Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

• a Lorentz transformation Λ must leave xp invariant because 2xp = (x + p) 2 −x 2 − p 2 :• derivatives:for examplex µ → x ′ µ = Λ νµ x ν (mit x ′2 = x 2 ) ⇐⇒ g µµ ′ = Λ νµ Λ ν′µ ′ g νν ′ (9)∂∂x µf(x) = ∂ µ xf(x) = ∂ µ f(x) ,Problem 1. Compute the partial derivative w. r. t. xfor constant four vectors a, b and p.Problem 2. Show that∂∂x µ f(x) = ∂ µf(x) (10)∂ µ x(xp) = ∂(x ν p ν )/∂x µ = p µ (11)∂ µ e −ipx , (a∂)(b∂)e −ipx , ∂ 2 e −ipx (12)∂ µ x µ = 4(NB: ∂ µ x µ = g νµ ∂x µ /∂x ν and g νµ = δ νµ ) and compute∂ 2 e −x2 /2(13a)(13b)1.3 Schrödinger Equation• Wave functions satisfy the Schrödinger equationi̷h d Ψ(t) = HΨ(t)dt (14a)with solution (for infinitesimal time intervals)Ψ(t + δt) = e −iH·δt/̷h Ψ(t)(14b)∴ scattering amplitude (infinite time intervals)〈A in→out = 〈out S in〉 = out(t2 ) e −iH·(t 2−t 1 )/̷h in(t 1 ) 〉 (15)• Problems with this approachlimt 1 →−∞t 2 →+∞– particle production and decay has been observed, but can not be describedby wave functions (without “2nd quantization”), because probability is conserved(“unitarity”)– Schrödinger equation (14) not manifestly Lorentz covariant– free single particle equationi̷h d dt Ψ(t) = 12mis manifestly not Lorentz covariant!3( ) 2 1∇i̷hc ⃗ Ψ(t) (16)