Second-class Systems Research Articles

Quantization of Constrained Systems as Dirac First Class versus Second Class: A Toy Model and Its Implications

A toy model (suggested by Klauder) was analyzed from the perspective of first-class and second-class Dirac constrained systems. First-class constraints are often associated with the existence of important gauge symmetries in a system. A comparison was made by turning a first-class system into a second-class system with the introduction of suitable auxiliary conditions. The links between Dirac’s system of constraints, the Faddeev–Popov canonical functional integral method and the Maskawa–Nakajima procedure for reducing the phase space are explicitly illustrated. The model reveals stark contrasts and physically distinguishable results between first and second-class routes. Physically relevant systems such as the relativistic point particle and electrodynamics are briefly recapped. Besides its pedagogical value, the article also advocates the route of rendering first-class systems into second-class systems prior to quantization. Second-class systems lead to a well-defined reduced phase space and physical observables; an absence of inconsistencies in the closure of quantum constraint algebra; and the consistent promotion of fundamental Dirac brackets to quantum commutators. As first-class systems can be turned into well-defined second-class ones, this has implications for the soundness of the “Dirac quantization” of first-class constrained systems by the simple promotion of Poisson brackets, rather than Dirac brackets, to commutators without proceeding through second-class procedures.

BRST analysis and BFV quantization of the generalized quantum rigid rotor

We identify a strong similarity among several distinct originally second-class systems, including both mechanical and field theory models, which can be naturally described in a gauge-invariant way. The canonical structure of such related systems is encoded into a gauge-invariant generalization of the quantum rigid rotor. We perform the BRST symmetry analysis and the BFV functional quantization for the mentioned gauge-invariant version of the generalized quantum rigid rotor. We obtain different equivalent effective actions according to specific gauge-fixing choices, showing explicitly their BRST symmetries. We apply and exemplify the ideas discussed to two particular models, namely, motion along an elliptical path and the [Formula: see text] nonlinear sigma model, showing that our results reproduce and connect previously unrelated known gauge-invariant systems.

First-Order Gauge-Invariant Generalization of the Quantum Rigid Rotor

A first-order gauge-invariant formulation for the two-dimensional quantum rigid rotor is long known in the theoretical physics community as an isolated peculiar model. Parallel to that fact, the longstanding constraints abelianization problem, aiming at the conversion from second- to first-class systems for quantization purposes, has been approached a number of times in the literature with a handful of different forms and techniques and still continues to be a source of lively and interesting discussions. Connecting these two points, we develop a new systematic method for converting second-class systems to first-class ones, valid for a family of systems encompassing the quantum rigid rotor as a special case. In particular, the gauge invariance of the quantum rigid rotor is fully clarified and generalized in the context of arbitrary translations along the radial momentum direction. Our method differs substantially from previous ones as it does not rely neither on the introduction of new auxiliary variables nor on the a priori interpretation of the second-class constraints as coming from a gauge-fixing process.

Obtaining gauge invariant actions via symplectic embedding formalism

AbstractThe concept of gauge invariance is one of the most subtle and useful concepts in modern theoretical physics. It is one of the Standard Model cornerstones. The main benefit due to the gauge invariance is that it can permit the comprehension of difficult systems in physics with an arbitrary choice of a reference frame at every instant of time. It is the objective of this work to show a path of obtaining gauge invariant theories from non‐invariant ones. Both are named also as first‐ and second‐class theories respectively, obeying Dirac's formalism. Namely, it is very important to understand why it is always desirable to have a bridge between gauge invariant and non‐invariant theories. Once established, this kind of mapping between first‐class (gauge invariant) and second‐class systems, in Dirac's formalism can be considered as a sort of equivalence. This work describe this kind of equivalence obtaining a gauge invariant theory starting with a non‐invariant one using the symplectic embedding formalism developed by some of us some years back. To illustrate the procedure it was analyzed both Abelian and non‐Abelian theories. It was demonstrated that this method is more convenient than others. For example, it was shown exactly that this embedding method used here does not require any special modification to handle with non‐Abelian systems.

Fractional Dirac bracket and quantization for constrained systems

So far, it is not well known how to deal with dissipative systems. There are many paths of investigation in the literature and none of them present a systematic and general procedure to tackle the problem. On the other hand, it is well known that the fractional formalism is a powerful alternative when treating dissipative problems. In this paper, we propose a detailed way of attacking the issue using fractional calculus to construct an extension of the Dirac brackets in order to carry out the quantization of nonconservative theories through the standard canonical way. We believe that, by using the extended Dirac bracket definition, it will be possible to analyze more deeply gauge theories starting with second-class systems.

