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.

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

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.

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.

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

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.

Quantum Teleportation Underlying this diagram is a tensor *-category. [B. Coecke, quant-ph/ ] 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. 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.

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.

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

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]

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.

Bohm trajectories Screen Slits Incident particles x t Barrier x t

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

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, , (1949)] Same as Bohm

Transport equation for This is Bohm’s quantum Hamilton-Jacobi equation. 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.

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

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

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

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

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

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

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

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.

Shadow Phase Spaces. [M. R. Brown & B. J. Hiley, quant-ph/ ] [B.Hiley, Quantum Theory:Reconsideration of Foundations, 2002, ] 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-commutative Algebraic structure. Shadow manifold Shadow manifold Shadow manifold Guillemin & Sternberg, Symplectic Techniques in Physics Abramsky & Coecke quant-ph/ Baez, quant-ph/ Covering space Sp(2n) ≈ Ham(2n) A general *-algebra Monoidal tensor *-category

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])