Narrow Results

An effective formalism for quantum constrained systems is presented which allows manageable derivations of solutions and observables, including a treatment of physical reality conditions without requiring full knowledge of the physical inner product. Instead of a state equation from a constraint operator, an infinite system of constraint functions on the quantum phase space of expectation values and moments of states is used. The examples of linear constraints as well as the free non-relativistic particle in parametrized form illustrate how standard problems of constrained systems can be dealt with in this framework.

We show that a nonlinear dynamical system in Poincaré–Dulac normal form (in ℝⁿ) can be seen as a constrained linear system; the constraints are given by the resonance conditions satisfied by the spectrum of (the linear part of) the system and identify a naturally invariant manifold for the flow of the "parent" linear system. The parent system is finite dimensional if the spectrum satisfies only a finite number of resonance conditions, as implied e.g. by the Poincaré condition. In this case our result can be used to integrate resonant normal forms, and sheds light on the geometry behind the classical integration method of Horn, Lyapounov and Dulac.

A new approach treating constrained systems based on the consistency condition of constraints is proposed in this paper. This method is simpler and more rigorous in mathematics. It is not necessary to introduce the concepts of weakly equal, strongly equal, Dirac brackets and Dirac conjecture. The procedure of our method is demonstrated by using Cawley's counterexample to Dirac's conjecture.

Motivated by the Hamilton–Jacobi approach of constrained systems, we analyze the classical structure of a four-dimensional superstring. The equations of motion for a singular system are obtained as total differential equations in many variables. The path integral quantization based on Hamilton–Jacobi approach is applied to quantize the system, and the integration is taken over the canonical phase space coordinates.

The dynamics of a (charged) pseudo-classical spinning particle, that evolves on a curved space, are addressed. Besides the Dirac approach, we give a consistent Hamiltonian analysis of the spinning particle without using the Dirac bracket. The link between our treatment and Dirac formalism is evidenced.

Canonical analysis of constraint structure of Einstein–Hilbert action in (1+1) dimensions possesses a mixed constrained model. By means of finite order BFT approach in the extended phase space, it converts to a fully gauge-invariant model. Embedded Hamiltonian in this extended phase space is independent of momenta similar to the classical Hamiltonian.

In this paper, BFT formalism of Proca model in noncommutative space is investigated. Considering that all theories with first class constraint are gauge theories, Proca model in noncommutative space is not a gauge theory in general due to the appearance of second class constraints in it. In present research, the Proca model is converted into a gauge theory using BFT approach by introducing several auxiliary variables which in turn manage to convert the second class constraints to first class ones. Consequently, we apply modified BFT that preserve the chain structure of constraints. Modified BFT has the benefit that it gives less number of independent gauge parameters and we obtain gauge generating function and infinitesimal gauge variation of fields in Proca model. As results, we investigate partition function of this model and embedded noncommutative Proca is ready to quantize in usual way.

In this paper, the gauge structure of non-Abelian Chern–Simons model is investigated. Non-Abelian Chern–Simons model, unlike first class constraints systems is not a gauge theory, primarily due to the appearance of two second class constraints in its algebra. These two second class constraints were converted into first ones using gauge unfixing formalism. The model Lagrangian and Hamiltonian were obtained, aiming to satisfy first class algebra and hence making it into a fully gauge theory. Partition function of the model was finally evaluated and presented here.

We analyze the constraints of Christ–Lee model by means of modified Faddeev–Jackiw formalism in Cartesian as well as polar coordinates. Further, we accomplish quantization à la Faddeev–Jackiw by choosing appropriate gauge conditions in both the coordinate systems. Finally, we establish gauge symmetries of Christ–Lee model with the help of zero-modes of the symplectic matrix.

We apply the Batalin–Fradkin–Fradkina–Tyutin formalism to a prototypical second-class system, aiming to convert its constraints from second class to first class. The proposed system admits a consistent initial set of second-class constraints and an open potential function providing room for feasible applications to field theory and mechanical models. The constraints can be arbitrarily nonlinear, broadly generalizing previously known cases. We obtain a sufficient condition for which a simple closed expression for the Abelian converted constraints and modified involutive Hamiltonian can be achieved. As an explicit example, we discuss a spontaneous Lorentz symmetry breaking vectorial model, obtaining the full first-class Abelianized constraints in closed form and the corresponding involutive Hamiltonian.

In this paper, we quantize the Friedberg–Lee–Pang–Ren (FLPR) model, using an admissible gauge condition, within the framework of modified Faddeev–Jackiw formalism. Further, we deduce the gauge symmetries and establish off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetries. We also show that the physical states of the theory are annihilated by the first class constraints which is consistent à la Dirac formalism.

In this work we derive the Hamiltonian formalism of the O(N) nonlinear sigma model in its original version as a second-class constrained field theory and then as a first-class constrained field theory. We treat the model as a second-class constrained field theory by two different methods: the unconstrained and the Dirac second-class formalisms. We show that the Hamiltonians for all these versions of the model are equivalent. Then, for a particular factor-ordering choice, we write the functional Schrödinger equation for each derived Hamiltonian. We show that they are all identical which justifies our factor-ordering choice and opens the way for a future quantization of the model via the functional Schrödinger representation.

