The problems of the non-archimedean analysis generated by quantum physics

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1.Analytic Functions over Non-Archimedean Fields Now we introduce spaces of analytic functions which will play a basic role in a theory of distributions.We study only the case of functions defined on balls with the centre at zero.It is much more simple to consider only the balls of radii R E r, because only such balls are the "natural balls." Denote by A' a complete non-Archimedean field ,CharK = 0 , |. IK is the valuation on K ; r = ~K~ = {a = E A'}.As usual Kn = A x ... x A. Denote by UR a ball UR(0) = {.r: R} in Ii".The function f : : UR -li, R E r, is said to be analytic if the series a converges uniformly on UR.Here as usual a = (03B11, ..., 03B1n), a; = 0, 1, 2, ..., and 03B1 = Xfl ....c~".By virtue of the non-Archimedean Cauchy criterion of convergence (1.1), it is necessary and sufficient that lim |f03B1|KR|03B1| = 0 (1.2) |03B1|~T o prove that this condition is necessary we use R E r.So there exists such aR e A-that |aR|K = R.The point XR = (aR, ..., aR) ~ UR.And if (1.1) converges in this point, then (1.2) holds.
The topology in the space A(UR, li ) (= AR) of the functions analytic on the ball UR is defined by the non-Archimedean norm = max03B1|f03B1|K R03B1.This is a non-Archimedean Banach space.
The function f : A" ta is entire if series ( l.1 ) converges on the ball UR for any R E f .The topology in the space of entire functions ,~(Ii", li )(= A) is defined by a system on non-Archimedean norms .
A sequence of entire functions converges in A if it converges uniformly on each ball UR.
There is no problem to see that this topology could be defined with the aid of arbitrary sequence of norms { .Rk } k= 1 , where limk~~ Rk = oo.It is a non-Archimedean Frechet space.Such type of topology is known as the topology of projective limit of Banach spaces A(Kn) = lim proj A(UR) R-oo Remark 1.1.A sequence of functions converges in A to f iff 0 for every norm ~.~R.
The function f is said to be analytic at zero if there exists R E r such that f E A(UR).
The space of functions analytic at zero, =E A0(Kn, Ii )(= A0) is provided with a topology of an inductive limit: A 0 ( K n ) = l i m indA(UR)

2.Analytic Distributions, Gauss Distributions
We choose the spaces of analytic functions ~4(A~) and as the spaces of test functions and the spaces of [(-linear continuous functionals and A'0(Kn) as the spaces of distributions (generalized functions).
As usual, it is convenient to use the symbol of an integral to denote the effect produced by a distribution on the test function.The following symbols are convenient for us Kn Thus,we write distributions of the class .4j; on the left side and distributions of the class A' on the right side to a test function.
Thus we have a non-Archimedean Laplace calculus ~1;, L ~t. ~o L , .~'. (2.I) The Laplace transformation has all standard properties of the usual Laplace transformation.
By the definition of a conjugate operator, we have Parseval's equality L(g)(y) (dy) = (2.2) ~n Kn Definition 2.2.The Gauss distribution on !in (with the mean value a E li" and the covariance matrir B) is ya,B with the Laplace trafisform = x) + (a~ x)} Ĩf we consider R instead of K, then we get ordinary Gaussian distribution for the matrix B>0.
To denote the integral with respect to the Gaussian distribution, we use the symbol a), (x -a))} dx.

Kn
The quadratic exponent multiplied to dz is only the symbol to denote the Gaussian distribu- tion But this symbol is sufficiently convenient in computations.Formally we can work with quadratic exponent as with usual density with respect to dz ,but we need to apply the Laplace transform to justify such computations because at the moment the symbol dz is not defined.This symbolic expression for the Gaussian distribution 'yQ,$ contains a convention on the unit normalization of the integral exp {-1 2 (B-1 (z -a), (x -a)) dx.

