313 research outputs found
How `hot' are mixed quantum states?
Given a mixed quantum state of a qudit, we consider any observable
as a kind of `thermometer' in the following sense. Given a source which emits
pure states with these or those distributions, we select such distributions
that the appropriate average value of the observable is equal to the
average Tr of in the stare . Among those distributions we find
the most typical one, namely, having the highest differential entropy. We call
this distribution conditional Gibbs ensemble as it turns out to be a Gibbs
distribution characterized by a temperature-like parameter . The
expressions establishing the liaisons between the density operator and
its temperature parameter are provided. Within this approach, the
uniform mixed state has the highest `temperature', which tends to zero as the
state in question approaches to a pure state.Comment: Contribution to Quantum 2006: III workshop ad memoriam of Carlo
Novero: Advances in Foundations of Quantum Mechanics and Quantum Information
with atoms and photons. 2-5 May 2006 - Turin, Ital
On a novel integrable generalization of the nonlinear Schr\"odinger equation
We consider an integrable generalization of the nonlinear Schr\"odinger (NLS)
equation that was recently derived by one of the authors using bi-Hamiltonian
methods. This equation is related to the NLS equation in the same way that the
Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use
the bi-Hamiltonian structure to write down the first few conservation laws. (b)
Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d)
Analyze solitons.Comment: 20 pages, 1 figur
Space-time Structures from Critical Values in 2D Quantum Gravity
A model for 2D Quantum Gravity is constructed out of the Virasoro group. To
this end the quantization of the abstract Virasoro group is revisited. For the
critical values of the conformal anomaly c, some quantum operators (SL(2,R)
generators) lose their dynamical content (they are no longer conjugated
operators). The notion of space-time itself in 2D gravity then arises as
associated with this kinematical SL(2,R) symmetry. An ensemble of different
copies of AdS do co-exist in this model with different weights, depending on
their curvature (which is proportional to \hbar^{2}) and they are connected by
gravity operators. This model suggests that, in general, quantum diffemorphisms
should not be imposed as constraints to the theory, except for the classical
limit.Comment: 22 pages, latex, no figures. Revised version with an effort in the
development of the underlying classical theory and the clarification of the
classical limit. To appear in Class. Quant. Gra
"Square Root" of the Proca Equation: Spin-3/2 Field Equation
New equations describing particles with spin 3/2 are derived. The non-local
equation with the unique mass can be considered as "square root" of the Proca
equation in the same sense as the Dirac equation is related to the
Klein-Gordon-Fock equation. The local equation describes spin 3/2 particles
with three mass states. The equations considered involve fields with spin-3/2
and spin-1/2, i.e. multi-spin 1/2, 3/2. The projection operators extracting
states with definite energy, spin, and spin projections are obtained. All
independent solutions of the local equation are expressed through projection
matrices. The first order relativistic wave equation in the 20-dimensional
matrix form, the relativistically invariant bilinear form and the corresponding
Lagrangian are given. Two parameters characterizing non-minimal electromagnetic
interactions of fermions are introduced, and the quantum-mechanical Hamiltonian
is found. It is proved that there is only causal propagation of waves in the
approach considered.Comment: 17 pages, corrections in Eqs. (50), (51
Spherical functions on the de Sitter group
Matrix elements and spherical functions of irreducible representations of the
de Sitter group are studied on the various homogeneous spaces of this group. It
is shown that a universal covering of the de Sitter group gives rise to
quaternion Euler angles. An explicit form of Casimir and Laplace-Beltrami
operators on the homogeneous spaces is given. Different expressions of the
matrix elements and spherical functions are given in terms of multiple
hypergeometric functions both for finite-dimensional and unitary
representations of the principal series of the de Sitter group.Comment: 40 page
A Lorentzian Signature Model for Quantum General Relativity
We give a relativistic spin network model for quantum gravity based on the
Lorentz group and its q-deformation, the Quantum Lorentz Algebra.
