Gauge fixing,BRS invariance and Ward identities for randomly stirred flows

The Galilean invariance of the Navier–Stokes equation is shown to be akin to a global gauge symmetry familiar from quantum field theory. This symmetry leads to a multiple counting of infinitely many inertial reference frames in the path integral approach to randomly stirred fluids. This problem is solved by fixing the gauge, i.e., singling out one reference frame. The gauge fixed theory has an underlying Becchi–Rouet–Stora (BRS) symmetry which leads to the Ward identity relating the exact inverse response and vertex functions. This identification of Galilean invariance as a gauge symmetry is explored in detail, for different gauge choices and by performing a rigorous examination of a discretized version of the theory. The Navier–Stokes equation is also invariant under arbitrary rectilinear frame accelerations, known as extended Galilean invariance (EGI). We gauge fix this extended symmetry and derive the generalized Ward identity that follows from the BRS invariance of the gauge-fixed theory. This new Ward identity reduces to the standard one in the limit of zero acceleration. This gauge-fixing approach unambiguously shows that Galilean invariance and EGI constrain only the zero mode of the vertex but none of the higher wavenumber modes.     

Slavnov-Taylor identities in Weinberg's renormalization scheme

Weinberg's renormalization scheme, although more cumbersome from the computational point of view, has a more immediate physical interpretation than 't Hooft's minimal renormalization scheme. It is expected to lead to smaller higher-order coefficients in a perturbative approach to QCD. However, it a priori violates the Slavnov-Taylor identities. A complete study of this problem is performed, both theoretically and for the practitioner's sake. The ambiguities in the choice of the tensorial basis of some of the QCD vertices, as well as the dependence in the gauge parameter are used for substantiating, eventually, the Slavnov-Taylor identities in this renormalization scheme.     

Gauge invariance and Nielsen identities

The one-loop contribution to the effective potential and mass are computed within the context of scalar electrodynamics for the class of generalR gauges in the \(\overline {MS} \) scheme. These calculations are performed in order to construct a non-trivial verification of the corresponding Nielsen identities within the context of the Higgs model. Some brief comments on the Coleman-Weinberg model are also included.     

Gauge identities and the Dirac conjecture

The gauge symmetries of a general dynamical system can be systematically obtained following either a Hamiltonian or a Lagrangean approach. In the former case, these symmetries are generated, according to Dirac's conjecture, by the first class constraints. In the latter approach such local symmetries are reflected in the existence of so called gauge identities. The connection between the two becomes apparent, if one works with a first order Lagrangean formulation. Our analysis applies to purely first class systems. We show that Dirac's conjecture applies to first class constraints which are generated in a particular iterative way, regardless of the possible existence of bifurcations or multiple zeroes of these constraints. We illustrate these statements in terms of several examples.     

The Hamiltonian formulation of the Euler equation and subsequent constraints on the properties of randomly stirred fluids

It is shown that the existence of a new integral of the motion for inviscid flow, evident in the hamiltonian formulation of the Euler equation, results in some important constraints on the properties of randomly stirred fluids.     

Gauge symmetry as symmetry of matrix coordinates

We propose a new point of view on gauge theories, based on taking the action of symmetry transformations directly on the space coordinates. Via this approach the gauge fields are not introduced at the first step, and they can be interpreted as fluctuations around some classical solutions of the model. The new point of view is connected to the lattice formulation of gauge theories, and the parameter of the non-commutativity of the coordinates appears as the lattice spacing parameter. Through the statements concerning the continuum limit of lattice gauge theories, the suggestion arises that the non-commutative spaces are the natural ones to formulate gauge theories at strong coupling. Via this point of view, a close relation between the large-N limit of gauge theories and string theory can be made manifest. Received: 16 June 2000 / Published online: 8 September 2000     

Gauge identities in dual field theory

In the dual and fermion proposed by Neveu, Schwarz and Ramond, we find gauge identities involving vertex parts and also off-shell supergauge identities. When their models are translated into the secondary quantized field theory, we no longer can impose the gauge identities as operator ones, since they would lead to a contradiction of the second-quantization rules. We therefore require weak forms of the gauge identities just in an analogous way to the Lorentz condition in quantum electrodynamics. It is then shown that the weak forms of the identities are consistent with the second-quantization rules and also with field equations.     

