Quantum Yang-Mills Research Articles

Gauge theories in the light-cone gauge

A canonical formulation, using equal-time commutation rules for canonically conjugate operator-valued fields, is given for quantum electrodynamics and Yang-Mills theory in the light-cone gauge. A gauge-fixing term is used that avoids all operator constraints by providing a canonically conjugate momentum for every field component. The theory is embedded in a space in which the light-cone gauge condition and all of Maxwell's equations hold. Interaction picture fields and the photon and gluon propagators in the light-cone gauge are evaluated for two alternate representations of the longitudinal and timelike components of gauge fields. One representation makes use of the entire momentum space to represent these gauge field components as superpositions of ghost annihilation and creation operators. The other uses only ghost excitations with {ital k}{sub 3}{gt}0 for the longitudinal modes of {ital A}{sub {ital i}}, but restricts the gauge-fixing field to ghost excitations with {ital k}{sub 3}{lt}0. It is shown that the former mode leads to a formulation that corresponds to the principal-value (PV) prescription for the extra pole in this gauge, the latter to the Mandelstam-Leibbrandt (ML) prescription. Nevertheless the underlying theory for these two cases is identical. In particular, for QED both modes give identical time evolution, within a physicalmore » subspace in which constraints are implemented, as does QED in the Coulomb gauge. It is therefore concluded that canonical formulations of the light-cone gauge cannot be a basis for preferring the ML to the PV prescription for the extra pole at {ital k}{sub 0}={ital k}{sub 3} in the light-cone propagator.« less

Two-dimensional topological Yang-Mills theory

Two-dimensional euclidean (topological) quantum Yang-Mills theory on the compact manifold in the Lorentz gauge is analysed in the framework of the covariant path-integral approach. The Nicolai map for the partition function and for the Wilson loop observables is explicitly given.

Topological Yang-Mills field theory as the stochastic quantization of Chern-Simons gauge theory.

I present a procedure of stochastic quantization of the Chern-Simons gauge field theory on a three-manifold for two cases in which the dimensions of a four-dimensional instanton moduli space are either zero or nonzero. The equivalence of the topological quantum Yang-Mills field theory and the expression of the stochastic quantization of the three-dimensional Chern-Simons gauge field theory is proved. The results of the stochastic quantization may be closely related to the generalized form of the Morse theory.

Quantum Yang-Mills theory on Riemann surfaces and conformal field theory

It is shown that Yang-Mills theory on a smooth surface, when suitably quantized, is a topological quantum field theory. This topological gauge theory is intimately related to two-dimensional conformal field theory. It is conjectured that all conformal field theories may be obtained from Yang-Mills theory on smooth surfaces.

The Three Avtars of the Chern-Simons Term in 3 Space-Time Dimensions

I discuss in detail the three avtars of the Chern-Simons (C-S) term in 2 + 1 dimensions i.e. (i) C-S term as gauge field mass term (ii) C-S term as a purely kinematic term (iii) gauge theories with purely C-S action. In the first case we find that because of the C-S term one has massive gauge quanta and still the theory is gauge invariant. Such a C-S term can be generated either by spontaneous symmetry breaking or by radiative corrections. The dramatic effect of this term is that the vortices of the abelian (or nonabelian) Higgs model now have finite, quantized charge and angular momentum. In the second case the C-S term is not really independent but can be expressed in terms of the basic quanta of the 0(3) nonlinear σ-model or CP1 model. In either case one finds that due to this term the soliton of the model has fractional spin and statistic interpolating between fermions and bosons. The relevance of this in the context of high-Tc, superconductivity is discussed in some detail. Finally the third avtar has to do with some recent work of Witten where he has shown that the Hilbert spaces of the quantum Yang-Mills theory with pure C-S action can be interpreted as the spaces of the conformal blocks in 1 + 1 dimensional conformal theories.

Quantum field theory and the Jones polynomial

It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS 3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.

Comment on “Topologically massive gauge theories”

In a recent paper by R. Pisarski and S. Rao concerning topologically massive quantum Yang-Mills theory, the expression of the P-even part of the non-Abelian gauge field selfenergy at one-loop order is shown to obey a consistency condition, which is not fulfilled by the formula originally presented by S. Deser, R. Jackiw, and S. Templeton. In this comment, I present a recalculation which agrees with Pisarski and Rao.

Quantum Yang-Mills theories in the manifestly conformal-covariant formalism

Quantum Yang-Mills theories are treated in the manifestly conformal-covariant way using a generalized version of the Dirac's 6-cone formalism. A new (nonlinear and nonlocal) conformal transformation law is systematically obtained for the vector and ghost fields.

Indefinite-Metric Quantum Field Theory of General Relativity. XX: Superalgebra Unifying Quantum Gravity and Quantum Yang-Mills Field in the Suzuki Gauge

The manifestly covariant canonical formalism of quantum gravity is extended to a coupled system of quantum gravity and a non-abelian gauge field in the Suzuki gauge, where the Suzuki gauge is a special gauge fixing in which the part corresponding to the Cartan subalgebra behaves like an abelian gauge theory. An indecomposable superalgebra is found which unifies the spacetime symmetry and the Cartan subalgebra without contradicting the no-go theorem.

