What is the precise role of non-commutativity in Quantum Theory?

B. J. Hiley.Theoretical Physics Research Unit, Birkbeck, University of

London, Malet Street, London, WC1E 7HX.[b.hiley@bbk.ac.uk]

Non-commutativity.We know

€

[X, P]= ih ⇒ ΔXΔP ≈ h.

The uncertainty principle:You cannot measure X and P simultaneously.

But is it just non-commutativity?

Rotations don’t commute in the classical world.

It is not non-commutativity per se

Eigenvalues.It is when we take eigenvalues that we get trouble.

€

X x = x xP p = p p

€

But not x, p

€

Sx x + = + x +Sz z + = + z +

€

But not x+, z +

But the symmetries are carried by operators and not eigenvalues.

X, P Heisenberg group

Sx, Sz Rotation groupThe dynamics is in the operators

€

i dAdt

= [A,H ] Heisenberg’s equation ofmotion.

Symbolism.Introduce symbols

andi j

To represent

€

i

€

j

And any operator can be written as

€

A = xij i∑ j

€

(1) i j →(2) i j → operator.

C

We know these satisfy

Thus we can form

ori i i j

Complex number Matrix.

Expectation values.

€

A ji =

i j

€

trace A = Aii∑ = i j

Call

€

trace MA = i j

But this is just

€

=M jii j

i i

€

= ψ A ψ

[Lou Kauffman Knots and Physics (2001)][Bob Coecke Växjö Lecture 2005]

Pure state

Mixed States and the GNS Construction.Can we write

€

A = tr(ρA) = Ψ A Ψ for mixed states?

Yes. You double everything!

€

ρ =. .. . ⎛ ⎝ ⎜

⎞ ⎠ ⎟→ Ψ =

.

.

.

.

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

€

AρB → A⊗ ˜ B ( )Ψand

€

A = trρA =

€

= Ψ A⊗A Ψ

Can be generalized to many particle systems.

[Bisch & Jones preprint 2004]

Planaralgebras.

Quantum Teleportation

Underlying this diagram is a tensor *-category.

[B. Coecke, quant-ph/0506132]

Input state. Entangled state.

Bell measurement.Output state.U

Elements of Left and Right ideals.1 two-sided object splits into 2 one-sided objects.

€

Aji = i j i j

Left ideal Right ideal

Algebraically the elements of the ideals are split byan IDEMPOTENT.

€

→ e with e2 = e

Left ideal

€

ˆ Ψ L = ˆ A e Right ideal

€

ˆ Ψ R = e ˆ A

i j j i ji

This are just spinors.

Examples.

Spinors are elements of a left ideal in Clifford algebra

Symplectic spinors are elements of left ideal in Heisenberg algebra

€

ˆ Ψ L ˆ Ψ R = ˆ X =t + z x − iyx + iy t − z ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Algebraic equivalent of wave function.

Everything is in the algebra.

Rotation Group.

k n

€

ˆ Ψ L ˆ Ψ R = i k n j

€

= ˆ X = i j

Symplectic Group.

Eigenvalues again.

Find in two ways.

(1) Diagonalising operator. Find spectra.

(2) Use eigenvector.i

€

X x = x x

€

i d ψdt

= H ψetc. and

But we also have

€

−i d ψdt

= ψ H

€

x X = x x etc. and

Anything new? Why complex? Why double?

i j

Now for something completely different!You can do quantum mechanics with sharply defined x and p!

€

Use ψ = Rexp[iS] in Schrödinger equation.Bohm model.

Real part gives:-

€

∂S∂t

+ ∇S( )2

2m+ ∇2R

2mR+V = 0

Quantum Hamilton-Jacobi

€

If p =∇S and E = −∂S∂t

this becomes

€

E = p2

2m+Q +V Conservation of

Energy.

New quality of energy Why?Quantum potential energy

[Bohm & Hiley, The Undivided Universe, 1993]

Probability.