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

It is shown that various Green function identities in quantized gauge theories may be viewed as arising from the local conformal groupSO(N+1, 1) of motions of theN-dimensional gauge fixing surface. Translations and rotations correspond respectively to the usual Slavnov-Taylorinvariance and anew ‘dual’ analogue. Dilations give rise to thecovariance identities, and in axial type gauges we obtain a closed form for the covariance relationships. It is shown that the generalized Slavnov-Taylor identities, and the BRST identities, are equivalent, as are their duals.     

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.     

Gauge symmetry and supersymmetric two-particle problem

An analogy between the removal of nonphysical relative time (or relative energy) in the supersymmetric two-particle problem and the account of local gauge invariance in supersymmetric quantum field theory is discussed. A group of gauge transformations for the Bethe-Salpeter amplitudes is suggested, the invariants of which are the relativistic three-dimensional (quasipotential) wave functions in the Logunov-Tavkhelidze approach. Subsidiary conditions imposed on the Bethe-Salpeter amplitudes in the Todorov approach are shown to be equivalent to appropriate gauge fixing.     

Clockwork SUSY: supersymmetric Ward and Slavnov-Taylor identities at workin Green's functions and scattering amplitudes

We study the cancellations among Feynman diagrams that implement the Ward and Slavnov-Taylor identities corresponding to the conserved supersymmetry current in supersymmetric quantum field theories. In particular, we show that the Faddeev-Popov ghosts of gauge and supersymmetries never decouple from the physical fields, even for abelian gauge groups. The supersymmetric Slavnov-Taylor identities provide efficient consistency checks for automatized calculations and can verify the supersymmetry of Feynman rules and the numerical stability of phenomenological predictions simultaneously.Received: 2 June 2003, Revised: 1 July 2003, Published online: 5 September 2003     

Gauge symmetry and the EMC spin effect

We emphasise the EMC spin effect as a problem of symmetry and discuss the renormalisation of theC=+1 axial tensor operators. This involves the generalisation of the Adler-Bell-Jackiw anomaly to each of these operators. We find that the contribution of the axial anomaly to the spin dependent structure functiong ₁(x, Q ²) scales at O(α_s). This means that the anomaly can be a largex effect ing ₁. Finally we discuss the jet signature of the anomaly.     

Gauge internal symmetry in extended supergravity

Extended supergravity theories with global O(2) and SO(3) internal symmetry have recently been constructed, and a mechanism which implements local O(2) and SO(3) gauge invariance is given here. The introduction of a minimal gauge coupling automatically leads to a $\text{spin}\text{-}\text{3}\text{2}$ mass and a cosmological term in order to preserve local supersymmetry. Local internal symmetry for a $\text{spin}\text{-}\text{3}\text{2}$ field is related to spontaneous breakdown of global supersymmetry. Perturbation theory results which confirm the physical consistency of the system are given.     

Gauge symmetry breaking in matrix models

We argue that some features of the standard model, in particular the fermion assignment and symmetry breaking, can be obtained in matrix model which describes noncommutative gauge theory as well as gravity in an emergent way. The mechanism is based on the presence of some extra (matrix) dimensions. These extra dimensions are different from the usual ones which give to a noncommutative geometry of the Grönewold-Moyal type, and are reminiscent of the Connes-Lott model, although the action is very different.     

Scaling properties of the two-dimensional randomly stirred Navier-Stokes equation

We inquire into the scaling properties of the 2D Navier-Stokes equation sustained by a force field with Gaussian statistics, white noise in time, and with a power-law correlation in momentum space of degree 2 - 2 epsilon. This is at variance with the setting usually assumed to derive Kraichnan's classical theory. We contrast accurate numerical experiments with the different predictions provided for the small epsilon regime by Kraichnan's double cascade theory and by renormalization group analysis. We give clear evidence that for all epsilon, Kraichnan's theory is consistent with the observed phenomenology. Our results call for a revision in the renormalization group analysis of (2D) fully developed turbulence.     

Gauge generators,transformations and identities on a noncommutative space

By abstracting a connection between gauge symmetry and gauge identity on a noncommutative space, we analyse star (deformed) gauge transformations with the usual Leibniz rule as well as undeformed gauge transformations with a twisted Leibniz rule. Explicit structures of the gauge generators in either case are computed. It is shown that, in the former case, the relation mapping the generator with the gauge identity is a star deformation of the commutative space result. In the latter case, on the other hand, this relation gets twisted to yield the desired map.