Path-integral representations for a scalar particle propagator in non-Abelian external backgrounds are derived. To this aim, we generalize the procedure proposed by Gitman and Schvartsman 1993 of path-integral construction to any representation of SU(N) given in terms of antisymmetric generators. And for arbitrary representations of SU(N), we present an alternative construction by means of fermionic coherent states. From the path-integral representations we derive pseudoclassical actions for a scalar particle placed in non-Abelian backgrounds. These actions are classically analyzed and then quantized to prove their consistency.
Starting from the Dirac equation in external electromagnetic and torsion fields we derive a path integral representation for the corresponding propagator. An effective action, which appears in the representation, is interpreted as a pseudoclassical action for a spinning particle. It is just a generalization of Berezin-Marinov action to the background under consideration. Pseudoclassical equations of motion in the nonrelativistic limit reproduce exactly the classical limit of the Pauli quantum mechanics in the same case. Quantization of the action appears to be nontrivial due to an ordering pro…
The propagator of a spinning particle in external Abelian field and in arbitrary dimensions is presented by means of a path integral. The problem has different solutions in even and odd dimensions. In even dimensions the representation is just a generalization of one in four dimensions (it has been known before). In this case a gauge invariant part of the effective action in the path integral has a form of the standard (Berezin-Marinov) pseudoclassical action. In odd dimensions the solution is presented for the first time and, in particular, it turns out that the gauge invariant part of the ef…
We study massive 1/2-spin particles in various external backgrounds keeping in mind applications to neutrino physics. We are mainly interested in massive Majorana (Weyl) fields. However, massive neutral Dirac particles are also considered. We formulate classical Lagrangian theory of the massive Weyl field in terms of Grassmann-odd two-component spinors. Then we construct the Hamiltonian formulation of such a theory, which turns out to be a theory with second-class constraints. Using this formulation we canonically quantize the massive free Weyl field. We derive propagators of the Weyl field an…
We present a path integral representation for massless spin one-half particles. It is shown that this gives us a super-symmetric, P-and T-non-invariant pseudoclassical model for relativistic massless spinning particles. Dirac quantization of this model is considered.
A generalization of the Lagrangian introduced earlier in [2011 {\it J. Phys. G} ${\bf 37}$ 105001] for a classical color spinning particle interacting with background non-Abelian gauge and fermion fields for purpose of considering a change in time of the spin particle degree of freedom, is suggested. In the case under consideration the spin degree of freedom is described by a commuting $c$-number Dirac spinor $ψ_α$. A mapping of this spinor into new variables: anticommuting pseudovector $ξ_μ$ and pseudoscalar $ξ_5$ commonly used in a description of the spin degree of freedom of a massive spin-…
The problem on mapping between two Lagrangian descriptions (using a commuting $c$-number spinor $ψ_α$ or anticommuting pseudovector $ξ_μ$ and pseudoscalar $ξ_5$ variables) of the spin degrees of freedom of a color spinning massive particle interacting with background non-Abelian gauge field, is considered. A general analysis of the mapping between a pair of {\it Majorana} spinors $(ψ_α,θ_α)$ ($θ_α$ is some auxiliary anticommuting spinor) and a real anticommuting tensor aggregate $(S,V_μ,\!\,^{\ast}T_{μν},A_μ,P)$, is presented. A complete system of bilinear relations between the tensor quantiti…
A generalization of the pseudoclassical action of a spinning particle in the presence of an anomalous magnetic moment is given. The leading considerations, to write the action, are gotten from the path integral representation for the causal Green's function of the generalized (by Pauli) Dirac equation for the particle with anomalous magnetic momentum in an external electromagnetic field. The action can be written in reparametrization and supergauge invariant form. Both operator (Dirac) and path-integral (BFV) quantization are discussed. The first one leads to the Dirac-Pauli equation, whereas …
It is known that actions of field theories on a noncommutative space-time can be written as some modified (we call them $θ$-modified) classical actions already on the commutative space-time (introducing a star product). Then the quantization of such modified actions reproduces both space-time noncommutativity and usual quantum mechanical features of the corresponding field theory. The $θ$-modification for arbitrary finite-dimensional nonrelativistic system was proposed by Deriglazov (2003). In the present article, we discuss the problem of constructing $θ$-modified actions for relativistic QM.…
We present a convenient method for deriving the transformation of the dilaton under T-duality in the path-integral approach. Subtleties arising in performing the integral over the gauge fields are carefully analysed using Pauli-Villars regularization, thereby clarifying existing ambiguities in the literature. The formalism can not only be applied to the abelian case, but, and this for the first time, to the non-abelian case as well. Furthermore, by choosing a particular gauge, we directly obtain the target-space covariant expression for the dual geometry in the abelian case. Finally it is show…
We make use of the properties of product integrals to obtain a surface product integral representation for the Wilson loop operator. The result can be interpreted as the non-abelian version of Stokes' theorem.
