We study model-theoretical structures for prototypical physical systems. First, a summary of the model theory of sheaves, adapted to the metric case, is presented. In particular, we provide conditions for a generalization of the generic model theorem to metric sheaves. The essentials of the model theory of metric sheaves appeared in the form of Conference Proceedings. We provide a version of those results, for the sake of completeness, and then build metric sheaves for physical systems in the second part of the paper. Specifically, metric sheaves for quantum mechanical systems with pure point and continuous spectra are constructed. In the former case, every fiber is a finite projective Hilbert space determined by the family of invariant subspaces of a given operator with pure point spectrum, and we also consider unitary transformations in a finite-dimensional space. For an operator with continuous spectrum, every fiber is a two sorted structure of subsets of the Schwartz space of rapidly decreasing functions that includes imperfect representations of position and momentum states. The imperfection character is parametrically determined by the elements on the base space and refined in the generic model. Position and momentum operators find a simple representation in every fiber as well as their corresponding unitary operators. These results follow after recasting the algebraic properties of the integral transformations frequently invoked in the description of quantum mechanical systems with continuous spectra. Finally, we illustrate how this construction permits the calculation of the quantum mechanical propagator for a free particle.
We study model-theoretical structures for prototypical physical systems. First, a summary of the model theory of sheaves, adapted to the metric case, is presented. In particular, we provide conditions for a generalization of the generic model theorem to metric sheaves. The essentials of the model theory of metric sheaves appeared in the form of Conference Proceedings. We provide a version of those results, for the sake of completeness, and then build metric sheaves for physical systems in the second part of the paper. Specifically, metric sheaves for quantum mechanical systems with pure point and continuous spectra are constructed. In the former case, every fiber is a finite projective Hilbert space determined by the family of invariant subspaces of a given operator with pure point spectrum, and we also consider unitary transformations in a finite-dimensional space. For an operator with continuous spectrum, every fiber is a two sorted structure of subsets of the Schwartz space of rapidly decreasing functions that includes imperfect representations of position and momentum states. The imperfection character is parametrically determined by the elements on the base space and refined in the generic model. Position and momentum operators find a simple representation in every fiber as well as their corresponding unitary operators. These results follow after recasting the algebraic properties of the integral transformations frequently invoked in the description of quantum mechanical systems with continuous spectra. Finally, we illustrate how this construction permits the calculation of the quantum mechanical propagator for a free particle.
We consider some generalization of the theory of quantum states, which is based on the analysis of long standing problems and unsatisfactory situation with the possible interpretations of quantum mechanics. We demonstrate that the consideration of quantum states as sheaves can provide, in principle, more deep understanding of some phenomena. The key ingredients of the proposed construction are the families of sections of sheaves with values in the category of the functional realizations of infinite-dimensional Hilbert spaces with special (multiscale) filtration. Three different symmetries, kinematical (on space-time), hidden/dynamical (on sections of sheaves), unified (on filtration of the full scale of spaces) are generic objects generating the full zoo of quantum phenomena.
We present in this paper a set of postulates for a physical system and deduce from these the main general features of the quantum theory of stationary states. Our theory is strictly operational in the sense that only the observables of the physical system are involved in the postulates. The collection of all bounded self-adj oint operators on a Hilbert space, which has previously been used as a mathematical model for the observables in quantum mechanics, satisfy the postulates, as do a variety of considerably more general mathematical structures. We show that for any two observables there exists a pure state of the system (this term being defined essentially as done by von Neumann and Weyl) in which they have different (expectation) values. Each observable has a spectral resolution, and a state of the system induces, in a natural way, a probability distribution on the range of spectral values of each observable. A number of observables are simultaneously observable if and only if they commute (although the product of two observables is not defined in general, a reasonable notion of commutativity can be introduced). Inasmuch as Hilbert space plays no role in our theory, our proofs are necessarily of a different character from the proofs of these results for the case of the system of all bounded selfadjoint operators. Actually, Hilbert space appears to be somewhat inadequate as a state space even for the latter system, in that there exist pure states of the system which cannot be represented in the usual way by rays of the Hilbert space. The postulates are partly algebraic and partly metric. The algebraic postulates require essentially that an observable can be multiplied by real numbers and raised to integral powers, and that any two observables can be added. It is assumed that the usual algebraic laws are satisfied so that (1) the observables can be treated like the elements of a linear space, and (2) the usual rules for dealing with polynomials in one variable with real coefficients remain valid when the variable is replaced by an observable. It is not assumed that two observables have a product. The metric postulates require that for each observable there be defined a kind of maximum numerical value, which plays the part of a norm, and has various properties in accord with its physical significance. While this norm is quite essential to the development of the theory, an interesting consequence of the theory is that the norm can be (uniquely) defined in a purely algebraic fashion. This shows that the objective features of a physical system,-the spectral values and probability distributions of the observables, and the pure states,-are completely determined by the algebra of observables, i.e., by the rules for addition, scalar multiplication, and powers, of observables.
