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Alternative (Bohmish) quantum formalism using Clifford algebras (B.J.Hiley) 
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#1
Nov1112, 07:35 PM

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PF Gold
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http://arxiv.org/abs/1211.2107
Process, Distinction, Groupoids and Clifford Algebras: an Alternative View of the Quantum Formalism B. J. Hiley (Submitted on 9 Nov 2012) In this paper we start from a basic notion of process, which we structure into two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford algebras respectively. We show how the orthogonal Clifford algebra, which include the Schrödinger, Pauli and Dirac formalisms, describe the classical lightcone structure of spacetime, as well as providing a basis for the description of quantum phenomena. By constructing an orthogonal Clifford bundle with a Dirac connection, we make contact with quantum mechanics through the Bohm formalism which emerges quite naturally from the connection, showing that it is a structural feature of the mathematics. We then generalise the approach to include the symplectic Clifford algebra, which leads us to a noncommutative geometry with projections onto shadow manifolds. These shadow manifolds are none other than examples of the phase space constructed by Bohm. We also argue that this provides us with a mathematical structure that fits the implicateexplicate order proposed by Bohm. Comments: 55 pages. 10 figures 


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Nov1812, 04:07 PM

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#3
Nov1812, 05:26 PM

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PF Gold
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#4
Nov1912, 02:14 AM

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Alternative (Bohmish) quantum formalism using Clifford algebras (B.J.Hiley)



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Nov1912, 09:21 AM

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#6
Nov1912, 10:48 AM

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The author seems to establish a Clifford algebra on the multiplication of "processes" between "points". He then maps the Clifford algebra into a vector space which has a certain dimension and metric. Is this a unique mapping into a vector space? Or is it only one representation of the Clifford algebra?



#7
Nov1912, 11:12 AM

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Process and Time, by Hiley and Fernandes
http://www.bbk.ac.uk/tpru/BasilHiley/PS25P&T3.pdf http://fqxi.org/data/forumattachmen..._OnlinePDF.pdf Being, Becoming and the Undivided Universe: A Dialogue Between Relational Blockworld and the Implicate Order Concerning the Uniﬁcation of Relativity and Quantum Theory Michael Silberstein ·W.M. Stuckey · Timothy McDevitt Received: 15 July 2011 / Accepted: 9 April 2012 © Springer Science+Business Media, LLC 2 


#8
Nov2012, 03:15 AM

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#9
Dec1012, 02:09 PM

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Quote:
"As I dig deeper into the mathematical structure that contains the mathematical features that the Bohm uses, Bohm energy, Bohm momentum, quantum potential etc. are essential features, as you imply, of a noncommutative phase space; strictly a symplectic structure with a noncommutative multiplication (the Moyalstar product). This product combines into two brackets, the Moyal bracket, (a*bb*a)/hbar and the Baker bracket (a*b+b*a)/2. The beauty of these brackets is to order hbar, Moyal becomes the Poisson and Baker becomes the ordinary product ab. Time evolution requires two equations, simply because you have to distinguish between 'left' and 'right' translations. These two equations are in fact the two Bohm equations produced from the Schrödinger equation under polar decomposition in disguised form. There is no need to appeal to classical physics at any stage. Nevertheless these two equations reduce in the limit order hbar to the classical Liouville equation and the classical HamiltonJacobi equation respectively. This then shows that the quantum potential becomes negligible in the classical limit as we have maintained all along. There are not two worlds, quantum and classical, there is just one world. It was by using this algebraic structure that I was able to show that the Bohm model can be extended to the Pauli and Dirac particles, each with their own quantum potential. However here not only do we have a noncommutative symplectic symmetry, but also a noncommutative orthogonal symmetry, hence my interests in symplectic and orthogonal Clifford algebras. In this algebraic approach the wave function is not taken to be something fundamental, indeed there is no need to introduce the wave function at all!. What is fundamental are the elements of the algebra, call it what you will, the Moyal algebra or the von Neumann algebra, they are exactly the same thing. This is algebraic quantum mechanics that Haag discusses in his book "Local Quantum Physics, fields, particles and algebra". Physicists used to call it matrix mechanics, but then it was unclear how it all hung together. In the algebraic approach there is no collapse of the wave function, because you don't need the wave function. All the information contained in the wave function is encoded in the algebra itself, in its left and right ideals which are intrinsic to the algebra itself. Where are the particles in this approach? For that we need Eddington's "The Philosophy of Science", a brilliant but neglected work. Like a point in geometry, what is a particle? Is it a hazy general bricklike entity out of which the world is constructed, or is it a quasilocal, semiautonomous feature within the total structureprocess? Notice the change, not thingsininteraction, but structureprocess in which any invariant feature takes its form and properties from the structureprocess that gives it subsistence. If an algebra is used to describe this structureprocess, then what is the element that subsists? What is the element of existence? The idempotent E^2=E has eigenvalues 0 or 1: it exists or it doesn't exist. An entity exists in a structureprocess if it continuously turns itself into itself. The Boolean logic of the classical world turns existence into a permanent order: quantum logic turns existence into a partial order of noncommutative E_i! Particles can be 'created' or 'annihilated' depending on the total overall process. Here there is an energy threshold, keep the energy low and it is the properties of the entity that are revealed through noncommutativity, these properties becoming commutativity to order hbar. The Bohm model can be used to complement the standard approach below the creation/annihilation threshold. Raise this threshold and then the field theoretic properties of the underlying algebras become apparent. All this needs a different debate from the usual one that seems to go round and round in circles, seemingly resolving very little. Basil." Source: http://stardrive.org/stardrive/index...tydebate.html 


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