Gauge-fixed Lagrangian Research Articles

Twistor formulation of a massive particle with rigidity

A massive rigid particle model in (3+1) dimensions is reformulated in terms of twistors. Beginning with a first-order Lagrangian, we establish a twistor representation of the Lagrangian for a massive particle with rigidity. The twistorial Lagrangian derived in this way remains invariant under a local U(1)×U(1) transformation of the twistor and other relevant variables. Considering this fact, we carry out a partial gauge-fixing so as to make our analysis simple and clear. We develop the canonical Hamiltonian formalism based on the gauge-fixed Lagrangian and perform the canonical quantization procedure of the Hamiltonian system. Also, we obtain an arbitrary-rank massive spinor field in (3+1) dimensions via the Penrose transform of a twistor function defined in the quantization procedure. Then we prove, in a twistorial fashion, that the spin quantum number of a massive particle with rigidity can take only non-negative integer values, which result is in agreement with the one shown earlier by Plyushchay. Interestingly, the mass of the spinor field is determined depending on the spin quantum number.

BRST-BV approach to continuous-spin field

Using BRST-BV approach, massless and massive continuous-spin fields propagating in the flat space are studied. For such fields, BRST-BV gauge invariant Lagrangian is obtained. The Lagrangian and gauge transformations are constructed out of traceless gauge fields and traceless gauge transformation parameters. Interrelation between the BRST-BV Lagrangian and the Lagrangian for the continuous-spin fields in metric-like approach is demonstrated. Considering the BRST-BV Lagrangian in the Siegel gauge, we get gauge-fixed Lagrangian which is invariant under global BRST and antiBRST transformations.

Novel symmetries in Christ-Lee model

We demonstrate that the gauge-fixed Lagrangian of the Christ-Lee model respects four fermionic symmetries, namely; (anti-)BRST symmetries, (anti-)co-BRST symmetries within the framework of BRST formalism. The appropriate anticommutators amongst the fermionic symmetries lead to a unique bosonic symmetry. It turns out that the algebra obeyed by the symmetry transformations (and their corresponding conserved charges) is reminiscent of the algebra satisfied by the de Rham cohomological operators of differential geometry. We also provide the physical realizations of the cohomological operators in terms of the symmetry properties. Thus, the present model provides a simple model for the Hodge theory.

The BRST-BV approach to massless fields adapted for the AdS/CFT correspondence

Using BRST-BV formulation of relativistic dynamics, arbitrary spin massless and massive field propagating in flat space and arbitrary spin massless fields propagating in AdS space are considered. For such fields, BRST-BV Lagrangians invariant under gauge transformations are obtained. The Lagrangians and gauge transformations are built in terms of traceless gauge fields and traceless gauge transformation parameters. The use of the Poincare parametrization of AdS space allows us to get simple BRST-BV Lagrangian for AdS fields. By imposing the Siegel gauge condition, we get gauge-fixed Lagrangian which leads to decoupled equations of motion for AdS fields. Such equations of motion considerably simplify the study of AdS/CFT correspondence. In the basis of conformal algebra, realization of relativistic symmetries on space of fields and antifields entering the BRST-BV formulation of AdS fields is obtained.

Long, partial-short, and special conformal fields

In the framework of metric-like approach, totally symmetric arbitrary spin bosonic conformal fields propagating in flat space-time are studied. Depending on the values of conformal dimension, spin, and dimension of space-time, we classify all conformal field as long, partial-short, short, and special conformal fields. An ordinary-derivative (second-derivative) Lagrangian formulation for such conformal fields is obtained. The ordinary-derivative Lagrangian formulation is realized by using double-traceless gauge fields, Stueckelberg fields, and auxiliary fields. Gauge-fixed Lagrangian invariant under global BRST transformations is obtained. The gauge-fixed BRST Lagrangian is used for the computation of partition functions for all conformal fields. Using the result for the partition functions, numbers of propagating D.o.F for the conformal fields are also found.

