You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<strong>Weak</strong> <strong>Measurements</strong>:<br />
Wigner-Moyal and Bohm in a<br />
New Light.<br />
Basil J. Hiley.<br />
www.bbk.ac.uk/tpru.<br />
Thursday, 2 February 2012
<strong>Weak</strong> <strong>Measurements</strong>.<br />
1. New type of quantum measurement.<br />
[Aharonov, and Vaidman, Phys. Rev., 41, (1990) 11-19].<br />
2. What do we measure?<br />
<strong>Weak</strong> values defined by<br />
A W (ψ,φ) = φ|A|ψ<br />
φ|ψ<br />
∈ C<br />
3. Will show that P W (ψ,x) is related to the Bohm momentum<br />
P 2 W (ψ,x)<br />
is related to the Bohm energy and quantum potential.<br />
[Leavens, Found. Phys., 35 (2005) 469-91]<br />
4. Will obtain expressions for weak values for spin-1/2 particles using Clifford algebras.<br />
5. Will indicate how to combine the Clifford with the Moyal algebra.<br />
6. Discuss what may lie behind the formalism-- Process.<br />
Acknowledgements<br />
Ernst Binz, Maurice de Gosson, Bob Callaghan and David Robson.<br />
Thursday, 2 February 2012
<strong>Weak</strong> measurements.<br />
Why the interest?<br />
Photon ‘trajectories’.<br />
Schrödinger particle ‘trajectories<br />
Experimental--Photons.<br />
[Kocsis, Braverman,Ravets,Stevens,Mirin, Shalm, Steinberg,<br />
Science 332, 1170 (2011)]<br />
[Prosser, IJTP , 15, (1976) 169]<br />
Theory--Schrödinger particle.<br />
[Philippidis, Dewdney and Hiley, Nuovo Cimento 52B, 15-28 (1979)]<br />
Real part of Schrödinger equation<br />
∂S<br />
∂t + (∇S)2<br />
2m + Q + V =0<br />
E B = − ∂S<br />
∂t<br />
P B = ∇S<br />
Thursday, 2 February 2012
<strong>Weak</strong> Measurement: the principle.<br />
H I = λ(t)Âs ˆB d<br />
|initial = |ψ s |Ψ d |final = e i R λ(t)Âs ˆB d dt |ψ s |Ψ d <br />
<br />
λ(t)dt =1<br />
|final = e iÂsb |ψ s |Ψ d <br />
Form the transition probability amplitude T φ−ψ = b d |φ s |final b d |Â|Ψ d = ÂΨ d(b)<br />
T φ−ψ = φ s |e iÂb |ψ s Ψ d (b) = (ib) n<br />
φ s<br />
n!<br />
|Ân |ψ s Ψ d (b) =φ s |ψ s (ib) n φ s |Ân |ψ s <br />
Ψ d (b)<br />
n! φ<br />
n=0<br />
n=0<br />
s |ψ s <br />
<br />
T φ−ψ = φ s |ψ s e ibA W<br />
+ <br />
(ib) n<br />
[A n W −A n W ] Ψ d (b)<br />
n!<br />
n=2<br />
small<br />
Post select with b = x T φ−ψ = φ s |ψ s e ixA W<br />
Ψ d (x)<br />
−x<br />
2<br />
Choose Ψ d (x) =exp<br />
and take the imaginary part of A W we find<br />
4(∆x) 2 <br />
T φ−ψ ∝ e −xA IW −x<br />
2<br />
exp<br />
4(∆x) 2 ∝ exp − <br />
x + 2(∆x) 2 2<br />
A IW<br />
4(∆x) 2<br />
Centre of Gaussian in x-space shifted by amount ∝A IW<br />
Centre of Gaussian in p-space shifted by amount ∝A RW<br />
Thursday, 2 February 2012<br />
[Duck, Stevenson and Sudarshan Phys. Rev, 40 (1989) 2112-7]
von Neumann measurement.<br />
