Feynman's Operational Calculus Research Articles

A Family of Banach Spaces Over ℝ∞

In T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016), the topology of [Formula: see text] was replaced with a new topology and denoted by [Formula: see text]. This space was then used to construct Lebesgue measure on [Formula: see text] in a manner that is no more difficult than the same construction on [Formula: see text]. More important for us, a new class of separable Banach spaces [Formula: see text], [Formula: see text], for the HK-integrable functions, was introduced. These spaces also contain the [Formula: see text] spaces and the Schwartz space as continuous dense embeddings. This paper extends the work in T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016) from [Formula: see text] to [Formula: see text].

The Feynman-Dyson view

This paper is a survey of our work on the mathematical foundations for the Feynman-Dyson program in quantum electrodynamics (QED). After a brief discussion of the history, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initial-value problem and construct the Dyson series. We show that the series is asymptotic, thus proving Dyson’s second conjecture for quantum electrodynamics. In addition, we show that the expansion may be considered exact to any finite order by producing the remainder term. This implies that every nonperturbative solution has a perturbative expansion. Using a physical analysis of information from experiment versus that implied by our models, we reformulate our theory as a sum over paths. This allows us to relate our theory to Feynman’s path integral, and to prove Dyson’s first conjecture that the divergences are in part due to a violation of Heisenberg’s uncertainly relations. As a by-product, we also prove Feynman’s conjecture about the relationship between the operator calculus and has path integral. Thus, providing the first rigorous justification for the Feynman formulation of quantum mechanics.

Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time Independent Setting

Via a general construction, we are able to establish a quite general and comprehensive stability theory for Feynman's operational calculus in the time independent setting. In particular, we are able to establish stability of the operational calculus with respect to general types of the time-ordering measures. While the domain of the operational calculus is somewhat restricted as compared to the "standard" version of the operational calculus (established by Jefferies and Johnson (Russ. J. Math. Phys. 8:153---171, 2001; Mat. Zametki 70:815---838, 2001; Adv. Appl. Clifford Algebras 11:239---264, 2001; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5:171---199, 2002), the advantages of this relatively minor domain restriction are significant in that the stability theory (with respect to the time-ordering measures) as it stands to this time is contained, essentially in its entirety, in the principle result of this paper, Theorem 2. Moreover, Theorem 2 allows immediate, and rather far-reaching extensions of the stability theory that, until now, have not been possible.

WEAK CONVERGENCE THEOREMS IN FEYNMAN'S OPERATIONAL CALCULI : THE CASE OF TIME DEPENDENT NONCOMMUTING OPERATORS

Feynman's operational calculus for noncommuting operators was studied by means of measures on the time inteval. And various stability theorems for Feynman's operational calculus were investigated. In this paper we see the time-dependent stability properties when the operator-valued functions take their values in a separable Hilbert space.

Feynman operator calculus: The constructive theory

In this paper, we survey progress on the Feynman operator calculus and path integral. We first develop an operator version of the Henstock–Kurzweil integral, construct the operator calculus and extend the Hille–Yosida theory. This shows that our approach is a natural extension of operator theory to the time-ordered setting. As an application, we unify the theory of time-dependent parabolic and hyperbolic evolution equations. Our theory is then reformulated as a sum over paths, providing a completely rigorous foundation for the Feynman path integral. Using our disentanglement approach, we extend the Trotter–Kato theory.

BLENDING INSTANTANEOUS AND CONTINUOUS PHENOMENA IN FEYNMAN'S OPERATIONAL CALCULI: THE CASE OF TIME DEPENDENT NONCOMMUTING OPERATORS

Feynman's operational calculus for noncommuting operators was studied via measures on the time interval. We investigate some prop- erties of Feynman's operational calculi which include a variety of blends of discrete and continuous measures in the time dependent setting.

Feynman disentangling of noncommuting operators and group representation theory

Feynman's method for disentangling noncommuting operators is discussed and applied to nonstationary problems in quantum mechanics, including the excitation ofa harmonic oscillator by an external force and/or by time-varying frequency; spin rotation in a time-varying magnetic field; the disentangling of an atom (ion) Hamiltonian in a laser field; a model with the hidden symmetry group of the hydrogen atom; and the theory of coherent states. The Feynman operator calculus combined with simple group-theoretical considerations allows disentangling the Hamiltonian and obtaining exact transition probabilities between the initial and final states of aquantum oscillator in analytic form without cumbersome calculations. The case of a D-dimensional oscillator is briefly discussed, in particular, in application to the problem of vacuum pair creation in an intense electric field.

Refined formulation of quantum analysis,q-derivative and exponential splitting

Quantum analysis is reformulated to clarify its essence, namely the invariance of quantum derivative for any choice of definitions of the differential df(A) satisfying the Leibniz rule. This formulation with use of the inner derivation ?A is convenient to study quantum corrections in contrast to the Feynman operator calculus. The present analysis can also be used to find a general scheme of constructing exponential product formulae of higher order. General recursive schemes are also reviewed with an emphasis to standard symmetric splitting formulae. Multiple integral representations of q-derivatives are derived using such general integral formulae of quantum derivatives as are expressed by hyperoperators. A simple explanation of the connection between quantum derivatives and q-derivative is also given.

EXTRACTING LINEAR FACTORS IN FEYNMAN'S OPERATIONAL CALCULI : THE CASE OF TIME DEPENDENT NONCOMMUTING OPERATORS

Disentangling is the essential operation of Feynman's operational calculus for noncommuting operators. Thus formulas which simplify this operation are central to the subject. In a recent paper the procedure for 'extracting a linear factor' has been established in the setting of Feynman's operational calculus for time independent operators and associated probability measures . While the setting just described is natural in many circumstances, it is not natural for evolution problems. There the measures should not be restricted to probability measures and it is worthwhile to allow the operators to depend on time. The main purpose for this paper is to extend the procedure for extracting a linear factor to this latter setting. We should mention that Feynman's primary motivation for developing an operational calculus for noncommuting operators came from a desire to describe the evolution of certain quantum systems.m systems.