The path integral for 3+1 abelian gauge theory is rewritten in terms of a real antisymmetric field allowing a dual action that couples the electric and magnetic currents to the photon and each other in a gauge invariant manner. Standard perturbative abelian quantum electrodynamics reemerges when the monopole current vanishes. For certain simple relationships between the monopole current and the electric current, the altered photon propagator can exhibit abelian charge confinement or develop mass, modeling effects believed to be present in non-abelian theories.
We present a derivation of abelian and non-abelian bosonization in a path integral setting by expressing the generating functional for current-current correlation functions as a product of a $G/G$-coset model, which is dynamically trivial, and a bosonic part which contains the dynamics. A BRST symmetry can be identified which leads to smooth bosonization in both the abelian and non-abelian cases.
We make use of product integrals to provide an unambiguous mathematical representation of Wilson line and Wilson loop operators. Then, drawing upon various properties of product integrals, we discuss such properties of these operators as approximating them with partial sums, their convergence, and their behavior under gauge transformations. We also obtain a surface product integral representation for the Wilson loop operator. The result can be interpreted as the non-abelian version of Stokes theorem.
It is argued that the massive non-Abelian gauge field theory without involving Higgs bosons may be well established on the basis of gauge-invariance principle because the dynamics of the field is gauge-invariant in the physical space defined by the Lorentz constraint condition. The quantization of the field can readily be performed in the Hamiltonian path-integral formalism and leads to a quantum theory which shows good renormalizability and unitarity.
We extend gauge symmetry of Abelian gauge field to incorporate quantum gauge degrees of freedom. We twice apply the Harada--Tsutsui gauge recovery procedure to gauge-fixed theories. First, starting from the Faddeev--Popov path integral in the Landau gauge, we recover the gauge symmetry by introducing an additional field as an extended gauge degree of freedom. Fixing the extended gauge symmetry by the usual Faddeev--Popov procedure, we obtain the theory of Type I gaugeon formalism. Next, applying the same procedure to the resulting gauge-fixed theory, we obtain a theory equivalent to the extend…
We present a recursive method to calculate the $α'$-expansion of disk integrals arising in tree-level scattering of open strings which resembles the approach of Berends and Giele to gluon amplitudes. Following an earlier interpretation of disk integrals as doubly partial amplitudes of an effective theory of scalars dubbed as $Z$-theory, we pinpoint the equation of motion of $Z$-theory from the Berends-Giele recursion for its tree amplitudes. A computer implementation of this method including explicit results for the recursion up to order $α'^7$ is made available on the website http://repo.or.c…
The cvariant path integral quantization of the theory of the scalar and spinor particles interacting through the abelian and non-Abelian Chern-Simons gauge fields is carried out and is shown to be mathematically ill defined due to the absence of the transverse components of these gauge fields. This is remedied by the introduction of the Maxwell or the Maxwell-type (in the non-Abelian case)term which makes the theory superrenormalizable and guarantees its gauge-invariant regularization and renormalization. The generating functionals are constructed and shown to be formally the same as those of …
The existence of several nilpotent Noether charges in the decoupled formulation of two-dimensional gauge theories does not imply that all of these are required to annihilate the physical states. We elucidate this matter in the context of simple quantum mechanical and field theoretical models, where the structure of the Hilbert space is known. We provide a systematic procedure for deciding which of the BRST conditions is to be imposed on the physical states in order to ensure the equivalence of the decoupled formulation to the original, coupled one.
