Extension of the York field decomposition to general gravitationally coupled fields

We extend the York decomposition analysis of the initial value constraints to general gravitationally coupled classical field theories. The decomposition is found to be particularly useful in solving the constraint equations for all theories of current physical interest. These include Einstein gravity or Einstein-Cartan (torsion) gravity coupled to the massive or massless version of the following: general scalar (including Klein-Gordon, Brans-Dicke, and Higgs), Dirac spin 1/2, Maxwell (Proca) and Yang-Mills (any gauge group). We show in detail how the program works for the general Yang-Mills field and for the Einstein-Cartan-Proca field.     

Generally covariant quantization and the Dirac field

Canonical Hamiltonian field theory in curved spacetime is formulated in a manifestly covariant way. Second quantization is achieved invoking a correspondence principle between the Poisson bracket of classical fields and the commutator of the corresponding quantum operators. The Dirac theory is investigated and it is shown that, in contrast to the case of bosonic fields, in curved spacetime, the field momentum does not coincide with the generators of spacetime translations. The reason is traced back to the presence of second class constraints occurring in Dirac theory. Further, it is shown that the modification of the Dirac Lagrangian by a surface term leads to a momentum transfer between the Dirac field and the gravitational background field, resulting in a theory that is free of constraints, but not manifestly hermitian.     

The Reeh–Schlieder Property for Quantum Fields on Stationary Spacetimes

We show that as soon as a linear quantum field on a stationary spacetime satisfies a certain type of hyperbolic equation, the (quasifree) ground- and KMS-states with respect to the canonical time flow have the Reeh–Schlieder property. We also obtain an analog of Borchers' timelike tube theorem. The class of fields we consider contains the Dirac field, the Klein–Gordon field and the Proca field. Received: 1 March 2000 / Accepted: 30 May 2000     

Boundary conditions as Dirac constraints

In this article we show that boundary conditions can be treated as Lagrangian and Hamiltonian constraints. Using the Dirac method, we find that boundary conditions are equivalent to an infinite chain of second class constraints, which is a new feature in the context of constrained systems. Constructing the Dirac brackets and the reduced phase space structure for different boundary conditions, we show why mode expanding and then quantizing a field theory with boundary conditions is the proper way. We also show that in a quantized field theory subjected to the mixed boundary conditions, the field components are non-commutative. Received: 16 October 2000 / Revised version: 8 January 2001 / Published online: 23 February 2001     

Conversion of second class constraints by deformation of Lagrangian local symmetries

For a theory with first and second class constraints, we propose a procedure for conversion of second class constraints based on deformation the structure of local symmetries of the Lagrangian formulation. It does not require extension or reduction of configuration space of the theory. We give examples in which the initial formulation implies a nonlinear realization of some global symmetries, therefore is not convenient. The conversion reveals hidden symmetry presented in the theory. The extra gauge freedom of conversed version is used to search for a parameterization which linearizes the equations of motion. We apply the above procedure to membrane theory (in the formulation with world-volume metric). In the resulting version, all the metric components are gauge degrees of freedom. The above procedure works also in a theory with only second class constraints presented. As an examples, we discuss arbitrary dynamical system of classical mechanics subject to kinematic constraints, O(N)$O(N)$-invariant nonlinear sigma-model, and the theory of massive vector field with Maxwell–Proca Lagrangian.     

Quantize Classical Field Theories with Boundaries Canonically

We study the problem of quantizing the classical fields with intrinsic second class constraints in a finite volume in this paper. To illustrate our idea clearly, we study the classical Schrodinger field in a finite volume. We work in the discrete version and take the discrete boundary conditions (BCs) as primary Dirac constraints, both Dirichlet and Neumann BCs are considered. We find it is possible to treat the BCs and intrinsic constraints on the same footing.     

Majorana-Oppenheimer Approach to Proca Field Equations

A Dirac-like equation for a massive field obeying the classical Proca equations of motion (PMO) is proposed in close analogy with Majorana’s construct for Maxwell electrodynamics. Its underlying algebraic structure is examined and a plausible physical interpretation is discussed. The behavior of the PMO equations in the presence of an external electromagnetic field is also investigated in the low energy limit, via unitary transformations similar to the Foldy-Wouthuysen canonical transformation for a Dirac fermion.     