h' n
This normalization differs from the standard real normalization constant 203C0 det B. But we cannot investigate the problem of the normalization on the basis of our definition of the Gaussian integral as the quadratic density is only convenient symbol in our computations.The separate question is a definition of a non-Archimedean analogue of IT.There were attempts to define a non-.archimedeanT, but these definitions were not connected in any way to our Gaussian integral.Consequently, M2k+1 = 0, = (2k -1)!!/2k.Using proposition 2.1, we get a formula for the calculation of the Gaussian integral of any entire function f (.r) = fnxn: dx -1 )II/2n.
Qp n :L et us introduce Hern1itian polynomials over the field K : : Hn(x) = ( -1 ) n e x 2 d n d x n e -x 2 and, as usual, KHn(x)Hm(x)e-x2 dx = 0, n ~ m, calculate KH3n(x)e-x2 dx.To do this, we take into account that the product of the test function Hn E A by the distribution v E A' is equal to the generalized derivative of v : (-1)n (v), Problem., ou the Gaussian distribution Let us restrict to the case Ia = Qp one dimensional case and a=0 .Hence B is the number in Qp 1.The Gaussian distribution 03B3B = 03B30,B was defined as the functional on the space of the entire analytical functions .The problem is to extend this functional to larger functional spaces.It was proved [13] that it would be impossible to extend this functional to the space of continuous functions on Zp .But this proof does not work in the case p = 2 .Is it possible to prove this fact for p = 2?
2.Schikhof's idea .Professor W.Schikhof propose to try to extend ~B on the space of Cl or C°° .At the moment ,there is no results in this direction.
3. The analytical function /(.c) is said to be 03B3B-negligible if = 0 for every analytical function .
Recently , it was proved by M.Endo and author that if p ~ 2 and B E ZJ' then f = 0. What is about p = 2 and another B?
The generalization of this problem is to consider an arbitrary distibution p and to try to find the class of ~-negligible functions.
4. All Gaussian measures are absolutely continuous in ordinary case .What is about p-adic case ?
3.Non-Archimedean Hilbert Space The quantization over non-Archimedean number fields must be based on the non-Archimedean analog of a Hilbert space.In the mathematical literature the concept of a non-Archimedean Hilbert space adequate for physical applications has not yet been worked out.The concept of orthogonality in non-Archimedean spaces is based not on an inner product but on a norm.Recall that the system of vectors {ej}j~J in the non-Archimedean normalized space E is said to be orthogonal if ~xjej ~ = max |xj |K ~ej for every finite set S C J and any xj ~ I{.The orthogonal system is called an orthogonal basis in E if x = ~ for any vector :r E E. In this case the space E is said to be orthogonalizable.By virtue of J.-P.Serre 's orthogonalization theorem, every discrete normalized non-Archimedean Banach space is orthogonalizable.' There is no canonical way of defining the inner product ( ,') in the orthogonalizable space E. Supposing that the vectors must be orthogonal not only with respect to the norm but also with respect to the inner product, we have (x,~'j= ~a~x~w here A, = This series converges if and only if (.rl~x ~a~ ~h' == 0. But the Banach space E consists of those ,c for which limj~~ |xj|K~ej~ = 0.If E r, then we can take as any elements of the field Is such that = Now if r, then it is impossible, in general, to find Aj E fi .It is natural to include the numbers Aj into the definition of a non-Archimedean Hilbert space.In what follows we shall consider only countable sets of indices. For the sequence A = (A,t) E Iv'~, 0 we set xa = {f = (fn) : the series 03A3 f2n03BBn converges}.
By virtue of the non-Archimedean Cauchy criterion we have In the space 'Hx we introduce a norm relative to which the base vectors (e~) = (bl ), are orthogonal, ~f~03BB = maxn The space is a non-Archimedean Banach space.On the space we introduce an inner product (., .)consistent with the length =03A3 f2n03BBn setting ( f, g)a = 03A3 fngn03BBn.The inner product (.,.):An inner product on the non-Archimedean linear space E is an arbitrary nondegenerated symmetric bilinear form (.,.) : E x E -K.It is evidently impossible to introduce an analog of the positive definiteness of a bilinear form.For instance, for the field of p-adic numbers any element 7 E Qp can be represented as 7 = (x, x)03BB, x E H03BB.
