First Class Constraints Research Articles

Relaxation of first-class constraints and the quantization of gauge theories: From “matter without matter” to the reappearance of time in quantum gravity

We make a conceptual overview of a particular approach to the initial-value problem in canonical gauge theories. We stress how the first-class phase-space constraints may be relaxed if we interpret them as fixing the values of new degrees of freedom. This idea goes back to Fock and Stueckelberg, leading to restrictions of the gauge symmetry of a theory, and it corresponds, in certain cases, to promoting constants of Nature to physical fields. Recently, different versions of this formulation have gained considerable attention in the literature, with several independent iterations, particularly in classical and quantum descriptions of gravity, cosmology, and electromagnetism. In particular, in the case of canonical quantum gravity, the Fock–Stueckelberg approach is relevant to the so-called problem of time. Our overview recalls and generalizes the work of Fock and Stueckelberg and its physical interpretation with the aim of conceptually unifying the different iterations of the idea that appear in the literature and of motivating further research.

On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC

Abstract In spatially non-compact homogeneous minisuperpace models, spatial integrals in the Hamiltonian and symplectic form must be regularised by confining them to a finite volume $V_o$, known as the \emph{fiducial cell}. As this restriction is unnecessary in the complete field theory before homogeneous reduction, the physical significance of the fiducial cell has been largely debated, especially in the context of (loop) quantum cosmology. Understanding the role of $V_o$ is in turn essential for assessing the minisuperspace description's validity and its connection to the full theory. In this work we present a systematic procedure for the field theory reduction to spatially homogeneous minisuperspaces within the canonical framework and apply it to a massive scalar field theory and gravity. Our strategy consists in implementing spatial homogeneity via second-class constraints for the discrete field modes over a partitioning of the spatial slice into countably many disjoint cells. The reduced theory's canonical structure is then given by the corresponding Dirac bracket. Importantly, the latter can only be defined on a finite number of cells homogeneously patched together. This identifies a finite region, the fiducial cell, whose physical size acquires then a precise meaning already at the classical level as the scale over which homogenenity is imposed. Additionally, the procedure allows us to track the information lost during homogeneous reduction and how the error depends on $V_o$. We then move to the quantisation of the classically reduced theories, focusing in particular on the relation between the theories for different $V_o$, and study the implications for statistical moments, quantum fluctuations, and semiclassical states. In the case of a quantum scalar field, a subsector of the full quantum field theory where the results from the ``first reduced, then quantised" approach can be reproduced is identified and the conditions for this to be a good approximation are determined.

Canonical analysis of linearized [formula omitted] gravity plus a Chern–Simons term

The Hamiltonian analysis for the linearized λR gravity plus a Chern–Simons term is performed. The first-class and second-class constraints for arbitrary values of λ are presented, and one physical degree of freedom is reported. The second-class constraints are removed, and the corresponding generalized Dirac brackets are constructed; then, the difference between theories with different values of λ is remarked.

Stückelberg-modified massive Abelian 3-form theory: Constraint analysis, conserved charges and BRST algebra

For the Stückelberg-modified massive Abelian 3-form theory in any arbitrary D-dimension of spacetime, we show that its classical gauge symmetry transformations are generated by the first-class constraints. We establish that the Noether conserved charge (corresponding to the local gauge symmetry transformations) is same as the standard form of the generator for the underlying local gauge symmetry transformations (expressed in terms of the first-class constraints). We promote these classical local, continuous and infinitesimal gauge symmetry transformations to their quantum counterparts Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations which are respected by the coupled (but equivalent) Lagrangian densities. We derive the conserved (anti-)BRST charges by exploiting the theoretical potential of Noether’s theorem. However, these charges turn out to be non-nilpotent. Some of the highlights of our present investigation are (i) the derivation of the off-shell nilpotent versions of the (anti-)BRST charges from the standard non-nilpotent Noether conserved (anti-)BRST charges, (ii) the appearance of the operator forms of the first-class constraints at the quantum level through the physicality criteria with respect to the nilpotent versions of the (anti-)BRST charges, and (iii) the deduction of the Curci–Ferrari-type restrictions from the straightforward equality of the coupled (anti-)BRST invariant Lagrangian densities as well as from the requirement of the absolute anticommutativity of the off-shell nilpotent versions of the conserved (anti-)BRST charges.