<strong>Weak</strong> measurement.<br />
<strong>Weak</strong> Experiment cont.<br />
Single measurement ---- collapse of the wave function.<br />
<br />
Ψ j → Ψ r<br />
j<br />
∝A IW<br />
Statistical measurement producing a phase shift in the distribution of final results.<br />
Actual experimental arrangement for photon trajectories.<br />
x<br />
|Ψ = |D polarizer |ψ path H = g ˆP x Ŝ 1 Ŝ 1 =(|HH| − |V V |)/2<br />
Thursday, 2 February 2012<br />
[Kocsis, Braverman,Ravets,Stevens,Mirin, Shalm, Steinberg,<br />
Science 332, 1170 (2011)]
<strong>Weak</strong> values.<br />
φ|A|ψ<br />
φ|ψ<br />
How do they appear in the formalism?<br />
[ Hiley quant-ph/1111.6536]<br />
Then<br />
ψ|A|ψ = ψ|φ j φ j |A|ψ where |φ j form a complete orthonormal set.<br />
ψ|A|ψ = ψ|φ j <br />
Post select<br />
φj |ψ<br />
φ j |A|ψ = φ j |A|ψ<br />
ρ j<br />
φ j |ψ<br />
φ j |ψ<br />
<strong>Weak</strong> value.<br />
Special case:<br />
If A|φ j = a j |φ j <br />
ψ|A|ψ = ρ j a j<br />
Well known result!<br />
Eigenvalue!<br />
Remember<br />
φ|A|ψ<br />
φ|ψ<br />
is a complex number.<br />
But what does it mean in general?<br />
It is clearly a transition probability amplitude.<br />
Thursday, 2 February 2012
<strong>Weak</strong> values when<br />
ˆP is involved.<br />
<br />
Consider x| ˆP |ψ(t) = x| ˆP |x x |ψ(t)dx = −i∇ x ψ(x, t)<br />
Write<br />
ψ(x, t) =R(x, t)e iS(x,t)<br />
then<br />
x| ˆP |ψ(t)<br />
x|ψ(t)<br />
= ∇ x S(x, t) − i∇ x ρ(x, t)/2ρ(x, t)<br />
with<br />
ρ(x, t) =|ψ(x, t)| 2<br />
Bohm velocity:<br />
Osmotic velocity:<br />
In terms of one expression<br />
⎡<br />
[[P W ]] ± =<br />
Bohm momentum.<br />
v B (x, t) =∇ x S(x, t)/m = 1<br />
2m<br />
v O (x, t) = 1<br />
2m<br />
⎣ ψ(t)|←−ˆP |x<br />
ψ(t)|x<br />
More simply for Schrödinger<br />
then<br />
[[P W ]] ± =<br />
∇ x ρ(x, t)<br />
ρ(x, t)<br />
osmotic momentum.<br />
⎡<br />
⎣ ψ(t)|←−ˆP |x<br />
ψ(t)|x<br />
= 1<br />
2m<br />
⎡<br />
⎣ ψ(t)|←−ˆP |x<br />
ψ(t)|x<br />
+ x|−→ˆP |ψ(t)<br />
x|ψ(t)<br />
⎤<br />
⎦<br />
⎤<br />
− x|−→ˆP |ψ(t)<br />
⎦<br />
x|ψ(t)<br />
⎤<br />
± x|−→ˆP |ψ(t)<br />
⎦ = ψ∗ (x, t) ←−ˆPx ψ(x, t) ± ψ ∗ (x, t) −→ˆPx ψ(x, t)<br />
.<br />
x|ψ(t)<br />
ρ(x, t)<br />
i<br />
ρ(x, t) [(∇ xψ ∗ (x, t))ψ(x, t) ∓ ψ ∗ (x, t)(∇ x ψ(x, t))]<br />
− iρ[[P xW ]] + = [∇ x ψ ∗ (x)]ψ(x) − ψ ∗ (x)[∇ x ψ(x)] = ψ ∗ (x) ←→ ∇ x ψ(x)<br />
− iρ[[P xW ]] − = [∇ x ψ ∗ (x)]ψ(x) + ψ ∗ (x)[∇ x ψ(x)] = ∇ x (ψ ∗ (x)ψ(x)) = ∇ x ρ(x).<br />
Thursday, 2 February 2012
Remark 1: Relation to Nelson.<br />
Mean forward derivative: Dx(t) = lim<br />
∆t→0 + E i<br />
x(t + ∆t) − x(t)<br />
∆t<br />
[Nelson, Phys. Rev., 150, (1966), 1079-1085.]<br />
Mean backward derivative:<br />
D ∗ x(t) =<br />
x(t) − x(t − ∆t)<br />
lim E i<br />
∆t→0 + ∆t<br />