Imaginary part of the Schrödinger equation gives

€

∂P∂t

+∇. P∇Sm

⎛ ⎝ ⎜ ⎞

⎠ ⎟= 0

Conservation of Probability.€

where P = R2

Start with quantum probability end with quantum probability.

Predictions identical to standard quantum mechanics.

Bohm trajectories

Screen

Slits

Incidentparticles

x

t

Barrier x

tBarrier

Wigner-Moyal Approach.

€

A = A X, P( )∫ f X, P,t( )dXdP = ψ ˆ A ψ

€

f X,P( ) = 1

2πψ ∗ X −ηh 2( )e− iηPψ∫ X +ηh 2( )dη

€

A X, P( ) = 1

2π( )2eiηP∫ ψ∗(X − ηh / 2) ˆ A ψ (X + ηh / 2)dηdX

€

ˆ A = 1

(2π )2

ψ * x, t( )∫ A X, P( )ei [ π ( ˆ x − X )+η ( ˆ p −P )]ψ (x,t )dηdπdXdPdx

Find probability distribution f (X, P, t) so that expectation value

Expectation value identical to quantum valueNeed relations

[C. Zachos, hep-th/0110114]

Problem: f (X, P, t) can be negative.

Bohm and Wigner-Moyal Different?

€

ρ(X)p n = P n f (X,P)dX = 12i ⎛ ⎝ ⎜ ⎞

⎠ ⎟∫n ∂

∂x1

− ∂∂x2

⎛ ⎝ ⎜

⎞ ⎠ ⎟n

ψ (x1 )ψ*(x2 )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥x1 =x2 =X

(A1.1)

€

p (x) = 12 i ψ *∇ψ − (∇ψ * )ψ[ ] =∇S (A1.6)

€

∂ρ∂t

+∇(ρ ∇Sm

) = 0 (A4.2)

NO!

Mean Moyal momentum

€

ψ(x,t) = ρ 12 exp[iS]Use

This is just Bohm’s

€

p =∇S

Transport equation for the probability

[J. E. Moyal, Proc. Camb. Soc, 45, 99-123, (1949)]

Same as Bohm

Transport equation for

This is Bohm’s quantum Hamilton-Jacobi equation.

p

€

∂∂t

ρp ( ) + ∂∂xi

ρpk

∂H ∂x1

⎛ ⎝ ⎜

⎞ ⎠ ⎟+ ρ ∂H

∂xki

∑ = 0 (A4.3)

€

∂∂xk

∂S∂t

+ H −∇ρ 8mρ ⎡ ⎣ ⎢

⎤ ⎦ ⎥= 0 (A4.5)

€

∂S∂t

+ H −∇ρ 8mρ = ∂S∂t

+ 1

2m(∇S)2 +V − 1

2m∇ 2R R = 0

€

p Transport equation for

Which finally gives

Moyal algebra is deformed Poisson algebra• Define Moyal product *

• Moyal bracket(commutator)

• Baker bracket (Jordan product or anti-commutator)

• Classical limit Sine becomes Poisson bracket.

• Cosine becomes ordinary product.

A,B{ }MB =A∗B−B∗A

ih =2A(X,P)sinh2

s ∂

∂X

r ∂ ∂P

−r ∂

∂X

s ∂ ∂P

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥ B(X,P)

A,B( )BB =A∗B+B∗A

2 =A(X,P)cosh2

s ∂

∂X

r ∂

∂P−

r ∂

∂X

s ∂

∂P⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥ B(X,P)

A∗B =A X,P( )eih2

s ∂ ∂x

r ∂ ∂p

−r ∂ ∂x

s ∂ ∂p

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥ B(X,P)

A,B{ }MB = A,B{ }PB +O(h2) ≈ ∂A

∂X∂B∂P

−∂A∂P

∂B∂X

⎡ ⎣ ⎢

⎤ ⎦ ⎥

A,B{ }BB =AB+O(h2)

Stationary Pure States.

