Quantum physics of an elementary system in de Sitter space
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Regular Article - Theoretical Physics
Quantum physics of an elementary system in de Sitter space A. Rabeiea Department of Physics, Razi University, Kermanshah, Iran
Received: 14 June 2012 / Revised: 27 July 2012 © Springer-Verlag / Società Italiana di Fisica 2012
Abstract We present the coherent states of a scalar massive particle on 1 + 3-de Sitter space. These states are vectors in Hilbert space, and they are labeled by points in the associated phase space. To do this, we use the fact that the phase space of a scalar massive particle on 1 + 3-de Sitter space is a cotangent bundle “T ∗ (S 3 )” which is isomorphic with the complex sphere “SC3 ”. Then by using the heat kernel on “SC3 ” that was presented by Hall–Mitchell, we construct our coherent states. At the end, by these states we quantize the classical kinetic energy on de Sitter space.
1 Introduction
spaces, on the basis of Thiemann complexification method [5], is isomorphic with SC3 and therefore we can construct our CS on this complex sphere instead of the cotangent space. To do this, we profit from the work of Hall–Mitchell [6]. In that work, B.C. Hall and J.J. Mitchell have presented the heat kernel (that is, the solution of heat equation) on a complex sphere. This heat kernel is connected with CS and we use this connection to create the CS on SC3 . These CS satisfy the resolution of the unity (see Sect. 4) and we quantize a classical observable on SC3 by the Berezin–Klauder–Toeplitz method (or anti-Wick quantization):
The de Sitter metric gμν is one of the solutions of Einstein’s equation [1]:
Of =
Gμν + Λgμν = −Tμν ,
where |Ψa , f (χ), μ(dχ) and Of are, respectively, the CS, a classical observable (a function in phase space), relevant measure in phase space and the quantum observable. Our paper is organized as follows: In Sect. 2 we discuss briefly the de Sitter group. Section 3 is devoted to construction of the phase space for our system. In Sect. 4 we create our CS, which is necessary for quantization. Finally in Sect. 5 we quantize directly the kinetic energy function on 1 + 3-de Sitter space. We expect that this quantity plays the role of Hamiltonian operator of harmonics oscillator in three dimensions.1
(1)
with positive constant cosmologic Λ and null energyimpulsion tensor, Tμν = 0. This metric is visualized by a hyperboloid embedded in a five-dimensional Minkowski space which is known as 1 + 3-de Sitter space-time. The associated symmetric group is SO0 (1, 4), or equivalently its universal covering, i.e. the symplectic Sp(2, 2) group which is described by 2 × 2 matrices with quaternionic coefficients. de Sitter classical mechanics is understood along with the traditional phase space approach. Studying the quantum physics associated to this phase space is very interesting for physicists. This paper is devoted to the construction of the coherent states (CS) and quantization on Hilbert space associated with the phase space of a scalar “massive” elementary system in de Sitter space. By the Kirillov orbit method [2–4], we find that the ph
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