The Coulomb interaction and the inverse Faddeev-Popov operator in QCD
We give a proof of a local relation between the inverse Faddeev-Popov operator and the non-Abelian Coulomb interaction between color charges.
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arXiv subjects
Kurt Haller
Publications and source records attributed to Kurt Haller.
Using gauge-invariant variables in QCD
The Weyl-gauge ($A_0^a=0)$ QCD Hamiltonian is unitarily transformed to a representation in which it is expressed entirely in terms of gauge-invariant quark and gluon fields. In a subspace of gauge-invariant states we have constructed that implement the non-Abelian Gauss's law, this unitarily transformed Weyl-gauge Hamiltonian can be further transformed and, under appropriate circumstances, can be identified with the QCD Hamiltonian in the Coulomb gauge. To circumvent the problem that this Hamiltonian, which is expressed entirely in terms of gauge-invariant variables, must be used with nonnormalizable complicated states that implement the non-Abelian Gauss's law, we demonstrate an isomorphism that materially facilitates the application of this Hamiltonian to a variety of physical processes, including the evaluation of $S$-matrix elements. This isomorphism relates the gauge-invariant representation of the Hamiltonian and the required set of gauge-invariant states to a Hamiltonian of the same functional form but dependent on ordinary unconstrained Weyl-gauge fields operating within a space of `standard' perturbative states. The fact that the gauge-invariant chromoelectric field is not hermitian has important implications for the functional form of the Hamiltonian finally obtained. When this nonhermiticity is taken into account, the `extra' perturbative terms in Christ and Lee's Coulomb-gauge Hamiltonian are natural outgrowths of the formalism. When this nonhermiticity is neglected, the Hamiltonian used in the earlier work of Gribov and others results.
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Gauge equivalence in QCD: the Weyl and Coulomb gauges
The Weyl-gauge ($A_0^a=0)$ QCD Hamiltonian is unitarily transformed to a representation in which it is expressed entirely in terms of gauge-invariant quark and gluon fields. In a subspace of gauge-invariant states we have constructed that implement the non-Abelian Gauss's law, this unitarily transformed Weyl-gauge Hamiltonian can be further transformed and, under appropriate circumstances, can be identified with the QCD Hamiltonian in the Coulomb gauge. We demonstrate an isomorphism that materially facilitates the application of this Hamiltonian to a variety of physical processes, including the evaluation of $S$-matrix elements. This isomorphism relates the gauge-invariant representation of the Hamiltonian and the required set of gauge-invariant states to a Hamiltonian of the same functional form but dependent on ordinary unconstrained Weyl-gauge fields operating within a space of ``standard'' perturbative states. The fact that the gauge-invariant chromoelectric field is not hermitian has important implications for the functional form of the Hamiltonian finally obtained. When this nonhermiticity is taken into account, the ``extra'' vertices in Christ and Lee's Coulomb-gauge Hamiltonian are natural outgrowths of the formalism. When this nonhermiticity is neglected, the Hamiltonian used in the earlier work of Gribov and others results.
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What can we learn about Gribov copies from a formulation of QCD in terms of gauge-invariant fields?
We review the procedure by which we implemented the non-Abelian Gauss's law and constructed gauge-invariant fields for QCD in the temporal (Weyl) gauge. We point out that the operator-valued transformation that transforms gauge-dependent temporal-gauge fields into gauge-invariant ones has the formal structure of a gauge transformation. We express the ``standard'' Hamiltonian for temporal-gauge QCD entirely in terms of gauge-invariant fields, calculate the commutation rules for these fields, and compare them to earlier work on Coulomb-gauge QCD. We also discuss multiplicities of gauge-invariant temporal-gauge fields that belong to different topological sectors and that, in previous work, were shown to be based on the same underlying gauge-dependent temporal-gauge fields. We relate these multiplicities of gauge-invariant fields to Gribov copies. We argue that Gribov copies appear in the temporal gauge, but not when the theory is represented in terms of gauge-dependent fields and Gauss's law is left unimplemented. There are Gribov copies of the gauge-invariant gauge field, which can be constructed when Gauss's law is implemented.
