**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Sun, 4 Apr 1999 13:45:39 +0300 (EET DST)*

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Below is tex file explaining p-adic integration, in particular the sum

formula: the explanation of the procedure would have required too much

typing.

MP

\documentstyle [12pt]{article}

\begin{document}

\newcommand{\vm}{\vspace{0.2cm}}

\newcommand{\vl}{\vspace{0.4cm}}

\newcommand{\per}{\hspace{.2cm}}

\section{p-Adic integration}

The concept of the p-adic definite integral can be defined for functions

$R_p\rightarrow C$ \cite{padrev} using translationally invariant Haar

measure

for $R_p$. In present context one is however interested in definining

a p-adic

valued definite integral for functions $f: R_p\rightarrow R_p$: target

and

source spaces could of course be also some some algebraic extensions of

the p-adic

numbers.

What makes the definition nontrivial is that the ordinary

definition as the limit of a Riemann sum doesn't seem to work: it seems

that Riemann sum

approaches to

zero in the p-adic topology since, by ultrametricity,

the p-adic norm of a sum is never

larger than the maximum p-adic norm for the summands.

The second

difficulty

is related to the absence of a well ordering for the p-adic numbers. The

problems are

avoided by

defining the integration essentially as the

inverse of the differentation and using

the canonical

correspondence to define ordering for the p-adic numbers.

More generally, the concepts of

the form, cohomology and homology

are crucially based on the concept of the boundary.

The concept of boundary reduces

to the concept of an ordered interval and canonical identification

makes it indeed possible to define this concept.

\subsection{Definition of definite integral in terms of integral

function concept and

canonical identification}

The definition of the

p-adic integral functions defining integration as inverse

of the differentation

is straightforward and one obtains just the generalization of

the standard

calculus. For instance, one has $\int z^n = \frac{z^{n+1}}{(n+1)}+ C$ and

integral of the Taylor series is obtained by generalizing this. One

must however notice that the concept of integration constant generalizes:

any

function $R_p\rightarrow R_p$ depending on a finite number of pinary

digits

only, has a vanishing derivative.

\vm

Consider next the definite integral. The absence of the well

ordering implies that the concept of the integration

range $(a,b)$ is not well defined as a purely p-adic concept. A

possible

resolution of the problem is based on the canonical identification.

Consider

p-adic

numbers $a$ and $b$. It is natural to define $a$ to be smaller than $b$ if

the

canonical images of $a$ and $b$ satisfy $a_R<b_R$. One must notice that

$a_R=b_R$ does not imply $a=b$, since the inverse of the canonical

identification

map is two-valued for the real numbers having a

finite number of pinary digits.

For

two p-adic numbers $a,b$ with $a<b$, one can define the integration

range

$(a,b)$ as the set of the p-adic numbers $x$ satisfying $a\leq x\leq b$ or

equivalently $a_R\leq x_R\leq b_R$. For a given value of $x_R$ with

a finite

number of pinary digits, one has two values of $x$ and $x$ can be made

unique

by requiring it to have a finite number of pinary digits.

\vm

One can define

definite integral $\int_a^b f(x)dx$ formally as

\begin{eqnarray}

\int_a^b f(x)dx&=& F(b)-F(a)\per ,

\end{eqnarray}

\noindent where $F(x)$ is

integral function obtained by allowing only ordinary integration constants

and

$b_R>a_R$ holds true. One encounters

however a problem, when $a_R=b_R$ and $a$ and $b$ are different. Problem

is

avoided if the integration limits are

assumed to correspond to p-adic numbers with

a finite number of pinary digits.

\vm

One could perhaps relate the possibility of the p-adic integration

constants

depending on

finite

number of pinary digits to the possibility to decompose integration range

$[a_R,b_R]$ as

$a=x_0<x_1<....x_n=b$ and to select in each subrange $[x_k,x_{k+1}]$ the

inverse images

of $x_k\leq x\leq x_{k+1}$, with $x$ having finite number of pinary

digits

in two

different

manners. These different choices correspond to different integration paths

and the value

of

the integral for different paths could correspond to the different

choices

of the p-adic

integration constant in integral function. The difference between a given

integration

path and

'standard' path is simply the sum of differences $F(x_k)-F(y_k)$,

$(x_k)_R= (y_k)_R$.

