We define multiplication and convolution of distributions and ultradistributions by introducing the notions of evaluation of distributions and integration of ultradistributions. An application is made to an interpretation of the Dirac formalism of quantum mechanics. The role of the Hilbert space of states is played by what is termed a Hermitian orthonormal system, and operators are replaced by the generalized matrices. We describe a simple example of one dimensional free particle and construct explicitly a representation of the Weyl algebra as the generalized matrices.
REFERENCES
1.
O.
Heaviside
, Proc. R. Soc. London
54
105
(1893
).2.
3.
4.
Théorie Des Distributions
(Hermann
, Paris
, 1951
), Vol. 2
.5.
6.
P.
Antosik
, J.
Mikusiński
, and R.
Sikorski
, Theory of Distributions: The Sequential Approach
(Elsevier Scientific
, Amsterdam
, 1973
).7.
I.
Richards
and H.
Youn
, Theory of Distributions: A Non-Technical Introduction
(Cambridge University Press
, Cambridge
, 1990
).8.
9.
I. M.
Gel’fand
and N. Ya.
Vilenkin
, Generalized Functions
(Academic
, New York
, 1964
), Vol. 4
.10.
11.
I. M.
Gel’fand
and G. E.
Shilov
, Generalized Functions
(Academic
, New York
, 1964
), Vols. 1
and 2.12.
13.
C.
Gasquet
and P.
Witomski
, Fourier Analyis and Applications
(Springer
, New York
, 1998
).14.
15.
B.
Nagel
, “Introduction to rigged Hilbert spaces (RHS)
,” Resonances: The Unifying Route Towards the Formulation of Dynamical Processes. Foundations and Applications in Nuclear, Atomic and Molecular Physics
, edited by E.
Brädas
and N.
Elander
, Lecture Notes in Physics
(Springer
, Berlin
, 1989
), Vol. 325
.© 2010 American Institute of Physics.
2010
American Institute of Physics
You do not currently have access to this content.