52 research outputs found
Superunitary Representations of Heisenberg Supergroups
Numerous Lie supergroups do not admit superunitary representations except the
trivial one, e.g., Heisenberg and orthosymplectic supergroups in mixed
signature. To avoid this situation, we introduce in this paper a broader
definition of superunitary representation, relying on a new definition of
Hilbert superspace. The latter is inspired by the notion of Krein space and was
developed initially for noncommutative supergeometry. For Heisenberg
supergroups, this new approach yields a smooth generalization, whatever the
signature, of the unitary representation theory of the classical Heisenberg
group. First, we obtain Schrodinger-like representations by quantizing generic
coadjoint orbits. They satisfy the new definition of irreducible superunitary
representations and serve as ground to the main result of this paper: a
generalized Stone-von Neumann theorem. Then, we obtain the superunitary dual
and build a group Fourier transformation, satisfying Parseval theorem. We
eventually show that metaplectic representations, which extend Schrodinger-like
representations to metaplectic supergroups, also fit into this definition of
superunitary representations.Comment: 59 pages. v2: section 6 (Proof of Stone-von Neumann theorem) and
section 7 (super unitary dual) were corrected and rewritte
Deformation Quantization for Heisenberg Supergroup
We construct a non-formal deformation machinery for the actions of the
Heisenberg supergroup analogue to the one developed by M. Rieffel for the
actions of R^d. However, the method used here differs from Rieffel's one: we
obtain a Universal Deformation Formula for the actions of R^{m|n} as a
byproduct of Weyl ordered Kirillov's orbit method adapted to the graded
setting. To do so, we have to introduce the notion of C*-superalgebra, which is
compatible with the deformation, and which can be seen as corresponding to
noncommutative superspaces. We also use this construction to interpret the
renormalizability of a noncommutative Quantum Field Theory.Comment: 49 page
Resurgent Deformation Quantisation
We construct a version of the complex Heisenberg algebra based on the idea of
endless analytic continuation. In particular, we exhibit an integral formula
for the product of resurgent operators with algebraic singularities. This
algebra would be large enough to capture quantum effects that escape ordinary
formal deformation quantisation.Comment: 28 pages, v2: published versio
Non-formal star-exponential on contracted one-sheeted hyperboloids
In this paper, we exhibit the non-formal star-exponential of the Lie group
SL(2,R) realized geometrically on the curvature contraction of its one-sheeted
hyperboloid orbits endowed with its natural non-formal star-product. It is done
by a direct resolution of the defining equation of the star-exponential and
produces an expression with Bessel functions. This yields a continuous group
homomorphism from SL(2,R) into the von Neumann algebra of multipliers of the
Hilbert algebra underlied by this natural star-product. As an application, we
prove a new identity on Bessel functions.Comment: 33 pages, 3 figures, v2: changes in section 3, references adde
Symmetries of noncommutative scalar field theory
We investigate symmetries of the scalar field theory with harmonic term on
the Moyal space with euclidean scalar product and general symplectic form. The
classical action is invariant under the orthogonal group if this group acts
also on the symplectic structure. We find that the invariance under the
orthogonal group can be restored also at the quantum level by restricting the
symplectic structures to a particular orbit.Comment: 12 pages, revised versio
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