Ghost Fields Research Articles

Path-integral quantization of electrodynamics in dielectric media

In the present paper we study the Faddeev-Popov path-integral quantization of electrodynamics in an inhomogeneous dielectric medium. We quantize all polarizations of the photons and introduce the corresponding ghost fields. Using the heat kernel technique, we express the heat kernel coefficients in terms of the dielectricity and calculate the ultraviolet divergent terms in the effective action. No cancellation between ghosts and `non-physical' degrees of freedom of the photon is observed.

Extra dimensions in superstring theory

It was earlier shown that an SO(9,1) θ α spinor variable can be constructed from RNS matter and ghost fields. θ α has a bosonic world-sheet super-partner λ α which plays the role of a twistor variable, satisfying λГ μλ = ∂x μ + iθΦ μ∂Θ . For Type IIA superstrings, the left-moving [ θ L α , λ L α ] and right-moving [ θ Rα , λ Rα ] can be combined into 32-component SO (10,1) spinors [ θ A , λ A ]. This suggests that λ AГ AB 11λ B = 2λ L αλ Rα can be interpreted as momentum in the eleventh direction. Evidence for this interpretation comes from the zero-momentum vertex operators of the Type IIA superstring and from consideration of D 0-branes. As in the work of Bars, one finds an SO(10,2) structure for the Type IIA superstring and an SO(9, 1) × SO(2, 1) structure for the Type IIB superstring.

Canonical global symmetry in the functional integral formalism of the system and conservation laws

Based on the phase-space path integral (functional integral) for a system with a regular or singular Lagrangian, the generalized Ward identities for phase space generating functional under the global transformation in phase space are derived respectively. The canonical Noether theorem at the quantum level is also established. It is pointed out that the connection between the symmetries and conservation laws in classical theories, in general,is no longer preserved in quantum theories. The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta as usually performed. Applying the present formulation to Yang-Mills theory, the quantal BRS conserved quantity and Ward-Takahashi identity for BRS tranformation are derived; the Ward identities for gaugeghost proper vertices and new quantal conserved quantity are also found. In comparison of quantal conservation laws with those one deriving from configuration-space path integral using the Faddeev-Popov(F-P) trick is discussed. A precise study of path-integral quantisation for a nonlinear sigma model with Hopf and Chern-Simons (CS) terms is reexamined. It has been shown that the angular momentum at the quantum level is equal to classical (Noether ) one. Applying our formulation to non-Abelian CS theory, the quantal conserved angular momentum of this system is obtained which differs from classical one in that one needs to take into account the contribution of angular momenta of ghost fields.

Symmetry properties and renormalizability of massive gauge theories in Delbourgo-Jarvis gauge

The massive Curci-Ferrari model in the nonlinear Delbourgo-Jarvis gauge can be renormalized analogously to Yang-Mills theories in linear gauges if besides the (anti-) Slavnov-Taylor and (anti-) Delduc-Sorella identies also the equations of motion for the auxillary field and the (anti-)ghost fields as well as the so-called local Lee identity is required.

Gauge invariance of quantum gravity in the causal approach

We investigate gauge invariance of perturbative quantum gravity without matter fields in the causal Epstein - Glaser approach. This approach uses free fields only so that all objects of the theory are mathematically well defined. The first-order graviton self-couplings are obtained from the Einstein - Hilbert Lagrangian written in terms of Goldberg variables and expanded to lowest order on the flat Minkowski background metric (linearized Einstein theory). Similar to Yang - Mills theory, gauge invariance to first order requires an additional coupling to fermionic ghost fields. For second-order tree graphs, gauge invariance generates four-graviton normalization terms, which agree exactly with the next order of the expansion of the Einstein - Hilbert Lagrangian. Gauge invariance of the ghost sector is then examined in detail. It is stressed that, despite some formal similarities, the concept of operator gauge invariance used in the causal method is different from the conventional BRS-invariance commonly used in the literature.

Consistency of the regularization of gauge theories by high covariant derivatives.

