ÙÐØØ 

i)ØÐÒ× 

 

= (Ei, p ÄÒÚÖ×ÓÒ×ÕÙÖÚØÙÖ×pi i) → (−Ei, −p 

 

d 4 xΨ ∗ π +×ÓÖØV Ψπ +ÒØÖe 

= d 4 xe +iEπ +t 

ÙÜÔÓÒ×ÐÔÖÖ ÄÒÖωÓÑÑÙÒÕÙÔÖÐÔÓØÒØÐÒØÖØÓÒ×ÖÔÖØØÒØÖÐ× ÆÓÙ×ÔÓÙÚÓÒ×ÑÒØÒÒØØÒÖÐ×ÓÒ×ÖØÓÒ×××Ù×Ù×ÐÒ 

−iωt −i(−E 

e e π−)t ∝ 2πδ(Eπ + − ω + Eπ−) ØÖØÓÒÑÖØÔÖÙÒÕÙÖÔÓØÒØÐA µËÐÓÒØÙÐÖÑÔÐÑÒØ ÓÙÔÐÑÒÑÐ∂ µ → ∂ µ +ieA µÒ×ÐÕÙØÓÒÃÐÒÓÖÓÒÓÒÓØÒØ 

(✷ + m 2 )Ψ = −ie(∂µA µ + A µ ∂µ)Ψ + e 2 (A µ ) 2 ÉÑÓÒØÖÖÐÕÙØÓÒ ÇÒÔÙØØÒØÖÙÒÔÖÑÖÔÔÖÓÜÑØÓÒÒÐ××ÒØ ØÐØÖÑÒ 

Ψ 

(A µ ) 2ØÒÖÑÔÐÒØÒ×ÐÕÙØÓÒ ÎÔÖ 

V = ie(∂µA µ + A µ 

∂µ) 

 

M = −i d 4 xΨ ∗ c [ie(∂µA µ + A µ 

 

∂µ)] Ψa

ØÙÓÒ×ÐÒØÖØÓÒÕÙÖÑÒ×ÓÒÒÐÐÔÖÔÖØ×ÙÔÖÑÖØÖÑÐ ×ÓÑÑ 

ØÑÔ×ÄÑÔÐØÙÚÒØ ÄØÖÑ×ÙÖ×ØÒÙÐ××ØÒÙÐÐÒÒÒ×Ð×ÔØÒ×Ð 

 

ÔÖØÙÐ×ÐÖ× ÔÖ××Ù×ØØÙØÓÒ×ÓÒØÓÒ×ÓÒΨaØΨcÔÖÐÙÖÜÔÖ××ÓÒÔÓÙÖ× 

ÚÐÕÙÖÚØÙÖØÖÒ×ÖØq µÓÒÒÔÖ 

ÇÒÒØÐÕÙÖÓÙÖÒØÑÔÖ 

×ÔÔÐÕÙÐÓÖ×ÕÙÓÒ×ØÒÔÖ×ÒÒØÖØÓÒÖÔÐÙÖ ×ØÙÒÜØÒ×ÓÒÙÓÒÔØÓÙÖÒØÓÒÒÙÔØÖ ÚÓÖ ÕÙ 

 

M = e 

 

−∞ − d 4 x(∂µΨ ∗ c)A µ Ψa 

d 4 xjµA µ 

 

d 4 x e −iqx A µ 

q µ = (pa − pc) µ 

d 4 xΨ ∗ c∂µA µ Ψa = Ψ ∗ cA µ Ψa| +∞ 

d 4 x[Ψ ∗ c(∂µΨa) − (∂µΨ ∗ c)Ψa]A µ 

= −i 

M = −ieNaNc(pa + pc)µ 

j µ = ie [Ψ ∗ c(∂ µ Ψa) − (∂ µ Ψ ∗ c)Ψa] 

temps c p 

3 

ÎÖØÜÙ×ÓÒ 

V 

a 

p 

×Ù××ÚÑÒØÔÖØ ÇÒÚÙØÖÖÐÚÓÐÙØÓÒÙÓÙÖÒØÐÖÐØÖÕÙØÖÒ×ÔÓÖØ ÐÔÓØÓÒÝÒØÙÒÖÒÙÐÐÈÓÙÖÖÓÒ 

1 

ÒÐ ÓÒ×ØÖÙØÐÐÑÒØÑØÖÙÓÙÖÒØÑÚÐÙÒØÖÐØØÒØÐØÐØØ 

j µ (ÔÓÒ) ≡ 〈c|j emµ 

|a〉 = 〈ÔÓÒ(pc)|j 

emµ 

|ÔÓÒ(pa)〉

ÆÓØÓÒ×ÕÙÐÓÙÖÒØÑ×ØÓÒ×ÖÚÐÖ×ØÐÑÑÕÙÐÐ ∂µj µ ØÙÐ×ÝÒØ×Ö×ÐØÖÕÙ×ÖÒØ×ÔÜÙÒÐØÖÓÒÚÒÒØÙÒ ÇÒÚÖÖÕÙÒ×ÐÒØÖØÓÒÐÐÓÙÖÒØÐÔÙØÑÔÐÕÙÖ×ÔÖ 

ÒÙØÖÒÓÐØÖÓÒÕÙ ÙØ ÄÓÖÑÙÓÙÖÒØÒØÐÔÓÙÖ×ØØ×ÖÔÖ×ÒØ×ÔÖ×ÓÒ×ÔÐÒ×× 

= 0 

 

ÁÐÖ×ØÜÔÖÑÖÐÑÔA µÖÔÖÐÙØÖÔÖØÙÐØÒØÖÓÙÖÐÖ ×ÙÜÔÖØÙÐ×ÓÑÑÓÒÐÑÒØÓÒÒÒ ×ÙÐØØÒ× ÓÒÔØ×ÐÐØÖÓÝÒÑÕÙÐ××ÕÙÐÓÙÖÒØÐÙØÖÔÖØÙÐÒÖ ÄÜÔÖ××ÓÒÒÐÓØØÖ×ÝÑØÖÕÙÙÒÚÙ×ÓÙÖÒØ× ÇÒ×Ò×ÔÖÒÓÙÚÙ× ÐÑÔA µ×ÓÙ×ÐÓÒØÖÒØ×ÕÙØÓÒ×ÅÜÛÐÐ 

∂µ∂ µ A ν = j ν 

ou ✷A ν = j ν ÇÒÓÔØÐÓÒØÓÒÙÄÓÖÒØÞ 

∂µA µ ÈÓÙÖÐÜÔÖ××ÓÒÙÓÙÖÒØÒÐÓÒÔÖ×ÝÑØÖÙÒÓÖÑ×ÑÐÖ 

ÄÑÓÑÒØØÖÒ×ÖØq×ØØÐÕÙ 

= 0 

ÇÒÖÑÖÕÙÕÙÐÔÒÒ×ÔØÓØÑÔÓÖÐÐÒ×ÐÜÔÖ××ÓÒÙÓÙÖÒØ ×ØÓÒØÒÙÒ×ÐØÖÑÐÜÔÓÒÒØÐÐØÕÙ✷ exp(iqx) ÇÒÒÙØÕÙÐÔÓØÒØÐA µÕÙ×Ø×Ø ×ØÐÓÖÑ 

×ÑÔÐÖÐ×ÒÓØØÓÒ×ÎÓÖÂÂ×ÓÒÐ××ÐÐØÖÓÝÒÑ×ÔÔÒÜÓÒÍÒØ×Ò ÓÙÖÒØÜÔÖÑÒÙÒØ×ÒØÙÖÐÐ×ØÖØÓÒÐ×ÕÙÐÑÒÐπØÔÖÑØ 

ÑÒ×ÓÒ× 

j µ ac = eNaNc(pa + pc) µ exp{−i(pa − pc)x} = eNaNc(pa + pc) µ exp{−iqx} 

j µ 

bd = eNbNd(pb + pd) µ exp{i(pd − pb)x} = eNbNd(pb + pd) µ exp{iqx} 

q µ = (pd − pb) µ = (pa − pc) µ 

A µ = − 1 

q2jµ bd 

= (−q 2 ) exp(iqx)

Ò×Ù×ØØÙÒØ Ø Ò× ÓÒÓØÒØÔÓÙÖÐÑÔÐØÙ 

M = +i 

bd = 

ÄÒØÖÐÓÒÒÙÒØÙÖ(2π) 4 ÐÕÙÒØØÑÓÙÚÑÒØØÐÒÖÄ×Æi×ÓÒØ×ØÙÖ×ÒÓÖÑÐ×ØÓÒ ×ÜÔÖ××ÓÒ××ÓÖ×ÔÖ×ØÙÖ×ÒÑØÕÙ×ÖØ×Ò×ÐÒÒÜ ÖÐØ×ÙÜÔÖØÙÐ×ÒØÖÒØ×Ø×ÓÖØÒØ×ÈÖÐ×ÙØ×ØÙÖ××ÔÖ××ÒØ 

δ(pc+pd−pa−pb)ÕÙÜÔÖÑÐÓÒ×ÖÚØÓÒ 

−pb)×ÔÖØÐÑÒØÐÓÒ×ÖÚØÓÒÙÕÙÖÑÓÑÒØ 

ØÒØÑÔÐØÑÒØÑ× ÄÖÑÑÝÒÑÒÔÓÙÖÐÑÔÐØÙÙ 

ÄÖδ(pc+pd −pa 

×ÓÒÑπ + 

 

d 4 xjacµ 

1 

q2jµ = ie 2 NaNbNcNd(pa + pc) µ (pb + pd)µ 

K + → π + K + 

1 

q2 

1 p 

c 

p 1 

d 

d 4 x e −i(pa−pc)x e i(pd−pb)x 

ie ( p a + pc)ν ig 

1 

1 

µν 

ie ( p + p 

ÖÑÑÝÒÑÒÔÓÙÖÐÙ×ÓÒÙÜÓ×ÓÒ× 

b d ) µ 

q 2 

pa pb 

ÄÑÔÐØÙÒÚÖÒØÄÓÖÒØÞÐÙ×ÓÒÑπ + K + → π + K +Ù ÔÖÑÖÓÖÖÒαÑ= e2 /4π×ÖØ×ÓÙ××ÓÖÑÐÝÒÑÒ 

−ig 

M = (−i)e(pa + pc)µ 

µν 

q2 

ÑÒÑÒÖÖÔØÙÖÐÑÒØ×ÐÑÒØ×Ò×ÙÒÖÔØÐÕÙÐÙ ÇÒÖÓÒÒØÒ× Ð×ÖÒØ×ÐÑÒØ×ÓÒ×ØØÙØ× ÊÝÒ 

(−i)e(pb + pd)ν 

ÐÙÖ

Spin 

Lignes externes 

0 

1 

1 

2 

1 

Lignes internes 

0 

1 

2 

1 

1 

, 

Vertex electrodynamique 

0−0 

1 

Description 

, 

Representation Facteur 

graphique multiplicatif 

Boson 

Fermion , 

entree (initial) 

sortie (final) 

Boson 

ou 

Antifermion 

sortie , (initial) 

entree (final) 

ou 

photon inclu 

Boson 

Fermion 

antiboson 

antiboson 

Boson massif 

Photon 

Particules ponctuelles 

−ie γ µ 

−ie p + p , µ 

( ) gαβ 

− p , αg µ 

β − p g 

β µ 

2 2 

ØÚÔ×ÓÒØ××ÔÒÙÖ×ÖÑÓÒØÒØÖÑÓÒeµ×ØÐÕÙÖÚØÙÖ ×ÓÒØÙØÐ××Ò×ÐÓÒ×ØÖÙØÓÒÐÐÑÒØÑØÖMÆÓØÓÒ×ÕÙÙÔ ÊÐ×ÔÓÙÖÐ×ÖÑÑ×ÝÒÑÒÄ×ØÙÖ×ÑÙÐØÔÐØ× 

1−1 

α 

ÔÓÐÖ×ØÓÒÙÒÓ×ÓÒ×ÔÒ 

 

p , 

p 

ou 

ou 

ou 

1 

u(p) 

− 

u(p) 

− 

v(p) 

v(p) 

ε µ 

ε µ * 

2 2 

− i /(k −m ) 

i 

i 

γ 

µ k µ + m 

k 2 − m 2 

k2 − m 2 

− g µν +k µ k ν 

i (− g µν / k 2 ) 

−ie ( p + , 

p ) µ 

/m 

2

ÙÖ ÄÔÔÐØÓÒÒ×ÐÙÄÓÖÒØÞ×ÖÐ×ÝÒÑÒ×ÙÚÒØ×ÚÓÖ 

ØÙÐ ÙÒØÙÖÔÓÙÖØÓÙØÓ×ÓÒÜØÖÒÒØÖÒØÓÙ×ÓÖØÒØØ×ÙÔÔÓ×ÔÓÒ ÔÖÑØÖØÖÓÙÚÖÐÖÐØÓÒ ÓÒÒØÖÓÙØ 

ÙÒÔÓØÓÒÒØÖÒ ÙÒØÖÑÔÔÔÓÙÖÕÙÚÖØÜÑÔÐÕÙÒØÔÖØÙÐ××ÔÒØ 

ÄÓÖÑ ÔÓÙÖÙØÖ×ÓÜÙÑ×Ð×Ö×ÙÐØØ×ÔÝ×ÕÙ××ÖÒØÐ×ÑÑ×Ä ÄØÙÖÕ2ÔÖÓ×ÔÔÐÐÑ××ÙÖÖÙÔÓØÓÒÚÖØÙÐ×ØÒÓÒÒÙÐ ÙÒÔÖÓÔØÙÖ Õ2ÚÒØÙÓÜÐÙÄÓÖÒØÞØØÓÖÑ×ÖØÖÒØ Õ2ÔÓÙÖÐÔÓØÓÒÙÄÓÖÒØÞ 

ØÙÖµνØÐÓÒÒÜÓÒÒØÖÐ×ÙÜÚÖØÜÑ ÄÔÖÓÐØØÖÒ×ØÓÒ×ØÓÒÔÖÓÔÓÖØÓÒÒÐÐ 

 

2 ËØÓÒÖÒØÐÐπ + ÈÐÓÒ×ÒÓÙ×Ò×ÐÖÖÒØÐÙÑØÒ×ÖÖÒØÐÐÓÖÑÙÐ ÑÐÖØÓÒ ×ÖÙØdσ(i → 

ÇÒÖØÖÓÙÚÐÖÐØÓÒ 1Ä×ØÓÒÖÒØÐÐÙ×ÓÒÒ×ÐÑ×ÖØÓÒ 

|p2)Ø×ÓÒ 

= ÓEi 

 

×ÐÔÖÓ××Ù××ØÐ×ØÕÙ(|p1| = ÔÓ×c = 

ÓÐÐÑÒØÑØÖÙÖÖ×ØÓÒÒÔÖ 

 

|Mfi| 2 = [(pa + pc)(pb + pd)] 2 

e 2 

q 2 

K + → π + K +Ò×Ð 

f) = 1 

|Mfi| 

4|p|Ei 

2 EbÍØÐ×ÒØÔÓÙÖÐÄÔ×ÓÒÓØÒØ 

dLips(pa + pb, {pj}) 

Ea + 

1 

dσ(i → f) = | Mfi | 

4 | p | Ei 

2 1 

(4π) 2 

| p | 

dΩ = 

Ei 

| Mfi | 2 

dΩ 

(8πEi) 2 

dσ 

dΩ 

(Ñ)= |Mfi| 2 

(8πEi) 2

ÃÒÑØ× ÇÒÒØÓÑÑÙÒÑÒØÐ×ÖÒÙÖ×ÒÚÖÒØ×ÚÓÖÐØÐÈ×ÓÙ× ËØÓÒÖÒØÐÐ×ÓÙ×ÓÖÑÒÚÖÒØ 

ÎÖÐ×ÅÒÐ×ØÑ 

 

ÒØÖÑ×ÚÖÐ×Ð×ØÓÒÖÒØÐÐ×ÖØ 

ÕÙÐÓÒÔÙØÚÐÓÔÔÖÒ 

 

ÇÒÔÙØÙ××ÜÔÖÑÖ|M| 2ÒÓÒØÓÒ×ØÙÇÒØÖÓÙÚÔÖØÖ 

 

 

 

2 

dσ 

dt 

s = (pa + pb) 2 = (pc + pd) 2 

t = (pa − pc) 2 = (pb − pd) 2 

u = (pa − pd) 2 = (pb − pc) 2 

s + t + u = m 2 a + m 2 b + m 2 c + m 2 d 

= 1 

64π 

dσ 1 

= 

dt 16π 

|M| 2 = e 4 

|M| 2 

(papb) 2 − (mamb) 2 

|M| 2 

[s − (ma + mb) 2 ][s − (ma − mb) 2 ] 

s − u 

t 

2 

= (4παem) 2 

 

s − u 

t

ÓÖÑÙÐÊÙØÖÓÖ 

mbÓÒØÖÓÙÚÐ 

≪ mbØ|p ÉÑÓÒØÖÖÕÙÒ×ÐÔÔÖÓÜÑØÓÒma b |≪ 

 

dσ 

= 

dΩ Rutherford 

1 α 

4 

2 

| pa | 2 

1 

sin 4 

(θ/2) ÑÔÐØÙ×Ù×ÓÒÑπ + π + → π + π + Øπ + π− → π + π− ÔÖÒÖÒÓÑÔØÐÖÔÒ×ÐÕÙÐÐ×ÙÜÔÖØÙÐ×ÒÐ××ÓÒØÒ× Ò×ÙÜÔÖØÙÐ×Ò×ÐÐÒ×ÖÔ×ÝÒÑÒÓÒÓØ ÚÓÖÙÖ Ò×Ð×ÙÜÓ×ÓÒ×ÒØÕÙ×ÐÑÔÐØÙÓØØÖ×ÝÑØÖÕÙÔÖ 

p 

c 

p d 

q 

p p 

a b 

, 

q 

p 

a p 

b ÖÑÑÒÔÓÙÖÐÙ×ÓÒπ + π + ÄÑÔÐØÙÙ×ÓÒ×ÖØ 

M(π + π + ) = (−i) 3 e 2 

 

(pa + pc)µ(pb + pd) µ 

(pb − pd) 2 + (pa + pd)µ(pb + pc) µ 

(pb − pc) 2 ÇÒÔÙØÙØÐ×ÖÖ×ÙÐØØÔÓÙÖÖÖÐÑÔÐØÙÙ×ÓÒπ + π− → π + π− Äπ −ØÒØÔÖ×ÓÑÑÐÒØÔÖØÙÐÓÒÐÙÔÔÐÕÙÐÖÐÐÒÚÖ×ÓÒÙ ×Ò×ÚØÙÖ× ÚØÙÖÙÖ ÇÒÒÐØØÒØÖÒØØÐØØ×ÓÖØÒØØÓÒØÙÙÒÒÑÒØ 

M(π + π − ) [pa, pb; pc, pd] ≡ M(π + π + 

) [pa, −pd; pc, −pb] 

p d 

p c

p a 

p c 

ÕÙÓÒÒ 

M(π + π − ) = (−i) 3 e 2 

p b 

p d 

π+ π − π + π + 

ÖÑÑÒÔÓÙÖÐÙ×ÓÒπ + π− . 

 

(pa + pc)µ(−pd − pb) µ 

(−pd + pb) 2 

+ (pa − pb)µ(−pd + pc) µ 

(−pd − pc) 2 

 

= 

= (−i) 3 e 2 

 

−(pa + pc)µ(pd + pb) µ 

(pb − pd) 2 + (pa − pb)µ(−pd + pc) µ 

(pa + pb) 2 

 

ÇÒÖÓÒÒØÒ×ÐÔÖÑÖØÖÑÐÑÔÐØÙÓØÒÙÔÓÙÖπ + K +ÓÖÑ×ÙÒ ÙÖ ÒÑÒØ×ÒÐÔÖ×ÒÙÒÖÒØÚÄÙÜÑØÖÑ ÓÖÖ×ÔÓÒÙÒÔÖÓ××Ù×ÒÒÐØÓÒÓÑÑÖÔÖ×ÒØÔÖÐÖÔÐ 

p d 

p c 

p a p b 

π π 

p b 

p a 

ÖÑÑÒÒÐØÓÒπ + π − → γ ∗ → π + π − 

 

p a 

p c 

p d 

p b

Ä×ÓÐÙØÓÒ×ÕÙØÓÒ×ÅÜÛÐÐÔÓÙÖÐÑÔEÐÓÒÑÐÖ× ÈÓØÓÒ×ÖÐ×ØÑ××× 

ÔÓÐÖ×ØÓÒÐÓÒ ÍÒÓÖÑÒÐÓÙÔÙØØÖÔÖ×ÔÓÙÖÐÑÔBÓÖØÓÓÒÐEØÐÜ ÔÖÓÔØÓÒÄÚØÙÖÙÒØeÓÒ×ØØÙÐ×Ò×ÐÕÙÐÐÓÒÜÔÖÑÐ ÙÖÙÒÔ×δÒØÖÐ×ÙÜÔÖÓØÓÒ×ÈÓÙÖÐÑÔÔÝ×ÕÙÓÒÔÖÒ Ä×ÑÔÐØÙ×E1ØE2×ÓÒØ×ÕÙÒØØ×ÓÑÔÐÜ×ÕÙÔÖÑØÒØÖÓ 

ÐÐ×Ø◦ÐÔÓÐÖ×ØÓÒ×ØÖÙÐÖÄÑÔÔÝ×ÕÙØÓÙÖÒÙØÓÙÖ ÐÔ×Ò×ØÔ×ÒÙÐÐÓÒÔÖÐÔÓÐÖ×ØÓÒÐÐÔØÕÙÒÔÖØÙÐÖ× 

0ÓÒÙÒÔÓÐÖ×ØÓÒÐÒÖÐÑÔ 

ÐÜzØÙÜÔÓÐÖ×ØÓÒ×ÖÙÐÖ×ÙØÖÓØ×ÓÒØÔÓ××Ð× 

ÐÔÖØÖÐÐÐ×ÓÐÙØÓÒËδ = Ó×ÐÐÒ×ÙÒÔÐÒÕÙØÙÒÒÐθ= tan 

ÓÖÑ×ÙÜÚØÙÖ×ÓÑÔÐÜ×ÓÖØÓÓÒÙÜ ÐÒÖ×ÓÖØÓÓÒÐ××ÐÓÒexØeyÇÒÔÙØÐØÖÒØÚÑÒØÒÖÙÒ× Ò×ÐÔÓÐÖ×ØÓÒÐÓÒ×ØÖØÐ×ÙÜÓÑÔÓ×ÒØ× ÇÒÔÖÐÐØ×ÔÓ×ØÚØÒØÚλ = 

e± = 1 Ä×ÓÐÙØÓÒÚÒØ 

√ (ex ± iey) ˆz ÕÙÐÓÒÓÑÔÐØÔÖÐÖØÓÒe0 

ÔÓÐÖ×ØÓÒ×ÖÙÐÖ×ÐØ×ÓÔÔÓ××Ò×ØØ×ÐÓÒÔÓÐÖ×Ö ÐÖÚÒØÜÔÖÑÖÐØØÔÓÐÖ×ØÓÒÓÑÑÙÒ×ÙÔÖÔÓ×ØÓÒÙÜ 

2 

ÓÑÔÓ×ÒØ× ÇÒ××ÓÒÖÐÑÒØÐÑÔÙÔÓØÓÒÙÔÓØÒØÐAÒ×ÐÙ 

+1Ð×ÓÑÔÓ×ÒØ× 0×ØÐÓÖÑ −1Ð× ÙÐÖÑÒØÚλ = ØÐÐÚλ = 

 

ÓÙÐÓÑÐ×ÓÐÙØÓÒÐÕÙØÓÒÔÖÓÔØÓÒ✷A = 

ÔÖÓÔÒØ×ÐÓÒÐÜÞÔÙØØÖÖØ 

E(t, z) = (exE1 + eyE2) exp(ikz − iωt) 

ÔÓÙÖÐÑÔÖÐ E(t, z) = E(ex ± iey) exp(ikz − iωt) 

Ex = E cos(kz − ωt) Ey = ∓E sin(kz − ωt) 

−1 (E2/E1)ÔÖÖÔÔÓÖØÐÜÜË 

±1 

 

= 

E(t, z) = (e+E+ + e−E−) exp(ikz − iwt) 

 

A = eN exp(ik · r − iωt)

Ò×ÐÖØÓÒkØÐÐÕÙ ÄÖÐØÓÒÖÔÖ×ÒØÙÒÓÒÔÓØÓÒÔÓÐÖ×ØÓÒe×ÔÖÓÔÒØ 

ÓÙ×ÙÖÙÒ×ÔÓÐÖ×ØÓÒ×ÖÙÐÖ× ÔÙÚÒØØÖÒ××ÙÖÙÒ×ÔÓÐÖ×ØÓÒ×ÐÒÖ×ÓÑÑÒ× ÓÒÚÒØÓÒÒÐÐÑÒØeÒÕÙÐÖØÓÒÙÑÔEË×ÓÑÔÓ×ÒØ× Ò×ÖÒÖ×ÓÒ 

k 

 

⎠ÖÔÖ×ÒØÙÒÔÓØÓÒÐØ 

⎠ 

ÐØÒÔÒÒØ× ×ÒØÐÓÒØÙÒÐ Ò×Ð×ÙÒÔÓØÓÒÑ××ÓÒÙÖØÙ××ÐÔÓ××ÐØÙÒÓÑÔÓ ÑÓÒØÖÕÙÐÓÒÒÙÜØØ×ÔÓÐÖ×ØÓÒ ÙÜØØ× ÄØØ⎛ 

ÔÖÓÔØÓÒÐÓÒÒ×Ð×ÐÒÖÐ×ÓÑÔÓ×ÒØ××ØÖÒÓÖÑÒØ ÇÒÔÙØÚÖÖÕÙ×ÐÓÒØÙÙÒÖÓØØÓÒÒÐθÙØÓÙÖÐÜÞ ÓÑÑ 

e(λ 

e 

e ′ y = ex ØÒ×ÕÙÒ×Ð×ÖÙÐÖÐÐ××ØÖÒ×ÓÖÑÒØÓÑÑ 

sin θ + ey cosθ 

ÇÒÖÓÒÒØÐÓÖÑÐÓÔÖØÙÖÖÓØØÓÒÙØÓÙÖÐÜÞÔÓÙÖÙÒ 

ÔÓØÓÒ Ð×ÙÒÔÖØÙÐÝÒØÙÒÑ××ØÐ×ÙÒÔÖØÙÐÑ××ÒÙÐÐÐ ×ÙØ×ØÖÔÖ×Ò×Ð×ÔÖÖÔ×ÕÙ×ÙÚÒØÒÓÒ×ÖÒØ×ÔÖÑÒØ 

exp(iJzθ)ÚÓÖï 

 

ÔÖØÙÐ×ÔÒÊz(θ) = 

 

· e = 0 ÉÚÖÖÐÔÓÙÖÐÙÓÙÐÓÑ∇A = 0. 

⎛ ⎞ 

⎛ 

1 

e(λ = +1) = ⎝ 0 ⎠ e(λ = −1) = ⎝ 

0 

⎞ 

1 

⎝ 0 

0 

⎛ 

0 

= 0) = ⎝ 1 

0 

0 

0 

1 

⎞ 

 

⎞ 

⎠ 

′ x = ex cosθ − ey sin θ 

e ′ z = ez 

e ′ + = e+ exp(i(+1)θ) 

e ′ − = e− exp(i(−1)θ) 

e ′ 0 = ez

ÉØÙÖÐÑ××ÓÒÖÝÓÒÒÑÒØÑÔÖÙÒÐØÖÓÒÒÓÒÖÐØÚ×ØÒ×ÙÒ ÑÔÖÐÐÐÑÒØÙÑÔÑÒØÕÙØÔÖÔÒÙÐÖÑÒØÑÔ ÉÑÔÓÙÖÐ×ÖÐØÚ×ØÖÝÓÒÒÑÒØ×ÝÒÖÓØÖÓÒ ÑÔÑÒØÕÙÖÝÓÒÒÑÒØÝÐÓØÖÓÒÉÙÐÐ×ØÐÔÓÐÖ×ØÓÒÐÓÒ 

e(λËÓÒ×ÔÐÒ×ÐÖÖÒØÐÖÔÓ×ÐÔÖØÙÐÓÒÐ× ÙÜØØ×ÐØλ ÄÔÖØÙÐÑ××Ú×ÔÒ −1ÓÒØÓÖÖ×ÔÓÒÖÙÒÚØÙÖÔÓÐÖ×ØÓÒ 

ÓÑÑÙÒÑÒØÐØØ×ØÖØÒ×ÙÒ×ÔÓÐÖ×ØÓÒÖÙÐÖ 

ÖØ×ÒÒex = 

ÚÐÓÒØÓÒÓÖØÓÓÒÐØ e ∗ (λ)e(λ ′ ) = δλλ ′ ÈÓÙÖÖÖÐÓØÒÑÓÙÚÑÒØÓÒ×ØÑÒÓÒ×ØÖÙÖÙÒÕÙÖÚØÙÖ 

e = (e 0 

(λ), e(λ)) Úe 0 µ Ö×ÙÐØØ×ØÚÐÐÒ×ØÓÙØÖÔÖØÑÓÒØÖÕÙÐÓÒØÖÓ×ÕÙÖÚØÙÖ× 

ÔÓÐÖ×ØÓÒÒÔÒÒØ×ÇÒÒÙÒÔÖØÙÐ×ÔÒ ÄÔÖØÙÐÑ××ÒÙÐÐØ×ÔÒ ÐÔÓØÓÒ 

eµ(λ) = 0 

ÄÔÖØÙÐ×ÔÖÓÔÒ×ÐÚÚØ××ØÓÒÒÔ×ÖÖÒØÐ ÖÔÓ×ÖÚÓÒ×Ð×ÓÐÙØÓÒÐÕÙØÓÒÓÒÐÖ✷A µ Ú k 2 

 

= 0 

(1, 0, 0) ey = (0, 1, 0) ez = (0, 0, 1) 

e(λ = +1) ≡ − 1 

√ (1, i, 0) 

2 

e(λ = 0) ≡ (0, 0, 1) 

e(λ = −1) ≡ − 1 

√ (1, −i, 0) 

2 

0Ò×Ð×Ý×ØÑÖÔÓ×ÐÔÖØÙÐÒ××Ý×ØÑÓÒÙ×× 

0)Ó = 

p = (M, 0, 0, 

p 

0×ÓÙ×ÐÓÖÑ 

= 

A µ = Ne µ exp(−ikx)

ØÕÙÐÓÒØÓÒÓÖØÓÓÒÐØ×ØØÓÙÓÙÖ×ÚÐÐÇÒÔÙØÑÓÒØÖÖÕÙ ØØÓÒØÓÒ×ØÒÐÓÙÑ×ÐÐ×Ø××ÓÙÓÜÐÙ ØÐ××ÓÒ×ÑÒØÒÒØÕÙÐÓÒÕÙÙÜØØ×ÔÓÐÖ×ØÓÒÒÔÒÒØ× 

ÖÐÓØ×ØÑ××ÒÙÐÐÄÙÄÓÖÒØÞ∂µA µ 0×ØÖÙØÔÖ 

= 

k µ 

eµ = 0 

ÐÙÄÓÖÒØÞ×ØÒÓÖ×Ø×Ø×ÐÓÒÖÑÔÐ 

A µ → A µ − ∂ µ χ 

ÔÓÙÖÚÙÕÙÐÓÒØÓÒ×ÐÖχ×ÓØÙÒ×ÓÐÙØÓÒÐÕÙØÓÒ 

✷χ = 0 

Ëχ×ØÐÓÖÑÜÔÜÚÒØ 

A µ → A µ − iαk µ exp(−ikx) = N(e µ + βk µ ) exp(−ikx) 

ÒÐÐÙÕÙÖÑÓÑÒØÐÔÖØÙÐ ÕÙ×ØÕÙÚÐÒØÒÖÐÚØÙÖÔÓÐÖ×ØÓÒÙÒÕÙÒØØÔÖÓÔÓÖØÓÒ 

ÇÒÔÙØ×ÖÖÒÖÔÓÙÖØÖÓÙÚÖÐÚÐÙÖβÕÙÒÒÙÐÐÐÓÑÔÓ×ÒØØÑÔÓ ÖÐÐÙÒÓÙÚÙÕÙÖÚØÙÖØØÓÒÓÒÖØÖÓÙÚÔÖØÖÐ ÈÙ×ÕÙ2 ÒÓÙÚÙÕÙÖÚØÙÖÔÓÐÖ×ØÓÒ×Ø×ØØÓÙÓÙÖ× 

ÔÓÐÖ×ØÓÒÒÔÒÒØ×ÊÑÖÕÙÞÕÙÖ×ÓÒÒÑÒØØÒØÔÖÕÙÐÓØ ÓÒØÓÒØÖÒ×ÚÖ×ÐØk ·e ÙÒÑ××ÒÙÐÐ(k 2 ÔÓØÓÒÖÐÕÙÖÑÓÑÒØØÔÓÐÖ×ØÓÒλÙÒÔÖÓ××Ù×Ù×ÓÒ Ò×Ð×ÖÐ×ÝÒÑÒÓÒÒÐÙØÐÔÓ××ÐØÖÔÖØÔÖÙÒ 

 

ÚÐÖÐØÓÒÓÖØÓÓÒÐØÐ×ÒÑÓÒ×ÚÒØÙØÕÙÐÓÒÙÒÕÙÖ ÚØÙÖ e ∗ (λ)e(λ ′ ) = −δλλ ′ ÄÒÓÖÑÐ×ØÓÒ×ØÐÑÑÕÙÔÓÙÖÐ×ÔÖØÙÐ××ÔÒ 

 

= 0) 

e µ → e µ + βk µ 

= 0ÕÙÜÔÖÑÙ××ÐÜ×ØÒÙÜØØ× 

A µ = Ne µ ÔÓÙÖÐÔÓØÓÒÒØÖÒØ (k, λ) exp(−ikx) 

A µ = Ne µ∗ ÔÓÙÖÐÔÓØÓÒ×ÓÖØÒØ 

(k, λ) exp(+ikx)

ÓÖÑ×ÕÙÓÒÔÖØÑÒØÒÒØ××ÓÐÙØÓÒ×ÐÕÙØÓÒÖÇÒÚÙï ÄÔÖÓÙÖ×ØÒÐÓÙÐÐÙØÐ×ÔÓÙÖÐÔÖØÙÐ×ÔÒ ÄÔÖØÙÐ×ÔÒ ï 

Ò×ØÓÙÖÒØÔÖÓÐØÔÓÙÖÐÒØÖØÓÒÑ Óω×ØÐÕÙÖ×ÔÒÙÖÖØÝÔÙÕÙØÓÒ ØØÝÔÚ ÈÖØÒØÐ×ØÖÙØÙÖÒÖÐÙÓÙÖÒØ ÔÓÙÖÐÒØÔÖØÙÐ ÓÒÔÙØÜÔÖÑÖÐ ÔÓÙÖÐÔÖØÙÐ ÐÙÐÔÖØÖ 

ÕÙÔÙØØÖÚÐÓÔÔÒ 

ÆÓÙ×ÓÔØÓÒ×ÐÑÑÒÓÖÑÐ×ØÓÒÕÙÔÓÙÖÐÔÖØÙÐ×ÔÒ ×ÓÒØÖÔÔÐ×Ò×ÐÙÖ ÓÒÚÒØÓÒ××ÓÒØÔÓ××Ð×Ä×ÖÐ×ÝÒÑÒÔÓÙÖÙÒÐØÖÓÒØÙÒÔÓ×ØÖÓÒ ÙØÖ× 

ÉÐÒØÖØÓÒÑÙÒÖÑÓÒ×ÔÒ ÈÓÙÖÓÒÒÖÙÒÜÑÔÐÐÑÔÐØÙÙ×ÓÒÑÙÒÐØÖÓÒÔÖÙÒ 

uiÅÓÒØÖÖÕÙÐÐØÙÖÑÓÒ×ØÓÒ×ÖÚ ×ÜÔÖÑÔÖÙÒÓÙÖÒØÚØÓ 

ÔÓÒÔÓ×Øe − ÄÔÖÓÔØÙÖ×ØÒÓÖÐÓÖÑ ÒÓÖÑÐ×ØÓÒØÐÖδÒÓÙ×ÓØÒÓÒ×ÐÒÐÓÙ Õ2ÍÒÓ×ÐÑÒ×Ð×ØÙÖ× 

ØscÖÔÐÙÖ 

Ò×ÐÙ×ÓÒÙÜÓ×ÓÒ×ØÒØÓÒÒÐÔÖ×ÒÙ×ÔÒÓÒÚÓØÒÖ ÄÓÒ×ØÖÙØÓÒÐ×ØÓÒÖÒØÐÐ×ØÔÐÙ×ÓÑÔÐÕÙÕÙ Ä×ØØ××ÔÒ×ÖÑÓÒ×ÒØÖÒØØ×ÓÖØÒØ×ÓÒØ×Ô×ÔÖÐ×Ò×sa 

dσsasc ∝ |Msasc| 2 •ËÐÓÒ×ÐØÖÓÒ×ÒÓÒÔÓÐÖ××ÓÒÓØÖÙÒÑÓÝÒÒ×ÙÖÐ×ØØ× 

 

ÕÙ××ÓÐÙØÓÒ××ÓÒØÐÓÖÑ 

Ψi×ÓÙ×ÐÓÖÑ 

Ψ = ω(p, s) exp(−ipx) 

 

4 d xΨfV 

j µ (e − ) = (−e)Ψfγ µ Ψi 

ÖÐÐÓÖÑufγ µ 

×ÔÒ Ø−1/2 

j µ (e − ) = (−e)NiNfu(pf, sf)γ µ u(pi, si) exp{i(pf − pi)x} 

π + → e−π +×ØÜÔÖÑÔÖ 

M = 

d 4 xj ν (e − ) gµν 

q2 jµ (π + ) 

Msasc = i(−i)eu(pc, sc)γµu(pa, sa) −igµν 

q2 (−i)e(pb + pd)ν

u− ( p c ,s c) p c 

ie 

u( p a ,s γµ 

igµν 

 

q2 

p 

a) a pb 

1 

e− π+ ÖÑÑÝÒÑÒÔÓÙÖÐÙ×ÓÒe −π + → e−π + 

ÐÒØÖÙÜØØ××ÔÒÔÓ××Ð×Ð×ÓÖØ×ÙÖÐ×ÕÙÐ×ÐÙØ×ÓÑÑÖ 

•ËÐØØÙÖ×ØÒ×Ò×ÐÐÔÓÐÖ×ØÓÒÓÒÔÓÙÖÕÙØØ×ÔÒ Ä×ØÓÒÒÓÒÔÓÐÖ××ØÓÒ 

dσNP = 1 

Ùï ÄÐÙÐÐÐÑÒØÑØÖÔÙØ×ÖÒÜÔÖÑÒØÜÔÐØÑÒØÐ× 

 

ÉØÙÖÐ×ÐÙ×ÓÒÙÜÔÖØÙÐ××ÔÒ ÓÒØÓÒ×ÓÒÐ×ÙÓÒÒ×ÙÔØÖ ÉÙÐÕÙ×ÜÑÔÐ××ÓÒØÓÒÒ× 

dσsasc 

2 

sa sc 

×ØÒÙÐÐÔÓÙÖÐÔÓØÓÒÖÐÑ×ÐÐ×ØÒÓÒÒÙÐÐÔÓÙÖÐÔÓØÓÒÚÖØÙÐÙ ÄÔÖÓÔØÙÖÙÒÔÖØÙÐÑ××Ú 

ÔÖÓÔØÙÖÊÔÔÐÓÒ××ÑØÕÙÑÒØÕÙÓÒØÑÒ ÔÖ 

✷A = j ∼ (q · q)A = j → A ∝ 1 

q2j Ó×ØÐÓÙÖÒØÑ ÃÐÒÓÖÓÒ ÓÒ×ÖÓÒ×ÑÒØÒÒØÙÒÓ×ÓÒÑ×××ÔÒ Ó××ÒØÐÕÙØÓÒ 

(✷ 

p d 

1 

ie ( p + p d ) 

b ν 

ÄÔÖÓÔØÙÖÔÓØÓÒÕÙ×ØÐÓÖÑ Õ2ÚÓÖ ÄÕÙÒØØq 2 

Ójh×ØÐ×ÓÙÖÓÙÖÒØÓ×ÓÒÕÙ 

 

+ m 2 )φ = jh → (−p 2 + m 2 )φ = jh

Ä×ÓÐÙØÓÒ×ØÐÓÖÑφ ∝ 

p2 − m2jh ÇÒ×ØÑÒÐÖÐ 

ÔÖÓÔØÙÖÙÒÓ×ÓÒ×ÔÒ Ñ××ÑØÑÔÙÐ×ÓÒÔ 

ÙÒÔÓÒÔÓ×ØÖÔÐÙÖ ÄÖÐØÓÒ ×ØÙØÐ×ÖÔÜÒ×Ð×ÐÙ×ÓÒÓÑÔØÓÒ×ÙÖ ÄÜÔÖ××ÓÒÙÔÖÓÔØÙÖÔÖ×ÒØ 

1 

−i 

p 2 − m 2 

 

−i ( p2 − m2) 

π+ 

ÖÑÑÐÙ×ÓÒÓÑÔØÓÒ×ÙÖÐπ + ÙÒÔÐÕÙÒÐÔÖØÙÐÒÚÒØÖÐÐØØÔÖÓÔÖØÔÖÑØÙÒ ÒØÓÒÐØÖÒØÚÐÑ×××ØÐÚÐÙÖÔ2ÙÔÐÙÔÖÓÔØÙÖ ÙØÖ××ÙØÐØ××ÓÒØÔÖ×ÒØ×Ò×ÐÔÖÓÔØÙÖÔÖØÙÐ××ÔÒ ÓÒ 

ÓÙ ÐÒÓÑÒØÙÖÒÔ2Ñ2ØÒØØÓÙÓÙÖ×ÐÈÓÙÖÐ×ÖÑÓÒ××ÔÒ 

ÔÖÓÔØÙÖÙÒÖÑÓÒ×ÔÒ Ñ××ÑØÑÔÙÐ×ÓÒÔ 

i /p + m 

p2 − m2 

ÎÓÖÔÖÜÖÔ

ÐÙÐÕÙÐÕÙ××ØÓÒ××dσ/dΩ ÓÒ×ÖÓÒ×ÐÔÖÓÐØÙ×ÓÒÐØÖÓÒÔÓÒe − ÐÙÐÖ |Msasc| 2 = Msasc(Msasc) ∗ ØÖÑÓÒÙÙÖÑØÕÙÓÒÒ ÈÓÙÖÙÒ×ÐÖÓÒÙÙÓÑÔÐÜØÓÒÙÙÖÑØÕÙÓÒÒØÄ 

= Óu = 

u † γ0 

e 2 

q 2 

2 

π + → e − π +ÇÒÓØ 

[u(pc, sc)γµu(pa, sa)(pb + pd) µ ] [...] † 

[u(pc, sc)γνu(pa, sa)(pb + pd) ν ] † = u † (pa, sa)γ † νu † (pc, sc) (pb + pd) ν = 

 

= u † (pa, sa)γ † νγ† 0u(pc, 

sc) (pb + pd) ν ÈÙ×ÕÙÜÖγ † νγ† † 

u (pa, sa)γ0γνu(pc, sc) ÒÓÒÐÙ×ÓÒÒ×ÐÐÙÐ ÓÒÓØÚÐÙÖ 

= [u(pa, sa)γνu(pc, sc)] 

1 

2 sa sc |Msasc| 2 = 

= 1 

 

e2 2 q2 2 

sa sc [u(pc, sc)γµu(pa, sa)(pb + pd) µ ] [u(pa, sa)γνu(pc, sc)(pb + pd)ν] = 

= 1 

 

e2 2 q2 2 

sa sc [u(pc, sc)γµu(pa, sa)][u(pa, sa)γνu(pc, sc)](pb + pd) µ (pb + pd) ν = 

= 1 

 

e2 2 q2 2 LµνT µν ÇÒÓÒÐÔÖÓÙØÙÜØÒ×ÙÖ× 

1 

| Msasc | 

2 

sa sc 

2 = 1 

2 e 

2 q2 ÐØÒ×ÙÖÐÔØÓÒÕÙÖÑÓÒ×ÔÒ ×Ø 

Lµν = 1 ÐØÒ×ÙÖÖÓÒÕÙÓ×ÓÒ×ÔÒ ×Ø 

 

[u(pc, sc)γµu(pa, sa)][u(pa, sa)γνu(pc, sc)] 

2 

sa sc 

0 = γ † ν γ0 = γ0(γ0γ † ν γ0) = γ0γν , ÓÒÙ×× 

2 

LµνT µν 

T µν = (pb + pd) µ (pb + pd) ν

ÔÐØ×ÙØÔÖ×ÙÒÔÙÐÖÓÒØÖÓÙÚ ÒØÖÐ×ÙÜÓÙÖÒØ××ØÒÕÙÒØÖÑ×ÙÐÙÐ ØØ×ÔÖØÓÒÕÙ×Ø××ÓÙØÕÙÐÓÒÒÙÒÕÙÑÒØÙÒÔÓØÓÒ ÈÓÙÖÐØÒ×ÙÖÐÔØÓÒÕÙÐÓÖÑ ÒÒØÖÓÙ×ÒØÐ×ÚÐÙÖ×Ü 

ÄÔÔÐØÓÒ×ØÓÖÑ×ØÖ×ÙÜÑØÖ×γÓÒÒ 

Ó: 

ÖÒØÐÐÒÙÐÖÒÑØØÒØÐÔÓÒÙÖÔÓ×Ø×Ò××ØÖÙØÙÖ ÄÚÐÓÔÔÑÒØÙÐÙÐÓÒÙØÐÜÔÖ××ÓÒ×ÙÚÒØÔÓÙÖÐ×ØÓÒ 

 

 

ÓÐÐ×ÓÒ ÇÒÒÐÐÑ××ÐÐØÖÓÒÕÙÔÖÑ×ÖÖ Ú| p 

ÄØÙÖ|p ′ ØÖÜÔÖÑ×ÓÙ×ÐÓÖÑÑÓÒØÖÖÓÑÑÜÖ 

ËÐÔÖØÙÐÐ×ØØÖ×ÐÓÙÖÔÖÖÔÔÓÖØÙÔÖÓØÐÓÒÔÙØÔÓ×Ö|p ′ ÉÔÖØÒØ×ØÒ×ÙÖ× Ø ØÙÖÒØÐÐÐÙÐdσ/dΩ Ò×Ð×ÐÙ×ÓÒe − ÖÑÓÒÕÙ×ÐØÒ×ÙÖÐØÖÓÒÕÙLµνØ×ÓÒÓÑÓÐÓÙÑÙÓÒÕÙMµνÄ Ö×ÙÐØØ×Ø 

 

×××ÒÒØ×Ò××ØÖÙØÙÖ ÎÓÖÔÖÜÖÒÒÜÓÙÖ § 

 

ssÐÒ ÔÔÐÓÑÑÙÒÑÒØ×ØÓÒÅÓØØÇÒØÖÓÙÚÙ××ÐÒÓØØÓÒ dσ 

Lµν = 1 

2 Tr 

 

/p = γ µ pµ. 

