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Lobachevskii Journal of Mathematics

Strong Projections in Hilbert Space and Quantum Logic

Marjan Matvejchuk, Elena Vladova

Published Oct 1, 2019• 2 citations
Publisher

Abstract

In the paper we study linear operators on complex Hilbert spaces which are strong real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We compare order properties of the orthogonal and of the strong real-orthogonal projections. We prove that the set of all strong real-orthogonal the projections in the complex space is the quantum pseudo logic. We also prove an analogy of Gleason’s theorem.

DOI
10.1134/s1995080219100196

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Posibilidades de generalización de las lógicas cuánticas

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The complete and atomic orthomodular lattice structure which is formed by the projectors (or equivalently, by the closed subspaces) of a complex and separable Hilbert space will be indicated in this paper by LG (H), where H is the Hilbert space. (We take operations and distinct elements for granted.) LG (H) is, classically considered, the “typical” logic of quantum mechanics, in the following sense: if H is the space associated with physical system S, then H’s rays, the ultramaximal operators, etc., represent typical properties of S, such as observable states and magnitudes. LG (H) is simply an element in model sets of more general theories such as the orthomodular lattice set, which we formalize in § 2, using a second-order logic. This generalization, which we call “formal” because it consists merely of giving more permissive axioms that facilitate the apparition of new structures other than LG (H), can yet be extended. 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For instance, in the theory of orthomodular lattices, besides Hilbert-space logics (or those based on other topological vectorial spaces), new models appear, such as finite or orthomodular lattices, for which the name “quantum logic” has little sense. In § 3 we study the modifications, and eventual generalizations, of the basic structure that generates quantum logic, i.e., the infinite-dimensional separable complex Hilbert space structure. A first way seems to be the generalizing of the field, and it is analyzed in § 3.1. In principle, it seems natural to admit that the field must contain some copy of the field of real numbers, or have at least an expressive force as great as theirs. This is why the use of real fields —and for a greater certainty, of complex fields— in the H spaces of quantum mechanics seems to be fully justified. They study of quaternions, carried on by Emch and others, but not clearly completed yet, could lead to an equivalent quantum mechanics, notwithstanding the non-commutativity of this field. Anyway, real, complex and quaternions seem to be only fields with a “natural” right to sustentate these spaces. The incompleteness of rational fields, and their undesirable properties as to dimension and measurability, cause them to be rejected. Other proposals are studied. Leaving aside Hensel’s bodies, that seem to be wholly different in nature, we are left with p-adics Qp and Galois’ fields as the only reasonable candidates. Know algebraic results are then invoked, according to which an extension or completion of these bodies would not lead to the desired results. Quite simply, we would not, save on trivial instances, have any logic at all. It is a well-known fact that by completing p-adic fields (let K be such a completion) we arrive at space (V, K) upon that field, which does not allow to define hermitian forms for dimensions greater than 4. Actually, no involutory anti-automorphism of the field (such as complex conjugation) permits to associate any quadratic form for a dimension greater than or equal to 5. Things are even worse in the case of Galois’ fields where limitative results begin to appear already in dimensions greater than 2. In § 3.2 we choose to accept the complex numbers body C as a canonic body, and see whether it is possible to generalize space into a class that need not be Hilbert’s. There are various possibilities: to consider Banach spaces, or else more or less general local convex spaces, and particularly, to focus analysis on Mackey spaces. For Banach spaces there appear Maczynski’s limitative results; for Mackey spaces, “analogous” results from Wilbur. (Cf. references at the end of the essay.) As a matter of fact, only infinite-dimensional spaces can be of any interest. In this case, if a Mackey space has an “acceptable” topology, i.e., a stronger one than the so-called weak convergence topology, and its LG (M) is really a logic, then M is once again a Hilbert space. Several of these results to the conclusion that we can hardly obtain a different logic (a more general or anyway more informative one) by taking the class LG (E) of some non-Hilbert space E. Rayski’s proposition about substituting Schwartz space for Hilbert space is also considered, but although such a substitution would modify actual work “within” the space in the interests of formalization, it is natural to expect the apparition of a logic LG (S), equivalent to former logics. Maybe the part special logics play in quantum mechanics will be entirely unnecessary when the axiomatizing of a physical system can be done in terms of some kind of spaces that leaves no doubt about its usefulness. [C.L.]

