Jacques Renaud

de Sitter waves and the zero curvature limit

We show that a particular set of global modes for the massive de Sitter scalar field (the de Sitter waves) allows us to manage the group representations and the Fourier transform in the flat (Minkowskian) limit. This is in opposition to the usual acceptance based on a previous result, suggesting the appearance of negative energy in the limit process. This method also confirms that the Euclidean vacuum, in de Sitter spacetime, is preferred as far as one wishes to recover ordinary quantum field theory in the flat limit.

Krein space quantization in curved and flat spacetimes

We re-examine in detail a canonical quantization method a la Gupta–Bleuler in which the Fock space is built over a so-called Krein space. This method has already been successfully applied to the massless minimally coupled scalar field in de Sitter spacetime for which it preserves covariance. Here, it is formulated in a more general context. An interesting feature of the theory is that, although the field is obtained by canonical quantization, it is independent of Bogoliubov transformations. Moreover, no infinite term appears in the computation of Tμν mean values and the vacuum energy of the free field vanishes: 0T000 = 0. We also investigate the behaviour of the Krein quantization in Minkowski space for a theory with interaction. We show that one can recover the usual theory with the exception that the vacuum energy of the free theory is zero.

Quantization of the Sphere With Coherent States

Current views link quantization with dynamics. The reason is that quantum mechanics or quantum field theories address to dynamical systems, i.e., particles or fields. Our point of view here breaks the link between quantization and dynamics: any (classical) physical system can be quantized. Only dynamical systems lead to dynamical quantum theories, which appear to result from the quantization of symplectic structures.

Fuzzy spheres from inequivalent coherent states quantizations

The existence of a family of coherent states (CS) solving the identity in a Hilbert space allows, under certain conditions, to quantize functions defined on the measure space of CS parameters. The application of this procedure to the 2-sphere provides a family of inequivalent CS quantizations based on the spin spherical harmonics (the CS quantization from usual spherical harmonics appears to give a trivial issue for the Cartesian coordinates). We compare these CS quantizations to the usual (Madore) construction of the fuzzy sphere. Due to these differences, our procedure yields new types of fuzzy spheres. Moreover, the general applicability of CS quantization suggests similar constructions of fuzzy versions of a large variety of sets.

The contraction of the SU(1,1) discrete series of representations by means of coherent states

The group SU(1,1) is a deformation of the Poincare group. This relationship is studied both at the classical level (coadjoint orbits) and at the quantum level (unitary representations). The contraction of the Lie algebras is written in such a way that the limit of coadjoint orbits, and hence of the classical mechanics, appears clearly. At the quantum level the representations are written on holomorphic functions Hilbert spaces and the contraction is realized by restricting these functions. It is shown that this restriction is a continuous operator. Moreover, using suitable coherent states, it is proved that the contraction extends to the representation of the whole enveloping algebras of the groups, hence it allows us to define the contraction of the quantum mechanics observables.

The indecomposable representation of SO0(2, 2) on the one-particle space of the massless field in 1 + 1 dimension

The group SO0(2, 2) is the finite-dimensional conformal group of the 1 + 1-dimensional Minkowski spacetimeM. We identify the indecomposable representation of SO0(2, 2) ≅ SO0(2, 1) × SO0(2, 1) that acts on the one-particle physical space of the massless scalar field onM. We accomplish this by realizing this space as a space ℋ of positive energy distributional solutions to the massless Klein-Gordon equation, on which the Klein-Gordon inner product is well defined and positive semi-definite. We then use the analyticity properties of these solutions in the forward tube to show that SO0(2, 2) acts naturally on ℋ, preserving the inner product. On right-moving solutions, one copy of SO0(2, 1) acts trivially, whereas the restriction of the representation to the other copy is the unique one-dimensional extension of the first term of the discrete series of representations of SO0(2, 1). Similar results hold for left-moving solutions.

Quantization of the nilpotent orbits in so(1,2)* and massless particles on (anti‐)de Sitter space–time

The nontrivial nilpotent orbits in so(1,2)*≂su(1,1)* are the phase spaces of the zero mass particles on the two‐dimensional (anti‐)de Sitter space–time. As is well known, the lack of global hyperbolicity (respectively, stationarity) for the anti‐de Sitter (respectively, de Sitter) space–time implies that the canonical field quantization of its free massless field is not uniquely defined. One might nevertheless hope to get the one‐particle quantum theory directly from an appropriate ‘‘first’’ quantization of the classical phase space. Unfortunately, geometric quantization (the orbit method) does not apply to the above orbits and a naive canonical quantization does not yield the correct result. To resolve these difficulties, we present a simple geometric construction that associates to them an indecomposable representation of SO0(1,2) on a positive semidefinite inner product space. It is shown that quotienting out its one‐dimensional invariant subspace yields the first term of the holomorphic discrete serie...

Relativistic harmonic oscillator and space curvature

Abstract Using simple group theoretical arguments we show how introducing a curvature parameter κ = ω c for a harmonic oscillator of mass m and frequency ω leads to an elementary entity of energy E 0 = mc 2 + 1 2 ℏω + O(κ) . This entity is a deformation of both an elementary vibration (rest energy 1 2 ℏω ) and an elementary particle (rest energy mc2).

Conformally invariant wave equation for a symmetric second rank tensor (“spin-2”) in a d -dimensional curved background

We build the general conformally invariant linear wave operator for a free, symmetric, second-rank tensor field in a

Wick ordering for coherent state quantization in 1+1 de Sitter space

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