How could one calculate p-adic integrals numerically?

Riemann sum gives the simplest numerical approach to the calculation of real integrals. Also p-adic integrals should allow a numerical approach and very probably such approaches already exist and "motivic integration" presumably is the proper word to google. The attempts of an average physicist to dig out this kind of wisdom from the vastness of mathematical literature however lead to a depression and deep feeling of inferiority. The only manner to avoid the painful question "To whom should I blame for ever imagining that I could become a real mathematical physicist some day?" is a humble attempt to extrapolate real common sense to p-adic realm. One must believe that the almost trivial Riemann integral must have an almost trivial p-adic generalization although this looks far from obvious.

1. A proposal for p-adic numerical integration

The physical picture provided by quantum TGD gives strong constraints on the notion of p-adic integral.

1. The most important integrals should be over partonic 2-surfaces. Also p-adic variants of 3-surfaces and 4-surfaces can be considered. The p-adic variant of Kähler action would be an especially interesting integral and reduces to Chern-Simons terms over 3-surfaces for preferred extremals. One should use this definition also in the p-adic context since the reduction of a total divergence to boundary term is not expected to take place in numerical approach if one begins from a 4-dimensional Kähler action since in p-adic context topological boundaries do not exist. The reduction to Chern-Simons term means also a reduction to cohomology and p-adic cohomology indeed exists.

   At the first step one could restrict the consideration to algebraic varieties - in other words zero loci for a set of polynomials P_i(x) at the boundary of causal diamond consisting of pieces of δ M⁴_+/-× CP₂. 5 equations are needed. The simplest integral would be the p-adic volume of the partonic 2-surface.

2. The numerics must somehow rely on the p-adic topology meaning that very large powers pⁿ are very small in p-adic sense. In the p-adic context Riemann sum makes no sense since the sum never has p-adic norm larger than the maximum p-adic norm for summands so that the limit would give just zero. Finite measurement resolution suggests that the analog for the limit Δ x→ 0 is pinary cutoff O(pⁿ)=0, n→ ∞, for the function f to be integrated. In the spirit of algebraic geometry one must asume at least power series expansion if not even the representability as a polynomial or rational function with rational or p-adic coefficients.

3. Number theoretic approach suggests that the calculation of the volume vol(V) of a p-adic algebraic variety V as integral should reduce to the counting of numbers for the solutions for the equations f_i(x)=0 defining the variety. Together with the finite pinary cutoff this would mean counting of numbers for the solutions of equations f_i(x) mod pⁿ=0 . The p-adic volume Vol(V,n) of the variety in the measurement resolution O(pⁿ)=0 would be simply the number of p-adic solutions to the equations f_i(x) mod pⁿ=0. Although this number is expected to become infinite as a real number at the limit n→ ∞, its p-adic norm is never larger than one. In the case that the limit is a well-defined as p-adic integer, one can say that the variety has a well-defined p-adic valued volume at the limit of infinite measurement resolution. The volume Vol(V,n) could behave like n_pⁿ and exist as a well defined p-adic number only if n_p is divisible by p.

4. The generalization of the formula for the volume to an integral of a function over the volume is straightforward. Let f be the function to be integrated. One considers solutions to the conditions f(x) =y, where y is p-adic number in resolution O(pⁿ)=0, and therefore has only a finite number of values. The condition f(x)-y=0 defines a codimension 1 sub-variety V_y of the original variety and the integral is defined as the weighted sum ∑_y y × vol(V_y), where y denotes the point in the finite set of allowed values of f(x) so that calculation reduces to the calculation of volumes also now.

2. General coordinate invariance

From the point of view of physics general coordinate invariance of the volume integral and more general integrals is of utmost importance.

1. The general coordinate invariance with respect to the internal coordinates of surface is achieved by using a subset of imbedding space-cooordinates as preferred coordinates for the surface. This is of also required if one works in algebraic geometric setting. In the case of projective spaces and similar standard imbedding spaces of algebraic varieties natural preferred coordinates exist. In TGD framework the isometries of M⁴× CP₂ define natural preferred coordinate systems.

