Fock Representation Research Articles

DUALITY IN INFINITE DIMENSIONAL FOCK REPRESENTATIONS

We construct and study in detail various dual pairs acting on some infinite dimensional Fock representations between a finite dimensional classical Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit decompositions of a Fock representation into a direct sum of irreducible isotypic subspaces with respect to the action of a dual pair, present explicit formulas for the highest weight vectors and calculate the corresponding highest weights. We further outline applications of these dual pairs to the study of tensor products of modules of such an infinite dimensional Lie algebra.

Polynomial Weyl Representations

For the canonical commutation relations in infinite dimensions, we offer an explicit direct construction of Weyl representations generated from the Fock representation by polynomial transformations of arbitrary degree, solving a problem posed by Proksch, Reents and Summers. Our solution employs new approaches to Hilbert–Schmidt polynomials and their Wick ordering.

New Bosonic Fock Representations of Affine-Virasoro Algebras**The project partly supported by Mathematical Center of State Education Commission of China

New bosonic Fock representations of the Virasoro algebra with nonzero centres are explicitly presented. Based on these, new Fock representations of affine-Virasoro algebras are constructed.

Noncoherent states having Poissonian statistics

A class of noncoherent states (NCS) with the same statistical properties of the coherent state (CS) is investigated. Contrarily to the CS, the NCS exhibits nonclassical properties which come from the difference between off-diagonal terms ρ n, n′ of the Fock's representation for the CS and the NCS. The phase-distribution and Wigner function for the NCS are also discussed.

Exact light-cone wavefunction representation of matrix elements of electroweak currents

The matrix elements of electroweak currents which occur in exclusive decays of heavy hadrons are evaluated in the non-perturbative light-cone Fock representation. In general, each semileptonic exclusive decay amplitude receives two contributions, a diagonal Δn = 0 parton-number-conserving amplitude and a Δn = −2 contribution in which a qaurk and an antiquark from the initial hadron Fock state annihilate to the leptonic current. The general formalism can be used as a basis for systematic approximations to heavy hadron decay amplitudes such as hard perturbative QCD contributions. We illustrate the general formalism using a simple perturbative model of composite hadrons. Our analysis demonstrates the occurrence of “zero-mode” endpoint contributions to matrix elements of the “bad” j − currents in the Drell-Yan frame when q + → 0.

Mixing and decay constants of pseudoscalar mesons: the sequel

We present further tests and applications of the new η– η′ mixing scheme recently proposed by us. The particle states are decomposed into orthonormal basis vectors in a light-cone Fock representation. Because of flavor symmetry breaking the mixing of the decay constants can be identical to the mixing of particle states at most for a specific choice of this basis. Theoretical and phenomenological considerations show that the quark flavor basis has this property and allows, therefore, for a reduction of the number of mixing parameters. A detailed comparison with other mixing schemes is also presented.

Field-theoretical approach to quantum mechanics in curved spacetimes

A general covariant formulation of quantum mechanics of a particle in the general Riemannian spacetime V1,3, which is based on asymptotics in c-1 of the one-particle sector of quantum field theory (QFT) and on the Born probabilistic interpretation of the resulting wavefunctions, is developed in the normal geodesic frame of reference. The operators of the momentum and the spatial position of the particle produced by the corresponding objects of QFT include relativistic corrections, and therefore differ generally from those which are postulated in canonical quantization of geodesic motion. In particular, the operators of coordinates do not commute as well as the operators of the conjugate momenta, except for the case of the Cartesian coordinates in the Minkowski spacetime or the exact non-relativistic limit c-1 = 0. The presented theory can be applied to study relativistic quantum-mechanical effects in the external gravitational field, provides the Fock representation in QFT in the general V1,3 with a particle interpretation and demonstrates a natural alternative to the canonical quantization of mechanics of a particle in a dynamically non-trivial case.

Strong anticommutativity of dirac operators on boson—fermion fock spaces and representations of a supersymmetry algebra

AbstractOn the boson—fermion Fock space F(H, K) associated with two complex Hilbert spaces H and K, there exists a family {Qs | S ∈ C(H, K)} of Dirac type operators, where C(H, K) is the set of densely defined closed linear operators from H to K. A theorem on the strong anticommutativity of two Dirac operators QS and QT is established. As an application, representations on F(H, K) of a supersymmetry algebra arising in a two—dimensional relativistic supersymmetric quantum field theory are discussed.

A description of the quantum superalgebraUq[sl(n+1|m)] via creation and annihilation generators

A description of the quantum superalgebra $U_q[sl(n+1|m)]$ and in particular of the special linear superalgebra $sl(n+1|m)$ via creation and annihilation generators (CAGs) is given. It provides an alternative to the canonical description of $U_q[sl(n+1|m)]$ in terms of Chevalley generators. A conjecture that the Fock representations of the CAGs provide microscopic realizations of exclusion statistics is formulated.

Crystal bases of quantum affine algebras and affine Kazhdan-Lusztig polynomials

We present a fast version of the algorithm of Lascoux, Leclerc, and Thibon for the lower global crystal base for the Fock representation of quantum affine sl_n. We also show that the coefficients of the lower global crystal base coincide with certain affine Kazhdan-Lusztig polynomials. It is known that the coefficients of the global crystal base are q-analogues of decomposition numbers for Specht modules of the Hecke algebra of type A_n, and that the coefficients of the affine Kazhdan-Lusztig polynomials are q-analogues of decomposition numbers for tilting modules for quantum sl_k. Thus our algorithm allows fast computation of these decomposition numbers.