We show that regularization of gauge theories by higher covariant derivatives and gauge invariant Pauli-Villars regulators is a consistent method if the Pauli-Villars vector fields are considered in a covariant in the regulating Pauli-Villars fields is pathological and the original Slavnov proposal in covariant Landau gauge is not correct because of the appearance of massless modes in the regulators which do not decouple when the ultraviolet regulator is removed. In such a case the method does not correspond to the regularization of a pure gauge theory but that of a gauge theory in interaction with massless ghost fields. This explains the problems pointed out by Martin and Ruiz in covariant Landau gauge. However, a minor modification of Slavnov method provides a consistent regularization even for such a case. The regularization that we introduce also solves the problem of overlapping divergences in a way similar to geometric regularization and yields the standard values of the $\beta$ and $\gamma$ functions of the renormalization group equations.

RR-dilaton interaction in a type IIA superstring

We analyse the interaction between the massless Ramond-Ramond states with a dilaton in a type II superstring. By constructing vertex operators for massless Ramond-Ramond states and computing their correlation functions with a dilaton we find the Ramond-Ramond part of the superstring low-energy effective action. Those RR terms appear in the action without the standard dilatonic factor (string coupling constant) as has been shown earlier on the basis of space-time supersymmetry. The geometrical interpretation of this fact is presented. Namely we argue that the spin operators in the RR vertices effectively decrease the Euler character of the worldsheet by 1 unit. As a result, the dilaton term in the worldsheet action has the form: ∼ ∫ d 2z e ΦΛF Λ where Λ and Λ are the (1, 0) and (0,1) spin operators for the ghost and matter fields, F is the RR field strength contracted with y-matrices. We also analyse the interaction of a dilaton with the massive RR states.

The antighost equation in N = 1 super-Yang-Mills theories

The antighost equation valid for usual gauge theories in the Landau gauge, is generalized to the case of N = 1 supersymmetric gauge theories in a supersymmetric version of the Landau gauge. This equation, which expresses the nonrenormalization of the Faddeev-Popov ghost field, plays an important role in the proof of the nonrenormalization theorems for the chiral anomalies.

Modified BFV quantization of membranes and BRST invariant boundary conditions on the ghost fields

The covariant BRST off-shell effective action for the open and closed membranes is obtained. Our result generalizes Fujikawa's on-shell effective action for membranes. We found boundary conditions on the ghost fields, which are necessary in order to obtain the off-shell nilpotency of the BRST charge for the open membrane. We also proved that these boundary conditions on the ghost fields are invariant under BRST transformations and are the generalization to the Kato - Ogawa boundary conditions for the open bosonic string.

𝒲-gauge structures and their anomalies: An algebraic approach

Starting from flat two-dimensional gauge potentials we propose the notion of 𝒲-gauge structure in terms of a nilpotent Becchi–Rouet–Stora (BRS) differential algebra. The decomposition of the underlying Lie algebra with respect to an SL(2) subalgebra is crucial for the discussion of conformal covariance, in particular the appearance of a projective connection. Different SL(2) embeddings lead to different 𝒲-gauge structures. We present a general soldering procedure which allows one to express zero curvature conditions for the 𝒲-currents in terms of conformally covariant differential operators acting on the 𝒲-gauge fields and to obtain, at the same time, the complete nilpotent BRS differential algebra generated by 𝒲-currents, gauge fields, and the ghost fields corresponding to 𝒲-diffeomorphisms. As illustrations we treat the cases of SL(2) itself and the two different SL(2) embeddings in SL(3), viz., the 𝒲 (1)3- and 𝒲 (2)3-gauge structures, in some detail. In these cases we determine algebraically 𝒲-anomalies as solutions of the consistency conditions and discuss their Chern–Simons origin.

