arXiv subjects

James Moffat

Publications and source records attributed to James Moffat.

5 recordsLinked to original sources

arXiv2018

Quantisation of Paths in Space-Time and Non-Perturbative Quantum Gravity

In previous work we discussed the quantization of paths in spacetime. Building on these ideas we have developed a mathematically coherent theory addressing a number of open questions concerning Loop Quantum Gravity. Our approach develops a discrete spacetime and shows that macroscopic spacetime is a renormalization limiting form. Weaving together a number of our previous results we then prove that quantum states invariant under either an external group of local diffeomorphisms of spacetime or by contrast quantum states invariant under the internal action of a compact Lie group are common in a well-defined sense. These form the building blocks of invariant fields and Lagrangians. A form of supersymmetry and noncommutative spacetime naturally emerges, which predicts a massless graviton and its companion gravitino.

gr-qc↗

arXiv2018

Factorial Unitary Representations of the Translational Group and the Supersymmetric Graviton

Unitary Representations corresponding to local shifts in reference frames are a key topic for progress in Quantum Gravity. The fibre bundle construct defined in our previous work in which quantum fields become liftings of; or sections through; a fibre bundle with base space curved space-time, continues to be the context for this paper. We investigate in more depth the subgroup T of the Poincare group consisting of translations of space-time as a gauge group of automorphisms. We define to be a representation of T as such automorphisms of the local fibre algebra A(x) which we assume to be isomorphic to a von Neumann algebra with trivial centre acting on a separable Hilbert space. Provided this group representation is weakly measurable, then we have previously proved that it is also norm continuous, and is implemented by a norm, hence weakly and strongly, continuous unitary representation. We discuss the dependence of this key result on the strange properties of Stonean spaces and extend our exploitation of Mackey Theory to give a more straightforward group theoretic proof. From such unitary representations a minimal form of Supersymmetry naturally emerges, which predicts the existence of both the mass zero graviton, and its gravitino partner.

math-ph↗

arXiv2017

Unitary Representations of the Translational Group Acting as Local Diffeomorphisms of Space-Time

We develop a new mathematical approach to diffeomorphism invariant quantum states for the quantisation of general field theories such as general relativity and modified gravity. Treating quantum fields as fibre bundles, we discuss operators acting on the fibre algebra that defines a Hilbert space. The algebras of two types of operators are considered in detail, namely the observables as generic physical variables and the quantum operators suitable for describing symmetries and transformations. We then introduce generalised quantum states of these operators and examine their properties. By establishing a link between the commutativity and group cohomology of the translational group as a subgroup of the Poincare group, we show that this leads to the construction of quantum states invariant under the action of the translational group, as the local gauge group of diffeomorphisms, with unitary representations.

math-ph↗

arXiv2017

Factorial Representations of Compact Lie Groups, Wigner Sets and Locally Invariant Quantum Fields

The fibre bundle construct defined in our previous work continues to be the context for this paper; quantum fields composed of fibre algebras become liftings of; or sections through; a fibre bundle with base space a subset of curved space-time. We consider a compact Lie group such as SU(n) acting as a local gauge group of automorphisms of each fibre algebra A(x). Compact Lie groups, represented as gauge groups acting locally on quantum fields, are key elements in electroweak and strong force unification. In our recent joint work we have focused on the translational subgroup of the Poincare group as the generator of local diffeomorphism invariant quantum states. Here we extend those algebraic non-perturbative approaches to address the other half of unification by considering the existence of quantum states of the fibre algebra A(x) invariant to the action of compact non-abelian Lie groups. Wigner sets are complementary to little groups and we prove they have the finite intersection property. Exploiting this then allows us to show that invariant states are common in the sense that the weakly closed convex hull of every normal (density matrix) state contains such an invariant state. From these results and our related research emerges the existence of a locally invariant density matrix quantum state of the field.

math-ph↗

arXiv2016

Ergodic Theory and the Structure of Noncommutative Space-Time

We develop further our fibre bundle construct of non-commutative space-time on a Minkowski base space. We assume space-time is non-commutative due to the existence of additional non-commutative algebraic structure at each point x of space-time, forming a quantum operator 'fibre algebra' A(x). This structure then corresponds to the single fibre of a fibre bundle. A gauge group acts on each fibre algebra locally, while a 'section' through this bundle is then a quantum field of the form {A(x); x in M} with M the underlying space-time manifold. In addition, we assume a local algebra O(D) corresponding to the algebra of sections of such a principal fibre bundle with base space a finite and bounded subset of space-time, D. The algebraic operations of addition and multiplication are assumed defined fibrewise for this algebra of sections. We characterise 'ergodic' extremal quantum states of the fibre algebra invariant under the subgroup T of local translations of space-time of the Poincare group P in terms of a non-commutative extension of entropy applied to the subgroup T. We also characterise the existence of T- invariant states by generalizing to the non-commutative case Kakutani's work on wandering projections. This leads on to a classification of the structure of the local algebra O(D) by using a 'T-Twisted' equivalence relation, including a full analysis of the T-type III case. In particular we show that O(D) is T-type III if and only if the crossed product algebra O(D)xT is type III in the sense of Murray-von Neumann.

math-ph↗