315 research outputs found
Ladder operators for isospectral oscillators
We present, for the isospectral family of oscillator Hamiltonians, a
systematic procedure for constructing raising and lowering operators satisfying
any prescribed `distorted' Heisenberg algebra (including the
-generalization). This is done by means of an operator transformation
implemented by a shift operator. The latter is obtained by solving an
appropriate partial isometry condition in the Hilbert space. Formal
representations of the non-local operators concerned are given in terms of
pseudo-differential operators. Using the new annihilation operators, new
classes of coherent states are constructed for isospectral oscillator
Hamiltonians. The corresponding Fock-Bargmann representations are also
considered, with specific reference to the order of the entire function family
in each case.Comment: 13 page
Bi-Hamiltonian representation of St\"{a}ckel systems
It is shown that a linear separation relations are fundamental objects for
integration by quadratures of St\"{a}ckel separable Liouville integrable
systems (the so-called St\"{a}ckel systems). These relations are further
employed for the classification of St\"{a}ckel systems. Moreover, we prove that
{\em any} St\"{a}ckel separable Liouville integrable system can be lifted to a
bi-Hamiltonian system of Gel'fand-Zakharevich type. In conjunction with other
known result this implies that the existence of bi-Hamiltonian representation
of Liouville integrable systems is a necessary condition for St\"{a}ckel
separability.Comment: To appear in Physical Review
Noncanonical Quantization of Gravity. I. Foundations of Affine Quantum Gravity
The nature of the classical canonical phase-space variables for gravity
suggests that the associated quantum field operators should obey affine
commutation relations rather than canonical commutation relations. Prior to the
introduction of constraints, a primary kinematical representation is derived in
the form of a reproducing kernel and its associated reproducing kernel Hilbert
space. Constraints are introduced following the projection operator method
which involves no gauge fixing, no complicated moduli space, nor any auxiliary
fields. The result, which is only qualitatively sketched in the present paper,
involves another reproducing kernel with which inner products are defined for
the physical Hilbert space and which is obtained through a reduction of the
original reproducing kernel. Several of the steps involved in this general
analysis are illustrated by means of analogous steps applied to one-dimensional
quantum mechanical models. These toy models help in motivating and
understanding the analysis in the case of gravity.Comment: minor changes, LaTeX, 37 pages, no figure
On pseudo-hyperk\"ahler prepotentials
An explicit surjection from a set of (locally defined) unconstrained
holomorphic functions on a certain submanifold of (Sp_1(C) \times C^{4n}) onto
the set HK_{p,q} of local isometry classes of real analytic
pseudo-hyperk\"ahler metrics of signature (4p,4q) in dimension 4n is
constructed. The holomorphic functions, called prepotentials, are analogues of
K\"ahler potentials for K\"ahler metrics and provide a complete
parameterisation of HK_{p,q}. In particular, there exists a bijection between
HK_{p,q} and the set of equivalence classes of prepotentials. This affords the
explicit construction of pseudo-hyperk\"ahler metrics from specified
prepotentials. The construction generalises one due to Galperin, Ivanov,
Ogievetsky and Sokatchev. Their work is given a coordinate-free formulation and
complete, self-contained proofs are provided. An appendix provides a vital tool
for this construction: a reformulation of real analytic G-structures in terms
of holomorphic frame fields on complex manifolds.Comment: 53 pages; v2: minor amendments to Def.4.1 and Theorem 4.5; a
paragraph inserted in the proof of the latter; V3: minor changes; V4: minor
changes/ typos corrected for journal versio
Vector coherent state theory of the generic representations of so(5) in an so(3) basis
For applications of group theory in quantum mechanics, one generally needs
explicit matrix representations of the spectrum generating algebras that arise
in bases that reduce the symmetry group of some Hamiltonian of interest. Here
we use vector coherent state techniques to develop an algorithm for
constructing the matrices for arbitrary finite-dimensional irreps of the SO(5)
Lie algebra in an SO(3) basis. The SO(3) subgroup of SO(5) is defined by
regarding SO(5) as linear transformations of the five-dimensional space of an
SO(3) irrep of angular momentum two. A need for such irreps arises in the
nuclear collective model of quadrupole vibrations and rotations. The algorithm
has been implemented in MAPLE, and some tables of results are presented.Comment: 20 pages, uses multirow.sty, submitted to J. Math. Phy
Conformally invariant wave-equations and massless fields in de Sitter spacetime
Conformally invariant wave equations in de Sitter space, for scalar and
vector fields, are introduced in the present paper. Solutions of their wave
equations and the related two-point functions, in the ambient space notation,
have been calculated. The ``Hilbert'' space structure and the field operator,
in terms of coordinate independent de Sitter plane waves, have been defined.
