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Journal of Mathematical Physics

Uniqueness result in the Segal--Weinless approach to linear Bose fields

Burak Kaya

Published Aug 1, 1979• 0 citations
OpenAlex

Abstract

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.

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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.
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.
This article discusses the quantization of linear Hamiltonian systems, a historically rich but under explored line of research. The key idea is that a classical linear Hamiltonian system induces on its phase space a compatible complex structure and scalar product, giving rise to a complex Hilbert space where classical dynamics becomes a one-parameter unitary group. Boson Fock quantization of this group then recovers, up to unitary equivalence, the results of canonical quantization. This expository overview traces the development of this framework from foundational works to modern symplectic perspectives, offering a case study in the dialogue between analysis, geometry, and physics.
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.
On the bosonic Fock space, a family of Bogoliubov transformations corresponding to a strongly continuous one-parameter group of symplectic maps R(t) is considered. Under suitable assumptions on the generator A of this group, which guarantee that the induced representations of CCR are unitarily equivalent for all time t, it is known that the unitary operator U_{nat}(t) which implement this transformation gives a prjective unitary representation of R(t). Under rather general assumptions on the generator A, we prove that the corresponding Bogoliubov transformations can be implemented by a one-parameter group U(t) of unitary operators. The generator of U(t) will be called a Bogoliubov Hamiltonian. We will introduce two kinds of Bogoliubov Hamiltonians (type I and II) and give conditions so that they are well defined.