Representation Of Field Operators Research Articles

Noncanonical quantization of gravity. II. Constraints and the physical Hilbert space

The program of quantizing the gravitational field with the help of affine field variables is continued. For completeness, a review of the selection criteria that singles out the affine fields, the alternative treatment of constraints, and the choice of the initial (before imposition of the constraints) ultralocal representation of the field operators is initially presented. As analogous examples demonstrate, the introduction and enforcement of the gravitational constraints will cause sufficient changes in the operator representations so that all vestiges of the initial ultralocal field operator representation disappear. To achieve this introduction and enforcement of the constraints, a well characterized phase space functional integral representation for the reproducing kernel of a suitably regularized physical Hilbert space is developed and extensively analyzed.

Ultralocal fields and their relevance for reparametrization-invariant quantum field theory

Reparametrization invariant theories have a vanishing Hamiltonian and enforce their dynamics through a constraint. We specifically choose the Dirac procedure of quantization before the introduction of constraints. Consequently, for field theories, and prior to the introduction of any constraints, it is argued that the original field operator representation should be ultralocal in order to remain totally unbiased toward those field correlations that will be imposed by the constraints. It is shown that relativistic free and interacting theories can be completely recovered starting from ultralocal representations followed by a careful enforcement of the appropriate constraints. In so doing all unnecessary features of the original ultralocal representation disappear. The present discussion is germane to a recent theory of affine quantum gravity in which ultralocal field representations have been invoked before the imposition of constraints.

Local non-abelian gauge symmetry of the Hubbard model

It is shown that the field operators of an electron system on a lattice can be decomposed into direct products of two kinds of operators acting in two separate Hilbert spaces. The Hilbert space of electron states thus becomes a direct product of two Hilbert spaces. By this fact a certain class of electron systems exhibits a formal separation of charge and spin degrees of freedom into two kinds of elementary excitations. A typical example of such a system is given by the Hubbard model. The separation of charge and spin resulting from the new representation of the field operators can be considered as a rigorous realization and generalization of an idea expressed by Anderson concerning the separation of spin and charge degrees of freedom in strongly correlated electron systems. The new representation of electron field operators implies the existence of a localU(2) gauge symmetry in the theory. The theory of superconductivity based on the Hubbard model is then represented by a non-abelian gauge field theory.

Negative binomial states of the field-operator representation and production by state reduction in optical processes.

Properties of the radiation fields in negative binomial states are investigated. The density matrix of the field is related to the density matrix ${\mathrm{\ensuremath{\rho}}}_{\mathit{c}}$ of the chaotic field via ${\mathit{a}}^{\mathit{s}}$${\mathrm{\ensuremath{\rho}}}_{\mathit{c}}$${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathit{s}}$. Various quasiprobability distributions and the thermofield representation for negative binomial states of the field are derived. The production of negative binomial distribution in a number of nonlinear processes is demonstrated.

Feynman path-integral representation of field operators and memory superoperators in a Liouville space.

The Feynman path-integral representation for memory superoperators is investigated. A physical interpretation of evolution superoperators in a Liouville space is given and shown to be closely related to the Feynman representation of quantum mechanics. From a general Huygens-like principle it is possible to obtain the Feynman path-integral representation for field operators. An application of the present formalism to a noninteracting many-boson system is provided.

Time-dependent mean field S-matrix theory

By using the path integral approach to many-body systems, we formulate a time-dependent mean field S-matrix theory of nuclear reactions. Many-body channel eigenstates are constructed by using projection techniques. In this way the S-matrix between the channel eigenstates is expressed as a superposition of S-matrix elements between wave-packet-like states localized in space and time. A field operator representation of the interaction picture S-matrix is derived which enables one to apply the path integral approach. Applying the stationary phase approximation to the path integral representation of the interaction picture S-matrix between the localized states an asymptotically constant time-dependent mean field approximation to this S-matrix is obtained. Finally, the S-matrix between the projected channel eigenstates is obtained by evaluating the integral, arising from the projections, over the space-time positions of the localized states in the stationary phase approximation. The stationary phase conditions select those localized states from the projected channel states for which the mean field values of energy and momentum coincide with their corresponding channel eigenvalues.

On the determination of the field operator representation in canonical model quantum field theory

In a canonical quantum field theory,i.e. based on the canonical commutation relations, the choice of an adequate field operator representation compatible with an assumed model of interaction is quite difficult to construct, unless the so-called vacuum expectation functional of the fieldE(f,g) = 〈Ω|U†(f,g)|Ω〉 is known. Here |Ω〉 is the vacuum state of the representation,U(f, g) is the Weyl operator of a neutral scalar field. In this paper a determination ofE(f, g) in a quartic self-interaction model is proposed as the invariant, positive and normalized function under the action of a system of partial differential functional operators, which form a representation of the Poincare-group generators.