(/p c + m)γµ(/p a + m)γν 

Lµν = 2 pcµpaν + pcνpaµ + (q 2 /2)gµν 

dσNP 

dΩ (e−π + α 

) = 

2 

4 | p | 2 sin 4 (θ/2) cos2 (θ/2) | p′ | 

| p | ≡ 

 

dσ 

dΩ ss 

|Ø|p ′ |Ð×ÕÙÒØØ×ÑÓÙÚÑÒØÐÐØÖÓÒÚÒØØÔÖ×Ð 

ÔÓÙÖÐÙ×ÓÒe − 

q 2 = −4 | p || p ′ | sin 2 |×ØÙÖÙÐÐÔÖØÙÐÐÑ××MÁÐÔÙØ 

(θ/2) 

| / | p 

| p ′ 

| 2 | p | 

= 1/ 1 + 

| p | M sin2 (θ/2) 

| 

|p| ≃ 

π + → e−π + 

µ − → e− µ −ÓÒÓØÓÒØÖØÖÙÜØÒ×ÙÖ× 

dΩ (e− µ − 

dσ 

) = 

dΩ 

dσNP 

ss 

 

1 − q2 tan2 θ/2 

2M2 

dΩ

ÒÓÒ×ÙÐÑÒØÔÖÐÙÖ×Ö×ÐØÖÕÙ×Ñ×ÐÑÒØÔÖÐÙÖ×ÑÓÑÒØ× ÑÒØÕÙ×M×ØÐÑ××ÙÑÙÓÒ 

θ/2×ØÙØÕÙÐÐØÖÓÒØÐÑÙÓÒÒØÖ××ÒØ 

×ÖØ ÜÔÖÑÒØÖÑ×ÚÖÐ×ÅÒÐ×ØÑ×ØÙØØ×ØÓÒ ÄØÖÑÒtan 

dσ 

dt (e− µ − ) = 2πα2 

t2 

 

ÝÔÔÔÓÔØÔ×ÓÒØÐ×ÑÓÙÐ××ÕÙÒØØ×ÑÓÙÚÑÒØÐÐØÖÓÒ Ò×ÐÖÖÒØÐÙÑÙÓÒÐ Ä×ØÓÒÖÒØÐÐÒÒÐØÓÒe + ÓÒÒÔÖ ØÓÙØ×Ð×Ñ×××ÓÒØØÒÐ×ØÓÒ×ØÐÓÒÐÔÖÓÙØÓÒ×Ø 

dσNP 

dΩ (e+ e − → µ + µ − ) = α2 

4q2(1 + cos2 

ØÐ×ØÓÒØÓØÐÔÖ 

θ) 

 

Úq 2 Ø 2E×Ù= √ ÒÙÐÖ ÉÙ×ØÖ ×ÙÖÐ×ÐÓÒ×ÖÚØÓÒÐÐØ ÔÖ×ÖÙÑÒØ×ÑÒ×ÓÒÒÐ×ÂÙ×ØÖÐÔÒÒ ÄÙ×ÓÒe + 

 

1 + u2 

s2 ÇÒØÖÓÙÚÙ×× dσ 

dy (e− µ − ) = 2πα2 

t2 s 1 + (1 − y) 2 

e− → µ + µ −Ò×ÐÑ 

= 4E 2×Ù 

1 

3 s = 

21.7Ò [E×Ù(GeV )] 2 

s 

e− → e + e−Ò×ÐÑØÐÓÒÙ×ØÓÒÒÔÖ σNP(e + e − → µ + µ − ) = 4πα2

dσNP 

dΩ (e+ e − → e + e − ) = α2 

 

4 1 + cos 

2s 

 

θ 

2 

sin 4 − θ 

2 

2 cos4 

θ 

2 

sin 2 + θ 

2 

1 + cos2 θ 

2 

ÉÔÓÙÖÕÙÓØØÖÐØÓÒ×ØÐÐ×ÖÒØÐÐØÒÙÔÓÙÖe + e− → 

µ + µ − Ð×ÒÓÙ×ÚÓÙÐÓÒ×ØÒÖÓÑÔØÐ×ØÖÙØÙÖ×ÔÖØÙÐ×ÁÐ×Ø ÂÙ×ÕÙÒÓÙ×ÚÓÒ××ÙÔÔÓ×ÐÔÓÒØÐÓÒÔÓÒØÙÐÉÙ×Ô××Ø ÄÓÒÔØÙØÙÖÓÖÑ 

ØÖÑÒÖÐØÐ×ØÖÙØÓÒ×Ö×ÐØÖÕÙ×ÐÒØÖÙÖÙÔÓÒ ÇÒÖÔÖ×ÒØÓÒÚÒØÓÒÒÐÐÑÒØØØÔÖÙÒ×ÕÙÔÐÒÐÓÙÚÖØÜ ÚÓÖÖÔÐÙÖ ÄÓÖÑÜÔÐØÐ×ØÖÙØÓÒÖ 

c p 

c 

q 

a 

ÐÒØÖÙÖÙÒÔÓÒØÙÒÖÓÒÒÒÖÐ×ØØÖ×ÐØÖÓÙÚÖ ÁÒØÖØÓÒÙÒÔÓÒÚ×ØÖÙØÙÖ 

pa 

π 

Ù×ÐÒÙÒÓÑÒÒØÐÒØÖØÓÒÓÖØÁÐ×ØÔÐÙ×ÓÑÑÓ ÐÒØÖØÓÒÑÖÔÖ×ÒØÔÖÙÒÔÓØÒØÐÎÓÒ×ØÒØÒ×ÐØÑÔ×ËÐÓÒ Ó×ÖÐÔÔÖÓÒÖØÑÒÒØÙÓÒÔØÙØÙÖÓÖÑËÙÔÔÓ×ÓÒ× 

ÔÓØÒØÐ ÐÑÔÐØÙØÖÒ×ØÓÒM×ØÔÖÓÔÓÖØÓÒÒÐÐÐØÖÒ×ÓÖÑÓÙÖÖÙ ÓÒ×ÖÐÔÖØ×ÔØÐØÕÙÐÓÒ×ÐÑØÙÒÐÙÐÙÔÖÑÖÓÖÖ 

 

M ∼ d 4 xΨ ∗ fV Ψi 

 

∼ d 3 xe −ipf ·x V (x)e ipi ·x 

∼ d 3 xV (x)e −iq·x Ó ∼ 

 

∼ V (q) 

q = pf − pi

×ØÖÙØÓÒÒ×ØÖ×ρ(x)ÔÖ ÙÔÓÒØÚÙÐ××ÕÙØÒÓÒÖÐØÚ×ØÐÔÓØÒØÐÎÜ×Ø××ÓÐ 

ÄÚÐÓÔÔÑÒØÙÐÙÐÓÒÒ 

∼ 

V (q) ∼ e2 

|q2 | F(q) ,Ú 

ÖÙØÞÖÓÙÒÓ×ÐÒØÖØÓÒØÙ ÉØÐÖÖ×ÙÐØØØØÖÜÖÈÓÙÖÚØÖÕÙÐÒØÖÐÒ××ÒØ 

ËÐ×ØÖÙØÓÒÖ×ÙÔÓÒ×Ø×ÙÔÔÓ××ÔÖÕÙÓÒÓÒÐÙØÕÙ 

|Ó×Ø 

ÒÔÒÕÙÙÑÓÙÐqÒÓÒ×ÖØÓÒ 

ØÖÒ×ÖØÚÒØÐÕÙÖÑÓÑÒØØÖÒ×ÖØq→qØÓÒÖØÖÓÙÚÒ×Ð ÚÑÒØÒÔ×Ð×ÝÑØÖ×ÔÖÕÙÒ×ÙÒØÖØÑÒØÖÐØÚ×ØÐÑÓÑÒØ Ä×ØÖÙØÓÒÖÐÒØÖÙÖÙÔÓÒÚÙÔÖÙÒÓ×ÖÚØÙÖÒÑÓÙ ØÖÑ1/q 2ÐÜÔÖ××ÓÒÙÔÖÓÔØÙÖÙÔÓØÓÒÚÙÔÖÑÑÒØÑÑ ÒØÖÐ×ÙÖÐ×ÔØÑÔ× ÉÑÓÒØÖÖÕÙÔÓÙÖÙÒÔÓØÒØÐØÝÔÙÛÐ×ØÖÙØÓÒÒ×Ø 

)ÕÙ×ØÒÔÖÙÒ 

ÔÖÐÒØÖØÓÒÓÖØØÙÖÕÙÖÔÖ×ÒØÐÜØÒ×ÓÒÐ×ÓÙÖØ×ÙÖ ÐÚÖØÜÒØÖØÓÒÚÐÔÓØÓÒÓÙÚÐ×ÙØÖ×Ó×ÓÒ×ÚØÙÖ× Ò×Ø×ÓÒ×ØØÙÒØ× Ð×ÙÓÒÔØØÙÖÓÖÑÐÝÓÒÙÒ×ØÖÙØÓÒ Ö×ÐØÖÕÙ×ÑÒØÕÙ×Ð× ÓÒÒ× 

ÓÖÑÓÒ×ÖÓÒ×Ð×ÙÔÓÒÔÖØÙÐ×Ò××ÔÒÄÚÖØÜÙÓÙÔÐ ×ÖÙÑÒØ××ÑÔÐ×ÔÖÑØØÒØÓÒÒÖÐ×ÖØÖ×ØÕÙ×ÙØÙÖ ØÙÖÓÖÑØ×ÝÑØÖ 

ÚÐÔÓØÓÒ×ØÖØÖ×ÔÖÐ×ÙÜÕÙÖÚØÙÖ×paØpcÄÓÙÖÒØÑ ÒØÖÐØØØÐØØ×ØÓÒÒÔÖ 

〈ÔÓÒ(pc)|j |ÔÓÒ(pa)〉 e(pc + pa) µ ÔÓÙÖØÒÖÓÑÔØÐ×ØÖÙØÙÖÐÔÖØÙÐÈÓÙÖÕÙÐØÓÐÒ×Ø×× ØØÜÔÖ××ÓÒ×ØÔÔÐÐÙÔÓÒ×ÙÔÔÓ×ÔÓÒØÙÐÇÒÑÖØÐÑÓÖ 

 

exp(iqx) 

V (x)ÚÖÒØÖÓÙÖÙÒÓÒØÖÒØÐÓÖÑe 

V (x) ∼ e 2 

 

F(q) 

d 3 x ′ ρ(x′ ) 

|x − x ′ | 

 

= 

d 3 xρ(x)e −iq·x 

−a|x′ 

F(q) → F(q 2 ) 

F(q2 ) → F(q2 )ØÓÒÓØÒØÙÒÓÖÑÒÚÖÒØF(q 2 Ñ×××ρ(r) = m2 exp(−mr) ÓÒÓØÒØÐØÙÖÓÖÑF(q 8π r 2 ÇÒÙÒÔÐÒq 2 = −m2 j µ (ÔÓÒ) = 

emµ 

∝ 

) = 1 

1+q 2 /m 2

ÕÙÖÚØÙÖ×ÒÔÒÒØ× ØÒÒÐÙØÕÙØÓÙØÑÓØÓÒÓÒ×ÖÚÐ×ØÖÙØÙÖÚØÓÖÐÐÙÓÙÖÒØ ÓÒÕÙÐÑÔÐØÙpcÒÔÙØØÖÓÒ×ØÖÙØÕÙÐ×ÙÜ Ö×ØÙÒÒÚÖÒØÄÓÖÒØÞÄÓÙÖÒØ 

ØÙ×ÐÖÒÔÒÒØ Ø ÙÔÓÒÒ×ÐØÖÒ×ØÓÒpa → 

ÓÅÑ××ÙÔÓÒ 

2ÔÖ 

pa + pc 

ÍÒÜÔÖ××ÓÒÒÚÖÒØÄÓÖÒØÞÔÓÙÖÐÓÙÖÒØÔÓÒÕÙÔÙØØÖÙØ ÒØÙÒØÐ×Ù×ØØÙØÓÒ 

ÆÓØÓÒ×ÕÙÐ×ÐÖ(pa + 2×ØÐq 

ÁÑÔÓ×ÓÒ×ÑÒØÒÒØÐÓÒ×ÖÚØÓÒÐÖÑÐÐÕÙØÓÒ ÓÒØÒÙØ 

ØÖÑÐ×ÓÒØÖÑØØÜÔÖ××ÓÒÔÖÓÒØÖÒ×ØÒÙÐÕÙÒ×ÐÐÑØ ÙÒÑÓÑÒØØÖÒ×ÖØÒÙÐÓÙÚÙÒÔÓØÓÒÒÓÒÚÖØÙÐÇÒÒØÖÐ ÔÔÐÕÙÐÜÔÖ××ÓÒÖÓØÓÒÒØÚÑÒØÔÓÙÖÐÔÖÑÖ ÄØÙÖqµÚÒØÐÖÚØÓÒÐÓÒÔÐÒÒ× ÄÔÖÓÙØ×ÐÖ 

∂µj 

ÓÒÐÙ×ÓÒÕÙÓÒÓØÚÓÖG(q 2 ÖÐ×ØÐ×ÙÐ×ÐÖF(q 2 ÐÓÒØÓÒÙÒØÙÖÕ2ÕÙÚÖØÜÑ ÄÖÐÝÒÑÒÔÓÙÖÙÒÓ×ÓÒ×ÔÒ Ú×ØÖÙØÙÖÖÕÙÖØ 

Ä×ØÓÒÖÒØÐÐÔÜ ×ØÓÖÖÖÔÖÐØÙÖ 

ÕÙÐÖØÓØÐÐÐÇÒ×ØÓÒÙØÒÓÖÑÐ×ÖÓÒØÐÐÕÙ ÑÒ×ÓÒÔÝ×ÕÙÐÔÖØÙÐÐÐÑØÕ→ÐÔÓØÓÒ×ÓÒÒÔÖÓØ ËÐÑÓÑÒØØÖÒ×ÖØÕ×ØÔÖÓÞÖÓÐÔÓØÓÒÒÒ×ØÔ× ÔÐÖÓÒÒØÖÐÜ×ØÒÙÒ×ØÖÙØÙÖÖÐÐÓÒÙÙÖÓÒØØÒØÐ 

F(q 2 

 

= 0) = 1 

|F(q 2 )| 2 

q 

q 2 = p 2 a + p 2 c − 2papc = 2M 2 − 2papc 

= pc − pa 

pc) 

e(pc + pa) µ → e F(q 2 )(pc + pa) µ + G(q 2 )q µ 

µ = 0 → qµj µ 

0ØÕÙÐØÙÖÓÖÑÖÚÒØÐÔÓÒ 

= 0 

) = 

) 

(pa + pc) 2 = p 2 a + p2 c + 2papc = 2M 2 + 2papc = −q 2 + 4M 2 

dσ 

dΩ = 

ÔÓÒØ× dσ 

|F(q 

dΩ 

2 )| 2

ÖÔ×ÓÖÖ××ÙÔÖÙÖ×ÐÔÖÓÐÑ×ÒÒ× 

ÄÐØÖÓÝÒÑÕÙÕÙÒØÕÙÉÔÖ×ÒØÕÙÐÕÙ×ÔÓÒØ×ÐØ×ÕÙ ×ØÓÒ ÚÖÒÙÐØÖÚÓÐØØÐÔÖÓÙÖÖÒÓÖÑÐ 

ÒÓÙ×ÚÓÙÐÓÒ××ÓÙÐÚÖ××ÓÙ× 

Ñ××ÔÖÓÒØÖ×ÕÙÒØØÑÓÙÚÑÒØ×ØØÖÑÒÔÖÐ×ÕÙÒØØ× ØÙÐÒÔÓØÓÒÚÖØÙÐÒ×ÐÖÔÐÙÖ ÐÙÐÙÖ Ä×ÖÔ×ÕÙÒÓÙ×ÚÓÒ×ØÙ×Ù×ÕÙÒÓÑÔÓÖØÒØÕÙÙÒ×ÙÐÔÖ 

ÑÓÙÚÑÒØ×ÔÖØÙÐ×ÜØÖÒ×ÒÖ×ÓÒ×ÐÓ×ÓÒ×ÖÚØÓÒÈÖ ÄÔÖØÙÐÒÒØÖÒÔÙØ×ÓÖØÖ×ÓÙ ÔÓÒÚÖØÙÐÒ× 

Ö×ÔÓ××Ð×ÓÒÒ ÜÑÔÐÒ×ÐÖÔÐÙÖ ÐÒØÖØÓÒ×ÙÖØÓÙ×Ð×ØØ×ÒØÖÑ 

Ú 

+∞ 

k ÕÙÜÔÐÕÙÐØÙÖ1 

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pØÙÐ×ØÀÊÀÑÓÙÖÍÒ×Ù ÑÒÓ×Ø×Ø16 × 1030 cm−2s−1Ä×ÙÜÜÔÖÒ×Ò×ØÐÐ××ÙÖÐÑÒ ×ÓÒØÍËØÀ 

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dσ 

dΩ = 

 

α2 4 | k | 2 sin 4 (θ/2) cos2 ′ 

| k | 

(θ/2) 1 − 

| k | 

q2 tan2 (θ/2) 

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ÅÓØØÔÓÙÖÙÒÒÓÝÙÐÑ××Å×Ò××ØÖÙØÙÖÒÓØÙ×× dσ 

dΩ ss 

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π ÓØØÖÑÙÐØÔÐÔÖÙÒØÙÖÓÖÖØÓÒB(q 2 ÇÒÓØÒØÒÐÑÒØÙÒÜÔÖ××ÓÒÐÓÖÑ 

) 

dσ 

dΩ = 

 

dσ 

× 

dΩ ss 

A(q 2 ) + B(q 2 ) tan 2 (θ/2) •×ÕÙÐÒÖØÖÒ×Ö×Ø×ÙÔÖÙÖÙÒÖØÒ×ÙÐÐÖÙÐÔÙØ ØÖÓÒ×ØØÙÔÖÙÒÖ×ÓÒÒÚÓÖÔÐÙ×ÙÖ×ÔÖØÙÐ××ØÐÙ×ÓÒ 

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)ÔÓÙÖÐØÖÑÑÒØÕÙ 

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0ÓÒ 

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ÔÓÙÖÐÔÖÓØÓÒF1(0) = ÔÓÙÖÐÒÙØÖÓÒF1(0) = 

Ð×ØÙÖ×A(q 2 Ò×ÐÐØØÖØÙÖÓÒØÖÓÙÚÙ××ÐÓÖÑÕÙÚÐÒØ ×ØÕÙÓÒÔÔÐÐÓÑÑÙÒÑÒØÐ×ØÓÒÊÓ×ÒÐÙØ Óτ = 

 

ÔÓÒØÙÐj µ (proton) = (+e)NN ′ u(p ′ , s ′ )γ µ u(p, s) exp{i(p ′ − p)x} 

j µ (proton) = (+e)NN ′ u(p ′ , s ′ 

) 

γ µ F1(q 2 ) + iκF2(q2 ) 

2M σµνqν 

u(p, s) exp{i(p ′ − p)x} 

2 σ µν = 1 

2 i [γµ .γ ν 

] ÉÕÙÚÙØσ µν 

dσ 

dΩ 

 

1 F2(0) = 1 

0 F2(0) = 1 

)ØB(q 2 )ÔÙÚÒØØÖÜÔÖÑ×ÒØÖÑ×F1(q 2 )ØF2(q 2 ) 

= (dσ 

dΩ )ss × [F 2 1 (q2 ) + τκ 2 F 2 2 (q2 )] + [2τ(F1(q 2 ) + κF2(q 2 )) 2 ] tan 2 (θ/2) 

−q 2 /4M 2

dσ 

dΩ 

dσ 

dΩ 

0.03 

0.02 

ss 

0.01 

= Β ( ) 

q2 q2 tan 2 pente 

 

A } ( ) 

0.2 0.4 0.6 0.8 1.0 

1 

2 θ ËØÓÒÖÒØÐÐÒÓÖÑÐ×ÐÙ×ÓÒe −pÒÓÒØÓÒ tan 2 (θ/2)ÔÓÙÖÙÒq 2Ü ÒÓÒÒÙÒÓÒÒÔÔÖÓÜÑØÓÒÐ×ØÓÒ ÚÐØÐÓÖÑÙÐÊÓ×ÒÐÙØÄÐÙÐÙÔÖÑÖÓÖÖÙÒ×ÙÐÔÓØÓÒ Î2Ä×ÔÓÒØ×ÜÔÖÑÒØÙÜÓÒÖÑÒØÐ 

dσ 

dΩ = 

2 dσ GE + τG 

× 

dΩ ss 

2 

M 

+ 2τG 

1 + τ 

2 M tan2 

Ó GE(q 2 ) = F1(q 2 ) + κ q2 

4M2F2(q 2 ) 

GM(q 2 ) = F1(q 2 ) + κF2(q 2 

) 

GE(q2 )ØGM(q 2 )×ÓÒØÔÔÐ×Ð×ØÙÖ×ÓÖÑÐØÖÕÙØÑÒØÕÙ 

 

θ 

2

Ë×ÔÓÙÖÐÔÖÓØÓÒG p 

E ÔÓÙÖÐÒÙØÖÓÒG n ÔÓÙÖ×ÚÐÙÖ×ÔÐÙ×ÔÖ×× ÓµÔØµÒ×ÓÒØÜÔÖÑ×ÒÑÒØÓÒÒÙÐÖµNÎÓÖÐØÐÈ 

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pe 

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Q 2 = 4 | k |2 sin 2 (θ/2) 

1 + 2|k| 

M sin2 

 

(θ/2) 

dΩ / 

dσ 

dΩ 

ssÒÓÒØÓÒtan 2 

µÔ 

(0) = 1 Gp 

M (0) = 1 + κ = 2.7928... = 

(0) = 1 + κ = −1.9130... = 

E (0) = 0 Gn M µÒ 

p → e + pÔÙØØÖÖØÐ× 

= (| k |, k) ≡ (| k |, | k |, 0, 0) 

pp = (M, 0, 0, 0) 

p ′ e = (| k′ |, k ′ )Ú k ′ = | k ′ | (cosθ, sin θ, 0) 

p ′ p = (E′ , p ′ ) 

| k ′ 

2 | k | 

| / | k |= 1 + 

M sin2 (θ/2) 

2 = (pe − p ′ e )2 = (| k | − | k ′ |) 2 − (k − k ′ ) 2 

≃ −2 | k || k ′ | (1 − cosθ) = −4 | k || k ′ | sin 2 

(θ/2) 

= −q2ÔÓÙÖ×ÖÖ××ÖÙ×ÒÒØÒÓÑ

ØÙÖ×ÓÖÑÑÒØÕÙØÐØÖÕÙÙÔÖÓØÓÒØÙÒÙØÖÓÒ

π + 

π + π0 π + 

π + 

p n p p p p 

ÔÓÒØÙÐÒØÓÙÖÔÖÙÒÒÙÔÓÒ×ÚÖØÙÐ×ÐÙÛÄ×Ö×ÙÐØØ× ÜÔÖÑÒØÙÜ×ÓÒØÒ×ÓÖÚØØÖÔÖ×ÒØØÓÒ ÁÑÐ×ØÖÙØÙÖÙÔÖÓØÓÒÓÑÑØÒØÐÐÙÒÒØÖÙÖ 

p n n p 

0.8 fm 

ÇÒÚÓØÕÙÐ×ØÔÓ××ÐÓÙÖ×ÙÖÐ×ÚÐÙÖ×ÐÒÖÙ×Ù|k|Ø ÐÒÐÙ×ÓÒθÓÒÓÒ×ÖÚÖÙÒÚÐÙÖÜQ 2×ØÐÔÖÓÙÖ ÙØÐ×ÔÓÙÖÓÒ×ØÖÙÖÐ×ÔÓÒØ×ÜÔÖÑÒØÙÜÐÙÖ Ç×ÖÚÓÒ×ÕÙ 

Q2ÔÖÒ×ÚÐÙÖÑÜÑÙÑÔÓÙÖθ= 180¦ 

Q 2ÑÜ= 4 | k |2 

1 + 2|k| ÈÓÙÖÐ×Ñ×ÙÖ×Ù×ÓÒ×ÙÖÐÒÙØÖÓÒÓÒÙØÐ×ÙÒÐÙØÖÙÑ ÐÕÙÇÒÓØÓÒ×ÓÙ×ØÖÖÐÓÒØÖÙØÓÒÙÔÖÓØÓÒØØÙÖÕÙÐÕÙ× ÓÖÖØÓÒ×ÒÙÐÖÖ× 

M 

dσ 

dΩ (Ò) = dσ 

dΩ () − dσ 

dΩ (Ô) ×ÑÔÐ ×ÜÔÖÒ×ÓÒØÑÓÒØÖÕÙÐ×ØÙÖ×ÓÖÑÔÖ×ÒØÒØÙÒÓÑÔÓÖØÑÒØ 

+ δ(ÒÙÐÖ×) 

G p 

E (q2 ) = G p 

M (q2 )/µp = G n M (q2 )/µn = G(q 2 ÓÙÔÐ 

) 

G(q 2 1 

) = 

(1 + Q2 /M 2 1 

= 

v )2 (1 − q2 /M 2 v )2 

 

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G n E(q 2 ) ≃ 0 

π 0

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ρ(r) = ρ0 exp(−rMv) 

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1 

Q2 

1 

Q4 e + p → e + XÊÔÔÐÓÒ×§ 

Q 2 /M 2 v) −1 

q) 

Q 2 = −q 2 ≃ 4 | k || k ′ | sin 2 ,Ú (θ/2) | 

me ≪ | k | 

Q 2 = −q 2 = −(p ′ − p) 2 

= −(p ′ ) 2 − p 2 + 2p ′ p 

= −W 2 − M 2 + 2E ′ ,Ú M E 

−1 exp(−rMv)ÓÒÙÖØÓØÒÙÙÒG(q 2 

k | − | k ′ |= ν 

′ − M = ν 

k2 

1 

Q4 2 ∼ 

, p ′ )

Ó 

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2 , ν) cos 2 (θ/2) + 2W1(Q 2 , ν) sin 2 (θ/2) 

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, ×ÓÒØÓÒ×ÒÚÖ×Ð×ÚÖÐ×Q 2Øν×ØÜÔÐØÖÒÙ×ÓÒÒÐ×ØÕÙ ×ÙÜÚÖÐ×Ò×ÓÒØÔ×ÖÑÒØÐ×ÜÔÖÑÒØÐÑÒØÐØÖÑ ÒØÓÒ×ÓÒØÓÒ××ØÖÙØÙÖW1ØW2ÑÒÕÙÐÓÒ××ÚÖÖkØθ ÒÑÒØÒÒØQ 2ØνÜ×ÐÔÖÑØ×ÔÖÖÐ×ÙÜÓÒØÖÙØÓÒ× Ù×ÓÒ×ÔÖÐÒ×ÑÐ×ÔÖØÓÒ×ÙÒÒÙÐÓÒËÐÔÖÓ××Ù×ÐÑÒØÖ×Ø ÐÙ×ÓÒÐ×ØÕÙ×ÙÖÙÒÔÖØÓÒÑ××mÐÙÒÖØÓÒfÐÑ×× ÊÚÒÓÒ×ÐÒØÖÔÖØØÓÒÙÔÒÓÑÒÓÑÑÐØÓÒÒÓÖÒØ 

fMÓÒÔÙØÔÓ×ÖÒÔÖÑÖÔÔÖÓÜÑØÓÒ ÙÔÖÓØÓÒm = 

Q 2 = 2mν = 2fMν ÇÒ×ØØÒÓÒÓ×ÖÚÖÙÒÓÖÖÐØÓÒÒØÖQ 2ØνÒ×ÐÖÓÒ ÙÁËØÚÓÖÐÔÒÒ×WiÖÙØÙÑÓÒ××ÝÑÔØÓÑØÕÙÑÒØ ÙÒ×ÙÐ×ÚÖÐ×ÈÖÓÒ×ÕÙÒØÓÒÔÙØÑØØÖÕÙQ 2ØνÜ×Ð ÔÖÓÐØÙ×ÓÒÐ×ØÕÙÐÑÒØÖ×ØÐÐÔÖÓÐØØÖÓÙÚÖ ÓÒØÓÒÙÖÔÔÓÖØ ÐÔÖØÓÒÑ××fMÒ×ÐÒÙÐÓÒÄØ×ØÖÙØÙÖÔÙØØÖÜÔÖÑÒ 

f = xÓÖÒ= x = Q 2 ×ÒØÒØÙÒÙ×ØÑÒØØÐÐ×ÐÒ ×ØÐÝÔÓØ×ØÙ×ÐÒÓÖÒÄ×ÓÒØÓÒ××ØÖÙØÙÖÔÖ ÕÙÒ 

/2Mν 

Q2 Ø → ∞ ν → ∞ x = Q2 

2Mν =Ü 

MW1(Q2 ÔÒÒ×ÑÔÐ×ÓÒØÓÒ××ØÖÙØÙÖÒÚÖ×ÐÚÖÐx(1/ωÒ×ÐÖ ÅÖÒØÐÈÝ×ÊÚÄØØ Ä×Ö×ÙÐØØ×ÜÔÖÒ×ØÙ×ÙËÄÓÒÖÑÒØÕÙÐÜ×ØÙÒ 

, ν) → 

 

F1(x) 

 

νW2(Q 2 , ν) → F2(x)

ØÐØÕÙÐ×ÓÒØÓÒ×F1(x)ØF2(x)Ö×ØÒØÒ×ÑÑÙÜØÖ×ÖÒ× ÚÐÙÖ×Q 2ØνÙÖ×Ø ÄØÖÑ×ÐÒ×ÒÔÖ×ÑÒØÕÙ×ÐÓÒÑÙÐØÔÐØÓÙØÚØÙÖ 

(E, p)ÔÖÙÒÓÒ×ØÒØ×ÓÒØÓÒ×Fi(x)Ö×ØÒØÒÒ× 

Ù×ØÑÒØØÐÐ×ÐÒÔÒÒÒx×ÓÒØÓÒ×F1Ø 

F2ÔÓÙÖÙÒ×ÖÑ×ÙÖ×Q 2 ÙÒÑÔÙÐ×ÓÒÖÒØÓÒÔÙØÒÒÐÖ×Ñ××ÄÔÓØÓÒ×ÓÒÚÓØÓÒ ×ÔÐÓÑÑÓ×ÖÚØÙÖØØÐÐØÖÓÒÒ×ÖÖÒØÐÐÔÖÓØÓÒ Ô××ÖÙÒÔÖÓØÐÜØÖÑÑÒØÖÔØ×ÔÓ×ÙÒØÑÔ×ØÖ×ÓÙÖØÇ ÊÝÒÑÒÓÒÒÙÒÒØÖÔÖØØÓÒÔÝ×ÕÙØØ×ÐÒÇÒ 

ÔÓÙÖÒØÖÖÖÙÒ×ÓÖØØ×ØÖÓÓ×ÓÔÕÙÐÚÓØÐ×ÔÖØÓÒ×ÕÙ ÆÓØÓÒ×ÕÙ×ÙÜÓÒØÓÒ×Ò×ÓÒØÔ×ÓÒÓÒÖÚÐ×ÓÒØÓÒ×Ò×Ù§ ÊÈÝÒÑÒÈÝ×ÊÚÄØØ 

ν 

> 1Î2

ØÙÖÓÖÑF2(x, Q2 ØÕÙÍÒØÙÖÑÙÐØÔÐØÒØÖÔÖÒØ×ØÔÔÐÕÙF2ÒØÒÖ ÐÔÐ×ÚÐÙÖ×xÖÔÖ×ÒØ×ÙÖÐÙÖ )ÙÒÙÐÓÒØÖÑÒÔÖÙ×ÓÒÒÐ× 

ÙØÙÒØÒ×ÐÔÖÓØÓÒÓÑÑÐ×Ò×ÔÓ×ØÓÒ×ÓÒÒ×ØÐÙÖØÙÒ ×ÔÖØÓÒ×ÇÒÑØÕÙÔÖØÓÒØÖÒ×ÔÓÖØÐÖØÓÒfÐÕÙÒØØ ÑÓÙÚÑÒØÙÔÖÓØÓÒØÕÙ×Ñ××ØÚm=fM×ØÒÐÐÔÖ ÖÔÔÓÖØQ 2 ÙØÖ×ÔÖØÓÒ×Ò×ÐÔÖÓØÓÒÇÒÑØÒÒÕÙÔÖ×ÐÒØÖØÓÒÐÔÖØÓÒ ÓÐÐ×ÓÒØfPÐÐÙÔÖØÓÒÔÖ×ÐÓÐÐ×ÓÒÐÑÔÙÐ×ÓÒÙÔÖØÓÒ×Ø 

0ÓÒÔÖÐÑ××ØÚÙØÕÙÔÖØÓÒ×ØÐÙÜ 

|)ÐÑÔÙÐ×ÓÒÙÔÖÓØÓÒÚÒØÐ 

: m ≃ ×ØÔÖ×ÕÙÐÖËÓØP = (| p |, 0, 0, | p 

fP + qØÐÐÕÙ(fP 

×ØÖ×ÓÒÒÐ×ÐÓÒÓÒ×ÖÐÔÖØÓÒÐÖØ×ÐÒÝÔ×ØØ×ÔÖØÓÒÕÙ× ÔÖØÓÒÒ×ÖÓØÔ×ÙÓÙÖ×ÐÓÐÐ×ÓÒÓÙÕÙÐÐÖ×ØÒÐÐÕÙ 

+ ÜØ×ÈÙ×ÕÙf 2P 2 = f2M 2 = m2 0ÓÒÓØÒØ 

≃ 

f = −q 2 /2Pq = Q 2 ÆÓØÓÒ×ÕÙØØÖÐØÓÒ×ØÔÔÐÐÙ××Ò×ÐÖÖÒØÐÙÐÓÖØÓÖ 

/2Mν ≡ x 

Öq 2ØPq×ÓÒØ×ÒÚÖÒØ×ÄÓÖÒØÞ 

 

q) 2 = m 2 ≃ 0ÇÒÓÒ×ÖÕÙÐÑ××ØÚÙ

P 

xP (1−x)P H 

a 

d 

r 

xP+q o 

ÖÑÑÐÙ×ÓÒÙÒÐÔØÓÒ×ÙÖÙÒÔÖØÓÒÕÙØÖÒ×ÔÓÖØ 

n 

ÙÒÖØÓÒxPÐÑÔÙÐ×ÓÒÙÔÖÓØÓÒ 

s 

ÖÓÒ×ØÓÒÚÓÖ§ÄÔÖÓ××Ù××ÖÓÙÐ×ÙÖÙÒÐÐØÑÔ×ØÖ× ØÖØÓÒÐ×ÔÖØÓÒ×ÖÓÖÑÒØ×ÖÓÒ×ÔÖÙÒÔÖÓ××Ù×ÓÑÔÐÕÙÔÔÐ ÙÒ×ÔÖØÓÒ×ÙÒÙÐÓÒÐ×ÙØÖ×ÔÖØÓÒ×ØÒØ××ÔØØÙÖ×ÔÖ×ÐÒ ÄÖÑÑÐÙÖÖÔÖ×ÒØÐÒØÖØÓÒÙÔÓØÓÒ×ÓÒÚ 

ØÒÙÒÖÖÐÙÖÙÔÖÓ××Ù×ÐÑÒØÖÙ×ÓÒ ÑØØÒØÕÙÐÔÖØÓÒ×ØÙÒÔÖØÙÐÖ×ÔÒ Ä×ÔÖØÓÒ×ØÓÖÕÙ×ÓÒÖÒÒØÐ×ÓÒØÓÒ××ØÖÙØÙÖ×ÓÒØØ×Ò Ä×ØÖÙØÙÖÙÔÖÓØÓÒ 

×ØÖÓÙÚÖÒ×ÐÔÖÓØÓÒÚÙÒÕÙÒØØÑÓÙÚÑÒØÓÑÔÖ×ÒØÖ ×ÝÑÓÐ×ÔÖqÓqÔÙØØÖÐÙÒ×ÕÙÖ×ÙÓÙ××ÖÐØÖÕÙ ÔÖØÓÒ×Ø 

ÒÔÒÒØ×ÔÖÓ××Ù×ÓÒØÐÔÙØØÖÐ×ÇÒÙØÐ×ÐÜÔÖ××ÓÒ Ð×ØÓÒÖÒØÐÐÙ×ÓÒÙÜÖÑÓÒ××ÔÒ zqα ÚÓÖ 

ÒÙÒØÐÖÙÔÓ×ØÖÓÒ×ØÒÕÙÔÖzqÁÐÙÒÔÖÓÐØq(x)dx 

Ò×ÐÖÐØÓÒØÐÐ×ÕÙ×ÙÚÒØÒ×ÔÖÖÔÐ×ØÓÒÚÒÙÕÙ 

ÒØÙÒØÐ×Ù×ØØÙØÓÒα 

ûÚÔÙ×ÓÒØÖÐØÚ×ÙÔÖØÓÒØÒ× 

→ 

 

 

Ð×ÚÖÐ×ÒÑØÕÙ×ˆs, ˆt, 

xPØ(x + dx)PØØÔÖÓÐØ×ØÙÒÔÖÓÔÖØÒØÖÒ×ÕÙÙÒÙÐÓÒ 

dˆσ 

dˆt = z2 q 

q 

2πα2 ˆt 2 

 

1 + û2 

ˆs 2

ÕÙÐ×ÚÖÐ×ÒÑØÕÙ××ØÙ×Ò×ÔÙ×ÓÒØÖÐØÚ×ÙÔÖÓØÓÒË ÐÓÒÒÐÐ×Ñ××××ÔÖØÓÒ×ÓÒÔÙØÑÓÒØÖÖÕÙÜÖ 

ˆs ≃ xs û ≃ xu ˆt = t = −Q 2 

dtØû 2 

ÑÒÒÖÐ×ØÓÒÓÙÐÑÒØÖÒØÐÐ Ò×ÐÙ×ÓÒÒÐ×ØÕÙÓÒ×ÓÒÙÜÚÖÐ×ÒÑØÕÙ×ØÓÒ×Ø 

ØÙÒÒØÒØÐ×ÔÖØÓÒ×ÙÜÕÙÖ×ÇÒÓØÒØ ÔÖ×ÒÓÑÔØÒÒ×ÖÒØÒ×ÐÔÖÓÐØq(x)dxÒ××Ù×Ò ÒØÖÒØ×ÙÖxØÒ×ÓÑÑÒØ×ÙÖÐ×ÔÖØÓÒ×ÔÖ×ÒØ×ØØ×ÓÑÑØÓÒ×Ø ÓÒØÖÙØÓÒ×Ñ××××Ø×ÙÔÔÓ×ÒÐÐÄ×ØÖÙØÙÖÒÔÖØÓÒ××Ø ÄÓÒØÓÒδ××ÙÖÐÐ×ØØÐÒØÖØÓÒÐÑÒØÖˆs + 

ÊÑÖÕÙÓÒ×ÕÙÐØÙÖ1/t 2 ÔÖÜÖ ÑØÓÒ×ÙÖÐ×ØÖÙØÙÖ×ØÓÒØÒÙÒ×Ð×ÓÑÑØÐÐ×ØÒÔÒÒØ QÓÐ×ÐÒÓÖÒÔÖØÖÐÓÒÔÙØÑÓÒØÖÖÕÙÚÓÖ § 

z 2 

ÕÙÐÓÒÓÒ×ÖÐ×ÕÙÖ×ÓÑÑ×ÔÖØÙÐ×ÔÓÒØÙÐÐ×ÇÒÓØÒØÔÖ ÄÖÐØÓÒØÐÐÒØÖÓ×××ØÙÒÓÒ×ÕÙÒÙØ 

qxq(x) 

ÒÓÑÖ×ÕÙÒØÕÙ×ÄÔÖÓØÓÒÕÙÖ×uØÙÒÕÙÖd ÙÖÖØÒ×ÓÒØÖÒØ×ÔÓÙÖÖÒØÖÕÙÐÒÙÐÓÒpÓÙnØÐ×ÓÒ× ÙÐÐ×ÒÙ×ÓÒÒÐ×ØÕÙÔÖÓÓÒÒ×ÙØÐ×ØÒ××ÖÒØÖÓ ÇÒ×ÒØÖ××ÜØÖÖÐ××ØÖÙØÓÒ×ÔÖØÓÒÕÙ×q(x)×ÓÒÒ×Ö ÜÑÔÐF2 = 

 

ÓÒ dˆσ dˆσ u2 

= = 

dˆt ˆs 2 s2 d2ˆσ dt(xdu) = z2 2πα 

q 

2 

t2 

1 + u2 

s2 0Ð 

 

δ(xs + xu + t) 

ˆt + û ≃ 

d2σ 2πα2 

= 

dtdu t2 

1 + u2 

s2 1 

dxδ(xs + xu + t) 

0 

 

z 

q 

2 qxq(x) 

= 1/Q4×ØÜØÖÙÖÐÒØÖÐØÓÙØÐÒÓÖ 

0 

1 

2xF1(x) = F2(x) = 

x[ 4 1 4 1 

u(x) + d(x) + u(x) + d(x) + ...] 

9 

dx(u(x) − u(x)) = 2 Ø 

9 

9 

q 

9 

1 

0 

dx(d(x) − d(x)) = 1

1/2Ó = ËÖÖÝÓÒÕÙ×ØØ×ÓÒ×Ó×ÔÒI3 

1 

B = dx 

0 

1 

3 (u(x) − u(x) + d(x) − d(x) + s(x) − s(x) + ...) = 1 

1 

 

1 

1 

I3 = dx (u(x) − u(x)) − (d(x) − d(x)) = 

2 2 

0 

1 

×q(x)ØØÙÖÙÒÙ×ØÑÒØÐÓÖÑÓØÒÙÙÜÚÐÙÖ×Ñ×ÙÖ× ×ÜÔÖ××ÓÒ××ÑÐÖ×ÔÙÚÒØØÖÖØ×ÔÓÙÖÐ×Ö×ØÖÒØ ÖÑ Ò×ÙØÐÙØÜÔÖÑÖÐ×ØÓÒÖÒØÐÐÙ×ÓÒÒØÖÑ ËØÖ 

2 

ØÐÖÐÜÔÖ××ÓÒ 

eXÓÒÔÙØ ØØ×ØÓÒÈÖÜÑÔÐÔÓÙÖÐÙ×ÓÒÒÐ×ØÕÙep → 

dσ 

dx (e − p) = 8πα2 

s 4 

1 

x [u(x) + u(x)] + d(x) + d(x) + s(x) + s(x) 

Q4 9 9 

 

nÙÒÚÙ×ÕÙÖ×ÑÒÔÓ×ØÙÐÖÕÙ 

Ð×ØÓÒÙ×ÓÒ×ÙÖÐÒÙØÖÓÒÔÙØ×ÖÖ Ä×ÝÑØÖ×Ó×ÔÒp − 

 

up(x) = dn(x), dp(x) = un(x), sp(x) = sn(x) ËÓÒØÒØÓÑÔØ×ÖÐØÓÒ×Ø×ÓÒÓÒÚÒØÕÙu=up d = dp s = sp 

dσ 

dx (e − n) = 8πα2 ÄÑÒ×ÑÒØÖØÓÒÓÒ××ØÒ××Ò×ÐÒÙÒÓ×ÓÒÐ ÅÒØÓÒÒÓÒ×ÕÙÐ×ØÖÙØÙÖÙÒÙÐÓÒÔÙØØÖ×ÓÒÐÑÒØÔÖÐ 

 

s 4 1 

x d(x) + d(x) + [u(x) + u(x) + s(x) + s(x)] 

Q4 9 

9 

ÐÚÓÖÔØÖ ØÝÔÒÙÐÓÒØÒØÒÙÐÓÒÒÖ×ÓÒÐÒØÙÖÔÖØÙÐÖÙÓÙÖÒØ ÖÐØÓÒÌÓÙØÓ×ÓÒ×ØÑÒÒØÖÓÙÖØÖÓ×ÓÒØÓÒ××ØÖÙØÙÖÔÖ ÏÄ×ØÓÒÖÒØÐÐÔÖÒÙÒÓÖÑÚÓ×ÒÐÐÐ Ù×ÓÒÒÐ×ØÕÙÔÖÓÓÒÒÙØÖÒÓνℓN 

ÄÖÒÕÙÒØØÓÒÒ×ÓÐÐØ×ÔÖÑ×ØÖÖ×ÖÒ×ÒÑÒØ× 

→ νℓXØνℓN → ℓXÓℓ=e, µ 

ÕÙØØÚ×ÓÒÔÖØÓÒÕÙÙÒÙÐÓÒ×ØÓÖÖØÒÔÖÑÖÔÔÖÓÜÑØÓÒ ÒÒØÖÓÙ×ÒØÐÓÑÔÓ×ÒØÐÙÓÒÕÙÙÒÙÐÓÒÉ×ØÙ××ÔÐÜ ×ÙÖÐ×ØÖÙØÙÖÒØÖÒÙÒÙÐÓÒÄÙÖÑÓÒØÖÙÒÔÖÑØÖ×ØÓÒ× ×ØÖÙØÓÒ×ÔÖØÓÒÕÙ×ÒÓÒØÓÒÐÚÖÐÜÄØÓÖÉÓÒÖÑ 

ÔÐÕÙÖÐÚÓÐØÓÒÙ×ÐÒÚ×Ð×ÙÖÐÙÖÐ×ÓÙÖ××ÙÖØØÙÖ

ÜÑÔÐÔÖÑØÖ×ØÓÒ×ÓÒØÓÒ×ÕÜÐÓÙÖÜÓÒÒ 

ÐÓÒØÖÙØÓÒ×ÐÙÓÒ× 

1) 

2) 

3) 

F 2 

F 2 

F 2 

x 

1/3 

F2 quarks de la mer 

ØÓÒ××ÙÚÒØ× ÁÒØÖÔÖØØÓÒÐÓÒØÓÒ×ØÖÙØÙÖÙÒÔÖÓØÓÒÒ×Ð××ØÙ ÕÙÖ×Ð×ÚÐÔÓ××ÐØÖÖ×ÕÙÖ×ÐÑÖ ÙÒ×ÙÐÕÙÖÔÖÔÖÓØÓÒ ÕÙÖ×ÐÖ× ÕÙÖ×Ð× 

4) 

quarks de valence 

x 

petit x 1/3 

 

1/3 

1 

x 

x

ÐÓÖÑ××ØÖÙØÓÒ×ÔÖØÓÒÕÙ×Ò×Ð×ÙÒ×ÙÐÕÙÖÔÖÖÓÒ ÖÔÖ×ÒØÒØÐÔÖØÓÒØÓÖÕÙÄÙÖÓÒÒÙÒÒØÖÔÖØØÓÒ ÕÙÖ×ÐÖ× ÖÖ×ÔÖ×ÕÙÖ×ÚÖØÙÐÐ×ÕÙÖ×ÐÑÖ ÕÙÖ×Ð×ÕÙÖ×ÚÐÒØÕÙÖ×Ð×ÚÐÔÓ××ÐØ 

ÄÒÒÐØÓÒe + ÙÔØÖÓÒÔÖ×ÒØÐÒÒÐØÓÒe + ÐÔØÓÒÒØÐÔØÓÒe + e − → γ ∗ → ℓℓ , Óℓ ÊÔÔÐÓÒ×ÕÙÐ×ØÓÒÖÒØÐÐÒ×ÐÑe + ÐÓÖÑÚÓÖ ØÐÓÒÐÖÓÒÓÖÑØÓÒÙÒÓ×ÓÒÒØÖÑÖÔÙØØÖÖØ×ÓÙ× 

ØÐ×ØÓÒÒØÖ×ÓÙ×ÐÓÖÑ 

Ú 

ÔÖÐÐØØÓÒ×ØÒ×Ò×ÐÐÔÓÐÖ×ØÓÒ×ÐÔØÓÒ×ÔÖÓÙØ× ÄÒÆÈÒÕÙÕÙÐ××ÙÜ×ÓÒØ×ÙÔÔÓ××ÒÓÒÔÓÐÖ××ØÕÙÐÔ ÊÔÔÐÓÒ×ÒÓÖÕÙÐ×ÖÐØÓÒ×ØÒ×ÓÒØÔ×ÔÔÐÐ×Ð 

q 

Ù×ÓÒe + ×ÔÖØÓÒ×ØÓÖÕÙ×ÓÒØØØ×Ø×ÜÔÖÑÒØÐÑÒØ×ÙÖÐÚÓe + ÇÒÔÙØÓÒ×ÖÖÐÜØÒ×ÓÒ×ÔÖØÓÒ×ÐÖØÓÒÒÒÐØÓÒ 

ÇÒØÒØÓÑÔØÐÖÖØÓÒÒÖÙÕÙÖÒØÙÒØÒÓÙÚÙ = Óq(q)×ÝÑÓÐ×ÙÒÕÙÖÒØÕÙÖÖÐØÖÕÙzqÚzq Ð×Ù×ØØÙØÓÒ×ÙÚÒØα 2 §ÜÔÖÑÒØÐÑÒØÐ×ÔÖØÓÒ×ÚÓ×ÒÒÐØÓÒÒÐÔØÓÒ×ØÒ ÐÙÔÖÐÑÒ×ÑÐÖÑÒØØÓÒÐÔÖÓ××Ù××ØØÙÒØÐÙ ÔÙÚÒØØÖÑ×ÒÚÒÚÐÓÖÑØÓÒÖÓÒ×ÐÕÙÐÐÐ×ÓÒÒÒØ ÊÔÔÐÓÒ×ÕÙÐ×ÕÙÖ×Ò×ÓÒØÔ×Ó×ÖÚÐ×ÓÑÑÔÖØÙÐ×ÐÖ×ÁÐ× 