\vm

This definition has several nice features:

a) The definition generalizes in an obvious manner to

the higher dimensional case.

The standard connection between integral function and definite integral

holds

true and in the higher-dimensional case the integral of

a total divergence

reduces to

integral over the boundaries of the

integration volume. This property guarantees

that

p-adic action principle leads to same field equations as its real

counterpart. It

this in fact

this property, which drops other alternatives from the consideration.

b) The basic results of the real integral calculus generalize as such to

the p-adic case. For instance, integral is a linear operation and

additive

as a set function.

\subsection{Riemann sum representation of the definite integral}

For numerical purposes it would be very advantageous to have a definition

of a p-adic definite integral based on Riemann sum. Real Riemann sum has

also the beautiful property that it reproduces the basic properties

of a definite integral crucial for variational principles: for instance,

the integral of a divergence reduces to a boundary integral. This in

turn would make it possible to define the concept of differential form

in p-adic context and to generalize differential geometry.

The concept of p-adic Riemannian metric would be more than a mere formal

concept since one could calculate

distances and volumes using numerical integration.

Variational principles, in particular absolute minimization of

K\"ahler action,

would become well defined

and ordinary Euler Lagrange field equations would

result from the stationarity requirement

of action. Note however that absolute

minimization for the real counterpart of K\"ahler action implies

that action must vanish for the absolute minima.

Also p-adic QFT in spacetime possessing effective p-adic topology

would make sense since the integral defining the action would

exist in a precise numerical sense.

\vm

It took a long time to realize

that the previous definition of integral might results from a

generalized Riemann sum! The idea is to induce

the division of the p-adic integration interval

to subintervals using the corresponding division for the

real image of the p-adic integration interval.

To see what is involved,

consider just the Riemann sum of a constant

function in the interval $[0,1]$: $\sum_n C \Delta x_n= C (x_N-x_1)=C$.

The sum has a finite value independent

of the number of division points rather than being zero

as one might naively expect! The reason is that

the p-adic norms of the intervals appearing in the division

are not constant and some of them do not approach zero!

For instance, for the interval $[0,1]$,

the p-adic norm for the increment

$\Delta_N=x_N-x_{N-1}$ associated with the interval $(x_{N-1},x_N=1]$

is always one rather than approaching to zero! Analogous

result holds for the end points of the intervals appearing in

subdivisions. Hence the sum approaches to a finite value.

\vm

Since continuous

functions behave as constant functions in sufficiently short

length scales, the results

of the real integration theory generalize to

p-adic context.

Important delicacies are associated with

the definition of the integration intervals. For instance, the integral

of a constant function $f(x)=C$ over a half-open interval $[0,1)$ is

$-Cp$

since the p-adic image of a real point $x_R$ approaches

$x=-p$ when $x_R$ approaches $x_R=1$ from below. On the other hand,

the integral over

the p-adic image of a closed interval $[0,1]$ is $C$.

*>From this it is clear that the existence of p-adic pseudo constants, that
*

is

non-constant functions with

vanishing p-adic derivative, corresponds to the various

decompositions of the integration region to half-closed or closed

intervals.

In real context this kind of decompositions are

of course equivalent.

Also the generalization

to finite- and infinite-dimensional cases is straightforward.

The essential difference between real and p-adic integration

is the replacement of real summation by p-adic summation.

p-Adic ultrametricity

implies automatic convergence of the Riemann sum. This makes

p-adic valued functional integral a very attractive manner

to avoid divergences in configuration space integration (see

chapter 'Mathematical building blocks' of this book).

