Dirac Brackets Research Articles

Alternative symplectic structures in second–class constrained systems

The classical description of second-class constrained systems using alternative pairs, comprising a symplectic structure and a primary Hamiltonian other than the canonical ones, is investigated. It is exhibited that the dynamical equations do not fix a unique pair and instead there may exist infinitely many of them whose difference is not exhausted by canonical transformations. The consequences on the symplectic structure of the reduced phase space are addressed, namely the effect of different pairs on the construction of Dirac bracket is explored. Since constrained systems can also be studied via the Faddeev–Jackiw symplectic analysis, the implementation of alternative pairs on this formalism is considered as well. It is shown that Dirac’s algorithm outcomes are independent of the type of alternative pairs considered here irrespectively to whether there are also first-class constraints in the system. Finally, some comments on the implications of our results in the path integral quantisation scheme are pointed out.

Basic Notions of Poisson and Symplectic Geometry in Local Coordinates, with Applications to Hamiltonian Systems

This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of the Dirac bracket on a symplectic manifold and provide a proof of the Jacobi identity on a Poisson manifold. A number of applications of the Dirac bracket are described: applications for proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of the reduction of a Hamiltonian system with known integrals of motion.

Spontaneous symmetry breaking in models with second-class constraints

In this work the spontaneous symmetry breaking in certain nonlinear theories with second-class constraints is explored. Using the Dirac's method we perform an analysis of the constraints and the counting of the degrees of freedom. The corresponding effective Hamiltonian is constructed explicitly. It is shown that on the surfaces where the effective Hamiltonian takes critical values the symplectic structure becomes degenerate. In particular, we demonstrate that under the condition of spontaneous symmetry breaking, which implies non trivial vacuum surfaces, second-class constraints behave as first-class ones on certain regions of the phase space, leading to undefined Dirac's brackets and to the modification of the number of degrees of freedom. As a physical consequence, these models can suffer from certain pathologies such as the existence of modes with an acausal propagation. Concrete examples where this phenomena occurs are described in detail.

Classical and quantum dynamics of a constrained particle on three-dimensional spaces of constant curvature: An algebraic approach on the superintegrable problem

In this study, we discuss a particle which is the constraint on a four-dimensional Euclidean space [Formula: see text]. In this respect and under the framework of Dirac’s approach, we analyze the classical and quantum dynamics of a particle constrained on an embedded surface in supersphere [Formula: see text], which has constant curvature and signature equal to [Formula: see text] and [Formula: see text], respectively. In fact, it is presented that the Cartesian components of momentum and position exhibit a proper description for the motion of the particle embedded in a generalized curved space. Thus, quantization of dynamics for a particle on the hypersurface [Formula: see text] derives an appropriate representation of momentum called geometric momentum as it depends on the mean curvature. The invariance algebra generated by the quantization process is an extension of [Formula: see text] with curvature parameters in both classical and quantum dynamics. Moreover, superintegrability appears in the deformed Hamiltonian [Formula: see text] arising from the Dirac brackets, in which [Formula: see text] of the underlying space behaves as the curvature (deformation) parameter.

On a geometric description of time-dependent singular Lagrangians with applications to biological systems

In this paper, we consider certain analytical features of a stochastic model that can explain competition among species and simultaneous predation on the competing species from a geometric perspective. This allows us to build a systematic description of models admitting singular Lagrangians. The model equations are shown to admit a Jacobi Last Multiplier which allows us to construct an appropriate Lagrangian. Due to the singular nature of the Lagrangian, the Hamiltonian formalism may be shown to exist in a submanifold of the cotangent space under certain minimal regularity conditions. In this communication, the Hamiltonian description of the model is constructed via the introduction of Dirac brackets and explicit results for the “Kill the winner” model and its reductions are presented.

Deformation quantization for systems with second-class constraints in deformed fermionic phase space

In order to quantize systems involving second-class constraints, one should use Dirac bracket instead of Poisson bracket. Furthermore, one can specify a star product in which the term linear in [Formula: see text] is proportional to the Dirac bracket. In this way an oscillator system in a deformed fermionic phase space is analyzed and the corresponding energy level and Wigner functions are evaluated according to scheme of deformation quantization. We also study the entanglement entropy induced by the deformation of the fermionic phase space.