ÖÓÒ×ÑÔÐÕÙÐÑ×ÒÓÙÚÖÙÒÔÔÖÐÐØØÓÒÓÑÔÐÜØ 

e− → γ∗ → qq 

e−ÚÔÖÓÙØÓÒÙÒÔÖ = e, µÓÙτ e−ÙÔÖÑÖÓÖÖ dσÆÈ dΩ (e+ e − → ℓℓ) = α2 

4q2(1 + cos2 

θ) 

σÆÈ(e + e − → ℓℓ) = 4πα2 

· 

3 

1 

s 

2 = 4E2×ÙØ2E×Ù= √ s 

e− → e + e− e− → 

µ + µ −Ò×ÙÒÐÖÔÐÒÒÖÐÔÒÒÒÖÐÒ1/s×ØÚÖ 

e + e − → γ ∗ → qq 

2/3ÓÙ 

→ z2 qα2Ò×Ð×ÖÐØÓÒ×Ø

ÖÕÙÖØÐÔÔÐØÓÒÖØÖ××ÐØÓÒÕÙØ×Ä×ÔÖ×ÑÙÓÒ××ÓÒØ ×ÔÖÖÖÐÔØÓÒ××ÒØÖÒ×××ÒÖÓÒ×××ÒØÐÐÑÒØ ÖÒÔÓÙÚÓÖÔÒØÖØÓÒÒ×ÐÑØÖÄ×ÔÖ×ØÙ×ÓÒØÔÐÙ×Ð× ×ÑÒØÖÓÒÒ××Ð×ÚÙÐÖÐØÚÐÓÒÙÙÖÚÐÔØÓÒØ×ÓÒ 

ÔÕÙÑÒØÐÔÖ×ÒÙÑÓÒ×ÖÓÒ×ÔÓÙÖÕÙÙÒÚÒÑÒØ×ÓØÐ×× ÚÒÑÒØ×ÙØÑÙÐØÔÐØÒ×Ð×ÜÔÖÒ×ÙÄÈÓÒÑÒØÝ ÓÑÑÒÒÐØÓÒÖÓÒÕÙÍÒÙØÖ×ÓÙÖÑÔÓÖØÒØÖÙØÓÒ ÔÓÒ×ÄÚÓÒÒÐØÓÒÒÖÓÒ×ÓÒÒÒ××ÒÔÖÒÔÐÑÒØ× 

e + 

γ 

e− ÖÑÑÙÔÖÓ××Ù×ÙÜÑÑ 

γ 

Ò×ÑØÖÐ×ÒØÒÖÓÒ×ÒÒÖÐÐÖØÓÒÒÖ×ÓÙ×ØÖØÙÜ ×ØÒÒÖÔÖÙÒ×ÔÖØÙÐ×ÒÒØ×Ð×ÙÜÔÓØÓÒ×ÒØÖ××ÒØ ÔÖ×ÒØÔÖÐÖÑÑÐÙÖ ×ØÓÒ×ØØÙÔÖÐÔÖÓ××Ù×ÓÑÑÙÒÑÒØÔÔÐÙÜÑÑÕÙ×ØÖ ÍÒÔÓØÓÒÖÝÓÒÒÑÒØÚÖØÙÐ 

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+ e ÁÒØÖÔÖØØÓÒÙÔÖÓ××Ù×νµ − → µ − ÔÓÒØÙÐÐ 

νeÔÖÙÒÒØÖØÓÒ 

+ 

dσ 

dΩ ≈ G2 ÇÖÐÔÖÒÔÐÙÒØÖØÚÓÖÔÐÙ×ÐÓÒÔÖ×ÖØÕÙÐ×ØÓÒ 

F 

ÖÓØÓÑÑ1/E 2ØÝÔÚÓÐÙØÓÒ×ØÒÕÙÓÒØÖÓÙÚÒ×× ÔÖÓ××Ù×ÉÔÖÜÒ×Ð×ØÓÒe + e− → µ + µ −ÚÓÖ σ = 4πα2 em 1 

3 E2 ÕÙÑÒÒØÖÓÙÖÙÒÒÓÙÚÐÐÓÒ×ØÒØÓÙÔÐÐgÖÒÙÖ×Ò× ÊÔÔÐÓÒ×ÕÙÒ×ØØÜÔÖ××ÓÒÐÓÒ×ØÒØαem×Ø×Ò×ÑÒ×ÓÒ 

ÓØÒØÐÓÒÒÔÒÒÒÒÖ×ÐÓÒØÒØÖÖÒÙÐÑ××ÙÓ×ÓÒ 

WÒÔÓ×ÒØ ÑÒ×ÓÒÒÐÓÙαemÄÑÓÐÐ×ÓÛÏÒÖËÐÑÑÓÒØÖÕÙÓÒ 

GF ∼ g 2 /M 2 

ÆÓØ×ÙÖÐÔÖÒÔÙÒØÖØ ÇÒÔÙØÖÖÐ×ØÓÒÓÑÑÙÒ×ÓÑÑ×ÓÒØÖÙØÓÒ×ÙÒ 

W 

∞ 

(2ℓ + 1) sin 

ℓ=0 

2 ÚÓÖÔÖÜÖ ØØÔÖØÓÒÔÙØØÖÖÒÙÔÐÙ×ÐÔÖÙÒÖ×ÓÒÒÑÒØÒÐÝ×ÑÒ×ÓÒÒÐÐ ÔØÖ 

δℓ 

ÒÓÑÖÒÒÓÒ×ÔÖØÐÐ× 

σ = 4π 

E 2 

e 

π2 E2

ÄÓÒØÓÒÙÒÖØ×ÜÔÖÑÔÖÐØÕÙÕÙÐÑÒØsin 2 ÐÒØÖØÓÒ×ØÔÓÒØÙÐÐÔÖÑØÖÑÔØÒÙÐ ÔØØÕÙ Ò×ÐÌÓÖÖÑ×ÙÐÐÓÒÔÖØÐÐℓ=0ÓÒØÖÙÖ δℓ×ØÔÐÙØ 

ÓÒÐÐÑØÙÒØÖ 

×ØÖÔÓÙÖ ×ØÚÓÐÔÖÕÙÒ △σ 

G 

ÖÒÙÏÙØÐ ÔÖÖÔ×ØÙÒÓÑÔÐÑÒØÙ§Ò×ÐÕÙÐÓÒÔÖ×ÒØÐÜÔ ÊØÓÙÖÐÚÓÐØÓÒÐÔÖØ ÊÔÔÐÓÒ×ÕÙØØÜÔÖÒÔÓÖØØ×ÙÖÐÑ×ÙÖ 

E 

Ð×ØÖÙØÓÒÒÙÐÖ×ÐØÖÓÒ×Ñ×ÐÓÖ×Ð×ÒØÖØÓÒβÒÓÝÙÜ 

Ä×ÙØÙÖ×ÓÒØØÖÓÙÚÕÙØØ×ØÖÙØÓÒÒÙÐÖÚÓÖÙÖ ×Ø×Ý 

 

α×ØÐÔÖÑØÖ×ÝÑØÖØÖÑÒÖ 

ÓˆJ×ØÐÚØÙÖÙÒØ×ÐÓÒJCo 

ÕÙÒ×ÓÒÖÒÒØÐÐØÐÐØÖÓÒ ÄÙ×ØÑÒØÐÓÖÑÐ×ØÖÙØÓÒÒÙÐÖÓ×ÖÚÓÒÒ Ö×ÙÐØØÓÒÙØÒØÖ××ÒØ×ÓÒ× 

p Ee×ÓÒØÐÕÙÒØØÑÓÙÚÑÒØØÐÒÖÐe e 

ØØ××ÔÒÒØÖÐ×ÒÓÝÙÜÒØÐØÒÐ×ÖØÖÓÙÚÒØÖÑÒØÒ×Ð ØÓÒ××ÔÒ××ÙÜÐÔØÓÒ×ØØØÖÔÐØ×ÔÒÐÒÝÔ×ÑÓÑÒØ Ò×ÙÒØÖÒ×ØÓÒÑÓÛÌÐÐÖØÐÐÕÙÐÐ××Ù×ÐÖÒ 

 

1 

 

△Ω 4π 

4π 1 

= 

E2 E2 σ = 4π 

E 2 sin2 δ0 < 4π 

E 2 

√ GF 

≈ 1 

∼ O (100Î) 

ÓÐØÔÓÐÖ×× 

60 Co(J P = 5 + ) → 60 Ni(J P = 4 + ) + e − + νe 

ÑØÖÕÙØÕÙÐÐ×ØÐÓÖÑ 

I(θ) ≈ 1 

π + α ˆJ · p e 

α = −1ÚÓÐØÓÒÈ 

− 

ÉÑÓÒØÖÖÕÙÐÓÒ×ÖÚØÓÒÐÔÖØÑÔÐÕÙα = 0 

Ee 

2 FE2 1 

> 

π2 E2

θ 

axe z 

J 

Co 

Pe e 

Co Ì×ØÚÓÐØÓÒÐÈÖØÒ×Ð×ÒØÖØÓÒβÙ60 

ÓˆSe×ØÐÚØÙÖÙÒØ×ÐÓÒSeÐ×ÔÒÐÐØÖÓÒ ÇÒÔÙØÒÜØÖÖÐÐØÑÓÝÒÒÐÐØÖÓÒ 

π/2 

I(θ)dθ − 0 

< λe− >= 

π 

π/2 I(θ)dθ 

π/2 

I(θ)dθ + 0 π 

π/2 I(θ)dθ = α| p−e | 

= α | β 

Ee− − 

e |= − | β−e 

| 

ÓÒØÑÓÒØÖÕÙ TÚÑ××ÓÒβÔÓ×ØÚ Úα = −1 ×ØÙ××ÑÐÖ×Ø××ÙÖ×ØÖÒ×ØÓÒ×G − 

< λe + >= + | βe + | Úα = 

ÓÖØÐÑÔÓÖØÔÖÐÔÖe − CoØÕÙÔÙØÙ×××ÖÖ νeÁÐ×Ò×ÙØÕÙÐ××ÔÒ××ÓÒØÐÒ××ÙÖÐ 

− ×ÔÒÙ60 I(θ) ≈ 1 + α ˆSe 

· pe E0 

+1 

60

ØÐÐ×ÜÔÖÒ×ÓÒØÔÓÖØÒÓÒ×ÙÐÑÒØ×ÙÖÐ××ÒØÖØÓÒ×βÖÓØÚ× Ö×ÐÖÒØ×ØÒÕÙ×ÒÐÝ×ÓÒØÓÒÖÑ×Ö×ÙÐØØ× ÈÖÐÐÙÖ×Ð×Ñ×ÙÖ×ÖØ×ØÒÔÒÒØ×ÐÔÓÐÖ×ØÓÒ×ÐÔØÓÒ× 

ÐÔØÓÒ×Ö×ØÒÚÖ×ÐÐØÐÙÖ×ÒÙØÖÒÓ×ÚÓÖ§ ÄÑ××ÙÑÙÓÒØÒØ ÄØÐÙ Ö×ÙÑÐÒ×ÑÐ×Ö×ÙÐØØ× 

1ÐÑÓÒØÖÕÙÐÐÑØÙÐØÖÖÐØÚ×ØÐÐØ× ÅÎÐÐØÖÓÒ×ÐØÓÒÒÔÖ×ÙÑÜÑÙÑÙ Ù×ÔØÖÙÒ|β e |⋍ 

Ì 

 

ÀÐØ×ÐÔØÓÒ×ℓ = 

ÖØÓÒ×ÖÓÒ×ÓÖÙÒÔÖÓ××Ù××ÑÐÔØÓÒÕÙØÐÕÙÐ×ÒØÖØÓÒ ÖÒÖÓÑÔØ×ØÖÒ×ØÓÒ×ÑÓÛÌÐÐÖÌØÐÚÓÐØÓÒÐÔ ÙÒÙØÖÓÒÇÒÔÙØÒ×ÙÒÔÖÑÖØÑÔ×ÑÓÖÓÑÑ×ÙØÐÓÖÑÙ ÁÐ×ÚÖÒ××ÖÑÓÖÐØÓÖÔÒÓÑÒÓÐÓÕÙÖÑÔÓÙÖ ÄÒØÖØÓÒV−A 

ÓÙÖÒØÐn → 

ÄÓÔÖØÙÖσØ×ÙÖÐ××ÔÒÙÖ×ψiÓÒÔÖÑØØÖÐÒÓÙÚÐÐÓÒÙÖ 

ØÓÒ×ÔÒÜÔÖÑÒØÐÑÒØÓÒØÖÓÙÚ 

 

Ñ×Ù×××ÙÖÐ××ÒØÖØÓÒ×ÑÙÓÒ× 

µ ± → e ± + νe(νe) + νµ(νµ) 

ℓÔØÓÒ× ℓ + ℓ− λ v/c −v/c 

νℓ νℓ 

F : ↑ ↑ 

 

△J=0 

p + e − + νe 

↑ ↓ 

 

S=0 

n → p + e− + νe 

GT : ↑ ↓ 

 

↑ ↑ 

 

△J=1 

S=1 

GGT 

GF 

e, µ, τ) 

M ∼ GF(ψ ∗ p ψn)(ψ ∗ e ψνe) 

M ∼ GGT(ψ ∗ p σψn)(ψ ∗ e σψνe) 

= 1.18 ± 0.03

ÇÒÔÙØÑÓÒØÖÖÕÙ Ø ×ÓÒØÐ×ÐÑØ×ÒÓÒÖÐØÚ×Ø××ÔÖÓÙØ× 

F : ∼ (ψpγ µ 

ψn)(ψeγµψνe) GT : ∼ (ψpγ µ γ 5 ψn)(ψeγµγ 5 ÓÖÑ×Ð×ÓÙÖÒØ×ÚØÓÖÐ×ØÜÙÜ 

×ÓÙÖÒØ××ÓÒØÖÔÖ×ÒØ×ÔÖÐ×ÖÑÑ×ÐÙÖÐØÙ 

ψνe) 

ÔÖÓÔØÙÖÔÓÒØÐÐØÒØÒÐÄ×ÔÖÓÙØ×ÓÙÖÒØ×ØÐ×ÕÙ Ø 

p e ν 

e p 

e 

ÕÙÚÐÒ×ÖÑÑ×Ð×ÒØÖØÓÒβÙÒÙØÖÓÒ 

W W 

n 

n 

νe V : (J V ) µ = ψγ µ ψ ≡ V µ 

A : (J A ) µ = ψγ µ γ 5 ψ ≡ A µ 

ÚÓÐØÓÒÐÔÖØÐÙØÓÒÚÓÖÙÒÑÐÒÓÙÖÒØ×ÔÖØÔÓ×ØÚ ÓÒÒÒØ×ÕÙÒØØ××ÐÖ×ÔÖØÔÓ×ØÚËÐÓÒÚÙØÒØÖÓÙÖÐ 

 

ÐÔÖÑØÖ×ØÓÒ×ÙÚÒØ Ó Ø×ÓÒØ×ÔÖÑØÖ×ÒÖ VØAAÔÖØÔÓ×ØÚØ 

ØÒØÚ×ØÕÙÓÒÓØÒØÚÐÑÐÒV−A 

ÒÜÔÐØÒØÓÒØÔÔÖØÖÐ×ÔÖÓÙØ×V 

M ≈ GF(ψpγ µ (1 − GGT ÄÓÙÖÒØÐÔØÓÒÕÙL µÓÒØÒØÐÔÖÓØÙÖ1 ×ÔÒÙÖÙÒÙØÖÒÓÒ×ÐØÓÒÒÐÓÑÔÓ×ÒØÙÖÓØÔÓÙÖÐν Ö×ÙÐØÒØÐÑÔÐØÓÒÐÒØÖØÓÒÓÖØÒØÖÐ×ÕÙÖ×ÔÖØÔÒØÙ ÔÖÓ××Ù× ÆÓØÞÕÙÒÓÒØÓÒÐÑØÖÕÙÓÔØÚÓÙ×ÔÓÙÚÞØÖÓÙÚÖÒ×ÐÐØØÖØÙÖ 

ÄÖØÒØÖÐ×ÚÐÙÖ×GFØGGTÑ×ÙÖ×ÔÙØØÖÜÔÐÕÙÓÑÑ 

ÔÖÓØÙÖÖØ1 

M ∼ G(aV pn + bA pn ) µ (cV eν + dA eν AÔÖØÒØÚÄÓÑÔÖ×ÓÒÚÐ×Ö×ÙÐØØ×ÜÔÖÑÒØÙÜÓÒÙØ 

)µ 

V 

γ5)ÕÙ××ÒØ×ÙÖÐ 

γ5)ψn) × (ψeγµ(1 − γ5)ψνe) 

GF 

Lµ 

(1 − 2 

2 

(1 + γ5)

ν µ 

e ν 

e ν e 

µ 

W W 

µ 

µ 

νe ÕÙÚÐÒ×ÖÑÑ×Ð×ÒØÖØÓÒµ − ÖØÓÒÙÑÙÓÒÙÖËÓÒÑÔÐØÙ×ØÔÖÓÔÓÖØÓÒÒÐÐÙÔÖÓÙØ ÙÜÓÙÖÒØ×ÐÔØÓÒÕÙ× ÓÒ×ÖÓÒ×Ò×ÙØÙÒÔÖÓ××Ù×ÔÙÖÑÒØÐÔØÓÒÕÙØÐÐÕÙÐ×ÒØ 

→ e + νe + νµ 

ÄÐÙÐØÐÐÓÒÙØÙØÙÜÐ×ÒØÖØÓÒ 

1 

τ = G2 µ m5 µ /192π3 ÒÒØÖÓÙ×ÒØÐ×ÚÐÙÖ×Ñ×ÙÖ×mµ = 105.65ÅÎØτ = 2.197 10−6×ÓÒ ÐÐGF 

ÔØØÖØÔÙØÐÑÒØØÖÜÔÐÕÙÒØÒÒØÓÑÔØÙÖØÖÑ ÚÐÒØ×ÕÙÖ×ÓÒ×ØØÙÒØ××Ò×Ð×ÙÜÒØÖØÓÒ×ÓÖØØÐÓ 

Gµ − GF 

= 0.028 ± 0.013 

ÙÒÚÖ×ÐØ ×ÓÖØÙÒÔÖÓÔÖØÖÑÖÕÙÐÙÓÙÔÐÒ×Ð×ÓÙÖÒØ×Ð×Ö××ÓÒ ÚÓÖ§ ÍÒÓ×ÔÖ×ÒÓÑÔØÐ×ÖØ×ÒØÖÐ×ÓÒ×ØÒØ×ÓÙÔÐÐÖ× 

GF 

ÓÙÔÐ[(eνe)Ú(pn)] ∼ ØØÔÖÓÔÖØÔÙØØÖÑ×ÒÔÖÐÐÐÚÐÐ×ÓÙÔÐ×ÑÔÖÓØÓÒ 

=ÓÙÔÐ[(eνe)Ú(µνµ)] ÔÖÓØÓÒÔÖÓØÓÒ−e + , µ + − µ +ÓÒ×ØÕÙ×ÓÙÔÐ××ÓÒØÒØÕÙ×Ò ÕÙÐÔÖÓØÓÒ×ÓØÙÒÔÖØÙÐÒÓÒÔÓÒØÙÐÐØØÐØ×ØÙÙØÕÙ ×ØÐÖØÓØÐÓÒ×ÖÚÐÓØÕÙ×ØÒÙÙÖ ÎÓÖÐÒÏÒØÖØÓÒ×ÒÀÐÖÄÌÖ×ØÓÐ§ ÇÒÐ××ØÔØØ×ÓÖÖØÓÒ×ÔÒÒØ(me/mµ) 2Ò×ÕÙÐ×ÓÖÖØÓÒ× ÙÒÙÐÓÒ 

ÖØÚ× ÎÖ××ÒÖÕÙÒÐÔÓØÓÒÒØÖÑÖÒÔÙØÔ××ÖÒÖÐ×ÓÒ×ØØÙÒØ× 

M ≈ Gµ[uνµγ µ (1 − γ5)uµ −][ue−γµ 

(1 − γ5)uνe] 

ÔÙØØÖÑÒÖÐÓÒ×ØÒØÓÙÔÐGµÄÚÐÙÖÓØÒÙ×ØØÖ×ÔÖÓ

+ + ... 

ÐÒÖÙÒÔÖÓØÓÒÔÖØÙÐÓÑÔÓ××ØÚÙÔÖÐ ÑØÙÖÐÒØÖØÓÒÑÓÑÑÙÒÔÖØÙÐÔÓÒØÙÐÐ 

p 

µ 

ÔÓÙÖØÓÙØÔÖØÙÐ×Ò×ÐÐÒØÖØÓÒÐÚÓÖÙÖ×ÚÐÙÖ×Ø ÑÑÐ×ÑÐÜ×ØÖÙÒÖÐÓÒ×ÖÚØÙÒÓÙÔÐÙÒÕÙ ÐÓÖÖ√ GF 

p ν µ 

e 

νe 

+ + ... 

e 

νe ÙÒÐÔØÓÒ ÍÒÒÙÐÓÒÚÙÔÖÐÒØÖØÓÒÐ×ØÐÓÐÑÒØ×ÑÐÖ 

n µ 

 

ÄÓÒÔØÙÒÚÖ×ÐØÐÒØÖØÓÒÐ×ØÚÖÚÖÒÔÖ×ÓÒ 

ÔÓÙÖÐ×ÑÐÐ×ÐÔØÓÒ×ÄÔÔÐØÓÒÙÜÑÐÐ×ÕÙÖ×ØÐÓØÙ§

ÑÔÐØÙÚÔÖÓÔØÙÖÓ×ÓÒÏÓÑÔÓÖØÑÒØ×× ÁÑÒÓÒ×ÙÒ×ÙÒÙØÖÒÓÑÙÓÒ×ÒØÖÔØÔÖÙÒÐÑØÖÐ ÒÖ ÐÓÙÖØÓÒ×ÖÓÒ×ÐÖØÓÒνµ + −×ØÙÒÐØÖÓÒÙÒ ÒØÖØÓÒÙÑÙÓÒ§ ÓÙØÓÑÕÙ ÆÓØÓÒ×ÕÙÐ×ØÙÒÔÖÓ××Ù×ÔÙÖÑÒØÐÔØÓÒÕÙÓÑÑÒ×Ð× ÔÖÓÒØÖÐÒÖØÓØÐÒÙÔÙØØÖÔÐÙ× ÐÚ×ÙÐ≃ mµc2ÒÓÒØÓÒÐÒÖÙ×Ù e − → µ − + νeÓe 

ÖÑÑνµe − → µ −νeÚÙÒÓ×ÓÒÒØÖÑÖWØÖÒ× ÄÔÔÐØÓÒ×ÖÐ×ÝÒÑÒÚÓÖ§ Ó×ÓÒWÚÖØÙÐÒØÖØÓÒ×ØÒ×ÙÚÒØÐÖÑÑÐÙÖ ÆÓÙ×ÔÓ×ØÙÐÓÒ×ÕÙÐÑÒ×ÑÐÖØÓÒ×ØÖÔÖÐÒÙÒ ÑÒÙÒÑÔÐØÙÐ 

ÔÓÖØÒØÐÕÙÒØØÑÓÙÚÑÒØq 

ÓÖÑM 

ÐÖÐØÖÕÙÕÙÒØÐÓÙÔÐÑ ×ØÙÒÖÐÕÙÔÖÑØÒÖÐÓÙÔÐÐÔÖÒÐÓ 

= Óui 

ÄÜÔÖ××ÓÒÙÔÖÓÔØÙÖ×ÑÐÐÓÑÔÖÒÖÔÖÑÓÖ ×ØÐÕÙÖÑÓÑÒØÙÓ×ÓÒÒ ×ØÐ×ÔÒÙÖÙÐÔØÓÒi 

g 

q MW×Ø×Ñ××≃Î2 ØÓÙØÓ××ÐÓÒ×ÔÐÒ×Ð×ØÙØÓÒÓq 2 

g2 

2 

 

uµ −γµ 

(1 − γ5) 

2 

uνµ 

−g µν + q µ q ν /M 2 W 

q 2 − M 2 W 

 

uνeγν 

(1 − γ5) 

2 

ue − 

≪ M 2 WØØÜÔÖ××ÓÒ×ÖÙØ

ÐÓÖÑ×ÑÔÐg µν /M 2 ÓÑÔÖ×ÓÒÚÐÑÔÐØÙÓÖÖ×ÔÓÒÒØÔÓÙÖÐÒØÖØÓÒÔÓÒØÙÐÐ 

WÒ××ÓÒØÓÒ×ÓÒÔÙØÖÔÐÙ××ÑÒØÐ 

MÔÓÒØ×ÐÓÒÔÓ× 

ÄÒØÖØÓÒÔÓÒØÙÐÐÔÓ×ØÙÐÒ×ÐØÓÖÖÑ×ÚÖÙ×ØÔÓ× ÇÒÚÓØÕÙ××ÒÖÐÑÔÐØÙM=⇒ √2 

ÎÐ×ÙÜÒØÖØÓÒ××ÓÒØÓÖÓÑÔÖÐ ØÖ×ÙØÒÖ≫ 

MÔÓÒØ= GF 

√2 uµ −γµ(1 

 

µν 

− γ5)uνµ g [uνeγν(1 − γ5)ue−] 

g2 8M2 = 

W 

GF 

ØÖÓÖÔÙ×ÕÙÒ×ÙÒÔÖÓ××Ù××ÒØÖØÓÒβ q2 ÑØÙÖ×ÕÙÐÖÒÒØÖÐÙÖ×ÓÒ×ØÒØ×ÓÙÔÐÁÐ×Ò×ÙØÕÙ ÐÒØÖØÓÒÐØÖÓÑÒØÕÙ×ØÙÔÐÙ×ÐÖÒÒØÖÐ×Ñ×××ÐÙÖ× ÇÒ×ÖÒÓÑÔØÙØÖÔÖØÕÙÐÖÒÒØÖÐÒØÖØÓÒÐØ 2 

×ÒØÖØÓÒÙÔÓÒÖÒÑÓ×ÔÙÖÑÒØÐÔØÓÒÕÙ× 

≃ ) ≪ M O(1ÅÚ2 W 

ØÐÈÚ×ÖÔÔÓÖØ×ÑÖÒÑÒØÑ×ÙÖ× ÄÔÓÒÖÙÜÑÓ××ÒØÖØÓÒÔÙÖÑÒØÐÔØÓÒÕÙ×ÚÓÖ 

π ± → µ ± νµ(νµ) : 0.999877 

→ e ± νe(νe) : 1.230 10−4 ×Ö×ÙÐØØ×ÓÒ×ØØÙÒØÙÒÔÔÐØÓÒÒØÖ××ÒØ×ÓÒØÖÒØ×Ö×ÙÐØÒØ Ô××ÔÓÒÐÓÒÔÖÖØÙÒÓÑÒÒÐÚÓÐØÖÓÒÕÙ×ÙÖÐÚÓ ×ÔÖÓÔÖØ×ÐØ×ÐÔØÓÒ×§ ÑÙÓÒÕÙ×ØÐ×ØÙØÓÒÒÚÖ×ÕÙÐÓÒÓ×ÖÚØÐØÓÖ×ÓÙÖÒØ× ÔÖÑÓÖ×ÙÖÐ×ÔÙÖ×ÓÒ×ÖØÓÒ×ÒÖØÕÙ××Ô 

ÖÔÔÓÖØ −AÔÖÑØÒÖÒÖÓÑÔØÐÐÔÖØ×ØÙÜ×ÒØÖØÓÒÒ×Ð 

V 

Γ(π → eν) 

Γ(π → µν) ≃ 

2 2 

me mπ − m 

· 

mµ 

2 e 

m2 π − m2 2 = 1.2 10 

µ 

−4 Ò×ÙÒØÐÔÖÓ××Ù× ÇÒÔÙØÐÓÑÔÖÒÖ×ÐÓÒ×ÖÖÐ×ØÖÙØÙÖÙÓÙÖÒØÐÑÔÐÕÙ 

ÓÒ×ÖÓÒ×ÔÖÜÑÔÐÐ×ÒØÖØÓÒÔÙÖÑÒØÐÔØÓÒÕÙÙÒπ +ÐÐ ×ØÖÔÖÙÒÓÙÖÒØÖÐÓÖÑ 

j µ = uℓγ µ 

 

1 − γ5 

uνℓ 

2 

≡ uℓγµ 

 

1 − γ5 1 − γ5 

uνℓ 

2 2 

 

1 + γ5 

≡ uℓ γ 

2 

 

hℓ=+1 µ 

=ÑÙÓÒÓÙÐØÖÓÒ 

 

 

1 − γ5 Óℓ uνℓ 

2 

 

hνℓ =−1

π 

e 00 11 

00 11 

+ + 

ÚØÓÒÙÑÓÑÒØÒÙÐÖ ×ÒØÖØÓÒÙÔÓÒÔÓ×ØØØ××ÔÒÑÔÓ××ÔÖÐÓÒ×Ö 

νe 

s + s 

e νe 

ÓÙÖÒØÑÔÐÕÙÙÒνℓÐØ Ô××ÔÒÐÓÒ×ÖÚØÓÒÐÓ×ÐÕÙÒØØÑÓÙÚÑÒØØÙÑÓÑÒØ ÒÙÐÖÑÔÓ×ÐℓØÙνℓØÖÒ×ÐÑÑØØÐØÙÖÍÒ ØÐÐÓÒÙÖØÓÒ 4Ó×ÔÐÙ×ÒØÖÖÐ×Ò×ÐÚÓÑÙÓÒÕÙ ØÙÒℓÐØ ÇÖÐÔÓÒÒÝÒØ 

ÄØÓÖÓ ÕÙÒ×ÐÚÓÐØÖÓÒÕÙÔÖÕÙÐÑÙÓÒ×ØÖÐØÚÑÒØÐÒØÔÖÖÔÔÓÖØ ÐÐØÖÓÒÙ×ÐÖÒ×Ñ×××ØÐÙ 

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s 

(s − M2 Z )2 + M2 ZΓ2 −)2×ØÐÖÖÐÒÖØÓØÐÒ×ÐÑZ 

(Ee + + Ee 

Γe + e−×ØÐÐÖÙÖÖØÖ×ØÕÙe + e− → ZÐÐÐZ→ e + e− Γ ff 

ff×ØÐÐÖÙÖÔÖØÐÐÐÚÓZ→ 

N f 

= C×ØÙÒØÙÖÓÙÐÙÖNC 

gV gA×ÓÒØÐ×ÓÒØ×ÓÙÔÐÙÓÙÖÒØÐÒÙØÖØÐÙ 

ff = Nf 

CGFM 2 Z 

6π √ (g 

2 

2 V + g2 A )(1 + δ) 

= 

ΓZ = Γe + e − + Γµ + µ − + Γτ + τ − + NνΓνν + Γ 

U D×ØÒÒ×ÐØÐÙ 

Γcc + Γ dd + Γss + Γ bb 

≡ 2Γ UU + 3Γ DD 

= 0.389... TeV 2 · nbarn

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• Γ= ± 

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ÕÙÖ×( u d )Ø( )ÓÖÑÒØÙÒÒ×ÑÐÔÖÐÐÐÙÜÙÜÑÐÐ×ÐÔØÓÒ×( Ø( νµ )ÐÓÖ×ÓÒÒÙ×Ä×ÙØÙÖ×ÓÒØÑ××ÙÖÖÓØÕÙÒÓÙÚÙÕÙÖ 

µ Ö×ÙÐØØØÓÒÖÑÔÖÐÑ×ÙÖÖØÐÚÓe + ÖÝÓÒÒÑÒØÑ×ÔÖÐÙÒ×ÔÖØÙÐ×××ÙÜ Ø×ÒÐ ËÄÐ×ÓÛÂÁÐÓÔÓÙÐÓ×ÄÅÒÈÝ×ÊÚ ×ÝÔÓØ××ÖÒÒØÔÐÙ×ÐÐØÕÙÙÙÒÑÒ×ØØÓÒÕÙÖÒÚØÒÓÖ 

 

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0.0023ÎÐÐÖÙÖØÓØÐØØÚÐÙÖ×ØØÖÐÒÐÝ× 

0.0021ÎÐÚÐÙÖÒØÖÐÐÑ×× 

• MZ = 91.1876 ± 

• ΓZ = 2.4952 ± 

• Γℓ + ℓ− = 0.08398 ± 0.00009ÎÐÐÖÙÖÔÖØÐÐÐÚÓÐÔØÓÒÕÙℓ + ℓ− , µ ± , τ ±ÚÓÖÙÖÓÒØØÐÓØ 

±0.0015ÎÐÐÖÙÖÐÒ×ÑÐ×ÚÓ×ÒÙØÖÒÓ×ÒÓÒÓ 

1.744 

• Γinv = 0.499 

Nν = 2.984 ± 0.008. 

ℓ− = 

104Ó×ÔÐÙ×ÐÕÙÐØÙÜÐÙÐÐÙÖÑÑÕÙÐ×××ÙÔÔÓ×Ö c νe 

s e ) 

e− ννγÓγ×ØÙÒÔÓØÓÒ 

→

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W 

µ 

s 

d 

u u 

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s 

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s 

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u ν µ 

d 

u 

W 

s 

d 

W W 

µ 

d 

s 

u ÖÑÑÐ×ÒØÖØÓÒK 0 → µ + µ −ÖÑÑ× ÐÓ×ÐÐØÓÒK 0 ←→ K 0 

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Ò×ÕÙÖ×uØcÓÙÔÐ×ÙÜÚÖØÜuWd, uWs, cWd 

ØcWs 

ÓÖÑ〈µ + µ − |M|K0 〉 ≈ g 4 ÓÐ×f(...)×ÓÒØ×ØÙÖ×ÒÑØÕÙ×ÐÙÐÐ×ÓÒØÒÒØÒØÖÙØÖ×Ð× 

sin θc cosθc [f(mu, MW, ...) − f(mc, MW, ...)] 

ÐÒÙÕÙÖuÐÙÙÕÙÖcÓÑÑÖÔÖ×ÒØÒ×ÐÙÖ Ò 

 

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←→ K 0ÐÐÙÐ×ØÔÐÙ×ÓÑÔÐÕÙÕÙ××Ù×ÔÖ 

 

u 

d ′ 

 

c 

s ′ 

MC = 

t 

b ′ 

cosθC sin θC 

− sin θC cosθC

ÖÖ×ÓÙ×ÐÓÖÑ ⎛ 

MCKM = 

⎜ 

⎝ 

c12 c13 s12 c13 s13e −iδ13 

ÒÜ×ÑÐÐ×ÒÖØÓÒ× 

δ13Ô×0 ≤ 

−s12c23 − c12s23s13e iδ13 c12c23 − s12s23s13e iδ13 s23c13 

s12s23 − c12c23s13e iδ13 −c12s23 − s12c23s13e iδ13 c23c13 

⎠ ⎞ 

⎟ 

Úcij cos θijsij = sin θij0 ≤ θij ≤ π 

j 2 

Ò×ÐÖØØÜØÒ×ÓÒÐÓÙÖÒØÐÖÕÙ×ÖÚØ 

0ÓÒÖØÖÓÙÚÓÑÑ×ÓÙ×Ò×ÑÐÐ× 

i, 

δ13 ≤ 2π 

→ ÆÓØÓÒ×ÕÙÐÐÑØθ13Øθ23 ÐÑÒØ×ÐÑØÖÓÒÒØÒØθ12θC 

j +µ 

W = (u c)γµ (1 − γ 5 ÚÒØ 

 

d 

)MC 

s 

j +µ 

W = (u c t)γµ (1 − γ 5 

ØÖÓ×ÕÙÖ×ÒÖÙÖ×ÇÒÔÙØÓÑÑÓÑÒØ×ÖÔÔÐÖÐ×ÓÙÔÐ×ÑÔÐÕÙ× ÁÐÓÒÒÐÙÙÒÑÒ×ÑÁÅÒÖÐ×ÙÜØÖÓ×ÕÙÖ××ÙÔÖÙÖ×ØÙÜ 

⎛ ⎞ 

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d 

)MCKM ⎝ s ⎠ 

b 

⎛ 

d 

⎝ 

′ 

s ′ 

b ′ 

⎞ ⎛ 

⎞ ⎛ ⎞ 

ÑØÖ×ØÙÒØÓÒ×ÖÐÐÕÙÐÐÓÒØÖÚÐÐÒÓÖÖÖÔÄ 

Vud Vus Vub d 

⎠ = ⎝ Vcd Vcs Vcb ⎠ ⎝ s ⎠ 

ØÐÈÜÔÓ×ÒØÐÐ××ÓÙÖ×ÒÓÖÑØÓÒ×ÙØÐ××Ä×ÐÑÒØ×Ð ÖØÒ×ÐÑÒØ×VijÔÙÚÒØØÖÖÒÙ×ÓÑÔÐÜ×ÔÖÐÓÜÙØÖÑ 

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Vtd Vts Vtb b 

ÓÒØÖÒØ×ÙÒØÖØÔÖÑØØÒØÑÐÓÖÖÐÓÒÒ××ÒÐÑÒØ×ÔÓÙÖ ×ØÐÓÖÑÖÓÑÑÒÒ×ÐØÐÈØÙÐÐÓÒØÖÓÙÚÙØÖ×ÓÖÑ×ÕÙÚ 1× ≃ ÐÒÐÝ×ÔÖÓ××Ù××ÒØÖØÓÒÓÒØÙÒÓÖÑ×ÑÔÐÖc13 

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| ×ÙÖÐ×ÚÐÙÖ×|Vij 

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ÓÑÔÐÜ××ÓÒØÑÓ××ÓÙ×ÐÓÔÖØÓÒÌØÈ ÕÙ×ÖÔÖÙØ×ÙÖÐ× 

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⎛ 

0.975 0.22 

⎞ 

0.003 

| Vij |= ⎝ 0.22 0.974 0.04 ⎠ 

0.01 0.04 0.999 

 

1.02 ± 0.22ÖÒ

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mπ 

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× 

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2 

mproton = mneutron 

 

1 

= 

0 

 

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1 

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mu,d... 

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ξ ′ ÈÜÒ×Ð×ÙÒÖÓØØÓÒÙØÓÙÖÐ×ÓÜÝÓÒÓØÒØÚÓÖ 

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= Uξ 

 

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ÌÖ(τi) = 

ÈÖÜÑÔÐ Ò×Ð×Ð×Ó×ÔÒÓÒÐ××ÒÒÖÐÑÒØÔÖÐ×ÐØØÖ×τ τi 

 

u 

= 

d 

| ξu | 2ÐÔÖÓÐØÕÙÐÕÙÖ×ÓØØÝÔu 

 

U =⇒ UI(ˆnθ) = exp{−iI · ˆnθ} 

U1 (ˆnθ) = exp{−i 

2 

1 

2 τ · ˆnθ} 

ξ ′ 

′ u 

= 

d ′ 

 

cosθ/2 − sin θ/2 u 

= 

sin θ/2 cosθ/2 d 

(ξ ′ ) † (ξ ′ ) = ξ † U † Uξ = ξ † 

ξ 

ξ ′ = ξ + δξ = (1 − iδθI · ˆn)ξ =⇒ (1 − i 

1 

I= 2 

1 

2 τ · ˆnδθ)ξ 

 

0 

[Ii, Ij] = iεijkIkÓε123,231,312 = 1Øε213,132,321 = −1 

 

1 

2 τi , 1 

2 τj 

 

1 

= iεijk 

2 τk

Ä×ÓÔÖØÙÖ×I 2ØIzÔÖÑØØÒØÐ××ÖÙÒØØÐÐÙÖ×ÚÐÙÖ× ÔÖÓÔÖ×Ä×I±ÔÖÑØØÒØØÖÒ×ØÖÒØÖÐ×ØØ×ÑÑI 2 ÈÓÙÖÒÓØÖÜÑÔÐÚ 

τ1 = 

1 

0 1 

1 0 

 

u = 1 

 

, τ2 = 

1 0 

0 −i 

i 0 

1 

 

1 0 

, τ3 = 

0 −1 

2 τ3 

2 0 −1 0 0 

 

1 

2 τ3 

 

d = 1 

 

1 0 0 

= 

2 0 −1 1 

1 

 

0 

= − 

2 −1 

1 

2 d 

τ± = 1 

2 (τ1 ÔÔÐÕÙÓÒ×ÙÜÚØÙÖ×× 

 

0 1 0 0 

± iτ2)Óτ+ = , τ− = 

0 0 1 0 

 

0 1 1 

0 0 1 0 

ÍÒÓÔÖØÙÖ×ÑÖ×ØÐÙÕÙÓÑÑÙØÚØÓÙØÒÖØÙÖÙÖÓÙÔ 

τ+u = 

= 0 , τ−u = 

= = d 

0 0 0 

1 0 0 1 

 

0 1 0 1 

0 0 0 

τ+d = 

= = u , τ−d = 

= 0 

0 0 1 0 

1 0 1 

ÓÒÓÒÔÙØÓÒ×ØÖÙÖ×ØØ×ÔÖÓÔÖ×I 2ØÙÒIjÓÜÓÜ×ØÒ 

 

I3|I, I3〉 = I3|I, I3〉 

2 (I+I− + I−I+) + I 2 3 = 1 

2 {I+, I−} + I 2 

3 ÈÓÙÖÙÒÖÔÖ×ÒØØÓÒÓÒÒIÐ×ÚÐÙÖ×ÔÖÓÔÖ×I 2×ÓÒØI(I 1)ÈÖ + ÜÑÔÐÓÒI 2 = 0ÔÓÙÖI= 0 1/2(1/2 + 1)ÔÓÙÖI = 1/2Ø 

 

= 1 

 

1 

2 

 

= 1 

2 u 

 

ÇÒÔÙØÚÖÖÕÙÒ×ËÍ ÐÓÔÖØÙÖI 2ÓÑÑÙØÚI1, I2ØI3 

[I 2 ÖI3 

, Ij] = 0 Ij = 1, 2, 3 

I 2 Ò×ËÍ |I, I3〉 = I(I + 1)|I, I3〉 

ÐÓÔÖØÙÖ×ÑÖI 2ÚÙØ 

C ≡ I 2 = 1

Ð×ÕÙÖ×uØdÇÒÒØÐÖÔÖ×ÒØØÓÒ ÊÔÖ×ÒØØÓÒ× ÔÖ ±1/2ÔÖÑØÖÔÖ×ÒØÖ Ø2 Ä×ÓÓÙÐØI = 1/2Ú××ÔÖÓØÓÒ×I3 

Ù2ÔÖ ÈÓÙÖÐÔÖÒØÕÙÖ×Ð×ØÒØÖ××ÒØÒÖÐÖÔÖ×ÒØØÓÒÓÒÙ 

 

ØÚÑÒØÓÒÔÙØÑÓÒØÖÖÖÓÑÑÜÖÕÙÐÓÙÐØ 

 

2 

×ØÖÒ×ÓÖÑÔÖÖÓØØÓÒÒ×Ð×ÔÐ×Ó×ÔÒÓÑÑÐÓÙÐØ×ÕÙÖ× 

′ ×ØÙÒ×ØÙØÓÒÔÖØÙÐÖÙÖÓÙÔËÍ ÐÐÒ×ÖØÖÓÙÚÔ×ÚÐ× 

= Uφ 

ÓÖÑØÓÒÙÒØÖUÒ×Ð×ÔÖ×ÒØÙÒÖÓØØÓÒÒ×Ð×Ô×Ó×ÔÒØ ÇÒÚÙÙÔØÖÕÙ×ÐÓÒØ×ÙÖÙÒ×Ý×ØÑÔÝ×ÕÙÙÒØÖÒ× ÄÖ×ÙÖËÍ ×Ó×ÔÒ ÖÓÙÔ×ËÍÆÓN 

×Ð×Ý×ØÑØÖÒ×ÓÖÑÒÔÙØÔ×××ØÒÙÖÙ×Ý×ØÑÒØÐÐÓÖ× 

> 

Ð×Ó×ÔÒ ØÖÔÖÓÕÙÑÒØÐÓÒÙØÙÒÐÓÓÒ×ÖÚØÓÒ××ÓUÈÓÙÖ 

×Ý×ØÑ×ÓÒØØ× ÑÔÐÕÙÐÓÒ×ÖÚØÓÒÐ×Ó×ÔÒÁÒÚÖ×ÑÒØÓÒÔÙØÑÔÖÑÖÙÒÖÓØØÓÒ Ù×Ý×ØÑÒ×Ð×ÔÐ×Ó×ÔÒ×Ò×ÕÙÐ×ÔÖÓÔÖØ×ÒÖØÕÙ× ÓÔØÐÓÒÚÒØÓÒÓÒÓÒËÓÖØÐÝÒ×ÐÐØØÖØÙÖÓÒØÖÓÙÚÙ××ÐÓÒÚÒØÓÒ ÄÓÔÖØÓÒÓÒÙ×ÓÒÖÒØÖÓÙØÙÒØÙÖÔ×ÖØÖÖÆÓÙ×ÚÓÒ× 

[H, 

ξ = ( u d ) 

φ 

ÓÔÔÓ×2 ≡ 

 

−d 

u 

 

= 

 

u 

2 ≡ 

d 

2 

 

d 

≡ 

−u 

 

d 

φ = 

−u 

 

 

[H, U] = 0 

I] = 0

ÈÖÜÑÔÐÑÒÓÒ×ÙÒÕÙÖ×ÓÐ×ØÙÖÔÓ× 

〈q|H0|q〉 = mq =Ñ××ÙÕÙÖq 

ËÐÓÒØÙÒÖÓØØÓÒÒÐδθÒ×Ð×Ô×Ó×ÔÒÓÒÓØÒØÐØØØÖÒ× ÓÖÑq ′ 

Å× |q 

〈q 

0ÓÒÙØÕÙ ÓÒ×[H0, I] = 

〈q ′ |H0|q ′ 〉 = mq ′ = 〈q|H0|q〉 ÓÒ×ÖÓÒ×ÑÑÙÒ×Ý×ØÑÕÙÖ{q}ÒÒØÖØÓÒÒ×ÐØØ×Ó×ÔÒ 

I3〉ËÐÀÑÐØÓÒÒÕÙÖØ×Ý×ØÑÓÑÑÙØÚIÓÒÓØÒØÐÑÑ 

= mq 

H0ÓÒÒÐÑ××Ù 

|I, ÒÖÔÓÙÖØÓÙ×Ð×ÑÑÖ×ÙÑÙÐØÔÐØ×Ó×ÔÒ|I, −I〉, 

ÈÖÜÑÔÐÔÓÙÖÐ×ØØ×ÖÙÔÓÒÓÒÔÖØÕÙ 

ËÐ×Ý×ØÑ×ØÙÖÔÓ×Ò×ÐÐÓÖØÓÖH = 

ÕÙ ÓÒÒÓÒ×ÙÒÓÒØÖÜÑÔÐÓÒ×ÖÓÒ×ÐÓÔÖØÙÖÖÐØÖÕÙQØÐ 

|Ò×ÑÐÔÖØÙÐ×〉 ÒÒÖÐ 

Q|Ò×ÑÐ×ÔÖØÙÐ×〉 = 

[Q, 

′ 

〉 = U|q〉 ≈ (1 − iI · ˆnδθ)|q〉 ′ |H|q ′ ÓÖ 

〉 ≈ 〈q|(1 + iI · ˆnδθ)H(1 − iI · ˆnδθ)|q〉 

(1 + x)H(1 − x) = H + xH − Hx + O(x 2 ÕÙÓÒÒ 

) 