We generalize the standard product integral formalism to incorporate Grassmann valued matrices and show that the resulting supersymmetric product integrals provide a natural framework for describing supersymmetric Wilson Lines and Wilson Loops. We use this formalism to establish the supersymmetric version of the non-Abelian Stokes theorem.
The non-Abelian tensor gauge fields take value in extended Poincaré algebra. In order to define the invariant Lagrangian we introduce a vector variable in two alternative ways: through the transversal representation of the extended Poincaré algebra and through the path integral over the auxiliary vector field with the U(1) Abelian action. We demonstrate that path integral formulation allows to fix the unitary gauge, derive scattering amplitudes in spinor representation and to define the gauge invariant Lagrangian in a curved space-time.
The functional integral computation of the various topological invariants, which are associated with the Chern-Simons field theory, is considered. The standard perturbative setting in quantum field theory is rewieved and new developments in the path-integral approach, based on the Deligne-Beilinson cohomology, are described in the case of the abelian U(1) Chern-Simons field theory formulated in S^1 x S^2.
This paper is a revised version of our recent publication Faber et al., Phys. Rev. D62 (2000) 025019, hep-th/9907048. The main revision concerns the expansion into group characters that we have used for the evaluation of path integrals over gauge degrees of freedom. In the present paper we apply the expansion recommended by Diakonov and Petrov in hep-lat/0008004. Our former expansion was approximate and in the region of the particular values of parameters violated the completeness condition by 1.4%. We show that by using the expansion into characters recommended by Diakonov and Petrov in hep-l…
We discuss the derivation of the path integral representation over gauge degrees of freedom for Wilson loops in SU(N) gauge theory and 4-dimensional Euclidean space-time by using well-known properties of group characters. A discretized form of the path integral is naturally provided by the properties of group characters and does not need any artificial regularization. We show that the path integral over gauge degrees of freedom for Wilson loops derived by Diakonov and Petrov (Phys. Lett. B224 (1989) 131) by using a special regularization is erroneous and predicts zero for the Wilson loop. This…
The construction of supersymmetric invariant integrals is discussed in a superspace setting. The formalism is applied to D=4, N=4 SYM and used to construct the F^2, F^4 and (F^5 + \del^2 F^4) terms in the effective action of coincident D-branes. The results are in agreement with those obtained by other methods. A simple derivation of the abelian \del^4 F^4 invariant is given and generalised to the non-abelian case. We also find some double-trace invariants. The invariants are interpreted in terms of superconformal multiplets: the F^2 and F^4 terms are given by one-half BPS multiplets, the (F^5…
Simple bosonic path integral representation for path ordered exponent is derived. This representation is used, at first, to obtain new variant of non-Abelian Stokes theorem. Then new pure bosonic worldline path integral representations for fermionic determinant and Green functions are presented. Finally, applying stationary phase method, we get quasiclassical equations of motion in QCD.
We investigate a new identity for Green functions using the higher covariant derivative regularization. It relates some coefficients in the vertex function of the matter superfield, in which one of external matter lines is not chiral. The calculation in the first nontrivial order (for the two-loop vertex function) reveals that the new identity is also valid for the non-Abelian Yang--Mills theory with matter fields. The new identity is shown to appear because three-loop integrals, defining the Gell-Mann--Low function are factorized into integrals of total derivatives.
The purpose of this paper is to investigate the gauge symmetry of classical field theories in integral formalism. A gauge invariant theory is defined in terms of the invariance of the physical observables under the coordinate transformations in principal bundle space. Through the detailed study on the property of non-Abelian parallel transportor under gauge transformations, we show that it is not generally a two-point spinor, i.e. an operator to be affected only by the gauge group elements at the two end points of the parallel transport path, except for the pure gauge situation, and therefore …
We study a model of massive photons with a parity invariant and non-local mass term. We identify a discrete symmetry of the classical equations of motion and show that this symmetry can be thought of as an electric-magnetic duality valid only in the presence of a non-zero photon mass. We explore this duality and discuss some features of the theory. We define notions of electric and magnetic charge that can be computed in any spatial domain as a volume integral of a density. These integrals can be transformed into integrals on the boundary. For instance, the electric charge is given by the sum …
We derive anomaly constraints for Abelian and non-Abelian discrete symmetries using the path integral approach. We survey anomalies of discrete symmetries in heterotic orbifolds and find a new relation between such anomalies and the so-called `anomalous' U(1).
