Feynman’s Functional Calculus and Stochastic Calculus of Variations

The unbounded Self-adjoint operators that strongly commute on a common dense subset of their domain commute pointwise. When the operators commute pointwise on the same dense subset, there is to guarantee that they will commute strongly. By imposing some conditions, we on the operators as well as the underlying space, we get pointwise commuting unbounded operators that commute strongly. This article shows that by suitably selecting two unbounded positive Self-adjoint operators with compact inverses we get a set of pointwise commuting self-adjoint operators that commute on common core. then prove that it strongly commutes on the same subspace. Keywords: Unbounded operators, Self-adjoint operators, Commutative operators DOI: 10.7176/MTM/10-8-03 Publication date: December 31 st 2020

Quantum mechanics is like a pot: it is almost indestructible and extremely rigid, but also very flexible because you can use any ingredients for your soup. Goran Lindblad Quantum mechanics Although our first four chapters have been very mathematical, quantum mechanics has never been very far away. Let us recall how the mathematical structure of quantum mechanics is usually summarized at the end of a first course in the subject. First of all the pure states are given by vectors in a Hilbert space. If that Hilbert space is finite dimensional it is simply the vector space equipped with a scalar product of the particular kind that we called a Hermitian form in Eq. (3.82). Actually a pure state corresponds to an entire equivalence class of vectors; this is usually treated in such a way that the vectors are normalized to have length one and afterwards vectors differing by an overall phase e io are regarded as physically equivalent. In effect then the space of pure states is the complex projective space; as always in this book n = N – 1. The notation used in quantum mechanics differs from what we have used so far. We have denoted vectors in by Z α , while in quantum mechanics they are usually denoted by a ket vector. We can think of the index α as just a label telling us that Z α is a vector, and then these two notations are in fact exactly equivalent.

Quantum mechanics is one of the two fundamental pillar of modern physics. The success of the theory can be found everywhere in our everyday life and essentially in every new product that we build. We just have to remember that every semiconductor chip usually uses a quantum behavior in an essential way, for example quantum tunneling, to work. Until now, none of the thousand of experiments realized have succeeded to contradicted or to find a problem with the predictions given by quantum mechanics. However, in spite of this incredible success, many profound questions are still open. For example, we have some problems understanding the measurement, the coherence and the decoherence process, as well as the interpretation of what the theory tell us about the world we live in (Schlosshauer, 2005). Among the possible ways of investigation that we have, we think that stressing the foundations of the theory at the level of the mathematical structure, on which the theory stands, could be a good way to understand why and how the theory works. The mathematical structure of quantum mechanics consists in Hilbert spaces defined over the field of complex numbers (Birkhoff & Von Neumann, 1936). The success of the theory has led a number of investigators, over many decades, to look for general principles or arguments that would lead quite inescapably to the complex Hilbert space structure. It has been argued (Stueckelberg, 1960; Stueckelberg & Guenin, 1961), for instance, that the formulation of an uncertainty principle, heavily motivated by experiment, implies that a real Hilbert space can in fact be endowed with a complex structure. The proof, however, involves a number of additional hypotheses that may not be so directly connected with experiment. In fact Reichenbach (Reichenbach, 1944) has shown that a theory is not straightforwardly deduced from experiments, but rather arrived at by a process involving a good deal of instinctive inferences. This was also pointed out more recently by Penrose (Penrose, 2005, p. 59);

In the absence of a satisfactory interpretation of quantum theory, physical law lacks physical basis. This paper reviews the orthodox, or Dirac–von Neumann interpretation, and makes explicit that Hilbert space describes propositions about measurement results. Kets are defined as conditional clauses referring to measurements in a formal language. It is seen that these clauses are elements of a Hilbert space, such that addition is logical disjunction, the dual space consists of consequent clauses, and the inner product is a set of statements in the subjunctive mood. The probability interpretation gives truth values for corresponding future tense statements when the initial state is actually prepared and the final state is to be measured. The mathematical structure of quantum mechanics is formulated in terms of discrete measurement results at finite level of accuracy and does not depend on an assumption of a substantive, or background, space–time continuum. A continuum of kets, ∣x〉 for x ∈ ℝ3, is constructed from linear combinations of kets in a finite basis. The inner product can be expressed either as a finite sum or as an integral. Discrete position functions are uniquely embedded into smooth wave functions in such a way that differential operators are defined. It is shown that the choice of basis has no effect on underlying physics (quantum covariance). The Dirac delta has a representation as a smooth function. Operators do not in general have an integral form. The Schrödinger equation is shown from the requirements of the probability interpretation. It is remarked that a formal construction of quantum electrodynamics (qed) avoiding divergence problems has been completed using finite dimensional Hilbert space. I conclude that quantum mechanics makes statements about the world with clear physical meaning, such that space is emergent from particle interactions and has no fundamental role.