We consider some generalization of the theory of quantum states and demonstrate that the consideration of quantum states as sheaves can provide, in principle, more deep understanding of some well-known phenomena. The key ingredients of the proposed construction are the families of sections of sheaves with values in the proper category of the functional realizations of infinite-dimensional Hilbert spaces with special (multiscale) filtrations decomposed into the (entangled) orbits generated by actions/representations of internal hidden symmetries. In such a way, we open a possibility for the exact description and reinterpretation of a lot of quantum phenomena.
In this chapter we take up systems whose state spaces form a continuum. These are, naturally, infinite dimensional vector spaces (often called Hilbert spaces). Unfortunately, we could not do any better than treating such systems by formal analogy with their finite dimensional counterparts. A rigorous treatment would require a background in functional analysis and is beyond the scope of the present work. Thus, we essentially lay out a dictionary for translating the mathematical ingredients and formulae, from discrete to continuum descriptions, motivated by plausibility arguments. In particular, we argue why states are now described by complex square integrable functions (called wave functions) and observables by Hermitian differential operators. We then move on to show how to quantize quantum systems with classical analogues, which comprise the bulk of real life physical systems. Here, we explicitly show how the most nontrivial component of the quantization, i.e., the description of the representation of the momentum operator (in position space) can be deduced from both, the de Broglie hypothesis, and the canonical commutation relation. The “particle in a box” problem is treated as an illustration. The parity operator, Ehrenfest's theorem and continuity conditions have been relegated to the problem section.
The purpose of the following paper, which is published in three parts, is to show the possibility of making consistent use of an indefinite metric in the space of states and of the probability interpretation of quantum mechanics. In the part I bilinear forms, which are invariant under the transformations of symmetry groups, are constructed. The construction of the fundamental metric tensor for the various types of representations is exemplified for the case of inhomogenious LORENTZ group. The results are also applicable for other groups. The representations of the finite and compact group are however normal. In these cases the use of an indefinite metric does not bring about a new look for the theory of representations. In the parts II and III besides invariance of bilinear forms further conditions are stated, which are sufficient for the propability interpretation. The existence of superselection rules by which the space of states breaks up into coherent sectors leads to two procedures. In the part II the facts within one coherent sector are solely studied. These results are important for the representations of those symmetry groups, which leave the coherent sectors invariant (e. g. the inhomogenious LORENTZ group). Each sector is divided by a cut into a subspace, the elements of which represent physical systems, and a rest. The representations in the „subspace of physical systems“ of these symmetry groups, which leave the sectors invariant, are unitary. In the part III the space of states as a whole is investigated. The symmetry group of isotopic spin for a simple case is discussed there as an example of mapping from one coherent sector to another. At the end a possible generalization for the dual vector of a vector is discussed.
In this paper we start with the development of a theory of presheaves on a lattice, in particular on the quantum lattice $\LL(\kH)$ of closed subspaces of a complex Hilbert space $\kH$, and their associated etale spaces. Even in this early state the theory has interesting applications to the theory of operator algebras and the foundations of quantum mechanics. Among other things we can show that classical observables (continuous functions on a topological space) and quantum observables (selfadjoint linear operators on a Hilbert space) are on the same structural footing.