Energy Gaps in a Spacetime Crystal

This paper presents an analysis of the band structure of a spacetime potential lattice created by a standing electromagnetic wave. We show that there are energy band gaps. We estimate the effect, and propose a measurement that could confirm the existence of such phenomena.Comment: 8 pages. 2 figure

Lax-Phillips Scattering Theory of a Relativistic Quantum Field Theoretical Lee-Friedrichs Model and Lee-Oehme-Yang-Wu Phenomenology

The one-channel Wigner-Weisskopf survival amplitude may be dominated by exponential type decay in pole approximation at times not too short or too long, but, in the two channel case, for example, the pole residues are not orthogonal, and the pole approximation evolution does not correspond to a semigroup (experiments on the decay of the neutral K-meson system support the semigroup evolution postulated by Lee, Oehme and Yang, and Yang and Wu, to very high accuracy). The scattering theory of Lax and Phillips, originally developed for classical wave equations, has been recently extended to the description of the evolution of resonant states in the framework of quantum theory. The resulting evolution law of the unstable system is that of a semigroup, and the resonant state is a well-defined funtion in the Lax-Phillips Hilbert space. In this paper we apply this theory to relativistically covarant quantum field theoretical form of the (soluble) Lee model. We show that this theory provides a rigorous underlying basis for the Lee-Oehme-Yang-Wu construction.Comment: Plain TeX, 34 page

On quaternionic functional analysis

In this article, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion $B^*$-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real $C^*$-algebras respectively. We will also give a Riesz representation theorem for quaternion Hilbert spaces and will extend two results of Kulkarni (namely, we will give the full versions of the Gelfand-Naimark theorem and the Gelfand theorem for quaternion $B^*$-algebras). On our way to these results, we compare, clarify and unify the term "quaternion Hilbert spaces" in the literatures.Comment: to appear in the Mathematical Proceedings of the Cambridge Philosophical Societ

Illumination by Taylor Polynomials

Let f(x) be a differentiable function on the real line R, and let P be a point not on the graph of f(x). Define the illumination index of P to be the number of distinct tangents to the graph of f which pass thru P. We prove that if f '' is continuous and nonnegative on R, f '' > m >0 outside a closed interval of R, and f '' has finitely many zeroes on R, then every point below the graph of f has illumination index 2. This result fails in general if f '' is not bounded away from 0 on R. Also, if f '' has finitely many zeroes and f '' is not nonnnegative on R, then some point below the graph has illumination index not equal to 2. Finally, we generalize our results to illumination by odd order Taylor polynomials.Comment: Minor modifications and correction

On the significance of a recent experiment demonstrating quantum interference in time

I comment on the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory, used by the authors in their analysis, cannot account for the results found, and therefore that this experiment has fundamental importance beyond the technical advances it represents. Some theoretical structures which consider the time as an observable, and thus could, in principle, have the required coherence in time, are discussed briefly, and the application of Floquet theory and the manifestly covariant quantum theory of Stueckelberg are treated in some detail. In particular, the latter is shown to account for the results in a simple and consistent way.Comment: 10 pages, plain TeX. Revision for clarity, reference to other candidate theorie

Compositions of Polynomials with Coefficients in a given Field

Let F and K be fields of characteristic 0, with F a subset of K. Let K[x] denote the ring of polynomials with coefficients in K. For p in K[x]\F[x], deg(p) = n, let r be the highest power of x with a coefficient not in F. We define the F deficit of p to be D_F(p) = n-r. For p in F[x], D_F(p) = n. Suppose that the leading coeffcients of p and q are in F, and that some coefficient of q(other than the constant term) is not in F. Our main result is that the F deficit of the composition of p with q equals the F deficit of q. This implies our earlier result: If p(q(x)) is in F[x]then p is in F[x] and/or q is in F[x]. We also prove similar results for compositions of the form p(q(x,y)), for the iterates of a polynomial, and for fields of finite characteristic, if the characteristic of the field does not divide the degree of p. Finally, If F and K are only rings, then we prove the inequality D_F(p(q(x))) >=D_F(q).Comment: Minor modifications and correction