Weak Convergence and Vector-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi

In this paper we present a theorem that establishes a relation between continuous, norm-bounded functions from a metric space into a separable Hilbert space and weak convergence of sequences of probability measures on the metric space. After establishing this result, it’s application to the stability theory of Feynman’s operational calculi will be illustrated. We will see that the existing time-dependent stability theory of the operational calculi will be significantly improved when the operator-valued functions take their values in , a separable Hilbert space.      

Weak Convergence and Banach Space-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi

In this paper we investigate the relation between weak convergence of a sequence \(\left\{ \mu_{n}\right\} \) of probability measures on a Polish space S converging weakly to the probability measure μ and continuous, norm-bounded functions into a Banach space X. We show that, given a norm-bounded continuous function f:S→X, it follows that \(\lim_{n\to\infty}\int_{S}f\, d\mu_{n}=\int_{S}f\, d\mu\)—the limit one has for bounded and continuous real (or complex)—valued functions on S. This result is then applied to the stability theory of Feynman’s operational calculus where it is shown that the theory can be significantly improved over previous results.     

Spectral Theory for Periodic Schrödinger Operators with Reflection Symmetries

Let H=–+V be defined on ^d with smooth potential V, such that In addition we assume that where This is a periodic Schrödinger operator with additional reflection symmetries. We investigate the associated Floquet operators H ^q , q[0,1] ^d . In particular we show that the associated lowest eigenvalues _q are simple if q=(q ₁ ,q ₂ ,,q _d ) satisfies q _j 1/2 for each j=1,2,,d. Supported by Ministerium für Bildung, Wissenschaft und Kunst der Republik ÖsterreichSupported by the European Science Foundation Programme Spectral Theory and Partial Differential Equations (SPECT)     

Euler’s Theorem for Homogeneous White Noise Operators

In this paper we introduce a new notion of λ ?order homogeneous operators on the nuclear algebra of white noise operators. Then, we give their Fock expansion in terms of quantum white noise (QWN) fields \(\{a_{t},\: a^{*}_{t}\, ; \; t\in \mathbb {R}\}\). The quantum extension of the scaling transform enables us to prove Euler’s theorem in quantum white noise setting.     

Inverse Spectral Problems for Schrödinger Operators

In this article we find some explicit formulas for the semi-classical wave invariants at the bottom of the well of a Schrödinger operator. As an application of these new formulas for the wave invariants, we improve the inverse spectral results proved by Guillemin and Uribe in [GU]. They proved that under some symmetry assumptions on the potential V(x), the Taylor expansion of V(x) near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schrödinger operator in \({\mathbb R^n}\) . We prove similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of V(x).     

The magic of Feynman’s QED: from field-less electrodynamics to the Feynman diagrams

For some time, even after the Feynman diagrams and rules were publicly known, the foundations of Feynman’s quantum electrodynamics remained mostly private. Its stupendous efficiency then appeared like magic to most of his competitors. The purpose of this essay is to reveal the hidden contrivances of this magic, in a journey from field-less electrodynamics to the Feynman diagrams.

     

Searching for a response: the intriguing mystery of Feynman’s theoretical reference amplifier

We analyze Feynman’s work on the response of an amplifier performed at Los Alamos and described in a technical report of 1946, as well as lectured on at the Cornell University in 1946–47 during his course on Mathematical Methods. The motivation for such a work was Feynman’s involvement in the Manhattan Project, for which the necessity emerged of feeding the output pulses of counters into amplifiers or several other circuits, with the risk of introducing distortion at each step. In order to deal with such a problem, Feynman designed a theoretical “reference amplifier”, thus enabling a characterization of the distortion by means of a benchmark relationship between phase and amplification for each frequency, and providing a standard tool for comparing the operation of real devices. A general theory was elaborated, from which he was able to deduce the basic features of an amplifier just from its response to a pulse or to a sine wave of definite frequency. Moreover, in order to apply such a theory to practical problems, a couple of remarkable examples were worked out, both for high-frequency cutoff amplifiers and for low-frequency ones. A special consideration deserves a mysteriously exceptional amplifier with best stability behavior introduced by Feynman, for which different physical interpretations are here envisaged. Feynman’s earlier work then later flowed in the Hughes lectures on Mathematical Methods in Physics and Engineering of 1970–71, where he also remarked on causality properties of an amplifier, that is on certain relations between frequency and phase shift that a real amplifier has to satisfy in order not to allow output signals to appear before input ones. Quite interestingly, dispersion relations to be satisfied by the response function were introduced.

