Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 4 Apr 1999 13:45:39 +0300 (EET DST)

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

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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
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
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
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
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
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
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.
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
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.
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
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}

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

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

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
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