UNDERLYING GAUGE SYMMETRIES OF SECOND-CLASS CONSTRAINTS SYSTEMS

Gauge-invariant systems in unconstrained configuration and phase spaces, equivalent to second-class constraints systems upon a gauge-fixing, are discussed. A mathematical pendulum on an (n-1)-dimensional sphere Sn-1 as an example of a mechanical second-class constraints system and the O(n) nonlinear sigma model as an example of a field theory under second-class constraints are discussed in details and quantized using the existence of underlying dilatation gauge symmetry and by solving the constraint equations explicitly. The underlying gauge symmetries involve, in general, velocity dependent gauge transformations and new auxiliary variables in extended configuration space. Systems under second-class holonomic constraints have gauge-invariant counterparts within original configuration and phase spaces. The Dirac's supplementary conditions for wave functions of first-class constraints systems are formulated in terms of the Wigner functions which admit, as we show, a broad set of physically equivalent supplementary conditions. Their concrete form depends on the manner the Wigner functions are extrapolated from the constraint submanifolds into the whole phase space.

Non-Abelian conversion and quantization of nonscalar second-class constraints

We propose a general method for deformation quantization of any second-class constrained system on a symplectic manifold. The constraints determining an arbitrary constraint surface are in general defined only locally and can be components of a section of a non-trivial vector bundle over the phase-space manifold. The covariance of the construction with respect to the change of the constraint basis is provided by introducing a connection in the ``constraint bundle'', which becomes a key ingredient of the conversion procedure for the non-scalar constraints. Unlike in the case of scalar second-class constraints, no Abelian conversion is possible in general. Within the BRST framework, a systematic procedure is worked out for converting non-scalar second-class constraints into non-Abelian first-class ones. The BRST-extended system is quantized, yielding an explicitly covariant quantization of the original system. An important feature of second-class systems with non-scalar constraints is that the appropriately generalized Dirac bracket satisfies the Jacobi identity only on the constraint surface. At the quantum level, this results in a weakly associative star-product on the phase space.

Operatorial quantization of the Born-Infeld Skyrmion model and hidden symmetries

The SU(2) collective coordinates expansion of the Born-Infeld\break Skyrmion Lagrangian is performed. The classical Hamiltonian is computed from this special Lagrangian in approximative way: it is derived from the expansion of this non-polynomial Lagrangian up to second-order variable in the collective coordinates. This second-class constrained model is quantized by Dirac Hamiltonian method and symplectic formalism. Although it is not expected to find symmetries on second-class systems, a hidden symmetry is disclosed by formulating the Born-Infeld Skyrmion %model as a gauge theory. To this end we developed a new constraint conversion technique based on the symplectic formalism. Finally, a discussion on the role played by the hidden symmetry on the computation of the energy spectrum is presented.

BFT EMBEDDING OF SECOND-CLASS SYSTEMS

The embedding procedure of Batalin, Fradkin, and Tyutin, which allows us to convert a second-class system into a first-class one, is pushed beyond the formal level. We study nonrelativistic as well as relativistic systems. We explicitly construct, in all cases, the variables of the converted first-class theory in terms of those of the corresponding second-class one. Moreover, we only conclude about the equivalence between these two different kinds of theories after comparing their respective spectra of excitations.

Hamiltonian embedding of the massive Yang-Mills theory and the generalized Stückelberg formalism

Using the general notions of Batalin, Fradkin, Fradkina and Tyutin to convert second-class systems into first-class ones, we present a gauge-invariant formulation of the massive Yang-Mills theory by embedding it in an extended phase space. The infinite set of correction terms necessary for obtaining the involutive constraints and Hamiltonian is explicitly computed and expressed in a closed form. It is also shown that the extra fields introduced in the correction terms are exactly identified with the auxiliary scalars used in the generalized Stückelberg formalism for converting a gauge non-invariant Lagrangian into a gauge-invariant form.

Extravariables in the BRST quantization of second-class constrained systems. Existence theorems

In this paper we show how the BRST quantization can be applied to systems possessing only second-class constraints through their conversion to some first-class ones starting with our method exposed in a previous paper [Nucl. Phys. B 456 (1995) 473]. Thus, it is proved that (i) for a certain class of second-class systems there exists a standard coupling between the variables of the original phase space and some extravariables such that we can transform the original system into a one-parameter family of first-class systems; (ii) the BRST quantization of this family in a standard gauge leads to the same path integral as that of the original system. The analysis is accomplished in both reducible and irreducible cases. At the same time, the Lagrangian action of the first-class family is obtained and its provenance is clarified. In this context, the Wess-Zumino action is also derived. The results from the theoretical part of the paper are exemplified in detail for the massive Yang-Mills theory and for massive abelian three-form gauge fields.

Batalin-Fradkin quantization of anomalous chiral gauge theories.

We quantize two-dimensional anomalous chiral gauge theories by following the Batalin-Fradkin prescription for converting second-class systems to first-class ones. Because of the chiral anomaly, the classical Poisson brackets of the Batalin-Fradkin scheme must be replaced by anomalous Poisson brackets in the fermionic formulation. With respect to these brackets the first-class constraints, the BRST charge, and the unitarizing Hamiltonian are explicitly constructed. The partition function is computed for different gauge choices. The unitary gauge and the conventional Faddeev-Popov-like gauges are analyzed in detail. In the latter gauges we obtain generalized Wess-Zumino-type actions.