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

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 Sx, SzRotation group The dynamics is in the operators Heisenberg’s equation of motion.

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

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

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

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

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

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

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

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

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

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

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

16. Transport equation for Transport equation for This is Bohm’s quantum Hamilton-Jacobi equation. 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. The * product is non-commutative [D. Fairlie and C. Manogue J. Phys A24, 3807-3815, (1991)] [C. Zachos, hep-th/0110114] Must have distinct left and right action. *-ganvalues. If we add and subtract Time dependent equations? Liouville Equation

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

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

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

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

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

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

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

26. Shadow Phase Spaces. Non-commutative quantum algebra implies no unique phase space. Project on to Shadow Phase Spaces. [M. R. Brown & B. J. Hiley, quant-ph/0005026] [B.Hiley, Quantum Theory:Reconsideration of Foundations, 2002, 141-162.] Quantum potential is an INTERNAL energy arising from projection into a classical space-time.

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

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