We consider a general non-Abelian renormalizable ${\cal N}=1$ supersymmetric gauge theory, regularized by higher covariant derivatives without breaking the BRST invariance, and calculate one-loop divergences for a general form of higher derivative regulator and of the gauge fixing term. It is demonstrated that the momentum integrals giving the one-loop $β$-function are integrals of double total derivatives independently of a particular choice of the higher derivative term. Evaluating them we reproduce the well-known result for the one-loop $β$-function. Also we find that the three-point ghost …
We introduce a new localization principle which is a generalized canonical transformation. It unifies BRST localization, the non-abelian localization principle and a special case of the conformal Duistermaat-Heckman integration formula of Paniak, Semenoff and Szabo. The heat kernel on compact Lie groups is localized in two ways. First using a non-abelian generalization of the derivative expansion localization of Palo and Niemi and secondly using the BRST localization principle and a configuration space path integral. In addition we present some new formulas on homogeneous spaces which might be…
The paper is a chapter of the above-mentioned book. It aims to give an expository presentation of author's version of the non-Abelian Stokes theorem in the framework of path-integral formalism.
The term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1- and 2-forms. So far, there have been two approaches to this subject. The differential picture uses non-Abelian 1- and 2-forms in order to generalize the connection 1-form of a conventional gauge theory to the next level. The integral picture makes use of curves and surfaces labeled with elements of non-Abelian groups and generalizes the formulation of gauge theory in terms of parallel transports. We recall how to circumvent the classic no-go theorems in or…
We present a quantized non-Abelian Berry phase for time reversal invariant systems such as quantum spin Hall effect. Ordinary Berry phase is defined by an integral of Berry's gauge potential along a loop (an integral of the Chern-Simons one-form), whereas we propose that a similar integral but over five dimensional parameter space (an integral of the Chern-Simons five-form) is suitable to define a non-Abelian Berry phase. We study its global topological aspects and show that it is indeed quantized into two values. We also discuss its close relationship with the nonperturbative anomalies.
The Lorentz-covariant quantization performed in the Hamiltonian path-integral formalism for massless non-Abelian gauge fields has been achieved. In this quantization, the Lorentz condition, as a constraint, must be introduced initially and incorporated into the Yang-Mills Lagrangian by the Lagrange undetermined multiplier method. In this way, it is found that all Lorentz components of a vector potential have thier corresponding conjugate canonical variables. This fact allows us to define Lorentz-invariant poisson brackets and carry out the quantization in a Lorent-covariant manner. Key words: …
The massive non-Abelian gauge fields are quantized Lorentz-covariantly in the Hamiltonian path-integral formalism. In the quantization, the Lorentz condition, as a necessary constraint, is introduced initially and incorporated into the massive Yang-Mills Lagrangian by the Lagrange multiplier method so as to make each temporal component of a vector potential to have a canonically conjugate counterpart. The result of this quantization is confirmed by the quantization performed in the Lagrangian path-integral formalism by applying the Lagrange multiplier method which is shown to be equivalent to …
When the path integral method of anomaly evaluation is applied to chiral gauge theories, two different types of gauge anomaly, i.e., the consistent form and the covariant form, appear depending on the regularization scheme for the Jacobian factor. We clarify the relation between the regularization scheme and the Pauli--Villars--Gupta (PVG) type Lagrangian level regularization. The conventional PVG, being non-gauge invariant for chiral gauge theories, in general corresponds to the consistent regularization scheme. The covariant regularization scheme, on the other hand, is realized by the genera…
We consider the $U(1)$ Chern-Simons gauge theory defined in a general closed oriented 3-manifold $M$; the functional integration is used to compute the normalized partition function and the expectation values of the link holonomies. The nonperturbative path-integral is defined in the space of the gauge orbits of the connections which belong to the various inequivalent $U(1)$ principal bundles over $M$; the different sectors of the configuration space are labelled by the elements of the first homology group of $M$ and are characterized by appropriate background connections. The gauge orbits of …
In this letter we describe an approach to the current algebra based in the Path Integral formalism. We use this method for abelian and non-abelian quantum field theories in 1+1 and 2+1 dimensions and the correct expressions are obtained. Our results show the independence of the regularization of the current algebras.