     

Einstein, Wigner and Feynman: From E=mc 2 to Feynman’s decoherence via Wigner’s little groups

The 20th-century physics starts with Einstein and ends with Feynman. Einstein introduced the Lorentz-covariant world with E=mc ². Feynman observed that fast-moving hadrons consist of partons which interact incoherently with external signals. If quarks and partons are the same entities observed in different Lorentz frames, the question then is why partons are incoherent while quarks are coherent. This is the most puzzling question Feynman left for us to solve. In this report, we discuss Wigner’s role in settling this question. Einstein’s E=mc ², which takes the form $E = \sqrt {m^2 + p^2 } $ , unifies the energy-momentum relations for massive and massless particles, but it does not take into account internal space-time structure of relativistic particles. It is pointed out Wigner’s 1939 paper on the inhomogeneous Lorentz group defines particle spin and gauge degrees of freedom in the Lorentz-covariant world. Within the Wigner framework, it is shown possible to construct the internal space-time structure for hadrons in the quark model. It is then shown that the quark model and the parton model are two different manifestations of the same covariant entity. It is shown therefore that the lack of coherence in Feynman’s parton picture is an effect of the Lorentz covariance.     

Hamiltonian Dynamics and Spectral Theory for Spin–Oscillators

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.     

Measure permutation formulas in Feynman’s operational calculi

In Jefferies-Johnson’s theory of Feynman’s operational calculi for noncommuting operators, the two operators T _{µ 1,µ 2} f(Ã, \(\tilde B\)) and T _{µ 2},µ₁ f(Ã, \(\tilde B\)) are not equal. Relationships between these two operators are given, i.e., “measure permutation formulas” in Feynman’s operational calculi are developed; they correspond to the “index permutation formula” in Maslov’s discretized version of Feynman’s operational calculus.     

Destructive interferences results in bosons anti bunching: refining Feynman’s argument

The photo-absorption cross sections for the dissociative ionization and dissociative excitation of H ₂ ⁺ are calculated for photon polarization parallel and perpendicular to the internuclear axis. The wavefunctions for the initial and final states are prepared using the Born-Oppenheimer approximation. For dissociative ionization, the cross section and angular distribution of photo-electrons are compared with those calculated with the fixed-nuclei approximation. For dissociative excitation, the cross sections for H?(N = 1～4) productions are shown.     

Spectral Gaps for Periodic Schrödinger Operators with Strong Magnetic Fields

We consider Schrödinger operators H^h=(ihd+A)*(ihd+A) with the periodic magnetic field B=dA on covering spaces of compact manifolds. Using methods of a paper by Kordyukov, Mathai and Shubin [14], we prove that, under some assumptions on B, there are in arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of the strong magnetic field h0.Acknowledgement I am very thankful to Bernard Helffer for bringing these problems to my attention and useful discussions and to Mikhail Shubin for his comments.     

On Some Sharp Spectral Inequalities for Schrödinger Operators on Semiaxis

In this paper we obtain sharp Lieb–Thirring inequalities for a Schrödinger operator on semiaxis with a matrix potential and show how they can be used to other related problems. Among them are spectral inequalities on star graphs and spectral inequalities for Schrödinger operators on half-spaces with Robin boundary conditions.     

Semiclassical Inverse Spectral Theory for Singularities of Focus–Focus Type

We prove, assuming that the Bohr–Sommerfeld rules hold, that the joint spectrum near a focus–focus singular value of a quantum integrable system determines the classical Lagrangian foliation around the full focus–focus leaf. The result applies, for instance, to ?-pseudodifferential operators on cotangent bundles and Berezin–Toeplitz operators on prequantizable compact symplectic manifolds.     

Celebrity Physicist: How the Press Sensationalized Einstein’s Search for a Unified Field Theory

In Einstein’s later years, from the late 1920s onward, his reputation in the physics community as an innovator had faded as he pursued increasingly unrealistic unified field theories. Yet from the perspective of the press, his image and ideas were still marketable. We will see how his various attempts to craft a unified field theory generated numerous headlines, despite their lack of experimental evidence or even realistic solutions. We will examine how Einstein’s “latest theory,” became a much sought-after commodity used to generate interest in books, magazines, and newspapers.     

Generalized measure permutation formulas in Feynman’s operational calculi

Measure permutation formulas in Feynman’s operational calculi for noncommuting operators give relationships between the two operators \(\mathcal{T}_{\mu 1,\mu 2} f\left( {\tilde A,\tilde B} \right)\) and \(\mathcal{T}_{\mu 2,\mu 1} f\left( {\tilde A,\tilde B} \right)\) . We develop generalized and iterated measure permutation formulas in the Jefferies-Johnson theory of Feynman’s operational calculi. In particular, we apply our formulas to derive an identity for a function of the Pauli matrices.     

Feynman’s Relativistic Electrodynamics Paradox and the Aharonov-Bohm Effect

An analysis is done of a relativistic paradox posed in the Feynman Lectures of Physics involving two interacting charges. The physical system presented is compared with similar systems that also lead to relativistic paradoxes. The momentum conservation problem for these systems is presented. The relation between the presented analysis and the ongoing debates on momentum conservation in the Aharonov-Bohm problem is discussed.