\subsection{Exact formula for the p-adic integral?}

An interesting feature of the p-adic Riemann sum is that

the dominating contributions come from the intervals whose p-adic lengths

do not approach zero at the limit of infinitely dense division.

This suggests that p-adic valued integral

could be in good approximation

calculated by including only the contributions coming from the

intervals, which do not shrink to zero with respect to

the p-adic norm at the limit of infinitely dense Riemann sum.

Since the integration intervals can be ordered hierarchically

with respect to the p-adic norm of the p-adic length of

the interval, this approximation method could be systematized to

give an expansion of the integral in powers of $p$. For large

values of $p$

(typically one has $p\simeq 10^{38}$ in applications!)

the lowest Riemann sum would give the dominating contribution

to the integral. In fact, one achieves much more than this: one can write

exact formula for the p-adic integral!

\vm

The points of the real axis, which are mapped to two p-adic points in

canonical identification correspond to zero length intervals

of real axis for which the p-adic interval spanned by

the image points has nonvanishing length.

These points correspond

to the points of the real axis, which can be represented

using pinary expansion containing finite number of pinary digits:

\begin{eqnarray}

x&= &\sum_{k\leq n} x_kp^{-k} \rightarrow \{x^+,x^-\}\per ,\nonumber\\

x^+ &=& \sum_{k\leq n} x_kp^{k}\per , \nonumber\\

x^- &=& \sum_{k<n}x_kp^k + (x_n-1)p^{n} - p^{n+1}\nonumber\\

&=&

\sum_{k<n}x_kp^k + (x_n-1)p^{n} +(p-1)(

p^{n+1}+p^{n+2}+.....)\per ,\nonumber\\

x^+-x^- &=& p^n+p^{n+1}\per .

\end{eqnarray}

\noindent An attractive possibility is that the Riemann sum containing

all

zero length real intervals provides an exact representation

of the p-adic valued integral for the {\it p-adic counterpart of a

real-continuous function}:

\begin{eqnarray}

I_f(a,b) &=& \int_a^b f(x)dx = \sum_k f(x_k) (x_k^+ -x_k^-)\nonumber\\

&=& \sum_n \left[\sum_{k(n)} f(x_{k(n)})\right] (p^n+p^{n+1})\per .

\end{eqnarray}

\noindent Here $k(n)$ labels the points $x$, which have

$n$ pinary digits in their pinary expansion.

This formula would give a closed expression of the p-adic integral

for the p-adic counterpart of real-continuous

function allowing an arbitrarily precise estimation of the integral

proceeding in powers of $p$ by including more and more pinary

digits to the expansion of the coordinates $x_k$ contributing to the

sum. For physically interesting primes this expansion converges

extremely rapidly by ultrametricity and one has strict upper bound for

the contribution of the terms of given order.

\vm

This formula holds true for half closed integration intervals

$(a,b]$ and is unique only for the functions having the symmetry property

\begin{eqnarray}

f(x_+)&=&f(x_-)\per .

\end{eqnarray}

\noindent For the p-adic counterparts

of real-continuous functions this property holds true but not for

p-adically analytic functions. Indeed, one can select in each

integration point

the value of function to be either $f(x_+)$ or $f(x_-)$. A

natural guess is that this nonuniqueness corresponds to the

appearence of a p-adic pseudoconstant in p-adic integral function.

If $f$ is the p-adic counterpart of

a real-continuous function, further nonuniqueness results from the

two-valuedness of the p-adic counterpart of $f(x)$ in canonical

identification.

*>From the point of view of numerics, a natural
*

manner to overcome this nonuniqueness is to

select the branch $f_+(x)$ with a finite number of pinary digits.

The construction of the p-adic counterpart

of the spacetime surface is based on

the mapping of only those points for which coordinates

have finite number of

pinary digits to their p-adic counterparts with finite number

of pinary digits and on the continuation of this discrete

point set to a smooth p-adic surface.