Looking for Carroll Particles in the Two-Time Spacetime

We make an attempt to describe Carroll particles with a non-vanishing value of energy (i.e., the Carroll particles which always stay in rest) in the framework of two-time physics, developed in the series of papers by I. Bars and his co-authors. In the spacetime with one additional time dimension and one additional space dimension, where one can localize the symmetry which exists between generalized coordinates and their conjugate momenta. Such a localization implies the introduction of the gauge fields, which, in turn, implies the appearance of some first-class constraints. Choosing different gauge-fixing conditions and solving the constraints, we obtain different time parameters, Hamiltonians, and, generally, physical systems in the standard one-time spacetime. We find a set of gauge fixing conditions which gives the description of a Carroll particle in the one-time world. We construct the quantum theory of such a particle using an unexpected correspondence between our parametrization and that obtained by Bars for the hydrogen atom in 1999.

Relativistic conic motion as a second-class dynamical system

We investigate relativistic motion along a general conic path under the influence of an open potential as a Dirac-Bergmann constrained dynamical model. The system turns out to exhibit a set of four second-class constraints in phase space which we fully explore obtaining a relativistic Poisson algebra generalizing previously known algebraic structures. With a convenient integration factor, the Euler-Lagrange differential equations can be worked out to its general solution in closed form. We perform the canonical quantization in terms of the corresponding Dirac brackets, applying the Dirac-Bergmann algorithm. The complete Dirac brackets algebra in phase space as well as its physical realization in terms of differential operators are explicitly obtained.

Infinite (continuous) spin particle in constant curvature space

We present a new particle model that generalize for constant curvature space an infinite spin particle in flat space. The model is described by commuting Weyl spinor additional coordinates. It proved that such a model is consistent only in external gravitational field corresponding to the constant curvature spaces. Full set of the first-class constraints in the de Sitter and anti-de Sitter spaces is obtained.

Quantization of Yang–Mills theory in de Sitter spacetime

In this paper, we analyze the quantization of Yang–Mills theory in the de Sitter spacetime. It is observed that the Faddeev–Popov ghost propagator is divergent in this spacetime. However, this divergence is removed by using an effective propagator, which is suitable for perturbation theory. To show that the quantization of Yang–Mills theory in the de Sitter is consistent, we quantize it using first-class constraints in the temporal gauge. We also demonstrate that this is equivalent to quantizing the theory in the Lorentz gauge.

Photon quantization in cosmological spaces

Canonical quantization of the photon—a free massless vector field—is considered in cosmological spacetimes in a two-parameter family of linear gauges that treat all the vector potential components on equal footing. The goal is setting up a framework for computing photon two-point functions appropriate for loop computations in realistic inflationary spacetimes. The quantization is implemented without relying on spacetime symmetries, but rather it is based on the classical canonical structure. Special attention is paid to the quantization of the canonical first-class constraint structure that is implemented as the condition on the physical states. This condition gives rise to subsidiary conditions that the photon two-point functions must satisfy. Some of the de Sitter space photon propagators from the literature are found not to satisfy these subsidiary conditions, bringing into question their consistency. Published by the American Physical Society 2024

Hamiltonian analysis for perturbative [formula omitted] gravity

The Hamiltonian analysis for the linearized λR gravity around the Minkowski background is performed. The first-class and second-class constraints for arbitrary values of λ are presented, and two physical degrees of freedom are reported. In addition, we remove the second-class constraints, and the generalized Dirac brackets are constructed; then, the equivalence between General Relativity and the λR theory is shown.

Dirac–Bergmann analysis and degrees of freedom of coincident f(Q)-gravity

We investigate the propagating degrees of freedom of f(Q)-gravity in a 4-dimensional space-time under the imposition of the coincident gauge by performing the Dirac–Bergmann analysis. In this work, we start with a top-down reconstruction of the metric-affine gauge theory of gravity based only on the concept of a vector bundle. Then, the so-called geometrical trinity of gravity is introduced and the role of the coincident GR is clarified. After that, we reveal relationships between the boundary terms in the variational principle and the symplectic structure of the theory in order to confirm the validity of the analysis for our studied theories. Then, as examples, we revisit the analysis of GR and its f(R∘)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f({\\mathop {R}\\limits ^{\\circ }})$$\\end{document}-extensions. Finally, after reviewing the Dirac–Bergmann analysis of the coincident GR and that of f(T)-gravity, we perform the analysis of coincident f(Q)-gravity. Under the imposition of appropriate spatial boundary conditions, we find that, as a generic case, the theory has five primary, three secondary, and two tertiary constraint densities and all these constraint densities are classified into second-class constraint density; the number six is the propagating degrees of freedom of the theory and there are no longer any remaining gauge degrees of freedom. We also discuss the condition of providing seven pDoF as a generic case. The violation of diffeomorphism invariance of coincident f(Q)-gravity make it possible to emerge such several sectors.

