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Journal of Mathematical Physics

Subsystems in physics described by bilinear maps between the corresponding vector spaces

Dirk Aerts

Published Apr 1, 1980• 11 citations
Publisher

Abstract

We show that whenever a physical system is composed of two subsystems, there exists a (σ1,σ2) -linear map between their generalized Hilbert spaces which describes this composition. As a consequence, subsystems of a physical system described by a generalized Hilbert space over a division ring K are always described by a generalized Hilbert space over a subdivision ring of K.

DOI
10.1063/1.524499

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We show that whenever a physical system is composed of two subsystems, there exists a (σ1,σ2) -linear map between their generalized Hilbert spaces which describes this composition. As a consequence, subsystems of a physical system described by a generalized Hilbert space over a division ring K are always described by a generalized Hilbert space over a subdivision ring of K.
Composition of Different and of Identical Systems It is a general tendency of physics to interpret physical systems as composed of simpler, more elementary subsystems. The question “What is it made of?” is a very tempting and popular one. Though this attitude does not reflect a logical necessity, it has proved to be successful in significant circumstances: we think, e.g., of macroscopic matter as composed of atoms and molecules, of molecules as composed of atoms, of atoms as composed of nuclei and electrons, and of nuclei as composed of protons and neutrons. Loosely speaking, one might expect that this attitude will be successful whenever the interaction that packs the constituent subsystems together is not strong enough to make them lose their identity. Here, we are concerned with the problem of constructing the quantum-mechanical formalism of a compound system knowing the formalisms of the constituent subsystems. Thus our first problem is to specify the Hilbert space of the compound system. For the sake of simplicity, let us restrict ourselves to the case of a system composed of only two subsystems. The prescriptions of quantum theory vary according to whether one is faced with the composition of identical or nonidentical subsystems.
Corresponding to any irreducible proposition system L in general quantum mechanics there is a division ring D with an anti-automorphism * and a vector space (V, D) over D with a definite sesquilinear form φ such that L is isomorphic to the set of φ closed subspaces of (V, D). The main task remaining in connecting the general quantum mechanics to the conventional quantum theory in a complex Hilbert space is to give physical arguments which force D to be the complex field. In this paper it is shown that if L admits a certain type of observable (together with other structure which seems to be physically justified), then D contains the real field as a subfield. Steps are then indicated that can be taken to move from the reals to the complexes or quaternions.
Abstract Any Hilbert space with composite dimension can be represented as a tensor product of Hilbert spaces of lower dimensions. This factorization makes it possible to decompose a quantum system into subsystems. We propose a model based on finite quantum mechanics for the constructive study of decompositions of an isolated quantum system into subsystems. To study the behavior of the composite systems resulting from the decompositions, we develop algorithms based on methods of computer algebra and computational group theory.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
The entangled states of composite quantum systems are well studied. The particle-like nature of these systems also means that, while entangled, they can be physically separated and measurements performed on the separated subsystems. Measurements on the separated subsystems A and B should pertain to vectors in the local Hilbert space of each subsystem, but to date it has not been clear how to elucidate the relevant states of the separated subsystems because it is not obvious how to resolve them from states given in the tensor product basis, except for the separable states. Here it is shown that the projections of any general entangled state that are detected by measurements on the separated subsystems can be obtained considering the corresponding (cosets of) states in the free vector space from which the tensor product space is defined. The result eliminates the need to invoke random collapse, and from this perspective nonlocality arises because of the way measurements on each separated subsystem projects possible measurement outcomes.