Crítica (México D F En línea)•Nov 9, 1979•0 citations
In the paper we study linear operators on complex Hilbert spaces which are strong real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We compare order properties of the orthogonal and of the strong real-orthogonal projections. We prove that the set of all strong real-orthogonal the projections in the complex space is the quantum pseudo logic. We also prove an analogy of Gleason’s theorem.
In this paper, we study linear operators on real and complex Euclidean spaces which are real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We note the difference between properties of real-orthogonal projections on real and on complex spaces. We can compare some partial order properties of orthogonal and of real-orthogonal projections. We prove that the set of all real-orthogonal projections in a finite-dimensional complex or real space is a quantum pseudo-logic.
In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. \nIn conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while \nthere exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is \ncompletely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators \nwith quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc \nHilbert spaces. Clearly, this property turns out to be crucial.
The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, and the quantum effects for a physical system can be described by the set E(H) of positive contractive operators on a complex Hilbert space H. In this note, by the techniques of operator block and spectral, we give the simpler representation of A∧P and obtained the new necessary and sufficient conditions for A∨P, for A∊S(H) and P∊P(H), where P(H) is the set of all orthogonal projection operators on H. In particular, we get that if A∨P exists, then A∨P∊E(H) for A∊E(H) and P∊P(H). In addition, we consider the relations between the existence of A∨B, A−∨B−, and A+∨B+, where A+, B+, A−, and B− are the positive and negative parts of A,B∊S(H).
We consider the standard quantum logic L(H){\mathcal L}(H)L(H) associated to a complex Hilbert space HHH, i.e. the lattice of closed subspaces of HHH together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic L(H){\mathcal L}(H)L(H)) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of L(H){\mathcal L}(H)L(H) and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.
This paper presents a generalization of quantum mechanics from conventional Hilbert space formalism to Banach space one. We construct quantum theory starting with any complex Banach space beyond a complex Hilbert space, through using a basic fact that a complex Banach space always admits a semi-inner product. Precisely, in a complex Banach space X\mathbb{X}X with a given semi-inner product, a pure state is defined by Lumer \cite{Lumer1961} to be a bounded linear functional on the space of bounded operators determined by a normalized element of X\mathbb{X}X under the semi-inner product, and then the state space S(X)\mathcal{S} (\mathbb{X})S(X) of the system is the weakly closed convex set spanned by all pure states. Based on Lumer's notion of the state, we associate a quantum system with a complex Banach space X\mathbb{X}X equipped with a fixed semi-inner product, and then define a physical event at a quantum state ω∈S(X)ω\in \mathcal{S}(\mathbb{X})ω∈S(X) to be a projection PPP (bounded operator such that P2=PP^2 =PP2=P) in X\mathbb{X}X satisfying the positivity condition 0≤ω(P)≤1,0 \le ω(P) \le 1,0≤ω(P)≤1, and a physical quantity at a quantum state ωωω to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at ω.ω.ω. The Born formula for measurement of a physical quantity is the natural pairing of operators with linear functionals satisfying the probability conservation law. A time evolution of the system is governed by a one-parameter group of invertible spectral operators determined by a scalar type operator with the real spectrum, which satisfies the Schrödinger equation. Our formulation is just a generalization of the Dirac-von Neumann formalism of quantum mechanics to the Banach space setting. We include some examples for illustration.
