Quantization Of non-Abelian Gauge Theories Research Articles

Quantization of non-Abelian gauge theory in graphene

A time-dependent strain induces non-Abelian gauge field in a sheet of graphene. We discuss the effective field theory of this system of graphene subjected to a general form of time-dependent strain. We study the modifications to this effective field theory as a result of breaking Lorentz symmetry down to its sub-group, [Formula: see text], employing the VSR formalism. As the effective theory describing the graphene system has gauge symmetry; to quantize this theory, we add a suitable ghost and gauge fixing terms to the original action. The resultant action is observed to be invariant under a BRST symmetry. We have studied the BRST symmetry of the resultant theory, and explicitly constructed the BRST transformations.

An all-order proof of the equivalence between Gribovʼs no-pole and Zwanzigerʼs horizon conditions

The quantization of non-Abelian gauge theories is known to be plagued by Gribov copies. Typical examples are the copies related to zero modes of the Faddeev–Popov operator, which give rise to singularities in the ghost propagator. In this work we present an exact and compact expression for the ghost propagator as a function of external gauge fields, in SU(N) Yang–Mills theory in the Landau gauge. It is shown, to all orders, that the condition for the ghost propagator not to have a pole, the so-called Gribovʼs no-pole condition, can be implemented by demanding a non-vanishing expectation value for a functional of the gauge fields that turns out to be Zwanzigerʼs horizon function. The action allowing to implement this condition is the Gribov–Zwanziger action. This establishes in a precise way the equivalence between Gribovʼs no-pole condition and Zwanzigerʼs horizon condition.

When do measures on the space of connections support the triad operators of loop quantum gravity?

In this work we investigate the question under what conditions Hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non-Abelian analogues of the electric flux. We give the problem a precise mathematical formulation and start its investigation. For the technically simple case of U(1) as gauge group, we establish a number of “no-go theorems” asserting that for certain classes of measures, the flux operators can not be represented on the corresponding Hilbert spaces. The flux-observables we consider, play an important role in loop quantum gravity since they can be defined without recurse to a background geometry and they might also be of interest in the general context of quantization of non-Abelian gauge theories.

GRIBOV PROBLEM FOR GAUGE THEORIES: A PEDAGOGICAL INTRODUCTION

The functional-integral quantization of non-Abelian gauge theories is affected by the Gribov problem at non-perturbative level: the requirement of preserving the supplementary conditions under gauge transformations leads to a nonlinear differential equation, and the various solutions of such a nonlinear equation represent different gauge configurations known as Gribov copies. Their occurrence (lack of global cross-sections from the point of view of differential geometry) is called Gribov ambiguity, and is here presented within the framework of a global approach to quantum field theory. We first give a simple (standard) example for the SU(2) group and spherically symmetric potentials, then we discuss this phenomenon in general relativity, and recent developments, including lattice calculations.

Canonical quantization of nonabelian gauge theories in two dimensions

The canonical quantization of vector and chiral nonabelian gauge theories is performed, using the bosonization procedure. The gauge current commutators are computed and a close relation between the vector and chiral interacting gauge theories is shown to exist for a particular value of the regularization dependent parameter a.

Path-integral quantization of constrained Hamiltonian dynamical system: A pedagogical example

The concept of constrained Hamiltonian dynamical systems and their quantization, introduced by Dirac, constitutes one of the most fundamental contributions to the development of the canonical formalism in recent times. Dirac’s concepts were later incorporated into the path-integral formulation of quantum mechanics by Faddeev, which played an important role in understanding some subtle aspects in the quantization of non-Abelian gauge theory. An example is given from nonrelativistic quantum mechanics which illustrates the formalism through simple analogy.

Operator quantization of non-abelian gauge theories in a completely fixed axial gauge

A consistent quantization of chromodynamics in a completely fixed axial gauge is carried out by using the Dirac bracket quantization procedure. The main results are: The translation of Dirac brackets into equal-time commutators is possible, without ambiguities, because of the absence of ordering problems. All equal-time commutators are compatible with constraints and gauge conditions holding as strong operator relations. All equal-time commutators are compatible with chromoelectric, chromomagnetic, and fermionic fields vanishing at spatial infinity. The colored gauge potentials A 0, a , A 1, a , and A 2, a are seen to develop a physically significant, although pure gauge, behavior at x 3 = ± ∞, as required by the presence of a nontrivial topological content. Poincaré invariance is satisfied without introducing in the Hamiltonian “extra” quantum mechanical potentials. The determinant of the Faddeev-Popov matrix does not depend upon the field variables.

Operator quantization of non-abelian gauge theories in a completely fixed axial gauge: T. J. M. Simões and H. O. Girotti. Instituto de Física, Universidade Federal do Rio Grande do Sul, 90000 Porto Alegre, RS, Brasil

Operator quantization of non-abelian gauge theories in a completely fixed axial gauge: T. J. M. Simões and H. O. Girotti. Instituto de Física, Universidade Federal do Rio Grande do Sul, 90000 Porto Alegre, RS, Brasil

Quantization of non-Abelian gauge theories in the Dirac-bracket formalism

We develop in detail the Dirac-bracket formalism for a non-Abelian gauge field coupled to fermions. The theory is then quantized in the Coulomb gauge by abstracting the equal-time commutators from the corresponding Dirac brackets. Our results, obtained by a significantly different method, coincide with those of Schwinger. The role of book-keeper played by the classical theory in the elaboration of the quantum dynamics is clarified.