2. The question whether the formula can give rise to a something proportional to the volume in the induced metric in the intersection of real and rational worlds interesting. One could argue that one must include the square root of the determinant of the induced metric to the definition of volume in preferred coordinates but this might not be necessary. In fact, p-adic integration is genuine summation whereas the determinant of metric corresponds density of volume and need not make no sense in p-adic context. Could the fact that the preferred coordinates transform in simple manner under isometries of the imbedding space (linearly under maximal subgoup) alone guarantee that the information about the imbedding space metric is conveyed to the formula?

3. Indeed, since the volume is defined as the number of p-adic points, the proposed formula should be invariant at least under coordinate transformations mediated by bijections of the preferred coordinates expressible in terms of rational functions. In fact, even more general bijections mapping p-adic numbers to p-adic numbers could be allowed since they effectively mean the introduction of new summation indices. Since the determinant of metric changes in coordinate transformations this requires that the metric determinant is not present at all. Thus summation is what allows to achieve the p-adic variant of general coordinate invariance.

4. This definition of volume and more general integrals amounts to solving the remaining coordinates of imbedding space as (in general) many-valued functions of these coordinates. In the integral those branches contribute to the integral for which the solution is p-adic number or belongs to the extension of p-adic numbers in question. By p-adic continuity the number of p-adic value solutions is locally constant. In the case that one integrates function over the surface one obtains effectively many-valued function of the preferred coordinates and can perform separate integrals over the branches.

A convenient iteration procedure is based on the representation of integrand f as sum ∑_kf_k of functions associated with different p-adic valued branches z_k=z_k(x) for the surface in the coordinates chosen and identified as a subset of preferred imbedding space coordinates. The number of branches z_k contributing is by p-adic continuity locally constant.

The function f_k - call it g for simplicity - can in turn be decomposed into a sum of piecewise constant functions by introducing first the piecewise constant pinary cutoffs g_n(x) obtained in the approximation O(pⁿ⁺¹)=0. One can write g as

g(x)= ∑ h_n (x) , h₀(x)=g₀(x) ,

h_n=g_n(x)-g_n-1(x) for n>0 .

Note that h_n(x) is of form g_n(x)= a_n(x)pⁿ, a_n(x) ∈ {0,p-1} so that the representation for integral as a sum of integrals for piecewise constant functions h_n converge rapidly. The technical problem is the determination of the boundaries of the regions inside which these functions contribute.

The integral reduces to the calculation of the number of points for given value of h_n(x) and by the local constancy for the number of p-adic valued roots z_k(x) the number of points for N₀ ∑_{k≥ 0} p^k= N₀/(1-p), where N₀ is the number of points x with the property that not all points y= x(1+O(p)) represent p-adic points z(x). Hence a finite number of calculational steps is enough to determine completely the contribution of given value to the integral and the only approximation comes from the cutoff in n for h_n(x).

4. Number theoretical universality

This picture looks nice but it is far from clear whether the resulting integral is that what physicist wants. It is not clear whether the limit Vol(V,n), n→ ∞, exists or even should exist always.

1. In TGD Universe a rather natural condition is algebraic universality requiring that the p-adic integral is proportional to a real integral in the intersection of real and p-adic worlds defined by varieties identified as loci of polynomials with integer/rational coefficients. Number theoretical universality would require that the value of the p-adic integral is p-adic rational (or algebraic number for extensions of p-adic numbers) equal to the value of the real integral and in algebraic sense independent of the number field. In the eyes of physicist this condition looks highly non-trivial. For a mathematician it should be extremely easy to show that this condition cannot hold true. If true the equality would represent extremely profound number theoretic truth.

   The basic idea of the motivic approach to integration is to generalize integral formulas so that the same formula applies in any number field: the specialization of the formula to given number field would give the integral in that particular number field. This is of course nothing but number theoretical universality. Note that the existence of this kind of formula requires that in the intersection of the real and p-adic worlds real and p-adic integrals reduce to the same rational or transcendentals (such as log(1+x) and polylogarithms).

2. If number theoretical universality holds true one can imagine that one just takes the real integral, expresses it as a function of the rational number valued parameters (continuable to real numbers) characterizing the integrand and the variety and algebraically continues this expression to p-adic number fields. This would give the universal formula which can be specified to any number field. But it is not at all clear whether this definition is consistent with the proposed numerical definition.

