Yang-Mills Mechanics Research Articles

Accurate Energy Spectrum for the Quantum Yang-Mills Mechanics with Nonlinear Color Oscillations

Yang-Mills theory as the foundation for quantum chromodynamics is a non-Abelian gauge theory with self-interactions between vector particles. Here, we study the Yang-Mills Hamiltonian with nonlinear color oscillations in the absence of external sources corresponding to the group $SU(2)$. In the quantum domain, we diagonalize the Hamiltonian using the optimized trigonometric basis expansion method and find accurate energy eigenvalues and eigenfunctions for one and two degrees of freedom. We also compare our results with the semiclassical solutions.

Light-cone Yang-Mills mechanics: SU(2) vs. SU(3)

We investigate the light-cone SU(n) Yang-Mills mechanics formulated as the leading order of the long-wavelength approximation to the light-front SU(n) Yang-Mills theory. In the framework of the Dirac formalism for degenerate Hamiltonian systems, for models with the structure groups SU(2) and SU(3), we determine the complete set of constraints and classify them. We show that the light-cone mechanics has an extended invariance: in addition to the local SU(n) gauge rotations, there is a new local two-parameter Abelian transformation, not related to the isotopic group, that leaves the Lagrangian system unchanged. This extended invariance has one profound consequence. It turns out that the light-cone SU(2) Yang-Mills mechanics, in contrast to the well-known instant-time SU(2) Yang-Mills mechanics, represents a classically integrable system. For calculations, we use the technique of Grobner bases in the theory of polynomial ideals.

On the homogeneous Gröbner basis for tensors

Algorithmic methods of commutative algebra based on the involutive and Grobner bases technique are efficient means for completion of equations governing dynamical systems to involution. At the same time, when working with high-dimensional tensor quantities, direct use of standard functions for calculating Grobner bases, which are built in computer algebra systems Maple and Mathematica, requires much memory. However, being multilinear forms, tensors admit special grading that makes it possible to classify polynomials in terms of their degree of homogeneity. With regard to this feature, we propose to use a special homogeneous Grobner basis, which allows us to avoid difficulties associated with large amount of computation. Such a basis is constructed step by step, as the degree of the polynomial grows. As an example, an algorithm for constructing the homogeneous basis in a finite-dimensional Hamiltonian system with many polynomial constraints (the so-called Yang-Mills mechanics) is presented.

A note on BRST quantization of SU(2) Yang-Mills mechanics

The quantization of SU(2) Yang-Mills theory reduced to 0+1 space-time dimensions is performed in the BRST framework. We show that in the unitary gauge A0=0 the BRST procedure has difficulties which can be solved by introduction of additional singlet ghost variables. In the Lorenz gauge Ȧ0=0 one has additional unphysical degrees of freedom, but the BRST quantization is free of the problems of the unitary gauge.

Euler-Calogero-Moser model fromSU(2)Yang-Mills theory

The relation between SU(2) Yang-Mills mechanics, originated from the 4-dimensional SU(2) Yang-Mills theory under the supposition of spatial homogeneity of the gauge fields, and the Euler-Calogero-Moser model is discussed in the framework of Hamiltonian reduction. Two kinds of reductions of the degrees of freedom are considered: due to the gauge invariance and due to the discrete symmetry. In the former case, it is shown that after elimination of the gauge degrees of freedom from the SU(2) Yang-Mills mechanics the resulting unconstrained system represents the ID_3 Euler-Calogero-Moser model with an external fourth-order potential. Whereas in the latter, the IA_6 Euler-Calogero-Moser model embedded in an external potential is derived whose projection onto the invariant submanifold through the discrete symmetry coincides again with the SU(2) Yang-Mills mechanics. Based on this connection, the equations of motion of the SU(2) Yang-Mills mechanics in the limit of the zero coupling constant are presented in the Lax form.

Instantons and Gribov copies in the maximally Abelian gauge

We calculate the Faddeev-Popov operator corresponding to the maximally Abelian gauge for gauge group SU(N). Specializing to SU(2) we look for explicit zero modes of this operator. Within an illuminating toy model (Yang-Mills mechanics) the problem can be completely solved and understood. In the field theory case we are able to find an analytic expression for a normalizable zero mode in the background of a single `t Hooft instanton. Accordingly, such an instanton corresponds to a horizon configuration in the maximally Abelian gauge. Possible physical implications are discussed.

Geometry of classical mechanics with non-Abelian gauge symmetry

The geometry of the gauge orbit bundle is considered in SO(3) Yang-Mills mechanics. It is shown that it can serve as a finite-dimensional model of the geometry of non-Abelian field theory with respect to properties such as the absence of a global gauge, the Gribov ambiguities, the convexity of the region within the corresponding horizon, and stratification. The connection and Lagrangian on the orbit space are obtained.

Hopf bifurcation in Yang–Mills mechanics

The periodic non-Abelian space-independent solutions to Yang–Mills SU(2) equations are studied. Hopf bifurcation is shown to appear. New analytic solutions generating a simple Abelian source are found and their stability is discussed. The source can be produced by a fermion field; a self-consistent solution of Yang–Mills–Dirac equations is presented.