Singular Lagrangians, Constrained Hamiltonian Systems and Gauge Invariance: An Example of the Dirac–Bergmann Algorithm

The Dirac–Bergmann algorithm is a recipe for converting a theory with a singular Lagrangian into a constrained Hamiltonian system. Constrained Hamiltonian systems include gauge theories—general relativity, electromagnetism, Yang–Mills, string theory, etc. The Dirac–Bergmann algorithm is elegant but at the same time rather complicated. It consists of a large number of logical steps linked together by a subtle chain of reasoning. Examples of the Dirac–Bergmann algorithm found in the literature are designed to isolate and illustrate just one or two of those logical steps. In this paper, I analyze a finite-dimensional system that exhibits all of the major steps in the algorithm. The system includes primary and secondary constraints, first and second class constraints, restrictions on Lagrange multipliers, and both physical and gauge degrees of freedom. This relatively simple system provides a platform for discussing the Dirac conjecture, constructing Dirac brackets, and applying gauge conditions.

Spinning particles, coadjoint orbits and Hamiltonian formalism

The extensive analysis of the dynamics of relativistic spinning particles is presented. Using the coadjoint orbits method the Hamiltonian dynamics is explicitly described. The main technical tool is the factorization of general Lorentz transformation into pure boost and rotation. The equivalent constrained dynamics on Poincare group (viewed as configuration space) is derived and complete classification of constraints is performed. It is shown that the first class constraints generate local symmetry corresponding to the stability subgroup of some point on coadjoint orbit. The Dirac brackets for second class constraints are computed. Finally, canonical quantization is performed leading to infinitesimal form of irreducible representations of Poincare group.

Dirac Procedure and the Hamiltonian Formalism for Cosmological Perturbations in a Bianchi I Universe

We apply the Dirac procedure for constrained systems to the Arnowitt-Deser-Misner formalism linearized around the Bianchi I universe. We discuss and employ basic concepts such as Dirac observables, Dirac brackets, gauge-fixing conditions, reduced phase space, physical Hamiltonian, canonical isomorphism between different gauge-fixing surfaces and spacetime reconstruction. We relate this approach to the gauge-fixing procedure for non-perturbative canonical relativity. We discuss the issue of propagating a basis for the scalar-vector-tensor decomposition as, in an anisotropic universe, the wavefronts of plane waves undergo a nontrivial evolution. We show that the definition of a gravitational wave as a traceless-transverse mode of the metric perturbation needs to be revised. Moreover there exist coordinate systems in which a polarization mode of the gravitational wave is given entirely in terms of a scalar metric perturbation. We first develop the formalism for the universe with a single scalar field and then extend it to the multi-field case. The obtained fully canonical formalism will serve as a starting point for a complete quantization of the cosmological perturbations and the cosmological background.

Barnich–Troessaert bracket as a Dirac bracket on the covariant phase space

The Barnich–Troessaert bracket is a proposal for a modified Poisson bracket on the covariant phase space for general relativity. The new bracket allows us to compute charges, which are otherwise not integrable. Yet there is a catch. There is a clear prescription for how to evaluate the new bracket for any such charge, but little is known how to extend the bracket to the entire phase space. This is a problem, because not every gravitational observable is also a charge. In this paper, we propose such an extension. The basic idea is to remove the radiative data from the covariant phase space. This requires second-class constraints. Given a few basic assumptions, we show that the resulting Dirac bracket on the constraint surface is nothing but the BT bracket. A heuristic argument is given to show that the resulting constraint surface can only contain gravitational edge modes.

Gauge fixing and constrained dynamics

We review the Dirac formalism for dealing with constraints in a canonical Hamiltonian formulation, discuss gauge freedom, and display constraints for gauge theories in a general context. We introduce the Dirac bracket and show that it provides a consistent method to remove any gauge freedom present. We discuss stability in evolution of gauge theories and show that fixing all gauge freedom is sufficient to ensure well-posedness for a large class of gauge theories. Electrodynamics provides examples of the methods outlined for general gauge theories. Future work will apply the formalism, and results derived here, to General Relativity.

