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arXiv:1812.04551

Uniqueness of Segal quantization for oscillating systems

Massimo Bertini, Sergio L. Cacciatori, Manuel Falchi Perna

Published Dec 11, 2018• 0 citations
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Abstract

We show that the Segal quantization of an arbitrary system of decoupled harmonic oscillators is unique in the sense that the one particle Hilbert space is completely determined by the requests of being a naturally complex symplectic space carrying a unitary realization of the dynamical evolution of the considered system.
DOI
10.48550/arxiv.1812.04551

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The kinematical description of a classical system having a linear phase space can be quantised if a complexification is given over the symplectic space that supports the description; indeed, a complexification determines a Segal quantisation. The question of the uniqueness of the Segal quantisation arises since the quantum descriptions constructed through different Segal quantisations need not be equivalent. If the classical description contains also a symmetry group (represented by linear symplectic transformations; this means that one deals with systems having linear dynamics only), the complexification to determine a Segal quantisation that also quantises the symmetry group. The authors show that this requirement determines uniquely the complexification, and therefore the Segal quantisation, provided the symmetry group fulfils a (real) irreducibility condition. They apply this uniqueness criterion to the free neutral scalar field.
We prove a theorem, which, while it fits naturally into the Segal--Weinless approach to quantization seems to have been overlooked in the literature: Let (D,sigma) be a symplectic space, and T (t) a one parameter group of symplectics on (D,sigma). Let (H, 2Im) be a complex Hilbert space considered as a real symplectic space, and U(t) a one-parameter unitary group on H with strictly positive energy. Suppose there is a linear symplectic map K from D to H with dense range, intertwining T (t) and U(t). Then K is unique up to unitary equivalence.
We prove a theorem, which, while it fits naturally into the Segal–Weinless approach to quantization seems to have been overlooked in the literature: Let (D,σ) be a symplectic space, and 𝒯 (t) a one parameter group of symplectics on (D,σ). Let (ℋ, 2Im〈⋅ ‖ ⋅〉) be a complex Hilbert space considered as a real symplectic space, and U(t) a one-parameter unitary group on ℋ with strictly positive energy. Suppose there is a linear symplectic map K from D to ℋ with dense range, intertwining 𝒯 (t) and U(t). Then K is unique up to unitary equivalence.
In Segal’s approach to linear Fermi quantum systems, a one particle picture with linear symmetries (e.g., with a linear dynamics) can be quantized very straightforwardly when a complexification is given for the (real linear) one particle picture. We examine how the symmetries that are embodied in the one particle picture can determine the structure of the family of the possible complexifications. Among other results, we prove that if the symmetries can be represented in a suitably irreducible way then the complexification is essentially unique. Also, when the one particle space is a generalization of the one defined by the Dirac equation, we prove that there are many complexifications, and inequivalent too, as they generate inequivalent representations of the canonical anticommutation relations; however, we find two criteria that single out the ’’physical’’ complexification. We use the general results we prove to discuss a few familiar models.
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