Canonical quantization of non-Abelian gauge theory

An essentially gauge invariant Hamiltonian formulation is given for a non-Abelian Yang-Mills system coupled to a fermion field. The Hamiltonian contains only unconstrained dynamical variables, which in the quantized version satisfy standard equal time commutation relations (for detail see Cronstrom C.: Ann. Phys. (N. Y.)134 (1981) 186).

On the light-cone quantization of non-abelian gauge theory

The implications of periodic boundary conditions in the light-cone quantization of non-abelian fields are studied. Formulation of the theory in the singularity-free case is presented. Some consequences of field singularities are discussed.

Quantization of non-abelian gauge theory. I. Gauge invariant Hamiltonian formulation

An essentially gauge invariant canonical Hamiltonian formulation is given for a non-Abelian Yang-Mills system coupled to a fermion field. The Hamiltonian contains only unconstrained dynamical variables, which in the quantum version satisfy canonical equal time commutation relations.

Quantizing gauge theories: Non-classical field configurations, broken symmetries and gauge copies

The quantization of non-Abelian gauge theories around arbitrary field configurations is considered. We find that the functional integral may be consistently expanded about any field which is sufficiently close to a constrained minimum. A unified treatment of gauge fixing, zero modes and constraints is presented in detail. All problems associated with gauge copies and similar double-counting phenomena are formally removed by further restrictions placed on the space of fluctuations; these restrictions have no effect on the perturbative expansion about a given field.

Quantization of non-Abelian gauge theories

It is shown that the fixing of the divergence of the potential in non-Abelian theories does not fix its gauge. The ambiguity in the definition of the potential leads to the fact that, when integrating over the fields in the functional integral, it is apparently enough for us to restrict ourselves to the potentials for which the Faddeev-Popov determinants are positive. This limitation on the integration range over the potentials cancels the infra-red singularity of perturbation theory and results in a linear increase of the charge interaction at large distances.

Canonical quantization of non-Abelian gauge theories in the axial gauge

We explore canonical quantization in the axial gauge, with special reference to the problems of (i) additional gauge fixing; and (ii) the infrared infinities which occur in eliminating the dependent variables. We show that the freedom inherent in (i) permits the removal of (ii), resulting in a Hamiltonian with no explicit infinities, generating the proper equations of motion. This does not mean, however, that all matrix elements of the Hamiltonian will be finite. In particular, the bare-vacuum-expectation value of $H$ still contains an infrared divergence.

A canonical and lorentz-covariant quantization of yang-mills theories

A Lorentz-covariant quantization of non-Abelian gauge theories is performed by using the canonical formalism. The Faddeev-Popov ghost is shown to be a consequence of two requirements: i) the vector states describing physical and longitudinal vector mesons should be orthogonal and ii) there should be no transition between physical and unphysical modes. The Slavnov-Taylor identities are derived. The metric of the vector space is studied.

Fermi quantization of non-Abelian gauge theory

The Yang-Mills field coupled to spinors is quantized, using a gauge-invariant Fermi addition to the Lagrangian, by introducing classical fictitious fields. Problems arising in the Fermi method are investigated. Use is made of the Schr\odinger representation, which may be seen as the differential equivalent of the path-integral method. In this representation the conditions selecting good states are very simple: Good states are gauge-invariant functionals of the spatial Yang-Mills potentials ${{b}_{\ensuremath{\alpha}}}^{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$. The divergence in the naive scalar product of states is removed by the Faddeev-Popov factorization, and also by the cutoff method. Both methods produce gauge- and Poincar\'e-invariant results. The Faddeev-Popov weight is expressed in a power series, which converges in the whole coupling-constant plane, by seeing this weight as a Fredholm determinant. The consistency problem of the Fermi method is resolved by allowing certain operators to be self-adjoint; it is shown that gauge transformations cannot be implemented unitarily outside the good state space. We investigate the question which operators raise charge multiplets of good states from the vacuum. It is shown that this is done by the Mandelstam-displaced operators $\ensuremath{\psi}$, $\overline{\ensuremath{\psi}}$, $B$, and $\ensuremath{\Pi}$, and also by the potentials ${b}_{\mathrm{g}}$ which represent gauge-equivalence classes of Yang-Mills potentials, if the ${b}_{\mathrm{g}}$ form a linear subspace of the configuration space. In a gauge-invariant theory involving gauge-variant operators, the energy-momentum operators, defined as displacement operators, cannot be gauge-invariant. However, the effect of these operators in $G$ is the same as that of certain gauge-invariant operators, and they can be used as the physical energy-momentum. Double commutators of these operators with the potentials are found to satisfy an identity which has a formal consequence for the mass of states raised from the vacuum by the potentials.

Lorentz Covariance and the Canonical Quantization of Non-Abelian Gauge Theories

One means of establishing the equivalence of the canonical and covariant quantization schemes for the massless, spin-1 non-Abelian gauge field is to demonstrate that these formalisms yield identical $S$-matrix elements. In this paper it is explicitly demonstrated that to second order in the coupling constant the canonical formalism yields a Lorentz-covariant $S$-matrix element for the Compton scattering of a nucleon and a quantum of this gauge field. The need to modify the canonical formalism in order to obtain a consistent description of higher-order processes is then briefly discussed.