[D. Fairlie and C. Manogue J. Phys A24, 3807-3815, (1991)] [C. Zachos, hep-th/0110114]

€

H (X,P)∗ f (X,P) = E1 f (X,P) and f (X,P)∗H (X, P) = E2 f (X,P)

€

H , f{ }MB = E2 − E1( ) f and H , f{ }BB = E2 + E1( ) f

€

∂f∂t

+ f , H{ }MB = 0 ⇒ ∂f∂t

+ f , H{ }PB = 0

The * product is non-commutative

€

A(X,P)∗B(X, P) ≠ B(X,P)∗A(X,P)

Must have distinct left and right action.

If we add and subtract

Time dependent equations?LiouvilleEquation

*-ganvalues.

The ‘Third’ Equation

[D. Fairlie and C. Manogue J. Phys A24, 3807-3815, (1991)]

€

H ∗ f = i2π dηe−iηpψ∗(x −η 2)∂ψ (x +η 2)

∂t∫

€

f ∗H = − i2π dηe−iηp ∂ψ∗(x −η 2)

∂t∫ ψ (x +η 2)

€

H ∗ f + f ∗H = i2π dηe−iηp ψ∗(x −η 2)∂ψ (x +η 2)

∂t−∂ψ∗(x −η 2)

∂tψ (x +η 2)

⎡ ⎣ ⎢

⎤ ⎦ ⎥∫

Need Left and Right ‘Schrödinger’ equations. Try

and

Difference gives Liouville equation.

€

∂f∂t

+ f , H{ }MB = 0 ⇒ ∂f∂t

+ f , H{ }PB = 0

Sum gives ‘third’ equation.

???

€

(h =1)

Third equation is Quantum H-J.

[B. Hiley, Reconsideration of Foundations 2, 267-86, Växjö, 2003]

ψ ∗∂ψ

∂t−∂ψ ∗

∂tψ

⎡ ⎣ ⎢

⎤ ⎦ ⎥ = 1

R(x+hη 2)∂R(x+hη 2)

∂t− 1

R(x−hη 2)∂R(x−hη 2)

∂t⎡ ⎣ ⎢

⎤ ⎦ ⎥ ψ ∗ψ

+i ∂S(x+hη 2)

S(x+hη 2) +∂S(x−hη 2)S(x−hη 2)

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ψ ∗ψ

€

∂S∂t

+ H = 0

€

ψ =ReiS /hSimplify by writing

Classical limit

€

H ∗ f + f ∗H = −2∂S∂t

f +O(h2 )

€

→ 2∂S∂t

f + H , f{ }BB = 0

Classical H-J equation

Same for Operators?

We have two sides.ji

and

Two symplectic spinors.

Operator equivalent of Wave Function

€

ˆ ψ L

= ˆ A ε

€

ˆ ψ R =ε ˆ B Operator equivalent of conjugate WF

Two operator Schrödinger equations

€

i ∂ ˆ ψ L∂t

= ˆ H ˆ ψ L and − i ∂ ˆ ψ R∂t

= ˆ ψ R ˆ H

The Two Operator Equations.

[Brown and Hiley quant-ph/0005026]

€

ir ∂ ∂t

ˆ ψ L ⎛ ⎝ ⎜

⎞ ⎠ ⎟ˆ ψ R + ˆ ψ L ˆ ψ R

s ∂ ∂t

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡ ⎣ ⎢

⎤ ⎦ ⎥= ˆ

r H ˆ ψ L( ) ˆ ψ R − ˆ ψ L ˆ ψ R ˆ

s H ( )