Solvable Analog of Quantum Chromodynamics

A two-dimensional gauge theory which can be coupled to massless fermions is considered. All the features expected in four-dimensional Yang-Mills theory and quantum chromodynamics are found: asymptotic freedom, confinement, and dynamical mass generation. These effects (including the nonperturbative ones) are computed explicitly and exactly.

Conformal invariance in quantum Yang-Mills theory

A new (nonlinear) realization of conformal transformations of nonabelian fields is found. These transformations (unlike the usual linear ones) leave the generating functional invariant and permit one to obtain a nontrivial quantum Yang-Mills theory. The gauge field dimension is shown to be canonical while the field tensor has an anomalous dimension. The invariant propagators are found. The possibility of the solution of the confinement problem is discussed.

An effective action describing long-range Yang-Mills theory

Using previously derived expressions for axial gauge propagators and vertex functions in the infrared limit of pure Yang-Mills theory, we show that at long-range quantum Yang-Mills theory is equivalent to a classical Yang-Mills theory in an anisotropic and inhomogeneous medium, the QCD vacuum. The properties of this classical theory are described and possible applications are outlined.

Quantum geometry and strings

In this paper we analyze the implications of Poliakov's definition of quantum geometry over Migdal's string model of QCD. We propose an ansatz to represent, in term of sums over random surfaces, the Schwinger functions of quantum Yang-Mills over loop space.

The principle of BRS symmetry: An alternative approach to Yang-Mills theories

Postulating BRS and anti-BRS symmetry of the quantum Yang-Mills lagrangian and of the measure of the functional integral, we prove gauge invariance and renormalizability and recover the classical theory from the quantum one. We also give a heuristic interpretation of the BRS and anti-BRS symmetry. The most general BRS and anti-BRS admissible gauges are linear in the gauge field, but may depend non-linearly on the ghost fields.

Quantum Non-Abelian Gauge Theory in Time-Dependent External Fields. II

We study quantum Yang-Mills fields under the influence of intensive time-dependent background vector potentials. The mode mixing formalism is constructed, which enables us to calculate the S-matrix elements. It is checked that the time-dependent external field cannot produce unphysical modes.

Operator Ordering and Feynman Rules in Gauge Theories

The question of operator ordering in the quantum Yang-Mills Hamiltonian is discussed for Coulomb gauge. Unconventional terms, first proposed by Schwinger, are found and their effect on the Coulomb gauge Feynman rules is analysed.

Point splitting in a curved space-time background: I. Gravitational contribution to the axial anomaly

A prescription is given for point splitting in a curved space-time background which is a natural generalization of that familiar in quantum electrodynamics and Yang-Mills theory. It is applied (to establish its validity) to the verification of the gravitational anomaly in the divergence of a fermion axial current. Notable features of the prescription are that it defines a point-split current that can be differentiated straightforwardly, and that it involves a natural way of averaging (four-dimensionally) over the directions of point splitting. The method can extend directly from the spin-1/2 fermion case treated to other cases, e.g., to spin-3/2 Rarita-Schwinger fermions.

Vacuum of the quantum Yang-Mills theory and magnetostatics

It is argued that since in asymptotically free Yang-Mills theories the quantum ground state is not controlled by perturbation theory, there is no a priori reason to believe that individual orbits corresponding to minima of the classical action dominate the Euclidean functional integral. To examine and classify the vacua of the quantum gauge theory, we propose an effective action in which the gauge field coupling constant g is replaced by the effective coupling g(t), t = ln[ F μν a) 2 μ 4 ] . The vacua of this model correspond to paramagnetism and perfect paramagnetism, for which the gauge field is F μν a = 0, and ferromagnetism, for which ( F μν a ) 2 = λ 2, i.e. spontaneous magnetization of the vacuum occurs. We show that there are no instanton solutions to the quantum effective action. The equations for a point classical source of color spin are solved, and we show that the field infrared energy becomes linearly divergent in the limit of spontaneous magnetization. This implies bag formation, and an electric Meissner effect confining the bag contents.

Quantization of the Yang-Mills Field

Recent efforts in quantum field theory have given rise to increased interest in nonlinear, gauge-type field theories. In this paper, we examine the Yang-Mills field, which is a theory of this type, which lies between electrodynamics and general relativity in complexity. The quantum Yang-Mills field is introduced to satisfy the requirement of invariance under isotopic phase transformations of the second kind. The theory is then put into its first-order form, so as to make it amenable to quantization by the methods of the Schwinger action principle. Quantization is then carried out for the two different gauge conditions; the first (which is the analog of the radiation gauge in electrodynamics) leads to a perturbation treatment of the constraints of the theory and the second to a rigorous solution of them. Two conditions for a consistent quantization are investigated. These are (1) the requirement that the Lagrange equations be identical to the Heisenberg equations of motion (the latter being evaluated using the field commutation relations) and (2) the requirement that the constraint equations be conserved in time (which leads to a condition on double commutators in the $q$-number theory). In the radiation gauge, these conditions are shown to be satisfied in lowest-order perturbation theory. In the second gauge used, they are rigorously satisfied. It is also shown for that gauge that the field Hamiltonian is positive definite when coupling to Fermion isotopic currents is ignored. No investigation of consistency conditions arising from Lorentz covariance requirements are made.