Electrodynamics with violations of Lorentz and U(1) gauge symmetries and their Hamiltonian structures* *Supported by the National Key Research and Development Program of China (2020YFC2201503, the Zhejiang Provincial Natural Science Foundation of China (LR21A050001, LY20A050002) and the National Natural Science Foundation of China (12275238, 11675143). B.-F. Li is supported by the National Natural Science Foundation of China (NNSFC) (12005186)

This study aims to investigate Lorentz/U(1) gauge symmetry-breaking electrodynamics in the framework of the standard-model extension and analyze the Hamiltonian structure for the theory with a specific dimension of Lorentz breaking operators. For this purpose, we consider a general quadratic action of the modified electrodynamics with Lorentz/gauge-breaking operators and calculate the number of independent components of the operators at different dimensions in gauge invariance and breaking. With this general action, we then analyze how Lorentz/gauge symmetry-breaking can change the Hamiltonian structure of the theories by considering Lorentz/gauge-breaking operators with dimension as examples. We show that the Lorentz-breaking operators with gauge invariance do not change the classes of the theory constrains and the number of physical degrees of freedom of the standard Maxwell electrodynamics. When U(1) gauge symmetry-breaking operators are present, the theories generally lack a first-class constraint and have one additional physical degree of freedom compared to the standard Maxwell electrodynamics.

Does a second-class primary constraint generate a gauge transformation? Electromagnetisms and gravities, massless and massive

In constrained Hamiltonian dynamics there are two views regarding how first-class constraints generate gauge transformations: individually or only in a certain combination, the Rosenfeld–Anderson–Bergmann–Castellani gauge generator. This gauge generator G preserves Hamilton’s equations and changes the canonical action at most by a boundary term; Hamiltonian’s equations are the Euler–Lagrange equations for the canonical action. Hence the canonical formalism is equivalent to the Lagrangian formalism, and indeed subsumed within it (in important examples) with many canonical momenta serving as auxiliary fields. G generates transformations basically equivalent to the usual 4-dimensional Lagrangian expressions, such as a 4-gradient in electromagnetism or a space–time coordinate transformation (on-shell) in General Relativity. It has been shown recently that separate first-class constraints lead to inequivalent observables between Proca non-gauge and Stueckelberg gauge massive electromagnetism. There is, however, widespread agreement that second-class constraints do not generate gauge transformations.Here it is shown that in such a sense as the first-class primary constraint in Maxwell’s theory generates a gauge transformation, the second-class primary constraint in Proca’s massive electromagnetism also generates a gauge transformation. Likewise the second-class primary constraints in various massive spin 2 relatives of General Relativity generate as much of a gauge transformation as do the corresponding first-class primary constraints in GR. Hence the view that first-class constraints typically generate gauge transformations individually faces a puzzle not faced by the gauge generator view.

Note on an extended chiral bosons system contextualized in a modified gauge-unfixing formalism

We analyze the Hamiltonian structure of an extended chiral bosons theory in which the self-dual constraint is introduced via a control α-parameter. The system has two second-class constraints in the non-critical regime and an additional one in the critical regime. We use a modified gauge-unfixing (GU) formalism to derive a first-class system, disclosing hidden symmetries. To this end, we choose one of the second-class constraints to build a corresponding gauge symmetry generator. The worked out procedure converts second-class variables into first-class ones allowing the lifting of gauge symmetry. Any function of these GU variables will also be invariant. We obtain the GU Hamiltonian and Lagrangian densities in a generalized context containing the Srivastava and Floreanini-Jackiw models as particular cases. Additionally, we observe that the resulting GU Lagrangian presents similarities to the Siegel invariant Lagrangian which is known to be suitable for describing chiral bosons theory with classical gauge invariance, however broken at quantum level. The final results signal a possible equivalence between our invariant Lagrangian obtained from the modified GU formalism and the Siegel invariant Lagrangian, with a distinct gauge symmetry.