The construction of the paper is based on the analyticity in the complexified
pseudo-Riemanian manifold, presented first by Bros et al.. Minkowskian limits
of these functions are analyzed. The relation between the ambient space
notation and the intrinsic coordinates is then studied in the final stage.Comment: 21 pages, LaTeX, some details adde
Higher-order-in-spin interaction Hamiltonians for binary black holes from source terms of Kerr geometry in approximate ADM coordinates
The Kerr metric outside the ergosphere is transformed into ADM coordinates up
to the orders and , respectively in radial coordinate and
reduced angular momentum variable , starting from the Kerr solution in
quasi-isotropic as well as harmonic coordinates. The distributional source
terms for the approximate solution are calculated. To leading order in linear
momenta, higher-order-in-spin interaction Hamiltonians for black-hole binaries
are derived.Comment: REVTeX4, 20 pages, typos corrected in Eq. (124) and (130
Partial separability revisited: Necessary and sufficient criteria
We extend the classification of mixed states of quantum systems composed of
arbitrary number of subsystems of arbitrary dimensions. This extended
classification is complete in the sense of partial separability and gives
1+18+1 partial separability classes in the tripartite case contrary to a former
1+8+1. Then we give necessary and sufficient criteria for these classes, which
make it possible to determine to which class a mixed state belongs. These
criteria are given by convex roof extensions of functions defined on pure
states. In the special case of three-qubit systems, we define a different set
of such functions with the help of the Freudenthal triple system approach of
three-qubit entanglement.Comment: v3: 22 pages, 5 tables, 1 figure, minor corrections (typos),
clarification in the Introduction. Accepted in Phys. Rev. A. Comments are
welcom
"Massless" vector field in de Sitter Universe
In the present work the massless vector field in the de Sitter (dS) space has
been quantized. "Massless" is used here by reference to conformal invariance
and propagation on the dS light-cone whereas "massive" refers to those dS
fields which contract at zero curvature unambiguously to massive fields in
Minkowski space. Due to the gauge invariance of the massless vector field, its
covariant quantization requires an indecomposable representation of the de
Sitter group and an indefinite metric quantization. We will work with a
specific gauge fixing which leads to the simplest one among all possible
related Gupta-Bleuler structures. The field operator will be defined with the
help of coordinate independent de Sitter waves (the modes) which are simple to
manipulate and most adapted to group theoretical matters. The physical states
characterized by the divergencelessness condition will for instance be easy to
identify. The whole construction is based on analyticity requirements in the
complexified pseudo-Riemanian manifold for the modes and the two-point
function.Comment: 33 pages, 3 figure
Covariant Affine Integral Quantization(s)
Covariant affine integral quantization of the half-plane is studied and
applied to the motion of a particle on the half-line. We examine the
consequences of different quantizer operators built from weight functions on
the half-plane. To illustrate the procedure, we examine two particular choices
of the weight function, yielding thermal density operators and affine inversion
respectively. The former gives rise to a temperature-dependent probability
distribution on the half-plane whereas the later yields the usual canonical
quantization and a quasi-probability distribution (affine Wigner function)
which is real, marginal in both momentum p and position q.Comment: 36 pages, 10 figure
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