The relation between the waveguide and overlap implementations of Kaplan's domain wall fermions

Recently, Narayanan and Neuberger proposed that the fermion determinant for a lattice chiral gauge theory be defined by an overlap formula. The motivation for that formula comes from Kaplan's five-dimensional lattice domain wall fermions. In the case that the target continuum theory contains 4 n chiral families, we show that the effective action defined by overlap formula is identical to the effective action of a modified waveguide model that has extra bosonic ghost fields. This raises serious questions about the viability of the overlap formula for defining chiral gauge theories on the lattice.

Non-local and local ghost fields in quantum correlations

Einstein Podolsky Rosen quantum correlations are discussed from the perspective of a ghost field introduced by Einstein. The concepts of ghost field, hidden variables, local reality and the Bell inequality are reviewed. In the framework of the correlated singlet state, it is shown that quantum mechanics can be cast in a way that has the form of either nonpositive and local ghost field or a positive and nonlocal ghost field.

Two-loop vacuum amplitudes in a non-Abelian gauge theory on S4

In a series of papers, the authors have developed a matrix-element approach to quantum field theory on S4, and used a complex l-plane method of performing the resulting angular momentum sums. The methods have been used to reproduce previous results for both one- and two-loop vacuum amplitudes for an Abelian gauge theory, including Dirac fermions and scalar bosons. In this paper we extend the formalism to non-Abelian gauge theory and show how the new complex l-plane methods allow the calculation of two-loop vacuum amplitudes involving ghost and gauge fields. We show how it is possible to overcome previous calculational limitations, resulting from the form of the ghost and gauge propagators, by performing the entire calculation in angular momentum space without reference to any particular configuration-space form of the propagators. We derive, for the first time, the two-loop interacting vacuum amplitude in a non-Abelian gauge theory on a curved background and give the required counterterm to renormalize the background gravitational action on S4.

Infrared Properties of the Hot Gauge Theory after Symmetry Breaking

It is shown that the fictitious infrared pole is eliminated from the hot gauge theory which acquires a new vacuum after the global gauge symmetry spontaneously broken. The nonzero W-condensate appears and leads to a screening of the chromomagnetic forces through the scenario with the standard magnetic mass. The strong infrared divergences which usually accompany the multi-loop pertur­ bative calculations with a hot non-Abelian gauge theory are an intrinsic property of SU(N)-gluodynamics and determine many of its peculiarities. Unfortunately, their essence remains unclear, although the infrared properties of the Yang-Mills theory at T-=1=0 have been intensively studied during recent years. At T-=1=0 the infrared problem in QCD is more pronounced since, unlike QFT, the leading infrared diver­ gences are power-like and most of the Green functions calculated perturbatively demonstrate the fictitious infrared pole when p4 = 0 and IPI ~ g 2 T. This singularity (originally found in papers 1) and 2)) is strongly sensitive to the gauge chosen although it cannot be removed through its redefinition and remains practically the same when the radiative corrections are taken into account. In the present paper we study the infrared properties of the hot SU(2)-model which acquires a new vacuum with the nonzero W-condensate after the global gauge symmetry spontaneously broken. Within this scenario the infrared fictitious pole being a signal of the infrared instability for the unbroken gauge theory is eliminated at the beginning and the standard magnetic mass arises already on the one-loop level. The broken gauge theory demonstrates the correct infrared properties but unfortu­ nately all its infrared limits are strongly gauge-dependent at least within a pertur­ bative expansion. This dependence seems not to be occasional and its appearance should be carefully investigated although it is necessary to remember (see, e.g., Ref. 3) for details) that all the standard nonperturbative infrared limits are gauge­ dependent as well. § 2. General formalism and definition Our formalism is built by using the standard Green function technique at T-=1=0 in the background gauge with an arbitrary parameter ~- The effective action W(A) is defined as a functional integral over periodic gauge and ghost fields

Stochastic Quantization of the Non-Abelian Chern-Simons Gauge Theory Based on a Kerneled Lagevin Equation and Faddeev-Popov Ghost Effects

We study the stochastic quantization of the non-Abelian Chern-Simons gauge theory based on a kerneled Langevin equation. We choose a non-singular kernel and show that our Langevin equation has a thermal equilibrium limit and reproduces the Faddeev-Popov ghost effects in the gauge invariant quantity automatically without introducing any ghost fields. We show that the 3-dimensional Chern-Simons gauge theory based on our Langevin equation and the 4-dimensional topological field theory give the same partition function and the same Green's function.