The measurement problem in quantum mechanics still appears to be an unresolved issue. Here we present a new quantum theory of measurement that overcomes many of the difficulties previously found. It is based on a consistent use of the linear superposition principle and distinguishes two aspects: recording and observation. A recording elicits the full interaction of the object quantum system with the quantum measuring apparatus. No wavefunction collapse is introduced. Statistics may appear at the observation of the recording only and depends on filtering processes. The theory presented here uses the existing mathematical structure of quantum mechanics but requires no ad hoc measurement postulates. Well-known paradoxical aspects in standard quantum mechanics, for instance, wave-particle duality, Schrodinger's cat, and Zeno effects do not appear in the current formulation. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 97: 670 - 678, 2004

We present and discuss selected issues related to the problem of representing in a machine-readable way data and knowledge about the quantum level of reality. In particular, we propose a method of creating an ontology for the quantum world. The method uses a mathematical structure of quantum mechanics. We apply the method to obtain a toy ontology corresponding to the Hilbert space formulation of quantum mechanics. We use the terms from the ontology to describe a simple quantum system. We also show how we can use the ontology to create semantic enhancements of scientific publications on quantum mechanics.

We now come to the mathematical foundation of quantum mechanics on phase space. In the two previous chapters, motivation for and some of the desired mathematical structure of quantum mechanics on phase space has emerged. In this chapter, we give a rigorous mathematical analysis of the origin, the variety, and the structure of quantum mechanics on phase space. To review, the discovered components of the structure of quantum mechanics on phase space are: (i) A symmetry group; (ii) A classical phase space, Γ; (iii) Representations of the symmetry group by unitary operators acting on the Hilbert space L2(Γ); (iv) Localization operators (P.V.M.’s) on these representations; (v) Harmonic decompositions of these representations into irreducible sub-representations; (vi) Intertwining operators between these irreducible sub-representations and the standard irreducible representations [as commonly used in physics and obtained by the Mackey-Frobenius method of induced representations.] The intertwined images of the (P.V.M.) localization operators become P.O.V.M.’s. (vii) Informational completeness criteria for these P.O.V.M.’s.

Quantum Darwinism posits that information becomes objective whenever multiple observers indirectly probe a quantum system by each measuring a fraction of the environment. It was recently shown that objectivity of observables emerges generically from the mathematical structure of quantum mechanics, whenever the system of interest has finite dimensions and the number of environment fragments is large [F. G. S. L. Brandão, M. Piani, and P. Horodecki, Nat. Commun. 6, 7908 (2015)NCAOBW2041-172310.1038/ncomms8908]. Despite the importance of this result, it necessarily excludes many practical systems of interest that are infinite dimensional, including harmonic oscillators. Extending the study of quantum Darwinism to infinite dimensions is a nontrivial task: we tackle it here by using a modified diamond norm, suitable to quantify the distinguishability of channels in infinite dimensions. We prove two theorems that bound the emergence of objectivity, first for finite mean energy systems, and then for systems that can only be prepared in states with an exponential energy cutoff. We show that the latter class includes any bounded-energy subset of single-mode Gaussian states.

We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e. the statement that, for typical large systems, every initial wave function ψ 0 from an energy shell is ‘normal’: it evolves in such a way that | ψ t 〉〈 ψ t | is, for most t , macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.

In the traditional Hilbert space quantum mechanics, (bounded or unbounded) selfadjoint operators are taken as physical observables. In the present work a more comprehensive approach is used: observables are represented as normalised positive operator measures. We develop the spectral representation theory of generally unbounded selfadjoint operators in this more general setting of positive operator measures, paving the way for important physical applications in the sequel.

In this chapter we shall prove the spectral theorem for self-adjoint (unbounded) operators A in Hilbert spaces. It says that for each A there exists a unique spectral measure E on a(A), so that A = ∫ t dE. In the sequel H is a complex Hilbert space.

Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

This survey is based on the works of Kiev mathematicians and others in the structure theory of pairs of bounded and unbounded self-adjoint operators that satisfy an algebraic relation.

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form \({-Au \in \partial\varphi(u)}\) as well ass dissipative initial value evolutions of the form \({-\dot{u}(t) - Au(t) + \omega{u}(t) \in \partial\varphi(l,u(l))}\) where \({\varphi}\) is a convex potential on an infinite dimensional space, A is a linear operator and \({\omega}\) is any scalar. The framework developed in the above mentioned paper reformulates these problems as \({0\in \bar{\partial}L(u)}\) and \({\dot{u}(t) \in \bar{\partial}L(t,u(t))}\) respectively, where \({\bar{\partial}L}\) is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term.

We give a functional analytical definition of the Maslov index for continuous curves in the Fredholm-Lagrangian Grassmannian. Our definition does not require assumptions either at the endpoints or at the crossings of the curve with the Maslov cycle. We demonstrate an application of our definition by developing the symplectic geometry of self-adjoint extensions of unbounded symmetric operators. We discuss continuous variations of the form $A_D+C_t$, where $A_D$ is a fixed self-adjoint unbounded Fredholm operator and $\{C_t\}$ a family of bounded self-adjoint operators. We extend the definition of the spectral flow to such families of unbounded operators in a purely functional analytical way. We then prove that the spectral flow is equal to the Maslov index of the corresponding family of abstract Cauchy data spaces.