〈q ′ |H|q ′ 〉 = 〈q|H|q〉 + 〈q|[H, I]|q〉 + O(δθ 2 

) 

|I, −I + 1〉, ..., |I, +I〉 

×Ý×ØÑ{q}ØØÓÙ×Ð×ÑÑÖ×ÙÑÙÐØÔÐØÓÒØÐÑÑÑ×× 

〈I, −I|H|I, −I〉 = 〈I, −I + 1|H|I, −I + 1〉 = ... = 〈I, +I|H|I, +I〉 

mI(I3 = −I) = mI(I3 = −I + 1) = ... = mI(I3 = +I) 

m(π + ) = m(π 0 ) = m(π − ),×Ð×ÝÑØÖ×Ó×ÔÒ×ØÔÖØ 

 

Ò×ÑÐÖ× 

 

 

I] = 0

ØÐ×ÑÑÖ×ÙÒÑÑÑÙÐØÔÐØ×Ó×ÔÒÓÒØ×Ö×ÖÒØ× 

Q|π + 〉 = +1|π + 〉 Q|π − 〉 = −1|π − 〉 Q|π 0 〉 = 0|π 0 〉 

ÙÒÓÒÒÖÐ×ÙÒØÖÑÐÀÑÐØÓÒÒÓÒØÒØÐÖÐØÖÕÙ ØÀÑÐØÓÒÒÔÙÒØÖÒÚÖÒØ×ÓÙ×Ð×ØÖÒ×ÓÖÑØÓÒ×Ò× 

ÓÒ× Ð×Ô×Ó×ÔÒ×ØÐ×ÙØÖÑÕÙÜÔÖÑÐÒØÖØÓÒÑHem 

ÖØÐ×Ý×ØÑ{q}ÒÒØÖØÓÒÓÒ 

∝ 

Q1Q2 

ØÐ×Ó×ÔÒ×ØÙÒ×ÝÑØÖÖ×ÔÖÐÒØÖØÓÒÑÈÖÜÑÔÐÐÑ×× 

Hint 

ÕÙÔÙØ×Ô××ÖÒ×{q}ÓÒ×ÖÓÒ×Ð×ÖÒ×Ñ××Ò×Ð× ×Ý×ØÑ×ÙÔÓÒØÙÓÒ ×ÙÜÔÓÒ×Ö×ÖÕÙÐÕÙ ÇÒÔÙØÖÖÕÙÒØÖØØÖ×ÙÖ×ÝÑØÖÈÓÙÖÓÒÒÖÙÒ ÐÑ××ÙÔÓÒÒÙØÖ 

ÈÓÙÖÐ×Ý×ØÑÙÔÓÒ 

ØØÖÒÔÙØØÖÙÖÒØ×ÓÒØÖÙØÓÒ× 

×ÓÒØÐ×Ö××ÙÜÕÙÖ××ØÒÑÓÝÒÒ〈R〉 

δud 

Ò×Ð×ÔÓÒ×ÐÓÒØÖÙØÓÒ×ØÒÙÐÐÖÓÒÓÒ×ÖÕÙÒ ÓÐ×µi×ÓÒØÐ×ÑÓÑÒØ×ÑÒØÕÙ×ØmiÐ×Ñ××××ÕÙÖ× 

ÐÒØÖØÓÒÐØÖÓ×ØØÕÙÔÖÓÔÓÖØÓÒÒÐÐQ1Q2/〈R〉ÓQ1ØQ2 ÐÒØÖØÓÒÑÒØÕÙÔÖÓÔÓÖØÓÒÒÐÐµ1, µ2 ÑÓÝÒÒÙÓÙÖ×ÙØÑÔ×ÐÓÒØÒÙÒÕÙÖuØd×ØÒØÕÙÔÓÙÖπ ± Øπ 0ÇÒÔÙØÒÐÓÖÐÓÒØÖÙØÓÒ Ä×ÓÒØÓÒ×ÓÒ×ÔÓÒ××ÓÒØÚÓÖÔÐÙ×ÐÓÒ Ò×ÕÙÚÒØÙÒØÖÑØ ÔÓÙÖÐ×ÓÒØÖÙØÓÒ×∝ Q1Q2 

π + Ø ≡ du 0 ≡ 1 Ä×ÚÐÙÖ×ÔÖÓÔÖ×Q×ÓÒØÜÔÖÑ×ÒÙÒØÐÖÙÔÓ×ØÖÓÒ 

 

√ (uu − dd) 

2 

= md − mu 

= HIF + Hem 

[Hint, I] = [HIF, I] + [Hem, I] 

 

δπ = m π + − m π 0 = 4.6ÅÎ 

π 

0 

= 0 

∝ Q1 

Q2 

m1 m2

ÐÙÐÓÒ×Ð×ÔÖÒÑØÑØÕÙQ1Q2 

〈π + |Q1Q2|π + 〉 = du 

Q1Q2 ÄÓÒØÖÙØÓÒ×ØÓÒÒÔÖ ×ÓÒØÓÖØÓÓÒÙÜ ÊÔÔÐÓÒ×ÕÙÐ×ØØ×|uu〉Ø dd 

ÔÖ ÓÒÓØÒØ〈R −1 ÓÒ×ÖÓÒ×ÑÒØÒÒØÐ×Ý×ØÑÙÓÒ 

ÐÖ× Ò××ÓÒÙÒÓÒØÖÙØÓÒ ÖÓÒÙÒÓÒØÒÙÖÒØÒÕÙÖ× 

K 

 

 

2 1 

 

du = + du 

2 

du = 

3 3 9 

〈π0 |Q1Q2|π0 〉 = 1 

 

uu − ddQ1Q2uu − dd = 

2 

1 

〈uu|Q1Q2|uu〉 + 

2 

dd 

 

Q1Q2 

1 

dd + 0 + 0 = − 

2 

4 

 

1 

− = − 

9 9 

5 

18 

π + 2 

: 

9 〈R−1 〉π, π 0 : − 5 

18 〈R−1 〉π 

 

2 5 

δπ = + 〈R 

9 18 

−1 〉π = 1 

2 〈R−1 〉π 

〈R−1 〉×ØÙÒÔÖÑØÖØÖÑÒÖÜÔÖÑÒØÐÑÒØÐδπÓÒÒ 

9.2ÅÎ 

〉π = 

δK = mK + − mK0 = −4ÅÎ 

ÓÒ 

+ Ñ×××mu ms md + ms 

 

δK 

ËÐÓÒÑØÕÙ〈R −1 ×ÓÒØÐ×ÓÖÖ×ÖÒÙÖ×ÄÑÑÔÔÖÓÔÓÙÖÐ×ÖÝÓÒ×ÓÒÒÐ× 

〉ØÅÎ 

 

×ØÑØÓÒ×δud = 4ÅÎØ〈R ÊÔÔÐÓÒÔÓ×c = 1 

Q1Q2 = 2 

3 

+ : su K 0 : sd 

1 

3 

= 2 

9 

−1 

3 

1 

3 

= −1 

9 

= mu + ms − md − ms + 2 

9 〈R−1 〉K − 1 

9 〈R−1 〉K 

= mu − md + 1 

3 〈R−1 〉K 

〉K ≈ 〈R−1 9.2ÅÎÓÒÓØÒØ 

〉π = 

−4 = mu − md + 1 

3 9.2 ; δud 7ÅÎ = md − mu ≃ 

−1

ÊÔÖ×ÒØØÓÒÑÒ×ÓÒ ËÍ1ÐÐ×ØÙØÐÔÓÙÖ ×Ó×ÔÒ ØØÖÔÖ×ÒØØÓÒÓÒÖÒÔÖÜÐ×ÓØÖÔÐØI = ÖÖÐ×Ý×ØÑ×ÔÓÒ×π + Ä×ÒÖØÙÖ××ÓÒØ 

⎠ 

ØÙÒØÐÔÖÓÙØÖØ 1ÔÙØ×ÙÖÐÖÔÖ×ÒØØÓÒÓÒÑÒØÐ Ò ÄÖÔÖ×ÒØØÓÒI = 

ÓÖÑ×ÔÖÐÓÙÔÐ×ÙÜ×ÔÒÙÖ×ψØφÐÓÖ×ÐÖÓØØÓÒRÓÒÒ ÇÒÔÙØÑÓÒØÖÖÜÖÕÙÐÓÖÑ×ÒÖØÙÖ××ØÓÖÒØ 

(ψ⊗φ)0,1Ò××ÒØÐ×ØØ××ÒÙÐØØØÖÔÐØ 

1 ÜÔÖÑÒØÖÑ×ÑÒ×ÓÒ×ÔÖÓÙØ×ÖØ2 ⊗ 

ÊÓØØÓÒ××ÔÒÙÖ× 

= Ú×Ú×ÒØ×Ψ0,1 

( 1 

2 ×ÑØÖÈÙÐ 

, π0 , π−ÖÔÖ×ÒØÔÖÐ× 

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 

1 0 0 

⎝ 0 ⎠ , ⎝ 1 ⎠, ⎝ 0 ⎠ 

0 0 1 

I1 = 1 

⎛ ⎞ 

0 1 0 

√ ⎝ 1 0 1 ⎠ , I2 = 

2 

0 1 0 

1 ⎛ ⎞ ⎛ ⎞ 

0 −i 0 1 0 0 

√ ⎝ i 0 −i ⎠ , I3 = ⎝ 0 0 0 

2 

0 i 0 0 0 −1 

1 

⊗ 

2 2 = 0 ⊕ 1 

2 = 1 ⊕ 3 

Ψ ′ 0,1 = U ↑ Ψ0,1 ≡ (U ↑ ψ ⊗ U ↑ φ)0,1 = (ψ ′ ⊗ φ ′ )0,1 

ψ = ÉÑÓÒØÖÖÕÙ×ÐÓÒÙØÐ×∼ 1 ⎛ ⎞ 

⎛ ⎞ 

−a + c 

a 

√ ⎝ i(a + c) ⎠ÐÔÐψ= ⎝ b ⎠ , 

2 

b 

c 

: ÓÒÓØÒØRθ ∼ 

ψ ↦→ ∼ 

ψ ′ 

= R(−θ) ∼ ÓÑÑÒØÓÒ×ØÖÙØÓÒÐ×ÓÒØÓÒ×ÓÒΨ0,1ÒÙØÖ×ØÖÑ×ÓÑ 

uØdÓÒÖ×ÔØÖÐÐÖËÍ 

ψ,ÚÊÐÑØÖÖÓØØÓÒÒ×Ê3 ÑÒØÖÖÒÖÜØÑÒØu, d, 

ÓÒ×ÖÓÒ×ÙÒ×Ý×ØÑqqÑ×ÓÒ 2 ⊗2Ò×Ò×ÔÖÒØ××Ý×ØÑ×

×ÔÒ ÓÒ ØØ×ÒÙÐØ≡ 1 

 

√ dd + uu = |0, 0〉 

2 

⎧ 

du = |1, 1〉 

ÉÚÖÖÕÙÐ×ØØ×ÙØÖÔÐØÓÒØÐÑÑ×ÝÑØÖÚ×Ú×ÐÒ 

⎪⎨ 

1√ dd − uu = |1, 0〉 ØØØÖÔÐØ≡ 2 

⎪⎩ 

|−ud〉 = |1, −1〉 

u ↔ −d, u ↔ d ÄÓÒ×ØÖÙØÓÒ×ØØ××ØÒÓÒ×ÖÒØÙÒØØÜØÖÑ(| I3 |= I) 

I−|I, I3〉 = 

I(I + 1) − I3(I3 − 1) |I, I3 − 1〉 

I+|I, I3〉 = I(I + 1) − I3(I3 + 1) |I, I3 + 1〉 ÚÐ×ÒØÓÒ×ÙÓÙÐØ( u d ) ÓÒ 

I+|d〉 = [ 3 1 + 4 4 ]12 

|u〉 = |u〉 

I+|u〉 = −I+|−u〉 = −1 

d = −d 

 

I+|u〉 = I+ d 

⎫ 

⎪⎬ 

I+ÒØÒÒØÓÑÔØ×ÖÐØÓÒ× ÔÖØÖ|−ud〉ÓÒÔÙØÓÒ×ØÖÙÖÐÒ×ÑÐÙØÖÔÐØÔÖÔÔÐØÓÒ 

⎪⎭ 

= 0 ×ÖÐØÓÒ×ÒÐÓÙ×ÔÙÚÒØ×ÖÖÔÓÙÖI− 

I+|1, −1〉 = √ 2|1, 0〉 

I+|1, 0〉 = √ 

⎫ 

ÁÑÔÓÙÖÐØØ×ÒÙÐØ 

⎪⎬ 

2|1, 1〉 

⎪⎭ 

I+|1, 1〉 = 0 

 

 

I+|0, 0〉 = 0 

ØÒÔÔÐÕÙÒØÐ×ÓÔÖØÙÖ×I±×ÓÙ×ÐÒÒÜÚÓÖ 

ØÐÒØÓÙÐØ( d −u )

I I3 Q J PÒØ 

du 0 − π + 

√1 (dd − uu) 

2 

0− π0 −ud 0− π− √1 (dd + uu) 0 

2 − ÌÊÔÖ×ÒØØÓÒ×ËÍ×Ó×ÔÒØ××ÓØÓÒ×ÔÓ××Ð×ÙÜÑ×ÓÒ× Ô×ÙÓ×ÐÖ×ÒÓÒØÖÒ×ÓÒÒÙ× 

η 

ØÕÙ×ØÐÙÖ×××ÓØÓÒ×ÔÓ××Ð×ÙÜÑ×ÓÒ×Ô×ÙÓ×ÐÖ×ÒÓÒØÖÒ× É×ÙØÖÐ×××ÓØÓÒ×ÔÖÓÔÓ××Ò×ÐØÐÙ ÓÒÒÙ× ÄØÐÙÓÒÒÐ×ÖÔÖ×ÒØØÓÒ×ËÍ×Ó×ÔÒÐÙÖ×ÒÓÑÖ×ÕÙÒ 

×ÝÑØÖÕÙ×ÙÔÓÒØÚÙÐ×ÚÙÖ ÇÒÔÙØÓÒ×ØÖÙÖÐÑÒØÐ×ÓÒØÓÒ×ÓÒÙÜÕÙÖ×qqÒ 

3⊕1ÇÒÓØÒØÙÒØØÒØ×ÝÑØÖÕÙØØØ× 

 

ÔÔÐÕÙÒØÐ×Ñ2⊗2 = 

⎧ 

1 

⎨ uu 

 

1 

√ (ud − du) √ (ud + du) 

2 ⎩ 2 

ÄÜØÒ×ÓÒÐ×ÝÑØÖ×Ó×ÔÒËÍ ËÍ Ð×ÝÑØÖÙÒØÖ Ð×ÝÑØÖÙÒØÖËÍ 

dd 

×Ø 

ÁÐ×ÒÓÖÖ×ÔÓÒÒØÙÙÒØØÐÓÒÒÙQ = 4/3, 1/3, −2/3 ! 

ÐÐÒØÖÓÙØÓÒÒ×ÐÑÓÐ×ÕÙÖ×ÙÒØÖÓ×ÑÕÙÖsÔÓÖØÙÖ

ØÖÒØSÒÖ 

 

u 

doublet ξ = 

ÍÒØØÔÙÖ×Ø×ÐÓÖ×ÖÔÖ×ÒØÔÖÐÙÒ×ÚØÙÖ×× 

d 

⎛ ⎞ 

u 

triplet ϕ = ⎝ d ⎠ ⇐⇒ ×Ó×ÔÒI+ØÖÒØS 

s 

ÄÓÖ×ÙÒÖÓØØÓÒÒ×Ð×ÔÓÑÒ×Ó×ÔÒØÖÒØ×ÝÑØÖÙÒØÖ ÐØØφ×ØÖÒ×ÓÖÑÓÑÑ 

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 

1 

0 

0 

|u〉 = ⎝ 0 ⎠ , |d〉 = ⎝ 1 ⎠ , |s〉 = ⎝ 0 ⎠ 

0 

0 

1 

∈ËÍ φ 

U 

 

U = exp −θˆn · λ 

ËÍ Ò×ËÍ ÓÒÙØÒÖØÙÖ×ÖÔÖ×ÒØ×ÔÖÑØÖ×ÐÐÅÒÒ ÓÒÚØØÖÓ×ÒÖØÙÖ×ÖÔÖ×ÒØ×ÔÖÐ×ÑØÖ×ÈÙÐÒ× 

)Ú 

≡ exp (−θˆn · F Ñˆn = 

2 

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 

⇐⇒ ×Ó×ÔÒI 

′ ×ØÙÒÑØÖ×ÙÒØÖØÕÙÐÓÒÔÙØÖÖ 

 

= Uφ 

8 

λ1 = 

λ4 = 

⎝ 

⎛ 

⎝ 

⎛ 

⎝ 

0 1 . 

1 0 . 

. . . 

0 . 1 

. . . 

1 . 0 

. . . 

. 0 −i 

. i 0 

⎠ , λ2 = ⎝ 

⎞ 

⎛ 

⎠ , λ5 = ⎝ 

⎞ 

0 −i . 

i 0 . 

. . . 

0 . −i 

. . . 

i . 0 

⎠ , λ8 = 1 √ ⎝ 

3 

λ7 = 

= ÕÙÐÓÒÓÑÔÐØÚλ0 2 ÒÓÑÖÕÙÒØÕÙØÒÓØËØÖÒ×ÐÖÐØÓÒ 

⎛ 

√ ⎝ 

3 

⎛ 

1 0 0 

0 1 0 

0 0 −2 

1 0 . 

0 −1 . 

⎠ , λ3 = ⎝ ⎠ 

. . . 

⎞ ⎛ 

. . 

⎞ 

. 

⎠ , λ6 = ⎝ . 0 1 

. 1 0 

⎞ 

⎠ 

⎞ 

1 0 0 

0 1 0 ⎠ 

0 0 1 

⎠ 

ÒÔ×ÓÒÓÒÖÚÐ×ØØ××ÐÖÔÖ×ÒØØÓÒÑÒ×ÓÒËÍ

Ä×ÔÓÒØ×ÖÔÖ×ÒØÒØ×ÞÖÓ×Ð×ÓÒØÔÓÙÖÓØ×ÓÙÐÒÖÐÔÖ×Ò× ×ÓÙ×ÖÓÙÔ×ËÍ Ò×ËÍ ÒÔÖØÙÐÖ×ÓÒÐ××ØÐÓÑÔÓ×ÒØ ØÖÒsÐØØφ λ1, 

ÔÖÑØØÒØÒÒÖÖÐ×ÓÙ××Ô×Ó×ÔÒÐ×ÝÑØÖÙÒØÖ 

≡ I3 ÓÒÐ×ÓÔÖØÙÖ×ÐÐI± 

ÔÐÙ×ÐÓÒÚÐ×ÓÔÖØÙÖ×ÐÐ ÑÑÓÒÔÖÐÍ×ÔÒØÎ×ÔÒÓÒØÐ×ÒØÓÒÒØÙØÚÚÒÖ 

= 

Ø 

 

ÇÒÔÙØ×ÑÒØÚÖÖÐÖÐØÓÒÖB ÔÓÙÖÙÒ×ÕÙÖ× ÇÒ 

Y ÖÒÖØÓÒØÓÒÓÔÖØÙÖÝÔÖÖÓY = 

ÔÔÐÕÙÓÒ×ÙÜØØ×uØsÔÖÜÑÔÐ 

ÖØÒ×ÙØÙÖ×Ò××ÒØÐÜÎ3Ò×Ð××ÒÓÔÔÓ×ÐÙÓÔØÕÙ×ÖÔÖ 

SØØÒÙ× ÕÙÓÒÒÒÐ×ÚÐÙÖ×B + 

 

ÙØ×ÙÖÐÒØÓÒÎ± ÖÔÔÐÙ××Y = 

S = −1ÔÓÙÖÐÕÙÖsØÔÓÙÖÐ×ÕÙÖ×uØdØÐÙ 

F1 = λ1 

λ2, λ3ÖÔÖ×ÒØÒØÐÔÖØ×Ó×ÔÒÙÖÓÙÔ 

2 ≡ I1 , F2 = λ2 

2 ≡ I2 , F3 = λ3 

2 

F1 ± iF2 

U± = F6 ± iF7 , V± = F4 ∓ iF5 

U3 = − 1 

2F3 

3 

+ 4F8 , V3 = − 1 

2F3 

3 

− 4F8 Y = 2 

√ F8 = 

3 2 ⎛ 

1 

1 1 

√ √ ⎝ 

3 3 2 

= 2 

√ F8 

3 

1 

−2 

⎞ 

⎠ = 1 

3 

⎛ ⎞ 

1 

Y |u〉 = Y ⎝ 0 ⎠ = 

0 

1 

⎛ 

0 

|u〉 , Y |s〉 = Y ⎝ 0 

3 

1 

Bar. + Str.Ò×ÐÖÐØÓÒ 

⎛ 

⎝ 

S 

 

B + 

⎞ 

1 

1 ⎠ 

−2 

⎞ 

⎠ = − 2 

3 |s〉

ÄÓÔÖØÙÖÖQ×ØÒÔÖ 

Q = 1 

2 Y + I3 → F8 

√3 + F3 = 1 

ÓÑÑÓÒ×ÝØØÒ 

⎠ 

 

 

 

1/2 

 

−1/2 

 

1/2 

 

1/2 

 

1/2 

−1/2 

ÌÄ×ÓÒ×ØÒØ××ØÖÙØÙÖËÍ √ 

3/2 

√ 

Ð×f k dijk×ÓÒØ×ÝÑØÖÕÙ× 

ij×ÓÒØÒØ×ÝÑØÖÕÙ×Ð× 

ÙÖÓÙÔÇÒ×ÖÔÖÒØØÓÒ×IÑÒ×ÓÒ× Ò×Ð×ËÍ ×ÔÒ ÕÙ 

φ 

 

= 

⎛ 

⎝ 

2 

3 

− 1 

3 

− 1 

3 

⎞ 

⎛ 

1 

⎝ 

6 

1 

⎞ 

⎠ + 

−2 

1 

⎛ 

1 

⎝ 

2 

i j k fk ij i j k dijk i j k dijk 

fk ij = −fk ji dijk = djik dijk = djik 

1/ √ 3 1/2 

1/2 −1/2 

1/2 −1/2 

1/ √ 3 −1/(2 √ 3) 

−1/2 −1/(2 √ 3) 

1/2 −1/(2 √ 3) 

1/ √ 3 −1/(2 √ 3) 

1/2 −1/ √ 3 

3/2 

′ ×ÓÒØ×ØÖÒØ×ÐÐÖËÍ I)φÔÖÑØÐØÙ×ÖÔÖ×ÒØØÓÒ× 

= (1 − iθˆn · 

[Ii, Ij] = iǫijkIk 

−1 

0 

⎞ 

⎠

Ò×Ð×ËÍ ×Øφ ′ F)ÕÙÔÖÑØØØØÙÓ××ÒØ 

= (1 − iθˆn · 

ÐÐÖËÍ 

[Fi, Fj] = if k ij Fk 

{Fi, Fj} = 1 

3 δij 

+ dijkFk Ä×f k ËÍ ×ÓÒØÓÒÒ×Ò×ÐØÐÙ Ò×ËÍ ijØdijk×ÓÒØÔÔÐ×Ð×ÓÒ×ØÒØ××ØÖÙØÙÖÙÖÓÙÔÐÙÖ×ÚÐÙÖ×Ò ØÓÔÖØÙÖ×ØÒÔÖ ÐÓÔÖØÙÖ×ÑÖØÒÔÖÐÖÐØÓÒ Ò× 

F 2 8 

≡ FiFi = 

i=1 

1 

2 {I+, I−} + I 2 1 

3 + 

2 {U+, U−} + 1 

2 {V+, V−} + F 2 

ØØÓÒ×ËÍ Ä×ÖÔÖ×ÒØØÓÒ×ËÍ 

8 

ÇÒÖ×ÓÒÒÔÖÒÐÓÚËÍ ÊÔÔÐÓÒ×ÕÙÒ×ËÍ ÐØØ×Ó×ÔÒ 

↔ÒÕÙÐÐÑÒØ∈ÁÁ 

ØÖÑÑ×ÔÓ×ÙÖ ÈÖØÒØÐÖÔÖ×ÒØØÓÒÓÒÑÒØÐ 0Ø ÑÒ×ÓÒ×Ø ÔÖÐÖÑÑÓÑÔÓ×ØÓÒ Ñ 1ÐÖÑÑÓÒÒ ÓÒÔÙØÓÖÑÖÐ× 

I3 I − 1, I ÖÔÖ×ÒØØÓÒ×I= ÈÓÙÖI = 

ÚÒØ×ÙØÖÙÓÑÔÓÖØÑÒØF 2ÖÓÒ×ÓÒÖØ×ÖÐ×ÖÔÖ×Ò 

↔ÒÕÙÐÖÔÖ×ÒØØÓÒÒÙ I3〉 Ù×Ý×ØÑ×ØÒÔÖÐ|I, I 

−1/2 +1/2 

2 ( 

) 

−1/2 +1/2 

2 2 

d u 

, 

= 

= 

representation 

ÖÑÑ×ÔÓ×ËÍ 

du ud 

ÐÓÖ×ÐØÓÒÙÜ×Ó×ÔÒ× 

pions 

fondamentale 

3 1 

+ 

η 

I3 −1 −1/2 0 1/2 1 

ØØ×ÖÔÖ×ÒØ×ÙÜÜØÖÑØ×ØÙÒØÖÙ×ÑÒØÓÖÖ×ÔÓÒÒØÄ× 

 

Dim

ØØ×ÖÙÔÓÒ×ÓÒØÐ×ÒØ×ÔÓÙÖ×ØØ×ÔÝ×ÕÙ×Ù×Ý×ØÑ 

ÕÙÖÒØÕÙÖÈÓÙÖI= 0ÓÒØØÖÔÖ×ÒØÐÓÖÒI3ÄÑ×ÓÒη 

×ØÐÒØÔÓÙÖØØØÔÝ×ÕÙ 

d 

U 3 

1 2 

1 

2 

1 

2 

I 3 

1 

2 

1 

2 

I 3 

Ä×ÖÔÖ×ÒØØÓÒ×ÓÒÑÒØÐ×ËÍ ×ÚÙÖ3Ø3Ò×Ð×Ý×ØÑ 

u d 

s 

V V 3 

3 

ÈÓÙÖÙÒÖÔÖ×ÒØØÓÒÓÒÒÚÓÖÔÐÙ×ÐÓÒÐ××ÖÒÔÖÐ× Ò×ËÍ ÓÒØÖÚÐÐÒ×ÙÒ×ÔÓÑØÖÕÙÑÒ×ÓÒÒÐ ÒÒ Ü×I3, U3, V3 

ÖÔÖI3ØÐØÖÒØSÓÙÙ××ÔÖI3ØÐÝÔÖÖY = B + S 

√ 

3 

F3 ≡ I3, F8 ≡ 

2 Y ÜÑÔÐ 

π + ×ÒÕÙÐÔÓÒÔÓ×Ø×ØÐÑÑÖÐÓØØ×ÕÙÖ×udsÒ×ØÓØØ 

ÇÒØÖÓÙÚÙ××ØØÜÔÖ××ÓÒÖØ×ÓÙ×ÐÓÖÑ 

+1ÆÓØÓÒ×ÕÙÓÒÒÔÙØÔ×Ð××Ö Ø 

≡ |octet qq, I = 1, I3 = +1, Y = 0〉 ÐÓÙÔÐÔÓ×ØÓÒY = 0 I3 = ÐÒØÓÒI = 

π + ×ÑÙÐØÒÑÒØ ÄÑÒ×ÓÒÐ×Ô×ØÐÙÒÓÑÖÒÖØÙÖ×ÕÙÔÙÚÒØØÖÓÒÐ×× 

 

 

 

 

≡ 

8, 

Y = 0, 

3, I3 = +1 

 

 

SU(3) SU(2) ÇÒÙØÐ×ÐÒÓØØÓÒËÍ×ÚÙÖÔÓÙÖÖÔÔÐÖÕÙÓÒ××ØÙÒ×Ð×ÔI3 − Y 

1 2 

F8)Ú ÒÓÑÖ×(I, I3, Y )ÓÙ(I, F3, 

3 

1 2 

1 2 

u 

1ÖÐÝÙÒÒÖ×Ò 

U 3 

1 2 

1 2 

3 

s 

1 2 

1 2

Ä×ÖÔÖ×ÒØØÓÒ×ÓÒÑÒØÐ×ËÍ×ÚÙÖ3Ø3ÔÙÚÒØ×ÜÔÖÑÖÖ ÇÒÔÙØÚÖÖÖÔÕÙÑÒØÕÙ 

V3ÙÖ ×ÓØÒ×Ð ÔÕÙÑÒØ×ÓØÒ×Ð×Ý×ØÑÜ×I3, U3, ×Ý×ØÑÜ×I3, Y 

d ( , , 0) 

3 

= B + SÙÖ 

u ( , 0 , ) 

1 1 

1 1 

2 2 

2 2 

Y = B + S 

I 3 

0 

1 1 

I 3 

Ä×ÖÔÖ×ÒØØÓÒ×ÓÒÑÒØÐ×ËÍ 

1 

1 

2 2 

2 2 

1 

3 

1 1 

ÒØÖÔÖÒØ×× ×ÚÙÖÒ×Ð×Ý×ØÑ 

1 1 

u ( 2, 

0, 

2) 

d ( 2 , , 2 0) 

2 

1 1 

s (0, 

, 

3 

2 2) 

U3)ÓÙÐØÐÖÐ×ÓÔÖØÙÖ× 

Ü×I3, Y = B + SÐ×ÚÐÙÖ×I3, U3, 

I3 = −(V3 + 

 

V3 + U3 = − 1 

2 F3 

 

3 

− 

4 F8 

 

+ − 1 

2 F3 

 

3 

+ 

4 F8 

V±ÓÒØÐØÓÒ×ÙÚÒØ 

 

= −F3 ≡ −I3 Ä×ÓÔÖØÙÖ×I±, U±, 

I3 + 1ØY → Y 

I3 − 1 2ØY → Y + 1 

I3 − 1 

×ØÙØÓÒÔÖÖÔÔÓÖØÐÐÚÓÕÙ×ÓÙ×ËÍ ÙÒØÖÔÐØÒØÔÖØÙÐ×ØÐÐ×ÓÖØÕÙÐ×ØÖÒ×ÓÖÑÓÑÑÐØÖÔÐØ× ÔÖØÙÐ×ËÐÓÒÓ×ØÖÖÐ×ØÖÔÐØ×ÓÖÖ×ÔÓÒÒØ×ÔÖ ÒÕÙÓÒÖÒÐ×ÖÔÖ×ÒØØÓÒ×3Ø3ÐÙØÖÐÚÖÐÖÒ ÁÐÒ×ØÔ×ÔÓ××ÐÖÖÒÖ 

2ØY → Y − 1 

Ø 

⎛ ⎞ 

⎛ ⎞ 

u 

u 

ψ ⎝ d ⎠ 

 

ϕ = ⎝ d ⎠ 

s 

s 

→ I+ØI3 → U+ØI3 V+ØI3 → 

2 

3 

1 

3 

3 

s (0, 

V3ÓÖÖ×ÔÓÒÒØ××ÓÒØÓÒÒ× 

1 

2 

, 

1 

2 

)

ÐÒ×ÑÐ×ÒÖØÙÖ×λiÙØÐ×ÖÔÓÙÖÐØÖÔÐØϕ×ÓØÒØÔÖ 

λi = Wλ ∗ W −1 ÓW = 

⎠ 

U 3 

u u 

1 2 

1 2 

su 

u 

⎛ 

⎝ 

du 

0 1 0 

−1 0 0 

0 0 −1 

V 

ÒØÕÙÖ×ØÕÙÖ× 

3 ÈÖÓÖÔÕÙÓÒ×ØÖÙØÓÒØØ×qq 

Ð×ÓÑÒ×ÓÒ× ÔÓÙÖÓÖÑÖÐ×ØØ×Ñ×ÓÒ×ÓÒ×ÖÓÒ×ÔÖÜÚÓÖ dÕÙÓÒÒ ÖÔÕÙÑÒØÓÒÔÙØØÙÖÐÔÖÓÙØÖØ3 ⊗ 

1/2, U3 = 

0, V3 = −1/2ÒØÓÙÖÓÒ×ÔÓÒØÙØÖÒÐ×ÒØÕÙÖ×u, s, 

I3 U3 V3 

uu = (−1 1 + 2 2 , 0 + 0 , 1 1 − ) = ( 0 , 0 , 0 ) 

2 2 

su = ( 0 + 1 

2 , 1 

1 1 

+ 0 , −1 − ) = ( 2 2 2 2 , 1 , −1 ) 2 

1 1 

1 

du = ( + , −1 + 0 , 0 − ) = ( 1 , −1 , −1 

2 2 2 2 2 2 ) ÓÒØÐ××ÓÑÑØ×ØÐÒØÖ×ÓÒØÓÙÔ×ÔÖÐ×ÖÒØ×ÓÑÒ×ÓÒ×ØØ× ØÇÒÔÙØÙ××ÖÐÔÖÓÙØÖØ3⊗3×ØÖÓÑÒÖÙÜÕÙÖ×ÐÒ×ÑÐ 

ÄÓÔÖØÓÒØÒØÖÔØÒÔÖØÒØ×ÕÙÖ×dØsÓÒÓØÒØÙÒÜÓÒ 

ÓÒÒÖÙÙÒØØÔÝ×ÕÙÑÒØÓ×ÖÚÐ 

ÎÓÖÔÖÜË×ÓÖÓÛÞÐÑÒØÖÝÔÖØÐÔÝ××ÂÏÐÝÒËÓÒ×Ô 

ÓÑÒÖ 

3 ÙÖÐÔÓÒØÓ×ØÖÓÙÚÐÕÙÖu = ×ØÖÒI3 

u 

I 3 

s 

⎞ 

d

ud 

U 3 

sd 

Ò×ÑÐ×ÖÔÖ×ÒØØÓÒ×ÒÓÒØÓØÒÙ×ÔÖÐÔÖÓÙØÖØ 

us ds 

V3 ÔÓ××Ð×ÚÓÖÙÖÄÔÓ×ØÓÒÒØÖÐÓÑÔÓÖØÙÒÒÖ×Ò 

3 ⊗ 3 

V±×ÒØØÖÒ×ØÖÙÒ×ÓÙ×ØØÐÙØÖÐÐÓÒÐÜÓÖÖ×ÔÓÒÒØÔÖ ØÒ×ÑÐÓÒ×Ò×ÔÖÐÔÖÓÙÖÙØÐ×Ò×Ð×ËÍ ssÈÓÙÖÖÖÒÖ ÇÒÔÖØ ØØ×ÔÙ×ÕÙÓÒØÖÓÙÚÒÔÓÒØÐ×ÓÑÒ×ÓÒ×uu, 

η 1〉Ò×ËÍ 0ÐÔÓÒÒÙØÖØÙÒÒØ 

dd, 

ÜÑÔÐI+|I, I3 = −I〉 ⇒ |I, I3 = −I + ×ÓÙ××ÔY = 0ÑÒÒÖÙÒØÖI = 

π 

η ≡ |0, 0〉 = 1 

ØÒ×ËÍ 0〉Ò×ËÍ ÇÒÚÓÒÐÖÖ 

1ÓÒÒÙÒ×ÓØÖÔÐØÔÓÒØÙÒ×Ó×ÒÙÐØ 

8⊕1ÓÒÒÙÒÓØØØÙÒ×ÒÙÐØÄ 

√ (dd + uu) 

2 ÄÓÑÔÓ×ØÓÒ2 ⊗ 2 = 3 ⊕ ÐÓÑÔÓ×ØÓÒ3⊗3 = 

|0, 

 

 

η1 ≡ |×ÒÙÐØ〉 ≡ 1 , 1 ≡ 1 ÇÒÖÚÒÖÙÜÓÖÑ×η1Øη8Ò×Ð§ÖÐØÓÒ×Ø 

 

√ (uu + dd + ss) 

3 

su 

du 

uu, 

dd, ss 

ÐØÓÒI+Ò×Ð U±ÓÙ ÙÒØØÜØÖÑÒÓÒÑÙØÓÒÔÔÐÕÙÐÙÒ×ÓÔÖØÙÖ×I±, 0 ≡ |1, 0〉 = 1 

√ (dd − uu) 

2 

dØsÓÑÑÐØØ ×ÒÙÐØÓØØÖÓÑÔÐØÑÒØ×ÝÑØÖÕÙÔÖÒu, 

I 3

↑ØËÍ ×Ó×ÔÒ 

×Ó×ÔÒ ÄØØÖÔÖ×ÒØÔÖØÔÖØÐÓØØËÍ ×ÚÙÖØÙØÖÔÐØ 

ÑÒ×ÓÒËÍ↑ 3 ÈÓÙÖØØØI±|1, 1〉 

ÐÓØØËÍ ÇÒÔÙØÓÒ×ØÖÙÖÙÒÓÑÒ×ÓÒÓÖØÓÓÒÐØ×ÒØÔÖØ ×ÚÙÖØÙ×ÒÙÐØ×Ó×ÔÒ 

π 

η8 = |8, Y = 0,1, I3 = 0〉 = 1 

8⊕1×ØÖÔÖ×ÒØÖÔÕÙÑÒØÒ×ÐÙÖ ÄÓÑÔÓ×ØÓÒ3⊗3 = 

= V±|1, 1〉 = U±|1, 1〉 = 0 

0 = |8, Y = 0,3, I3 = 0〉 = 1 

 

√ (dd − uu) 

 

2 

√ (dd + uu − 2ss) 

6 

3 3 

8 

1 

×ÖÓÙÔ× ÇÒÔÙØÑÓÒØÖÖÕÙØÓÙØ×Ð×ÖÔÖ×ÒØØÓÒ×ËÍ×ÚÙÖÓÖÖ×ÔÓÒÒØ ÁÐ×ØÓÑÑÓÒÚÙÐÒÖÐ×ØÓÒÖÓÙÖÖÙÐÒÐØÓÖ ÓÑÔÓ×ØÓÒ3 ⊗ 3 = 8 ⊕ 1 

ÜÓÒ×ØÔÐÙ×ÒÖÐÑÒØÐ×ÓÖÑ× )ÚÔÖÑØÖÓÒÚÜØÖÒÐ× ××ØÖÙØÙÖ×ÓÑØÖÕÙ×Ò×ÐÔÐÒ(I3, Y 

✑ 

❚ 

❚ 

❚ 

❚ 

❚ 

❚ 

✑✑✑✑ 

◗ Õ 

◗ 

◗◗◗✔ 

✔ 

✔ 

✔ 

✔ 

✔ Ô 

8,1> 

8,3> 1,1>

ÔÓÙÖØØÒÖÙÒÜØÖÑÙÓÑÒÓÒÚÜÔÖØÖÙÒÙØÖÜØÖÑ ÕÙÒØÐÒÓÑÖÔ×ÕÙÐÙØÖ×ÐÓÒÐ×Ü×V3Î×ÔÒØI3Á×ÔÒ ÇÒÔÙØÓÒ Ð××ÖÙÒÖÔÖ×ÒØØÓÒÔÖÙÜÒÓÑÖ×ÔØÕÕÙÒ 

Á ÕI+φmax = 0 

· 

· ←− · 

ÌËÍ 

· ←− · ←− · 

· ←− · ←− · ←− · 

φmax ր 

ÒÔÖØÒØÐØØÑÜÑÐÓÒÔÙØÓÒÖ2I×ÙØ×ÚÐÓÔÖØÙÖÐÐ Ò×ËÍ ×ÙÐÐÜI3×ØÓÒÖÒÒ×ÐØÐÙÓÒÓÒÒÐ× = ÕÒÓÒØÓÒÁÕÙÒÓÒÔÖØφmaxÓI3 Imax = I 

= ÚÐÙÖ×ÕÒÓÒØÓÒ×ÚÐÙÖ×ÁÕÙÒÓÒÔÖØφmaxÓI3 Imax = I ÚÒØ×ØÖÓÙÚÖÒ×ÐÚÐ2I + 1Ñ×ÙØÓÒÒ(I−) 2I+1 ÐÔÖÓÔÖØ Ò×ËÍ ÐÑÑÔÖÓÙÖÔÙØØÖÔÔÐÕÙ×ÙÕÙÐÙØÖ×ÓÒÒÖ 

U3ØV3ÙÔÐÒÚÓÖÙÖÄØØÑÜÑÐ×ØÐÙÕÙ 

φmax = 0 ×ÐÓÒÐ×Ü×I3, I+φmax = U+φmax = V+φmax = 0 ÖÙÐÓÒÙÒÔ×ØÓÒØÔ×ÙØ×(V−) p Õ φmaxÔÖØÖÐ×ÐÓÒÔÔÐÕÙ Ó×ÐÓÔÖØÙÖI−ÓÒ×ØÒÓÙÚÙÒ×ÐÚ(I−) q+1 (V−) p ÄØØφmax×ØÒÔÖ 

φmax = 0. 

I3 = 1 

(p + q) = Imax 

2 3 

Y = 1 Ò×ËÍ 

⎫ 

⎬ 

max ⎭ 

(p − q) = Y 3 ÐÚÐÙÖÔÖÓÔÖI 2×Ø 

〈I 2 〉 = I(I + 1) = I 2 + I = (I max 

3 ) 2 + I max 

3 ÓI max 

3 ×ØÐÚÐÙÖÔÖÓÔÖI3ÐØØφmax 

ËÐÓÒÔÔÐÕÙφmaxÔ Ó×ÐÓÔÖØÙÖV−ÓÒ×ØÖÓÙÚÒ×ÐÚÇÒ

Ð××ØÓÒÒ×ËÍ 

ÑÒ×ÓÒ ÔÕ 〈F 

2 〉 

 

1 

3 

3 

8 

6 

10 

Ò×ËÍ ÓÒÔÙØÑÓÒØÖÖÕÙ 

〈F 2 〉 = (I max 

3 ) 2 + 2I max 

3 + 3 

ÌÎÐÙÖ×ÔÖÓÔÖ×F 2ÒÓÒØÓÒ(p, q) 

4 (Y max ) 2 ÒÙØÐ×ÒØ ÓÒÓØÒØ 

〈F 2 〉 = 1 

3 (p2 + pq + q 2 

) + p + q ÄØÐÙÓÒÒÐ×ÚÐÙÖ×ÔÖÓÔÖ×F 2ÒÓÒØÓÒ×ÚÐÙÖ×(p, ÇÒÚÓØÕÙÐÐ×Ò×ÔÒØÔ×ÓÑÔÐØÑÒØÐÖÔÖ×ÒØØÓÒÓÑÑØØ q) Ð×ÔÓÙÖI 2ÔÖÜF 2 (3) = F 2 

(3) = 4/3

Ò×ÕÙÓÒÐÖÔÔÐÒ×ÐÒØÖÓÙØÓÒ§ ÄÑÓÐ×ØØÕÙ×ÕÙÖ× ÐÜ×ØÙÒÓÒÒÓÖÖ× ÔÓÒÒÒØÖÐÐ××ØÓÒÒJ P×ÖÓÒ×Ó×ÖÚ×ØÐ×ÖÔÖ×ÒØØÓÒ× ÌÓÙØÓ×Ð×ÝÑØÖÙÒØÖÒ×ØÕÙÖÓ××ÖÑÒØÖ×ÔØÔÜÐ×Ñ××× ÖÓÒ×ÔÔÖØÒÒØÙÒÑÙÐØÔÐØËÍ ÖÖÙØÐ×ÙÖÓÙÔËÍ ×ÚÙÖÓÒÒÔÖ×ÒØÒØÒÓØÐ× 

ÖÕÙÐÑ××Ù×Ý×ØÑÖÓØÕÙÒÐØØÜØØÓÒÖÐÒ ÕÙÐ×ÔÖÓÔÖØ×ÙÖÓÒ×ÓÒØÓÒØÓÒ×ÙÓÒØÒÙÒÕÙÖ×ÚÐÒÐ ÖÒ×ÒØÖÐÐ×Ò×ÓÒ×ØÑÒÓÑÔÐØÖÐÑÓÐÒÑØØÒØ ÓÒÙÖØÓÒ×ÔØÐØÐØØ×ÔÒÖÐØ×ÖÒÖ×ÈÖÜÓÒÓÒ× 

×ÚÙÖÔÓÙÖÐ×ØØ×Ð×qqØqqqÓÙqqq 

ÐØØÜØØÓÒÓÖØÐÄ ÙÑÒØÒØ Ø 

q(q I I3 S B Q Y = 2(Q − I3) 

u(u) ±1/2 ±1/3 ±2/3 ±1/3 

ÕÙÖÒØÕÙÖØÖÒ ÌÆÓÑÖ×ÕÙÒØÕÙ×Ø××ÕÙÖ×ÒØÕÙÖ×ÐÖ×ØÙ 

d(d) ∓1/2 ±1/3 ∓1/3 ±1/3 

s(s) ∓1 ±1/3 ∓1/3 ∓2/3 

Ñ×ÓÒÕÙ×ØÖÝÓÒÕÙ×ÓÐÒØÖÔÖØØÓÒÒ×ÐÖ×ÖÔÖ×ÒØØÓÒ× ÈÓÙÖ×ÑÔÐÖÐÜÔÓ×ÒÓÙ×ÒÖ×ØÓÒ×ÙÜÕÙÖ×ÒØÕÙÖ××ØÖÓ× 

ËÍ ÔÐÙ×ÒÓÙ×ÒÓÙ×ÓÒÒØÖÓÒ××ÙÖÖØÒ××ØÙØÓÒ×ÔÖØÙÐÖ×ØØ× 

Ä××Ý×ØÑ×Ð×qqÑ×ÓÒ× ×ÚÙÖÒ×ØÔ×ØÖÚÐ ×ÚÙÖ×u(u), d(d) 

×ÚÙÖ×ÔÙØ××ØÙÖÒ×ÙÒ×ÒÙÐØÓÙÒ×ÙÒÓØØ×ÝÑØÖÙÒØÖ ÕÙ×ÜÔÖÑÔÖÐÓÑÔÓ×ØÓÒ Ä×ÝÑØÖËÍ×ÚÙÖÔÖØÕÙÐ×Ý×ØÑÐqqÓq×ØÐÙÒ×ØÖÓ× 

ÈÔØÖ ÄÜØÒ×ÓÒÙÒÕÙØÖÑ×ÚÙÖÕÙÖ×ØÔÖ×ÒØÙ§ ÉÙÖÑÓÐ ÎÓÖÙ××ÐØÐ 

 

3 ⊗ 3 = 1 ⊕ 8 

s(s)ÓÒÒÖÔÔÐÐÐ×ÒÓÑÖ×ÕÙÒØÕÙ×Ò×ÐØÐÙ

ØØÔÖØÓÒ×ØÚÖÔÖÐØÙ×Ñ×ÓÒ×ØÐÙÖ×ØØ×ÜØ×ÈÖ 

0ÓÙ ÄÑÓÐÒÔÖÚÓØÔ× ÈÓÙÖÐØØ×ÒÙÐØÙ×Ý×ØÑqqÓÒI = ÐÓØØÓÒI = 

ÉÒ×ÕÙÐÐ×ÚÓ××ÒØÖØÓÒÙØÐÖÖÖÚÒØÙÐ×Ñ×ÓÒ× ØÓÙÓÙÖ×ØÒÖÙØÙÙ× 

2××ÒØÖÒØÒ×ÐÚÓ× ØØ×Ó×ÔÒI = ØØÖÒØÝÔÖÖS(Y ) ÜÑÔÐÐÖÖÒØ×Ð×Ó×ÔÒI = 2××ÒØÖÒØÒ×Ð×ÚÓ×π 

ÒØÔÖÐÐÐ×↑↓ÓÙÔÖÐÐÐ×↑↑×ÔÒÓØØÖÓÑÒÚÐÑÓÑÒØ 

2Ð×Ý×ØÑqqÔÙØÓÒ× 

Lz×ÑÑÖ×ÐÔÖqqÔÓÙÖÓØÒÖÐÑÓÑÒØÒÙÐÖ 

ÄÕÙÖÐÒØÕÙÖ×ØÙÒÖÑÓÒ×ÔÒ1 ØÖÓÙÚÖÒ×ÐØØ×ÒÙÐØ×ÔÒS= 0, ×ÔÒS = 

ØÖÑ×ÐÙÒ×ÖÔÔÓÖØÒØÐÖÔÖ×ÒØØÓÒËÍ ×ÒÓÖÑØÓÒ××ÖØÖÓÙÚÒÓÒÒ×Ò×ÐÒÓØØÓÒ×ÔØÖÓ×ÓÔÕÙÙ×ÙÐÐ ÄÓÒØÓÒÓÒÙ×Ý×ØÑÔÙØØÖÒÓÑÑÐÔÖÓÙØÙÜ ×ÚÙÖÐÙØÖÙÜÓÒÙÖ 

ÓÖØÐÖÐØL, 

ØÓÒ××ÔÐØ×ÔÒÓÖÐÐ ÓÑÔÓ×ÒØ ËÍ×ÚÙÖ 

ÔÖÜÐÓÒØÓÒÓÒÙÒÑ×ÓÒπ +ÔÙØ×ÒÖÔÖ 

|π + ÍÒÓ×ÓÒÒ×I3ØYÐÖÐØÖÕÙ×ØÜÖÐØÓÒ 

ÖÐØÓÒ ÑÑ 

〉 = |8, Y = 0, I = 1, I3 = 1〉 |0, 0, 0, 0〉 

ØÖÒ×ØØÖÒ×ÓÒÒÙ×ÚÐ×ÒÓÑÖ×ÕÙÒØÕÙ×Ø×ØØÖÙ×ØÐ ØØ×Ñ×ÓÒÕÙ× Ò×ÐØÐÙ ÔÖ×ÖÖÔÔÐÒÓÙ×ÚÒÓÒ×ÙÔÖÓÐÑÒØÖÔÖØØÓÒÖØÒ× ÓÒÜØÖØÐØÐÈÙÒ×ÖÑ×ÓÒ×ÒÓÒ 

ÙÒÓ×ÓÒÒLÐÔÖØÙ×Ý×ØÑ×ØØÖÑÒÔÖP = 

ÓÑÒ×Ñ×××ÓÒÖÒÁÐ×ØÖÔÔÒØØÖÓÙÚÖÒ×ÐÓÐÓÒÒI = 

= ÒÖÐÑÒØÙÜØØ×ÒÙØÖ×I3 PCÓÒÒØÚ ×Ñ×××ÚÓ×Ò×ÆÓØÓÒ×ÕÙÙÒØÖÓ×ÑØØÒÙØÖ×ØÓÒØÒÙÑÔÐØÑÒØ Ò×ÐÓÐÓÒÒI = 1ÔÓÙÖÐÑÑJ PCÊÔÔÐÓÒ×ÕÙËÍ×ÚÙÖ×ÓÒ ×ÖÒÙÖ××ÓÒØÙØÐ××ÚÖØÔÖÑØØÖÐÚÖØÓÙØÑÙØÐÙÖ×ÙØ ÁÐ×ØÓÙÖÒØ×ÒÖÐØÖÒØØÐ×ÔÒÔÖÐÑÑÐØØÖSÐÓÒØÜØÒ×ÐÕÙÐ Ø 

 

ÔÖØÐÑÒØØÖÓ×ØØ×ÒÙØÖ×I3 = 

0ÓÙ 0ÔÓÙÖÐ×ØØ× 

0 S = Y = 

S = Y = −1, 3 

= ∓2, ∓3, ... 

