Kurt Haller

Quantum gauge equivalence in QED

We discuss gauge transformations in QED coupled to a charged spinor field, and examine whether we can gauge-transform the entire formulation of the theory from one gauge to another, so that not only the gauge and spinor fields, but also the forms of the operator-valued Hamiltonians are transformed. The discussion includes the covariant gauge, in which the gauge condition and Gausss law are not primary constraints on operator-valued quantities; it also includes the Coulomb gauge, and the spatial axial gauge, in which the constraints are imposed on operator-valued fields by applying the Dirac-Bergmann procedure. We show how to transform the covariant, Coulomb, and spatial axial gauges to what we call “common form,” in which all particle excitation modes have identical properties. We also show that, once that common form has been reached, QED in different gauges has a common time-evolution operator that defines time-translation for states that represent systems of electrons and photons. By combining gauge transformations with changes of representation from standard to common form, the entire apparatus of a gauge theory can be transformed from one gauge to another.

Gauss’s law and gauge-invariant operators and states in QCD

In this work, we prove a previously published conjecture that a prescription we gave for constructing states that implement Gauss`s law for {open_quotes}pure glue{close_quotes} QCD is correct. We also construct a unitary transformation that extends this prescription so that it produces additional states that implement Gauss`s law for QCD with quarks as well as gluons. Furthermore, we use the mathematical apparatus developed in the course of this work to construct gauge-invariant spinor (quark) and gauge (gluon) field operators. We adapt this SU(3) construction for the SU(2) Yang-Mills case, and we consider the dynamical implications of these developments. {copyright} {ital 1997} {ital The American Physical Society}

Integer quantization of the Chern-Simons coefficient in a broken phase

Abstract We consider a spontaneously broken nonabelian topologically massive gauge theory in a broken phase possessing a residual nonabelian symmetry. Recently there has been some question concerning the renormalization of the Chern-Simons coefficient in such a broken phase. We show that, in this broken vacuum, the renormalized ratio of the Chern-Simons coupling to the gauge coupling is shifted by 1 4π times an integer, preserving the usual integer quantization condition on the bare parameters.

QUARK CONFINEMENT AND COLOR TRANSPARENCY IN A GAUGE-INVARIANT FORMULATION OF QCD

We examine a nonlocal interaction that results from expressing the QCD Hamiltonian entirely in terms of gauge-invariant quark and gluon fields. The interaction couples one quark color-charge density to another, much as electric charge densities are coupled to each other by the Coulomb interaction in QED. In QCD, this nonlocal interaction also couples quark color-charge densities to gluonic color. We show how the leading part of the interaction between quark color-charge densities vanishes when the participating quarks are in a color singlet configuration, and that, for singlet configurations, the residual interaction weakens as the size of a packet of quarks shrinks. Because of this effect, color-singlet packets of quarks should experience final state interactions that increase in strength as these packets expand in size. For the case of an SU(2) model of QCD based on the ansatz that the gauge-invariant gauge field is a hedgehog configuration, we show how the infinite series that represents the nonlocal interaction between quark color-charge densities can be evaluated nonperturbatively, without expanding it term-by-term. We discuss the implications of this model for QCD with SU(3) color and a gauge-invariant gauge field determined by QCD dynamics.

Implementing Gauss's law in Yang-Mills theory and QCD☆

Abstract We construct a transformation that transforms perturbative states into states that implement Gausss law for ‘pure gluonic’ Yang-Mills theory and QCD. The fact that this transformation is not and cannot be unitary has special significance. Previous work has shown that only states that are unitarily equivalent to perturbative states necessarily give the same S-matrix elements as are obtained with Feynman rules.

Gauss's law, gauge invariance, and long-range forces in QCD

Abstract We use a unitary operator constructed in earlier work to transform the Hamiltonian for QCD in the temporal ( A 0 = 0) gauge into a representation in which the quark field is gauge-invariant, and its elementary excitations - quark and antiquark creation and annihilation operators - implement Gausss law. In that representation, the interactions between gauge-dependent parts of the gauge field and the spinor (quark) field have been transformed away and replaced by long-range non-local interactions of quark color charge densities. These long-range interactions connect SU(3) color charge densities through an infinite chain of gauge-invariant gauge fields either to other SU(3) color charge densities, or to a gluon “anchor”. We discuss possible implications of this formalism for low-energy processes, including confinement of quarks that are not in color singlet configurations.