We propose a new procedure to embed second class systems by introducing Wess–Zumino (WZ) fields in order to unveil hidden symmetries existent in the models. This formalism is based on the direct imposition that the new Hamiltonian must be invariant by gauge-symmetry transformations. An interesting feature in this approach is the possibility to find a representation for the WZ fields in a convenient way, which leads to preserve the gauge symmetry in the original phase space. Consequently, the gauge-invariant Hamiltonian can be written only in terms of the original phase-space variables. In this situation, the WZ variables are only auxiliary tools that permit to reveal the hidden symmetries present in the original second class model. We apply this formalism to important physical models: the reduced-SU(2) Skyrme model, the Chern–Simons–Proca quantum mechanics and the chiral bosons field theory. In all these systems, the gauge-invariant Hamiltonians are derived in a very simple way.

One approach for solving mechanical problems of constrained systems using the Hamilton–Jacobi formulation is examined. The Hamilton–Jacobi function is obtained in the same manner as for regular systems. This is used to determine the solutions of the equations of motion for constrained systems.

The aim of the present paper is to describe the symmetry structure of a general gauge (singular) theory, and, in particular, to relate the structure of gauge transformations with the constraint structure of a theory in the Hamiltonian formulation. We demonstrate that the symmetry structure of a theory action can be completely revealed by solving the so-called symmetry equation. We develop a corresponding constructive procedure of solving the symmetry equation with the help of a special orthogonal basis for the constraints. Thus, we succeed in describing all the gauge transformations of a given action. We find the gauge charge as a decomposition in the orthogonal constraint basis. Thus, we establish a relation between the constraint structure of a theory and the structure of its gauge transformations. In particular, we demonstrate that, in the general case, the gauge charge cannot be constructed with the help of some complete set of first-class constraints alone, because the charge decomposition also contains second-class constraints. The above-mentioned procedure of solving the symmetry equation allows us to describe the structure of an arbitrary symmetry for a general singular action. Finally, using the revealed structure of an arbitrary gauge symmetry, we give a rigorous proof of the equivalence of two definitions of physicality condition in gauge theories: one of them states that physical functions are gauge-invariant on the extremals, and the other requires that physical functions commute with FCC (the Dirac conjecture).

The relation between conformal generators and Magueijo–Smolin Doubly Special Relativity term, is achieved. Through a dimensional reduction procedure, it is demonstrated that a massless relativistic particle living in a d-dimensional space, is isomorphic to the one living in a d+2 space with pure Lorentz invariance and to a particle living in a AdS_d+1 space. To accomplish these identifications, the conformal group is extended and a nonlinear algebra is obtained. Finally, because the relation between momenta and velocities is known through the dimensional reduction procedure, the problem of position space dynamics is solved.

We extend the Shirafuji model for massless particles with primary space–time coordinates and composite four-momenta to a model for massive particles with spin and electric charge. The primary variables in the model are the space–time four-vector, four scalars describing spin and charge degrees of freedom as well as a pair of Weyl spinors. The geometric description proposed in this paper provides an intermediate step between the free purely twistorial model in two-twistor space in which both space–time and four-momenta vectors are composite, and the standard particle model, where both space–time and four-momenta vectors are elementary. We quantize the model and find explicitly the first-quantized wave functions describing relativistic particles with mass, spin and electric charge. The space–time coordinates in the model are not commutative; this leads to a wave function that depends only on one covariant projection of the space–time four-vector (covariantized time coordinate) defining plane wave solutions.

Time boundary terms usually added to action principles are systematically handled in the framework of Dirac's canonical analysis. The procedure begins with the introduction of the boundary term into the integral Hamiltonian action and then the resulting action is interpreted as a Lagrangian one to which Dirac's method is applied. Once the general theory is developed, the current procedure is implemented and illustrated in various examples which are originally endowed with different types of constraints.

The energy eigenvalues of the quantum particle constrained in a surface of the sphere of D dimensions embedded in a R^D+1 space are obtained by using two different procedures: in the first, we derive the Hamiltonian operator by squaring the expression of the momentum, written in Cartesian components, which satisfies the Dirac brackets between the canonical operators of this second-class system. We use the Weyl ordering prescription to construct the Hermitian operators. When D=2 we verify that there is no constant parameter in the expression of the eigenvalues energy, a result that is in agreement with the fact that an extra term would change the level spacings in the hydrogen atom; in the second procedure it is adopted the non-Abelian BFFT formalism to convert the second-class constraints into first-class ones. The non-Abelian first-class Hamiltonian operator is symmetrized by also using the Weyl ordering rule. We observe that their energy eigenvalues differ from a constant parameter when we compare with the second-class system. Thus, a conversion of the D-dimensional sphere second-class system for a first-class one does not reproduce the same values.

There is a review of the main mathematical properties of system described by singular Lagrangians and requiring Dirac–Bergmann theory of constraints at the Hamiltonian level. The following aspects are discussed:

• ⁽ⁱ⁾the connection of the rank and eigenvalues of the Hessian matrix in the Euler–Lagrange equations with the chains of first- and second-class constraints;
• ⁽ⁱⁱ⁾the connection of the Noether identities of the second Noether theorem with the Hamiltonian constraints;
• ⁽ⁱⁱⁱ⁾the Shanmugadhasan canonical transformation for the identification of the gauge variables and for the search of the Dirac observables, i.e. the quantities invariant under Hamiltonian gauge transformations.