2 

3 

K ± π ±ØÐÐÒØ×Ð×Ó×ÔÒI = ± π ± 

±2 

0ÓÙÒ×Ð×ØØ×ØÖÔÐØ× ÒØ×ÐØÖÒØS= 

JzÙ×Ý×ØÑ×ÓÒ×ÔÒÔÓÙÖÐÓ×ÖÚØÙÖÔÐÙÖÝÒØÖÄÒ×ÑÐ 

0ÓÙ×ÐÓÒÕÙÐ××ÔÒ××ÕÙÖ×ÓÒ×ØØÙÒØ××ÓÒØ 

Sz = 

1Sz = −1, 

J, 

 

2S+1LJ 

 

|Ñ×ÓÒ〉 =⇒ |qq〉 = |L, Lz, J, Jz〉 

(−1) L+1ÚÓÖ 

0ÐÑÑÖÒÙØÖÙØÖÔÐØ 

0 

Y = 0ÒØ×ÔÓÙÖÙÒJ S = Y =

2S+1LJ JPC ÖÓÒ 

du, uu, dd su, sd uu, dd, ss 

I = 1 I = 1 Ñ××ÅÎ℄ 

I = 0 2 

1S0 0−+ π K η η ′ 

3S1 1−− ρ K∗ ω 

1P1 1 +− b1 K1B 

3P0 0 ++ a0 K∗ 0 

f0(1370) 

3P1 1 ++ 

a1 K1A f1(1285) f1(1420) 

3P2 2 ++ a2 K∗ 2 f2(1270) 

1D2 2−+ π2 3D1 1−− ρ K∗ ×ØØ× ÌØØ×Ð×Ù×Ý×ØÑqqÑ×ÓÒ×ÒÓÒØÖÒ×ØØÖÒ×××Ò× 

ω(1650) 

√3 (uu+dd+ 

 

 

φ 

h1(1170) 

f0(1710) 

f ′ 2 (1525) K2 

 

ØÖÕÙÖØÙÒÜÑÒÔÐÙ×ÔÔÖÓÓÒØØÖÜÑÔÐÒÓÙ×ÓÒ×ÖÓÒ×Ð ×ØØ××ØÑÑØÐÐ×ÙÜÙØÖ×ÔÖÓÒØÖÒ×ØÔ×ÚÒØ ×Ó×ÔÒÓÒØÒÙÒÕÙÖ×1 √2 (uu+dd)Ð×ÒÙÐØËÍ×ÚÙÖ1 ss)ØÐÑÑÖÒÙØÖÐÓØØ1 √6 ×ØÙØÓÒ×Ñ×ÓÒ×Ô×ÙÓ×ÐÖ×J PC Å×ÓÒ×Ô×ÙÓ×ÐÖ× 

ÐÙÖÑ××ØÐÙÖÓÒØÒÙÒÕÙÖ× ÇÒÖÔÔÐÐÒ×ÐØÐÙÐÙÖÔÔÖØÒÒÙÜÖÔÖ×ÒØØÓÒ×ËÍ ÄÑ××Ù×Ý×ØÑqqÔÙØ×ÜÔÖÑÖ×ÓÙ×ÐÓÖÑ ×ÚÙÖ 

m = 〈qq|H0|qq〉 = M( 2S+1 ÑÓÐ 1×ØÔÖ×ÒØÒ×ÐØÐÈÔ Ó ÉÙÖ 

 

LJ) + mq + mq Ä×ØÙØÓÒ×Ñ×ÓÒ××ÔÒJ > 

J PC = 1 −−Ð×ÙÜÔÖÑÖ×ÐÒ×ÙØÐÙ 

−2ss)Ä××ÒØÓÒÙÔÖÑÖ 

(uu+dd 

= 0−+ØÐÐ×Ñ×ÓÒ×ÚØÙÖ×

H0×ØÐÓÔÖØÙÖÑ×× 

M( 2S+1 ÓÖØÐÓÒÑØÕÙØØÓÒØÖÙØÓÒ×ØÒÔÒÒØÐ×ÚÙÖÙÕÙÖ ÈÓÙÖ×ÑÔÐÖÐÖØÙÖÒ×ÕÙ×ÙØÓÒÔÓ× 

LJ)×ØÐÓÒØÖÙØÓÒÔÒÒØ×ØØ××ÔÒØÑÓÑÒØ 

Ò×ØØÒÓØØÓÒÐ×Ñ××××ØØ×ÙØÐÙ×ÖÚÒØ 

mq = mqm0ÔÓÙÖÐ×ÕÙÖ×uØd msÔÓÙÖÐÕÙÖs 

M( 1 

S0) M0 

mπ = M0 + 2m0 

mK = M0 + m0 + ms 

 

1√6 

mη8 = (uu + dd − 2ss) H0 

1 

 

√ (uu + dd − 2ss) 

6 

= M0 + 1 

⎧ 

⎪⎨ 

6 ⎪⎩ 〈uu|H0|uu〉 + 

 

=2m0 

dd ⎫ 

⎪⎬ 

 

H0 dd + 〈uu|H0 

dd + ..... + 4〈ss|H0|ss〉 

⎪⎭ 

=2m0 =0 

=8ms 

= M0 + 2 

3 m0 + 4 

3 ms ÒÓÑÒÒØØÓÒÓØÒØÐÖÐØÓÒÐÐÅÒÒÇÙÓ 

mη8 = 1 

3 (4mK 

− mπ) 

 

1√3 

mη1 = (uu + dd + ss) H0 

1 

√ (uu + dd + ss) 

3 

= M0 + 1 

3 (4m0 ÇÒÔÙØÐÑÒØÒØÖÖÙÒÖÐØÓÒÒØÖmη1ØÐ×Ñ×××Ñ×ÓÒ×Ó×Ö Ú× 

ÔÙØÒÒÖÖÐÑÐÒ×ØØ×ÔÔÖØÒÒØÙ×ÒÙÐØØÐÓØØØØ ÑÐÒÔÙØØÖÖØÐÙÒÑØÖÖÓØØÓÒÊ ÄÓÑÔÓ×ÒØ×ÒØÖØÓÒ×Ö×ÔÓÒ×ÐÐÚÓÐØÓÒ×ÝÑØÖËÍ ×ÚÙÖ 

+ 2ms) 

ÒØÖÑ×Ñ×××ÙÖÖÄÖÙÑÒØØÓÒ×ØÕÙÐ×Ó×ÓÒ××ÓÒØÖØ×ÔÖÐÕÙØÓÒ Ò×ÐØÐÈØÒ×ÐÐØØÖØÙÖÓÒØÖÓÙÚØØÖÐØÓÒØÐÐ×ÕÙ×ÙÚÒØÖØ× 

 

′ η η1 cosθps sin θps η1 

ÐÑÒØÖÝÈÖØÐ×Ð×ÐÐÈÙÐÓÑÔÔØ ÔÖØÙÐÖÔÓÙÖÐ×Ñ×ÓÒ×Ô×ÙÓ×ÐÖ×ÎÓÖ×ÙØÔÖÜÑÔÐÌÐÅÓÐ×Ó ÃÐÒÓÖÓÒÓÐÑ×××ØÔÖ×ÒØ×ÓÙ××ÓÖÑÕÙÖØÕÙ ÆÓÙ×ÒÜÔÐØÓÒ×Ô×ØØÖÐØÓÒÖÐÐÓÒÙØ×Ö×ÙÐØØ×ÔÙ×Ø××ÒØ×Ò 

= R = 

η η8 − sin θps cosθps η8

ËÍ×ÚÙÖ ÑÙÐØÔÐØ ÅÎ℄ ÅÎ℄ Î2℄ ÓÒØÒÙÒÕÙÖ× Ñ2 ØØ×Ñ××Ñ∆ÑÑÑπ 

π0 √2 1 (uu − dd) 

π + 

du 

π− ud 

{8} K + K 

su 

− K 

us 

0 sd 

K 0 

ds 

{8} η8 ≃ 612 1 

√6 (uu + dd − 2ss) 

ÑÐÒ 

{1} 

 

η1 ≃ 

 

886 

 

√1 (uu + dd + ss) 

3 

m18 ≃ 155 

η sin −η1 θps + η8 cosθps 

{8}Ø{1} 

η ′ cos η1 θps + η8 sin θps 

θps ≃ −23¦ ÌËÖÑ×ÓÒ×Ô×ÙÓ×ÐÖ×J P = 0−ÆÓØÓÒ×ÕÙ×Ð×Ñ××× ≃ 

−11¦ Óθps×ØÐÒÐÑÐÒÔÓÙÖÐ×Ñ×ÓÒ×Ô×ÙÓ×ÐÖ× 

×ÓÒØÖÑÔÐ×ÔÖÐÙÖÖÖÒ×Ð×ÖÐØÓÒ×ÓÒÓØÒØθps

ÈÖÖÔÔÓÖØÐ×η, η ′ÐÑØÖ×Ñ××××ØÓÒÐ 

η1ÔÖ×ÙØÐØÑÐÒÓÒ 

 

Ó ÈÖÓÒØÖÒ×Ð×η8, 

ÖÒØ×ÖÐØÓÒ×ÒØÖÐ×Ñ×××ØÐÒÐÑÐÒØÐÐ×ÕÙ ÔÖ×ÑÙÐØÔÐØÓÒ×ÑØÖ×ØÐ×ØÓÒØÖÑØÖÑÓÒÔÙØÓØÒÖ 

 

m1 + m8 = mη + mη ′ 

m 2 18 = m 2 81 = m1 · m8 − mη · mη ′ 

Ò×ÐÓÖÖ×ÙÚÒØm8m1 

ÑÐÒθps×ÒÙØÐ×ÖÐØÓÒ×ÓÙÐ×Ö×ÙÐØØ××ÓÒØ Ä×Ñ×××ÒÓÒÒÙ××ÓÒØØÖÑÒ×Ð×Ñ×××Ñ×ÙÖ×ÒÔÖÓÒØ Øm18ÄÒÐ 

′ − m8 

ÓÒÒ×Ò×ÐØÐÙ 

Å×ÓÒ×ÚØÙÖ× ÆÓØÓÒ×ÕÙÐ×Òθps×ØÓÒØÓÒÙ×Òm18ÆÓÙ×ÚÓÒ×Ñ×ÕÙ 

×ØÓÒ×ÒÕÙÖ×Ù×ÒÙÐØØÐÓØØ×ÓÒØÐ×ÑÑ×Ñ×ÐÓÒÙÖØÓÒ ÔÖÑØÖÐÓÖÖ×ÔÓÒÒÒØÖÐ×ÙÜ×Ö×Ñ×ÓÒ×Ä×ÓÑÔÓ ×ÔÒÓÖÐÐ×ØÖÒØÙ××ÚÓÒ×ÒÓÙ×ÙØÐ××ÖØÖ×ÖÒØ×ÔÓÙÖ ÄØÐÙ×ØÐÔÒÒØÙØÐÙØÐÓÑÔÖ×ÓÒÐÒÐÒ 

×ÒÖ×ØØ× Ä×Ñ××××Ñ×ÓÒ×ÚØÙÖ××ÓÒØ×ÙÔÖÙÖ×ÙÜÑ××××Ñ×ÓÒ×Ô×Ù Ó×ÐÖ×ÕÙÒÕÙÕÙÒ×ÐÖÐØÓÒM( 3 ËÍ×ÚÙÖ ÈÓÙÖÐÐÖÐÖØÙÖÓÒÖØm1Øm8ÔÓÙÖÐ×Ñ×××Ù×ÒÙÐØØÐÓØØ ÎÓÖØÐÈÔØÖ ÉÙÖÅÓÐ H0 

H0 

η ′ 

η 

η1 

η8 

m1 m18 

m81 m8 

 

= 

 

= 

 

mη ′ 0 

0 mη 

m1 m18 

 

= R † · 

m81 m8 

tan 2 θps = m8 − mη 

mη 

η ′ 

η 

η1 

 

mη ′ 0 

0 mη 

η8 

 

· R 

m18 

tanθps = 

mη ′ − m8 m18 

0Ò×ÙÚÒØÐÖÓÑÑÒØÓÒÐØÐÈ 

m18 < 

= m8 − mη 

S1) = M1 > M( 1 S0) =

ËÍ ÑÙÐØÔÐØ ×ÚÙÖ ÅÎ℄ ÅÎ℄ Î2℄ ÓÒØÒÙÒÕÙÖ 

 

Ñ2 ØØ×Ñ××Ñ∆ÑÑÑπ 

ρ0 1 

 

{8} 

ρ + du 

ρ − ud 

K ∗+ 

K ∗− 

K ∗0 

K ∗0 

√2 (uu − dd) 

{8} ω8 ≃ 938 √6 1 (uu + dd − 2ss) 

ÑÐÒ 

{1} 

 

ω1 

m18 

φ 

ω 

≃ 884 

≃ 

112 

√ 

1 

(uu + dd + ss) 

3 

−ω1 

ω1 

ÌËÖÑ×ÓÒ×ÚØÙÖ×J P M0Ä×Ñ×××ÒÓÒÒÙ×ØÐÒÐÑÐÒθv×ÓÒØØÖÑÒ×Ò×ÙÚÒØÐ ÖÑÔÐ×ÔÖÐÙÖÖÖÒ×Ð×ÖÐØÓÒ×ÓÒÓØÒØθv ≃ 

 

{8}Ø{1} 

su 

us 

sd 

ds 

sin θv + ω8 cosθv 

cosθv + ω8 sin θv 

θv ≃ 36¦ 

= 1−ÆÓØÓÒ×ÕÙ×Ð×Ñ××××ÓÒØ 39¦

ÑÑ×ÑÕÙÙÔÖÚÒØØÒÓÒÚÒÒØÖÐ××Ù×ØØÙØÓÒ××ÙÚÒØ× 

K0 =⇒ K∗0(896) ; π0 =⇒ ρ0 Ò× 

η ′ ÎÓÖÐ×Ö×ÙÐØØ×Ò×ÐØÐÙ Ò× Ø 

ÁÐÔÔÖØÒ×Ð×ÔÖ×ÒØÕÙÓÒ×ÔÔÖÓÐ×ØÙØÓÒÙÑÐÒ 

=⇒ ω ; η =⇒ φ 

ÐÓθÐ= 35.3¦cosθÐ= 2 

3sin θÐ= 1 

 

 

|φ〉 ≈ − 

1 

 

√ 1 

(uu + dd + ss) 

3 3 + 

 

 

1 

√ 2 

(uu + dd − 2ss) = −|ss〉 

6 3 

3Ø 

×Ý×ØÑss××ÒØÖÒÓÒ×ØÒÔÓÒ×Ò×ÐÔÖÓÔÓÖØÓÒ ÇÒÒØÖÓÙÚÙÒÓÒÖÑØÓÒÒ×ÐØÕÙÐÑ×ÓÒφ××ÒØÐÐÑÒØ 

ÄÑ××ÙφÔ××ÔÙÐ×ÙÐ×ÒØÖØÓÒÒKKÕÙÜÔÐÕÙ Êφ → 

ÓÒÒØÙÖÈÓÙÖÜÔÐÕÙÖÐÖÔÔÓÖØÑÖÒÑÒØÐ ÙÒÔÔÐØÓÒÐÖÐÇÁ ÔÖÙÒÖÑÑÓÒÒØÙÖ×ÙÖÐÔÖÓ××Ù×ÖÔÖÙÒÖÑÑ 

4.5MeVÄ×ÑÒ×Ñ×Ð×ÒØÖØÓÒÙφ×ÓÒØ ÕÙÔÖ×ÖØÐÔÖÓÑÒÒÙÔÖÓ××Ù×Ö 

Êφ → 

ρπÓÒÔÓ×ØÙÐÐÒØÖÚÒØÓÒÙÒÑÒ×ÑÓÒÙÖÖÒØÐÙØÝÔ 

ÐÐÖÙÖÓ×ÖÚΓφ ≃ 

Ò×ÓÑÒ×ÓÒ×ÖÒØ×ÖÔÔÐÓÒ×ÐÑØ×ÚÙÖ× qÖÝÓÒ×ØÒØÖÝÓÒ× 

ÚÓφ → 

ÔÖÐÓÑÔÓ×ØÓÒ 

q)ÔÙØ×ÔÖ×ÒØÖ ÕÙ×ÜÔÖÑ Ä××Ý×ØÑ×Ð×q q qØq Ä×ÝÑØÖËÍ×ÚÙÖÔÖØÕÙÐ×Ý×ØÑqqq(q q 

3 ⊗ 3 ⊗ 3 = 3 ⊗ (6 ⊕ 3) = 10 ⊕ 8 ⊕ 8 ′ Ò×ÕÙ×ÙØÓÒÑØÕÙÐ×ØÖÓ×ÕÙÖ××ÓÒØÒ×ÐØØÜØØÓÒÐÔÐÙ× 

⊕ 1 

ËÇÙÓÛÂÁÞÙÈÝ×ÄØØ ËÙÔÔÐÈÖÓÌÓÖÈÝ× 

×ÔÓ××Ðn = Ä×ØØ×ÙÙÔÐØ{10} 

 

 

 

|ω〉 ≈ − 

1 √ (uu + dd + ss) 

3 

KK) ≃ 83% 

ρπ, ρππ) ≃ 16% 

2 

φ → KK → ρπÖÑÑÐÙÖ 

1, L = 0 

3 + 

 

 

1 √ (uu + dd − 2ss) 

6 

q 

 

1 

3 

= 1 

√ 2 (uu + dd)

a) b) 

K 

s 

u 

K 

s 

ρ 

u 

d 

π + 

+ 

u 

d 

u 

ÖÑÑφ→KKÚÓÖ×ÔÖÐÖÐÇÁÖÑÑ 

ρπÚÓÖ×ÔÖÐÖÐÇÁ 

s s s s 

φ φ 

φ → 

ρ 

s 

K 

φ{ 

s 

K 

d 

π + 

+ 

u 

} d 

ÚÖØÙÐ 

ρπÓKK×ØÙÒØØÒØÖÑÖ 

} u 

Ä××ÔÒ××ÕÙÖ××ÓÒØÐÒ×Ð×Ý×ØÑ×ØÓÒÒ×ÐØØ×ÝÑØÖÕÙ 

ÖÑÑφ→KK → 

×ÔÒ|↑↑↑〉 = J = 3 ÓJZÔÙØÔÖÒÖÐ×ÚÐÙÖ×− , JZ 2 3 

ÙÒÙÖ×ÝÑØÖÕÙÔÖÖÔÔÓÖØ×ÙÜÜ×Ä×ØØ××ØÙ×ÙÜ×ÓÑÑØ× ÄÙÖ ÖÔÖ×ÒØÐ×ÖÝÓÒ×Ó×ÖÚ×ØÐÙÖÓÒØÒÙÒÕÙÖ×Ò× 

YÄ×ÒØÖÝÓÒ×ÓÖÖ×ÔÓÒÒØ××ÖÒØÖÔÖ×ÒØ×ÔÖ 

 

1 1 3 − , , 2 2 2 2 Ð×Ý×ØÑÜ×I3,

ÙØÖÒÐ×ÓÒØÑÒ×ØÑÒØ×ÝÑØÖÕÙ×ÒØÖÑÐ×ÚÙÖ×ÕÙÖ×ÔÖ ÓÒØÒÙØÐÒ×ÑÐ×ØØ×ÐÖÔÖ×ÒØØÓÒ{10}ÓÚÒØÐØÖÓÒÒ× 

ÐÙÖ ÐÙØÓÑÔÖÒÖÕÙ 

ddu =⇒×ÝÑ(ddu) = 1 ØÑÑ 

√ (ddu + dud + udd) 

3 

uds =⇒×ÝÑ(uds) = 1 

√ [(uds + usd) + (dus + dsu) + (sud + sdu)] 

6 

Y 

ddd ddu duu uuu 

1 

dds dus 

0 

uus 

I 3 

Σ* Σ* 

Σ* 

I 3 Σ (1385) 

0 

dss uss 

Ξ* Ξ* 

 

1 

Ξ (1530) 

sss Ω 

2 

Ω (1672) 

ÙÔÐØJ P = 3+ØÖÝÓÒ×ÓÖÖ×ÔÓÒÒØ× 

ÔÖØÖ×ØØ××ØÙ×ÙÜ×ÓÑÑØ×ÙØÖÒÐÚÓÖÙÖ Ä×ÓÔÖØÙÖ×I±ÔÙÚÒØØÖÙØÐ××ÔÓÙÖÓÒ×ØÖÙÖÐ×ØØ×ÒØÖÑÖ× ÔÖÜÑÔÐ 

2 

I−(uuu) ≈ I−(u)uu + uI−(u)u + uuI−(u) 

= 1 

ÑÒØ×ÝÑØÖÕÙ×ÓÒØÔÖÓ×ÖØ×ÔÖÐÔÖÒÔÜÐÙ×ÓÒÈÙÐÒ×Ð× 

√3×ØÒØÖÓÙØÔÓÙÖ××ÙÖÖÐÒÓÖÑÐ×ØÓÒÐÓÒØÓÒÓÒ 

ÔÖ×ÒØÐÒØ×ÝÑØÖÐÓÐ×ØÖ×ØÙÖÔÖÐÓÒØÓÒÙÒÓÑÔÓ×ÒØ ÊÐÚÓÒ×ÕÙ×ØØ×ÖÑÓÒ×ÓÒØÐÓÒØÓÒÓÒ×ØÓÑÔÐØ 

√ (duu + udu + uud) 

ËÍ 

3 

Ä×ÕÙÖ××ÓÒØÔÓÖØÙÖ×ÙÒÖÓÙÐÙÖ×Ù×ÔØÐÔÖÒÖØÖÓ× ÓÙÐÙÖÒØ×ÝÑØÖÕÙ 

ÄØÙÖ1 

ÚÐÙÖ×ÖÒØ× ÓÖÑÐÐÑÒØÓÒÔÙØ×ÒÖ×ØÖÓ×ÓÙÐÙÖ×ÔÖÐÙÖ× 

Y 

0 ∆ ∆ ∆ ∆ 

+ ++ 

0 + 

ÄÓÒÔØÐÖÓÙÐÙÖ×ÕÙÖ××ÖÚÐÓÔÔÙ§ 

 

∆ 

(1232)

ÓÒÙÖØÓÒ×Ù×Ý×ØÑqqqÔÖØ×ÔÖËÍ ×ÚÙÖ

ÖÝÓÒ×ÓÒÔÖ×ÖØÕÙ |ÓÑÔÓ×ÒØËÍ ÒØÐ×ÔÖÜÑÔÐÐÐ××ØÖÓ×ÓÙÐÙÖ×ÖÖÒÐÙÈÓÙÖÐ×ØØ×qqq 

×ØØ××ÓÒØÓÒ××ÒÙÐØ×ËÍ ÓÙÐÙÖ ×ØÖÒÓÐÓÖ× ÓÙÐÙÖ〉 = |ÒØ×ÝÑÖ〉 

Ò×ÙØÕÙÖÐ××Ù×ØØÙØÓÒ×ÕÙØ×Ò× ÈÓÙÖÐ×ØØ×ÒØÖÝÓÒ×ÓÒÒØÐ×Ö×ÓÙÐÙÖÓÑÔÐÑÒØÖ× 

(ÖÖ) (ÖÖ)] 

×ØØ×ØÕÙÖÑÖÓÒÖÔÒÒØÐÔÖ×ÖÔØÓÒÕÙÐ×ØØØ× ÒÓÐÓÖ×ÕÙÓÒÙØÖÖÐÙÖÓÑÔÓ×ÒØËÍ ÆÓØÓÒ×ÕÙÐ××Ý×ØÑ×Ð×qqÑ×ÓÒ×ÔÔÒØÐÓÒØÖÒØÐ ÓÙÐÙÖ×ÓÙ×ÐÓÖÑ 

ØÖÕÙÙÖ Ò×ÐØØ×ÒÙÐØÐÓÑÔÓ×ÒØËÍ ×ÚÙÖ×ØÓÑÔÐØÑÒØÒØ×ÝÑ ÄØØÙ×ÒÙÐØ{1} 

 

É ÑÓÒØÖÖÕÙÐØØ×ÒÙÐØËÍ 0×ØÒ 

ÒÓÒÒÖÐ×ÒÓÑÖ×ÕÙÒØÕÙ× ÖÖÙÒÒØÔÓÙÖ×ÒÙÐØËÍ ×ÚÙÖÒ×ÐØÐÈØ ×ÚÙÖÙ×Ý×ØÑqqqÚL = 

ÄÓÑÔÓ×ÒØËÍ ×ÚÙÖÙÒ×ÝÑØÖÑÜØÖÐØÚÑÒØÙÜÔÖ× 

Ä×ÓÔÖØÙÖ×I±ÔÖÑØØÒØØÖÒ×ØÖÙÒØØÐÙØÖÐÐÓÒÐÜ 

q2q3Øq1q3ÇÒÔÙØÓÒ×ØÖÙÖÒ×ÚÓÖÙÖ ØÖÓ×ÓØØ× ÕÙÖ×q1q2 = ÚÖÖÒÔÖÒÒØÐ×ØÖÑ×ÓÖÖ×ÔÓÒÒØ××ÓØØ×ÕÙψ13 I3ÐÓØØÈÖÜÑÔÐÔÖØÒØÐØØϕ= 1 

√ (ud − du)dÒ×ÐÓØØψ12 

2 

= 

= 1 b,ÙÑÑØØÖÕÙÓÒ×Ö×ÐØÖÕÙ××Ò×ÓÔÔÓ××ÁÐÒÝ 

√ [(ÖÖ) + + 

6 

r g, 

|ÓÑÔÓ×ÒØËÍ ÓÙÐÙÖqq〉 = 

= 1 √ 3 (rr + gg + bb) 

|ÓÑÔÓ×ÒØËÍ ×ÚÙÖqqq〉 =⇒ÒØ×ÝÑ(uds) = 

= 1 ØÖØÔÖÐ×ØØ×ØÕÙÖÑÖÓÑÔØØÒÙ×ÔÖ×ÖÔØÓÒ××ÙÖËÍ 

√ [u(sd − ds) + d(us − su) + s(du − ud)] 

6 

couleur 

Ä×ØØ××ÓØØ×{8}Ø{8 ′ ψ23Øψ13ÓÒØÙÜ×ÙÐÑÒØ×ÓÒØÒÔÒÒØ×ØÚÑÒØÓÒÔÙØ 

} 

ψ12, 

ψ12 + ψ23

ÓÒ 

ÓÒØÓÒÓÒØÓØÐÙ×Ý×ØÑ×ØÖÓ×ÕÙÖ×ÓØØÖÒØ×ÝÑØÖÕÙ ÄÓÒ×ØÖÙØÓÒÐÓÑÔÓ×ÒØ×ÔÒÓÖÐÐÒ×ØÔ×ØÖÚÐÊÔÔÐÓÒ×ÕÙÐ 

I+[(ud − du)d] = I+[ud − du]d + (ud − du)I+[d] 

= (I+[u]d + uI+[d] − I+[d]u + dI+[u])d + (ud − du)u 

= (0 + uu − uu + 0)d + (ud − du)u 

 

= (ud − du)u 

×ÝÑØÖÕÙ |ÓÑÔËÍ 

|ÓÑÔËÍ ×ÝÑØÖÕÙ ×ÝÑÑÜØ 

|q1q2q3〉 = |ÓÑÔ×ÔÐÄ〉 · |ÓÑÔ×ÔÒ〉 · 

×ÚÙÖ〉 · 

 

×ÑÙÐØÒÙ×ÔÒØÐ×ÚÙÖÄ×ØÖÙØÙÖÐÓÑÔÓ×ÒØ×ÔÒÓÖÐÐ×Ø ÒØ×ÝÑØÖÕÙ 

antisym. 

? 

 

ÓÒØÖÑÒÔÖÐ×ÝÑØÖÑÜØÐÓÑÔÓ×ÒØËÍ ØØÓÒØÖÒØÑÔÐÕÙÕÙÐÓÒØÓÒÓÒ×ÓØ×ÝÑØÖÕÙÔÖÐÒ ×ÚÙÖÁÐÐÙ×ØÖÓÒ× 

· 

ÓÙÐÙÖ〉 

ØØÓÖÖÐØÓÒÒÖ×ÓÒÒÒØÒÓÙÚÙ×ÙÖÐØØϕ= 1 

duÙÒÓÑÔÓ×ÒØÒØ×ÝÑØÖÕÙÒ×ÔÒ××ÕÙÖ× du)dÐÓØØ 

 

√ (ud − 

2 ψ12Ò×ÙÒÔÖÑÖØÑÔ×ÓÒ××ÓÐÔÖØÒØ×ÝÑØÖÕÙÒ×ÚÙÖ×ud − 

Ð××ÓØÓÒ××Ù×ÒÓÒÒØÙÒ×ØÖÑ×ÖÓØ ÒÒÒØÐÔÓ×ØÓÒ×ÙÜÕÙÖ×ÊÑÖÕÙÓÒ×ÔÐÙ×ÕÙÐÐÓÒÒ Ä××ÓØÓÒÒ×ÖÐ××ØÑÒ×ØÑÒØ×ÝÑØÖÕÙÕÙÓÒÔÙØÚÖÖ ÙÒÓÒØÖÙØÓÒÒÙÐÐÙ×ÔÒÙ×Ý×ØÑÒ×ÙÒ×ÓÒØÑÔ×ÓÒÓÑÔÐØ Ð 

(ud − du) ⊗ (↑↓ − ↓↑) = u ↑ d ↓ −u ↓ d ↑ −d ↑ u ↓ +d ↓ u ↑ 

ÐÔÓ×ØÓÒÙØÖÓ×ÑÕÙÖÔÖÖÔÔÓÖØÙÜÙÜÙØÖ×ÄÜÔÖ××ÓÒÒÐ ×ÝÑØÖ×ÐØØÓØÒÙÒØÙÒØÒ×ÙÒ×ØÖÑ×ÙÒÔÖÑÙØØÓÒ ÓÑÔÓÖØ ØÖÑ×ÕÙÔÙÚÒØØÖÔÖØÐÐÑÒØÖÖÓÙÔ×ÓÑÑ×ÙØ 

1 [−2(d ↑ d ↑ u ↓) − 2(d ↑ u ↓ d ↑) − 2(u ↓ d ↑ d ↑) 

↓Ò×ÙÒØÖÓ×ÑØÑÔ×ÓÒ ØÖÓ×ÑÕÙÖÑÙÒ×ÓÒ×ÔÒd ↑ÓÙd 

 

√18××ÙÖÐÒÓÖÑÐ×ØÓÒÐØØÄ×ÔÒÙ×Ý×ØÑ×ØØÖÑÒ ÄØÙÖ1 ÔÖÐ×ÔÒÙØÖÓ×ÑÕÙÖJ = 

√ 18 

+u ↑ d ↑ d ↓ +u ↑ d ↓ d ↑ +d ↓ u ↑ d ↑ 

+d ↑ u ↑ d ↓ +d ↑ d ↓ u ↑ +d ↓ d ↑ u ↑] 

1/2, JZ = ±1/2

ψ12ÙÖ ÔÖ××ÓÒ×ÒØÕÙ×ÑÑÖÑÑÖÔÓÙÖ×ÙÜÓØØ×ÚÖÖØØÖ ÄÑÑÑÖÔÙØØÖÑÔÖÙÒØÔÓÙÖÐ×ÙØÖ×ÑÑÖ×ÐÓØØ Ò×ÕÙÔÓÙÖÐ×ÑÑÖ×ÐÓØØψ23ÇÒÓØÒØ×Ü ÜÖÒ×Ð×ÓØØ×{8}Ø{8 ′ ÔÖ××ÓÒ×ØÐÐ×ÔÓÙÖÐÒ×ÑÐ×ÑÑÖ×ÐÓØØÖÝÓÒÕÙ×ØÖÓÙÚÒØ Ò×ÐÐØØÖØÙÖ×ÔÐ×ÆÓØÓÒ×ÕÙ×ÜÔÖ××ÓÒ××ØÖÓÙÚÒØ×ÓÙÚÒØ ÓÑÔÓ×ÒØ××ÔÒÒØÖÓÙØ×ØÐ×ÝÑØÖ×ØÓÒÕÙØØÙÄ×Ü 

}Ò× ×ÓÒÓÒÒØÙÒÓ×Ð× 

ÓÒØÖÓÙÚ ÖØ××ÓÙ×ÙÒÓÖÑÖØÓÒÐ×ÔÐÙ×ÓÑÑÓÔÓÙÖÐÜÔÐÓØØÓÒÈÖÜÑÔÐ ÖØÓÑÑ×ÙØ 

×ØØ×ØÕÙÖÑÖÑÒÔÖÖÕÙÐ×ÖÝÓÒ×ÒØÖÝÓÒ×ÒÓÒ ÒÓÒÐÙ×ÓÒÐÔÔÐØÓÒ×ÖÐ××ÝÑØÖÙÒØÖØ×ÖÐ×Ð Ó 

ØÖÒ×ØØÖÒ×Ò×ÐØØÓÒÑÒØÐ××ØÙÒØÒ×ÐÙÔÐØ×ÔÒ 

 

1/2ÙÖ 

×ÙÖÐ×ÖÐ×ÐØÓÖ×ÖÓÙÔ×ÄÓÑÒ×ÓÒ××ÝÑØÖ×ËÍ ËÍ ×ÚÙÖ⊗ËÍ ×ÔÒ 

ØËÍ ÄÔÖÓÙÖÖØÙ§ ×ÔÒÒÒÖÐ×ÝÑØÖËÍ×ÚÙÖ×ÔÒÄ×ØØ××ÐÖÔÖ×Ò ÔÙØØÖØÒÙØÓÖÑÐ×Ò×ÔÔÙÝÒØ ×ÚÙÖ 

ØÒ×ÐÓØØ×ÔÒJ = 

ØØÓÒ×ÓÒØ 

J = 3/2ÙÖ 

8.10.11 =⇒ 1 

 

↓ ↑ ↑ ↑ ↑ ↓ ↑ ↓ ↑ 

√ −2 + + 

6 u d d u d d u d d 

 

↓ ↑ ↑ 

≡ √3 [u ↓ d ↑ d ↑ + d ↑ d ↑ u ↓ + d ↑ u ↓ d ↑] 

u d d ×ÝÑØÖ1 

u ↑ = 

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 

1 

0 0 0 

⎜ 0 ⎟ ⎜ 

⎟ ⎜ 1 ⎟ ⎜ 

⎟ ⎜ 0 ⎟ ⎜ 

⎟ ⎜ 0 ⎟ 

⎜ 0 ⎟ ⎜ 

⎟ 

⎜ 0 ⎟ , u ↓ = ⎜ 0 ⎟ ⎜ 

⎟ 

⎜ 

⎟ ⎜ 0 ⎟ , d ↑ = ⎜ 1 ⎟ ⎜ 

⎟ 

⎜ 

⎟ ⎜ 0 ⎟ , d ↓ = ⎜ 0 ⎟ 

⎜ 

⎟ ⎜ 1 ⎟ , 

⎟ 

⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠ 

0 

0 0 0 

⎛ ⎞ ⎛ ⎞ 

0 

0 

⎜ 0 ⎟ ⎜ 

⎟ ⎜ 0 ⎟ 

ÈÝ××ÑÖÍÒÚÖ×ØÝÈÖ××ÌÐ ÎÓÖÔÖÜÏÅ×ÓÒØÊÈÓÐÐÖËÝÑÑØÖÝÈÖÒÔÐ×ÒÐÑÒØÖÝÈÖØÐ 

⎜ 

s ↑ = ⎜ 0 ⎟ ⎜ 

⎟ 

⎜ 0 ⎟ , s ↓ = ⎜ 0 ⎟ 

⎜ 

⎟ ⎜ 0 ⎟ 

⎝ 1 ⎠ ⎝ 0 ⎠ 

 

 

0 

1

Y 

ddu duu 

1 

1 

dds 

uus 

Σ Σ Σ 

I 

3/ 2 

1/2 

1/2 

3/ 

3 

I Σ (1195) 

0 

3 

2 

Λ (1116) 

Λ 

dss uss 

Ξ Ξ 

1 

1 

Ξ (1318) 

0 

dus 

+ 

 

0 

1 

1 

0 

0 

ÇØØJ P = 1+ØÖÝÓÒ×ÓÖÖ×ÔÓÒÒØ× 

×ÝÑØÖÄØÐÙ ØØ××ØÖÙÖÒØÖÖÒØ×ÑÙÐØÔÐØ×ÕÙ××ØÒÙÒØÔÖÐÙÖ×ÔÖÓÔÖØ× Ø×ËÍ Ò×Ð×ÙÒ×Ý×ØÑØÖÓ×ÕÙÖ×ÒØÕÙÖ×ÓÒÔÙØÓÒ×ØØÙÖ ×ÚÙÖØËÍ×ÔÒ×ÓÑÒÒØÇÒÓØÒØÐÓÑÔÓ×ØÓÒ×ÙÚÒØ ÓÑÔÖÒÖÓÑÑÒØÐ××ÝÑØÖ××ÔÖ 

2 

6 ⊗ 6 ⊗ 6 = 56S ⊕ 70M ⊕ 70 ′ ×ÝÑØÖÑÜØ MØÐ×ÓÑÔÓ×ÓÑÑ×ÙØ 

SØ 

M + 20A ÓS×ÒÓÑÔÐØÑÒØ×ÝÑØÖÕÙAÓÑÔÐØÑÒØÒØ×ÝÑØÖÕÙØM ÄÔÐØÓÑÔÐØÑÒØ×ÝÑØÖÕÙ×ØÓÖÑÐ×ÔÖÓÙØ×S ⊗ 

M ⊗ 

 

56 = 10 J = 3 

ÓÑÔÓ×ÒØÓÑÑ×ÙØ ÇÒÐÙ××ÓÐÙÔÐØØÐÓØØ×ÖÝÓÒ×ÓÒÒÙ×Ò×ÐØØÜØØÓÒ Ä ÔÐØÓÑÔÐØÑÒØÒØ×ÝÑØÖÕÙØÐÔÐØ×ÝÑØÖÑÜØ× 

 

1 

⊕ 8 

2 2 

L = 0 

 

3 1 

××ÝÑØÖ××ÓÙ×ÒØ×ÓÔÔÓ××Ò×Ð×ÔÖ×ÕÙÖ×ÓÒ×ØØÙÒØ×× 

20 = 1 ⊕ 8 

2 2 

ÖÒØ×ÑÙÐØÔÐØ×ØÖÓÙÚÒØ×ØØÓÒ×Ò×Ð×Ö×ÓÒÒ×ÖÝÓÒÕÙ× Ä′ÔÐØÐÑÑ×ØÖÙØÙÖÕÙÐÔÐØØÒ×Ò×ØÒÙÕÙÔÖ 

 

1 3 1 1 

70 = 10 ⊕ 8 ⊕ 8 ⊕ 1 

2 2 2 2 

 

...ÇÒ××ØÙÐÓÖ×Ò×ÐÖ ÓÖÖ×ÔÓÒÒØÙÜØØ×ÜØØÓÒL = 1, 2, 

n 

Y 

p 

N 

(939)

ÓÒØÓÒÓÒ O(3)ÓO(3)ÓÒÖÒÐÔÖØ×ÔÐÐ 

ËÝÑØÖËÍ×ÚÙÖ 

×ÖÔÖ×ÒØØÓÒ×ËÍ⊗ 

ËÝÑØÖËÍ×ÔÒ S 

Ì ×ÝÑØÖ×ËÍ ËÝÑØÖ×ËÍ×ÚÙÖ×ÔÒÙ×Ý×ØÑ ×ÚÙÖØËÍ ×ÔÒÆÓØÓÒ×ÕÙÐÒÜ×ØÔ×ØØÓÑÔÐØÑÒØ ÕÙÖ×ÒÓÒØÓÒ× 

 

ÒØ×ÝÑØÖÕÙ×ÔÒÔÓÙÖÐ×Ý×ØÑÕÙÖ×Ò×ÐØØÓÒÑÒØÐL 

ÔÔÐØÓÒ×ÒÙÑÖÕÙ×ÙÑÓÐ×ØØÕÙ×ÕÙÖ× 

= 

ÖÓÑÓÑÒØ×ÑÒ×Ð×Ñ×ÓÒ×Ô×ÙÓ×ÐÖ×(J P ØÐ×Ñ×ÓÒ×ÚØÙÖ×(J P ÇÒØÙÒÖØÓÙÖÙ§ØÔÐÙ××ÔÕÙÑÒØÙØÖÑM( 2S+1 ÐÜÔÖ××ÓÒÐÑ××Ù×Ý×ØÑqqÖÐØÓÒÒ×Ð××Ñ×ÓÒ× LJ)Ò× 

ÙÜÖ×ÓÙÐÙÖ ÐÓÒÙÖØÓÒ××ÔÒ×ÙÕÙÖØÐÒØÕÙÖÒ×ÉÐ×ØÖÔÖ×ÒØ ÓÑÑÐÖ×ÙÐØØÐÒØÖØÓÒÖÓÑÓÑÒØÕÙÒØÖÐ×ÑÓÑÒØ×××Ó× ØÖÑ×ÖÙØÐØ 

 

Ô×ÙÓ×ÐÖ×ØÚØÙÖ×(L = 

ÍÒÖÐØÓÒÒÐÓÙÖØÐÒØÖØÓÒÐØÖÓÑÒØÕÙÒØÖÐÑÓÑÒØÑÒØÕÙ 

Ó: mq · mq 

×ØÖÙØÙÖÒÎÓÖÙÒÐÚÖÔÝ×ÕÙØÓÑÕÙÙÔØÖ×ÙÖÐ×ØÖÙØÙÖÝÔÖÒÙ ×ÔØÖÓÔØÕÙ αe×ØÐÓÒ×ØÒØ 

 

ÐÐØÖÓÒØÐÙÙÔÖÓØÓÒÒ×ÐØÓÑÝÖÓÒÒ××α = 

0 

M A 

S(J = 3) 

S M A 

2 

M(J = 1 

) M S, M, A M 

2 

= 1 − ) 

0, S = J = 0Ø 

∆EÖÓÑÓ= 8πα 

| ψ(0) | 

3 

2 < sq · sq > 

α = αqq = 4 

3αs =ÓÒ×ØÒØÓÙÔÐÓÖØ 

, αs 

ψ(rq, = rq)ÓÒØÓÒÓÒÙ×Ý×ØÑqqÔÖ×ÐÓÖÒ(rq rq = 0) 

= 0 − )

mq(mq)Øsq(sq)Ñ××Ø×ÔÒÐÒØÕÙÖÕÙÖ 

Ñ×ÓÒ JP Ñ××Ñ×ÙÖÑ×××ØÚ×∆EÖÓÑÓ ×ÓÒ×ØØÙÒØ× 

, I [Î] 

π 0− , 1 ≃ 2m0 − 3 Kqq 

4 m2 0 

K(K) 0− ≃ , 1/2 m0 + ms −3 Kqq 

4 m0ms 

ρ 1− ≃ 1 Kqq 

, 1 2m0 

4 m2 0 

K∗ (K∗ ) 1− ≃ 

×ÐÓÒÐÑÓÐ×ØØÕÙ×ÕÙÖ× Ì ÓÒØÖÙØÓÒ×ÐÑ××Ñ×ÓÒ×Ô×ÙÓ×ÐÖ×ØÚØÙÖ× 

1 Kqq 

, 1/2 m0 + 

ÄÓÙÔÐ×ÔÒÑÓÝÒÚÙØ 

ms 4 m0ms 

< sq · sq > = 1 

 

J(J + 1) − 

2 

1 

 

1 

+ 1 − 

2 2 1 

 