The isomorphic relation divides the class of Hilbert spaces into equivalence classes, we shall define the equivalence class of Hilbert spaces by some coordinate representative H03BB.
Example 3.1.Let A = (1) and = ( 2" ).The spaces H03BB and H belong to the same class of equivalence for the field li = Qp, p ~ 2, and to different classes for the field Ii = Q2.
We shall also use a non-Archimedean Hilbert space over the field of complex p-adic numbers Cp in some physical models.There is a possibility to compute all square roots ~/Aã nd this is why we need not to consider these coefficients at all.Thus, we need only to define the standard sequence space : = ~ f = ( f,l ), f" E Cp : the series converges}.
But there is a serious difference with the usual complex case , because there is not any involution on the field Cp.This is why the only possibility to define an inner product on is the following one (f, g) = 03A3fngn.It is the hard problem in our further physical considerations that this inner product is valued in Cp and not in Qp. .Problems on the Hilbert space Proposition 4.2.Let f, g E A. Then (f, g) = 03A3 03B103B12|03B1|.In order to prove (~.~.), it is sufficient to use Theorem ~.l and Proposition 4.1.
In the space we introduce a norm relative to which the Hermitian polynomials are orthogonal: The completion of the space of entire functions with respect to this norm is called the space of functions square summable by the Gaussian distribution v and is denoted by v).Proposition 4.3 The inner product (4.5) is continuous on L2(Kn, v) and the Cauchy-Buniakovski inequal- ity (3.1) holds true.The triplet v), (~, ~), (j ~ ») is a non-Archimedean complex Hilbert space of the class H(03B1!2|03B1|).Remark 4.1.Thus, in a non-Archimedean case all entire functions are square summable with respect to canonical Gaussian distribution and all square summable functions are analytical.
Problems on L2-theory 1.What is about L2-space for the Gaussian measure 03B303B1,B '?In particular, a = 0, B E Qp?
2.What are the conditions to B1 and B~ for the corresponding L,,-spaces to isomorphic ?We introduce a functional space U(Kn, Z) = (f(x) = : So(x) E A(Kn, Z)}.
The topology in che space U is induced from the space A of entire functions by the isomorphism I : U -A, I( f )i,r) = f(x)e|x|2.The space Cj is a reflexive non-Archimedean Fréchet space.
We choose this space as the space of test functions and U' as the space of generalized functions.The space of generalized functions C~' is isomorphic to the space A', I' : A' --~ U'.Definitioii 5.1.The Lebesgue distribution on the non-Archimedean space h'~ is a gen- eralized function dx = I'(v) E L~'( la", Z).
For the Lebesgue distribution rlx to act on the test function f ~ U, we use the integral notation (dx, f = f(x) dz.
h' n Note the obvious properties of the integral with respect to the Lebesgue distribution dz.The completion of the space Ii ", Z according to this norm is called the space of functions square summable by the Lebesgue distribution and is denoted by we shall use notations (5.1) and (5.2) for the inner product (5.3) and the square of the length (5.4).

The isomorphism .J :
A is continued to the isomorphism of the Hilbert space J : : v~.In particular.the following nested Hilbert space appears: Z) C L2(Kn, dx) C L'2(Kn, dx) C Z). Remark 5.1.Thus, in a non-Archimedean case, all functions square summable by the Lebesgue distribution d.z: are analytic.
Problems on the Lebesgue distribution 1.The Lebesgue distribution is an analogue of the uniform distribution.But the Volken- born distribution is also the analogue of the uniform distribution .What is the connection ? 2. We can also introduce Lebesgue distribution on the basis of an arbitrary Gaussian distribution, 03B3B.What is the connection between Lebesque distributions ,corresponding to different Gaussian distributions.
In applications I am also interested very much in the following question : Is it possible to introduce on Q such metric or topology t that every sequence An converging with respect to the real metric on Q will converge with respect to t and every Bn converging with respect to the p-adic metric will converge with respect to t "? V'hat the completion of Q with respect tot?
This work was realized on the basis of the Alexander von Humboldt-Fellowship.