With these construct a forward velocity b(x, t) and backward velocity b ∗ (x, t) :<br />
[b(x, t)+b ∗ (x, t)]/2 =v B (x, t) = ∇ xS(x, t)<br />
m<br />
b(x, t) − b ∗ (x, t) =v O (x, t) = 1<br />
2m<br />
∇ x ρ(x, t)<br />
ρ(x, t)<br />
[Bohm and Hiley, Phys. Reps, 172, (1989), 92-122.]<br />
Thursday, 2 February 2012
Remark 2: Relation to Energy-Momentum Tensor.<br />
∂L<br />
T µν = −<br />
∂(∂ µ ψ) ∂ν ψ +<br />
∂L <br />
∂(∂ µ ψ ∗ ) ∂ν ψ ∗<br />
Take the Schrödinger Lagrangian: L = − 1<br />
2m ∇ψ∗ · ∇ψ + i 2 [ψ∗ (∂ t ψ) − (∂ t ψ ∗ )ψ] − V ψ ∗ ψ.<br />
and find<br />
T 0µ = − i 2 [(∂µ ψ ∗ )ψ − ψ ∗ (∂ µ ψ)] = i 2 [ψ∗←→ ∂ µ ψ]=−ρ∂ µ S<br />
Recalling that<br />
P B = ∇S<br />
and<br />
E B = −∂ t S<br />
Then we find<br />
ρP jB = ρ∂ j S = −T 0j = − ρ 2 [[P jW]] +<br />
ρE B = −ρ∂ t S = −T 00 = − ρ 2 [[P tW ]] +<br />
Bohm momentum.<br />
Bohm energy.<br />
This generalises to the Pauli and Dirac particles<br />
The Bohm kinetic energy.<br />
[Hiley and Callaghan, Fond. Phys 42 (2012) 192-208,<br />
math-ph:1011.4031 and 1011.4033]<br />
1<br />
2 [[P W 2 ]] + = (∇ x S(x)) 2 − ∇2 xR(x)<br />
= PB 2 + Q.<br />
R(x)<br />
<br />
1<br />
2i [[P W 2 ]] − = ∇ 2 ∇x ρ(x)<br />
xS(x) +<br />
∇ x S(x).<br />
ρ(x)<br />
Quantum potential<br />
[Leavens, Found. Phys., 35 (2005) 469-91]<br />
[Wiseman, New J. Phys., 9 (2007) 165-77.]<br />
Thursday, 2 February 2012
Multiplication of phase space functions defined by<br />
Let<br />
<strong>Weak</strong> values from Moyal algebra.<br />
a(x, p) b(x, p)<br />
where<br />
f(x, p) be the density matrix in (x, p) representation, viz. f(x, p) =<br />
<br />
i<br />
=exp<br />
2 ( ∂ ← →<br />
x ∂p − ∂ ← →<br />
p ∂x )<br />
<br />
[ Moyal, Proc. Camb. Phil. Soc. 45, (1949), 99-123].<br />
[Baker, Phys. Rev. 6 (1958) 2198-2206.]<br />
ρ(x − y/2; x + y/2)e −ipy dy<br />
p f(x, p) =<br />
<br />
p − i <br />
→<br />
∂ x<br />
f(x, p) f(x, p) p = f(x, p) p + i <br />
←<br />
and<br />
∂ x<br />
2<br />
2<br />
Now form<br />
(f p + p f)<br />
[p, f(x, p)] BB := = pf(x, p)<br />
2<br />
Baker bracket<br />
(f p − p f)<br />
[p, f(x, p)] MB := = ∇ x f(x, p)<br />
i<br />
Moyal bracket [x, p] MB =1<br />
To make contact with configuration space we must form a marginal;<br />
<br />
<br />
[p, f(x, p)] BB dp = pf(x, p)dp = ρ(x) p (x)<br />
Moyal momentum<br />
Using<br />
ρ(x) p n (x) =<br />
1<br />
2i<br />
n<br />
∂<br />
∂x 1<br />
−<br />
∂ n<br />
<br />
ψ(x 1 )ψ (x 2 )<br />
∂x 2 x 1 =x 2 =x<br />
(A1.1)<br />
we find<br />
p (x) = ∇ x S(x)<br />
(A1.6)<br />
P B (x) =<br />
<br />
[p, f(x, p)] BB dp/ρ(x)<br />
Bohm Momentum<br />
Finally<br />
∇ x ρ(x)<br />
2ρ(x)<br />
=<br />
<br />
<br />
[p, f(x, p)] MB dp /ρ(x)<br />
Osmotic Momentum<br />
Thursday, 2 February 2012
Kinetic energy.<br />