1 

+ 1 

2 2 

⎧ 

⎨ − 

= 

⎩ 

3 

ÐÒØÖØÓÒÖÓÑÓÑÒØÕÙÒ×Ð××ÓØÖÔÐØ××Ñ×ÓÒ×πØρØÐ× ÄØÐÙ ÓÒÒÐ×ÓÒØÖÙØÓÒ××Ñ×××ØÚ×ÓÒ×ØØÙØÚ×Ø 

4ÔÓÙÖJ = 0 

1 ÔÓÙÖJ = 1 

4 

ÒÓÑÔÓ×ÒØÐ×Ñ×××Ò×ÐÓØØÚØÙÖØÐÓØØÔ×ÙÓ×ÐÖ 

mρ(770) − mπ(140) =⇒ Kqq 

m2 

≃ 0.630GeV 

0 

mK∗(890) − mK(490) =⇒ Kqq Ó 

 

 

≃ 0.400GeV 

m0 · ms 

ms 

≃ 1.6 

m0 

 

×ÓÓÙÐØ××Ñ×ÓÒ×KØK ∗ÇÒÖÔÖ×ÐÒÓØØÓÒÙ§ÔÐÙ×ÔÓÙÖ ÐÐÖÐÖØÙÖÓÒÔÓ×32π 9 αs | ψ(0) | 2 = Kqq

ÒÓÑÔÖÒØÐ×Ñ×××ÐÒØÖÙÖ×ÓØØ× 

mK(490) − mπ(140) = ms − m0 − 3 

 

Kqq 

− 

4 m0ms 

Kqq 

m2 

0 

≃ 0.350 GeV 

m ∗ K (890) − mρ(770) = ms − m0 + 1 

 

Kqq 

− 

4 m0ms 

Kqq 

m2 ÓÒÓÑÒÒØ Ø 

 

0 

≃ 0.120 GeV 

ms − m0 ≃ 0.180 GeV 

Ñ×ÓÒ×ÐÐ×ÒÓÚÒØÔ×ØÖÓÒÓÒÙ×ÚÐ×Ñ×××ÓÙÖÒØ×ÒØÖÒØ Ò×ÐÓÒ×ØÖÙØÓÒÙÄÖÒÒÉ ÆÓØÓÒ×ÕÙÐ×ØÑ×××ØÚ× ÓÒ×ØØÙØÚ××ÕÙÖ×Ò×Ð× 

m0 ≃ 0.300 GeV (ÕÙÖ×u, d) 

 

ms ≃ 0.480 GeV (ÕÙÖs) 

ÖÓÑÓÑÒØ×ÑÒ×Ð×ÖÝÓÒ×ÙÙÔÐØ(J P = 3 )Ø + 

2 ÐÓØØ(J P = 1 

ØÒÖÐ×ÓÑÑ q)ÓÒØÐÙØØÖÑÒÖÐ×ÓÒØÖÙØÓÒ×ÐÒØÖØÓÒÖÓÑÓÑÒØÕÙ 

q2×ØÖÔÖ×ÒØÔÖÙÒÜÔÖ××ÓÒÙØÝÔ 

+ 

) 2 Ò×××Ý×ØÑ×ØÖÓ×ÕÙÖ×ÒØÕÙÖ×ÓÒÔÙØÓÖÑÖØÖÓ×ÔÖ×q 

q (qÄÓÒØÖÙØÓÒÙÒÔÖq1 = Ò×ÐÕÙÐÐα=αqq 2 

3αs, ÄØÐÙ ÑÓÒØÖÐ×ÓÒØÖÙØÓÒ××Ñ×××ØÚ×ØÐÒØÖ 

=ÓÒ×ØÒØÓÙÔÐÓÖØ 

αs ØÓÒÖÓÑÓÑÒØÕÙØÓØÐÒ×ÐÙÔÐØJ P = 3 

= 2 

1+ 

2 ÇÒÔÓ×16π 9 αs | ψ(0) | 2 Ò×Ð×ÙÙÔÐØÐÚÐÙØÓÒ×ÓÙÔÐ××ÔÒ×Ø×ÑÔÐÖ 

= Kqq 

ØØÒØÓÒÔÖØÙÐÖÖÐÖ×ÙÐØØÖÙÒÑÙÐØÔÐØ×Ó×ÔÒÐÙØÖ ÔÓÙÖÓØÒÖÐ∆EÖÓÑÓÓÖÖ×ÔÓÒÒØ ÒÕÙÓÒÖÒÐÓØØÐÚÐÙØÓÒ×ÓÙÔÐ××ÔÒÖÕÙÖØÙÒ 

4ØÐ×ÙØÔÓÒÖÖÔÖÐÒÚÖ××Ñ×××ÓÒÖÒ× 

2ÒØÖαqqØαqq×ÖÙ×ØÙ§ 

1 

sqi ·sqj >= 

 

ÄØÙÖ1 

 

+ØÐÓØØJ P ÔÙ×ÕÙÐ×ØÖÓ××ÔÒ××ÓÒØÐÒ×ÈÓÙÖÙÒ×ÔÖ×ÕÙÖ×ÓÒ

ÖÝÓÒ ×ÓÒ×ØØÙÒØ× ∆EÖÓÑÓ 

J 

 

[Î] 

Ω Ξ 3 τ 

N 

3 

3 

1 

Λ 

+ 

×ÐÓÒÐÑÓÐ×ØØÕÙ×ÕÙÖ× Ì ÓÒØÖÙØÓÒ×ÐÑ×××ÖÝÓÒ×ÙÙÔÐØØÐÓØØ 

τ , 1 ≃ 

Ξ 

ÕÙ×ÖÔÖÙØ×ÙÖÐ∆EÖÓÑÓÓÖÖ×ÔÓÒÒØØØÖÜÑÔÐÒÓÙ×ÜÑÒÓÒ× ÓÖÑ×ÙÜÕÙÖ×ÐÖ× 

×ÖÑÒØÒ×ÐØØ×ÝÑØÖÕÙ×Ó×ÔÒIÔÖ=1ØÒ×ÐØØ×ÝÑØÖÕÙ 

1)ÐÔÖÓÖÑ×ÙÜÕÙÖ×ÐÖ××ØÒ× 

4ÙØÖÔÖØ 

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∆ 

P , IÑ××Ñ×ÙÖÑ×××ØÚ× 

3+ 

3 , ≃ 3m0 

2 2 

+ 

, 1 ≃ 2 

2m0 + ms 

m0 + 2ms 

3ms 

3m0 

+ 1 , 2 2 ≃ ≃ + 

, 0 2 

+ 1 , 2 2 

1+ 

, 0 2 

1 

2 

1+ 

1 , 2 

≃ ≃ 2m0 + ms 

m0 

×ÔÒÓ < 

 

i= 1 

< s1 · ss > + < s2 · ss >= −3 4 

s1 · s2 >= 1 

= 0 

m0ms 

 

JΣ(JΣ + 1) − 3 

2 

1 

 

1 

+ 1 = − 

2 2 3 

4 

− 1 

4 

= −1 

Kqq 

m 2 0 

+ 1 

4 

+ 1 

4 

Kqq 

m 2 s 

Kqq 

m 2 0 

Kqq 

m 2 0 

+ 1 

4 

+ 1 

4 

Kqq 

m 2 0 

Kqq 

m 2 s 

Kqq 

m2 0 

Kqq 

m2 s

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ÒÐ×ÒÑÓÑÒØÓÖØÐL = 

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µ 

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Kqq 

m0ms 

+ 1 

4 

Kqq 

m2 0 

< s1 · s2 >= −3 4 < s1 · ss > + < s2 · ss >= −3 4 

Ω(JP = 3 

2 

µqi ≃ Qqi 

e 

e 

m0 ≃ 0.360 GeV ; ms ≃ 0.540 GeV 

Kqq 

m2 0 

ms/m ≃ 1.5 

≃ 0.200 GeV ; Kqq 

m0ms 

P 

3 + 4 = 0Ø∆EÖÓÑÓ= − 

≃ 0.135 GeV 

= 1+ 

) 2 

B = 

µ qi , µ qi = µqi · σqi 

i=1 

3 Kqq 

4 m2 0

+1ÓÒÓØÒØ 

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ØÖÓÙÚÔÖÜÑÔÐ ÔÖ×ÓÒÚÓÖØÐÈÚÐ×ÚÐÙÖ××Ñ×××ÓÒ×ØØÙØÚ× 2mpcÖÒÙÖÓÒÒÙÚÙÒÖÒ ÓÒ 

= ÒÙÒØÑÒØÓÒÒÙÐÖµN 

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ÄÔÔÐØÓÒÐÔÐÙ××ÑÔÐÓÒÖÒÐÝÔÖÓÒΩ −ÔÙ×ÕÙÐ×ØÙÒ×Ý× ØÑØÖÓ×ÕÙÖ×sÙÜ×ÔÒ×ÐÒ×J P + 

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(µu)Z = + 3 e 

2 2muc , (µd)Z = − 1 e 

3 2mdc , (µs)Z = − 1 

3 

(µu)Z = 2 

3 

e 

µN ≃ 1.7µN ; (µd)Z = − 1 

3 

e 

2msc , 

mp 

mp 

µN ≃ −0.9µN 

mu 

md 

(µs)Z = − 1 mp 

µN ≃ −0.6µN 

3 ms 

+1〉×ÝÑÓÐ×ÐÓÒØÓÒÓÒÒÓÖÑÐ×ËÍ×ÚÙÖ×ÔÒ 

3 

(µB)Z = 〈B ↑| (µqi)Z|B ↑〉 

i=1 

Ó|B ↑〉 = |B, σZ = 

= 3 

2 

(µΩ)Z = 3(µs)Z = − mp 

µN ≃ −1.8µN 

ms 

= 1 

ËÍ×ÚÙÖ×ÔÒ 

2 

|n ↑〉 = 1 

 

↓ ↑ ↑ ↑ ↑ ↓ ↑ ↓ ↑ 

√ −2 + + 

6 u d d u d d u d d 

 

↓ ↑ ↑ 

= 

u d d 

1 √ (u ↓ d ↑ d ↑ + d ↑ d ↑ u ↓ + d ↑ u ↓ d ↑) 

3 

 

2 1√3 

[−(µu)Z + (µd)Z + (µd)Z]

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u 

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Ò×Ð××ÖÒØ×ÝÔÖÓÒ×ÐÓØØJ P ÔÖÓÔÖØ××ÝÑØÖ×ÙØ×Ù§ ÑÓÑÒØÑÒØÕÙ ÔÓÙÖØÐÖÐ×ÜÔÖ××ÓÒ×ÐÙÖ 

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ÙØÐÙ×ÓÒØÓÒØÓÒ××ÚÐÙÖ××Ñ×××ÓÒ×ØØÙØÚ××ÕÙÖ×ÓÒ ×ØÐ××ÓÒØÓÒ×Ò×Ò×ÐØÐÙ ÓÖÖ×ÔÓÒÒØ×ÚÓÖØÐÈÄ×ÔÖØÓÒ×ÒÙÑÖÕÙ×ØÖÓ×ÑÓÐÓÒÒ Ä×ÔÖØÓÒ×ÙÑÓÐ×ØØÕÙ×ÕÙÖ×ÔÓÙÖÐÒ×ÑÐ×ÖÝÓÒ× 

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×ØÖÚÑÓÝÒÒÖÐØÚÑÒØÐÓÒÙ≫ 10−20 ØÓÒÐÓÙÐØÖÓÑÒØÕÙ 

s)Ö××ÒØÖÒØÔÖÒØÖ 

 

1 

(µn)Z = 3 √6 

+ 

2 

{2 2 

 

1 

√3 

2 

[−(µu)Z + 2(µd)Z] 

2 1 

√3 [(µu)Z + (µd)Z − (µd)Z] 

2 1 

+ √3 [(µu)Z − (µd)Z + (µd)Z]} 

= − 1 

3 (µu)Z + 4 

3 (µd)Z 

ddÚ×ÔÓ×ÔÔÖÓÔÖ×ÓÒÒ×ÔÖÐØÐÓÒØ× ddØ · |↑↑〉 

|↑〉 u |↑↓ + ↓↑〉 

 

1 

, +1 

2 2 n = 

 

2 

 

1 

, −1 

2 2 3 u |1, +1〉 dd − 

 

1 

 

1 

, +1 

2 2 3 u |1, 0〉 dd 

0.51Î 

3 [(µu)Z + 0]×ÓØÙØÓØÐ− 

(µp)Z = − 1 

3 (µd)Z + 4 

3 (µu)Z 

1 

3 (µu)Z + 4 

3 [−(µu)Z +2(µd)Z +2(µd)Z] 

= 1 

2 

3 (µd)Z 

md ≃ 0.33Î, ms ≃

ÖÝÓÒ ÓÒØÖÙØÓÒ××ÕÙÖ× ÚÐÙÖ×ÔÖØ× ÚÐÙÖ×Ñ×ÙÖ× ÑÓÑÒØÑÒØÕÙÔÓÐÖÒÙÒØµN 

 

 

2.793.... 

−1.913.... 

 

2.458 

 

−1.160 

−1.250 

Ì 

−0.6507 

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3Ò×ÐÚÐÙÖ 

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p 

n 

4 

3 (µu)Z − 1 

3 (µd)Z 

4 

3 (µd)Z − 1 

3 (µu)Z 

Λ (µs)Z 

+ 4 Σ 3 (µu)Z − 1 

3 (µs)Z 

0 2 

Σ 3 (µu)Z + 2 

3 (µd)Z − 1 

3 (µs)Z 

− 4 Σ 3 (µd)Z − 1 

3 (µs)Z 

0 4 Ξ 3 (µs)Z − 1 

3 (µu)Z 

Ξ 

− 4 

3 (µs)Z − 1 

Ω − 3(µs)Z 

3 (µd)Z 

−0.613 

−2.02 

± (2.9 10 −8 ) 

± (5 10 −7 ) 

± 0.004 

± 0.010 

± 0.025 

± 0.014 

± 0.0025 

± 0.05 

= 1 

2

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3 

1 

2 

1 

3 

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2 

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3 

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3 

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3 

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3 

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3 

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3 

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d, sÓÙu, d, s 

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ϕ = ⎜ d ⎟ 

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us 

cc 

ss 

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dcc 

C 

scc 

1 

2 

3 

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ddc udc 

uuc 

dsc ssc 2 

usc 

udd uud 

Y 

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×ÒØÖØÓÒ×Ð××Ñ×ÓÒ×D 0ØD 0 s×ÓÒØÙÒÖÒÒØÖØÔÖÓÑ ÔÖ×ÓÒÐÐÙK 0 ÑÐÒØØ×ÚÓÖ§ Ä×ÔÖÓÔÖØ×ÙÖÑÓÒÙÑ×Ý×ØÑcc×ÖÓÒØÜÑÒ×Ù§ ÇÒÓÒ×ÖÐ×ØÙØÓÒ×ÖÓÒ×ÓÖÑ×ÙÑÓÒ×ÙÒÕÙÖb(b)Ø ÜØÒ×ÓÒÐÓØØÓÑÒ××ËÍ×ÚÙÖ 

×ÓÒØÖ×ÔÖÐ×ÖÐ×Ð×ÝÑØÖËÍ×ÚÙÖÒÖØÙÖ× ÙÒÔÐØÄ×Ý×ØÑqqqÖÝÓÒ×ÓÖÑÙÒÑÙÐØÔÐØ ÄØØÒÖÕÙ×ØÙÒÚØÙÖÒÕÓÑÔÓ×ÒØ×ÓÒØÐ×ØÖÒ×ÓÖÑØÓÒ× Ä×ØØ×Ù×Ý×ØÑqqÑ×ÓÒ× ÓÑÔÖÒÒÒØÙÒ×ÒÙÐØËÍ×ÚÙÖØ ØØ×ÕÙ× 

ÙÒÓÙÕÙÖÔÐÙ×ÐÖu, d, 

BØØÒÙ×ØÙÒÖØÒÒÓÑÖÐÙÖ×ØØ×ÜØ×ÔÖÓÒØÖ×ÙÐ×Ð× ÓÒÓÒÒØÙÒÓ×Ð×ÔÒÒØÖÓÙØ ÄÒÚ×ØØÓÒÑÒÔÙ×ÔÖÑ×ÐÒØØÓÒØÓÙ×Ð×Ñ×ÓÒ× M× ÓÑÔÓ×ÓÑÑ125 = ×ÓÒØ××ÓÖÙ35SÔÐØÙÜ×ÔÒJ = 

ÖÝÓÒ×Λ 0 b×Ý×ØÑudb), b (dsb)×ÓÒØÖÓÒÒÙ×ØÐÈ ÑÔÓÖØÒØÕÙÒ×ËÍ×ÚÙÖ Ä××ÔØÖ×Ñ×××ÑÓÒØÖÒØÕÙÐÖ×ÙÖ×ÝÑØÖ×ØÙÓÙÔÔÐÙ× ÉÖÖÖÐ×Ñ×××ÙÑ×ÓÒB ±ØÙΛ 0 bÒ×ÐØÐÈÐ×ÓÑÔÖÖ ÙÜÑ×××ÙK ±ØÙΛÒÙÖÐÑ××ØÚÓÒ×ØØÙØÚÙÕÙÖ 

JP = 1 

2 

JP = 1 

2 

= 3 

2 

= 3 

+ 

2 

ÖÝÓÒ×ÖÑ×Σ ++ cØΣ 0 

Ξ 

(D 0 )ØÐÐ×× (K 0 ) 

(K 0 )ÖÐÐ×ÓÒÒÒØÐÑÒØÐÙÙÔÒÓÑÒ 

s, cÓÙu, d, s, c 

= 

35S ⊕ 40M ⊕ 40 ′ M ⊕ 10AÄ×ÖÝÓÒ××ÔÒJ 3 

2 

1 2Ù40MÔÐØ40MØ40 ′ 

0 b 

(usb)ØΞ −

ÄØÙ××ÒØÖØÓÒ×Ð×ÙB ±ØÙB ± ÚÐÙÖ××ÐÑÒØ×VubØVcbÐÑØÖÃÅcÔÖÑ×ØÖÑÒÖÐ× 

Ä×ÔÖÓÔÖØ×××ÒØÖØÓÒ×Ð×ÙB 0 ÒÖÐØÓÒÚ ÐÐ××Ñ×ÓÒ×K 0 s )ÄÙÖØÙ×Ø ÈÒ××ÔÖÓ××Ù×Ð×ÆÓØÓÒ×ÕÙÐ×ØÙØÓÒ×ØÔÐÙ×ÚÓÖÐÙÒÔÓÒØ ÙÒÑÔÓÖØÒÑÙÖÓÑÑØ×ØØÓÒÒÐÚÓÖ§ÐÚÓÐØÓÒ 

0ÖÐ×ÒØÙÖ ÚÙÜÔÖÑÒØÐÚÐ×Ñ×ÓÒ×B 0ÕÙÚÐ×Ñ×ÓÒ×D 

0 

(B 0 )ØÙB (K 0 ) D0 (D 0 )ØD 0 s (D0 

Ä×ÔÖÓÔÖØ×ÙÓØØÓÑÓÒÙÑ×Ý×ØÑbb×ÓÒØÜÔÓ××Ù§ Ò×ÐÜÔÖÒÝÒØÓÒÙØÐÓÙÚÖØÙÕÙÖtÌÚØÖÓÒÄ ×ØÙÖÒÔ×ÒÓÖØÜÔÐÓÖ ÜØÒ×ÓÒÙØÓÔ ××ÒØÖØÓÒ××ØÑÙÜÖÓÒÒ××Ðmb ≫ mc 

ÖÑÐ×ÑÒ×Ñ×ÔÖÓÙØÓÒÓÑÒÒØ××ÓÒØ 

Ó Ø ØÓÔÓÒÙÑ 

qq → tt gg → tt 

t → W − Ø b t → W + b Ä×Ó×ÓÒ×W ±×ÓÒØÒØ×ÚÐÙÖ×ÔÖÓÙØ××ÒØÖØÓÒÐÔØÓÒÕÙ ØÖÓÒÕÙÄÕÙÖb×ØÖÓÒÒÙÚ××ÒØÙÖØÖØÖ×ØÕÙ ÇÒÔÔÐÐÓÑÑÙÒÑÒØÕÙÖÓÒÐ×ØØ×Ð×Ø×Ò××ÚÙÖ×ÚÙÖ Ä×ÕÙÖÓÒ 

×Ý×ØÑccÐÓØØÓÑÓÒÙÑÐ×Ý×ØÑbbØÐØÓÔÓÒÙÑÐ×Ý×ØÑttÜÔÖ 

bÓÙtÄÖÑÓÒÙÑ×ØÐ Óq(q)ÓÒÖÒÙÒ×ÕÙÖ×ÐÓÙÖ×c, ÑÒØÐÑÒØÓÒÓ×ÖÚÐÙÖÓÖÑØÓÒÒ×Ð×ÓÐÐ×ÓÒ×e + e− ppÓÙpÒÓÝÙ 

×ÚÙÖ×ÕÙÖ ÙÜÙØ×ÒÖ××ÓÙ×ÐÓÖÑØÖÓØ×Ô×Ö×ÓÒÒÒ×Ð××ØÓÒ× ×ØÓØÐØÔÖØÐÐÚÓÖÔÖÜÐÙÖÀ×ØÓÖÕÙÑÒØ×Ó×Ö ÚØÓÒ×ÓÒØÓÙÙÒÖÐÑÙÖÒ×ÐÓÒÖÑØÓÒÐÜ×ØÒ×ÖÒØ× Ò×ÙÒ×ÖÔØÓÒÒÓÒÖÐØÚ×ØÙÒ×Ý×ØÑqq×ØÖØÖ×ÔÖ×ÓÒ 

, 

0ÓÙ SÓL×ØÐÑÓÑÒØÓÖØÐÖÐØØSÐ×ÔÒ ÑÓÑÒØÒÙÐÖJ = L + ØÓØÐS = = (−1) ÄÔÖØÙ×Ý×ØÑ×ØÓÒÒÔÖPqq L+1Ø ×ÓÒÙ×ÓÒÖÔÖCqq = (−1) L+SÒÓÑÖÙÜØØ×ÕÙÖÓÒ ÔÙÚÒØØÖÓÖÑ×ÑÔÐÕÙÒØÖÒØ×ÑÒ×Ñ×ÖØÓÒÈÖÜÑÔÐ Ò×Ð×ÓÐÐ×ÓÒ×e + e−ÐÑÒ×ÑÓÖÑØÓÒÓÑÒÒØ×ØÐÒÙÒ ÚJ PC = 1−−ÒÖ×ÕÙÒØÕÙ×ÙÔÓØÓÒÒ×ÐØÐÈ×ØØ××ÓÒØ ΥØθÒÓÒØÓÒ××ÚÙÖ×ÓÒÖÒ× 

×Ò×ÔÖÐ××ÝÑÓÐ×ψ(ou J/ψ) 

s (B0 

s )×ÓÒØÑØØÖ 

ÔÓØÓÒÚÖØÙÐÒ×ÐÚÓsÕÙÓÒÒÒ××Ò×ØØ×ÕÙÖÓÒ

∆E [eV] 

10 −5 

X 

10 −4 

X 

10 

0 

10 

1 

2 S0 

3 

16 

5.1eV 

α2 mec M1 

M1 

2 

M1 

3 

2 S1 

3 

1 S1 

7 

12 

2 

α 4 

1 P1 

m ec 

−4 

8.4.10 eV 

+ − 

+ + 

+ + 

+ + 

1 0 1 2 J PC 

0 − + 1 

1 S0 

0 

2 γ 

− − 

1 Ä×ÒÚÙÜÒÖÙÔÓ×ØÖÓÒÙÑ×Ý×Øe + e−ÒÓÒØÓÒÐÙÖ JPCÖØÖ×ØÕÙÄ×ÐÒ×ÒØÖØÐÐ×ÒÒØÐ×ØÖÒ×ØÓÒ×ÓÑÒÒØ× Ð×ÜØØÓÒÑÇÒØÙ×ÐÒÓØØÓÒÐ×ÔØÖÓ×ÓÔØÓÑÕÙ 

N2S+1 0ØÒØØØÒÙ× 

LJ ÄØØ3S1Ñ×ÓÒÚØÙÖ×ØÐÔÐÙ×ÓÒÒØÐ×ØØ×ÚL > ÔÖÐØÐÖÖÖÑÓÑÒØÒÙÐÖÈÓÙÖ×ÚÐÙÖ×J PCÖÒØ× 

ÇÒÔÙØÓÒ×ØÖÙÖÙÒÑÓÐÙÕÙÖÓÒÙÑ××ÙÖÐÑÓÐÙÔÓ×ØÖÓ 

0Ö×ÔØÚÑÒØ ...×ÓÒ×ØÒ× ... ×ÓÒ×ØÒ×ÐØØØÖÔÐØ×ÔÒØL>0, ηc, ηb, ...Øhc, hb, ÐØØ×ÒÙÐØ×ÔÒØL = 0ØL > ÒÙÑ×Ý×ØÑe + e−ÒÓÒÒÙÒØÓÖÉÒÔÖÑÖÔÔÖÓÜÑØÓÒÓÒ ÔÙØÖÖÖÐ×ÒÚÙÜÒÖ×ØØÓÒÒÖ××Ý×ØÑÔÓÙÖÙÒÔÙØ× ÔÓØÒØÐÓÙÐÓÑÒVem = −α rÓα=e 2 /c×ØÐÓÒ×ØÒØ×ØÖÙØÙÖÒ 

1 −−ÐØÐÈÑÔÐÓÙÒÒÓØØÓÒ×ÔØÖÓ×ÓÔÕÙÖÒχc, χb, 

ËÐÓÒÐØÐÈ ÙÙÒØØηbÒÒÓÖØÐÖÑÒØÒØ 

 

E1 

E1 

2 

2 

3 P0 

2 

3 P1 

2 

3 P2

V [GeV] 

5 

0 

−5 

−10 

0 

0.5 1 1.5 2 2.5 3 

dominance de Kr 

r [fm] 

4 

dominance de 1 

3 r 

αs 

−15 

ÔÓÖØÓÒÐÙÖÚÙÒÐÐÐØØÐÒÖÒ××× ÈÙØ×ÔÓØÒØÐÙÕÙÖÓÒÙÑVQCD(r)ÖÔÕÙ×ØÙÒ 

×ÒÚÙÜ×ÓÒØÖÔÖ×ÒØ×ÔÖÐÖÐØÓÒÓÖ 

−20 

2 

2 mec 1 

EN = −α 

4 N2 Ó 

ÐÒÓÑÖÕÙÒØÕÙÓÖØÐ 

...Ò×ÔØÖÓ×ÓÔN×ØÔÔÐÐÒÓÑÖÕÙÒØÕÙÔÖÒÔÐ1×Ø 

ÓÖØÐØÔÖØ×ÒÚÙÜÒÖ×ÒNØØÔÖØÓÒ×ØÒ×ÓÖ 

N 

ÚÐÓ×ÖÚØÓÒÚÓÖÙÖ 

= 1, 2, 

N 

ÔÖÐÒØÖÓÙØÓÒÖÒØ×ØÖÑ×ÓÖÖØ×VemÄÓÙÔÐ×ÔÒÓÖØ ØØÖÐØÓÒÓÒØÒØÑÔÐØÑÒØÐ×ÓÒØÖÙØÓÒ×ÜØØÓÒ×ÖÐØ ÇÒÔÙØÖØÐÖÙÒÓÖÔÐÙ××Ø××ÒØ 

= n + Ln = 1, 2, ...×ØÐÒÓÑÖÕÙÒØÕÙÖÐL=0 N − 

Ð×ÓÖÖØÓÒ××ÒÚÙÜÙ×VLSØVSS×ÓÒØÓÑÔÖÐ×ÒÑÔÓÖØÒØ ×ÒÚÙÜÐØÖÓÒÕÙ×ÐØÓÑØ×ÒÚÙÜÒÙÐÓÒÕÙ×ÙÒÓÝÙÇÒ Ð×ÖØÖÓÙÚÚÐÔÓ×ØÖÓÒÙÑØÚÐÕÙÖÓÒÙÑ×ÐÓÒÐØÓÖÉ 

SØÐÓÙÔÐ×ÔÒ×ÔÒVSS×ÓÒØÒÓÒÒÙ×Ò×Ð×ÖÔØÓÒ 

ÐÓÖÖ 

VLS ≈ L · 

∆E ≃ α 4 mec 2 · 1 

N3 ÎÓÖÔÖÜÑÔÐÖÖÒ

masse [GeV] 

4.5 

4.0 

3.5 

ηc(2S) 

ψ(4415) 

ψ(4160) 

ψ(4040) 

ψ(3770) 

ψ (2S) 

γ γ 

χ c1(1P) 

seuil 

DD 

χ c2 (1P) 

− + − − + − + + + + + + 

0 1 1 0 1 2 JPC γ 

h c(1P) 

χc 0 

ππ η, π 

ψ(1S) 

γ γγ 

hadrons 

hadrons 

ηc 

3.0 

0 

γ * 

hadrons 

(1P) 

hadrons 

J hadrons 

(1S) 

γ* 

ØØ×ÒÙÐØ×ÔÒ×ØÖØX(nL)ÙÒØØØÖÔÐØXJ(nL)Ón×ØÐÒÖ ÐÒ××ÒÕÙÒØÐ×ØÖÒ×ØÓÒ×ÓÑÒÒØ×ØØÒÙ×ÄÒÓØØÓÒ×Ø ÐÐÐØÐÈÕÙ×Ò×ÔÖÐÒÓØØÓÒ×ÔØÖÓ×ÓÔÒÙÐÖÙÒ 

hadrons hadrons radiatif 

ÒÙÑÐÐØÖÓÒØÐÔÓ×ØÖÓÒÔÓÙÚÒØ×ÒÒÐÖÒÔÓØÓÒ×ÖÐ×ÄØÓÑÒ 

ÕÙÒØÕÙÖÐLÐÒÖÕÙÒØÕÙÓÖØÐÖÔÔÐN = n + L J = L + S 

×ÓÖÖØÓÒ××ØÒÐÙ×Ò×Ð×ÔØÖÖÔÖ×ÒØÐÙÖØÓÒÝ 0)ÙÔÓ×ØÖÓ 

S = 0ÓÙ1) ÁÐ×ÝÓÙØÙÒÓÖÖØÓÒ×ÔÕÙØÒØÐ×ÒÚÙÜ1S(L − ÓÒÒÜÔÐØÑÒØÐ×ÔÖØÓÒÒØÖÐ×ÒÚÙÜ1 3S1Ø1 1 Ð×ØÖÒ×ØÓÒ×ÑÓÑÒÒØ××ÓÒØÒÕÙ××ÙÖÐÙÖÐÐ×ÓÒØØÐÓØ É×ØØÖ×ÓÒØÓÙØÙÑÓÒ×ÔÓÙÖÐ×ÔÖÑÖ×ÒÚÙÜ Ñ×ÙÖ××ÔØÖÓ×ÓÔÕÙ×ÙØÔÖ×ÓÒÐÓÖÚÐ×ÔÖØÓÒ× Ä×ÒÚÙÜ×ÙÔÖÙÖ×ÙÔÓ×ØÖÓÒÙÑ××ÜØÒØÔÖÑ××ÓÒÔÓØÓÒ× 

S0 

Ä×ÔÖØÓÒÑ×ÙÖ×ÒÚÙÜ2 1S0Ø1 1 ÙÒÚÐÙÖÔÔÖÓÜÑØÚαÐS0×ØÎÓÒÔÙØÒØÖÖ 

2 

2 mec 

 

1 

 

Ä×ÒÚÙÜÒÖÙÖÑÓÒÙÑÒÓÒØÓÒÐÙÖJ PCÄ× 

ÎÒÓÒ×ÒÑÒØÒÒØÐÔÖØÓÒ×ÒÚÙÜÒÖÙÕÙÖÓÒÙÑ 

 

E2 − E1 = −α 

4 

− 1 

4 

α = 

 

16 5.1 [eV ] 

3 0.51 · 106 

[ eV ] 

γ 

, Ó 

= 1 

136.9

ØØcc JPC Ñ×× ÐÖÙÖ 

 

[MeV ] [MeV ] 

 

 

ÐÙÖ ÌÅ×××ØÐÖÙÖ×ÒØÖÒ×ÕÙ×ØØ×ÒØ×ÙÖÑÓÒÙÑ ØÐÈ ÄÒÓØØÓÒ×ÔØÖÓ×ÓÔÕÙÙØÐ××ØÒÒ×ÐÐÒ 

Ò×ÐÖÐÔÔÖÓÜÑØÓÒÒÓÒÖÐØÚ×ØÊÑÖÕÙÓÒ×ÕÙØØÔÔÖÓÜ ÑØÓÒ×Ù×ØÙØÕÙÐ×ÓÒ×ØØÙÒØ×Ù×Ý×ØÑ×ÓÒØÐÓÙÖ×(mc ≃ 

1.8 GeV/c2 , mb ≃ 5.3 GeV/c2 , mt ≃ 175 GeV/c2 ÒØÖØÓÒqqÚÓÖÖÐØÓÒ ÇÒÙÐÓ×ÓÒÔÖ×ÒØÖÙÒÓÖÑÔÐÙ×ÐÔÓÙÖÐÔÓØÒØÐ ) 

VQCD = − 4 Ó αs 

+ Kr , 

3 r 

ηc(1S) 0−+ ηc(2S) 0 −+ J/psi(1S) 1 −− ψ(2S) 1 −− 

hc(1P) 1 +− χc0(1P) 0 ++ χc1(1P) 1 ++ χc2(1P) 2 ++ 

αs×ØÐÓÒ×ØÒØÓÙÔÐÓÖØ 

K×ØÙÒÔÖÑØÖÑÔÖÕÙ 

3×ÖÙ×ØÙ§ 

ÚÒØÒ×ÐÑÓÐÙÔÓ×ØÖÓÒÙÑ×ÓÖÖØÓÒ× ÔÖÑØ××ÙÖÖÐÓÒÒÑÒØ×ÕÙÖ×ÚÓÖÙÖ ØÐÒÙÒÐÙÓÒÐ×ÓÒØÖÑÔÖÔÓÒÖÒØÙÜÖÒ××ØÒ× ÊÔÔÐÓÒ×ÕÙÐÔÖÑÖØÖÑ ÓÑÒÒØÙÜÓÙÖØ××ØÒ×Ö ×ÓÒØÔÔÓÖØ××ÓÙ× ÓÑÑÙÔÖ 

ÐØÙÖ4

ØØbb JPC Ñ×× ÐÖÙÖ 

 

[MeV ] [MeV ] 

Υ(1S) 1−− 

Υ(2S) 1−− 

Υ(3S) 1−− 

 

 

ÌÅ×××ØÐÖÙÖ×ÒØÖÒ×ÕÙ×ØØ×ÒØ×ÙÓØØÓÑÓÒÙÑ 

ÐÙÖ ØÐÈ ÄÒÓØØÓÒ×ÔØÖÓ×ÓÔÕÙÙØÐ××ØÒÒ×ÐÐÒ 

ØÖÒ×ØÓÒ×ÔÖØ×ÔÖÐÑÓÐ×ÕÙÖ×ÔÓÙÖÐ××Ý×ØÑ×ÙÖÑÓÒÙÑ ÒØÖÒ×ÕÙ×Ñ×ÙÖ×ÔÓÙÖÐ×ÖÒØ×ØØ×ÒØ×ØÐÈ ØÙÓØØÓÑÓÒÙÑÄ×ØÐÙÜØÓÒÒÒØÐ×Ñ×××ØÐÖÙÖ× ÑÔÖÕÙÑÒØÄ×ÙÖ×ØÑÓÒØÖÒØÐ×ÒÚÙÜÒÖØÐ× 

ÐÓÖÖÙÔÓÙÖÒØÐ×ÔÖÑÖ×ÒÚÙÜÜØØÓÒØÓÙØÙÑÓÒ× ÑÓÐØÓÖÕÙÔÖÑØÖÔÖÓÙÖÐ×ÒÚÙÜÓ×ÖÚ×ÙÒÔÖ×ÓÒ ÇÒÔÙØÚÐÙÖÐÓÒ×ØÒØÓÙÔÐÓÖØαs(E)ÔÖØÖÐ×ÔÖØÓÒ Ä 

Ñ×ÙÖ×ÒÚÙÜÒÖ2 3S1Ø1 3S1ØÒÙØÐ×ÒØÐÖÐØÓÒ Ó 

 

ηb(1S) 0 −+ 

χb0(1P) 0 ++ χb1(1P) 1 ++ χb2(1P) 2 ++ χb0(2P) 0 ++ 

χb1(2P) 1 ++ 

χb2(2P) 2 ++ 

ÐÓÖÑÓÙÔÐ×VLSØVSSÐÐ×ÑÒÒØ×ÔÖÑØÖ×ÕÙ×ÓÒØÙ×Ø×

Masse [GeV] 

11.0 

10.5 

10.0 

η b (3S) 

ηb 

(2S) 

hadrons 

I (11019) 

I (10865) 

I (10580) 

I (3S) 

hadrons 

I (2S) 

hadrons 

γ 

γ 

hb 

hb 

η 

I 

b 

− + − − + − 

1 1 

+ + + + + + 

0 0 1 2 JPC γ γ 

9.5 

(1S) 

(1S) 

Ä×ÒÚÙÜÒÖÙÓØØÓÑÓÒÙÑÒÓÒØÓÒÐÙÖJ PCÄ× ÙØÐ×ÚÓÖÐÜÔÐØÓÒÓÒÒÒ×ÐÐÒÐÙÖ ÐÒ××ÒÕÙÒØÐ×ØÖÒ×ØÓÒ×ÓÑÒÒØ×ØØÒÙ×ÈÓÙÖÐÒÓØØÓÒ 

b0 

b1 

Seuil BB 

χ 

χ 

(2P) b0(2P) b1(2P) b2(2P) 

χ 

γ 

(1P) 

χ (1P) 

9 

9 

9 9 

9 

9 9 

9 

9 99 

9 

9 

9 

χ (1P) 

χ (1P) 

a) b) 

K 

s 

u 

K 

s s 

φ 

J/ ψ 

+ 

u 

s 

M1 q1 M2 q2 

q2 M3 

q3 q3 

q1 

ÖÑÑ×ÓÑÒÒØ×Ð×ÒØÖØÓÒØØ×ÙÖÑÓÒÙÑ 

c 

c 

ØØ×J PC = 1−−×ØÙ×Ù××Ù×Ù×ÙÐDDÖÑÑÓÒÒØØØ× JPC = 1−−×ØÙ×Ù××ÓÙ×Ù×ÙÐDDÖÑÑÓÒÒØÚÒ ÐÙÓÒ×ÚÖØÙÐ×MiÑ×ÓÒ×ÒÓÒÖÑ× 

 

b2

4 

3αs×Ø×Ù×ØØÙÖαÐÒÖÓÖÖ×ÔÓÒÒØÐÑ××ÙÖÑÓÒÙÑ αs(mJ/ψ) ≃ 3 ÐÒÖÑ××ÙÓØØÓÑÓÒÙÑÐÐÙÐÓÒÒ 

1 

2 

16 3.69 − 3.10 

≃ 1. 

4 3 1.8 

αs(mΥ) ≃ 3 

ÔÖÑÖ×ÒÚÙÜÙ×ÔØÖ××ØÙÒØÙ××ÓÙ×Ù×ÙÐ×ÒØÖØÓÒÒ ÙÜÖÓÒ×ÔÓÖØÙÖ×Ð×ÚÙÖÕÙÖÐÓÙÖÓÒÖÒÒ×ÐÖÑÓÒÙÑ ÄÑÓÐÙÕÙÖÓÒÙÑÖÚÐÙÒÖØÖ×ØÕÙÔÖÓÔÖ×Ý×ØÑÐ× 

1 

2 

16 10.02 − 9.46 

≃ 0.5 

4 3 5.3 

×ÓÒØÒ××ÓÙ×Ù×ÙÐ×ÒØÖØÓÒÒBBÚÓÖÙÖÈÓÙÖ× ÚÓÖÙÖ 

ÖÑÑÓÒÒØÚÓÖÙÖØÓÒÙÒÔÖÓ××Ù×ÓÖØÑÒØÒ ÚÔÖÓÙØÓÒÖÓÒ×ÐÖ×ÄÑÒ×ÑÒ××ÓÒ×ÑÔÐÕÙÙÒ ÒÚÙÜÐ×ÒØÖØÓÒÔÙØØÖ×ÓØÑÚÑ××ÓÒÔÓØÓÒ××ÓØÓÖØ Ð×ÒÚÙÜL 

ÖÐÇÁÚÓÖ§ÇÒÓÑÔÖÒ×ÐÓÖ×ÔÓÙÖÕÙÓ×ØØ×ÕÙÖÓÒ 

= 0, N = n < 3×ÓÒØÒ××ÓÙ×Ù×ÙÐ×ÒØÖØÓÒÒDD ØÒ×ÐÓØØÓÑÓÒÙÑÐ×ÒÚÙÜL = 0, N = n < 4 

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= 0 −− , 0 +− , 1 −+ , ...) 

Aℓ = (gℓ − 2)/2 = 0.5 

π αeff ,Ó αeff = α(q2 )×ØÐÓÒ×ØÒØÓÙÔÐØÚÖÙÒÒÒÓÙÔÐÒÓÒ×ØÒØ 

e−ÒØÖÑÖÉÔÖØÕÙÚÓÖÔÔÒ 2 αem 

) ≃ 

1 − αem 

 

q2 ln 3π 4m2 , Ó: e 

αem = α(q2 → 0) = e2 /4πǫ0cÐÓÒ×ØÒØ×ØÖÙØÙÖÒ 

(2me) 2 ≃ 1 < q2 < 400 [ÅÎ] 2

Ë×Ò×ÖÒ×ÚÐÙÖ×q 2×ÓÒØÔÓ××Ð××ÓÖÖØÓÒ×ÓÖÖ××Ù ÔÖÙÖ××ÓÒØÒØÖÓÙÖÕÙØÒÒÒØÓÑÔØÖÑÑ×ÔÐÙ×ÙÖ×ÓÙÐ× ØÓÙÐ×ÖÑÓÒ×ÔÐÙ×ÐÓÙÖ×ÕÙÐÐØÖÓÒÐÔØÓÒ×ØÕÙÖ×ËÓÒÒ ÖÕÙÐ×ØÖÑ×ÐÓÖØÑÕÙ×ÓÑÒÒØ×ÓÒÓØÒØÐÖÐØÓÒÔÔÖÓÜÑ ØÚ×ÙÚÒØα(q 2 

q0×ØÙÒÚÐÙÖÖÖÒÖØÖÖ 

) ≃ 

ÇÒÚÓØÕÙÒ×ÉÐÓÒ×ØÒØÓÙÔÐØÚÖÓØÐÒØÑÒØÒÓÒ ØÓÒq 2ÔÖÜÑÔÐα(0) = 

ÊÔÖ×ÒØØÓÒ×ÑØÕÙÐØÖÒÙØÓÙÖÙÒÔÖØÙÐ ÔÓÖØÒØÙÒÖÐØÖÕÙ 

α(q2 0) 

1 − α(q2 0 ) 

3π 

i 

q2 >4m2 Q 

i 

2 

q2 i ln 4m2 mi×ÓÒØÐ×Ö×ØÑ××××ÖÑÓÒ×ÓÒÖÒ× 

, Ó: i 

Qi, 

sonde 

αem ≃ 1/137, α(mW ≃ 80Î) ≃ 1/128 

e− e + 

e + 

e− ~ ~ 

particule de particule 

charge +e 

nue 

+e 

+ 

+ 

− 

+ 

− 

+ 

− 

− 

− 

+ + 

+ 

− 

− − 

r 

r− 

r− 

− 

r 

gg 

~ 9 

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r 

− 

− 

− 

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rg 

rg 

9 

rg 

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br 

r 

r 

quark de charge 

de couleur r 

− 

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r 

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r 

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Nc ÓÙÐÙÖ×ÕÙÖ 

q0ÚÐÙÖÖÖÒq ÎÙÕÙB > ÓÒ×ØÒØÓÙÔÐØÚαs(q 2 )ÑÒÙÕÙÒq 2ÖÓØÄÙÖ 

100Î ÑÓÒØÖÐÚÖØÓÒÓ×ÖÚÒ×ÐÒØÖÚÐÐ1 

α s(q) 0.3 

0.2 

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q [GeV] 

αs(q2 (ÓÙΛQCD)ØÐÕÙ 

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1 

− 

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0)

ÒÒ×ÖÒØÒÓÙÚÙÔÖÑØÖÒ×ÓÒÓØÒØÐÓÖÑ×ÑÔÐ 

αs(q 2 

ÐÒÖÑ×ÒÙÈÖÜÑÔÐÔÓÙÖÐÒÖÕÙÚÐÒØÐÑ××ÙÓ×ÓÒ 

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) ≃ 

−24ÅÎ 

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B 

ÊÐÚÓÒ×Ð×ÖØÖ×ØÕÙ××ÐÐÒØ×ÐÚÓÐÙØÓÒαs(q 2 •ÙÜÐ×ÑÓÑÒØ×ØÖÒ×ÖØq≤200ÅÎ×ØÖÖÒ×ØÒ 

) ÖÐØÚ×ÕÙÖ×r = q ≥ •ÙÜÖÒ×ÑÓÑÒØ×ØÖÒ×ÖØq≫Î×ØÖÙÜÐ××ØÒ× ÓÒÒÑÒØÓ×ÖÚ×ÕÙÖ× ÕÙ×ÖØÔÖÐØ 

1ÕÙØÔÔÖØÖÐ×ÕÙÖ×ÓÑÑÕÙ× 

1 Ñαs 

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ÖÐØÚ×r ≪ Ñαs ÐÖ×ÓÒÒÑÒØÐÐÑØq → 

ÐÓÖÑV (r) r (αs×ÙÔÔÓ×ÜØÕÙÓÒ×Ù×ØØÙαsÙÒÓÒ×ØÒØ ÓÙÔÐØÚÖÔÖ×ÒØÔÖÜÑÔÐÔÖÓÒÔÙØÖÖ 

ÚÖÐÒØÑÒØØÐÔÒÒ 

Ó: 2 200ÅÎ) 

ÎÓÖÔÖÜÐØÐÈ×ÓÙ×ÉÙÒØÙÑÖÓÑÓÝÒÑ× 

rÖÓ××ÒÐÐÙÖÓÙÐÓÑÒÒ 

rÖÓ××ÒÐÒÖ ÚÖÖÔÑÒØÒÖÓ×ÓÑÑ ÈÓÙÖr ≪ÑØRΛ/r ≫ 1ÐØÙÖ1/ ÒrV (r)×ØÓÑÒÔÖÐØÙÖ1 ÈÓÙÖr≫ÑØRΛ/r ≪ ÐØÙÖ1/ 

 

≈ 

π− 

Λ (5) 216 +25 

B ln 

1 

q 2 

Λ 2 

s, c, b) 

≃ 0.61 

αs(Z)1/{0.61 ln[(91.2 · 10 3 /216) 2 ]} ≃ 0.117 

≪ 

∝ αs 

c 

≥ 

V (r) ≈ 1 1 

r ln q 2 ≈ 

Λ 

1 1 

r ln RΛ r 

RΛ = c 

Λ ≃ 1Ñ×Λ ≃ 

ln 

ln 

r2ÕÙÓÒÒÙÒÔÒÒV (r) 

∞ αs(q 2 ) → 0×ØÕÙÓÒÔÔÐÐ

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ØÖÒ×ØÓÒÖÑÑÙÔÖÑÖÓÖÖÒ×ØÖÓ×ÔÖÓ××Ù×ÑÔÐÕÙÒØ× ÕÙÖ×ÒØÕÙÖ×Ø×ÐÙÓÒ×ÄÒØÖØÓÒÑÒØÖÕÙÖ××ØÐ×× ÓÑÔØ××ÑÐØÙ×Ø×ÖÒ×ÒØÖÐ××ØÙØÓÒ×ÒÉØÉ 