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A Gauge-Invariant Color Charge in QCD
A gauge-invariant color-charge operator is defined and related to an integral of the gauge-invariant chromoelectric field over a closed surface. We discuss the case of a surface all of whose points are a macroscopic distance from a system of quarks and gluons which it entirely surrounds. When this system of quarks and gluons forms a hadron or an object composed of hadrons, such as a nucleus, it is argued that the gauge-invariant color charge enclosed within this surface must vanish and the system of hadrons in the interior of the surface must be a color singlet.
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Gauge-invariant quark and gluon fields in QCD: dynamics, topology, and the Gribov ambiguity
We review the implementation, in a temporal-gauge formulation of QCD, of the non-Abelian Gauss's law and the construction of gauge-invariant gauge and matter fields. We then express the QCD Hamiltonian in terms of these gauge-invariant operator-valued fields, and discuss the relation of this Hamiltonian and the gauge-invariant fields to the corresponding quantities in a Coulomb gauge formulation of QCD. We argue that a representation of QCD in terms of gauge-invariant quantities could be particularly useful for understanding low-energy phenomenology. We present the results of an investigation into the topological properties of the gauge-invariant fields, and show that there are Gribov copies of these gauge-invariant gauge fields, which are constructed in the temporal gauge, even though the conditions that give rise to Gribov copies do not obtain for the gauge-dependent temporal-gauge fields.
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Gauge-invariant fields in the temporal gauge, Coulomb-gauge fields, and the Gribov ambiguity
We examine the relation between Coulomb-gauge fields and the gauge-invariant fields constructed in the temporal gauge for two-color QCD by comparing a variety of properties, including their equal-time commutation rules and those of their conjugate chromoelectric fields. We also express the temporal-gauge Hamiltonian in terms of gauge-invariant fields and show that it can be interpreted as a sum of the Coulomb-gauge Hamiltonian and another part that is important for determining the equations of motion of temporal-gauge fields, but that can never affect the time evolution of ``physical'' state vectors. We also discuss multiplicities of gauge-invariant temporal-gauge fields that belong to different topological sectors and that, in previous work, were shown to be based on the same underlying gauge-dependent temporal-gauge fields. We argue that these multiplicities of gauge-invariant fields are manifestations of the Gribov ambiguity. We show that the differential equation that bases the multiplicities of gauge-invariant fields on their underlying gauge-dependent temporal-gauge fields has nonlinearities identical to those of the ``Gribov'' equation, which demonstrates the non-uniqueness of Coulomb-gauge fields. These multiplicities of gauge-invariant fields --- and, hence, Gribov copies --- appear in the temporal gauge, but only with the imposition of Gauss's law and the implementation of gauge invariance; they do not arise when the theory is represented in terms of gauge-dependent fields and Gauss's law is left unimplemented.
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Dynamics and topology of the gauge-invariant gauge field in two-color QCD
A nonlinear integral equation that is responsible for the implementation of the non-Abelian Gauss's law is applied to an investigation of the topological features of two-color QCD and to a discussion of their relation to QCD dynamics. We also draw a parallel between the nonuniqueness of the solutions of the equations that govern the gauge-invariant gauge field and Gribov copies.
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Topology of the gauge-invariant gauge field in two-color QCD
We investigate solutions to a nonlinear integral equation which has a central role in implementing the non-Abelian Gauss's Law and in constructing gauge-invariant quark and gluon fields. Here we concern ourselves with solutions to this same equation that are not operator-valued, but are functions of spatial variables and carry spatial and SU(2) indices. We obtain an expression for the gauge-invariant gauge field in two-color QCD, define an index that we will refer to as the ``winding number'' that characterizes it, and show that this winding number is invariant to a small gauge transformation of the gauge field on which our construction of the gauge-invariant gauge field is based. We discuss the role of this gauge field in determining the winding number of the gauge-invariant gauge field. We also show that when the winding number of the gauge field is an integer $\ell{\neq}0$, the gauge-invariant gauge field manifests winding numbers that are not integers, and are half-integers only when $\ell=0$.