The one--loop effective action for the case of a massive scalar loop in the background of both a scalar potential and an abelian or non--abelian gauge field is written in a one--dimensional path integral representation. From this the inverse mass expansion is obtained by Wick contractions using a suitable Green function, which allows the computation of higher order coefficients. For the scalar case, explicit results are presented up to order O(T**8) in the proper time expansion. The relation to previous work is clarified.
It has been conjectured that the Abelian projection of QCD is responsible for the confinement of color. Using a gauge independent definition of the Abelian projection which does {\it not} employ any gauge fixing, we provide a strong evidence for the Abelian dominance in Wilson loop integral. In specific we prove that the gauge potential which contributes to the Wilson loop integral is precisely the one restricted by the Abelian projection.
We investigate the relationship between colour confinement and the gauge independent Cho-Duan-Ge Abelian decomposition. The decomposition is defined in terms of a colour field $n$; the principle novelty of our study is that we have defined this field in terms of the eigenvectors of the Wilson Loop. This establishes an equivalence between the path ordered integral of the non-Abelian gauge fields with an integral over an Abelian restricted gauge field which is tractable both theoretically and numerically in lattice QCD. We circumvent path ordering without needing an additional path integral. By …
Relevant physical models are described by singular Lagrangians, so that their Hamiltonian description is based on the Dirac theory of constraints. The qualitative aspects of this theory are now understood, in particular the role of the Shanmugadhasan canonical transformation in the determination of a canonical basis of Dirac's observables allowing the elimination of gauge degrees of freedom from the classical description of physical systems. This programme was initiated by Dirac for the electromagnetic field with charged fermions. Now Dirac's observables for Yang-Mills theory with fermions (wh…
It is proposed an integral formulation of classical Yang-Mills equations in the presence of sources, based on concepts in loop spaces and on a generalization of the non-abelian Stokes theorem for two-form connections. The formulation leads in a quite direct way to the construction of gauge invariant conserved quantities which are also independent of the parameterization of surfaces and volumes. Our results are important in understanding global properties of non-abelian gauge theories.
We study different aspects the worldline path integrals with gauge fields using quantum computing. We use the Variational Quantum Eigensolver (VQE) and Evolution of Hamiltonian (EOH) quantum algorithms and IBM QISKit to perform our computations. We apply these methods to the path integral of a particle moving in a Abelian and non-Abelian background gauge field associated with a constant magnetic field and the field of a chromo-magnetic field. In all cases we found excellent agreement with the classical computation. We also discuss the insertion of vertex operators into the worldline path integ…
The general properties of the abelian projection are reviewed. We derive the explicit expression for the abelian functional integral for U(1) abelian theory which corresponds to the abelian projection of the SU(2) gluodynamics. The numerical results for the temperature dependence of the monopole condensate are presented.
We discuss a new approach to strong coupling expansion and dual representations for non-abelian lattice gauge theories. The Wilson gauge action is decomposed into a sum over "abelian color cycles" (ACC), which are loops around plaquettes visiting different colors at the corners. ACCs are complex numbers and thus commute such that a dual representation of a non-abelian theory can be obtained as in the abelian case. We apply the ACC approach to SU(2) and SU(3) lattice gauge theory and exactly rewrite the two partition sums in a strong coupling series where all gauge integrals are known in closed…
Despite the fact that the integral form of the equations of classical electrodynamics is well known, the same is not true for non-abelian gauge theories. The aim of the present paper is threefold. First, we present the integral form of the classical Yang-Mills equations in the presence of sources, and then use it to solve the long standing problem of constructing conserved charges, for any field configuration, which are invariant under general gauge transformations and not only under transformations that go to a constant at spatial infinity. The construction is based on concepts in loop spaces…