Looking for Carroll particles in two time spacetime

We make an attempt to describe Carroll particles with a nonvanishing value of energy (i.e. the Carroll particles which always stay in rest) in the framework of two time physics, developed in the series of papers by I. Bars and his coauthors. In the spacetime with one additional time dimension and one additional space dimension one can localize the symmetry which exists between generalized coordinate and their conjugate momenta. Such a localization implies the introduction of the gauge fields, which in turn implies the appearance of some first-class constraints. Choosing different gauge-fixing conditions and solving the constraints one obtain different time parameters, Hamiltonians, and generally, physical systems in the standard one time spacetime. In this way such systems as nonrelativistic particle, relativistic particles, hydrogen atoms, and harmonic oscillators were described as dual systems in the framework of two time physics. Here, we find a set of gauge fixing conditions which provides as with such a parametrization of the phase-space variables in the two time world which gives the description of Carroll particle in the one time world. Besides, we construct the quantum theory of such a particle using an unexpected correspondence between our parametrization and that obtained by Bars for the hydrogen atom in 1999. Published by the American Physical Society 2024

A note on the Hamiltonian structure of transgression forms

By incorporating two gauge connections, transgression forms provide a generalization of Chern-Simons actions that are genuinely gauge-invariant on bounded manifolds. In this work, we show that, when defined on a manifold with a boundary, the Hamiltonian formulation of a transgression field theory can be consistently carried out without the need to implement regularizing boundary terms at the level of first-class constraints. By considering boundary variations of the relevant functionals in the Poisson brackets, the surface integral in the very definition of a transgression action can be translated into boundary contributions in the generators of gauge transformations and diffeomorphisms. This prescription systematically leads to the corresponding surface charges of the theory, reducing to the general expression for conserved charges in (higher-dimensional) Chern-Simons theories when one of the gauge connections in the transgression form is set to zero.

Gauge-Invariant Lagrangian Formulations for Mixed-Symmetry Higher-Spin Bosonic Fields in AdS Spaces

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux Y(s1,…,sk) with k≥2 rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a k=2 Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic sp(2k) algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the su(1,1) algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for Y(s1,s2) is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin (s1,s2) is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin (1,1) is obtained.

FLPR model: Nilpotent (anti-)co-BRST symmetries

We demonstrate the existence of a set of novel off-shell nilpotent and absolutely anticommuting continuous symmetry transformations, within the framework of the Becchi-Rouet-Stora-Tyutin (BRST) formalism, which are over and above the usual off-shell nilpotent and absolutely anticommuting (anti-)BRST transformations that are respected by the quantum version of the classical first-order Lagrangian for the Friedberg-Lee-Pang-Ren (FLPR) model that describes the motion of a single non-relativistic particle of unit mass (moving under the influence of a general rotationally invariant spatial two-dimensional potential). We christen these novel set of fermionic transformations as the (anti-)co-BRST transformations because the gauge-fixing terms remain invariant under them. We derive the conserved and off-shell nilpotent (anti-)BRST and (anti-)co-BRST charges and comment on the physicality criteria with respect to them where we establish the presence of the operator forms of the first-class constraints (of the original classical gauge theory) at the quantum level which is consistent with the Dirac-quantization conditions.

Constraints, symmetry transformations and conserved charges for massless Abelian 3-form theory

We demonstrate the existence of the first-class constraints on the massless Abelian 3-form theory which generate the classical gauge symmetry transformations for this theory in any arbitrary D-dimension of spacetime. We write down the explicit expression for the generator in terms of these first-class constraints. Using the celebrated Noether theorem, corresponding to the gauge symmetry transformations, we derive the Noether conserved current and conserved charge. The latter is connected with the first-class constraints of the theory in a subtle manner as we demonstrate clearly in our present investigation. We comment on the first-class constraints within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the conserved (anti-)BRST charges are the generalizations of the above generator for the classical gauge symmetry transformation. The standard Noether conserved (anti-)BRST charges are found to be non-nilpotent. We derive the nilpotent versions of the (anti-)BRST charges. One of the interesting observations of our present endeavor is the result that only the nilpotent versions of the conserved (anti-)BRST charges lead to the annihilation of the physical states by the operator form of the first-class constraints at the quantum level which is consistent with the Dirac quantization condition for the systems that are endowed with any kind of constraints. We comment on the existence of the Curci-Ferrari (CF) type restrictions from different theoretical angles, too.

Canonical analysis and modified Faddeev-Jackiw approach for the Jackiw-Teitelboim model in two dimensions

This article presents an analysis of the constraints of the Jackiw-Teitelboim model in two dimensions via the canonical analysis using the Dirac algorithm and modified Faddeev-Jackiw (FJ) approach. The analysis primarily focuses on the identification of constraints, gauge transformations, counting of physical degrees of freedom, and the generalized FJ brackets and Dirac’s brackets. To ensure gauge symmetry within the symplectic formalism and maintain consistency with the Dirac formalism, we employ the Montani-Wotzasek method, which effectively utilizes the zero modes of the symplectic matrix. Additionally, the Poincaré symmetry and diffeomorphisms in the model are identified. Finally, we present the equivalence between the generalized FJ and Dirac brackets when all the second-class constraints are treated as zero equations.