The complete and atomic orthomodular lattice structure which is formed by the projectors (or equivalently, by the closed subspaces) of a complex and separable Hilbert space will be indicated in this paper by LG (H), where H is the Hilbert space. (We take operations and distinct elements for granted.) LG (H) is, classically considered, the “typical” logic of quantum mechanics, in the following sense: if H is the space associated with physical system S, then H’s rays, the ultramaximal operators, etc., represent typical properties of S, such as observable states and magnitudes. LG (H) is simply an element in model sets of more general theories such as the orthomodular lattice set, which we formalize in § 2, using a second-order logic. This generalization, which we call “formal” because it consists merely of giving more permissive axioms that facilitate the apparition of new structures other than LG (H), can yet be extended. In § 2 we use infinitarian logics, where a numerable number of conjunctions and quantificators is permitted, to express the theory of orthomodular partially ordered sets (posets). That formulation uses essentially the logic called (omega-1, omega-1), and a conjecture is raised about proving it cannot be realized within a weaker logic. The way to prove this, it is suggested, should be analogous to the way the indefinability of well-ordering is proved. Such generalizations are analogous to those obtained when we pass, e.g., from the field of real numbers to any commutative, Archimedian, etc., field; then to a commutative ring with and unity, and so on. They are, therefore, formal, in the sense that (1) they do not “say” much about the initial structure which motivated the generalization and that (2) in generalizing, we allow the “entrance” of new structures that have little to do with the original motivation. For instance, in the theory of orthomodular lattices, besides Hilbert-space logics (or those based on other topological vectorial spaces), new models appear, such as finite or orthomodular lattices, for which the name “quantum logic” has little sense. In § 3 we study the modifications, and eventual generalizations, of the basic structure that generates quantum logic, i.e., the infinite-dimensional separable complex Hilbert space structure. A first way seems to be the generalizing of the field, and it is analyzed in § 3.1. In principle, it seems natural to admit that the field must contain some copy of the field of real numbers, or have at least an expressive force as great as theirs. This is why the use of real fields —and for a greater certainty, of complex fields— in the H spaces of quantum mechanics seems to be fully justified. They study of quaternions, carried on by Emch and others, but not clearly completed yet, could lead to an equivalent quantum mechanics, notwithstanding the non-commutativity of this field. Anyway, real, complex and quaternions seem to be only fields with a “natural” right to sustentate these spaces. The incompleteness of rational fields, and their undesirable properties as to dimension and measurability, cause them to be rejected. Other proposals are studied. Leaving aside Hensel’s bodies, that seem to be wholly different in nature, we are left with p-adics Qp and Galois’ fields as the only reasonable candidates. Know algebraic results are then invoked, according to which an extension or completion of these bodies would not lead to the desired results. Quite simply, we would not, save on trivial instances, have any logic at all. It is a well-known fact that by completing p-adic fields (let K be such a completion) we arrive at space (V, K) upon that field, which does not allow to define hermitian forms for dimensions greater than 4. Actually, no involutory anti-automorphism of the field (such as complex conjugation) permits to associate any quadratic form for a dimension greater than or equal to 5. Things are even worse in the case of Galois’ fields where limitative results begin to appear already in dimensions greater than 2. In § 3.2 we choose to accept the complex numbers body C as a canonic body, and see whether it is possible to generalize space into a class that need not be Hilbert’s. There are various possibilities: to consider Banach spaces, or else more or less general local convex spaces, and particularly, to focus analysis on Mackey spaces. For Banach spaces there appear Maczynski’s limitative results; for Mackey spaces, “analogous” results from Wilbur. (Cf. references at the end of the essay.) As a matter of fact, only infinite-dimensional spaces can be of any interest. In this case, if a Mackey space has an “acceptable” topology, i.e., a stronger one than the so-called weak convergence topology, and its LG (M) is really a logic, then M is once again a Hilbert space. Several of these results to the conclusion that we can hardly obtain a different logic (a more general or anyway more informative one) by taking the class LG (E) of some non-Hilbert space E. Rayski’s proposition about substituting Schwartz space for Hilbert space is also considered, but although such a substitution would modify actual work “within” the space in the interests of formalization, it is natural to expect the apparition of a logic LG (S), equivalent to former logics. Maybe the part special logics play in quantum mechanics will be entirely unnecessary when the axiomatizing of a physical system can be done in terms of some kind of spaces that leaves no doubt about its usefulness. [C.L.]