Form<br />
p 2 f(x, p) = [p 2 − ip −→ ∇ x − 1 4<br />
−→ 2<br />
∇ x ]f(x, p).<br />
and<br />
f(x, p) p 2 = f(x, p)[p 2 + ip ←−<br />
∇ x − 1 4<br />
←−<br />
∇ x<br />
2]<br />
Now form the Baker bracket<br />
[p 2 ,f(x, p)] BB = p 2 f(x, p) − 1 4 ∇2 xf(x, p).<br />
We need to find the marginal<br />
<br />
[p 2 ,f(x, p)] BB dp = ρ(x) p 2 (x) − 1 4 ∇2 xρ(x).<br />
with<br />
p 2 (x) = (∇ x S(x)) 2 + 1 4<br />
∇ 2 xρ(x)<br />
ρ(x)<br />
− ∇2 xR(x)<br />
R(x)<br />
So that<br />
<br />
Bohm Kinetic Energy<br />
[p 2 ,f(x, p)] BB dp/ρ(x) = (∇ x S(x)) 2 −∇ 2 xR(x)/R(x) = P 2 B + Q. [Remember m=1/2.]<br />
To complete the story, the Moyal bracket gives<br />
<br />
[p 2 ,F(x, p)] MB dp/ρ(x) = ∇ 2 xS(x) +<br />
<br />
∇x ρ(x)<br />
∇ x S(x).<br />
ρ(x)<br />
Quantum Potential<br />
[Leavens, Found. Phys., 35 (2005) 469-91]<br />
Thursday, 2 February 2012
Summary so far.<br />
We are interested in the values that can be found experimentally using weak measurements.<br />
We have seen how these weak values are related to the Bohm momentum, Bohm energy and the quantum potential<br />
ρP jB = ρ∂ j S = −T 0j , ρE B = −ρ∂ t S = −T 00<br />
The separation of the real from the imaginary parts of weak values achieved by forming brackets<br />
Then, for example<br />
[[P W ]] ± =<br />
ψ(t)|<br />
←<br />
P |x<br />
ψ(t)|x<br />
→<br />
± x| P |ψ(t)<br />
x|ψ(t)<br />
and<br />
P B = − 1 2 [[P xW ]] + E B = − 1 2 [[P tW ]] +<br />
In the Moyal algebra these brackets are replaced by the Baker and Moyal brackets<br />
[[P W ]] + ⇒<br />
[p, f(x, p)] BB =<br />
(f p + p f)<br />
2<br />
and<br />
[[P W ]] − ⇒<br />
[p, f(x, p)] MB =<br />
(f p − p f)<br />
i<br />
What about spin and relativity?<br />
Thursday, 2 February 2012
Spin and Clifford Algebras.<br />
We could use standard approach with matrices.<br />
Neater to use Clifford algebras.<br />
One single mathematical structure with a natural hierarchy<br />
Generating<br />
elements.<br />
C (2,4)<br />
Conformal {1, e 0<br />
, e 1<br />
, e 2<br />
, e 3<br />
, e 4<br />
, e 5<br />
}<br />
Twistors {ω, π}<br />
C (1,3)<br />
C (3,0)<br />
C (0,1)<br />
Dirac<br />
Pauli<br />
Schrödinger<br />
Klein-Gordon.<br />
{1, e 0<br />
, e 1<br />
, e 2<br />
, e 3<br />
}<br />
{1, γ 0<br />
, γ 1<br />
, γ 2<br />
, γ 3<br />
}<br />
{1, e 1<br />
, e 2<br />
, e 3<br />
}<br />
{1, σ 1<br />
, σ 2<br />
, σ 3<br />
}<br />
{1, e 1<br />
}<br />
{1, i }<br />
General element of algebra A(x) = g(x)e K where e K = e i ◦ e j ◦ ···◦ e n and i
Replace<br />
Replacement of kets and bras by elements of the algebra.<br />
x|ψ<br />
by<br />
Φ L (x) =φ L (x)<br />
ψ|x<br />
Replace by ΦL (x) = φ L (x)<br />
Clifford reversion<br />
element of min left ideal<br />
idempotent 2 = <br />