Canonical quantization of the generalized axial gauge

Abstract The incompatibility of the constraint A3=0 with canonical commutation rules is discussed. A canonical formulation is given of QED and QCD in the axial gauge with n1=n2=0, n3=α and n0=β, where α and β are arbitrary real numbers. A Hilbert space is established for the perturbative theory, and a propagator is derived by obtaining an expression for the interaction picture gauge fields, and evaluating the vacuum expectation value of its time-ordered products in the perturbative vacuum. The propagator is expressed in terms of the parameter γ= α β and is shown to reproduce the light cone gauge propagator when γ=1, and the temporal gauge propagator when γ=0, accommodating various prescriptions for the spurious propagator pole, including the Mandelstam-Leibbrandt and principal value prescriptions. When γ→∞, the generalized axial gauge propagator leads to an expression for the propagator in the A3=0 gauge, though in that case the order in which the integration over k0 is performed, and the limit γ→∞ is taken, affects the resulting expression. Another Hilbert space is established, in which the constraints that include all interactions are implemented in a time independent fashion. It is pointed out that this Hilbert space, and the Hilbert space of the perturbative theory are unitarily equivalent in QED, but they cannot be unitarily equivalent in QCD. Implications of this fact for the nonperturbative states of QCD are discussed.

Canonical quantization of spontaneously broken topologically massive gauge theory

In this paper we investigate the canonical quantization of a non‐Abelian topologically massive Chern–Simons theory in which the gauge fields are minimally coupled to a multiplet of scalar fields in such a way that the gauge symmetry is spontaneously broken. Such a model produces the Chern–Simons–Higgs mechanism in which the gauge excitations acquire mass both from the Chern–Simons term and from the Higgs–Kibble effect. The symmetry breaking is chosen to be only partially broken, in such a way that in the broken vacuum there remains a residual non‐Abelian symmetry. We develop the canonical operator structure of this theory in the broken vacuum, with particular emphasis on the particle‐content of the fields involved in the Chern–Simons–Higgs mechanism. We construct the Fock space and express the dynamical generators in terms of creation and annihilation operator modes. The canonical apparatus is used to obtain the propagators for this theory, and we use the Poincare generators to demonstrate the effect of L...

Canonical operator theory of non-Abelian gauge fields

SummaryNon-Abelian gauge theory in a manifestly covariant gauge is formulated as a theory of canonical field operators and embedded in an indefinite metric space. A gauge-fixing field is included and every field component has a nonvanishing adjoint momentum with which it has canonical commutation (or anticommutation) relations. Faddeev-Popov fields are represented as scalar fermion fields with ghost particle excitations. Feynman rules are derived from the canonical formulation. A discussion is given of the relation between the existence of a subsidiary condition and the existence of « pure gauge » states that dynamically detach from observable states.RiassuntoSi formula teoria non abeliana di gauge in una gauge chiaramente covariante come una teoria di operatori di campo canonici inserita in uno spazio metrico indefinito. Si include un campo che fissa la gauge e ogni componente del campo ha un impulso aggiunto che non si annulla con il quale ha relazioni canoniche di commutazione (anti-commutazione). I campi di Faddeev-Popov sono rappresentati come campi scalari fermionici con eccitazioni di particelle fantasma. Si derivano le regole di Feynman dalla formulazione canonica. Si discute la relazione tra l’esistenza di una condizione sussidiaria e l’esistenza di stati di «gauge puro» che si distaccano dinamicamente da stati osservabili.РеэюмеФормулируется неабелева калибровочная теория в явно ковариантной калибровке, как теория канонических полевых операторов, и внедряется в пространство с индефинитной метрикой. Обраэуется калибровочное фиксированное поле и каждая полевая компонента имеет ненулевой сопряженный импульс, причем между компонентой поля и сопряженным импульсом сушествует каноническое соотнощение коммутации (или антикоммутации). Поля Фаддеева-Попова представляются как скалярные фермионные поля с воэбуждениями « частиц-духов ». Иэ канонической формулировки выводятся правила Фейнмана. Обсуждается свяэь между сушествованием дополнительного условия и сушествованием « чисто калибровочных » состояний, которые динамически раэщединены от наблюдаемых состояний.

Operator gauge transformations in quantum electrodynamics

Abstract An operator transformation is formulated which transforms electromagnetic potentials and spinor fields from one gauge to another. Commutation rules are transformed under this gauge transformation and, in particular, change from the commutation rules of the Lorentz gauge to those of the Coulomb gauge when the appropriate gauge transformation is made. The equations of motion are also correspondingly transformed. The transformation is used to generate commutation rules and equations of motion for the null-plane gauge.