\vm

To verify that this formula is exact, one must calculate the

p-adic derivative of the integral function $I_f(a,x)$

with respect to the end point $x$ and show that it equals to $f(x)$:

$$ \frac{dI_f(a,x)}{dx}= f(x)\per .$$

\noindent The calculation of the derivative reduces to the calculation

of

the limiting value

$$lim_{\Delta x\rightarrow 0}\frac{I_f(x,x+\Delta x)}{\Delta x}\per , $$

\noindent which by continuity from right reduces to the calculation of the

limit

$$f(x) \times lim_{\Delta x\rightarrow 0}\frac{I_1(x,x+\Delta x)}{\Delta

x}\per .$$

\noindent Hence, if the integral formula holds true for a constant

function

$f(x)=1$

and gives $I_1(a,x)=x-a$, the proof follows.

By translational invariance, one can restrict

the consideration to an interval starting at origin so that the claim

reduces to proving that following identity holds true:

$$ I_1(0,x) = \sum_n \left[\sum_{x_k(n)\in (0,x]}\right]

(p^n+p^{n+1}) = x\per .$$

\vm

The proof of the identity is achieved by a direct calculation.

\noindent a) The first observation is that $I_1(0,x)$ is fractal:

$$ I_1(0,p^kx) = p^kI_1(0,x)\per .$$

\noindent This gives $I_1(0,p^k)= p^kI_1(0,1)$, which already

suggests that the desired formula holds.

\vm

\noindent b) Additivity of the summation formula together with

the translational invariance

implies that the sum for a given value $x=\sum x_kp^k$ can be written

as a sum

$$ I_1(0,\sum_k x_kp^k) = \sum_k x_k I_1(0,p^k)= \sum_k x_kp^k I(0,1)

=xI(0,1)\per .$$

\noindent Therefore the result follows if one can show that

$I(0,1)=1$ holds true.

i) To the lowest order in $p$ $I_1(0,1)$ is simply

$I_1(0,1) = 1+p $

so that the remaining terms in the sum should sum up to $-p$.

ii) Next order gives to the sum $p-1$ terms having value $p(1+p)$

so that one has $I_2(0,1) = 1+p^{3}$.

iii) In next order there are $p$ intervals with each interval giving

$(p-1)$ terms with value $p^2(1+p)$ and one

has

$$I_3(0,1)= I_2(0,1) + p(p-1)p^2(1+p) = 1+p^5\per .$$

iv) In n:th order there are $p^{n-2}$ intervals

with each interval giving $p(p-1)$ terms having value $p^{n-1}(1+p)$

so that one has

$$\begin{array}{l}

I_n(0,1)= I_{n-1}(0,1)+(p-1)p^{2n-3}(1+p)\\

= 1+p + (p+p^3+p^5+...p^{2n-3})(p^2-1) = 1+p^{2n-1}

\per .\\

\end{array} $$

\noindent At the limit $n\rightarrow \infty$ one obtains the desired

result.

\vm

Thus p-adic integral

allows exact representation in terms of the function values

and the representation is extremely rapidly converging for large values

of $p$. What is interesting is that p-adic integral for the p-adic

counterpart

of a real-continuous function gives rise to p-adically differentiable

function

having p-adic fractal as its real counterpart.

The formula generalizes in an obvious manner to

many- and infinite-dimensional cases. In these cases

only few points in the integration volume give significant

contribution to the integral formula since

each contribution involves product of powers $(p^n+p^{n+1})$ associated

with various coordinate variables. In infinite-dimensional case it could

even happen that all terms correspond to infinite power of $p$.

In quantum TGD one must calculate ratios

of configuration space integrals and

in these ration large (possibly infinite) powers of $p$ cancel each

other.

\vm

It is important to notice that sum formula makes sense,

when upper and lower limits of integration correspond to

the p-adic images $b_+$ and $a_-$ of real numbers $b$ and $a$ having

finite number of pinary digits. This means that p-adic integral

$I(a,x)$ as a function of its upper limit is

defined for a discrete dense subset of $R_p$ only. An

interesting question is whether it might be possible to

continue $I(a,x)$ to a p-adically differentiable function

defined in the entire p-adic axis.