Pauli<br />
x|Ψ P →<br />
<br />
ψ1 (x)<br />
ψ 2 (x)<br />
Φ L (x) =<br />
g 0 (x)+ <br />
g i (x)e jk <br />
i, j, k cyclic<br />
g ∈ R<br />
Dirac<br />
x|Ψ D →<br />
⎛ ⎞<br />
ψ 1 (x)<br />
⎜ψ 2 (x)<br />
⎟<br />
⎝ψ 3 (x) ⎠<br />
ψ 4 (x)<br />
2g 0 =(ψ1 ∗ + ψ 1 ) 2e 123 g 3 =(ψ1 ∗ − ψ 1 ) = (1+e 3 )/2<br />
2g 2 =(ψ2 ∗ + ψ 2 ) 2e 123 g 1 =(ψ2 ∗ − ψ 2 )<br />
Φ L (x) =<br />
g 0 (x)+ <br />
g i (x)e µν + g 5 (x)e 5 µ
P B = − 1 2 [[P W ]] +<br />
Bohm momentum and energy for Pauli particle.<br />
now becomes<br />
ρP B (x) =−iΦ L (x) ←→ ∇ x<br />
ΦL (x) =−i<br />
<br />
(∇ x Φ L (x)) Φ <br />
L (x) − Φ L (x) ∇ xΦL (x)<br />
Φ L (x) =φ L (x)<br />
we choose the idempotent =(1+e 3 )/2<br />
Pauli current = convection part + rotation part<br />
The Bohm momentum comes from the convection part using<br />
Φ L (x) =<br />
g 0 (x)+ <br />
g i (x)e jk <br />
ρP B = −e 123 [g 0 ∇ x g 3 − g 3 ∇ x g 0 + g 2 ∇ x g 1 − g 1 ∇ x g 2 ].<br />
Using the conversion g(x) → ψ(x)<br />
and<br />
ψ j (x) = R j (x)e iS j(x)<br />
ρP B (x) = ρ 1 (x)∇ x S 1 (x) + ρ 2 (x)∇ x S 2 (x) = − 1 2 [[P W ]] +<br />
ρE B (x) = ρ 1 (x)∂ t S 1 (x) + ρ 2 (x)∂ t S 2 (x) = − 1 2 [[P tW ]] +<br />
Bohm Momentum<br />
Bohm Energy<br />
Bohm, Schiller and Tiomno spin-1/2 model.<br />
P B =(∇S + cos θ∇φ)/2<br />
E B = −(∂ t S + cos θ∂ t φ)/2<br />
They are the same if you use<br />
Ψ =<br />
<br />
ψ1<br />
ψ 2<br />
<br />
=<br />
<br />
cos(θ/2) exp(iφ/2)<br />
exp(iψ/2)<br />
i sin(θ/2)exp(−iφ/2)<br />
[Bohm, Schiller, and Tiomno, Nuovo Cim. Supp. 1, (1955) 48-66 and 67-91].<br />
Thursday, 2 February 2012
Bohm kinetic energy for spin-1/2 particle.<br />
Then as shown in Hiley and Callaghan, the Bohm kinetic energy is<br />
− m[[P 2 W ]] + = P 2 B (x) + [2(∇ xW (x) · S(x)) + W 2 (x)] = P 2 B + Q.<br />
Spin of particle<br />
S = i(φ L e 3<br />
φL )<br />
and<br />
ρW = ∇ x (ρS)<br />
[Hiley, and Callaghan, Found. Phys. 42, (2012) 192 and Maths-ph: 1011.4031]<br />
BST values:<br />
Q = − 1<br />
2m<br />
∇ 2 R<br />
R<br />
+[(∇θ)2 +sin 2 θ(∇φ) 2 ]/8m<br />
[Bohm, Schiller, and Tiomno, Nuovo Cim. Supp. 1, (1955) 48-66 and 67-91].<br />
Thursday, 2 February 2012
Spin in the Moyal Algebra.<br />
Central to the the Clifford algebra approach is the Clifford density element.<br />
Then<br />
ρ(x) = Φ L (x) ΦL (x)<br />
= Tr[Aρ(x)] = Tr[AΦ L (x) ΦL (x)] = Tr[Φ L (x)A Φ L (x)] = ψ|Â|ψ<br />
Generalise ρ(x 1 ,x 2 ) = Φ L (x 1 )Ξ R (x 2 )<br />
Now we are in a position extend this to phase space using a Wigner-Moyal transformation<br />
<br />
ρ M (x, p) = F (x, p) =2π Φ L (x 1 )Ξ R (x 2 )e ipy dy Cross-Wigner function.<br />
where x = (x 2 + x 1 )/2 and y = x 2 − x 1<br />
[de Gosson, Symplectic Metods in Harmonic Analysis]<br />
Recall for Pauli Φ L (x) =<br />