Arbitrary spin conformal fields in (A)dS

Totally symmetric arbitrary spin conformal fields in (A)dS space of even dimension greater than or equal to four are studied. Ordinary-derivative and gauge invariant Lagrangian formulation for such fields is obtained. Gauge symmetries are realized by using auxiliary fields and Stueckelberg fields. We demonstrate that Lagrangian of conformal field is decomposed into a sum of gauge invariant Lagrangians for massless, partial-massless, and massive fields. We obtain a mass spectrum of the partial-massless and massive fields and confirm the conjecture about the mass spectrum made in the earlier literature. In contrast to conformal fields in flat space, the kinetic terms of conformal fields in (A)dS space turn out to be diagonal with respect to fields entering the Lagrangian. Explicit form of conformal transformation which maps conformal field in flat space to conformal field in (A)dS space is obtained. Covariant Lorentz-like and de-Donder like gauge conditions leading to simple gauge-fixed Lagrangian of conformal fields are proposed. Using such gauge-fixed Lagrangian, which is invariant under global BRST transformations, we explain how the partition function of conformal field is obtained in the framework of our approach.

Off-shell nilpotent (anti-)BRST symmetries for a free particle system on a toric geometry: Superfield formalism

We derive the off-shell nilpotent as well as absolutely anti-commuting (anti-)BRST symmetry transformations, within the framework of superfield approach to BRST formalism, for a free particle system constrained to move on a torus. We also construct the most appropriate gauge-fixed Lagrangian which respects the (anti-)BRST symmetry transformations.

A Renormalizable Theory of Quantum Gravity: Renormalization Proof of the Gauge Theory of Volume Preserving Diffeomorphisms

Inertial and gravitational mass or energy-momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory are shown to obey a BRST-type symmetry. The related Zinn-Justin-type equation restricting the corresponding quantum effective action is established. This equation limits the infinite parts of the quantum effective action to have the same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. The inner space integrals occurring in the quantum effective action which are divergent due to the gauge group's infinite volume are shown to be regularizable in a way consistent with the symmetries of the theory demonstrating as a byproduct that viable quantum gauge field theories are not limited to finite-dimensional compact gauge groups as is commonly assumed.

Nonquadratic gauge fixing and ghosts for gauge theories on the hypersphere

It has been suggested that using a gauge fixing Lagrangian that is not quadratic in a gauge fixing condition is most appropriate for gauge theories formulated on a hypersphere. We reexamine the appropriate ghost action that is to be associated with gauge fixing, applying a technique that has been used for ensuring that the propagator for a massless spin-two field is transverse and traceless. It is shown that this nonquadratic gauge fixing Lagrangian leads to two pair of complex Fermionic ghosts and two Bosonic real ghosts.

Evolutionary quantum cosmology in a gauge-fixed picture

We study the classical and quantum models of a flat Friedmann-Robertson-Walker space-time, coupled to a perfect fluid, in the context of the consensus and a gauge-fixed Lagrangian frameworks. It is shown that, either in the usual or in the gauge-fixed actions, the evolution of the Universe based on the classical cosmology represents a late time power law expansion, coming from a big-bang singularity in which the scale factor goes to zero for the standard matter, and tending towards a big-rip singularity in which the scale factor diverges for the phantom fluid. We then employ the familiar canonical quantization procedure in the given cosmological setting to find the cosmological wave functions in the corresponding minisuperspace. Using a gauge-fixed (reduced) Lagrangian, we show that it may lead to a Schr\"odinger equation for the quantum-mechanical description of the model under consideration, the eigenfunctions of which can be used to construct the time dependent wave function of the Universe. We use the resulting wave function in order to investigate the possibility of the avoidance of classical singularities due to quantum effects by means of the many-worlds and ontological interpretation of quantum cosmology.

A Chern–Simons gauge-fixed Lagrangian in a ‘non-canonical’ BRST approach

This paper presents a possible path which starts from the extended BRST Hamiltonian formalism and ends with a covariant Lagrangian action, using the equivalence between the two formalisms. The approach allows a simple account of the form of the master equation and offers a natural identification of some ‘non-canonical’ operators and variables. These are the main items which solve the major difficulty of the extended BRST Lagrangian formalism, i.e., the gauge-fixing problem. The algorithm we propose applies to a non-Abelian Chern–Simons model coupled with Dirac fields.

