Let X be a locally convex real or
complex vector space. Denote by L(X) the space of linear continuous operators on X.
The oscillations of the particle in the zeropoint field may be considered as complex rotations in
complex vector space. The local particle harmonic oscillator is analysed in the complex vector formalism considering the algebra of complex null vectors.
In this framework, C is the
complex vector space, [G.sub.n] is the Clifford algebra or geometric algebra associated with the n-dimensional real space [V.sup.n], and [dot in a circle]: ([Real part] x [cross product]) is the new generalized geometric product.
Complex numbers are sets that represent possible physical states and form an abstract
complex vector space of growth and increment.
A quantum bit, or qubit, is a unit vector in a two-dimensional
complex vector space for which a particular basis, denoted by {|0>, |1>}, has been fixed.
In fact, let L be any nondegenerate inner product on a real or
complex vector space. We claim that L restricted to any generic subspace is also nondegenerate.
The aim of this article is to explore the structure of the photon in
complex vector space. To understand the structure of the photon, the electromagnetic field is expressed as a complex vector and the total energy momentum even multivector is developed in section 2.
In [2], Abardia and Bernig studied projection bodies in
complex vector spaces: The real vector space V of real dimension n is replaced by a
complex vector space W of complex dimension m and the group SL(V) = SL(n, R) is replaced by the group SL(W,C) = SL(m,C).
By a Hermitian structure on [C.sup.2] we mean any Hermitian product (definite or indefinite) G on the
complex vector space [C.sup.2].
If A [member of] [0, +[infinity]), we denote by [H.sub.b(s,m(r;q))] ([B.sub.1/A](0)) the
complex vector space of all f [member of] H ([B.sub.1/A] (0)) such that [[??].sup.j]f (0) [member of] [P.sub.(s,m(r;q))] ([sup.j]E), for all j [member of] N and
Beginning chapters cover mathematical preliminaries (complex numbers and
complex vector spaces), and present the basic architecture of quantum computing.