Infinitesimal Fourier Transformation for the Space of Functionals

The purpose is to formulate a Fourier transformation for the space of functionals, as an infinitesimal meaning. We extend R to ( ∗R) under the base of nonstandard methods for the construction. The domain of a functional is the set of all internal functions from a ∗-finite lattice to a ∗-finite lattice with a double meaning. Considering a ∗-finite lattice with a double meaning, we find how to treat the domain for a functional in our theory of Fourier transformation, and calculate two typical examples. 0. Introduction Recently many kinds of geometric invariants are defined on manifolds and they are used for studying low dimensional manifolds, for example, Donaldson’s invariant, Chern-Simon’s invariant and so on. They are originally defined as Feynman path integrals in physics. The Feynman path integral is in a sense an integral of a functional on an infinite dimensional space of functions. We would like to study the Feynman path integral and the originally defined invariants. For the purpose, we would be sure that it is necessary to construct a theory of Fourier transformation on the space of functionals. For it, as the later argument, we would need many stages of infinitesimals and infinites, that is, we need to put a concept of stage on the field of real numbers. We use nonstandard methods to develop a theory of Fourier transformation on the space of functionals. Feynman([F-H]) used the concept of his path integral for physical quantizations. The word ′′physical quantizations′′ has two meanings : one is for quantum mechanics and the other is for quantum field theory. We usually use the same word ′′Feynman path integral′′. However the meanings included in ′′Feynman path integral′′ are two sides, according to the above. One is of quantum mechanics and the other is of quantum field theory. To understand the Feynman path integral of the first type, Fujiwara([F]) studied it as a fundamental solution, and 1 Hida([H]), Ichinose, Tamura([Ic],[I-T]) studied it from their stochastical interests and obtained deep results, using standard mathematics. In stochastic mathematics, Loeb([Loe]) constructed Loeb measure theory and investigated Brownian motion that relates to Itô integral([It]). Anderson([An]) deloped it. Kamae([Ka]) proved Ergodic theory using nonstandard analysis. From a nonstandard approach, Nelson([Ne]), Nakamura([Na1],[Na2]) studied Schrödinger equation, Dirac equation and Loo([Loo1],[Loo2]) calculated rigidly the quantum mechanics of harmonic oscillator. It corresponds to functional analysis on the space of functions in standard mathematics. On the other hand, we would like to construct a frame of Feynman path integral of the second type, that is, a functional analysis on the space of functionals. Our idea is the following : in nonstandard analysis, model theory, especially non-wellfounded set theory([N-O-T]), we can extend R to ∗R furthermore a double extension ( ∗R), and so on. For formulation of Feynman path integral of the first type, it was necessary only one extension ∗R of R in nonstandard analysis([A-F-HK-L]). In fact there exists an infinite in ∗R, however there are no elements in ∗R, that is greater than images of the infinite for any functions. The same situation occurs for infinitesimals. Hence we consider to need a further extension of R to construct a formulation of Feynman path integral of the second type. If the further extension satisfies some condition, the extension ( ∗R) has a higher degree of infinite and also infinitesimal. We use these to formulate the space of functionals. We would like to try to construct a theory of Fourier transformation on the space of functionals and calculate two typical examples of it. Historically, for the theories of Fourier transformations in nonstandard analysis, in 1972, Luxemburg([Lu]) developed a theory of Fourier series with ∗-finite summation on the basis of nonstandard analysis. The basic idea of his approach is to replace the usual ∞ of the summation to an infinite natural number N . He approximated the Fourier transformation on the unit circle by the Fourier transformation on the group of Nth roots of unity. Gaishi Takeuti([T]) introduced an infinitesimal delta function δ, and Kinoshita([Ki]) defined in 1988 a discrete Fourier transformation for each even ∗-finite number H(∈ ∗R) : (Fφ)(p) = ∑ −H 2 ≤z<H 2 1 H exp(−2πip 1 H z)φ( 1 H z), called ”infinitesimal Fourier transformation”. He developed a theory for the infinitesimal Fourier transformation and studied the distribution space deeply, and proved the same properties hold as usual Fourier transformation of L(R). Especially saying, the delta function δ satisfies that δ, δ, ... , √ δ, ... are also hyperfunctions as their meaning, and Fδ = 1, Fδ = H , Fδ = H, ... , F √ δ = 1 √ H , ... . In 1989, Gordon([G]) independently defined a generic, discrete Fourier transformation for each infinitesimal ∆ and ∗-finite number M , defined by (F∆,M φ)(p) = ∑ −M≤z≤M ∆exp(−2πip∆z)φ(∆z). He studied under which condition the discrete Fourier transformation F∆,M approximates the usual Fourier transformation F for L(R). His proposed condition is (A′) of his notation : let ∆ be an infinitely small and M an infinitely large natural number such that M ·∆ is in2 finitely large. He showed that under the condition (A′) the standard part of F∆,M φ approximates the usual Fφ for φ ∈ L(R). One of the different points between Kinoshita’s and Gordon’s is that there is the term ∆exp(−2πip∆M)φ(∆M) in the summation of their two definitions or not. We mention that both definitions are same for the standard part of the dicrete Fourier transformation for φ ∈ L(R) and Kinoshita’s definition satisfies the condition (A′) for an even infinite number H if ∆ = 1 H , M = H 2 2 . We shall extend their theory of Fourier transformation for the space of functions to a thery of Fourier transformation for the space of functionals. For the purpose of this, we shall represent a space of functions from R to R as a space of functions from a set of lattices in an infinite interval [ − 2 , H 2 ) to a set of lattices in an infinite interval [ −′ 2 , H ′ 2 ) . We consider what H ′ is to treat any function from R to R. If we put a function a(x) = x(n ∈ Z), we need that H 2 is greater than ( H 2 )n , and if we choose a function a(x) = e, we need that H ′ 2 is greater than e H 2 . If we choose any infinite number, there exists a function whose image is beyond the infinite number. Since we treat all functions from R to R, we need to put H ′ 2 as an infinite number greater than any infinite number of ∗R. Hence we make [ −′ 2 , H ′ 2 ) not in ∗R but in ( ∗R), where ( ∗R) is a double extension of R, that is, H ′ is an infinite number in ( ∗R). First we shall develop an infinitesimal Fourier transformation theory for the space of functionals, and secondly we calculate foundamental two examples for our infinitesimal Fourier transformation. In our case, we define an infinitesimal delta function δ satisfies that Fδ = 1, Fδ = H ′H 2 , Fδ = H ′2H 2 , ... , F √ δ = H ′− 1 2 H2 , ... , that is, Fδ, Fδ, ... are infinite and F √ δ, ... are infinitesimal. These are a functional f and an infinite-dimensional Gaussian distribution g where st(f(α)) = exp (