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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 QED. In addition, we show that the expansion may be considered exact to any finite order by producing… (More)

In this paper, we construct a parallel image of the conventional Maxwell theory by replacing the observer-time by the proper-time of the source. This formulation is mathematically, but not physically, equivalent to the conventional form. The change induces a new symmetry group which is distinct from, but closely related to the Lorentz group, and fixes the… (More)

- T. L. GILL
- 2006

Let Ω be an open domain of class C 3 contained in R 3 , let (L 2 [Ω]) 3 be the real Hilbert space of square integrable functions on Ω with values in R 3 , and let H[Ω] be the completion of the set, ˘ u ∈ (C ∞ 0 [Ω]) 3 | ∇ · u = 0 ¯ , with respect to the inner product of (L 2 [Ω]) 3. A well-known unsolved problem is the construction of a sufficient class of… (More)

- T. L. GILL
- 2003

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit… (More)

In this paper, using the theory of fractional powers for operators, we construct the most general (analytic) representation for the square-root operator of relativistic quantum theory. We allow for arbitrary, but time-independent, vector potential and mass terms. Our representation is uniquely determined by the Green's function for the corresponding… (More)

This paper is a progress report on the foundations for the canonical proper-time approach to relativistic quantum theory. We first review the the standard square-root equation of relativistic quantum theory, followed by a review of the Dirac equation, providing new insights into the physical properties of both. We then introduce the canonical proper-time… (More)

- TEPPER L. GILL
- 2015

The purpose of this note is to show that, if B is a uniformly convex Banach, then the dual space B has a " Hilbert space representation " (defined in the paper), that makes B much closer to a Hilbert space then previously suspected. As an application, we prove that, if B also has a Schauder basis (S-basis), then for each A ∈ C[B] (the closed and densely… (More)

- TEPPER L. GILL
- 2015

In this note, we introduce a new class of separable Banach spaces, SD p [R n ], 1 p ∞, which contain each L p-space as a dense continuous and compact embedding. They also contain the nonabso-lutely integrable functions and the space of test functions D[R n ], as dense continuous embeddings. These spaces have the remarkable property that, for any multi-index… (More)

- T. L. GILL
- 2006

In this paper, we survey recent progress on the constructive theory of the Feynman operator calculus. We first develop an operator version of the Henstock-Kurzweil integral, and a new Hilbert space that allows us to construct the elementary path integral in the manner originally envisioned by Feynman. After developing our time-ordered operator theory we… (More)