The Algebraic Way: Professor Basil J. Hiley leads me to his noncommutative algebra paper revealing Nonlocality as the truth of reality

Dear Drew,

In my approach there is no need to refer to any wave function.  Each individual process is described by the non-commutative elements of the phase-space algebra itself.  Classical physics uses a commutative phase-space algebra.  Classical physics has the Poisson brackets as a vital part of the description.  What we have to understand is how that bracket emerges from the non-commutative structure.  Now the non-commutative algebra contains two types of bracket, a commutator or Lie bracket (or Lie product to give it its proper mathematical name) and an anti-commutator or Baker bracket ( known as the Jordan product).  The Lie bracket becomes the Poisson bracket as we go to the classical limit, while the Jordan product becomes the normal inner product.  In symbols (AB + BA)/2 —> AB.  The Jordan product is the most neglected product in the whole discussion of the foundations of quantum mechanics.  

This is not a 'cheat answer’.  It's what you have to understand if you really want to the relation between quantum and classical physics.  The clearest discussion of this issue is in the paper I have attached.

I hope you find it helpful.

Basil Hiley.

 

The Algebraic Way 

 

 It is the spin that led me to the idea of no ‘waves’.  One of the important things which keeps the ‘wave’ idea alive is the notion of ‘phase’, of interference, but in the case of non-relativistic spin there two ‘phases’— R(1)exp[iS(1)] and the second component R(2)exp[iS(2)].  So how do you think of two ‘phases’ in interference phenomena; in Dirac there are four 'phases’.  OK so you can think of 'four waves’.  Why ‘four’, but what has happened to the simple idea of interference?  The whole picture has suddenly become more complicated and confused.  Four waves to be ‘collapsed’?  The story seems bogus to me, but that is just my opinion.  The simple picture loses it appeal, like the epicycles of Ptolemy.  They work but nobody now believes in that story.

Of course you have to explore the algebraic way and see if it makes more sense.  I do and to me it fits more naturally into the mathematical scheme I am developing.  By the way it is not just me.  Look, for example, at Rudolf Haag’s book, “Local Quantum Physics, Fields, particles and Algebras.”  Chapter III onwards is the area I have been studying. But there is much more interesting ideas being developed i.e. Alan Connes “Noncommutative Geometry."

Basil Hiley.

 The assumption that Fψ(x, p) is a probability density then
opens up a debate as to the validity of the whole approach. However
we will show that Fψ(x, p) is not a probability distribution, but the
kernel of a density matrix which is not necessarily positive definite or
even real. Thus it is the interpretation of Fψ(x, p) being a probability distribution that is not valid, not the method in which it arises, so we can follow Feynman [17] and use equation (7) as a valid way to evaluate the quantum expectation values without worrying about the appearance of negative values of Fψ(x, p). We need to remember that we are dealing with a non-commutative structure and not simply averaging over classical coordinates.

Some Notes on Quantum Information in Spacetime

• Ignazio Licata(

  • ISEM, Palermo and
  • Bari Polytechnic

  )

Aug 6, 2020

4 pages

Published in:

• Entropy 22 (2020) 8, 864

• Published: Aug 6, 2020

In particular, considering the four-dimensional dynamics as the explication (in a Bohmian sense) of a De Sitter non-perturbative vacuum offers an improvement of Hartle–Hawking proposal in quantum cosmology and a solution to the informational paradox in the BH [27 – 29 ]. This line of reasoning is also promising for an event-based reading of Quantum Mechanics [30].
For a long time, holography and emergentism appeared as two styles of explanation irreconcilable with respect to the locality, but an emergency of time could offer new perspectives with a duality between imaginary time and real time, in a diachronic/synchronic complementarity [31–33].

 Hiley, B. J. and Callaghan, R. E., The Clifford Algebra Approach to
Quantum Mechanics B: The Dirac Particle and its relation to the Bohm
Approach, (2010) arXiv: 1011.4033.
[28] Hiley, B. J., Process, Distinction, Groupoids and Clifford Algebras: an
Alternative View of the Quantum Formalism, in New Structures for
Physics, ed Coecke, B., Lecture Notes in Physics, vol. 813, pp. 705-750,
Springer 2011.
[29] Hiley, B. J., and Callaghan, R. E., Clifford Algebras and the Dirac-
Bohm Quantum Hamilton-Jacobi Equation. Foundations of Physics,
42 (2012) 192-208.

 the orthogonal Clifford algebra used to describe the spin and relativistic
properties of quantum phenomena [29]. A detailed discussion of how one
chooses these ideals will be found in that paper. ...In a series of papers Hiley and Callaghan [26, 27, 29] have shown how the orthogonal Clifford algebras can be used to describe the spin and relativistic properties of quantum systems

 Moyal’s contribution was to show how the algebra generalised classical
statistics to a non-commutative statistics that emerges from a more general
non-commutative probability theory [15]. By recognising this generalisation,
we have shown that the Wigner function emerges from a representation of
the kernel of the density matrix. We argue that it is therefore incorrect to
regard this kernel as a probability density. Furthermore this fact explains
why the negative values of the Wigner function present no difficulty