Weak Measurements:

Weak Measurements:
Wigner-Moyal and Bohm in a
New Light.
Basil J. Hiley.
www.bbk.ac.uk/tpru.
Thursday, 2 February 2012

Weak Measurements.
1. New type of quantum measurement.
[Aharonov, and Vaidman, Phys. Rev., 41, (1990) 11-19].
2. What do we measure?
Weak values defined by
A W (ψ,φ) = φ|A|ψ
φ|ψ
∈ C
3. Will show that P W (ψ,x) is related to the Bohm momentum
P 2 W (ψ,x)
is related to the Bohm energy and quantum potential.
[Leavens, Found. Phys., 35 (2005) 469-91]
4. Will obtain expressions for weak values for spin-1/2 particles using Clifford algebras.
5. Will indicate how to combine the Clifford with the Moyal algebra.
6. Discuss what may lie behind the formalism-- Process.
Acknowledgements
Ernst Binz, Maurice de Gosson, Bob Callaghan and David Robson.
Thursday, 2 February 2012

Weak measurements.
Why the interest?
Photon ‘trajectories’.
Schrödinger particle ‘trajectories
Experimental--Photons.
[Kocsis, Braverman,Ravets,Stevens,Mirin, Shalm, Steinberg,
Science 332, 1170 (2011)]
[Prosser, IJTP , 15, (1976) 169]
Theory--Schrödinger particle.
[Philippidis, Dewdney and Hiley, Nuovo Cimento 52B, 15-28 (1979)]
Real part of Schrödinger equation
∂S
∂t + (∇S)2
2m + Q + V =0
E B = − ∂S
∂t
P B = ∇S
Thursday, 2 February 2012

Weak Measurement: the principle.
H I = λ(t)Âs ˆB d
|initial = |ψ s |Ψ d |final = e i R λ(t)Âs ˆB d dt |ψ s |Ψ d
λ(t)dt =1
|final = e iÂsb |ψ s |Ψ d
Form the transition probability amplitude T φ−ψ = b d |φ s |final b d |Â|Ψ d = ÂΨ d(b)
T φ−ψ = φ s |e iÂb |ψ s Ψ d (b) = (ib) n
φ s
n!
|Ân |ψ s Ψ d (b) =φ s |ψ s (ib) n φ s |Ân |ψ s
Ψ d (b)
n! φ
n=0
n=0
s |ψ s
T φ−ψ = φ s |ψ s e ibA W
+
(ib) n
[A n W −A n W ] Ψ d (b)
n!
n=2
small
Post select with b = x T φ−ψ = φ s |ψ s e ixA W
Ψ d (x)
−x
2
Choose Ψ d (x) =exp
and take the imaginary part of A W we find
4(∆x) 2
T φ−ψ ∝ e −xA IW −x
2
exp
4(∆x) 2 ∝ exp −
x + 2(∆x) 2 2
A IW
4(∆x) 2
Centre of Gaussian in x-space shifted by amount ∝A IW
Centre of Gaussian in p-space shifted by amount ∝A RW
Thursday, 2 February 2012
[Duck, Stevenson and Sudarshan Phys. Rev, 40 (1989) 2112-7]

von Neumann measurement.
Weak measurement.
Weak Experiment cont.
Single measurement ---- collapse of the wave function.
Ψ j → Ψ r
j
∝A IW
Statistical measurement producing a phase shift in the distribution of final results.
Actual experimental arrangement for photon trajectories.
x
|Ψ = |D polarizer |ψ path H = g ˆP x Ŝ 1 Ŝ 1 =(|HH| − |V V |)/2
Thursday, 2 February 2012
[Kocsis, Braverman,Ravets,Stevens,Mirin, Shalm, Steinberg,
Science 332, 1170 (2011)]

