62 research outputs found
Wigner Oscillators, Twisted Hopf Algebras and Second Quantization
By correctly identifying the role of central extension in the centrally
extended Heisenberg algebra h, we show that it is indeed possible to construct
a Hopf algebraic structure on the corresponding enveloping algebra U(h) and
eventually deform it through Drinfeld twist. This Hopf algebraic structure and
its deformed version U^F(h) are shown to be induced from a more fundamental
Hopf algebra obtained from the Schroedinger field/oscillator algebra and its
deformed version, provided that the fields/oscillators are regarded as
odd-elements of the super-algebra osp(1|2n). We also discuss the possible
implications in the context of quantum statistics.Comment: 23 page
An Unfolded Quantization for Twisted Hopf Algebras
In this talk I discuss a recently developed "Unfolded Quantization
Framework". It allows to introduce a Hamiltonian Second Quantization based on a
Hopf algebra endowed with a coproduct satisfying, for the Hamiltonian, the
physical requirement of being a primitive element. The scheme can be applied to
theories deformed via a Drinfeld twist. I discuss in particular two cases: the
abelian twist deformation of a rotationally invariant nonrelativistic Quantum
Mechanics (the twist induces a standard noncommutativity) and the Jordanian
twist of the harmonic oscillator. In the latter case the twist induces a Snyder
non-commutativity for the space-coordinates, with a pseudo-Hermitian deformed
Hamiltonian. The "Unfolded Quantization Framework" unambiguously fixes the
non-additive effective interactions in the multi-particle sector of the
deformed quantum theory. The statistics of the particles is preserved even in
the presence of a deformation.Comment: 9 pages. Talk given at QTS7 (7th Int. Conf. on Quantum Theory and
Symmetries, Prague, August 2011
On the Construction and the Structure of Off-Shell Supermultiplet Quotients
Recent efforts to classify representations of supersymmetry with no central
charge have focused on supermultiplets that are aptly depicted by Adinkras,
wherein every supersymmetry generator transforms each component field into
precisely one other component field or its derivative. Herein, we study
gauge-quotients of direct sums of Adinkras by a supersymmetric image of another
Adinkra and thus solve a puzzle from Ref.[2]: The so-defined supermultiplets do
not produce Adinkras but more general types of supermultiplets, each depicted
as a connected network of Adinkras. Iterating this gauge-quotient construction
then yields an indefinite sequence of ever larger supermultiplets, reminiscent
of Weyl's construction that is known to produce all finite-dimensional unitary
representations in Lie algebras.Comment: 20 pages, revised to clarify the problem addressed and solve
Inequivalent -graded brackets, -bit parastatistics and statistical transmutations of supersymmetric quantum mechanics
Given an associative ring of -graded operators, the number of
inequivalent brackets of Lie-type which are compatible with the grading and
satisfy graded Jacobi identities is . This follows
from the Rittenberg-Wyler and Scheunert analysis of "color" Lie (super)algebras
which is revisited here in terms of Boolean logic gates. The inequivalent
brackets, recovered from mappings, are
defined by consistent sets of commutators/anticommutators describing particles
accommodated into an -bit parastatistics (ordinary bosons/fermions
correspond to bit). Depending on the given graded Lie (super)algebra, its
graded sectors can fall into different classes of equivalence expressing
different types of (para)bosons and/or (para)fermions. As a first application
we construct and -graded quantum Hamiltonians which
respectively admit and inequivalent multiparticle quantizations
(the inequivalent parastatistics are discriminated by measuring the eigenvalues
of certain observables in some given states). As a main physical application we
prove that the -extended, supersymmetric and superconformal quantum
mechanics, for , are respectively described by
alternative formulations based on the inequivalent graded Lie (super)algebras.
These numbers correspond to all possible "statistical transmutations" of a
given set of supercharges which, for , are accommodated into a
-grading with (the identification is ). In the
simplest setting (the -particle sector of the de DFF deformed
oscillator with spectrum-generating superalgebra), the -graded
parastatistics imply a degeneration of the energy levels which cannot be
reproduced by ordinary bosons/fermions statistics.Comment: 57 pages, 16 figure
Bosonic Super Liouville System: Lax Pair and Solution
We study the bosonic super Liouville system which is a statistical
transmutation of super Liouville system. Lax pair for the bosonic super
Liouville system is constructed using prolongation method, ensuring the Lax
integrability, and the solution to the equations of motion is also considered
via Leznov-Saveliev analysis.Comment: LaTeX, no figures, 11 page
On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields
In this paper we discuss off-shell representations of N-extended
supersymmetry in one dimension, ie, N-extended supersymmetric quantum
mechanics, and following earlier work on the subject codify them in terms of
certain graphs, called Adinkras. This framework provides a method of generating
all Adinkras with the same topology, and so also all the corresponding
irreducible supersymmetric multiplets. We develop some graph theoretic
techniques to understand these diagrams in terms of a relatively small amount
of information, namely, at what heights various vertices of the graph should be
"hung".
We then show how Adinkras that are the graphs of N-dimensional cubes can be
obtained as the Adinkra for superfields satisfying constraints that involve
superderivatives. This dramatically widens the range of supermultiplets that
can be described using the superspace formalism and organizes them. Other
topologies for Adinkras are possible, and we show that it is reasonable that
these are also the result of constraining superfields using superderivatives.
The family of Adinkras with an N-cubical topology, and so also the sequence
of corresponding irreducible supersymmetric multiplets, are arranged in a
cyclical sequence called the main sequence. We produce the N=1 and N=2 main
sequences in detail, and indicate some aspects of the situation for higher N.Comment: LaTeX, 58 pages, 52 illustrations in color; minor typos correcte
Twist Deformation of Rotationally Invariant Quantum Mechanics
Non-commutative Quantum Mechanics in 3D is investigated in the framework of
the abelian Drinfeld twist which deforms a given Hopf algebra while preserving
its Hopf algebra structure. Composite operators (of coordinates and momenta)
entering the Hamiltonian have to be reinterpreted as primitive elements of a
dynamical Lie algebra which could be either finite (for the harmonic
oscillator) or infinite (in the general case). The deformed brackets of the
deformed angular momenta close the so(3) algebra. On the other hand, undeformed
rotationally invariant operators can become, under deformation, anomalous (the
anomaly vanishes when the deformation parameter goes to zero). The deformed
operators, Taylor-expanded in the deformation parameter, can be selected to
minimize the anomaly. We present the deformations (and their anomalies) of
undeformed rotationally-invariant operators corresponding to the harmonic
oscillator (quadratic potential), the anharmonic oscillator (quartic potential)
and the Coulomb potential.Comment: 20 page
Second Hopf map and supersymmetric mechanics with Yang monopole
We propose to use the second Hopf map for the reduction (via SU(2) group
action) of the eight-dimensional N=8 supersymmetric mechanics to
five-dimensional supersymmetric systems specified by the presence of an SU(2)
Yang monopole. For our purpose we develop the relevant Lagrangian reduction
procedure. The reduced system is characterized by its invariance under the N=5
or N=4 supersymmetry generators (with or without an additional conserved BRST
charge operator) which commute with the su(2) generators.Comment: Final version. To appear in Phys. Rev.
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