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Gauge-Invariant QCD - an Approach to Long-Range Color-Confining Forces
We discuss the transformation of the QCD temporal-gauge Hamiltonian to a representation in which it can be expressed as a functional of gauge-invariant quark and gluon fields. We show how this objective can be realized by implementing the non-Abelian Gauss's law, and by using the mathematical apparatus developed for that purpose to also construct gauge-invariant quark and gluon fields. We demonstrate that, in the transformed QCD Hamiltonian, the interactions of pure-gauge components of the gauge field with color-current densities are replaced by nonlocal interactions connecting quark color-charge densities to each other and to `glue'-color. We discuss the nonperturbative evaluation of these nonlocal interactions, which are non-Abelian analogs of the Coulomb interaction in QED, and we explore their implications for QCD in the low-energy regime.
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Nonlocal color interactions in a gauge-invariant formulation of QCD
We construct a set of states that implement the non-Abelian Gauss's law for QCD. We also construct a set of gauge-invariant operator-valued quark and gluon fields by establishing an explicit unitary equivalence between the Gauss's law operator and the `pure glue' part of the Gauss's law operator. This unitary equivalence enables us to use the `pure glue' Gauss's law operator to represent the entire Gauss's law operator in a new representation. Since the quark field commutes with the `pure glue' Gauss's law operator, it is a gauge-invariant field in this new representation. We use the unitary equivalence of the new and the conventional representations to construct gauge-invariant quark and gluon fields in both representations, and to transform the QCD Hamiltonian in the temporal gauge so that it is expressed entirely in terms of gauge-invariant quantities. In that form, all interactions between quark fields mediated by `pure gauge' components of the gluon field have been transformed away, and replaced by a nonlocal interaction between gauge-invariant color-charge densities. This feature --- that, in gauge-invariant formulations, interactions mediated by pure gauge components of gauge fields have been replaced by nonlocal interactions --- is shared by many gauge theories. In QED, the resulting nonlocal interaction is the Coulomb interaction, which is the Abelian analog of the QCD interaction we have identified and are describing in this work. The leading term, in a multipole expansion, of this nonlocal QCD interaction vanishes for quarks in color-singlet configurations, suggesting a dynamical origin for color confinement; higher order terms of this multipole expansion suggest a QCD mechanism for color transparency. We also show how, in an SU(2) model, this nonlocal interaction can be evaluated nonperturbatively.
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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 {\em 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.
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Gauss's law, gauge invariance, and long-range forces in QCD
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 Gauss's 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.
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S-matrix elements for gauge theories with and without implemented constraints
We derive an expression for the relation between two scattering transition amplitudes which reflect the same dynamics, but which differ in the description of their initial and final state vectors. In one version, the incident and scattered states are elements of a perturbative Fock space, and solve the eigenvalue problem for the `free' part of the Hamiltonian --- the part that remains after the interactions between particle excitations have been `switched off'. Alternatively, the incident and scattered states may be coherent states that are transforms of these Fock states. In earlier work, we reported on the scattering amplitudes for QED, in which a unitary transformation relates perturbative and non-perturbative sets of incident and scattered states. In this work, we generalize this earlier result to the case of transformations that are not necessarily unitary and that may not have unique inverses. We discuss the implication of this relationship for Abelian and non-Abelian gauge theories in which the `transformed', non-perturbative states implement constraints, such as Gauss's law.
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Gauss's Law, Gauge-Invariant States, and Spin and Statistics In Abelian Chern-Simons Theories
We discuss topologically massive QED --- the Abelian gauge theory in which (2+1)-dimensional QED with a Chern-Simons term is minimally coupled to a spinor field. We quantize the theory in covariant gauges, and construct a class of unitary transformations that enable us to embed the theory in a Fock space of states that implement Gauss's law. We show that when electron (and positron) creation and annihilation operators represent gauge-invariant charged particles that are surrounded by the electric and magnetic fields required by Gauss's law, the unitarity of the theory is manifest, and charged particles interact with photons and with each other through nonlocal potentials. These potentials include a Hopf-like interaction, and a planar analog of the Coulomb interaction. The gauge-invariant charged particle excitations that implement Gauss's law obey the identical anticommutation rules as do the original gauge-dependent ones. Rotational phases, commonly identified as planar `spin', are arbitrary, however.
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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 `pure glue' 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.
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Implementing Gauss's law in Yang-Mills theory and QCD
We construct a transformation that transforms perturbative states into states that implement Gauss's 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.
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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 Poincaré generators to demonstrate the effect of Lorentz boosts on the particle states.
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