Weak values.
φ|A|ψ
φ|ψ
How do they appear in the formalism?
[ Hiley quant-ph/1111.6536]
Then
ψ|A|ψ = ψ|φ j φ j |A|ψ where |φ j form a complete orthonormal set.
ψ|A|ψ = ψ|φ j
Post select
φj |ψ
φ j |A|ψ = φ j |A|ψ
ρ j
φ j |ψ
φ j |ψ
Weak value.
Special case:
If A|φ j = a j |φ j
ψ|A|ψ = ρ j a j
Well known result!
Eigenvalue!
Remember
φ|A|ψ
φ|ψ
is a complex number.
But what does it mean in general?
It is clearly a transition probability amplitude.
Thursday, 2 February 2012

Weak values when
ˆP is involved.
Consider x| ˆP |ψ(t) = x| ˆP |x x |ψ(t)dx = −i∇ x ψ(x, t)
Write
ψ(x, t) =R(x, t)e iS(x,t)
then
x| ˆP |ψ(t)
x|ψ(t)
= ∇ x S(x, t) − i∇ x ρ(x, t)/2ρ(x, t)
with
ρ(x, t) =|ψ(x, t)| 2
Bohm velocity:
Osmotic velocity:
In terms of one expression
⎡
[[P W ]] ± =
Bohm momentum.
v B (x, t) =∇ x S(x, t)/m = 1
2m
v O (x, t) = 1
2m
⎣ ψ(t)|←−ˆP |x
ψ(t)|x
More simply for Schrödinger
then
[[P W ]] ± =
∇ x ρ(x, t)
ρ(x, t)
osmotic momentum.
⎡
⎣ ψ(t)|←−ˆP |x
ψ(t)|x
= 1
2m
⎡
⎣ ψ(t)|←−ˆP |x
ψ(t)|x
+ x|−→ˆP |ψ(t)
x|ψ(t)
⎤
⎦
⎤
− x|−→ˆP |ψ(t)
⎦
x|ψ(t)
⎤
± x|−→ˆP |ψ(t)
⎦ = ψ∗ (x, t) ←−ˆPx ψ(x, t) ± ψ ∗ (x, t) −→ˆPx ψ(x, t)
.
x|ψ(t)
ρ(x, t)
i
ρ(x, t) [(∇ xψ ∗ (x, t))ψ(x, t) ∓ ψ ∗ (x, t)(∇ x ψ(x, t))]
− iρ[[P xW ]] + = [∇ x ψ ∗ (x)]ψ(x) − ψ ∗ (x)[∇ x ψ(x)] = ψ ∗ (x) ←→ ∇ x ψ(x)
− iρ[[P xW ]] − = [∇ x ψ ∗ (x)]ψ(x) + ψ ∗ (x)[∇ x ψ(x)] = ∇ x (ψ ∗ (x)ψ(x)) = ∇ x ρ(x).
Thursday, 2 February 2012

Remark 1: Relation to Nelson.
Mean forward derivative: Dx(t) = lim
∆t→0 + E i
x(t + ∆t) − x(t)
∆t
[Nelson, Phys. Rev., 150, (1966), 1079-1085.]
Mean backward derivative:
D ∗ x(t) =
x(t) − x(t − ∆t)
lim E i
∆t→0 + ∆t
With these construct a forward velocity b(x, t) and backward velocity b ∗ (x, t) :
[b(x, t)+b ∗ (x, t)]/2 =v B (x, t) = ∇ xS(x, t)
m
b(x, t) − b ∗ (x, t) =v O (x, t) = 1
2m
∇ x ρ(x, t)
ρ(x, t)
[Bohm and Hiley, Phys. Reps, 172, (1989), 92-122.]
Thursday, 2 February 2012

