Physical Fock Space Research Articles

A gauge theory of quantum gravity: quantization of the non-interacting field

The non-interacting field belonging to a new SO(1,3) gauge field theory equivalent to general relativity is canonically quantized in the Lorentz gauge and the physical Fock space for non-interacting gauge particles is constructed. To assure both Lorentz covariance and positivity of the norm and energy expectation value for physical states restrictions needed in the construction of the physical Fock space are put consequentially on state vectors, and not on the algebra of creation and annihilation operators as usual—alltogether providing the second step in consistently quantizing gravitation

TENSIONLESS STRINGS: PHYSICAL FOCK SPACE AND HIGHER SPIN FIELDS

We study the physical Fock space of the tensionless string theory with perimeter action, exploring its new gauge symmetry algebra. The cancellation of conformal anomaly requires the space–time to be 13-dimensional. All particles are massless and there are no tachyon states in the spectrum. The zero mode conformal operator defines the levels of the physical Fock space. All levels can be classified by the highest Casimir operator W of the little group E(11) for massless particles in 11-dimensions. The ground state is infinitely degenerated and contains massless gauge fields of arbitrary large integer spin, realizing the irreducible representations of E(11) of fixed helicity. The excitation levels realize CSR representations of little group E(11) with an infinite number of helicities. After inspection of the first excitation level, which, as we prove, is a physical null state, we conjecture that all excitation levels are physical null states. In this theory the tensor field of the second rank does not play any distinctive role and therefore one can suggest that in this model there is no gravity.

Finite Chern-Simons matrix model - algebraic approach

We analyze the algebra of observables and the physical Fock space of the finite Chern-Simons matrix model. We observe that the minimal algebra of observables acting on that Fock space is identical to that of the Calogero model. Our main result is the identification of the states in the l-th tower of the Chern-Simons matrix model Fock space and the states of the Calogero model with the interaction parameter nu=l+1. We describe quasiparticle and quasihole states in the both models in terms of Schur functions, and discuss some nontrivial consequences of our algebraic approach.

Quantization of the electromagnetic field on static space–times

A quantization of the electromagnetic four-potential in the Lorentz gauge on static space–times with compact Cauchy surfaces is developed herein. For the classical problem, the field solution is obtained by use of a Hilbert space approach in which time evolution reduces to a unitary mapping of Cauchy data. For the quantum problem, a representation of the CCRs on a Fock space is constructed. The gauge condition is imposed as a constraint on state vectors and determines a physical subspace. The field equations are recovered on the physical Fock space (Gupta–Bleuler method).

Infinite abelian subalgebra of W(sl( n))

A representation theoretical construction of the conservation laws of affine Toda type systems is described. The construction employs the completely degenerate representations of the extended conformal algebras W(sl( n)). The conserved charges are shown to generate an infinite-dimensional abelian subalgebra of W(sl( n)). Different characterizations of this subalgebra are obtained: As space of physical Fock space operators with dihedral symmetry, as constants of commuting flows of quantum KdV-type equations and as subalgebra of the sl( n) singlets in affine sl ( n) level-1 modules. The existence of the subalgebras is established for low-rank cases by means of an algorithmic Fock space procedure.

ABELIZED VIRASORO OPERATORS, GEOMETRY AND THE STRING VERTEX OPERATOR

In this article, we review recent work on constrained Hamiltonian systems with infinite-dimensional constraint algebras. The work is based on an extension of the constraint algebra to include subsidiary (gauge-fixing) constraints and a central element, forming a larger, closed algebra. This extension leads to several immediate results, at both a practical and a conceptual level: 1) A simple method is found for abelizing the original constraints. 2) This leads naturally to a “generalized vielbein” relating Abelian and non-Abelian quantities. 3) The vielbein apparatus makes transparent the full global symmetry of the extended phase space associated with the Batalin, Fradkin and Vilkovisky formalism. An interesting duality also arises between quantities with “Abelian” and “non-Abelian” indices. String theory provides the most important and straightforward application of these general methods. For this special case, one finds that the vielbeins are the Fourier components of the string vertex operator. The vertex operator describes the physical process of photon emission, and determines the physical modes of the string by way of the DDF operators. Due to the relation between the vertex operator and the vielbein, the dual abelized Virasoro operators simplify the usual discussion of the physical Fock space. The correspondence between these two fundamental objects, the vertex and the vielbein, thus provides a geometric understanding of the usual algebraic approach to string theory.

Indefinite metric and other-than-Feynman propagators as tools in a relativistic two-body problem

By embedding the usual physical Fock space P of a simple field theory in a larger non-Hilbert representation space R, a formalism is obtained permitting the use of Δ̄ propagators instead of the Feynman propagators ΔF to represent the internal particle lines in Feynman graphs. The new formalism is subsequently applied to a Bethe–Salpeter equation in the g2 approximation of its kernel and an exact solution of the latter is obtained exhibiting much simpler analytical features compared to its standard counterpart—the so-called Wick equation.

Composite particles in nonrelativistic many-body theory: Foundations and statistical mechanics

A new fundamental theory of composite particles in nonrelativistic many-body systems is developed. The theory is constructed making use solely of the physical Fock space for the basic elementary particles which comprise the many-body system. In this manner exchange symmetry in the elementary particles is exact. Physical composite particle creation and annihilation operators are introduced and these operators satisfy exact Bose (Fermi) commutation (anticommutati$\stackrel{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}{\mathrm{o}}$n) relations depending on whether the composites correspond to bosons or to fermions. Commuting physical occupation number operators for composite particles are then constructed in the usual manner from the composite creation and annihilation operators. These number operators are highly dressed in terms of the basic elementary particle operators. Finally, the foundations of statistical mechanics for bound composite particles are formulated in terms of the appropriate occupation number operators.

The Problem of P+ = 0 Mode in the Null-Plane Field Theory and Dirac's Method of Quantization

The null-plane quantization is studied with the emphasis on the P+=O mode, by using Dirac's quantization for constrained systems. This mode is eliminated from the Hilbert space and the physical vacuum can be defined in a kinematical way. It enables us to con­ struct the physical Fock space kinematically. Poincare invariance is also studied in detail.

Equal-Time Commutation Relations for Heisenberg Fields in the Lee Model

>~ 4 > has been successfully applied to problems in both high energy physics5> and the area of many-body problems, particularly in superconductivity 6> and ferromagnetism. 7> In this paper we apply this method to the solvable Dirac­ Lee model which, unlike the pair model considered in Ref. 3), has mass and coupling constant renormalizations. The equal-time commutation relations among Heisenberg fields are derived and not assumed. Further we show that the wave­ function renormalization constant is determined from microcausality. This result sheds light on the question why in Ref. 3) the existence of the~ bound state was found to be closely connected with microcausality. The self-consistent method exploits the duality between basic fields and observed particles concretely exhibited in the presence of interactions. The basic Heisenberg fields obey nonlinear field equations which reflect the laws of nature, while the physical particles are the ones that appear in observations. These physical fields obey free field equations and define the physical Fock space. The field equations for the Heisenberg fields are to be regarded as operator equations in this physical Fock space. The Heisenberg field is related to the physical field by means of a mapping known as the dynamical map.4> The set ?f physical fields is taken to form an irreducible operator ring. Therefore, any operator of the Fock space can be written as a sum of normal products of physical flelds. The ability of this method to predict new particles rests on this postulate. The asymptotic condition may tell us that the physical fields are not complete. The completion of this set of fields then requires the existence of other particles at the level of observation and should be incorporated. 8>·8> The success of this method therefore rests on the proper choice of the set of physical fields.