Dirac procedure and the Hamiltonian formalism for cosmological perturbations in a Bianchi I universe

We apply the Dirac procedure for constrained systems to the Arnowitt–Deser–Misner formalism linearized around the Bianchi I universe. We discuss and employ basic concepts such as Dirac observables, Dirac brackets, gauge-fixing conditions, reduced phase space, physical Hamiltonian, canonical isomorphism between different gauge-fixing surfaces and spacetime reconstruction. We relate this approach to the gauge-fixing procedure for non-perturbative canonical relativity. We discuss the issue of propagating a basis for the scalar-vector-tensor decomposition as, in an anisotropic universe, the wavefronts of plane waves undergo a nontrivial evolution. We show that the definition of a gravitational wave as a traceless-transverse mode of the metric perturbation needs to be revised. Moreover there exist coordinate systems in which a polarization mode of the gravitational wave is given entirely in terms of a scalar metric perturbation. We first develop the formalism for the universe with a single minimally coupled scalar field and then extend it to the multi-field case. The obtained fully canonical formalism will serve as a starting point for a complete quantization of the cosmological perturbations and the cosmological background.

Dirac’s analysis and Ostrogradskii’s theorem for a class of second-order degenerate Lagrangians

This paper analyzes the constraint structure of a class of degenerate second-order particle Lagrangian that includes chiral oscillator, noncommutative oscillator, and two examples from reduced topologically massive gravity. For even-dimensional configuration spaces with maximal nondegeneracy, Dirac bracket is defined solely by coefficient field of highest derivative whereas for odd dimensions almost all fields may contribute. Ostrogradskii’s theorem on energy instability is discussed. Results of Dirac analysis are used to identify ghost degrees of freedom. Translational symmetries are used to construct first-order variational formalisms for oscillator examples, thereby making them ghost-free.

Quantum cosmology in f(Q) theory

We use Dirac’s method for the quantization of constrained systems in order to quantize a spatially flat Friedmann–Lemaître–Robertson–Walker spacetime in the context of f(Q) cosmology. When the coincident gauge is considered, the resulting minisuperspace system possesses second class constraints. This distinguishes the quantization process from the typical Wheeler–DeWitt quantization, which is applied for cosmological models where only first class constraints are present (e.g. for models in general relativity or in f(R) gravity). We introduce the Dirac brackets, find appropriate canonical coordinates and then apply the canonical quantization procedure. We perform this method both in vacuum and in the presence of matter: a minimally coupled scalar field and a perfect fluid with a linear equation of state. We demonstrate that the matter content changes significantly the quantization procedure, with the perfect fluid even requiring to put in use the theory of fractional quantum mechanics in which the power of the momentum in the Hamiltonian is associated with the fractal dimension of a Lévy flight. The results of this analysis can be applied in f(T) teleparallel cosmology, since f(Q) and f(T) theories have the same degrees of freedom and same dynamical constraints in cosmological studies.

The relativistic spherical top as a massive twistor

We prove the equivalence between two traditional approaches to the classical mechanics of a massive spinning particle in special relativity. One is the spherical top model of Hanson and Regge, recast in a Hamiltonian formulation with improved treatment of covariant spin constraints. The other is the massive twistor model, slightly generalized to incorporate the Regge trajectory relating the mass to the total spin angular momentum. We establish the equivalence by computing the Dirac brackets of the physical phase space carrying three translation and three rotation degrees of freedom. Lorentz covariance and little group covariance uniquely determine the structure of the physical phase space. The Regge trajectory does not affect the phase space but enters the equations of motion. Upon quantization, the twistor model produces a spectrum that agrees perfectly with the massive spinor-helicity description proposed by Arkani-Hamed, Huang and Huang for scattering amplitudes for all masses and spins.

