Download presentation

Presentation is loading. Please wait.

Published byFrida Daley Modified about 1 year ago

1
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]

2
Non-commutativity. We know 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

3
Eigenvalues. It is when we take eigenvalues that we get trouble. But the symmetries are carried by operators and not eigenvalues. X, P Heisenberg group S x, S z Rotation group The dynamics is in the operators Heisenberg’s equation of motion.

4
Symbolism. Introduce symbols and i j To represent And any operator can be written as C We know these satisfy Thus we can form or i i i j Complex number Matrix.

5
Expectation values. ij ij Call ij But this is just ij ii [Lou Kauffman Knots and Physics (2001)] [Bob Coecke Växjö Lecture 2005] Pure state

6
Mixed States and the GNS Construction. Can we write for mixed states? Yes. You double everything! and Can be generalized to many particle systems. [Bisch & Jones preprint 2004] Planar algebras.

7
Quantum Teleportation Underlying this diagram is a tensor *-category. [B. Coecke, quant-ph/0506132] Input state. Entangled state. Bell measurement. Output state. U

8
Elements of Left and Right ideals. 1 two-sided object splits into 2 one-sided objects. ij i j Left ideal Right ideal Algebraically the elements of the ideals are split by an IDEMPOTENT. Left ideal Right ideal i j j i j i This are just spinors.

9
Examples. Spinors are elements of a left ideal in Clifford algebra Symplectic spinors are elements of left ideal in Heisenberg algebra Algebraic equivalent of wave function. Everything is in the algebra. Rotation Group. k n ik nj ij Symplectic Group.

10
Eigenvalues again. Find in two ways. (1) Diagonalising operator.Find spectra. (2) Use eigenvector. i etc. and But we also have etc. and Anything new?Why complex?Why double? i j

11
Now for something completely different! You can do quantum mechanics with sharply defined x and p! in Schrödinger equation. Bohm model. Real part gives:- Quantum Hamilton-Jacobi this becomes Conservation of Energy. New quality of energy Why? Quantum potential energy [Bohm & Hiley, The Undivided Universe, 1993]

12
Probability. Imaginary part of the Schrödinger equation gives Conservation of Probability. Start with quantum probability end with quantum probability. Predictions identical to standard quantum mechanics.

13
Bohm trajectories Screen Slits Incident particles x t Barrier x t

14
Wigner-Moyal Approach. Find probability distribution f (X, P, t) so that expectation value Expectation value identical to quantum value Need relations [C. Zachos, hep-th/0110114] Problem: f (X, P, t) can be negative.

15
Bohm and Wigner-Moyal Different? NO! Mean Moyal momentum Use This is just Bohm’s Transport equation for the probability [J. E. Moyal, Proc. Camb. Soc, 45, 99-123, (1949)] Same as Bohm

16
Transport equation for This is Bohm’s quantum Hamilton-Jacobi equation. Transport equation for Which finally gives

17
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.

18
Stationary Pure States. [D. Fairlie and C. Manogue J. Phys A24, 3807-3815, (1991)] [C. Zachos, hep-th/0110114] The * product is non-commutative Must have distinct left and right action. If we add and subtract Time dependent equations? Liouville Equation *-ganvalues.

19
The ‘Third’ Equation [D. Fairlie and C. Manogue J. Phys A24, 3807-3815, (1991)] Need Left and Right ‘Schrödinger’ equations. Try and Difference gives Liouville equation. Sum gives ‘third’ equation. ???

20
Third equation is Quantum H-J. [B. Hiley, Reconsideration of Foundations 2, 267-86, Växjö, 2003] Simplify by writing Classical limit Classical H-J equation

21
Same for Operators? We have two sides. j i and Two symplectic spinors. Operator equivalent of Wave Function Operator equivalent of conjugate WF Two operator Schrödinger equations

22
The Two Operator Equations. [Brown and Hiley quant-ph/0005026] Sum Quantum Liouville Difference New equation

23
The Operator Equations. Wigner-MoyalQuantum Where is the quantum potential?

24
Projection into a Representation. [Brown and Hiley quant-ph/0005026] Project into representation using Still no quantum potential Choose Conservation of probability Quantum H-J equation. Out pops the quantum potential

25
The Momentum Representation. Trajectories from the streamlines of probability current. Choose But now Possibility of Bohm model in momentum space. Returns the x p symmetry to Bohm model.

26
Shadow Phase Spaces. [M. R. Brown & B. J. Hiley, quant-ph/0005026] [B.Hiley, Quantum Theory:Reconsideration of Foundations, 2002, 141-162.] 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.

27
General structure. Shadow phase spaces Non-commutative Algebraic structure. Shadow manifold Shadow manifold Shadow manifold Guillemin & Sternberg, Symplectic Techniques in Physics 1990. Abramsky & Coecke quant-ph/0402130 Baez, quant-ph/0404040 Covering space Sp(2n) ≈ Ham(2n) A general *-algebra Monoidal tensor *-category

28
The Philosophy. Non-commutative Algebraic structure. Implicate order. Holomovement Shadow manifold Shadow manifold Shadow manifold Possible explicate orders. [D. Bohm Wholeness and the Implicate Order (1980])

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google