What must be done is to first continue the p-adic counterpart

of $f(x)$ from its values known

for p-adic argument with finite number of pinary digits to a p-adically

differentiable function. If the real counterpart of $f$ satisfies

some differential equation, it is natural to require that

the same differential

equation is satisfied by $f$.

That this is possible is suggested by the possibility of

p-adic pseudo constants, which implies nondeterminism and

suggests strongly that one can fix the solution

of a p-adic differential equation in all points $x$ having finite number

of pinary

digits. In fact, this kind of continuation procedure

will be used to define the p-adic counterparts of real spacetime surfaces

as p-adically differentiable surfaces.

In the similar manner, one can define the integral function $I(a,x)$

as a function satisfying differential equation $dI(a,x)/dx= f(x)$

and clearly coinciding with the function given by the sum formula at

the points $x$ possessing finite number of pinary digits.

\subsection{Change of integration variables}

The generalized Jacobi determininant associated with the coordinate

transformattion

$\{x^k\}\rightarrow \{y^l=f^l(x)\}$ is obtained by forming the ratio

of the products of coordinate differentials at summation points

$$J= \frac{\prod_k (x^k_+-x^k_-)}{\prod_k (y^k_+-y^k_-)}

\per .$$

\noindent The technical problems are related to the

fact that the image points $y^k$ do not necessary have finite number of

digits in their pinary expansion. It seems that in this kind of

situation $y^l(x)$ must be replaced with the nearest $y^l(x)$

having finite number of pinary digits.

\subsection{Is the generalization of the measure theory possible?}

An interesting question is whether the p-adic valued integral could

in some sense satisfy the basic axioms of measure

theory. Ordinary measure $\mu$ is real valued

function in the subsets of

a given set $X$ such that the measure of $X$ equals to one, the measure

of

an empty set vanishes, the measure of a disjoint union is the sum over

the measures for the component sets in union and

measure is monotonically increasing set function: $\mu (A)\leq \mu (B)$

for $A\subset B$. The last axiom does not make sense for p-adic

valued measure but could make sense for its real counterpart

under canonical identification.

\vm

p-Adic valued measure in the subsets of p-adic numbers is obtained by

mapping

the intervals of real axis to p-adics by canonical identification: the

images

of these intervals and their unions form by definition the set of

measurable

sets. The real counterpart of the p-adic measure under canonical

identification

defines a candidate for a real valued measure. Monotonicity however

fails if the measure of disjoint union of intervals is {\it p-adic sum

for the measures for the intervals}:

$$\left[\mu (\cup_i I_i)\right]_R= \left[\sum_i \mu (I_i)\right]_R \per

.$$

\noindent For instance, the real counter part

for the p-adic sum of p-adic lengths for $p$ segments

$(n_k,n_k+1]$ is $1/p<1$.

One could however define

the real valued measure by requiring that the real counterpart

of the measure for a p-adic image

of a disjoint union of intervals

is the {\it sum of the real counterparts

for the measures of intervals}.

$$\left[\mu (\cup_i I_i)\right]_R= \sum_i \left[\mu (I_i)\right]_R\per

.$$

\noindent In this manner monotonicity is achieved.

In two-dimensional case situation however changes. The real counterpart

for the area of a parallelogram

$0 \leq x\leq mp, 0<y \leq np$ is given by $(mnp^2)_R$ and ceases

to be monotonically increasing function of $mn$, when $mn>p$ holds

true.

\vm

Note that it is not possible to

interpret the integral of an arbitrary p-adic function $f$ in

terms of a measure.

For instance, the real counterpart $y_R$ of the integral

$y=\int_0^x x dx=x^2/2$

is not monotonically increasing function of $x_R$ as it would

be in real

context: this is obviously due to the special features

of p-adic summation.

\end{document}

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