26,117 research outputs found
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 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
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 -algebras are equivalent to the category of
real vector spaces, the category of real Hilbert spaces and the category of
real -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 -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
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
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