Quantization of pure gravitational plane waves

Pure gravitational plane waves are considered as a special case of spacetimes with two commuting spacelike Killing vector fields. Starting with a midisuperspace that describes this kind of spacetimes, we introduce gauge-fixing and symmetry conditions that remove all non-physical degrees of freedom and ensure that the classical solutions are plane waves. In this way, we arrive at a reduced model with no constraints and whose only degrees of freedom are given by two fields. In a suitable coordinate system, the reduced Hamiltonian that generates the time evolution of this model turns out to vanish, so that all relevant information is contained in the symplectic structure. We calculate this symplectic structure and particularize our discussion to the case of linearly polarized plane waves. The reduced phase space can then be described by an infinite set of annihilation and creation like variables. We finally quantize the linearly polarized model by introducing a Fock representation for these variables.

Deforming maps for Lie group covariant creation and annihilation operators

Any deformation of a Weyl or Clifford algebra 𝒜 can be realized through a “deforming map,” i.e., a formal change of generators in 𝒜. This is true in particular if 𝒜 is covariant under a Lie algebra g and its deformation is induced by some triangular deformation Uhg of the Hopf algebra Ug. We propose a systematic method to construct all the corresponding deforming maps, together with the corresponding realizations of the action of Uhg. The method is then generalized and explicitly applied to the case that Uhg is the quantum group Uhsl(2). A preliminary study of the status of deforming maps at the representation level shows in particular that “deformed” Fock representations induced by a compact Uhg can be interpreted as standard “undeformed” Fock representations describing particles with ordinary Bose or Fermi statistics.

Boundary Exchange Algebras and Scattering on the Half Line

Some algebraic aspects of field quantization in space-time with boundaries are discussed. We introduce an associative algebra, whose exchange properties are inferred from the scattering processes in integrable models with reflecting boundary conditions on the half line. The basic properties of this algebra are established and the Fock representations associated with certain involutions are derived. We apply these results for the construction of quantum fields and for the study of scattering on the half line.

Irreducible representations of the Heisenberg algebra in Krein spaces

The representations of the Heisenberg algebra in Krein spaces, more generally in weakly complete inner product spaces, are classified under general regularity and irreducibility conditions. Besides the Fock representation, two other types appear; one with negative, the other with a two-sided discrete spectrum of the number operator.

Hyperspherical Symmetry of Hydrogenic Orbitals and Recoupling Coefficients among Alternative Bases

Fock's representation of momentum space hydrogenic orbitals in terms of harmonics on the hypersphere ${S}^{3}$ of a four-dimensional space is extended to classify alternative bases. These orbitals are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. Because of the correspondence between the ${S}^{3}$ manifold and the $\mathrm{SU}(2)$ group, new sum rules are established which are of relevance for the connection, not only among hydrogen atom orbitals in different bases, but also among the usual vector coupling coefficients and rotation matrix elements.

Quantum spin dynamics (QSD)

An anomaly-free operator corresponding to the Wheeler - DeWitt constraint of Lorentzian, four-dimensional, canonical, non-perturbative vacuum gravity is constructed in the continuum. This operator is entirely free of factor-ordering singularities and can be defined in symmetric and non-symmetric form. We work in the real connection representation and obtain a well defined quantum theory. The action of the Wheeler - DeWitt constraint on spin-network states is by annihilating, creating and rerouting the quanta of angular momentum associated with the edges of the underlying graph while the ADM energy is essentially diagonalized by the spin-network states. We argue that the spin-network representation is the `nonlinear Fock representation' of quantum gravity, thus justifying the term `quantum spin dynamics (QSD)'. This paper is the first in a series of seven papers with the title `quantum spin dynamics (QSD)'.

Functional Representations for Fock Superalgebras

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.

Quantum-state reconstruction in the one-atom maser

We propose a reconstruction scheme for the quantum-mechanical state of a field inside a microwave cavity in the Fock representation. It will be shown that two experimental steps are required for this purpose: (i) the quantum state under consideration has to be shifted in phase space, and (ii) it has to be examined by state-selective detection of two-level atoms after a resonant interaction with the field. The method is examined in a computer simulation including measurement errors.

Unitary implementations of one-parameter squeezing groups

For the case of infinitely many photon (Boson) modes we investigate the unitary implementability of a class of symplectic one-parameter groups (more exactly, of the associated groups of Bogoliubov automorphisms on the CCR algebra) in the Fock representation and in representations of the CCR algebra, which are symplectically related and inequivalent to Fock. Furthermore, the existence of the associated (squeezing) quadratic Hamiltonians is discussed. Finally, applications in the theory of quantization in QED and in the Luttinger model are pointed out.

QUANTUM SYMMETRIES AND THERMAL FIELD DYNAMICS

In Thermal Field Dynamics, thermal states are obtained from restrictions of vacuum states on a doubled field algebra. It is shown that the suitably doubled Fock representations of the Heisenberg algebra do not need to be introduced by hand but can be canonically handed down from deformations of the extended Heisenberg bialgebra. The relationship between quantum symmetries and doublings is discussed.