g 0 (x)+ <br />
g i (x)e jk <br />
Replaces spinor ket.<br />
Ξ R (x) =<br />
G 0 (x) − <br />
G i (x)e jk <br />
Replaces spinor bra.<br />
Use these expressions in the cross-Wigner function.<br />
Thursday, 2 February 2012
Easier to use the momentum representation.<br />
Details of Spin in the Moyal Algebra.<br />
F (x, p) =<br />
Now with p = (p 2 + p 1 )/2 and ∆p = p 2 − p 1<br />
and Ξ L (p) = ξ(p) =<br />
γ 0 (p) + <br />
γ K (p)e K , where the γ(p) s are Fourier transforms of the g(x)s.<br />
<br />
Again choose =(1+e 3 )/2,<br />
taking just the 1 term and then define 2F (x, p) := ξ L (p 2 ) 1 ξ L (p 1 )e −ix∆p d∆p<br />
<br />
So that 2F (x, p) = [γ 0 (p 2 )γ 0 (p 1 ) + γ 1 (p 2 )γ 1 (p 1 ) + γ 2 (p 2 )γ 2 (p 1 ) + γ 3 (p 2 )γ 3 (p 1 )]e −ix∆p d∆p.<br />
1<br />
2π<br />
<br />
Ξ L (p 1 ) ΞL (p 2 )e −ix∆p d∆p.<br />
Now form<br />
p F (x, p) = pF (x, p) − i 2 ∇ xF (x, p) F (x, p) p = pF (x, p) + i 2 ∇ xF (x, p)<br />
and<br />
Baker bracket<br />
[p, F ] BB = pF (x, p),<br />
Moyal bracket [p, F ] MB = ∇ x F (x, p).<br />
Use the Moyal relation<br />
F (x, p) = 1<br />
2π<br />
<br />
<br />
M(p 1 ,p 2 )δ p − p <br />
2 − p 1<br />
e ix(p 2−p 1 ) dp 1 dp 2<br />
2<br />
where<br />
M(p 1 ,p 2 ) = 1 2 [φ 0(p 1 )φ ∗ 0 (p 2) + φ 1 (p 1 )φ ∗ 1 (p 2) + φ 2 (p 1 )φ ∗ 2 (p 2) + φ 3 (p 1 )φ ∗ 3 (p 2)]<br />
<br />
so that we find<br />
[p, F ] BB dp = ρ 1 (x)∂ x S 1 (x) + ρ 2 (x)∂ x S 2 (x). Baker bracket Bohm momentum.<br />
<br />
[p 2 ,F(x, p)] BB dp = ρ 1 (x)(∇ x S 1 (x)) 2 + ρ 2 (x)(∇ x S 2 (x)) 2 − R 1 (x)∇ 2 xR 1 (x) − R 2 (x)∇ 2 xR.<br />
Thursday, 2 February 2012<br />
Bohm KE<br />
Quantum Potential
Time Development Equations.<br />
-ganvalues<br />
H(x, p) f(x, p) =E 1 f(x, p) f(x, p) H(x, p) =E 2 f(x, p)<br />
or<br />
Time development<br />
Difference<br />
Sum<br />
H f =<br />
i<br />
2π<br />
<br />
ψ ∗ (x 2 )[∂ t ψ(x 1 )]e ipy dy<br />
and<br />
f H = −i<br />
2π<br />
∂f<br />
∂t +[f,H] MB =0 In the limit O() → Classical Liouville eqn.<br />
<br />
i<br />
ψ ∗ (x 2 ) ←→ ∂ t ψ(x 1 )e ipy dy +[f,H] BB =0<br />
π<br />
f(x, p)E B (x, p) =<br />
i <br />
ψ ∗ (x 2 ) ←→ ∂ t ψ(x 1 )e ipy dy<br />
2π<br />
<br />
[∂ t ψ ∗ (x 2 )]ψ(x 1 )e ipy dy<br />
NB<br />
x =(x 2 + x 1 )/2<br />
y =(x 2 − x 1 )<br />
In the limit O() →<br />
∂S<br />
∂t + H =0<br />
Hamilton-Jacobi eqn.<br />
Take the marginal as before<br />
∂f(x, p)<br />
∂t<br />
+[f,H] MB =0<br />
is equivalent to<br />
∂ρ(x)<br />
∂t<br />
+[ρ, H] − =0<br />
Quantum Liouville eqn.<br />
2f(x, p)E B (x, p)+[f,H] BB =0<br />
is equivalent to 2ρ(x)E B (x)+[ρ,H] + =0<br />
If<br />
H = p2 (x)<br />
2m<br />
+ V (x)<br />
then<br />
∂S<br />
∂t + (∇S)2<br />
2m<br />
+ Q + V =0 Quantum Hamilton-Jacobi eqn.<br />
