Faddeev-Popov Determinant Research Articles

Superholomorphic anomalies and supermoduli space

Superholomorphic structure on supermoduli space is constructed. Variations with respect to supermoduli of the effective action and the Faddeev-Popov determinant in the superstring are calculated. One finds a superholomorphic anomaly which cancels in the critical dimension and leads to a superholomorphic splitting of the quantum measure on supermoduli.

Explicit formulae for one-, two-, three- and four-loop string amplitudes

Abstract In the critical dimension d = 26 the p-loop scattering amplitude for tachyons with momenta kα in the bosonic string theory can be expressed through a finite-fold integral over the space Mp of moduli of Riemann surfaces of genus p: ∫ D g ab D x μ exp (− 1 2 M 2 ∫ d 2 ξ√gg ab ∂ a x μ ∂ b x μ ) α Π ∫ d 2 ξ α √g(ξ α ) exp [ik α x(ξ α ) ] ∼ M p ∫ d ν p ( det N 0 det ' Δ 0 ) 13 det ' Δ −1 ( α Π ∫ d 2 ξ α α,β Π exp [−( 1 2M 2 ) k α k β G(ξ α , ξ β )]) . In the r.h.s. of this formula the two-dimensional metric is supposed to have the conformal form g ab = ϱ(ξ) δ ab ; Δ j = −ϱ j−1 ∂ϱ −j ∂ stands for the Laplace operator ∂ + ∂ acting on j-differentials f(ξ, ξ ) dξ j ; N j are scalar product matrices of zero modes of the operators i.e. of holomorphic j-differentials: det N j  det (mn) ∫ ƒ m (ξ) d ξ j ƒ n ( ξ ) d ξ j [ϱ(ξ, |ϱ ) d ξ d ξ ] j−1 = det (mn) ∫ϱ 1−j ƒ m ƒ n d 2 ξ . The functional determinant det' Δ0 comes from integration over the fields xμ (two-dimensional scalars), while det'· Δ−1 is the Faddeev-Popov determinant arising due to fixation of the conformal gauge for gab (the parameters of general coordinate transformations are vectors, i.e. j = −1 differentials). Through GR the regularized Green function on the Riemann surface is denoted.

Derivation of an expression for a Hamiltonian functional integral in theories with first and second class constraints

A Hamiltonian functional integral in theories with first and second class constraints (both stationary and nonstationary) is derived in a simple and consistent manner. The gauge conditions need not be in involution and may contain the time explicitly. In contrast to other studies, much attention is devoted to the proof of the Hamiltonicity of the theory on the physical submanifold Gamma* of the phase space Gamma. It is shown that ..delta../sup -1/, where D is the Faddeev-Popov determinant, is simply the volume element of the subspace Gamma = Gamma/Gamma* in noncanonical coordinates determined by the constraints and gauge conditions. It is proved that the expression for the functional integral is invariant under finite transformations of the gauge conditions.

The geometrical interpretation of the Faddeev–Popov determinant in Polyakov’s theory of random surfaces

In the context of Polyakov’s theory the geometrical approach, in which the functional measure is defined as a formal volume ∞-form, is developed. The special version of the Fubini theorem on manifolds is derived, which provides the geometrical interpretation of the Faddeev–Popov method. The conformal gauge is studied within this framework. As a result, the explicit form of functional measure in effective Liouville quantum field theory is obtained.

Magnetic response of the Yang-Mills vacuum

A steady external chromatic current of uniform magnitude is used to perturb the vacuum state of a Yang-Mills field. In a gaussian approximation, the Faddeev-Popov determinant and leading higher-order magnetic terms are found to give strongly diamagnetic effects. It is shown that for this problem, the hypersphere S 3 provides some remarkable simplicities relative to the more conventional rectangular box T 3, and allows a careful examination of infra-red properties. On S 3, a uniform current generates a uniform magnetic field, and the Landau states in such a field can be expressed in a simple closed form.

Efficient numerical techniques for perturbative lattice gauge theory computations

We discuss a set of methods and numerical tools, which are useful for a computer based approach to perturbative calculations in lattice gauge theory. The topics considered include the automatic generation of gluon vertex programs, a derivation of the Faddeev-Popov determinant on lattices with boundary, the use of a partially finite lattice with twisted boundary conditions as an infrared cutoff without zero modes, and finally the numerical extrapolation of lattice Feynman diagrams to the continuum limit. As an illustration of the methods we describe their implementation in the computation of the on-shell improved lattice action at weak coupling.

New ghost-free infrared-soft gauges.

We discuss a new class of infrared-soft, Gribov copy-free gauges in which ghosts decouple. The gauges are natural for gauge theory, allowing a geometric interpretation of the Faddeev-Popov determinant in terms of the gauge-fixing flow on the space of gauge transformations. When a Nicolai-Langevin map is available, the new gauges are seen to be equivalent to Zwanziger's stochastic gauge fixing.