On the boundary conditions as Dirac constraints

In this paper, the canonical quantization of singular Lagrangian defined in a finite volume is discussed by studying a 1 + 1 dimensional free Schrödinger field. We take the boundary conditions (BCs) as Dirac constraints, and show that those BCs as well as the intrinsic constraints (which are introduced by the singularities of Lagrangian) form the second class constraints. The quantization is performed canonically.Received: 30 August 2004, Revised: 2 October 2004, Published online: 17 December 2004PACS: 11.25-W, 04.60.D, 11.10.E     

Superspace mechanics as the classical limit of superfield theory

The semi-classical superspace pseudomechanics of Casalbuoni and others is shown to be the classical limit of a scalar superfield theory by a WKB type of approximation. By avoiding second class constraints, the relationship between the mechanics and the field theory is clarified. The spin1/2sector is then examined to find the (semiclassical) WKB limit of the Dirac equation.     

The reduced phase space of an open string in the background B-field

The problem of an open string in background B-field is discussed. Using the discretized model in details we show that the system is influenced by an infinite number of second class constraints. We interpret the allowed Fourier modes as the coordinates of the reduced phase space. This enables us to compute the Dirac brackets more easily. We prove that the coordinates of the string are non-commutative at the boundaries. We argue that in order to find the Dirac bracket or commutator algebra of the physical variables, one should not expand the fields in terms of the solutions of the equations of motion. Instead, one should impose a set of constraints in suitable coordinates. PACS 11.10.Ef, 04.60.Ds     

Generalized Noether theorems in canonical formalism for field theories and their applications

A generalization of Noether's first theorem in phase space for an invariant system with a singular Lagrangian in field theories is derived and a generalization of Noether's second theorem in phase space for a noninvariant system in field theories is deduced. A counterexample is given to show that Dirac's conjecture fails. Some preliminary applications of the generalized Noether second theorem to the gauge field theories are discussed. It is pointed out that for certain systems with a noninvariant Lagrangian in canonical variables for field theories there is also a Dirac constraint. Along the trajectory of motion for a gauge-invariant system some supplementary relations of canonical variables and Lagrange multipliers connected with secondary first-class constraints are obtained.     

Singular mechanics and Landau two-fluid model of superfluidity

We present a theoretical treatment of the Landau two-fluid model of superfluidity in liquid helium by means of the Dirac formalism. We introduce hydrodynamic considerations in a natural way by means of Lagrange multipliers. All constraints in phase space, in Dirac's sense, are second class and, as a consequence, the Dirac bracket differs strongly from the Poisson bracket. We calculate the Dirac bracket of the canonical variables, putting special interest on the density and the momentum density of the system. Our results generalize the results given by Dzyaloshinskii and Volovik and correct other published results.     

Finite BRST transformation and constrained systems

We establish the connection between the generating functional for the first class theories and the generating functional for the second class theories using the finite field dependent BRST (FFBRST) transformation. We show this connection with the help of explicit calculations in two different models. The generating functional of the Proca model is obtained from the generating functional of the Stueckelberg theory for massive spin 1 vector field using FFBRST transformation. In the other example we relate the generating functionals for gauge invariant and gauge variant theories for a self-dual chiral boson.     

Improved Canonical Quantization Method of Self Dual Field

In this paper,the improved canonical quantization method of the self dual field is given in order to overcome linear combination problem about the second class constraint and the first class constraint number maximization problem in the Dirac method.In the improved canonical quantization method,there are no artificial linear combination and the first class constraint number maximization problems,at the same time,the stability of the system is considered.Therefore,the improved canonical quantization method is more natural and easier accepted by people than the usual Dirac method.We use the improved canonical quantization method to realize the canonical quantization of the self dual field,which has relation with string theory successfully and the results are equal to the results by using the Dirac method.     

New realizations of observables in dynamical systems with second class constraints

In the Dirac bracket approach to dynamical systems with second class constraints observables are represented by elements of a quotient Dirac bracket algebra. We describe families of new realizations of this algebra through quotients of the original Poisson bracket algebra. Explicit expressions for generators and brackets of the algebras under consideration are found.     

Hamiltonian formulation of the supermembrane

The hamiltonian formulation of the supermembrane theory in eleven dimensions is given. The covariant split of the first and second class constraints is exhibited, and their Dirac brackets are computed. Gauge conditions are imposed in such a way that the reparametrizations of the membrane with divergence-free 2-vectors are unfixed.