General covariant gauge fixing for massless spin-two fields

The most general covariant gauge fixing Lagrangian is considered for a spin-two gauge theory in the context of the Faddeev-Popov procedure. In general, five parameters characterize this gauge fixing. Certain limiting values for these parameters give rise to a spin-two propagator that is either traceless or transverse, but for no values of these parameters is this propagator simultaneously traceless and transverse. Having a traceless-transverse propagator ensures that only the physical degrees of freedom associated with the tensor field propagate, and hence it is analogous to the Landau gauge in electrodynamics. To obtain such a traceless-transverse propagator, a gauge fixing Lagrangian which is not quadratic must be employed; this sort of gauge fixing Lagrangian is not encountered in the usual Faddeev-Popov procedure. It is shown that when this nonquadratic gauge fixing Lagrangian is used, two fermionic and one bosonic ghosts arise. As a simple application we discuss the energy-momentum tensor of the gravitational field at finite temperature.

BRST Symmetric Gaugeon Formalism for the Higgs Model

We reinvestigate Yokoyama’s gaugeon formalism for the spontaneously broken Abelian gauge theory. Within the framework of the covariant linear gauges, we give a general gaugefixing Lagrangian which includes the gauge field, the Goldstone mode, the multiplier B-field and Yokoyama’s gaugeon fields (as well as Faddeev-Popov ghosts). As special choices of the values of the gauge-fixing parameters, our theory includes the usual covariant gauges and Rξ-like gauges. Although some of the gauge-fixing parameters can be shifted by the q-number gauge transformation, the ξ parameter cannot be shifted in any of the Rξ-like gauges.

The AdS5× S5superstring in light-cone gauge and its Bethe equations

We use the uniform light-cone gauge to derive an exact gauge-fixed Lagrangian and light-cone Hamiltonian for the Green-Schwarz superstring in AdS5xS5. We then quantize the theory perturbatively in the near plane-wave limit, and compute the leading 1/J correction to a generic string state from the rank-1 subsectors. These investigations enable us to propose a new set of light-cone Bethe equations for the quantum string. The equations have a simple form and yield the correct spinning string and flat space limits. Finally, we clarify the notion of closed sectors in string theory by proving the existence of perturbative effective string Hamiltonians which are direct analogues of (all loop) dilatation operators in the dual N=4 gauge theory.

ON THE BV QUANTIZATION OF GAUGE GRAVITATION THEORY

Quantization of gravitation theory as gauge theory of general covariant transformations in the framework of Batalin–Vilkoviski (BV) formalism is considered. Its gauge-fixed Lagrangian is constructed.

A superspace formulation of the BV action

We show that the BV (Batalin-Vilkovisky) action, formulated with a generalized BRST symmetry (including the shift symmetry), is also invariant under a generalized anti-BRST transformation (where the antifields are the parameters of the transformation), when the gauge fixing lagrangian is both BRST and anti-BRST invariant. We show that for a general gauge fixing lagrangian, the BV action can be written in a manifestly generalized BRST invariant manner in a superspace with one Grassmann coordinate whereas it can be expressed in a manifestly generalized BRST and anti-BRST invariant manner in a superspace with two Grassmann coordinates when the gauge fixing lagrangian is invariant under both BRST and anti-BRST transformations.

BRST symmetry in the Schrödinger picture.

We show that the effective Lagrangian including the gauge-fixing and ghost terms of the non-Abelian gauge theories can be derived in the functional Schr\"odinger picture by using the residual symmetry of the gauge-fixed Lagrangian. This residual gauge symmetry is shown to be equivalent to the well-known Becchi-Rouet-Stora-Tyutin symmetry.

A spin-two gauge theory

We formulate a Lagrangian for a free massless spin-two field hμν that is invariant under the transformation hμν→hμν + ∂μfν + ∂νfμ This is then coupled to a massive scalar field [Formula: see text]. In two dimensions the classical Lagrangian for hμν vanishes identically, and hence the kinetic term in the effective Lagrangian for hμν reduces to the gauge fixing Lagrangian. The theory is renormalizable in two dimensions provided we mix all operators of the form [Formula: see text]. One-loop two-point functions are computed using operator regularization.

On the quantization of string field theory

The quantization of free, gauge-fixed and covariant open-string field theory is examined. It is noted that the BRST symmetry of the world sheet is not exactly relevant for second quantization. An extra local symmetry in the auxiliary sector requires a gauge-fixing condition, which then breaks the BRST symmetry. The remaining BRST symmetries are identified and are shown to contain the “physical” BRST symmetry, which corresponds to the BRST symmetry found in the usual Yang-Mills theories. The gauge-fixing and ghost lagrangian is written as an anticommutator of the sum of the surviving BRST charges and a bilinear in field operators.