We propose a combinatorial model for the path integral given by an integral
over suitable representations of this algebra. This generalises the state sum
models for the case of the four-dimensional rotation group previously studied
in gr-qc/9709028.
As a technical tool, formulae for the evaluation of relativistic spin
networks for the Lorentz group are developed, with some simple examples which
show that the evaluation is finite in interesting cases. We conjecture that the
`10J' symbol needed in our model has a finite value.Comment: 22 pages, latex, amsfonts, Xypic. Version 3: improved presentation.
Version 2 is a major revision with explicit formulae included for the
evaluation of relativistic spin networks and the computation of examples
which have finite value
A Class of Coupled KdV systems and Their Bi-Hamiltonian Formulations
A Hamiltonian pair with arbitrary constants is proposed and thus a sort of
hereditary operators is resulted. All the corresponding systems of evolution
equations possess local bi-Hamiltonian formulation and a special choice of the
systems leads to the KdV hierarchy. Illustrative examples are given.Comment: 8 pages, late
Rigorous steps towards holography in asymptotically flat spacetimes
Scalar QFT on the boundary at null infinity of a general
asymptotically flat 4D spacetime is constructed using the algebraic approach
based on Weyl algebra associated to a BMS-invariant symplectic form. The
constructed theory is invariant under a suitable unitary representation of the
BMS group with manifest meaning when the fields are interpreted as suitable
extensions to of massless minimally coupled fields propagating in the
bulk. The analysis of the found unitary BMS representation proves that such a
field on coincides with the natural wave function constructed out of
the unitary BMS irreducible representation induced from the little group
, the semidirect product between SO(2) and the two dimensional
translational group. The result proposes a natural criterion to solve the long
standing problem of the topology of BMS group. Indeed the found natural
correspondence of quantum field theories holds only if the BMS group is
equipped with the nuclear topology rejecting instead the Hilbert one.
Eventually some theorems towards a holographic description on of QFT in
the bulk are established at level of algebras of fields for strongly
asymptotically predictable spacetimes. It is proved that preservation of a
certain symplectic form implies the existence of an injective -homomorphism
from the Weyl algebra of fields of the bulk into that associated with the
boundary . Those results are, in particular, applied to 4D Minkowski
spacetime where a nice interplay between Poincar\'e invariance in the bulk and
BMS invariance on the boundary at is established at level of QFT. It
arises that the -homomorphism admits unitary implementation and Minkowski
vacuum is mapped into the BMS invariant vacuum on .Comment: 62 pages, amslatex, xy package; revised section 2 and the
conclusions; corrected some typos; added some references; accepted for
pubblication on Rev. Math. Phy
Solutions of Podolsky's Electrodynamics Equation in the First-Order Formalism
The Podolsky generalized electrodynamics with higher derivatives is
formulated in the first-order formalism. The first-order relativistic wave
equation in the 20-dimensional matrix form is derived. We prove that the
matrices of the equation obey the Petiau-Duffin-Kemmer algebra. The
Hermitianizing matrix and Lagrangian in the first-order formalism are given.
The projection operators extracting solutions of field equations for states
with definite energy-momentum and spin projections are obtained, and we find
the density matrix for the massive state. The -matrix Schrodinger
form of the equation is derived, and the Hamiltonian is obtained. Projection
operators extracting the physical eigenvalues of the Hamiltonian are found.Comment: 17 pages, minor corrections, published versio
Abelian Gauge Theory in de Sitter Space
Quantization of spinor and vector free fields in 4-dimensional de Sitter
space-time, in the ambient space notation, has been studied in the previous
works. Various two-points functions for the above fields are presented in this
paper. The interaction between the spinor field and the vector field is then
studied by the abelian gauge theory. The U(1) gauge invariant spinor field
equation is obtained in a coordinate independent way notation and their
corresponding conserved currents are computed. The solution of the field
equation is obtained by use of the perturbation method in terms of the Green's
function. The null curvature limit is discussed in the final stage.Comment: 10 pages, typos corrected, reference adde
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