Remark 2: Relation to Energy-Momentum Tensor.
∂L
T µν = −
∂(∂ µ ψ) ∂ν ψ +
∂L
∂(∂ µ ψ ∗ ) ∂ν ψ ∗
Take the Schrödinger Lagrangian: L = − 1
2m ∇ψ∗ · ∇ψ + i 2 [ψ∗ (∂ t ψ) − (∂ t ψ ∗ )ψ] − V ψ ∗ ψ.
and find
T 0µ = − i 2 [(∂µ ψ ∗ )ψ − ψ ∗ (∂ µ ψ)] = i 2 [ψ∗←→ ∂ µ ψ]=−ρ∂ µ S
Recalling that
P B = ∇S
and
E B = −∂ t S
Then we find
ρP jB = ρ∂ j S = −T 0j = − ρ 2 [[P jW]] +
ρE B = −ρ∂ t S = −T 00 = − ρ 2 [[P tW ]] +
Bohm momentum.
Bohm energy.
This generalises to the Pauli and Dirac particles
The Bohm kinetic energy.
[Hiley and Callaghan, Fond. Phys 42 (2012) 192-208,
math-ph:1011.4031 and 1011.4033]
1
2 [[P W 2 ]] + = (∇ x S(x)) 2 − ∇2 xR(x)
= PB 2 + Q.
R(x)
1
2i [[P W 2 ]] − = ∇ 2 ∇x ρ(x)
xS(x) +
∇ x S(x).
ρ(x)
Quantum potential
[Leavens, Found. Phys., 35 (2005) 469-91]
[Wiseman, New J. Phys., 9 (2007) 165-77.]
Thursday, 2 February 2012

Multiplication of phase space functions defined by
Let
Weak values from Moyal algebra.
a(x, p) b(x, p)
where
f(x, p) be the density matrix in (x, p) representation, viz. f(x, p) =
i
=exp
2 ( ∂ ← →
x ∂p − ∂ ← →
p ∂x )
[ Moyal, Proc. Camb. Phil. Soc. 45, (1949), 99-123].
[Baker, Phys. Rev. 6 (1958) 2198-2206.]
ρ(x − y/2; x + y/2)e −ipy dy
p f(x, p) =
p − i
→
∂ x
f(x, p) f(x, p) p = f(x, p) p + i
←
and
∂ x
2
2
Now form
(f p + p f)
[p, f(x, p)] BB := = pf(x, p)
2
Baker bracket
(f p − p f)
[p, f(x, p)] MB := = ∇ x f(x, p)
i
Moyal bracket [x, p] MB =1
To make contact with configuration space we must form a marginal;
[p, f(x, p)] BB dp = pf(x, p)dp = ρ(x) p (x)
Moyal momentum
Using
ρ(x) p n (x) =
1
2i
n
∂
∂x 1
−
∂ n
ψ(x 1 )ψ (x 2 )
∂x 2 x 1 =x 2 =x
(A1.1)
we find
p (x) = ∇ x S(x)
(A1.6)
P B (x) =
[p, f(x, p)] BB dp/ρ(x)
Bohm Momentum
Finally
∇ x ρ(x)
2ρ(x)
=
[p, f(x, p)] MB dp /ρ(x)
Osmotic Momentum
Thursday, 2 February 2012

Kinetic energy.
Form
p 2 f(x, p) = [p 2 − ip −→ ∇ x − 1 4
−→ 2
∇ x ]f(x, p).
and
f(x, p) p 2 = f(x, p)[p 2 + ip ←−
∇ x − 1 4
←−
∇ x
2]
Now form the Baker bracket
[p 2 ,f(x, p)] BB = p 2 f(x, p) − 1 4 ∇2 xf(x, p).
We need to find the marginal
[p 2 ,f(x, p)] BB dp = ρ(x) p 2 (x) − 1 4 ∇2 xρ(x).
with
p 2 (x) = (∇ x S(x)) 2 + 1 4
∇ 2 xρ(x)
ρ(x)
− ∇2 xR(x)
R(x)
So that
Bohm Kinetic Energy
[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.]
To complete the story, the Moyal bracket gives
[p 2 ,F(x, p)] MB dp/ρ(x) = ∇ 2 xS(x) +
∇x ρ(x)
∇ x S(x).
ρ(x)
Quantum Potential
[Leavens, Found. Phys., 35 (2005) 469-91]
Thursday, 2 February 2012