Euclidean Maxwell theory in the presence of boundaries. II

-function regularization is applied to complete a recent analysis of the quantized electromagnetic field in the presence of boundaries. The quantum theory is studied by setting to zero on the boundary the magnetic field, the gauge-averaging functional, and hence the Faddeev--Popov ghost field. Electric boundary conditions are also studied. On considering two gauge functionals which involve covariant derivatives of the 4-vector potential, a series of detailed calculations shows that, in the case of flat Euclidean 4-space bounded by two concentric 3-spheres, one-loop quantum amplitudes are gauge independent and their mode-by-mode evaluation agrees with the covariant formulae for such amplitudes and coincides for magnetic or electric boundary conditions. By contrast, if a single 3-sphere boundary is studied, one finds some inconsistencies, i.e. gauge dependence of the amplitudes.

Stochastic Quantization of the Non-Abelian Chern-Simons Gauge Theory Based on a Kerneled Lagevin Equation and Faddeev-Popov Ghost Effects

We study the stochastic quantization of the non-Abelian Chern-Simons gauge theory based on a kerneled Langevin equation. We choose a non-singular kernel and show that our Langevin equation has a thermal equilibrium limit and reproduces the Faddeev-Popov ghost effects in the gauge invar­ iant quantity automatically without introducing any ghost fields. We show that the 3-dimensional Chern-Simons gauge theory based on our Langevin equation and the 4-dimensional topological field theory give the same partition function and the same Green's function.

Redundancy of the quantum level gauge condition

The manifold Γ̂ defined by the equations of motion (EM) of the gauge and ghost fields w.r.t. the gauge-fixed action is regarded as a supercanonical line bundle over the manifold Γ defined by the EM of the gauge fields only w.r.t. the classical action. In this language the BRST operator Q is a local section of the bundle Γ̂. Using this we give a local expression for Q as being, on the other hand, the nilpotent exterior derivative on Γ with the ghost field Ψ as its generator. This fiber bundle setup allows us to prove that any ‘‘second level’’ gauge condition, i.e., a gauge condition on Γ̂ is equivalent to a gauge on the base manifold Γ and thus does not break the BRST symmetry of the quantized theory.

Generalized Lagrangian master equations

We discuss the geometry of the Lagrangian quantization scheme based on (generalized) Schwinger-Dyson BRST symmetries. When a certain set of ghost fields are integrated out of the path integral, we recover the Batalin-Vilkovisky formalism, now extended to arbitrary functional measures for the classical fields. Keeping the ghost reveals the crucial role played by a natural connection on the space of fields.

Local cohomology in gauge theories, BRST transformations and anomalies

We introduce a geometric framework needed for a mathematical understanding of the BRST symmetries and chiral anomalies in gauge field theories. We define the BRST bicomplex in terms of local cohomology using differential forms on the infinite jet bundle and consider variational aspects of the problem in this cohomological context. The adjoint representation of the structure group induces a representation of the infinite dimensional Lie algebra g of infinitesimal gauge transformations on the space of local differential forms, with respect to which the BRST bicomplex is defined using the Chevalley-Eilenberg construction. The induced coboundary operator of the associated cohomology H ∗ loc ( g ) is the BRST operator s. With this we derive the classical BRST transformations of the vector potential A and the ghost field η as sA = dη+[A, η] , and sη = - 1 2 [η, η] . Moreover the ghost field η is identified with the canonical Maurer-Cartan form of the infinite dimensional Lie group G of gauge transformations. We give a homotopy formula on the BRST bicomplex and with the introduction of Chern-Simon type forms we derive the associated descent equations and show that the non-Abelian anomalies, which satisfy the Wess-Zumino consistency condition, represent cohomology classes in H 1 loc ( g ) .