SYMPLECTIC REDUCTION FOR YANG–MILLS ON A CYLINDER

An account of the Lagrangian and Hamiltonian dynamics of the pure Yang–Mills system is presented. This framework is applied to the case of (1+1)-dimensional cylindrical spacetime. Hamiltonian dynamics on the space of connections over a circle is often identified with dynamics on the cotangent bundle of the gauge group by means of the holonomy. In support of this procedure we show that the symplectic structure for Hamiltonian dynamics for connections on a circle is identifiable with the natural symplectic structure on the cotangent bundle of the gauge group.

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Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back

SciPost Physics ◽

10.21468/scipostphys.10.6.125 ◽

2021 ◽

Vol 10 (6) ◽

Author(s):

Aldo Riello

Keyword(s):

Degrees Of Freedom ◽

Symplectic Structure ◽

Vector Fields ◽

Symplectic Reduction ◽

Yang Mills ◽

Hamiltonian Vector Fields ◽

Space Extension ◽

Region I ◽

Abelian Case ◽

Electric Flux

I develop a theory of symplectic reduction that applies to bounded regions in electromagnetism and Yang--Mills theories. In this theory gauge-covariant superselection sectors for the electric flux through the boundary of the region play a central role: within such sectors, there exists a natural, canonically defined, symplectic structure for the reduced Yang--Mills theory. This symplectic structure does not require the inclusion of any new degrees of freedom. In the non-Abelian case, it also supports a family of Hamiltonian vector fields, which I call ``flux rotations,'' generated by smeared, Poisson-non-commutative, electric fluxes. Since the action of flux rotations affects the total energy of the system, I argue that flux rotations fail to be dynamical symmetries of Yang--Mills theory restricted to a region. I also consider the possibility of defining a symplectic structure on the union of all superselection sectors. This in turn requires including additional boundary degrees of freedom aka ``edge modes.'' However, I argue that a commonly used phase space extension by edge modes is inherently ambiguous and gauge-breaking.

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EXTENSION OF THE POINCARÉ GROUP AND NON-ABELIAN TENSOR GAUGE FIELDS

International Journal of Modern Physics A ◽

10.1142/s0217751x10051050 ◽

2010 ◽

Vol 25 (31) ◽

pp. 5765-5785 ◽

Cited By ~ 12

Author(s):

GEORGE SAVVIDY

Keyword(s):

Gauge Group ◽

Gauge Transformation ◽

Space Time ◽

Gauge Fields ◽

Poincaré Group ◽

Matrix Representations ◽

Poincare Group ◽

Yang Mills ◽

The Matrix ◽

Transversal Plane

In the recently proposed generalization of the Yang–Mills theory, the group of gauge transformation gets essentially enlarged. This enlargement involves a mixture of the internal and space–time symmetries. The resulting group is an extension of the Poincaré group with infinitely many generators which carry internal and space–time indices. The matrix representations of the extended Poincaré generators are expressible in terms of Pauli–Lubanski vector in one case and in terms of its invariant derivative in another. In the later case the generators of the gauge group are transversal to the momentum and are projecting the non-Abelian tensor gauge fields into the transversal plane, keeping only their positively definite spacelike components.

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Symplectic structure of isospin particles in Yang-Mills fields

10.2172/10159194 ◽

1992 ◽

Author(s):

P. Oh

Keyword(s):

Symplectic Structure ◽

Yang Mills

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Classical Yang-Mills fields with non-compact symmetry and an arbitrary compact gauge group

Classical and Quantum Gravity ◽

10.1088/0264-9381/5/11/513 ◽

1988 ◽

Vol 5 (11) ◽

pp. 1513-1513

Author(s):

J-P Antoine ◽

M Jacques

Keyword(s):

Gauge Group ◽

Yang Mills ◽

Compact Gauge Group

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New soliton solutions of anti-self-dual Yang-Mills equations

Journal of High Energy Physics ◽

10.1007/jhep10(2020)101 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

Masashi Hamanaka ◽

Shan-Chi Huang

Keyword(s):

Gauge Group ◽

Darboux Transformation ◽

Domain Walls ◽

Soliton Solutions ◽

Real Space ◽

Explicit Calculation ◽

Yang Mills ◽

Action Density ◽

New Type ◽

Exact Soliton Solutions

Abstract We study exact soliton solutions of anti-self-dual Yang-Mills equations for G = GL(2) in four-dimensional spaces with the Euclidean, Minkowski and Ultrahyperbolic signatures and construct special kinds of one-soliton solutions whose action density TrFμνFμν can be real-valued. These solitons are shown to be new type of domain walls in four dimension by explicit calculation of the real-valued action density. Our results are successful applications of the Darboux transformation developed by Nimmo, Gilson and Ohta. More surprisingly, integration of these action densities over the four-dimensional spaces are suggested to be not infinity but zero. Furthermore, whether gauge group G = U(2) can be realized on our solition solutions or not is also discussed on each real space.

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GARCH in spinor field

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819500993 ◽

2019 ◽

Vol 16 (07) ◽

pp. 1950099

Author(s):

Richard Pincak ◽

Kabin Kanjamapornkul

Keyword(s):

Time Series ◽

Gauge Group ◽

Spinor Field ◽

Mirror Symmetry ◽

Euclidean Plane ◽

Financial Time Series ◽

Equivalent Form ◽

Yang Mills ◽

Noise Trader ◽

Financial Time

We extend generalized autoregressive conditional heteroscedastic (GARCH) errors in the Euclidean plane of the scalar field to the tensor field and to the spinor field [Formula: see text], the so-called spinor garch, S-GARCH. We use the model of S-GARCH to explain the stylized fact in financial time series, the so-called volatility cluster, by using hyperbolic coordinate with induced complex lag of delay time scale in mirror symmetry concept. As the result of this theory, we obtain an equivalent form of Yang–Mills equation for financial time series as the interaction between the behavior of traders, the so-called, fundamentalist, chatlist and noise trader, by using volatility in spinor field with invariant of the gauge group [Formula: see text], the so-called modeling of the financial market in icosahedral supersymmetry gauge group.

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