Summary so far.
We are interested in the values that can be found experimentally using weak measurements.
We have seen how these weak values are related to the Bohm momentum, Bohm energy and the quantum potential
ρP jB = ρ∂ j S = −T 0j , ρE B = −ρ∂ t S = −T 00
The separation of the real from the imaginary parts of weak values achieved by forming brackets
Then, for example
[[P W ]] ± =
ψ(t)|
←
P |x
ψ(t)|x
→
± x| P |ψ(t)
x|ψ(t)
and
P B = − 1 2 [[P xW ]] + E B = − 1 2 [[P tW ]] +
In the Moyal algebra these brackets are replaced by the Baker and Moyal brackets
[[P W ]] + ⇒
[p, f(x, p)] BB =
(f p + p f)
2
and
[[P W ]] − ⇒
[p, f(x, p)] MB =
(f p − p f)
i
What about spin and relativity?
Thursday, 2 February 2012

Spin and Clifford Algebras.
We could use standard approach with matrices.
Neater to use Clifford algebras.
One single mathematical structure with a natural hierarchy
Generating
elements.
C (2,4)
Conformal {1, e 0
, e 1
, e 2
, e 3
, e 4
, e 5
}
Twistors {ω, π}
C (1,3)
C (3,0)
C (0,1)
Dirac
Pauli
Schrödinger
Klein-Gordon.
{1, e 0
, e 1
, e 2
, e 3
}
{1, γ 0
, γ 1
, γ 2
, γ 3
}
{1, e 1
, e 2
, e 3
}
{1, σ 1
, σ 2
, σ 3
}
{1, e 1
}
{1, i }
General element of algebra A(x) = g(x)e K where e K = e i ◦ e j ◦ ···◦ e n and i

Replace
Replacement of kets and bras by elements of the algebra.
x|ψ
by
Φ L (x) =φ L (x)
ψ|x
Replace by ΦL (x) = φ L (x)
Clifford reversion
element of min left ideal
idempotent 2 =
Pauli
x|Ψ P →
ψ1 (x)
ψ 2 (x)
Φ L (x) =
g 0 (x)+
g i (x)e jk
i, j, k cyclic
g ∈ R
Dirac
x|Ψ D →
⎛ ⎞
ψ 1 (x)
⎜ψ 2 (x)
⎟
⎝ψ 3 (x) ⎠
ψ 4 (x)
2g 0 =(ψ1 ∗ + ψ 1 ) 2e 123 g 3 =(ψ1 ∗ − ψ 1 ) = (1+e 3 )/2
2g 2 =(ψ2 ∗ + ψ 2 ) 2e 123 g 1 =(ψ2 ∗ − ψ 2 )
Φ L (x) =
g 0 (x)+
g i (x)e µν + g 5 (x)e 5 µ

P B = − 1 2 [[P W ]] +
Bohm momentum and energy for Pauli particle.
now becomes
ρP B (x) =−iΦ L (x) ←→ ∇ x
ΦL (x) =−i
(∇ x Φ L (x)) Φ
L (x) − Φ L (x) ∇ xΦL (x)
Φ L (x) =φ L (x)
we choose the idempotent =(1+e 3 )/2
Pauli current = convection part + rotation part
The Bohm momentum comes from the convection part using
Φ L (x) =
g 0 (x)+
g i (x)e jk
ρP B = −e 123 [g 0 ∇ x g 3 − g 3 ∇ x g 0 + g 2 ∇ x g 1 − g 1 ∇ x g 2 ].
Using the conversion g(x) → ψ(x)
and
ψ j (x) = R j (x)e iS j(x)
ρP B (x) = ρ 1 (x)∇ x S 1 (x) + ρ 2 (x)∇ x S 2 (x) = − 1 2 [[P W ]] +
ρE B (x) = ρ 1 (x)∂ t S 1 (x) + ρ 2 (x)∂ t S 2 (x) = − 1 2 [[P tW ]] +
Bohm Momentum
Bohm Energy
Bohm, Schiller and Tiomno spin-1/2 model.
P B =(∇S + cos θ∇φ)/2
E B = −(∂ t S + cos θ∂ t φ)/2
They are the same if you use
Ψ =
ψ1
ψ 2
=
cos(θ/2) exp(iφ/2)
exp(iψ/2)
i sin(θ/2)exp(−iφ/2)
[Bohm, Schiller, and Tiomno, Nuovo Cim. Supp. 1, (1955) 48-66 and 67-91].
Thursday, 2 February 2012