3. There is also an intuitive expectation in an apparent conflict with the number theoretic universality. The existence of the limit for a finite number p-adic primes could be interpreted as mathematical realization of the physical intuition suggesting that one can assign to a given partonic 2-surface only a finite number of p-adic primes. Indeed, quantum classical correspondence combined with the p-adic mass calculations suggests that the partonic 2-surfaces assignable to a given elementary particle in the intersection of real and p-adic worlds corresponds to a finite number of p-adic primes somehow coded by the geometry of the partonic 2-surface.

   One way out of the difficulty is that the functions - say polynomials - defining the surface have as coefficients powers of eⁿ. For given prime p only the powers of e^p exist p-adically so that only the primes p dividing n would be allowed. The transcendenteals of form log(1+px) and their polylogarithmic generalizations resulting from integrals in the intersection of real and p-adic worlds would have the same effect. Second way out of the difficulty would be based on the condition that the functional integral over WCW ("world of classical worlds") converges. There is a good argument stating that the exponent of Kähler action reduces to an exponent of integer n and since all powers of n appear the convergence is achieved only for p-adic primes dividing n.

The equivalence of the proposed numerical integral with the algebraic definition of p-adic integral motivated by the algebraic formula in real context expressed in terms of various parameters defining the variety and the integrand and continued to all number fields would be such a number theoretical miracle that it deserves italics around it:

For algebraic surfaces the real volume of the variety equals apart from constant C to the number of p-adic points of the variety in the case that the volume is expressible as p-adic integer.

The proportionality constant C can depend on p-adic number field , and the previous numerical argument suggests that the constant could be simply the factor 1/(1-p) resulting from the sum of p-adic points in p-adic scales so short that the number of the p-adic branches z_k(x) is locally constant. This constant is indeed needed: without it the real integrals in the intersection of real and p-adic worlds giving integer valued result I=m would correspond to functions for which the number of p-adic valued points is finite.

The statement generalizes to apply also to the integrals of rational and perhaps even more general functions. The equivalence should be considered in a weak form by allowing the transcendentals contained by the formulas have different meanings in real and p-adic number fields. Already the integrals of rational functions contain this kind of transcendentals.

The basic objection that number of p-adic points without cannot give something proportional to real volume with an appropriate interpretation cannot hold true since real integral contains the determinant of the induced metric. As already noticed the preferred coordinates for the imbedding space are fixed by the isometries of the imbedding space and therefore the information about metric is actually present. For constant function the correspondence holds true and since the recipe for performing of the integral reduce to that for an infinite sum of constant functions, it might be that the miracle indeed happens.

The proposal can be tested in a very simple manner. The simplest possible algebraic variety is unit circle defined by the condition x²+y²=1.

1. In the real context the circumference is 2π and p-adic transcendental requiring an infinite-dimensional algebraic extension defined in terms of powers of 2π. Does this mean that the number of p-adic points of circle at the limit n→ ∞ for the pinary cutoff O(pⁿ)=0 is ill-defined? Should one define 2π as this integral and say that the motivic integral calculus based on manipulation of formulas reduces the integrals to a combination of p-adically existing numbers and 2π? In motivic integration the outcome of the integration is indeed formula rather than number and only a specialization gives it a value in a particular number field. Does 2π have a specialization to the original p-adic number field or should one introduce it via transcendental extension?

2. The rational points (x,y)=(k/m,l/m) of the p-adic unit circle would correspond to Pythagorean triangles satisfying k²+l²= m² with the general solution k= r²-s², l=2rs, m=r²+s². Besides this there is an infinite number of p-adic points satisfying the same equation: some of the integers k,l,m would be however infinite as real integers. These points can be solved by starting from O(p)=0 approximation (k,l,m)→ (k,l,m)~ mod~ p== (k₀,l₀,m₀). One must assume that the equations are satisfied only modulo p so that Pythagorean triangles modulo p are the basic objects. Pythagorean triangles can be also degenerate modulo p so that either k₀,l₀ or evenm₀ vanishes. Note that for surfaces xⁿ+yⁿ=zⁿ no non-trivial solutions exists for xⁿ,yⁿ,zⁿ<p for n> 2 and all p-adic points are infinite as real integers.

