arXiv subjects
Publications and source records attributed to Hideki Omori.
Consider the problem "Give the equation of the conceptional rotations in ${\mathbb R}^3$ without using the parameter expressing individual rotations", just as the conceptional motion of constant velocity along straight lines (Galiley motions) is expressed by an elementary differential equation. In this we try to give an answer to the topic related with the above issue.
Thinking back the long history of physics, we see that the calculation used by physicists was nothing but the ordinary calculus. Another word, physicists have never wrote theories beyond the basic axioms of the calculus. This is not to declare of the victory of calculus or algebraic topology. On the contrary, we are thinking that every theory of mathematical physics must suggest new frontier of ordinary calculus, which are never viewed by classical geometers. Weyl algebras or Heisenberg algebras are naturally involved in slightly extended systems of the algebra of ordinary calculus, and are supported by the classical notion of phase spaces on which the general mechanics are based. The theory of deformation quantizations gives a notion of quantization of "phase space". To explain its essence in brief we proposed in the previous note the notion of $μ$-regulated algebra. In this series, we have introduced elements, called "vacuums" to consider the state vectors and the configuration spaces within the world of extended algebra of calculus with various expressions. We have found several strange elements, called polar elements, and an extended notions of vacuums, which were called pseudo-vacuums in our paper. These are not established notions in mathematical physics, but we are thinking that these must propose new frontier for mathematical physics. We are thinking that vacuums and pseudo-vacuums are not unique, but the function algebra of the configuration spaces must be an algebra similar to the Frobenius algebra defined by vacuums. The point in this note is that to obtain classical pictures one has often to restrict the expression parameters, and there are two essentially different expression parameters.
The Weyl algebra (W_{2m}[h]; *) is the algebra generated by u=(u_1,...,u_m,v_1,.....,v_m) over C with the fundamental commutation relation [u_i,v_j]=-ihδ_{ij}, where h is a positive constant. The Heisenberg algebra (\Cal H_{2m}[nu];*) is the algebra given by regarding the scalar parameter h in the Weyl algebra W_{2m}[h] to be a generator nu which commutes with all others.
This is a noncommutative version of the previous work entitled "Deformation Expression for Elements of Algebras (I)." In general in a noncommutative algebra, there is no canonical way to express elements in univalent way, which is often called "ordering problem". In this note we discuss this problem in the case of the Weyl algebra of 2m-generators. By fixing an expression, we extends Weyl algebra transcendentally. We treat *-exponential functions of linear forms, and quadratic forms of crossed symbol under generic expression parameters.
In this note, we are interested in the *-version of various special functions. Noting that many special functions are defined by integrals involving the exponential functions, we define *-special functions by similar integral formula replacing exponential functions by *-exponential functions.
In this note, we mainly consider the extended Weyl algebra of two generators (u,v), that is, the algebra generated by u,v with the fundamental commutation relation. Weyl algebra is realized on the space of polynomials of u and v by defining various product depending on a symmetric matrix K called the expression parameter. Via such expressions and ordinary calculus one can treat various transcendental elements such as *-exponential functions and elements obtained by integrations.
In a noncommutative algebra there is no canonical way to express elements in univalent way, which is often called "ordering problem". In this note we give product formula of the Weyl algebra in generic ordered expression. In particular, the generic product formula of *-exponential functions of quadratic forms will be given.
This paper presents a preliminary version of the deformation theory of expressions of elements of algebras. The notion of *-functions is given. Several important problems appear in simplified forms, and these give an intuitive bird's-eye of the whole theory.
Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is studied as an example of bundle-like objects with discordant (sogo) transition functions, which suggests us to treat movable branching singularities.
We propose a relatively new notion of two-valued elements, which arises naturally in constructing the star exponential functions of the quad-ratics in the Weyl algebra over the complex number field. This notion enables us to describe the group like objects of the set of star exponential functions of quadratics in the Weyl algebra.
Ideas from deformation quantization are applied to deform the expression of elements of an algebra. Extending these ideas to certain transcendental elements implies that $\frac{1}{i\h}uv$ in the Weyl algebra is naturally viewed as an indeterminate living in a discrete set $\mathbb{N}{+}{1/2}$ {\it or} ${-}(\mathbb{N}{+}{1/2})$ . This may yield a more mathematical understanding of Dirac's positron theory.