Gravitational SL(2, \u211d) algebra on the light cone

In a region with a boundary, the gravitational phase space consists of radiative modes in the interior and edge modes at the boundary. Such edge modes are necessary to explain how the region couples to its environment. In this paper, we characterise the edge modes and radiative modes on a null surface for the tetradic Palatini-Holst action. Our starting point is the definition of the action and its boundary terms. We choose the least restrictive boundary conditions possible. The fixed boundary data consists of the radiative modes alone (two degrees of freedom per point). All other boundary fields are dynamical. We introduce the covariant phase space and explain how the Holst term alters the boundary symmetries. To infer the Poisson brackets among Dirac observables, we define an auxiliary phase space, where the SL(2, ℝ) symmetries of the boundary fields are manifest. We identify the gauge generators and second-class constraints that remove the auxiliary variables. All gauge generators are at most quadratic in the fundamental SL(2, ℝ) variables on phase space. We compute the Dirac bracket and identify the Dirac observables on the light cone. Finally, we discuss various truncations to quantise the system in an effective way.

Toverline{T} -deformed fermionic theories revisited

We revisit Toverline{T} deformations of d = 2 theories with fermions with a view toward the quantization. As a simple illustration, we compute the deformed Dirac bracket for a Majorana doublet and confirm the known eigenvalue flows perturbatively. We mostly consider those Toverline{T} theories that can be reconstructed from string-like theories upon integrating out the worldsheet metric. After a quick overview of how this works when we add NSR-like or GS-like fermions, we obtain a known non-supersymmetric Toverline{T} deformation of a mathcal{N} = (1, 1) theory from the latter, based on the Noether energy-momentum. This world- sheet reconstruction implies that the latter is actually a supersymmetric subsector of a d = 3 GS-like model, implying hidden supercharges, which we do construct explicitly. This brings us to ask about different Toverline{T} deformations, such as manifestly supersymmetric Toverline{T} and also more generally via the symmetric energy-momentum. We show that, for theories with fermions, such choices often lead us to doubling of degrees of freedom, with potential unitarity issues. We show that the extra sector develops a divergent gap in the “small deformation” limit and decouples in the infrared, although it remains uncertain in what sense these can be considered a deformation.

Faddeev–Jackiw quantization of the higher derivative Chern–Simons gauge theory in the first-order formalism

We analyse the physical constraints of the higher derivative Chern–Simons gauge model by means of Faddeev–Jackiw symplectic approach in the first-order formalism. Within such framework, we systematically determine the zero-mode structure of the corresponding symplectic matrix. In addition, we calculate the Faddeev–Jackiw quantum brackets by choosing appropriate gauge-fixing conditions and evaluate the determinant of the non-singular symplectic matrix as well as the transition-amplitude. Finally, we present a detailed Hamiltonian analysis using Dirac–Bergmann algorithm method and show that the Dirac brackets coincide with the FJ brackets when all the second-class constraints are treated as zero equations.

A canonical Hamiltonian analysis of Podolsky’s generalized electrodynamics in first-order formalism

We give a canonical Hamiltonian analysis of Podolsky’s generalized electrodynamics by introducing two sets of new variables which help us transform the Lagrangian into an equivalent first-order formalism. After eliminating the unphysical sector, we calculate the physical degrees of freedom of the higher derivative system and obtain the Dirac brackets in the reduced phase space. Then with the aid of the first-class constraints, we construct the independent gauge generator which is closely connected with the BRST charge and the BRST-invariant Hamiltonian. Finally, by choosing appropriate gauge-fixing fermion, we evaluate the path integral of this higher derivative constrained system in BRST quantization scheme with the generalized Lorenz gauge condition.

BFV quantization of the nonprojectable ( 2+1 )-dimensional Hořava theory

We show that the Batalin-Fradkin-Vilkovisky (BFV) quantization scheme can be implemented in the nonprojectable ($2+1$)-Ho\ifmmode \check{r}\else \v{r}\fi{}ava theory. This opens the possibility of imposing more general gauge conditions in the quantization of this theory. The BFV quantization is based on the canonical formalism, which is suitable to incorporate the measure associated to the second-class constraints that the theory has. Special features of the Hamiltonian density and the matrix of second-class constraints allow that the system be involutive in terms of Dirac brackets, which is a nontrivial requisite for implementing the BFV formalism. We present the Becchi-Rouet-Stora-Tyutin symmetry transformations in the canonical variables. The theory is of rank one in the classification introduced by Fradkin and Fradkina. The originally called relativistic gauge-fixing conditions of the BFV formalism can be implemented in the nonprojectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava theory, extended to nonrelativistic forms. We show that the nonlocal gauge condition introduced in the projectable theory can be included among these gauges.