Bohm kinetic energy for spin-1/2 particle.
Then as shown in Hiley and Callaghan, the Bohm kinetic energy is
− m[[P 2 W ]] + = P 2 B (x) + [2(∇ xW (x) · S(x)) + W 2 (x)] = P 2 B + Q.
Spin of particle
S = i(φ L e 3
φL )
and
ρW = ∇ x (ρS)
[Hiley, and Callaghan, Found. Phys. 42, (2012) 192 and Maths-ph: 1011.4031]
BST values:
Q = − 1
2m
∇ 2 R
R
+[(∇θ)2 +sin 2 θ(∇φ) 2 ]/8m
[Bohm, Schiller, and Tiomno, Nuovo Cim. Supp. 1, (1955) 48-66 and 67-91].
Thursday, 2 February 2012

Spin in the Moyal Algebra.
Central to the the Clifford algebra approach is the Clifford density element.
Then
ρ(x) = Φ L (x) ΦL (x)
= Tr[Aρ(x)] = Tr[AΦ L (x) ΦL (x)] = Tr[Φ L (x)A Φ L (x)] = ψ|Â|ψ
Generalise ρ(x 1 ,x 2 ) = Φ L (x 1 )Ξ R (x 2 )
Now we are in a position extend this to phase space using a Wigner-Moyal transformation
ρ M (x, p) = F (x, p) =2π Φ L (x 1 )Ξ R (x 2 )e ipy dy Cross-Wigner function.
where x = (x 2 + x 1 )/2 and y = x 2 − x 1
[de Gosson, Symplectic Metods in Harmonic Analysis]
Recall for Pauli Φ L (x) =
g 0 (x)+
g i (x)e jk
Replaces spinor ket.
Ξ R (x) =
G 0 (x) −
G i (x)e jk
Replaces spinor bra.
Use these expressions in the cross-Wigner function.
Thursday, 2 February 2012

Easier to use the momentum representation.
Details of Spin in the Moyal Algebra.
F (x, p) =
Now with p = (p 2 + p 1 )/2 and ∆p = p 2 − p 1
and Ξ L (p) = ξ(p) =
γ 0 (p) +
γ K (p)e K , where the γ(p) s are Fourier transforms of the g(x)s.
Again choose =(1+e 3 )/2,
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
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.
1
2π
Ξ L (p 1 ) ΞL (p 2 )e −ix∆p d∆p.
Now form
p F (x, p) = pF (x, p) − i 2 ∇ xF (x, p) F (x, p) p = pF (x, p) + i 2 ∇ xF (x, p)
and
Baker bracket
[p, F ] BB = pF (x, p),
Moyal bracket [p, F ] MB = ∇ x F (x, p).
Use the Moyal relation
F (x, p) = 1
2π
M(p 1 ,p 2 )δ p − p
2 − p 1
e ix(p 2−p 1 ) dp 1 dp 2
2
where
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)]
so that we find
[p, F ] BB dp = ρ 1 (x)∂ x S 1 (x) + ρ 2 (x)∂ x S 2 (x). Baker bracket Bohm momentum.
[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.
Thursday, 2 February 2012
Bohm KE
Quantum Potential

