22 research outputs found
Phase Space Quantum Mechanics on the Anti-De Sitter Spacetime and its Poincar\'e Contraction
In this work we propose an alternative description of the quantum mechanics
of a massive and spinning free particle in anti-de~Sitter spacetime, using a
phase space rather than a spacetime representation. The regularizing character
of the curvature appears clearly in connection with a notion of localization in
phase space which is shown to disappear in the zero curvature limit. We show in
particular how the anti-de~Sitter optimally localized (coherent) states
contract to plane waves as the curvature goes to zero. In the first part we
give a detailed description of the classical theory {\it \a la Souriau\/}. This
serves as a basis for the quantum theory which is constructed in the second
part using methods of geometric quantization. The invariant positive K\"ahler
polarization that selects the anti-de~Sitter quantum elementary system is shown
to have as zero curvature limit the Poincar\'e polarization which is no longer
K\"ahler. This phenomenon is then related to the disappearance of the notion of
localization in the zero curvature limit.Comment: 37 pgs+3 figures (not included), PlainTeX, Preprint CRM-183
On the Supersymplectic Homogeneous Superspace Underlying the OSp(1/2) Coherent States
In this work we extend Onofri and Perelomov's coherent states methods to the
recently introduced coherent states. These latter are shown to be
parametrized by points of a supersymplectic supermanifold, namely the
homogeneous superspace , which is clearly identified with a
supercoadjoint orbit of by exhibiting the corresponding equivariant
supermoment map. Moreover, this supermanifold is shown to be a nontrivial
example of Rothstein's supersymplectic supermanifolds. More precisely, we show
that its supersymplectic structure is completely determined in terms of
-invariant (but unrelated) K\"ahler -form and K\"ahler metric on
the unit disc. This result allows us to define the notions of a superK\"ahler
supermanifold and a superK\"ahler superpotential, the geometric structure of
the former being encoded into the latter.Comment: 19 pgs, PlainTeX, Preprint CRM-185
Projectively and conformally invariant star-products
We consider the Poisson algebra S(M) of smooth functions on T^*M which are
fiberwise polynomial. In the case where M is locally projectively (resp.
conformally) flat, we seek the star-products on S(M) which are SL(n+1,R) (resp.
SO(p+1,q+1))-invariant. We prove the existence of such star-products using the
projectively (resp. conformally) equivariant quantization, then prove their
uniqueness, and study their main properties. We finally give an explicit
formula for the canonical projectively invariant star-product.Comment: 37 pages, Latex; minor correction
Geometric Quantization on the Super-Disc
In this article we discuss the geometric quantization on a certain type of
infinite dimensional super-disc. Such systems are quite natural when we analyze
coupled bosons and fermions. The large-N limit of a system like that
corresponds to a certain super-homogeneous space. First, we define an example
of a super-homogeneous manifold: a super-disc. We show that it has a natural
symplectic form, it can be used to introduce classical dynamics once a
Hamiltonian is chosen. Existence of moment maps provide a Poisson realization
of the underlying symmetry super-group. These are the natural operators to
quantize via methods of geometric quantization, and we show that this can be
done.Comment: 17 pages, Latex file. Subject: Mathematical physics, geometric
quantizatio
Large N limit of SO(N) gauge theory of fermions and bosons
In this paper we study the large N_c limit of SO(N_c) gauge theory coupled to
a Majorana field and a real scalar field in 1+1 dimensions extending ideas of
Rajeev. We show that the phase space of the resulting classical theory of
bilinears, which are the mesonic operators of this theory, is OSp_1(H|H
)/U(H_+|H_+), where H|H refers to the underlying complex graded space of
combined one-particle states of fermions and bosons and H_+|H_+ corresponds to
the positive frequency subspace. In the begining to simplify our presentation
we discuss in detail the case with Majorana fermions only (the purely bosonic
case is treated in our earlier work). In the Majorana fermion case the phase
space is given by O_1(H)/U(H_+), where H refers to the complex one-particle
states and H_+ to its positive frequency subspace. The meson spectrum in the
linear approximation again obeys a variant of the 't Hooft equation. The linear
approximation to the boson/fermion coupled case brings an additonal bound state
equation for mesons, which consists of one fermion and one boson, again of the
same form as the well-known 't Hooft equation.Comment: 27 pages, no figure
Supercoherent States, Super K\"ahler Geometry and Geometric Quantization
Generalized coherent states provide a means of connecting square integrable
representations of a semi-simple Lie group with the symplectic geometry of some
of its homogeneous spaces. In the first part of the present work this point of
view is extended to the supersymmetric context, through the study of the
OSp(2/2) coherent states. These are explicitly constructed starting from the
known abstract typical and atypical representations of osp(2/2). Their
underlying geometries turn out to be those of supersymplectic OSp(2/2)
homogeneous spaces. Moment maps identifying the latter with coadjoint orbits of
OSp(2/2) are exhibited via Berezin's symbols. When considered within
Rothstein's general paradigm, these results lead to a natural general
definition of a super K\"ahler supermanifold, the supergeometry of which is
determined in terms of the usual geometry of holomorphic Hermitian vector
bundles over K\"ahler manifolds. In particular, the supergeometry of the above
orbits is interpreted in terms of the geometry of Einstein-Hermitian vector
bundles. In the second part, an extension of the full geometric quantization
procedure is applied to the same coadjoint orbits. Thanks to the super K\"ahler
character of the latter, this procedure leads to explicit super unitary
irreducible representations of OSp(2/2) in super Hilbert spaces of
superholomorphic sections of prequantum bundles of the Kostant type. This work
lays the foundations of a program aimed at classifying Lie supergroups'
coadjoint orbits and their associated irreducible representations, ultimately
leading to harmonic superanalysis. For this purpose a set of consistent
conventions is exhibited.Comment: 53 pages, AMS-LaTeX (or LaTeX+AMSfonts
N=1, D=3 Superanyons, osp(2|2) and the Deformed Heisenberg Algebra
We introduce N=1 supersymmetric generalization of the mechanical system
describing a particle with fractional spin in D=1+2 dimensions and being
classically equivalent to the formulation based on the Dirac monopole two-form.
The model introduced possesses hidden invariance under N=2 Poincar\'e
supergroup with a central charge saturating the BPS bound. At the classical
level the model admits a Hamiltonian formulation with two first class
constraints on the phase space , where the
K\"ahler supermanifold is a minimal
superextension of the Lobachevsky plane. The model is quantized by combining
the geometric quantization on and the Dirac quantization with
respect to the first class constraints. The constructed quantum theory
describes a supersymmetric doublet of fractional spin particles. The space of
quantum superparticle states with a fixed momentum is embedded into the Fock
space of a deformed harmonic oscillator.Comment: 23 pages, Late
On representation of the t-J model via spin-charge variables
We show that the t-J Hamiltonian is not in general reduced to H(S,f), where S
and f stand for independent ([S,f]=0) SU(2) (spin) generators and spinless
fermionic (hole) field, respectively. The proof is based upon an identification
of the Hubbard operators with the generators of the su(2|1) superalgebra in the
degenerate fundamental representation and ensuing SU(2|1) path integral
representation of the partition function.Comment: 15 pages, latex, no figure