Functional-integral approach to Parisi-Wu stochastic quantization: Abelian gauge theory

The method of Parisi and Wu of quantizing gauge theories (stochastic quantization) is reformulated using path integrals. We first review how the gauge fixing enters through the initial condition of the associated Langevin equation. We then prove, nonperturbatively, how the contribution of the Faddeev-Popov determinant is naturally generated by the Fokker-Planck dynamics without ever having to introduce it by hand. We restrict ourselves in this paper to Abelian gauge theories with nonlinear gauge fixing.

Estimation of the Faddeev-Popov spectral density and determinant

We construct a simple approximation for the Faddeev-Popov determinant in non-abelian gauge models by using sum rules for the density of ghost states. Values are presented for the spectral density and for the determinant, as calculated for the SU(2) gauge model formulated on the hypersphere S 3.

Generalized Slavnov-Taylor, BRST and covariance identities from the geometry of the gauge surface

We establish the equivalence between the extended BRST invariances, and the conventional Slavnov-Taylor transformations together with a new “dual” analogue. However, the latter (a non-local gauge transformation, generating anA-dependent translation of the gauge-fixing surface) isnot an invariance of the Faddeev-Popov determinant, contrary to the published claim.

Renormalization of gauge theories: Non-linear gauges

The renormalization action for gauge theories quantized with non-linear gauge conditions is written down in its most general form. A natural interpretation of the well-known quartic ghost term generated by renormalization is given in terms of a generalized Faddeev-Popov determinant.

Faddeev-Popov Zeros and Confinement of Color in a Hyperspherical Gauge Model

A new rotationally invariant Hamiltonian method, formulated on a four-dimensional hyperspherical surface, is proposed for the numerical study of quantum gauge field models. It is shown that in the Coulomb gauge it is sufficient, as well as necessary, to restrict transverse potentials to the zero-free domain of the Faddeev-Popov determinant. Numerical studies of an SU(2) model support Gribov's suggestion that the zeros provide a natural way to understand the origin of confinement.

Gribov ambiguities from the bifurcation theory viewpoint

The connection between singularities of the Faddeev–Popov determinant and the local gauge degeneracy is discussed. A criterion relating the two phenomena is given. It is proved that a whole neighborhood of the local gauge copy is filled with copies of transverse potentials.

The operators governing quantum fluctuations of Yang-Mills multi-instantons onS 4 and their Seeley coefficients

We give explicit expressions for the Seeley coefficients of the fluctuation operator and the operator that appears in the Faddeev-Popov determinant, which arise in the calculation of quantum fluctuations around Yang-Mills multi-instantons.

Strong evidence that Gribov copying does not affect the gauge theory functional integral

We give arguments to the effect that if the Faddeev-Popov determinant is defined properly as the Jacobian determinant of a transformation from a set of gauge-fixing coordinates to a set of group coordinates, the usual functional integral prescription for the quantization of gauge field theories gives the correct result even in the case of Gribov copying. An immediate corollary is that, for a large class of gauges, the gauge-fixing surface intersects every orbit of the group at least once.

The geometrical interpretation of the Faddeev-Popov determinant

We give a geometrical interpretation of the Faddeev-Popov determinant in Yang-Mills theories by comparing it to the natural metric on the orbit space.

Renormalization of Yang-Mills theory developed around an instanton

We define the perturbation theory of Yang-Mills theory developed around an instanton. This needs for a proper treatment of the zero modes. The generalized Faddeev-Popov determinant mixes the contributions of the gauge, translations and dilatations zero modes; this difficulty necessitates the introduction of new ghosts. The existence of Feynman rules and a proper renormalization scheme are settled. Slavnov identities are proved; they allow us to show the independence of the theory from the parameters introduced to deal with the zero modes. At last, fermions are introduced and the framework is extended to QCD.

The Schrödinger equation in the theory of non-Abelian massless fields

The Schrodinger equation for the state vectors in the gauge theory of non-Abelian massless fields is discussed. The energy operator H has a most simple form, being expressed in terms of 3-dimensional potentials Aja (j = 1, 2, 3, space index of the gauge group) which are in general not transverse. However, the spectrum of such an operator H is wider than the spectrum of the physical states. It is shown that all eigenstates of H can be classified according to the representations of the group Y, the generators of which coincide with the covariant derivatives of the electric field. The vectors of the physical states form the unit representation of Y and depend only on the transverse (in the 3-dimensional sense) fields Bja. The fields Bja are connected with the fields Aja by a gauge transformation. By means of this transformation, the energy operator Hr for the physical states is obtained from the operator H. Our expression for Hr is different from corresponding operator proposed by Schwinger. The latter as is shown below is incorrect. The operator Hr is singular at that surface in the space of fields B where the Faddeev-Popov determinant Z is equal to zero. Near the surface Z = 0 it is possible to interpret Hr as an energy operator with a singular repulsive potential. The question of the boundary condition at Z = 0 is discussed.

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.

Faddeev-Popov procedure and Gribov copies

In simple examples, we find that gauges with Gribov copying are valid gauges. One need only compute the Faddeev-Popov determinant Δ exactly and procede. Δ provides the correct measure for the copies. A cut-off procedure keeping only the first copy is seen to work as well. In quantum chromodynamics, we point out a class of gauges, the field-strength gauges, for which Δ is easily computed.