€

With ˆ ρ = ˆ ψ L ˆ ψ R

€

i ∂ ˆ ρ ∂t

+ ˆ ρ , ˆ H [ ] −= 0

€

ir ∂ ∂t

ˆ ψ L ⎛ ⎝ ⎜

⎞ ⎠ ⎟ˆ ψ R − ˆ ψ L ˆ ψ R

s ∂ ∂t

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡ ⎣ ⎢

⎤ ⎦ ⎥= ˆ

r H ˆ ψ L( ) ˆ ψ R + ˆ ψ L ˆ ψ R ˆ

s H ( )

€

ˆ ψ L = ˆ R ei ˆ S ε and ˆ ψ R = εei ˆ S ˆ R

€

ˆ ρ ∂ˆ S

∂t+ 1

2ˆ ρ , ˆ H [ ] +

= 0

Sum

Quantum Liouville

Difference

New equation

The Operator Equations.

€

i ∂ ˆ ρ ∂t

+ ˆ ρ , ˆ H [ ] −= 0

€

ˆ ρ ∂ˆ S

∂t+ 1

2ˆ ρ , ˆ H [ ] +

= 0

€

2∂S∂t

f + f , H{ }BB = 0€

∂f∂t

+{ f , H}MB = 0

Wigner-Moyal Quantum

Where is the quantum potential?

Projection into a Representation.

[Brown and Hiley quant-ph/0005026]

€

i dP(a)dt

+ ˆ ρ , ˆ H [ ] − a= 0

€

P(a)∂S(a)∂t

+ 12 ˆ ρ , ˆ H [ ] + a

= 0

€

ˆ H = ˆ p 2

2m+ K ˆ x 2

2€

∂P∂t

+∇. P∇Sm

⎛ ⎝ ⎜ ⎞

⎠ ⎟= 0

€

∂Sx

∂t+ 1

2m∂Sx

∂x ⎛ ⎝ ⎜ ⎞

⎠ ⎟2

+ Kx 2

2− 1

2mRx

∂2Rx

∂x 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟= 0

Project into representation using

€

Pa = a a

Still no quantum potential

Choose

€

Px = x x

Conservation of probability

Quantum H-J equation.

Out pops thequantum potential

The Momentum Representation.

Trajectories from the streamlines of probability current.€

∂Sp

∂t+ p 2

2m+ K

2∂Sp

∂p ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

− K2Rp

∂2Rp

∂p 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟= 0

€

∂Pp

∂t+

∂j p

∂p= 0

€

j p = − p ∂( ˆ ρ V ( ˆ x ))∂x

p

Choose

€

Pp = p p

€

x = −∂Sp

∂p ⎛ ⎝ ⎜

⎞ ⎠ ⎟But now

Possibility of Bohm model in momentum space.

Returns the x p symmetry to Bohm model.

Shadow Phase Spaces.

[M. R. Brown & B. J. Hiley, quant-ph/0005026][B.Hiley, Quantum Theory:Reconsideration of Foundations, 2002, 141-162.]

x p

pr = Re(ψ*Pψ) xρ = Re(ψ*Xψ)

OR

Non-commutative quantum algebra implies no unique phase space.

Project on to Shadow Phase Spaces.

Quantum potential is an INTERNAL energy arising from

projection into a classical space-time.

General structure.

Shadow phase spaces

Non-commutativeAlgebraic structure.

Shadowmanifold

Shadowmanifold

Shadowmanifold

€

B = H + Qψ

€

B = H + Qφ

€

B = H + Qη

Guillemin & Sternberg, Symplectic Techniques in Physics 1990.Abramsky & Coecke quant-ph/0402130Baez, quant-ph/0404040

Covering space Sp(2n) ≈ Ham(2n)

A general *-algebra

Monoidal tensor *-category

The Philosophy.

Non-commutativeAlgebraic structure.

Implicate order.

Holomovement

Shadowmanifold

Shadowmanifold

Shadowmanifold

€

B = H + Qψ

€

B = H + Qφ

€

B = H + Qη

Possible explicate orders.[D. Bohm Wholeness and the Implicate Order (1980])