ØÄÚÐÓÔÔÑÒØÓÑÔÐØ×ÐÙÐ××ØÖÓÙÚÒ×Ð×ÐÚÖ××ÔÐ×× ØØÖÔÔÐØÓÒÓÒÑÓÒØÖÔÖ×ÓÑÑÒØ×ÓÒ×ØÖÙØÐÑÔÐØÙ 

Ù×ÓÒÓÖØÙÒÕÙÖ×ÙÖÙÒÙØÖÕÙÖ×ÚÙÖ 

ÕÙÓÒ×ÕÙÐÖÑÑÖÓ×ÐÙÖÒÔ×ØÖÔÖ×ÒÓÑÔØ ÄÔÖÓ××Ù××ØÖÔÖ×ÒØÔÖÐÖÑÑÐÙÖÊÑÖ ÖÒØ 

cØaÐÔÒÒÓÙÐÙÖ ÔÙ×ÕÙÔÖÝÔÓØ×Ð××ÚÙÖ××ÕÙÖ××ÓÒØÖÒØ×ØÐÐ×ÕÙ ÊÔÔÐÓÒ×ÕÙÐ×ÓÒØÓÒ×uØvÓÒØÒÒÒØÐÔÒÒ×ÔØÓØÑÔÓÖÐÐÐ×ÓÒØÓÒ× 

µØν×ÓÒØ× ÓÑÔÓ×ÒØ×ÔÓÙÖcÓÑÔÓ×ÒØ×ÔÓÙÖaÄ×λ α×ÓÒØÐ× ÑØÖ×ÐÐÅÒÒÐ×f αβγ×ÓÒØÐ×ÓÒ×ØÒØ××ØÖÙØÙÖËÍ ÎÓÖÔÖÜÑÔÐÖÔÔÒÜ ×ÖØÙÒÖÑÑÓÒÙÖÖÒØ×Ð×ÕÙÖ×ØÒØÐÑÑ×ÚÙÖ 

ggÔÓÙÖ×ÕÙÖ×ÐÖÖØØÒ×ÐØØ×ÒÙÐØ×ÔÒ Ö ÔØÖÓÐÓÒØÖÓÙÚÙÒØÖØÑÒØÙ 

 

u + d → u + dÄÒÓØØÓÒ×ØÐÐÐÙÖ 

ÔÖÓ××Ù×ÒÒÐØÓÒqq → 

2 1ØÐÑÔÐÓÐÑØÓ×ÔÖØÙÖ 

1ØÐÖÓÙÖ×ÙÒ 

) ≪ 

) ≥ 

b (r, g, b)

lepton 

antilepton 

photon 

α,µ β,ν 

propagateur gluon 

−ig µ δ 

q q 

: 

q2 

αβ 

−ig µ 

ν 

q2 propagateur photon ν : 

vertex 

)( 

+ 

)( − 

entrant 

sortant 

entrant 

sortant 

entrant 

sortant 

propagateur lepton − 

antilepton 

γ 

QED QCD 

q 

: 

: 

: 

: 

: 

: 

: 

u( 

p) 

u( 

p) 

v( 

p) 

v( 

p) 

εµ 

( p) 

ε µ ( p) * 

. 

i( + m ) 

q 

q 

2 − 

m 2 

: − i e γ µ 

quark 

antiquark 

gluon 

propagateur 

antiquark 

vertex qqg 

entrant 

sortant 

entrant 

sortant 

entrant 

sortant 

quark − 

∼ 

( ( 

) 

: 6666 

∼ 

( ( 

) 

: 

6666 

: 

: 

q : 

u( p)c 

u( p)c 

v( p)c 

v( p)c 

εµ ( p)a α 

ε µ ( * p)a α* 

( 2 

i q+mq) 

q −m 2 

− f αβγ gs [gµ (k −k ν 1 2) λ+g λ(k −k ν 2 3) µ+g µ ( λ k3−k1) ν] 

f SU( 

3) 

αβγ 

pas de vertex γ γ γ 

vertex ggg 

k2 

γ,λ : 

k 

k3 

α,µ 

1 

ÙÔÖÑÖÓÖÖÒÉØÒÉ ÓÑÔÖ×ÓÒ×ÖÐ×ÝÒÑÒÔÓÙÖÐÚÐÙØÓÒÖÑÑ× 

= cste de structure d’ couleur 

 

)))))6666 

( 

β, ν 

6 666 

(666∼ 

∼ 

) 

( ( 

(666∼ 

∼ 

( α,µ 

6 666 

: 

: 

∼ 

: 

/− 

/− 

i 

− gs 

2 λα γ µ

Ò××ÔØÓØÑÔÓÖÐ× u(i)ciÔÓÙÖÐÐÖÐÖØÙÖ αØβ×Ò×ÓÙÐÙÖÔÐÙ×ÓÒÔÓ× 

u(pi)ci, ..., ≡ 

a) b) 

p , c 3 3 

p , c 4 4 4 3 

ÕÙÖ ÖÑÑ×ÙÔÖÑÖÓÖÖÖÔÖ×ÒØÒØÐÙ×ÓÒÓÖØÕÙÖ 

q 

ÇÒÒØÐ×ÓÙÖÒØ×ÕÙÖ× 

p, c p , c 

1 1 2 2 1 2 

j µα 

1 = [u(3)c † 

3][−i gs 

j νβ 

2 = [u(4)c † 

4][−i gs ÐÔÖÓÔØÙÖÙÐÙÓÒ αβ 

igµνδ 

− 

q2 ØÐÑÔÐØÙØÖÒ×ØÓÒ 

−iMqq = j µα 

αβ 

igµνδ 

1 − 

q2 

j νβ 

×ÔÒÙÖ×ÓÙÐÙÖÙØÖÔÖØÐÑÔÐØÙ×ÖØ ÒÖÖÓÙÔÒØÐ×ÔÖÓÙØ××ÔÒÙÖ××ÔØÓØÑÔÓÖÐ×ÙÒÔÖØØÙÜ 

2 

Mqq = −g 2 1 

s 

q2 [u(3)γµ u(1)] [u(4)γµu(2)] 1 

4 (c† 3λαc1)(c † ÇÒÒØg 2 s = 4παs, αs×ØÐÓÒ×ØÒØÓÙÔÐÓÖØ, 

fqq = 1 

4 (c† 3λ α c1)(c † 

4λ α c2),ÐØÙÖÓÙÐÙÖ 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

2 λαγ µ 

][u(1)c1] 

2 λβγ ν 

][u(2)c2] 

4 λα c2)

Ä×ÓÑÑØÓÒ×ÙÖÐÒÓÙÐÙÖα×ØÑÔÐØ 

ÓÒ×ÕÙÒØÐ×ÖÐØÓÒ× 

 

8 

ÖÒØÐÐ× ÓÙÐÙÖ×ÙÜÕÙÖ×ÒÒØÖØÓÒ×ØÖ×ÓØÐÒØØÖÔÐØÓÑÒ×ÓÒ× ÇÒÚÙÙ§ÕÙÐÚÐÙÖÙØÙÖfqq×ØÓÒØÓÒÐÓÒÙÖØÓÒ ×ÓÒØÔÔÐÐ×ÙÜ×ØÓÒ×× 

ÒØ×ÝÑØÖÕÙ××ÓØÐ×ÜØØÓÑÒ×ÓÒ××ÝÑØÖÕÙ×ÇÒÒØÐ ÐÙÐÔÓÙÖÐÔÖÑÖ×ÔÖÙÒÚÓÖÔØÙÓÒ×ØØÖÜÑÔÐÐ ÐÙÐÔÓÙÖÙÒØØØÝÔÕÙÙ×ÜØØrrÒÔÖØÒØÐÜÔÖ××ÓÒfqqÒ× ÇÒc1 = 

ÍÒÓÙÔÓÐÙÜÑØÖ×λαÐÐÅÒÒÚÓÖ ×ÙÐ×λ3Øλ8ÓÒØÙÒÒØÖÒÔÓ×ØÓÒ ÈÖÓÒ×ÕÙÒØ ØÔÔÖØÖÕÙ 

ÒÐÒÒÓÒÖÐØÚ×ØÓÒÚÙÕÙÐ×ÒÔÓ×ØfqqÓÖÖ×ÔÓÒÙÒ ÔÓØÒØÐÒØÖØÓÒÖÔÙÐ× Ù×ÓÒÓÖØÙÒÕÙÖ×ÙÖÙÒÒØÕÙÖ×ÚÙÖ ÖÒØ 

3 

ÔÙØÖÖÑÐ Ð××ÚÙÖ××ÓÒØÖÒØ×ÔÖÝÔÓØ×ÚÐÒÓØØÓÒÐÙÖ ÔÖÐÖÑÑÐÙÖÄÖÑÑ×ØÒÓÖÔÙ×ÕÙ ÓÒ ÇÒÙÒÔÖÓ××Ù×ÙØÝÔu + 

−iMqq = [u(3)c † 

3 ] 

 

 

1Ô×ØÙÖÓÙÐÙÖÈÖ 

 

α=1 

geØfqq = 

ÄÖÐØÓÒÖÔÖ×ÒØÙ××ÐÑÔÐØÙÙ×ÓÒÑe − + µ − → 

e− + µ −×ÐÓÒÔÓ×gs = 

⎛ 

c2 = c3 = c4 = ⎝ 

1 

0 

0 

Ó ⎞ 

⎠ 

f {6} 

qq = 1 

⎡ ⎛ ⎞⎤ 

⎡ ⎛ ⎞⎤ 

8 1 

1 

⎣(100)λα ⎝ 0 ⎠⎦ 

⎣(100)λα ⎝ 0 ⎠⎦ 

= 

4 

α=1 0 

0 

1 

4 

f {6} 1 

= 

qq 

4 [λ11 3 λ11 3 + λ11 8 λ11 8 

8 

(λ 11 

α=1 

α λ 11 

 

1 1 1 

] = (1)(1) + √3 √3 = + 

4 

1 

α ) 

d → u + dÓÒØÐÑÒ×Ñ×ØÖÔÖ×ÒØ 

 

−i gs 

[v(2)c † 

2] 

 

2 λαγ µ 

 

−i gs 

2 λβγ ν 

 

−igµνδ 

[u(1)c1] 

αβ 

q2 

[v(4)c4]

a) b) 

p , c p , c 

3 3 

9 

9 

9 

9 

9 

9 

4 4 

3 4 

9 

9 

9 

9 

ÒØÕÙÖ ÖÑÑ×ÙÔÖÑÖÓÖÖÖÔÖ×ÒØÒØÐÙ×ÓÒÓÖØÕÙÖ 

q 

p, c p , c 

1 1 

2 2 

1 2 

ÐÑÔÐØÙ×ÓÙ×ÐÓÖÑ ÒÖÖÓÙÔÒØÐ×ØÓÖ××ÔÒÙÖ×ÓÑÑÙÔÖÚÒØÓÒÔÙØÖÖ 

Mqq = −g 2 1 

s 

q2[u(3)γµ u(1)][v(2)γµv(4)] 1 

4 (c† 3λ α c1)(c † 

fqq1 

4 (c† 3λαc1)(c † 

2λα ×ÓÑÑØÓÒ×ÙÖÐÒαÑÔÐØ 1Ò× ØØÖÐØÓÒ 

c4) ËÐÓÒ×Ù×ØØÙgs → ge, αs → αØÐÓÒÔÓ×f = ÖÔÖ×ÒØÐÑÔÐØÙÙ×ÓÒÑe − + µ + → e− + µ + ØÙÖÓÙÐÙÖfqq ÓÙÐÙÖÓÙÒ×ÐÐÐÓØØÓÙÐÙÖØÓÒÒÓÒÒÐ×ÚÐÙÖ×Ù ÇÒÚÙÙ§ÕÙÐ×Ý×ØÑqqÔÙØØÖÒ×ÐÓÒÙÖØÓÒÙ×ÒÙÐØ 

ÒÒÐØÓÒÓÖØÙÒÒØÕÙÖØÙÒÕÙÖÑÑ ×ÚÙÖ 

. 

ÒÙÜÐÙÓÒ×ÚÓÖÙÖ ÌÖÓ×ÖÑÑ×ÙÔÖÑÖÓÖÖÓÒØÖÙÒØÙÜÔÖÓ××Ù×ÒÒÐØÓÒ ÈÓÙÖÐÖÑÑÓÒ 

−iMa = [v(2)c † 

2 ][−igs 

2 λβγ µ ][ε ∗ 4µ aβ∗ 4 ] 

 

i(/q + m) 

q2 − m2 

[−i gs 

2 λαγ ν ][ε ∗ 3νaα∗ ÊÔÔÐÐÓÒØβ 2ÚÙØαs/2 ÊÔÔÐÐÙÖ ØÐÒÓØØÓÒÖ /p = γ µ pµ 

Ú 

g 2 s4παs 

2λ α c4) 

3 ][u(1)c1]

a) b) 

p , ε , c 3 3 3 

p , ε , c 4 4 4 

α,µ 

β,ν 

) 

99 

99 

p,c 

1 1 

) 

q 

c) 

( 

9 

9 

3 

9 

9 

p , c 

9) 

2 2 

α,µ 

9 

9 

9 

9 

9 

9 

3 

) 4 

) 

α,µ 

9 

9 

γ,λ 

ÖÑÑ×ÙÔÖÑÖÓÖÖÖÔÖ×ÒØÒØÐÒÒÐØÓÒÓÖØ 

q 

δ,σ 

1 2 

qq → gg 

p3 

q2 − m2p 2 1 − 2p1p3 + p2 3 − m2 ÓÒÖÖÓÙÔÒØÐ×ØÙÖ×ÓÙÐÙÖÒÒÜÔÖ××ÓÒ ÐÙÓÒ×Ò×Ñ×× 

= −2p1p3 

Ma = −g 2 1 

s v(2)[/ε 

p1p3 

∗ 

4 (/p − 

1 /p + m)/ε 

3 ∗ 

3 ]u(1)1 

8 (aα∗ 3 aβ∗ 4 )(c† 2λβλ α 

ÔÓÙÖÐÖÑÑ ÓÒ×ÑÐÖÑ×ÒÒÚÖ×ÒØÐÓÖÖ×ÑØÖ×λÓÒÔÙØÖÖ 

βÑÔÐØ 

c1) ×ÓÑÑØÓÒ×ÙÖÐ×Ò×α, 

Mb = −g 2 1 

s v(2)[/ε 

p1p4 

∗ 

3 (/p − 

1 /p + m)/ε 

4 ∗ 

4 ]u(1)1 

8 (aα∗ 3 a β∗ 

4 )(c † 

2λ α λ β ×ÑÔÐØÙ×MaØMbÓÒØÐÙÖ×ÓÖÖ×ÔÓÒÒØ×Ò×ÐÒÒÐØÓÒÑ c1) 

e + e− ØÐ×ØÙÖ×ÓÙÐÙÖ 

γγ×Ò×Ø geØÓÒÐ×× 

→ → ÓÒØÐ×Ù×ØØÙØÓÒgs 

Å×qp1 − 

( 

9 

9 

9 

β,ν 

9 

9 

9 

9 

( 

( 

9 999 

q 

β,ν 

1 2 

) 

4

ÐÙÓÒ×Ò×ÖÔÔÓÖØÒØ×ÙØÐÙÖ ÄÖÑÑ ×ÖØÖ×ÔÖÐÔÖ×ÒÙÒÚÖØÜØÖÓ× ÓÒÔÙØÖÖ 

−iMc = [v(2)c † 

2][−i gs 

2 λδ γσ][u(1)c1] 

 

−i gσλ δ δγ 

q 2 

ÔÐÙ× Ó 

Å×qp3 + 

×ÓÑÑØÓÒ×ÙÖÐ×Ò×ÓÙÐÙÖÑÔÐØ ØØÑÔÐØÙMcÒÔ×ÓÖÖ×ÔÓÒÒØÒ×ÐÒÒÐØÓÒe + ØÙÖØÚØØÔÔÖÓÓÒÖÒÐÓÑÔÓÖØÑÒØØÖ×ÓÙÖØ×ØÒ Ä×ÔÔÐØÓÒ×××Ù×××ØÙÒØÒ×ÐÖÐØÓÖÉÔÖ ÊÑÖÕÙ× 

ÒÖÚÖ×ÐÐÑØÐÐÖØ×ÝÑÔØÓÑØÕÙØÓÒ×ÔÖÑ×ÔÖÓ ××Ù×ÔÖÑØØÒØØ×ØÖÐ×ÔÖØÓÒ×ØÓÖÕÙ×ÐÙ×ÓÒÒÐ×ØÕÙ ×ØÖÐÙÖ××ÒØÐ×ÔÖÓ××Ù×ÖÒØÖÒ×ÖØ ÐÒØÖØÓÒqq qq ÔÖÓÓÒÐÒÒÐØÓÒe + ÖÓÒÖÓÒÖÒØÖÒ×ÖØÒÖÐ×ÒØÖØÓÒØØ×ÕÙÖÓ ÒÙÑÖØÒ×ÒØÖÙÜÓÒØØÐÓØÔÖ×ÒØØÓÒ×Ò×Ð×ÔØÖ× 

ÒÔÖØÙÐÖÐÔÖÓÐÑÙÓÒÒÑÒØ×ÕÙÖ×ÒØÖÒ×ÐÖ ÔÖÒØ×ÙÓÙÖ× 

ÑÓÐ×ÔÒÓÑÒÓÐÓÕÙ×ÚÓÖÔØÖØÒ×ÐÙÐØÓÖÉ ÒÓÒÔÖØÙÖØÚØÓÖÙ×ÙÖÖ×ÙÖÒÖ×ÙØÒ×ØÔ×ÓÖ ÄØÙÙÓÑÔÓÖØÑÒØÐÓÒÙ×ØÒÐÒØÖØÓÒqqÓÙqqØ 

 

{−gsf αβγ [gµν(−p3 + p4)λ + gνλ(−p4 − q)µ + gλµ(q + p3)ν]}[ε µ∗ 

3 a α∗ 

3 ][εν∗ 4 aβ∗ 4 ] 

p4 

q22p3p4 ε ∗ 3p3 = ε ∗ 4p4 = 0ÓÒØÓÒÄÓÖÒØÞ 

1 

Mc = ig 2 s v(2)[(ε 

p3p4 

∗ 3ε∗4 )(/p − 

4 /p ) − 2(p4ε 

3 ∗ 3 )ε∗4 + 2(p3ε ∗ 4 )ε∗3 ]u(1)fαβγa α∗ 

3 aβ∗ 4 (c† 2λγc1) 

e − → γγ 

e −ÒÐÔØÓÒ×ØÖÓÒ×ÙØÒÖÐ×ÓÐÐ×ÓÒ×

Ä×ØÓÖ×ÙÐØÓÖÐØÖÓÐ 

Ï 

ÄÓÒÔØÁÒÚÖÒÂÙÁÂÚÒØÙÒÀÖÑÒÒÏÝÐ ÕÙ××ÝÙÒÖÐÖÚØØÓÒØÐÐØÖÓÑÒØ×ÑÒ×ÙÒØÓÖ ÁÒØÖÓÙØÓÒÓÒÔØÒÚÖÒÙ 

Ò×ÔÖ×ÓÒÔØ×ÓÑØÖÕÙ×ÐÖÐØÚØÒÖÐÀÏÝÐÒØÖÓÙØÙÒ ØÙÖÐÐÙÕÙÑÓÐÓÐÑÒØÐÓÑØÖËÓØÙÒÓÒØÓÒ 

ÈÖÓÒØÖ×ÐÙÒØÑ×ÙÖÐÙÚÖÙÒÔÓÒØÐÙØÖ ØÖÓÖÖÔÖÙÒØÙÖS(x)ÕÙÔÒÙÐÙ ÓØ 

×Ø××ÒØÔÖÓÐÑÒÓ×ÖÚÒØÐÕÙÚÐÒ ÎÓØÄÓÒÓÒÒØÀÏÝÐÒØÖÓÙÚÖÒØÙÒ×ÓÐÙØÓÒ 

µ×Ò××Ù× 

ÖØÙÖÐ××ÕÙ ÖØÙÖÕÙÒØÕÙ 

ÀÏÝÐÖÖÓÖÖ×ÔÓÒÖS µÙÑÔÑA 

ÇÒÑÓÒØÖÕÙ Ò×Ð§ ÓÒÓÑÑÒÔÖÓÒÒÖÐ×ÒØÓÒÐÁÂÒØÓÖÉ 

ÈÖÐ×ÙØÓÒÒÚÖ×ÐÖ×ÓÒÒÑÒØÒ×ÙÔÔÓ×ÒØÐÁÂÚÐÐÓÒÒÙØ Ð×ÐÓ×ÐÑÁÂ=⇒ÕÅÜÛÐÐ=⇒É 

ÕÅÜÛÐÐ=⇒ÁÂ 

ØØÔÔÖÓÔÙØ×ÒÖÐ×ÖÙØÖ×ØÝÔ×ÒØÖØÓÒ×ÔÖÜ ÁÂËÍ =⇒É 

Ô×ÐÓÒØÓÒÓÒ 

ØÄÇÄÐÐ×××ØÒÙÒØÔÖÐÔÖÑØÖÖØÖ×ÒØÐÒÑÒØ ÇÒÓÒ×ÖÙÜØÝÔ××ÝÑØÖÙÐÙÒØÄÇÄØÐÙØÖ ÓÙÐÙÖ) 

f(x)ÔÒÒØÐÒÖÓØxËÐÓÑØÖÙÒÐÐÙÒÓÖÑ∀x 

f(x + dx) = f(x) + ∂ µ f(x)dxµ 

f(x + dx) = f(x) + ∂ µ f(x)dxµ[1 + S ν 

dxν] 

= f(x) + (∂ µ + S µ )f(x)dxµ + 0(dx 2 ) 

p µ → p µ − eA µ ⇐⇒ i(∂ µ + ieA µ 

) ÓÒÐÀÏÝÐÙÖØÓÒØÓÒÒÒÔÝ×ÕÙÕÙÒØÕÙ×S µ → ieA µ

ËÝÑÄÓÐ ËÝÑÐÓÐ =ÓÒ×ØÒØ ψ(x) 

ψ(x) → ψ ′ (x)exp iα(x) ×ÖÓÙÔ×ÓÒÔÖÐÒÚÖÒ×ÓÙ×ÐÖÓÙÔU(1)×Ô×× ÇÒ×ØÕÙÐ×ÖÔØÓÒÔÝ×ÕÙÙÒ×Ý×ØÑÐÖÒ×ØÔ×ØÔÖÐ ÒÑÒØÐÔ×ÐÓÐ×Ù×ÔØÓÐÓÕÙÒÐÒÐØÓÖ 

ψ(x)Óα = α(x) 

A µ → A µ − ∂ µ Ð××ÒÚÖÒØÐ×ÖÔØÓÒÙÔÒÓÑÒÓÒ×Öf(x)×ØÙÒÓÒØÓÒ ÖØÖÖ×ÓÓÖÓÒÒ×ÄÁÂÒÑ×ØÑÓÒØÖÒÓ×ÖÚÒØÕÙÐ× 

ÕÙØÓÒ×ÙÑÓÙÚÑÒØ ÑÔ×EØB×ÓÒØÒ×Ò×Ð×ØØØÖÒ×ÓÖÑØÓÒÕÙ×ÖØ×ÙÖÐ× ÒÒÓØØÓÒÓÚÖÒØÐ×ÖÒÙÖ× 

f(x) 

Ø µν ν µ µ ν 

= ∂ A − ∂ A 

F µν = 1 

2 εµναβαβ ×ÓÒØÒ×Ò×Ð×ÐØÖÒ×ÓÖÑØÓÒ 

Ä×ÕÙØÓÒ×ÅÜÛÐÐ×ÓÒØÖØ××ÙÖÐ×µνØF µν 

 

Ú(E) 

Ä×Ù×ØØÙØÓÒ 

ÖÓØ(B) 

Ò×ÓÒÒ 

 

ÖÓØ(E) 

ËÐÓÒÓÔØÐÙÄÓÖÒØÞ 

Ú(B) 

ÇÒÔÓ×4π c = 1. 

→ ψ ′ (x)exp iα Óα ψ(x) 

ÁÒÚÖÒÙÒØÓÖÉ ËA µ×ØÐÑÔÑÓÒ×ØÕÙÐØÖÒ×ÓÖÑØÓÒÙ 

 

ν 

→ ∂µµν 

= j 

→ ∂µF µν 

= 0 

✷A µ − ∂ µ (∂νA ν ) = j µ 

∂νA ν 

= 0

ÓÒÓØÒØÙÒÓÖÑÐÃÐÒÓÖÓÒÔÓÙÖÙÒÔÖØÙÐÑ××ÒÙÐÐ 

✷A µ = 0 

ÓÒÓÖÖ×ÔÓÒÒØ×ÖØÕÈÖÓ 

0ÄÕÙØÓÒ 

(✷ + M 2 )A µ − ∂ µ (∂νA ν ) = j µ 

µ ÇÒÔÙØ×Ù×ÔØÖÕÙÐÓÒ×ÖÚØÓÒÙÓÙÖÒØÑØÒÔÖØÙÐÖÐÓÒ×Ö 

= 0 ÚØÓÒÐÖj 0ÒÐ×ÒÙÒ×ÓÙÖ×ÓØÙÒÓÒÓÙÙÒÙØÖ 

ÙÒØÖÚÐWÈÓÙÖÒÒÐÖØØÖÐÙØÒÖÙÒØÖÚÐ−WÔÖ ÖÐØÖÕÙ×ØÓÒ×ÖÚËÙÔÔÓ×ÓÒ×ÕÙÓÒÖÙÒÖqÒÒÒØ ÓÒÒØÐÒÚÖÒÙØÓÙØÓ×ØØÓÒÒÜÓÒÒ×ØÔ×ÑÑØ 

ÓÒ×ÖÚØÓÒÐÒÖËÐÖq×ØÖÒx1ØÔÐÒx2Ø×Ð ÅÒØÓÒÒÓÒ×ÐÖ×ÓÒÒÑÒØÈÏÒÖ ÔÓÙÖÑÓÒØÖÖÕÙÐ 

Ü×ØÙÒÔVÒØÖx1Øx2Ð×Ý×ØÑÒÙÒÒÖqVÙØÓØÐ 

ÐÓÒÐÙ×ÓÒÕÙÐÖqÒÔÙØÔ×ØÖÖÐÐÒÔÙØØÖÕÙÓÒ×ÖÚ ÄÔÖÒÔÐÓÒ×ÖÚØÓÒÐÒÖØÐÜ×ØÒÔÓØÒØÐ×ÑÒÒØ 

E 

ÉÔÙØÓÒÓÒØÓÙÖÒÖÖ×ÓÒÒÑÒØ ÈÓÙÖÑØØÖÒÚÒÐØÐÁÂÔÐÓÒ×ÒÓÙ×Ò×ÐÔÔÖÓÜÑØÓÒ 

+W 

ÒÓÒÖÐØÚ×ØÄÕÙØÓÒËÖÒÖÒÔÖ×ÒÙÒÑÔÑ×Ø 

ÓÒ×ÖÓÒ×ÐØÖÒ×ÓÖÑØÓÒÙ 

Ó: 

ÄÖÐØÓÒ×ØÒÚÖÒØ×ÓÙ×ÐØÖÒ×ÓÖÑØÓÒÙA µ ⇒ A µ − 

∂ µ ÕÙÒ×ØÔ×ÒÚÖÒØÙÓÒÐÒÚÖÒÙÑ×ØÖÐÙØ ÕÙÐÑ××ÙÔÓØÓÒ×ØÒÙÐÐÚÓÖÔØÖ 

f(x)ÁÐÒÒ×ÖØÔ×Ò××ÐÔÓØÓÒÚØÙÒÑ××M = 

Ñ ÓÙØÓÒ×ÕÙÐ×ÕÙØÓÒ×ØÓÒØÒÒÒØÐÓÒ×ÖÚØÓÒÙÓÙÖÒØ 

∂µj 

+qV −W qV 

 

1 

2m (−i∇ + qA)2 

+ qV ψ(x, t) = i ∂ψ 

∂t (x, t) 

A → A ′ 

= A + ∇f 

V → V ′ = V − ∂f 

 

∂t 

f = f(x, t)

ËÐÓÒÒ×Ö Ò× ÐÕÙØÓÒËÖÒÖÔÓÙÖ×ÓÐÙØÓÒÐ ÓÒØÓÒψ ′ØÐÐÕÙ | ψ ′ 

(x, t) |=| ψ(x, t) | , ÖÓÒÚÙØÕÙÐÒ×ØÔÖÓÐØ×ÓØÓÒ×ÖÚÓÒψ ′ØψÒ ØÖÒ×ÓÖÑØÓÒÙÓÑÔÐØ×ØÓÒ ÖÒØÕÙÔÖÐÙÖÔ×ÇÒÚÖÜÖÕÙØØÔ×ÚÙØexp{iqf}Ä 

A → A ′ = A + ∇f 

V → V ′ = V − ∂f 

∂t 

ψ → ψ ′ ÓÖÑÓÚÖÒØÐØÖÒ×ÓÖÑØÓÒ t)ÓÒÐØÖÒ×ÓÖÑØÓÒÙ×ØÐÓÐËÓÙ× ×ÖØ = exp{iqf}ψ ÊÑÖÕÙÞÕÙf = f(x, 

A µ → A ′µ = A µ − ∂ µ f 

ψ → ψ ′ 

= exp{iqf}ψ 

ÄÜØÒ×ÓÒÙØÖØÑÒØÖÐØÚ×Ø×ÖÜÑÒÙ§ Ò×ÐÔÖÖÔÔÖÒØÐÕÙØÓÒËÖÒÖ×ØÔÖ×ÓÑÑ ØÓÒÐØÓÖÐÑ ÄÒÚÖÒÙÓÑÑÔÖÓÖÑÑÓÒ×ØÖÙ 

ØÓÒ×ØÒÚÖÒØ×ÓÙ×ÐØÖÒ×ÓÖÑØÓÒÙ ÔÓÒØÔÖØÆÓÙ×ÚÓÒ×ÚÖÕÙÐÝÒÑÕÙÓÒØÒÙÒ×ØØÕÙ ÄÓÒØÓÒÓÒ×Ø 

=Ô×ÐÓÐ Ù§ÓÒÒÚÖ×ÖÖ×ÓÒÒÑÒØ ÓÒÑÒÖÕÙÐØÓÖ×ÓØÒÚÖÒØ×ÓÙ×ÐØÖÒ×ÓÖÑØÓÒÙ 

ÑÒÓÙÐÖÓÒØ ÓÒÒÙÖÕÙÐ×Ý×ØÑÖØ×ØÓÖÑÒØÒÒØÖØÓÒÐ×Ý×ØÑ 

VÄ×ÕÙØÓÒ×ÔÖÓÔØÓÒÚÒØÖØÓÒ 

∼ 

ÖÞÐ×ÐÜÔÖÒÓÙÒÚÙÒÖÒÙÜØÖÓÙ×ÚÓÖÙÖ ÇÒÓÒÓØÒØÙØÚÑÒØÕÙÙÒÒÑÒØÐÔ×ÐÓÐÐÓÒØÓÒ ØÒØØÔÖÙÒÑÔA, 

 

ÓÒψ → 

ÑÓÔÖÐÒÑÒØÐÔ×ÐÓÐÓÒØÓÒÙÐÙ 

ψ(x, t) → ψ ′ (x, t) = exp{iqf(x, t)}ψ(x, t) 

ψe iα(x)×ØÒÓÑÔØÐÚÐ×ÖÔØÓÒÙÒ×Ý×ØÑÐÖÓÒ×

Ψ1 

Ψ 

ËÑÐÜÔÖÒÓÙÒ 

Ψ2 

ÄÓÒØÓÒψ×ÙÖÐÖÒ×ØÐ×ÙÔÖÔÓ×ØÓÒÓÖÒØ×ÙÜÓÒØÖÙØÓÒ× 

ψ1Øψ2ÄÒØÒ×ØÐØ×ÙÖÐÖÒ×ØÔÖÓÔÓÖØÓÒÒÐÐ| ψ | 2 

| ψ | 2 =| ψ1 + ψ2 | 2 = | ψ1 | 2 + | ψ2 | 2 +2Reψ ∗ 1ψ2 = | ψ1 | 2 + | ψ2 | 2 ÓÒ×ØÒØ×Ò×ÐØÖÖÐÖ×ÙÐØØÔÝ×ÕÙÈÖÓÒØÖÐ×ØÜÐÙÒØÖÓÙÖ 

+2 | ψ1 || ψ2 | cosδ 

Ò××Ý×ØÑÐÖÙÒÔ×ÐÓÐe iα(x,t)×Ò×ØÖÐÓÖÒÒØÖ 

ψ1Øψ2ÐÒÚÖÒÙÐÓÐ ÐÖÁÐÙØÒ××ÖÑÒØÒØÖÓÙÖÙÒÑÔÕÙ×××ÙÖÐÔÖØÙÐ Ò×ØÔ×ÙÒ×ÝÑØÖÙ×Ý×ØÑ 

ÙØÖÑÒØØÒØÖÑÐÁÂÐÓÐÐ×ÚÖÑÔÓ××Ð×ØÒÙÖÐØ 

ÑØÑØÕÙ×Ä×ÝÑØÖÐÓÐ×ØÐÐÙÖ×ÒØÖ××ÒØÒÐÐÑÑ ÑÒÖÐ×ÙÒ×ÝÑØÖÐÓÐÕÙÒÓÙ×ÔÖÑØØÖÒÖÐ×ÓÙØÐ× ÚÒØÓÒ×ÖÖÐ×ØÙØÓÒÙÒ×ÝÑØÖÙÐÓÐÓÒÚÜ 

ÌÖÒ×ÓÖÑØÓÒÙÐÓÐ 

ÙÒÑÔÓÖ×ÐÙÙÒÒÑÒØÔ×ψ 

ÇÒÒØÖÓÙØÓÑÑÙÒÑÒØÐÖÐÔÖØÙÐqÒ×ÐØÙÖÔ× 

qθÓ 

=ÓÒ×ØÒØ 

α ÒÔÓ×ÒØα = 

ψ → ψ ′ = e iqθ ØØØÖÒ×ÓÖÑØÓÒÙ×ØÔÔÐÐÙÜÕÙØÓÒ×ÖÐÖØ Ú =ÓÒ×ØÒØ 

ψ θ 

Óδ×ØÐÖÒÔ×ÒØÖψ1Øψ2ÁÐ×ØÐÖÕÙÓÒÔÙØÑÙÐØÔÐÖ ×ÑÙÐØÒÑÒØψ1Øψ2ÔÖÙÒØÙÖÔ×ÓÑÑÙÒe iαÓα 

ÑÔÒØÖØÓÒ 

∈Ê= 

ÒÚÖÒÙÐÓÐ⇐⇒ 

ËÓØÐØÖÒ×ÓÖÑØÓÒÙÐÓÐ 

ψ → ψ ′ = e iα Ú ψ 

ÃÐÒÓÖÓÒÐÖ

ÓÒ×ÖÓÒ×ÓÒÒÖÐÐØÖÒ×ÓÖÑØÓÒÓÒØÒÙÔÖÑØÖ 

Uθ ≡ e iq1θ Óq 

∈ÖÓÙÔU(1) 

Uθ 

Uθ : ψ → ψ ′ ÇÒÙ××ÐØÖÒ×ÓÖÑØÓÒÒÒØ×ÑÐ = Uθψ 

=Ö 

θ =ÔÖÑØÖ 

1 =ÒÖØÙÖ 

 

ψ → (1 + iqθ)ψ ÄÒÚÖÒÐÑÐØÓÒÒÙ×Ý×ØÑ〈ψ|H|ψ〉 = 〈ψ ′ |H|ψ ′ ÑÙØØÚØqØH 〉ÑÔÐÕÙÐÓÑ 

ÑÓÙÚÑÒØ ÕÙ×ØÕÙÚÐÒØÖÕÙÐÖq×ØÓÒ×ÖÚq×ØÙÒÓÒ×ØÒØÙ 

[H, q] = 0, 

ÈÐÙ×ÓÑÑÙÒÑÒØÓÒÜÔÐÓØÐÁÂÐÓÐÙÄÖÒÒ 

L(x, t) ≡ L(x) = L(ψ(x), ∂ µ ψ(x)) 

ÇÒÖÕÙÖØÕÙL(x, t)×ÓØÒÚÖÒØ×ÓÙ×ÐØÖÒ×ÓÖÑØÓÒψ → 

∂(∂ µ ψ) δ(∂µ 

ψ) Ñ×: δ(∂ µ ψ) = iqθ∂µψ Ö: ∂(∂ µ ψ) iqθ∂µψ ∂ψÔÖ×ÓÒÜÔÖ××ÓÒØÖÐÕÙÐÖÄÖÒ 

ÊÑÔÐÓÒ×∂L 

 

ÖÝÓÒÕÙÝÔÖÖÈÓÙÖËÍÆÓÒÙÖÙÒ×ØÙØÓÒ×ÑÐÐÔÖÜËÍ qÔÙØØÖÙÒÖÑÓÙØÓÙØÙØÖÒÓÑÖÕÙÒØÕÙØÓÒ×ÖÚÖÒÓÑÖ 

ÚÓÖ§ ÓÙÐÙÖ 

 

 

ψ + δψ 

0 = δL = ∂L ∂L 

δψ + 

∂ψ 

∂µθ 

0 = ∂L ∂L 

iqθψ + 

∂ψ 

 

∂ 

0 = iθ 

∂x µ 

∂L 

∂(∂ µ ∂L 

qψ + 

ψ) ∂(∂ µ ψ) q∂µψ 

0 = iθ ∂ 

∂x µ 

 

∂L 

∂(∂ µ ψ) qψ 

= 0 Ó: 

ψ + iqθψ =

ÇÒÒØÐÓÙÖÒØ××ÓÐØÖÒ×ÓÖÑØÓÒÙ 

j µ = iq ∂L 

∂(∂µψ) ψ 

ÄÖÐØÓÒÜÔÖÑÕÙÐÚÖÒÑÒ×ÓÒÒÐÐ 

∂µj µ 

= 0 ×ØÖÕÙÐÓÙÖÒØj µ×ØÓÒ×ÖÚØ×Ø××ÓÐ×ÝÑØÖU(1). ÇÒÙÒÔÔÐØÓÒÙØÓÖÑÆÓØÖ 

∃ÙÒÓÙÖÒØÓÒ×ÖÚ ÄÄÖÒÒÙÒÔÖØÙÐÖÐÖL=ψ(iγ µ ×ØÒÚÖÒØ×ÓÙ×ÐØÖÒ×ÓÖÑØÓÒψ→e iqθ ØÓÒÙÓÙÖÒØ 

ψÔÔÐÕÙÓÒ×Ò××ÐÒ 

j µ = iq ∂L 

∂(∂µψ) ψ = iq(ψiγµ )ψ = −qψγ µ 

ÇÒÚÑÒØÒÒØÖÐ×ÖÐÔÖÓÖÑÑÒÒÓÒÙ§ ÌÖÒ×ÓÖÑØÓÒÙÐÓÐ 

ψ 

ÄØÖÒ×ÓÖÑØÓÒ 

ÇÒÓØÒØÐÓÙÖÒØÖÓÒ×ÖÚ××Óq 

ÙÐÓÐ×Øψ(x) ′ ÄÖÒØψ×ØØÔÖÐÒÑÒØÔ× 

ÇÒÚÓØÕÙ∂µψ ′ ÖÚDµØÐÐÕÙ ÓÙÃÐÒÓÖÓÒÐÖ×ÈÓÙÖÖØÐÖØØÓÑÔØÐØÐÙØØÖÓÙÚÖÙÒÓÖÑ 

(Dµψ) ′ → e iqθ(x) 

 

 

Dµψ 

∃ÙÒ×ÝÑØÖÓÒØÒÙÐÓÐ 

⇕ 

∂µ − m)ψÔÖÜÑÔÐ 

→ e iα(x) ψ(x) = e iqθ(x) ψ(x) 

∂µψ ′ (x) → e iqθ(x) 

⎡ 

⎤ 

⎣∂µψ(x) + iqψ(x)∂µθ(x) ⎦ 

 

E 

= eiqθ(x) ∂µψÕÙ×ØÒÓÑÔØÐÚÐ×ÕÙØÓÒ×Ö 

ÇÒÐÓØÒØÒÒ××ÒØÐÖÚÓÚÖÒØ 

Dµ = ∂µ + iqAµ

ÓAµ×ØÙÒÑÔÙØÐÕÙ 

Aµ(x) → A ′ 

µ(x) = Aµ(x) − ∂µθ(x) ØÚÑÒØDµ → (Dµψ) ′ = D ′ µ ψ′ 

= (∂µ + iqA ′ µ )ψ′ = (∂µ + iqAµ − iq∂µθ)e iqθ ψ 

= ∂µ(e iqθ ψ) + iqAµe iqθ ψ − iqe iqθ ∂µθψ 

= e iqθ ∂µψ + iq∂µθe iqθ ψ + iqAµe iqθ ψ − iqe iqθ ∂µθψ 

= e iqθ (∂µ + iqAµ)ψ 

= e iqθ ÓÑÑ×Ö 

ÓÒÓØÒØÐÓÖÑÚÒØÖØÓÒÑ ÒÔÔÐÕÙÒØÐÒØÓÒÙÄÖÒÒ Dµψ 

L = ψ(iγ µ 

(∂µ + iqAµ) − m)ψ 

= ψ(iγ µ ∂µ − m)ψ − qψγ µ ÄÔÖØÒØÖØÓÒ×Ø 

ψAµ 

LÒØ= −qψγ µ ψAµ = j µ 

Aµ Ój µ×ØÐÓÙÖÒØÕÙÚÒØÐ×ÝÑØÖU(1)ÙÄÖÒÒ 

•ÒÖ×ÙÑÑÒÖÐÒÚÖÒÙÐÓÐÑÔÓ×ÐÑÓØÓÒ ÐÀÑÐØÓÒÒØÙÄÖÒÒÐÖ×ÔÖÐÒØÖÓÙØÓÒÙÒÑÔÙ 

A µÐÔÙØ×ÖÔÖÐ×Ù×ØØÙØÓÒ ∂ µ → ∂ µ + iqA µ ≡ D µ 

•ËÓÒÓÙØÒ×ÐØÖÑÕÙÖØÐÔÖÓÔØÓÒ×ÔÓØÓÒ×ÐÖ× 

−1 ×Ò×Ñ×× ÄÔÔÐØÓÒÐÕÙØÓÒÙÐÖÄÖÒÑÒÐÕÙØÓÒÓÒÙÒÑÔ 

4 

✷A µ − ∂ µ (∂νA ν ) = j µ 

L = ψ(iγ µ ∂µ − m)ψ 

ÓD µ×ØÐÖÚÓÚÖÒØÆÐÁÂÑÔÐÕÙÐÓÙÔÐÑÒÑÐ µνÓÒÓØÒØÐÄÖÒÒØÓØÐ 

FµνF 

L = ψ(iγ µ ∂µ − m)ψ + j µ Aµ − 1 

4 FµνF µν

ÊÑÖÕÙÞÕÙÒ×Ð×ÙÒÑÔÑ××ÓÒÙÖØÙÒØÖÑ✷ + m2 )A µ ×ØÒ×ÓÑÔÐØ ÕÙÒ×ØÔ×ÒÚÖÒØÙ 

×ÓÒÔÐÙ×ÙÒÑÔÙÒÔÖØÒØÙÄÖÒÒÔÔÖÓÔÖ ÍÒÖ×ÓÒÒÑÒØ×ÑÐÖ×ÔÔÐÕÙÒ×Ð×ÖÔØÓÒÙ×Ý×ØÑÙÒÓ Ä×ÖÔØÓÒÙ×Ý×ØÑÙÒÖÑÓÒÔÐÙ×ÙÒÑÔÙÒÒØÖØÓÒ 

ÇÒØÙ×ÔØÖØÖÒ×ÓÖÑØÓÒ×ÐÓÐ×ØÝÔËÍÆ ÒÖÐ×ØÓÒ×ØÖÒ×ÓÖÑØÓÒ×ÙÐÓÐ× 

Ð×ÒÒÖÙÒ×ÔÆÑÒ×ÓÒ×ÊÔÔÐÓÒ×ÒÙÜÜÑÔÐ× 

−1ÔÖÑØÖ×Ò×ÐÖÔÖ×ÒØØÓÒÓÒÑÒØÐ ÓËÍÆ×ØÙÒÖÓÙÔÆ2 

p 

= 

n ØÖÒ×ÓÖÑØÓÒψ ′ = UψÚU = exp(iθi ÈÙÐ τi/2)ÓÐ×τi×ÓÒØÐ×ÑØÖ× 

⎠×ÓÒØÐ×ÔÖÑØÖ×ØÖÒ×ÓÖÑØÓÒ 

⎛ ⎞ 

θ1 

θ ≡ ⎝ θ2 

θ3 ËU×ØÙÒ×ÝÑØÖÙ×Ý×ØÑÐÓÖ×Ð×Ó×ÔÒI ≡ τ 2×ØÓÒ×ÖÚI 2ØI3×ÓÒØ ÓÒ×ÖÚ× 

⎛ ⎞ 

⎛ ⎞ 

ÜÑÔÐ: ËÍ(2)×Ó×ÔÒ: ψ 

ÜÑÔÐ Ø : ËÍ(3)×ÚÙÖ: ⎠ ËÍ(3)ÓÙÐÙÖ: ψ ØÖÒ×ÓÖÑØÓÒψ ′ = UψÚU exp(iαiλi/2)ÓÐ×λi×ÓÒØÐ×ÑØÖ× •ÓÒ×ÖÓÒ×Ð×ÙÒ×ÝÑØÖËÍ ÒÖÐ×ØÓÒ×ØÖÒ×ÓÖÑØÓÒ×ÙÐÓÐ× ØÓÖ×ÒØÅÐÐ× 

ÐÐÅÒÒØα1, α2, 

 

×ÖØ ÑØØÓÒ×ÕÙÐ×ÔÖÑØÖ×αÔÒÒØÐÔÓ×ØÓÒxÒ×Ð×ÔØÑÔ× 

gθ(x)ÐØÖÒ×ÓÖÑØÓÒ 

 

ØÒØÖÓÙ×ÓÒ×ÙÒÖgØÐÐÕÙα=α(x) = 

ψ 

ψ = 

ψ1 

α8×ÓÒØÐ×ÔÖÑØÖ×ÐØÖÒ×ÓÖÑØÓÒ 

u 

r 

ψ ≡ ⎝ d 

≡ ⎝ g ⎠ 

s 

b 

= 

..., 

ψ2 

 

→ ψ ′ 

= exp iα τ 

 

 

′ ψ 1 

ψ = exp(iα · T)ψ = 

2 

ψ ′ 2 

′ 

= exp(igθ · T)ψ

ÐØÓÖØÖÒÚÖÒØ×ÓÙ× ÁÒØÖÓÙ×ÓÒ×ÐÖÚÓÚÖÒØÖ 

§ ÓÒÑÔÓ× 

 

ËÒ×ÔÖÒØÙÖ×ÓÒÒÑÒØÚÐÓÔÔÔÓÙÖÐ×ÝÑØÖU(1) 

Dµ Ð×W µ×ÓÒØ ÑÔ×Ù×Ò×Ñ×× 

ÙÒÓÒÒÖÐÐ×ÒÖØÙÖ×ËÍÆÓÆÒÓÑÑÙØÒØÔ×Ò ÄØÙÖÑÙÐØÔÐØg×ØÙØÕÙÐÖÓÙÔËÍ ×ØÒÓÒÐÒ 

3 ) ËÐÓÒÑÒÕÙDµψ → 

ÓÒ×ÕÙÒÐÒØÖÓÙØÓÒÐÖÚÓÚÖÒØD µÒ×ÐÕÙØÓÒÖ 

ÓÙÒ×ÐÄÖÒÒL = ÚÔÖÓÙÖ×ØÖÑ×Ògψγ µ µÕÙÖÔÖ×ÒØÒØÐÒØÖØÓÒ×ÑÔ× 

Tψ)·W 

W µØÙÓÙÖÒØj 

ÉØÐÖÐÜÔÖ××ÓÒ×ØÖÑ×ÙØÓÒØÖØÓÒ ÒØÖØÓÒÙØÝÔWWWØWWWWÚÓÖÐÙÖ ÖÇÒØÖÓÙÚÔÐÙ××ØÖÑ×ÙØÓ 

•ÇÒÔÙØØÖØÒØÔÔÐÕÙÖ×Ò×ÙØÖÐÔÖÓÙÖÐÒØÖØÓÒÐ Ò×ÒØÐ××ÓØÓÒ{W1 W2 ÔÖÓÐÑÙØÕÙÐØÖÑÐÑ××Ò×ÐÄÖÒÒ∼m 2 •ÓÒ×ÖÓÒ×Ð×Ð×ÝÑØÖËÍ ÙÐØÖÙÖÑÒØÓÑÑÒØÖÑÖÐ ÐÓÖÖ Î×ØÒÓÑÔØÐÚÐÒÚÖÒÙÇÒÚÖÖ 

