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Journal of Physics A Mathematical and General

On the uniqueness of the Segal quantisation of linear Bose systems

Franco Gallone, Antonio Sparzani

Published Jun 1, 1981• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/14/6/012

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Journal of Physics A Mathematical and General•Oct 1, 1983•7 citations
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.
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.
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.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.
A real symplectic flow may be viewed as being pseudo-unitary, provided there exists a symplectic complex structure J which commutes with it. The author determines the class of dynamical systems for which J exists. Such systems may be quantised algebraically but the state space may have indefinite metric. This procedure admits a larger class of classical systems than does Segal's algebraic quantisation but there are problems in interpreting the indefinite metric and these are listed and discussed.