A phase cell approach to Yang–Mills theory. II. Analysis of a mode

Federbush, Paul; Williamson, C.

doi:10.1063/1.527495

Cited by 11 publications

(12 citation statements)

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“…This approach has yielded major successes, including, a proof of the ultraviolet stability of the partition function [52] in three spacetime dimensions. Similar to Ba laban, the second programme [57][58][59][60][61][62], due to Federbush, yields a continuum limit of the lattice gauge theory as an inductive limit of block-spin renormalisations. The final approach [63] studies pure Yang-Mills in the continuous case, but in the presence of an infrared cutoff.…”

Section: Introductionmentioning

confidence: 99%

Quantum Yang-Mills theory: An overview of a program

Milsted

Osborne

2018

Phys. Rev. D

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We present an overview of a programme to understand the low-energy physics of quantum Yang-Mills theory from a quantum-information perspective. Our setting is that of the hamiltonian formulation of pure Yang-Mills theory in the temporal gauge on the lattice. Firstly, inspired by recent constructions for Z/2Z lattice gauge theory, in particular, Kitaev's toric code, we describe the gauge-invariant sector of hilbert space by introducing a primitive quantum gate: the quantum parallel transporter. We then develop a nonabelian generalisation of laplace interpolation to present an ansatz for the ground state of pure Yang-Mills theory which interpolates between the weak-and strong-coupling RG fixed points. The resulting state acquires the structure of a tensor network, namely, a multiscale entanglement renormalisation ansatz, and allows for the efficient computation of local observables and Wilson loops. Various refinements of the tensor network are discussed leading to several generalisations. Finally, the continuum limit of our ansatz as the lattice regulator is removed is then described. This paper is intended as an abstract for an ongoing programme: there are still many open problems.

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Section: Introductionmentioning

confidence: 99%

Quantum Yang-Mills theory: An overview of a program

Milsted

Osborne

2018

Phys. Rev. D

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“…We would like the (isolated) modes, in the small field region, to have their infinite volume form, as in [2,3]:…”

Section: Preliminaries Notation and Definitionsmentioning

confidence: 99%

“…We now turn to a block mode m at level r. In the figure, as in the last case, e(m) in gBM is the home edge of m, v the pinning vertex v = v(m). Ife' and e" in the figure are both in ga, and parallel to e(m) as indicated, then e' and e" are the two home We now have the A field assignments at the r level for both kinds of pure modes, zero except for the values in (2:3) and (2:4): [2,3] present a continuum A field of this mode consistent with these assignments, as well as assignments to each edge: (The continuum field does not minimize the "quadratic" terms in So, the "free action", but rather this "free action" in the "approximation" that all the gr's are equal: This is technically easier0…”

Section: The Isolated Pure Modementioning

confidence: 99%

“…The "small field modes" of [2,3] are the starting point for defining our two types of mode excitations. These make their contribution to the total field in a manner chosen to meet the gauge invariant coupling principle.…”

Section: O Introductionmentioning

confidence: 99%

“…The isolated fields of modes will be easily extracted from the analysis in [2,3], yielding all necessary estimates: The brunt of our efforts in this paper will be to define and estimate the isolated fields of chunks, an effort continuing into the next paper in this series: The touchiest task is the definition of a gauge for the isolated chunk field: Construction of a gauge involves us with topological obstructions, in re3 (G) [we have assumed rq (G) = 0, and rc z (G) = 0 for free ], obstructions that would not appear in three dimensions: Our choice of a gauge is one that has possible isolated point singilarities:…”

Section: O Introductionmentioning

confidence: 99%

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A phase cell approach to Yang-Mills theory

Federbush

1988

Commun.Math. Phys.

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Variables are chosen to describe the continuum Yang-Mills fields, a discrete set of group valued variables. These are group elements associated to the sequence of lattice field theory configurations realizing the continuum field: The field is "laid down" inductively: At each inductive step one of three types of "field excitations" makes its contribution to the total field. These are either "pure modes", "averaging correction modes", or "chunks". The pure modes are small field excitations, as studied in previous papers in this series [2, 3]. The averaging correction modes are small excitations added to make sure the block spin transformation is satisfied at each edge: The chunks, encompassing most of our difficulties, are large field excitations, Topological obstructions in n3(G) must be dealt with in defining a gauge choice for each chunk. The laying down process is complex, but fiendishly clever, ensuring a principle of "gauge invariant coupling": Each group valued variable is either the "amplitude" of a pure mode or an "internal variable" in a chunk, The amplitude of an averaging correction mode is a dependent variable, a function of the (independent) variables used to describe the field. The (independent) variables herein defined are those whose mutual interaction will later be inductively decoupled in defining the phase celt cluster expansion (of course treating the variables of each chunk as a unit).

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Yang–Mills for Probabilists

Chatterjee

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Dedicated to friend and teacher Raghu Varadhan on the occasion of his 75 th birthday.Abstract. The rigorous construction of quantum Yang-Mills theories, especially in dimension four, is one of the central open problems of mathematical physics. Construction of Euclidean Yang-Mills theories is the first step towards this goal. This article presents a formulation of some of the core aspects this problem as problems in probability theory. The presentation begins with an introduction to the basic setup of Euclidean Yang-Mills theories and lattice gauge theories. This is followed by a discussion of what is meant by a continuum limit of lattice gauge theories from the point of view of theoretical physicists. Some of the main issues are then posed as problems in probability. The article ends with a brief review of the mathematical literature.

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A phase cell approach to Yang–Mills theory. II. Analysis of a mode

Abstract: Properties of a mode announced in a previous paper are proved. This involves some complicated calculations in linear algebra, and observation of the structure of a function of several complex variables.

Cited by 11 publications

References 2 publications

Quantum Yang-Mills theory: An overview of a program

Quantum Yang-Mills theory: An overview of a program

A phase cell approach to Yang-Mills theory

Yang–Mills for Probabilists

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