Effects of absolute continuity in Feynman’s operational calculus

We investigate the effects of having an absolute continuity relation between the time-ordering measures in Feynman's operational calculus. In particular, we obtain some theorems concerning the formation of functions of several noncommuting operators or operator-valued functions under specific absolute continuity assumptions on the time-ordering measures.

Stability Properties of Feynman's Operational Calculus for Exponential Functions of Noncommuting Operators

Stability properties of Feynman's operational calculus are addressed in the setting of exponential functions of noncommuting operators. Applications of some of the stability results are presented. In particular, the time-dependent perturbation theory of nonrelativistic quantum mechanics is presented in the setting of the operational calculus and application of the stability results of this paper to the perturbation theory are discussed.

Foundations for relativistic quantum theory. I. Feynman’s operator calculus and the Dyson conjectures

In this paper, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initial-value problem and construct the Dyson series. We show that the series is asymptotic, thus proving Dyson’s second conjecture for quantum electrodynamics. In addition, we show that the expansion may be considered exact to any finite order by producing the remainder term. This implies that every nonperturbative solution has a perturbative expansion. Using a physical analysis of information from experiment versus that implied by our models, we reformulate our theory as a sum over paths. This allows us to relate our theory to Feynman’s path integral, and to prove Dyson’s first conjecture that the divergences are in part due to a violation of Heisenberg’s uncertainly relations.

Feynman's Operational Calculus for a Sequential Operator-Valued Function Space Integral

We study Feynman's operational calculus and change of scale for a sequential operator-valued function space integral as a bounded linear operator from L 2(R) into itself.

Path Integrals, Fourier Transforms, and Feynman's Operational Calculus

The disentangling process is the key to Feynman's operational calculus for noncommuting operators. The main result of his heuristic calculations deals with disentangling an exponential factor. We use the Wiener and Feynman integrals to make this disentangling (or time-ordering) mathematically rigorous in the case where the analytic functions from earlier work are replaced by Fourier transforms of complex-valued measures.

Feynman's Operational Calculus and Evolution Equations

We give an exposition and an extension of the ideas of Feynman's time-ordered operational calculus for noncommuting operators. Various directions for 'disentangling' functions of such operators are provided by measures on the time intervals in question. We concentrate especially on exponentials of sums of noncommuting operators and prove that the unique solution of a broad class of evolution equations is given by the time-dependent operators arrived at by disentangling such exponential expressions.

Noncommutative operations on Wiener functionals and Feynman's operational calculus

The authors recently introduced a family { A t t⩾0} of Banach algebras of functionals on Wiener space and showed that, for every F in A t the functional integrals K λ t ( F) exist and are given by a perturbation expansion which serves to “disentangle,” in the sense of Feynman's operational calculus, the operator K λ t ( F). When λ is purely imaginary, K λ t ( F) is the analytic operator-valued Feynman integral of F. In this paper, noncommutative operations ∗ and + on Wiener functionals are defined which enable the authors to provide a precise and rigorous interpretation of certain aspects of Feynman's operational calculus for noncommuting operators. In particular, the authors show that if FϵA t1 and G ϵ A t2 then F ∗G ϵA t1+t2 and K λ t 1 + t 2 (F ∗ G) = K λ t 1 (F)K λ t 2 (G) . Hence, the product of operators which can be disentangled using their recent results can itself be disentangled. The noncommutative operations, the disentangling algebras { A t}, and the functional integrals possess a rich interlocking structure.

Operator method for the perturbative solution of ordinary differential equations

We adapt the Feynman operator calculus to the Lie operators connected with the solution of ordinary, coupled, first order differential equations, particularly those arising from non-linear oscillations or from interactions between modes. Combining this calculus with the method of averaging we obtain approximate solutions which are algebraic and exponential functions of the appropriate expansion parameter rather than power series of this parameter. In addition to illustrating the method for the damped harmonic oscillator and for the van der Pol oscillator we present an approximate solution of the Lotka-Volterra problem of two competing species. The solutions are compared numerically with those of a variant of the Bogoliubov-Krylov-Mitropolsky (BKM) method of averaging. The bulk of the analytic calculations both for the operator algorithm and for the BKM method have been automated.

Intramolecular Nonradiative Transitions in the ``Non-Condon'' Scheme

In this paper we consider the nonradiative intramolecular decay of a large molecule utilizing Born-Oppenheimer wavefunctions as a zero order basis, and bypassing the conventional Condon approximation for the calculation of the electronic coupling matrix matrix elements. The electronic adiabatic wavefunctions are expanded in terms of the Wigner-Brillouin perturbation series in the weak electronic-vibrational coupling limit. We have applied a generalized version of Feynman's operator calculus to derive general expressions for the nonradiative decay probability of a statistical harmonic molecule characterized by displaced potential surfaces. Numerical calculations were performed for the decay of the vibrationless excited electronic state in the ``non-Condon'' scheme. The numerical data for the decay rate in a two electronic level system in the weak electronic vibrational coupling limit exceed the results obtained invoking the Condon approximation by 2–3 orders of magnitude; the exact correction factor depends on the molecular parameters, and, in particular, is roughly proportional to the square of the (frequency normalized) electronic energy gap. Finally, the relevant off resonance coupling terms in the adiabatic representation are shown to be appreciably smaller than the near resonance coupling terms, demonstrating the superiority of the adiabatic basis over the crude adiabatic basis in describing electronic relaxation processes.