ÐÖÓÙÐÙÖ ÊÔÔÐÓÒ×ÕÙÐÓÒØÓÒÓÒψÙÒÕÙÖÓÑÔÓÖØÙÒÔÖØÔÒÒØ ÓÙÐÙÖ 

 

ÚÓÖÔÖÜÖ§ 

qψγ µ ψAµÐÒØÖØÓÒÑ 

µ 

= ∂µ + igT · W µ Ó (x), 

W µ ≡ (W µ 

1 , W µ 

2 , W µ θ)DµψÓÒØÖÓÙÚÐÒÐÓÙ 

exp(igT · 

W µ → (W µ (x) − ∂ µ θ(x) − gθ(x) · W µ 

(x) 

(iγµD µ − m)ψ = 0 

[−i(Dµψ)γ µ − mψ]ψ 

= ψγ µ TψÚÙÒÓÙÔÐÙÒÕÙgÇÒÐÒÐÓÙ 

W3} ⇒ {W ± Z}ÌÓÙØÓ×ÓÒ×ÙÖØÙÒ 

, 

W µ WµÑ 

⎛ 

|ÓÙÐÙÖ〉 = ⎝ 

r 

g 

b 

⎞ 

⎛ 

⎠ ⇒ ⎝ 

ψ1 

ψ2 

ψ3 

⎞ 

⎠

a) b) c) 

W W W 

ÖÑÑ×ÒØÖØÓÒ×ÚÐÑÔWÒØÖØÓÒÚÙÒ 

W W 

ÓÙÖÒØÖÑÓÒØ ÙØÓÒØÖØÓÒ× W W W 

ÒØÖØÓÒ×ÓÒØÐÓÖÑ Ð×ÐÙÓÒ×ÕÙÒØÖ××ÒØÚÐ×Ö×ÓÙÐÙÖ×ÕÙÖ×Ä×ØÖÑ× 

gsψγ µλα 

2 ψAα = gsj α ÔÕÙ×ÙÒÖÓÙÔÒÓÒÐÒ×ÓÒØÐ×ÒØÖØÓÒ×ØÐÙÓÒ×§ ÔÐÙ×ÓÒ×ØÖÑ×ÒØÖØÓÒÙÑÔÙÚÐÙÑÑØÝ ÒÖ×ÙÑ 

Aα 

ÒØÖ××ÒØÚÐ×ÖÑÓÒ×ØÒØÖÙÜ×Æ ÄÚÖ×ÓÒÐÓÐÐÑÑ×ÝÑØÖÙËÍÆÖÒØØÕÙÐ× ÑÔ×ÙÓÒØÐ×ÕÙÒØ×ÓÒØ×Ó×ÓÒ××Ò×Ñ×××Ó×ÓÒ× Ö××ÓÒØÓÒ×ÖÚ×Ä×ÓÙÖÒØ×ÑÓÙÐÙÖ ×Ø×ÓÒØÐÕÙ 

Ò×ÐÝÔÓØ×ÙÒ×ÝÑØÖÙÐÓÐËÍÆÓÒÓØÒØÆ2 

ØÓÒÓÒØÒÙØ∂ µ jα ØÖÒÔÖÑÔ×ÐÖÓ×ÓÒ×À×Ø 

α×Ø××ÓÙÜÖ×ÒÕÙ×ØÓÒ 

ÙÐØÓÖØÔÖÓÒ×ÕÙÒØÐÚÐØÐÔÖÓÙÖ ×ØÐÔÓ××ÐÚÓÖ×ÑÔ×ÙÑ××××Ò×ØÖÙÖÐÒÚÖÒ Ó×ÓÒ×ÓÐ×ØÓÒ 

µ ≡ 0 

ÒØÓÒÒØÖÓÙØ×ÑÔ×ÔÓÙÖÓÒÒØÖ×ÔÓÒØ×Ð×ÔØÑÔ× ××ØÒ×ÖØÖÖÑÒØÖÒ×ÐÚØÙÖÐÓÒÒÜÓÒÓØÚÓÖÙÒ ÓÐ×ÓÒØÓÒ×ÓÒÓÒØ×Ô××ÖÒØ××ÔÓÒØ×ÔÓÙÚÒØ×ØÖÓÙÚÖ ÍÒÖ×ÓÒÒÑÒØÙÖ×ØÕÙÑÒÔÖÑÓÖÙÒÖÔÓÒ×ÒØÚ 

ÔÓÖØÒÒÓÙÒÑÔÙÑ××ÒÙÐÐ ÊÔÔÐgs = √ =ÓÒ×ØÒØÓÙÔÐÓÖØ 

4παs, αs 

ÄÁÂÒÓÙ×ÓÒÙØÒØÖÓÙÖÑÔ×Ù×Ò×Ñ××A α , α = 1, ...8

×ØÙÒØÖÒÐÜØØÓÒÙÑÐÙÕÙ×ÓÔÔÓ×ÙÒÑÒØÄ Ñ××ØÚÐÓÖ×ÙÔ××Ò×ÙÒÑÐÙÓ×ÓÙÖÒØ×ÔÙÚÒØ×ØÐÖ ×ÐÑØ×ØÐÙÙ×ÙÔÖÓÒÙØÙÖÕÙÜÔÙÐ×ÐÑÔB×ÓÒÒØÖÙÖ ÌÓÙØÓ×ÓÒ×ÜÑÔÐ×ÒÔÝ×ÕÙÙ×ÓÐÓÐÔÓØÓÒÔÖÒÙÒ 

ÑÔÑ×× ÄÒØÖÔÖØØÓÒØØ×ØÐ×ÙÚÒØÒ×ÐÙÄÓÖÒØÞÓÒ ÓÒMÔÓØÓÒ→ ∞ ÓÒÓØÒØ✷A = 

(✷ + M 2 ËÙÔÔÓ×ÓÒ×ÕÙÓÒÖÖÔÒØÖÖÑÐÙÔÖÙÒÑÔBÄÖÔÓÒ× 

)A = 0 

Ä×ÓÐÙØÓÒÐÕÙØÓÒ×ØÙØÝÔ 

ÙÑÐÙ×ÜÔÖÑÔÖ∇ × ËÐÓÒÔÖÒÐÖÓØØÓÒÒÐØØÜÔÖ××ÓÒÓÒÓØÒØÚ∇ × 

ÞÓ×ÓÒ×ÓÒÒÒØÐÙÙÒÓÙÖÒØÑÖÓ×ÓÔÕÙÐÐ×ÔÙÚÒØØÖØÓÙØ× Ò××ÓÙ××ØÑÔÖØÙÖÖØÕÙTcÄ×ÔÖ×ÓÓÔÖÓÒ×ØØÙÒØÙÒ ÍÒÜÑÔÐØØ×ØÙØÓÒ×ØÐÐÙÒÓÖÔ××ÙÔÖÓÒÙØÙÖÑÒØÒÙ 

Ò×ØØÜÔÖ××ÓÒQ = 2ÒÙÒØe 2Óns×ØÐÒÓÑÖe −×ÙÔÖÓÒÙØÙÖÔÖÙÒØÚÓÐÙÑÇÒÓØÒØ ÐÕÙØÓÒÄÓÒÓÒÒÔÖÒÒØÐÖÓØØÓÒÒÐ 

ÄÐÓÒÙÙÖÔÒØÖØÓÒ×Ø 

nsÑÒÙÖÙ×ÕÙÑÒØØÒ 

0)ÚÓÖÙÖ 

≃ ÈÓÙÖns ÓÖÖ×ÔÓÒÙÒÑ××M = ×ØÐÚÙ××Ù××ØÑÔÖØÙÖÖØÕÙTc, ÓÒ×ÕÙÒÐÐÓÒÙÙÖÔÒØÖØÓÒλÖÒØ(M ≃ ÇÒÔÓ×4π 

✷A = jËÐÓÙÖÒØj×ØÖÔÖÐÔÖ×ÒÙÒÑÔA×ÓÒ×j = −M2A −M2AÕÙ×ØÕÙÚÐÒØÐÕÙØÓÒÔÖÓÔØÓÒÙÒ B = j = (−M 2 B 

A) 

A = 

∇ 2 B = −M 2 

B 

B = B0 exp −Mx ÖØÖ×ÔÖÐÐÓÒÙÙÖÔÒØÖØÓÒ 

Ò×ÐÑÑØØÕÙÒØÕÙ|p〉 

λ = 1/M 

j = (−Q 2 /m) | φ | 2 

A 

m = 2meÒÙÒØc | φ | 2 = 

ns 

∇ × j = − 22 ns 

2me 2 B 

λ = M −1 

2me 

= 

22 2 

ns 

4.1028 m−3 , 2me ≃ 5.1014 m−1ÓÒÓØÒØλ≃10 −8 20ÎËÐØÑÔÖØÙÖÙ×ÙÔÖÓÒÙØÙÖ 100ÕÙ 

m ≃ 

c/λ ≃ 

c = 1

10 8 

[m −1 ] 

M 

ÚÓÐÙØÓÒÐÑ××ØÚÙÔÓØÓÒÒÓÒØÓÒÐØÑÔÖ ØÙÖÙÒÑÐÙ×ÙÔÖÓÒÙØÙÖ 

Τ 

Τ 

ÈÓÙÖØÖÒ×ÔÓ×ÖÙÒØÐÐ×ØÙØÓÒÒ×ÐÓÒØÜØÙÒØÓÖÙÓÒ 

c 

×ØÑÒÔÓ×ØÙÐÖÐÜ×ØÒÒ×ØÓÙØÐ×ÔÙÒÑÔ×ÐÖÑÔ Ó×ÓÒ××Ù×ÔØÐÒÒÖÖÙÒÓÙÖÒØÖÒØÐÓÒ× ÔÖ×ÓÓÔÖÒÓÒ×ÕÙÒÐ×ÑÔ×ÙØÐ×ÑÔ×ÖÑÓÒ× ÔÙÚÒØÕÙÖÖÙÒÑ×× ÒÓÒÒÙÐÐ ÑÔÖÒÓÒØÖÒÔÝ×ÕÙ×ÔÖØÙÐ×ÐÑÔÐØÙÑÓÝÒÒÒ×ÐÚ×Ø ÑÔÔÖ×ÒØÙÒÖØÖ×ØÕÙÕÙÓÒÒÖØÖÓÙÚÒ×ÙÙÒÙØÖ 

ÆÑÙØ×ÙÖÐ×ÐØÓÖÒÞÙÖÄÒÙÒ×ÙÔÖÓÒÙØÚØ Ä×ÖÔØÓÒØÓÖÕÙÙÒØÐÑÔØÐÓÖÔÖØÖÙÒ×Ù×ØÓÒ 

〈0|φ|0〉 = 0 

ÓÑÑÕÙÒØÑÔØÓÙØÓ××ÔÖÑÖ×Ö×ÙÐØØ×ÑÒÒØØÖÓÒÖÑ×ÚÓÖ Ä×ÜÔÖÒ×ÙÄÈÁÁÙÊÆÓÒØÓÙÖÒÙÒ×ÒÐØÒÙÚÒÑÒØ×ÒØ× 

ÊÔÔÐ ÐÓÙÖÒØ××ÓÙÒÑÔ×ÐÖφ×ØÓÒÒÔÖj µ (φ) = 

iq[φ∗ (∂ µ φ) − (∂ µ φ∗ )φ] 

ØÐÈ ÙÒÀ×Ó×ÓÒÈÖØÐÄÒ×ØÒ×

a) 

φ 1 

v (φ1 , φ2) 

φ 2 

v (φ1 , φ2) 

Cercle Vmin ( r = µ/λ) 

+ iφ2V (φ) ÐÐÙÖÙÔÓØÒØÐÙÑÔÓÑÔÐÜφ=φ1 ÔÓÙÖµ 2 > 0V (φ)ÔÓÙÖµ 2 < 0 ÓÒ×ÖÓÒ×ÐÑÔ×ÐÖÓÑÔÐÜφ∈φ = ÔÓ×ÒØ×φ1Øφ2ØÒØÖÐÐ×ÄÄÖÒÒÜÔÖÑÒÓÒØÓÒφ, φ∗Ø ÐÙÖ×ÖÚ××ÖØ L = (∂ µ φ)(∂ µ φ) ∗ ÄØÖÑÙÔÓØÒØÐÔÙØØÖÜÔÐØ×ÓÙ×ÐÓÖÑ 

− V (φ) 

V (φ) = −µ 2 φ ∗ φ + λ 2 (φ ∗ φ) 2 

= − 1 

2 µ2 (φ 2 1 + φ 2 2) + 1 

4 λ2 (φ 2 1 + φ 2 2) 2 µØλ×ÓÒØ×ÓÒ×ØÒØ×ÔÓ×ØÚ× ,Ó 

ÖÝÓÒ ÙÖ φ2)ÐÐÐÙÖÙÒÓÒÓÙØÐÐÚÓÖ 

φ2)ÙÒÑÜÑÙÑÐÓÐÓÖÖ× 

φ2)××ØÙ×ÙÖÐÖÐ Ò×ÐÔÐÒÓÑÔÐÜφ1, φ2, V (φ1, ÐÓÖÒφ1 = φ2 = 0, V (φ1, ÔÓÒÒØÙÒØØÒ×ØÐÄÐÙ×ÑÒÑV (φ1, 

 

| φÑÒ|= φ2 1 + φ2 µ 

2 = 

λ ≡ f ×Ø×Ø×Ø ÁÐÖÔÖ×ÒØÐÒ×ÑÐ×ØØ×ÕÙÐÖ×ØÐÈÓÙÖ×ØØ×ÐÖÐØÓÒ 

√ 

2 

〈0|φÑÒ|0〉 = f √ = 0 

2 

b) 

φ 1 

φ 2 

1 √ 2 (φ1 + iφ2)Ð×ÓÑ

ÓÖÑ ÇÒÔÙØÔ××ÖÙÒ×ØØ×ÕÙÐÖÐÙØÖÔÖÐØÖÒ×ÓÖÑØÓÒ ÐÓÐU(1) = φÚ= f √ e 

2 iα ÐÓÖ×ÕÙÐÔÓØÒØÐØÐÄÖÒÒ×Ø×ÝÑØÖÕÙÒÚÖÒØ×ÓÙ×ÐØÖÒ×ÓÖ ËÐÓÒÔÖÒÙÒ×ØØ×ÕÙÐÖÙ×ÖÐÔÔÖØ××ÝÑØÖÕÙ 

ÑØÓÒU(1)ÇÒÙÒ×ØÙØÓÒ×ÝÑØÖÖ×ÓÒØÙ×××ÝÑØÖ ×ØÒØÒØÓÒÒÐÐÑÒØÕÙÓÒÑ×ÙÒ×ÒÒØÚÒØÐØÖÑÒµ 2 ÐÐÐÙÖÙÒÓÒÑÔÓÖÚÓÖÙÖ ÐÜÔÖ××ÓÒ φ2)ÙÖØÙ ËÓÒÚØÑ×ÙÒ×ÒÔÓ×ØV (φ1, ÚÙÒÑÒÑÙÑÒφ1 = ÄÚÐÙÖµ 2 ××ÝÑØÖÕÙ× ×ÓÐÙØÓÒÙÒØØÕÙÐÖ×ÝÑØÖÕÙØÐÐØØ×ÕÙÐÖÒÖ× ÊÚÒÓÒ×ÙÒ×ØÙØÓÒ×ÝÑØÖÖ×ÒÓÒ×ÖÒØÐØØ 

Óη(x)Øχ(x)×ÓÒØ×ÑÔ××ÐÖ×ØÐ×ÕÙ 

 

ÚÓ×ÒÙÒÔÓÒØÐÜÖÐÙÔÐÒÚÓÖÙÖ 

λØ 

ËÓÒÒ×ÖÒ×ÐÜÔÖ××ÓÒÙÄÖÒÒ ÓÒÓØÒØ 

0ÇÒ××ØÙÓÒÙ ÒØÒÒØÓÑÔØ = ÇÒÔÙØ×ÖÔÖ×ÒØÖη(x)ÓÑÑÙÒÙØÙØÓÒÖÐÙØÓÙÖφ1 = χ(x)ÓÑÑÙÒÙØÙØÓÒÞÑÙØÐÙØÓÙÖφ2 

ÉÓÒ×ØÖÙÖÐÜÔÖ××ÓÒÓÑÔÐØÙÄÖÒÒÚÓÖÕÙL(η)×ØØÚ ÑÒØ××ÝÑØÖÕÙÐÓÖ×ÕÙL(φ)×Ø×ÝÑØÖÕÙ ÇÒÒÜÔÐØÕÙÐ×ØÖÑ××ÒØ×ÔÓÙÖÐÔÖ×ÒØ×Ù××ÓÒ 

ÄØÖÑÒη(x)ÖÔÖ×ÒØÐÓÒØÖÙØÓÒÙÒÑÔ×ÐÖÚ×ÐÑÒØ 

= Ñ××ÕÙ(mη 2µ 2 Ò×ÐÐØØÖØÙÖÓÒØÖÓÙÚÔÐÙ××ÓÙÚÒØØÖÑÚÐ×ÒÔÓ×ØÑ×ÓÒÓ×Ø ×ØÕÙÓÒÒÓÑÑÐÑÔ×Ó×ÓÒ×À×Ä 

) ÒÖµ 2 

e iαÓα∈Ê×ØÙÒÔÖÑØÖÖÐ×ØØ××ÓÒØÓÒÐ 

φ2 = 0ÄØØÕÙÐÖ×ØÐÒÕÙ×ØÓÒ×ÖØ×ÝÑØÖÕÙ×ÓÙ×U(1)××Ù× 

= 0×ØÓÒÙÒÔÓÒØÖØÕÙÓ×ØÙÐØÖÒ×ØÓÒÒØÖÐ 

φ(x) = 1 

 

 

µ 

√ + η(x) + iχ(x) 

2 λ 

〈0|ηÑÒ|0〉 = 0Ø〈0|χÑÒ|0〉 = 0 µ 

 

1 

L = 

2 (∂µη)(∂ µ η) − 1 

2 (2µ2 )η 2 

 

1 

+ 

2 (∂µχ)(∂ µ 

 

χ) + ... 

< 0

Vmin 

µ/λ 

Im φ 

= φ 2 

φ 

χ 

V 

η 

Re φ = φ1 ÕÙÓÒÔÔÐÐÐÑÔÓ×ÓÒ×ÓÐ×ØÓÒ ØÖÑÒχ(x)ÖÔÖ×ÒØÐÓÒØÖÙØÓÒÙÒÑÔ×ÐÖ×Ò×Ñ×× ÇÒÔÙØÓÑÔÖÒÖÒØÙØÚÑÒØÖ×ÙÐØØÒÑÒÒØÙÒÔØØÔÐ ×Ø VÑÒ(Re(φ), Im(φ)) = VÑÒ(φ1, φ2). 

ØÒÒØÐÐÑÒØÒÓØÖÒ ÑÒØÔÖÖÔÔÓÖØÐÖÓÒVÑÒÙÖ ÐÒÖÒØÕÙÓÐÔÖ×ÒÙÒØÖÑÑ××ÐÓÖ×ÕÙ×ÔÐÖ ÁÐÙØÑÒØÒÒØÜÑÒÖÐ×ÓÒ×ÕÙÒ×ÐÒØÖØÓÒÒØÖ×ÑÔ× ×ÔÐÖÖÐÑÒØÓØ 

Ó×ÓÒ×ØÙÒÑÔÙ×ØÐÓØÙÔÖÖÔ×ÙÚÒØ Ö×ÙÖÐ×ÝÑØÖÙÐÓÐÑÒ×Ñ 

ÑÓÐÓÐ×ØÓÒÕÙÓÒÚÒØÔÖ×ÒØÖÚÐØÓÒÙÒÒØÖØÓÒ Ò××ÓÖÑÐÔÐÙ××ÑÔÐÐÑÒ×ÑÀ××Ø××ÒØÐÐÑÒØÐ À× 

ÙÐÓÐÕÙÑÔÓ×ÐÒØÖÓÙØÓÒÙÒÑÔÙAµÚÐ×Ù×Ø ÐØÖÓÑÒØÕÙÐÐ×ØÑÒÔÖÐÔÔÐØÓÒÙÔÖÒÔÙÜ ÔÓ×Ù§ÓÒÖÕÙÖØÐÒÚÖÒÙÄÖÒÒ×ÓÙ×ÙÒØÖÒ×ÓÖÑØÓÒ 

ÑÓÐ×ØÙÒÐÐÙ×ØÖØÓÒÙÒØÓÖÑÒÖÐÂÓÐ×ØÓÒÆÙÓÚÓÑÒØÓ iqAµÒ×ÓÖÒÒØÙÜØÖÑ×Ð×ÔÐÙ××ÒØ×ÔÓÙÖÐ 

→ ØÙØÓÒ∂µ ∂µ + 

ÈÏÀ×ÈÝ×ÄØØ ØÈÝ×ÊÚÄØØ

ÔÖ×ÒØÜÔÓ×ÓÒÓØÒØÐÜÔÖ××ÓÒÙÄÖÒÒ×ÙÚÒØ 

 

Lη,χ,Aµ = 

Ñ××ÕÙη(x)ØÐÑÔ×Ò×Ñ××χ(x)ÓÐ×ØÓÒÔÐÙ×ÓÒØÖÓÙÚÐ ÇÒÖØÖÓÙÚÐ×ÓÒØÖÙØÓÒ××ÑÔ×ÖÒØ ×ØÖÐÑÔ ÉÓÒ×ØÖÙÖÐÜÔÖ××ÓÒÓÑÔÐØÙÄÖÒÒLη,χ,Aµ 

ÓÒØÖÙØÓÒÙÒÑÔÙÑ××ÕÙAµ(Ñ××qµ λ )ØØ×ØÙØÓÒÒÓÙÚÐÐ 

ÓØØÖÓÙÚÖÙÒÓÑÔÒ×ØÓÒÔÖÐÐÙÖ×ÐÐÔÙØØÖÑ×ÒÚÒÒ 

AµÔÙÖÑÒØØÖÒ×ÚÖ×ÐÙÔÖØÕÙÖØÙÒÔÓÐÖ×ØÓÒÐÓÒØÙÒÐÒ ÚÒÒØÑ××ÕÙÓÒÙÒÖÐÖØ×ÙÔÔÐÑÒØÖØÖÓ××ÑÒØ ÔÓ×ÙÒÔÖÓÐÑØÓÙÒØÙÒÓÑÖÖ×Ù×Ý×ØÑÒØÐÑÔ 

ÓÖÑ ÖÒ××ÒØÐÑÔφ(x)ÒØÖÑ××ÓÒÑÓÙÐØ×Ô××ÓÙ×Ð 

H(x)Øξ(x)×ÓÒØ××ÐÖ×ÖÐ×ÙÜÑÓÝÒÒ×Ò×ÐÚÒÙÐÐ× Ó 

×ÖÐØÙÖÔ×ÕÙÒØÖÒÐÐÑÒØÓÒÙÑÔξ(x)ÓÐ×ØÓÒ ÈÖÙÒØÖÒ×ÓÖÑØÓÒÙÕÙØ×ÓÙ×ËÍLÓÒ×ÖÖÒÓÑÔÒ 

 

Ò×ÐÜÔÖ××ÓÒÙÄÖÒÒ×ØÕÙÓÒÔÔÐÐÐÑÒ×ÑÀ× 

φ(x) 

Ò×ØØÙÔÖØÙÐÖÔÔÐÓÑÑÙÒÑÒØÙÙÒØÖÐÜÔÖ××ÓÒ ÙÑÔφ(x)×Ø 

 

φ ØÐÐÙÑÔAµA ′ µ(x) = Aµ(x) + 1 

q µ 

∂µξ(x) 

λ 

 

+ 

 

1 

2 (∂µη)(∂ µ η) + 1 

2 (2µ2 )η 2 

 

 

− 1 

4 FµνF µν + 1 

2 

Fµν = ∂µAν − ∂νAµ 

+ 

 

qµ 

2 AµA 

λ 

µ 

 

= 1 

µ 

 

√ + H(x) exp i 

2 λ ξ(x) 

µ 

′ = 1 

√ 2 

 

µ 

+ H(x) 

λ 

λ 

1 

2 (∂µχ)(∂ µ χ) 

+ ... , Ó

H 

H 

H 2 AµA µ 

A 

A 

AµA µ H 

H 3 H 4 

H 

H H 

H 

H 

H H 

ÖÑÑ×ÒØÖØÓÒÒØÖÑÔÀ×ØÑÔA µØ ÑÔÀ×ÚÐÙÑÑ ÄÄÖÒÒ ÚÒØ 

LH,Aµ = 

1 

 

 

ÄÄÖÒÒLH,AµÖØÓÖÖØÑÒØÐ×ÔØÖÑ×××ØØÒÙÁÐÖÔÖ 

ÒØÖÐ×ÑÔ×ÚÓÖÙÖ ×ÒØÙÜÔÖØÙÐ×Ñ××Ú×ÒØÖ××ÒØ×ÐÓ×ÓÒ×ÐÖÀ×Ñ×× Ä×ÙØÖ× 

ÉÓÒ×ØÖÙÖÐÜÔÖ××ÓÒÙÄÖÒÒLH,Aµ 

ÐÐÐÙ×ØÖÓÒ×Ù§ 

ØØÒÐÝ×ÔÙØØÖØÒÙÙØÖ××ÝÑØÖ×ÙÕÙU(1)ÒÓÙ× ØÖÑ×ÐÖÐØÓÒÖÚÒØÐÓÒ×ØÒØ1 

mH = 2µ 2ØÐÓ×ÓÒÙÚØÓÖÐAµÑ××mAµ 

+ 

+ 

2 (∂µH)(∂ µ H) + 1 

2 (2µ2 )H 2 

 

− 1 

4 FµνF µν + 1 

 

qµ 

2 λ 

2 q µ 

HAµA 

λ 

µ + 1 

2 

AµA µ 

 

H 

2 q2H 2 AµA µ − λµH 3 − 1 

4 λ2H 4 + 1 

4 

= 

µ 4 

4 

A 

A 

µ 4 

λ2 qµ 

λ 

λ2ÔÖ×Ð×ÒØÖØÓÒ×

ÄØÓÖÐØÖÓÐÏ Ä×Ó×ÔÒØÐÝÔÖÖÐ×Ð×ÝÑØÖËÍ 

ÒÜÔÓ×ÖÐ×××ØÒÖÔÔÐÖÐ×ÔÖØÓÒ××ÐÐÒØ××Ò×ÒØÖÖÒ× ×ØÐ×ÐÙÐ ØÚÓÕÙÒ×ÓÙÖ×ÒÔÖØÙÐÖÙÔØÖÆÓØÖÓØ×Ø ÄÒÓÑÐØÓÖÐØÖÓÐÑÓÐÐ×ÓÛÏÒÖËÐÑ 

ÄÒÚÖÒÙ×ØÙÒÔÖÓÔÖØÒØÖÐ×ØÓÖ×ÑÔ×ÕÙÒ 

L 

ØÕÙ×ÔÖÕÙÐÐÖÒØØÕÙÐ×ÖÒÙÖ×ÐÙÐ×ÓÒØ×ÚÐÙÖ×Ò× ÄØÓÖÉÒ×ØÙÒÜÑÔÐÓÒÚÙ§ ÕÙÐÄÖÒÒLQED×Ø 

ÇÒ×ÔÓ××ÐÓÖ×ÐÕÙ×ØÓÒÕÙÐ×ØÐÖÓÙÔ×ÝÑØÖÙÖÐÚÒØ 

ÐÓÙÖÒØÐÒÙØÖÑÑÕÙÐÓÙÖÒØÑØÓÙÐ×ÓÑÔÓ×ÒØ×Ù Ò×Ð×ÐÒØÖØÓÒÐÍÒÖÔÓÒ×Ø×ÙÖÔÖËÄÐ×ÓÛ ÕÙÐÓÑÔÓ×ÒØÙÖÐØ ÖØÖÓØÖÐØ±1ÇÒÒØÖÓÙØÐÓÒÔØ×Ó×ÔÒÐÒ Ò×ÓÒÒØ×ÙÖÐÓ×ÖÚØÓÒÕÙÐ×ÓÙÖÒØ×Ð×Ö×ÒØÒØ ×ÖÑÓÒ×ÐÔØÓÒ×ØÕÙÖ×ÐÓÖ×ÕÙ 

×ØÖÒ×ÓÖÑØÓÒ×ÙÒÖÓÙÔ×ÝÑØÖËÍLÒ×Ð×Ô×Ó×ÔÒÐ ÙÖ×ÙÒÑÐÐÖÑÓÒ××ÓÒØÓÒ×Ö×ÓÑÑÓÖÑÒØÙÒÓÙÐØ 

T3)ÓÒØÐ×ÓÑÔÓ×ÒØ×Ti×ÓÒØÐ×ÒÖØÙÖ× 

iεijkTkÖÓÙÔÒÓÒÐÒÄ×ÑÑÖ× ÔÖÐÚØÙÖÓÔÖØÙÖT(T1, T2, Ä×TiÒÓÑÑÙØÒØÔ×[Ti, Tj] = 

ÓÙÐØ×ØÖÒ×ÓÖÑ×ÓÙ×ËÍ LÓÑÑ 

×Ð×§ ÌÝÔÕÙÑÒØÐ×ÑÔÐØÙ×ØÖÒ×ØÓÒÒ×ÐÐÙÐ×ÔÖØÙÖØÓÒ××ÓÒØÖÒÓÖÑÐ ÊÔÔÐÓÒ×ÕÙÐ×ÓÑÔÓ×ÒØ×fL,RÙÒÖÑÓÒ×f×ÓØÒÒÒØÔÖÐÓÔÖØÓÒÔÖÓ 

g×ØÙÒÖÐÔÖÒÐÓÚÐÖÐØÖÕÙÙÒØe 

ÙÜÙÜÙØÖ×ÑÐÐ× 

ÈÓÙÖ×ÑÔÐÖÐÖØÙÖÓÒ××ØÐÑØÐÔÖÑÖÑÐÐÐ×ÖÔØÓÒÔÙØØÖØÒÙ ØÓÒfL,R = 

× U(1)Y 

ÒÚÖÒØ×ÓÙ×ÐØÖÒ×ÓÖÑØÓÒÙ 

U(1) = exp[iqα(x)] 

×Ó×ÔÒÐØÐÕÙ 

T = 1 

2 , T3 

 

+1/2 

= 

−1/2 

T = 1 

νe 

e − 

 

u 

, ...ÓÙ 

d 

L 

′ 

χ ′ L ⇒ exp[igθ(x) · T] χL,Ó 

νe 

χL = 

e− 

u 

, ...,ÓÙ 

d ′ 

2τÓÐ×τi×ÓÒØÐ×ÑØÖ×ÈÙÐ 1 γ5)fÚLÔÓÙÖÐØØÊÔÓÙÖÖØ 

2 (1 ∓ 

 

L 

 

, ... 

L 

, ...

ÓÖÑÒØÙÒ×ÒÙÐØ×Ó×ÔÒÐØÐÕÙ Ä×ÑÑÖ×ÖÓØÖ×ÐÑÑÑÐÐÖÑÓÒ××ÓÒØÓÒ×Ö×ÓÑÑ 

T = 0 ; T3 = 0 : e − 

R , ... , uR , d ′ ÁÐ××ØÖÒ×ÓÖÑÒØ×ÓÙ×U(1)YWÓÑÑ 

ÓÙ 

R , ... 

ØÖØÓÒÓÖØÖÐØÓÒ ÄÖÒÙÖYW×ØÒÔÖÒÐÓÚÐÝÔÖÖÒØÖÓÙØÒÒ 

YW×ØÐÝÔÖÖÐ 

R 

ÇÒÓÒ×ÒÒ×ÐØÐÙ Ð×ÚÐÙÖ×ÒÙÑÖÕÙ××ÒÓÑÖ×ÕÙÒ 

YW 

Ù××Ó×ÙÖÓÙÔËÍ ÒØÖØÓÒÄÑÓÐÐ×ÓÛÏÒÖËÐÑÒØÖÓÙØØÖÓ×ÑÔ× Ä×ØÖÒ×ÓÖÑØÓÒ×ÐÓÐ× T3ØYW××Ò×ÙÜÖÒØ×ØØ×ÖÑÓÒ× 

LØÓÖÑÒØÙÒØÖÔÐØ×Ó×ÔÒÐÚØÙÖ Ø ÑÔÐÕÙÒØÐÜ×ØÒÑÔ× ØÕÙ×Ø×T, 

ØÙÒÑÔÙ××ÓÙÖÓÙÔU(1)YWÓÖÑÒØÙÒ×ÒÙÐØ×Ó×ÔÒ 

Ð×ÐÖ 

⎠ 

T = 0 ; T3 = 0 : (B µ 

ÑÓÒ×ÙÖ× Ä×ÝÑØÖÓÑÒËÍ ×Ò×Ñ×× ÇÒÔÙØ×ÐÓÖ×ÖÖÐ×ÜÔÖ××ÓÒ××ÖÚ×ÓÚÖÒØ×ÈÓÙÖÐ×Ö 

U(1)YWØÒØ×ÙÔÔÓ×ÔÖØ×ÑÔ××ÓÒØ 

) 

L × 

D 

2 Bµ ÖÓØÖÐÙØÖÒ×ÙÒÑÑÑÐÐÖÑÓÒ×ÊÔÔÐÓÒ×ÔÖÐÐÙÖ×ÕÙÐ×ÒÙØÖÒÓ× ÂÙ×ÕÙÔÖÙÚÙÓÒØÖÖÐÒÝÔ×ØÖÒ×ØÓÒÐ×ÒØÔ××ÖÙÒÑÑÖ 

ÔÖÖÙÒÔÖØØÙÜ×Ó×ÔÒØÝÔÖÖÐ×ÙØÖÔÖØÐÒÝÒØÖÙÜ Ò×ÓÒØÓ×ÖÚ×ÕÙÒ×ÐØØÙÖ§ ÕÙÙÒ×ÑÐØÙÓÒ×ØÖÙØÓÒÓÖÑÐÐ ÊÑÖÕÙÓÒ×ÕÙÐÒÜ×ØÙÙÒÓÒÒÜÓÒÔÝ×ÕÙÒØÖÐ×ÓÒÔØ××Ó×ÔÒØÝ ÆÓØÓÒ×ÕÙÒ×ÐÑÓÐÐÔÓØÓÒØÐÐÙÓÒÒÓÒØÔ×ØØÖÙØ×Ó×ÔÒÐ 

χ ′ R =⇒ exp[ig′ θ(x) YW 

uR , d ′ R , ... 

χR = e − 

, ... 

g ′×ØÙÒÙØÖÖÐ 

2 

⎧ 

⎨ +1 

T = 1, T3 = −1 

⎩ 

0 

= Q − T3 

: 

2 ]χR ,Ó 

µ = ∂ µ + ig τ 

2 · W µ + ig ′YW 

⎛ 

W 

⎝ 

µ 

1 

W µ 

2 

W µ 

⎞ 

3

ØØ×ÖÑÓÒ 

νe, νµ, ντ 

e − 

L , µ− 

L 

e − 

R , µ− 

R 

, τ − 

L 

, τ − 

uL, cL, tL 

R 

É T 

+ 2 

3 

T3 YW 

d ′ L , s′ L , b′ L 

−1 1 − 3 2 

1 + 2 

1 

3 

uR, cR, tR + 2 

+ 3 

4 

3 

d ′ R , s′ R , b′ R 

−1 − 3 

2 

ØØÖÙ×ÙÜÖÒØ×ØØ×ÐØ×ÖÑÓÒ× Ì ÎÐÙÖ××ÒÓÑÖ×ÕÙÒØÕÙ××Ó×ÔÒØÝÔÖÖÐ× 

3 

ØÔÓÙÖÐ×ÖÑÓÒ×ÖÓØÖ×ÒÓÒØ×ÔÖËÍ L 

D µ = ∂ µ + ig ′YW 

2 Bµ ÔÖ××ÓÒÙÄÖÒÒÒÚÖÒØÔÖÒÐÓÖÑ×ÙÚÒØ ËÐÓÒÔÖÒÒÓÑÔØÐ×ØÖÑ×ÒÖÒØÕÙ×ÑÔ×ÙÐÜ 

 

L = χLγµ ∂ µ − ig τ 

2 · W µ − ig ′YW 

2 Bµ 

 

χL 

 

+ χRγµ ∂ µ − ig ′YW 

2 Bµ 

 

χR − 1 

4 W µν · W µν − 1 ÄÜÑÒ×ÓÑÔÓ×ÒØ×ÒØÖØÓÒ×ÖÑÓÒ×ÙÖ×ØÖÓØÖ×Ú 

µÑÒÒØÖ××ÒØ×ÙØÓÒ×ÐÐ×ÚÐÑÔ 

µν 

BµνB 

4 Ð×ÑÔ×W µØB 

1 

2 

1 

2 

1 

2 

1 

2 

1 

2 

1 

2 

+ 1 

3

W µÔÙÚÒØØÖÖØ× 

−igj µ · W µ 

1√2 

= −ig (j + µ W µ+ + j − µ W µ− ) + j 3 Ó 

 

µ3 

µ W , 

τ 

j µ = χLγµ 2 χL , 

j ± µ = j 1 µ ± ij 2 µ , 

W µ± = 1 √ (W 

2 µ1 ∓ iW µ2 ) ÐÐ×ÚÐÑÔB µÓÒØÐÓÖÑ 

jYW 

′ µ 

−ig 

2 Bµ = −ig ′ [(j em 

µ − j3 µ )Bµ Ó 

] , 

j YW 

µ = χLγµYWχL + χRγµYWχR , 

j em 

µ = eχ ØÓÙØÓ×ÓÒÒ×ØÒÙÔ×ÙÔÖÑÖÓÙÔÓÐÕÙÔÙØÔÖÓÚÒÖÙ ØÖÑÓÒØ ÇÒÖÓÒÒØÒ×ÐÔÖÑÖÔÖØÙØÖÑÖÓØ ÙØÓÒÐÒØÖØÓÒÔÖÓÚÒÒØ×ÓÙÖÒØ×Ð×Ö×Ä×ÓÒÔÖØ ÐÓÒØÖ 

ÓÙÖÒØÐÒÙØÖØÙÓÙÖÒØÐØÖÓÑÒØÕÙÄÑÓÐÏËÒÓÙ× ÖÔÖ×ÒØÐ×ÓÒØÖÙØÓÒ×ÓÙÖÒØ×ÒÙØÖ× 

LγµQχL + eχRγµQχR ÔÔÓÖØÐÐÖÒ××ÖÔÖÓÔÓ×ÐÓÒ×ÖÕÙÐ×ÑÔ×W µ3 , B µ ×ÓÒØ×ÓÑÒ×ÓÒ×ÐÒÖ×ØÓÖØÓÓÒÐ××ÑÔ×Z µ , A µ 

 

µ3 W 

B µ 

 

µ 

cosθW sin θW Z 

= 

− sin θW cosθW A µ ÚÓÖ§ θW×ØÐÒÐÑÐÒÐØÖÓÐÔÔÐÓÑÑÙÒÑÒØÐÒÐÏÒÖ 

 

, Ó: 

Ò×Ù×ØØÙÒØÐ×ÜÔÖ××ÓÒ×W µ3ØB µÒØÖÑ×A µ , Z µÒ× 

 

Ø ØÒÖÖÓÙÔÒØÐ×ÔÒÒ×ÒA µ ÓÒØÖÙØÓÒ×ÓÙÖÒØÒÙØÖ× 

ÇÒÖÓÒÒØÑÒØÒÒØÒ× ÐÓÖÑÙ×ÙÐÐÙÓÙÔÐÑ 

ÇÒØÙ×ÐÖÐØÓÒYW 2 T3ÒØÒÒØÓÑÔØÙØÕÙT3 ÖÑÓÒÖÓØÖ 

0ÔÓÙÖÐ 

 

ËÐÓÒ×ÖÔÔÓÖØÙÜÒÓØØÓÒ×Ù§ÓÒgγ = 

−iej em 

−i[(g sin θW − g ′ cosθW)j 3 µ + g ′ cosθWj em 

µ ]A µ −i[(g cosθW + g ′ sin θW)j 3 µ − g′ sin θWj em 

µ ]Zµ 

µ Aµ×ÐÓÒÔÓ× g = e 

sin θW 

; g ′ = e 

, Z µÓÒÓØÒØÔÓÙÖÐ× 

cosθW 

= Q + = 

e, g = gW ±, g ′ = gZ sin θW

ØÔÖÓÒ×ÕÙÒØ 

ÌÒÒØÓÑÔØ ÐØÖÑÓÙÔÐÐÒÙØÖÔÙØ×ÖÖ 

g 

×ÓÙ×ÐÓÖÑ 

ÇÒÚÓØÔÔÖØÖÒ××ÓÒÒÜÓÒ×ÒØÑ×ÒØÖÐÓÙÖÒØÐÒÙØÖ Ð×ØÑ×ÔÖØÓÒ×ÙÑÓÐÏËÓÒØØØ×Ø×Ú×Ù×ÓÖÑ× 

µÒ×ÕÙÒØÖÐ×ÓÒ×ØÒØ×ÓÙÔÐ ÕÙÐ×ÕÙÒØ×ÑÔ×Ð×Ó×ÓÒ×W ±ØZÓÒØØØÖÓÙÚ×Ñ×××ÚÓÖ ÙÑÓÐÒ×Ò×ÔÖÒØÐÔÖÓÙÖÖØÙÜ§ØØØ×ÓÒ ØÔØÐÓØÙÔÖÖÔ×ÙÚÒØ ÔØÖÌÓÙØÓ×ÓÒ×ØÙÑÓÒ×ÒÔÖÒÔÓÑÑÒØÖÑÖÙØ 

ÇÒÕÙØÖÑÔ×ÙÓÒØØÖÓ×ÓÚÒØØÖÖÒÙ×Ñ××××ÓØÐ× Ò×ÑÀ× ËÝÑØÖÐÓÐÖ×ËÍ L⊗Í YÑÔÐØÓÒÙÑ 

Ó×ÓÒ×W ±××Ó×ÙÜÓÙÖÒØ×Ð×Ö×ØÐÓ×ÓÒ××ÓÙÓÙÖÒØ ÐÖØØ×××ÙÖÖÔÐÙ×ÕÙÙÒ×ÝÑØÖÙÖ×ØÒÓÒÖ×ØÐÐ ÐÒÙØÖÁÐÙØÓÒÒØÖÓÙÖÙÒÑÔ×ÐÖÚÙÑÓÒ×ØÖÓ×Ö× ×ÓÖØÕÙÙÒÑÔÙ×Ò×Ñ×××Ù××ØÔÓÙÖÖÔÖ×ÒØÖÐÔÓØÓÒÒ× ÙÒÓÙÐØËÍ ÐÑÓÐÏËÓÒÒØÖÓÙØÙÒÑÔ×ÐÖÓÑÔÐÜ×ØÖÒ×ÓÖÑÒØÓÑÑ 

L 

T = 1 

2 , T3 

ØÖ×ÔØÚÑÒØÐÙÖÝÔÖÖÐ 

= 

, Ó: 

ÆÓØÓÒ×ÕÙÑÔ×Ø×ÙÔÔÓ×ÓÖÑÖÙÒ×ÒÙÐØÓÙÐÙÖ 

2×ØÜÔÖÑÒÙÒØε0cÒÙÑÖÕÙÑÒØÓÒ ÚÙØÓÒYW = 

 

g2Øg ′2×ÓÒØ×Ò×ÑÒ×ÓÒe 

j NC 

−i 

g 

′ 

g 

= tan θW 

 

1 1 

+ 

g2 g ′2 = 1 

e2 (j 

cos θW 

3 µ − sin 2 θWj em 

µ )Z µ = −i 

cos θW 

j NC 

µ = j3 µ − sin 2 θWj em 

µ 

µØÐÓÙÖÒØÐØÖÓÑÒØÕÙj em 

g 

j NC 

µ Z µ ,Ó: 

 

+ 

+1/2 φ = √2 1 (φ1 + iφ2) 

φ = 

−1/2 φ0 = 1 

φ4×ÓÒØ××ÐÖ×ÖÐ× 

√ (φ3 + iφ4) 

2 

φ1, ... , 

φ + φ0ÓÒØ×Ö×ÐØÖÕÙ× , 

+1. 

g2 /4π ≃ 1/30 ; g ′2 /4π ≃ 1/100 ; e2 /4π ≃ 1/137 ; sin 2 90Î 

0.231ÔÓÙÖÐÒÖÕÙÚÐÒØ 

θW ≃ 

MZ ≃

ÑÔ×Ø×ÙÔÔÓ×ÚÓÖÙÒÑÓÝÒÒÒÓÒÒÙÐÐÒ×ÐÚØÓÒØÐ ÓÜφ1 = λ ) 

ÚÓ×ØÓÒÖÒØÐ×ÓÓÙÐØÒØÖÑ××ÓÒÑÓÙÐØ×Ô× Ò×ÙÚÒØÐÔÖÓÙÖÖØÙ§ÓÒÚÐÓÔÔφ(x)ÙØÓÙÖÐØØ 

 

 

ÐÚ 

, Ó: 

Ä×ØÖÓ×Ö×ÐÖØÒ×ÐÖ××ÖØÖÓÙÚÒØÒ×Ð×ÓÑÔÓ×ÒØ× ÈÖÙÒÓÜÙÕÙØÙÙÒØÖÐØÙÖÔ××ØÓÑÔÒ× 

ξ3(x)Ó×ÓÒ×ÓÐ×ØÓÒ ÓÒÐÑÒÖÐ×ØÖÓ×ÓÑÔÓ×ÒØ×ξ1(x), ξ2(x), ÔÓÐÖ×ØÓÒÐÓÒØÙÒÐ×ÑÔ×W ±ØZÒ×ØØÙÐÄÖÒÒ 

 

φ2 = φ4 = 0 ; φ3 = µ 

〈0|φ|0〉 = φÚ= 1 

√ 

2 

0µ 

λ 

φ(x) = 1 

0µ 

iξ(x) · τ 

√ exp 

2 + H(x) 

λ 2 µ 

ξ3(x)]×ÓÒØ×ÑÔ×ÖÐ×ÑÓÝÒÒ×ÒÙÐÐ×Ò× 

 

λ 

H(x)Ø[ξ1(x), ξ2(x), 

×ØÐÓÖÑ LH,W ±µ ,Z µ ,A µ = 

(τ1, τ2, τ3)×ÓÒØÐ×ÑØÖ×ÈÙÐ 

+ 

1 

2 (∂µH)(∂ µ H) + 1 

2 (2µ2 )H 2 

 

− 1 

2 (FW −)µν(FW +) µν + 1 

2 

 

g µ 

λ 

2 (W−)µ(W+) ν 

 

+ − 1 

4 ZµνZ µν + 1 

 

µ 

2 (g 

2 2λ 

2 + g ′2 )ZµZ µ 

 

+ − 1 

 

′µν 

FµνF + ....... , Ó: 

4 

Fµν ∂µAν − ∂νAµ 

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Ш ЪЬС ЭФ Ы Ф Х ЦЬ СЪ Ы - Laboratoire de Physique des Hautes ...

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