   The Pythagorean condition would give a constraint between higher powers in the expressions for k,l and m. The challenge would be to calculate the number of this kind of points. If one can choose the integers k-(k mod p) and l-(l mod p) freely and solve m-(m mod p) from the quadratic equations uniquely, the number of points of the unit circle consisting of p-adic integers must be of form N₀/(1-p). At the limit n→ ∞ the p-adic length of the unit circle would be in p-adic topology equal to the number of modulo p Pythagorean triangles (r,s) satisfying the condition (r²+s²)²<p. The p-adic counterpart of 2π would be ordinary p-adic number depending on p. This definition of the length of unit circle as number of its modulo p Pythagorean points also Pythagoras would have agreed with since in the Pythagorean world view only rational triangles were accepted.

3. One can look the situation also directly solving y as y=+/- (1-x²)^1/2. The p-adic square root exists always for x=O(pⁿ), n>0. The number of these points x is 2/(1-p) taking into account the minus sign. For x=O(p⁰) the square root exist for roughly one half of the integers n∈ {0,p-1}. The number of integers (x²)₀ is therefore roughly (p-1)/2. The study of p=5 case suggests that the number of integers (1-(x²)₀)₀∈ {0,p-1} which are squares is about (p-1)/4. Taking into account the +/- sign the number of these points by N₀≈ (p-1)/2. In this case the higher O(p) contribution to x is arbitrary and one obtains total contribution N₀/(1-p). Altogether one would have (N₀+2)/(1-p) so that eliminating the proportionality factor the estimate for the p-adic counterpart of 2π would be (p+3)/2.

4. One could also try a trick. Express the points of circle as (x,y)=(cos(t),sin(t)) such that t is any p-adic number with norm smaller than one in p-adic case. This unit circle is definitely not the same object as the one defined as algebraic variety in plane. One can however calculate the number of p-adic points at the limit n→ ∞. Besides t=0, all p-adic numbers with norm larger than p^-n and smaller than 1 are acceptable and one obtains as a result N(n)= 1+ p^n-1, where "1" comes from overall important point t=0. One has N(n)→ 1 in p-adic sense. If t=0 is not allowed the length vanishes p-adically. The circumference of circle in p-adic context would have length equal to 1 in p-adic topology so that no problems would be encountered (numbers exp(i2π/n) would require algebraic extension of p-adic numbers and would not exist as power series).

   The replacement of the coordinates (x,y) with coordinate t does not respect the rules of algebraic geometry since trigonometric functions are not algebraic functions. Should one allow also exponential and trigonometric functions and their inverses besides rational functions and define circle also in terms of these. Note that these functions are exceptional in that corresponding transcendental extensions -say that containing e and its powers- are finite-dimensional?

5. To make things more complicated, one could allow algebraic extensions of p-adic numbers containing roots U_n=exp(i2π/n) of unity. This would affect the count too but give a well-defined answer if one accepts that the points of unit circle correspond to the Pythagorean points multiplied by the roots of unity.

6. p-Adic thermodynamics for measurement resolution?

The proposed definition is rather attractive number theoretically since everything would reduce to the counting of p-adic points of algebraic varieties. The approach generalizes also to algebraic extensions of p-adic numbers. Mathematicians and also physicists love partition functions, and one can indeed assign to the volume integral a partition function as p-adic valued power series in powers Z(t)=∑ v_ntⁿ with the coefficients v_n giving the volume in O(pⁿ)=0 cutoff. One can also define partition functions Z_f(t)= ∑ f_ntⁿ, with f_n giving the integral of f in the same approximation.

Could this kind of partition functions have a physical interpretation as averages over physical measurements over different pinary cutoffs? p-Adic temperature can be identified as t = p^1/T, T=1/k. For p-adically small temperatures the lowest terms corresponding to the worst measurement resolution dominate. At first this sounds counter-intuitive since usually low temperatures are thought to make possible good measurement resolution. One can however argue that one must excite p-adic short range degrees of freedom to get information about them. These degrees of freedom correspond to the higher pinary digits by p-adic length scale hypothesis and high energies by Uncertainty Principle. Hence high p-adic temperatures are needed. Also measurement resolution would be subject to p-adic thermodynamics rather than being freely fixed by the experimentalist.

For details see the new chapter Motives and Infinite Primes or the article with same title.