Time Development Equations.
-ganvalues
H(x, p) f(x, p) =E 1 f(x, p) f(x, p) H(x, p) =E 2 f(x, p)
or
Time development
Difference
Sum
H f =
i
2π
ψ ∗ (x 2 )[∂ t ψ(x 1 )]e ipy dy
and
f H = −i
2π
∂f
∂t +[f,H] MB =0 In the limit O() → Classical Liouville eqn.
i
ψ ∗ (x 2 ) ←→ ∂ t ψ(x 1 )e ipy dy +[f,H] BB =0
π
f(x, p)E B (x, p) =
i
ψ ∗ (x 2 ) ←→ ∂ t ψ(x 1 )e ipy dy
2π
[∂ t ψ ∗ (x 2 )]ψ(x 1 )e ipy dy
NB
x =(x 2 + x 1 )/2
y =(x 2 − x 1 )
In the limit O() →
∂S
∂t + H =0
Hamilton-Jacobi eqn.
Take the marginal as before
∂f(x, p)
∂t
+[f,H] MB =0
is equivalent to
∂ρ(x)
∂t
+[ρ, H] − =0
Quantum Liouville eqn.
2f(x, p)E B (x, p)+[f,H] BB =0
is equivalent to 2ρ(x)E B (x)+[ρ,H] + =0
If
H = p2 (x)
2m
+ V (x)
then
∂S
∂t + (∇S)2
2m
+ Q + V =0 Quantum Hamilton-Jacobi eqn.
[Bohm, Phys. Rev., 85 (1952) 166-179; and 85 (1952), 180-193\.
Thursday, 2 February 2012

Summary of Algebraic Time Evolution Equations.
Moyal algebra
Phase space
Quantum algebra
Configuration space
Notice in the general form there is no quantum potential.
The QP appears ONLY in a representation
In the x-representation you get Bohm’s Q(x).
In the p-representation you get another QP--Q(p)
In Moyal take the x-marginal.
[M. R. Brown & B. J. Hiley, quant-ph/0005026]
Thursday, 2 February 2012

Mathematical Structure.
Orthogonal Clifford
+ Moyal
Symplectic Clifford.
[Crumeyrolle Orthogonal and Symplectic Clifford Algebras, (1990)]
Take marginals
Configuration space
Position
Configuration space
Momentum
Order
Classical
phase space
Thursday, 2 February 2012

Physics behind these Algebras?
Basic structure ρ φψ (x 1 ,x 2 ) → f(x, p)
Transition ⇒ Process “Phase” space
(x, p)
x 1
x 2
y
ρ M (x, p) =2π
Φ L (x 1 )Ξ R (x 2 )e ipy dy
Quantum particle not a ‘rock’ but a ‘blob’.
Structure process.
[Bohm, Proc. Int. Conf. on Elementary Particles, Kyoto, 252-287, (1965)]
t s t Only combine if t = s a b
groupoid
s
orthogonal and symplectic
product ⇒ order of succession and the order of co-existence ⇒ addition gives the algebra.
Combine Clifford and Moyal algebras
⇒
Non-commutative geometry
Orthogonal Clifford and symplectic Clifford algebras.
[Hiley, Lecture Notes in Physics, vol. 813, pp. 705-750, Springer (2011)].
[Lizzi, Non-commutative spaces, Springer Lecture Notes in Physics 774, 2009]
Thursday, 2 February 2012

Consequences of Non-commutative Structure.
Changes from both sides Inner automorphism T AT −1
Evolution in τ
A τ = T (τ)A τ0 T (τ) −1
If
T =exp[iHτ] then for small τ
i (A τ − A 0 )
τ
=(HA 0 − A 0 H)
⇒
i ˙ A =[A, H]
Heisenberg eqn.
Just Bohm’s ‘folding’ and ‘unfolding’.
T 1
T 2
T 1
T 2
[T 1
,T 2
]
[Hiley, Lecture Notes in Physics, vol. 813, pp. 705-750, Springer (2011)].
Thursday, 2 February 2012

Overarching Philosophy.
[Bohm, D., Wholeness and the Implicate Order, 1980.]
The deeper structure gives rise to a non-commutative phase space geometry.
In this context non-commutativity ⇒ not all orders can be made explicit together.
Implicate order
⇒
algebra
Project out explicate orders
ψ(x)
or
φ(p)
Quantum world
Classical world
Thursday, 2 February 2012