[Bohm, Phys. Rev., 85 (1952) 166-179; and 85 (1952), 180-193\.<br />
Thursday, 2 February 2012
Summary of Algebraic Time Evolution Equations.<br />
Moyal algebra<br />
Phase space<br />
Quantum algebra<br />
Configuration space<br />
Notice in the general form there is no quantum potential.<br />
The QP appears ONLY in a representation<br />
In the x-representation you get Bohm’s Q(x).<br />
In the p-representation you get another QP--Q(p)<br />
In Moyal take the x-marginal.<br />
[M. R. Brown & B. J. Hiley, quant-ph/0005026]<br />
Thursday, 2 February 2012
Mathematical Structure.<br />
Orthogonal Clifford<br />
+ Moyal<br />
Symplectic Clifford.<br />
[Crumeyrolle Orthogonal and Symplectic Clifford Algebras, (1990)]<br />
Take marginals<br />
Configuration space<br />
Position<br />
Configuration space<br />
Momentum<br />
Order <br />
Classical<br />
phase space<br />
Thursday, 2 February 2012
Physics behind these Algebras?<br />
Basic structure ρ φψ (x 1 ,x 2 ) → f(x, p)<br />
Transition ⇒ Process “Phase” space<br />
(x, p)<br />
x 1<br />
x 2<br />
y<br />
<br />
ρ M (x, p) =2π<br />
Φ L (x 1 )Ξ R (x 2 )e ipy dy<br />
Quantum particle not a ‘rock’ but a ‘blob’.<br />
Structure process.<br />
[Bohm, Proc. Int. Conf. on Elementary Particles, Kyoto, 252-287, (1965)]<br />
t s t Only combine if t = s a b<br />
groupoid<br />
s<br />
orthogonal and symplectic<br />
product ⇒ order of succession and the order of co-existence ⇒ addition gives the algebra.<br />
Combine Clifford and Moyal algebras<br />
⇒<br />
Non-commutative geometry<br />
Orthogonal Clifford and symplectic Clifford algebras.<br />
[Hiley, Lecture Notes in Physics, vol. 813, pp. 705-750, Springer (2011)].<br />
[Lizzi, Non-commutative spaces, Springer Lecture Notes in Physics 774, 2009]<br />
Thursday, 2 February 2012
Consequences of Non-commutative Structure.<br />
Changes from both sides Inner automorphism T AT −1<br />
Evolution in τ<br />
A τ = T (τ)A τ0 T (τ) −1<br />
If<br />
T =exp[iHτ] then for small τ<br />
i (A τ − A 0 )<br />
τ<br />
=(HA 0 − A 0 H)<br />
⇒<br />
i ˙ A =[A, H]<br />
Heisenberg eqn.<br />
Just Bohm’s ‘folding’ and ‘unfolding’.<br />
T 1<br />
T 2<br />
T 1<br />
T 2<br />
[T 1<br />
,T 2<br />
]<br />
[Hiley, Lecture Notes in Physics, vol. 813, pp. 705-750, Springer (2011)].<br />
Thursday, 2 February 2012
Overarching Philosophy.<br />
[Bohm, D., Wholeness and the Implicate Order, 1980.]<br />
The deeper structure gives rise to a non-commutative phase space geometry.<br />
In this context non-commutativity ⇒ not all orders can be made explicit together.<br />
Implicate order<br />
⇒<br />
algebra<br />
Project out explicate orders<br />
ψ(x)<br />
or<br />
φ(p)<br />
Quantum world<br />
Classical world<br />
Thursday, 2 February 2012