Monday, July 11, 2011

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 Pi(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 fi(x)=0 defining the variety. Together with the finite pinary cutoff this would mean counting of numbers for the solutions of equations fi(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 fi(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 npn and exist as a well defined p-adic number only if np 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 Vy of the original variety and the integral is defined as the weighted sum ∑y y × vol(Vy), 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.

3. Numerical iteration procedure

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

The function fk - 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 gn(x) obtained in the approximation O(pn+1)=0. One can write g as

g(x)= ∑ hn (x) , h0(x)=g0(x) ,

hn=gn(x)-gn-1(x) for n>0 .

Note that hn(x) is of form gn(x)= an(x)pn, an(x) ∈ {0,p-1} so that the representation for integral as a sum of integrals for piecewise constant functions hn 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 hn(x) and by the local constancy for the number of p-adic valued roots zk(x) the number of points for N0k≥ 0 pk= 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 hn(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 ep 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.

5. Can number theoretical universality be consistent with the proposed numerical definition of the p-adic integral?

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 zk(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+ pn-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 Un=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.

A question inspired by this example is whether the values of p-adic integrals as p-adic numbers could be determined by the few lowest powers of p with higher order contribution giving something proportional to an infinite power of p.

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)=∑ vntn with the coefficients vn giving the volume in O(pn)=0 cutoff. One can also define partition functions Zf(t)= ∑ fntn, with fn 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 = p1/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.

7 Comments:

At 10:30 AM, Blogger ThePeSla said...

Matti,

We all seem to have similar concerns and directions again.

The curl concept in physics and the simple at a small region integration of the boundary- and in view of our new concepts of multidimesions- has to be generalized a great deal more than it has even for supersymmetry -see Gauss Bonnet theorem.

I had a problem today with the series concept of bundling positive or negative regions- this may have useful explorations and all of this certainly applies to thermodynamics.

I doubt you should soul search about becoming a theoretical physicist for you definitely are one already regardless of employment or training. Some imagine they want to be this or that but never see the reality of it.

 
At 8:36 PM, Anonymous matpitka@luukku.com said...

In this posting the question what about the possibility of p-adic numeric integration.

The basic problem reason to my worries is that mathematicians usually express themselves in Bourbaki, a language incomprehensible for even mathematicians. Only rarely- such as in Strings 2011- they talk in such a manner that the physicist can understand motivations and goals and even something about what has been done and how. This is a pity.

I am not quite sure whether my worries about identity as theoretical physicist- or to put it precisely- mathematical physicist- did not contain some tongue in cheek. The fact is however that the immensity of the mathematical world as compared to shortness of life span is devastating.

 
At 8:13 AM, Blogger ThePeSla said...

Matti :-)

I had to look up what Bourbaki was but I agree with your analysis and that of the evaluations of it in the link. I find several areas of which I would be in direct disagreement with its style- for us the inclusion of topology and for me in particular the possibility of pictures.

Lubos has a most interesting post today (for we are all three concerned with this issue) and it is a different language again as mine where certainly such languages are incomprehensible to ourselves (few can be fluent in more than one language).

Your persistence to our purpose will lead to a long life bypassing the third part of our gene. The 1100 or so maths are wide but not devastating. There is time enough for one language of these notions and the wideness is life giving just as much as our stable relationships with others.

A little bit of worry is a good thing, but in moderation.

Last night looking at this problem again I see as the same sort of frontier notions I may have shifted my thoughts on such summations and clearly each of you theoreticians here have a part of the ingredients in the new physics bigger picture.

The PeSla

 
At 6:56 PM, Anonymous matpitka@luukku.com said...

TGD has reached now the stage in which it begins to contribute more and more to mathematics in form of conjectures. I would compare the situation to that when Newton's differential and integral calculus began to emerge. Integral calculus is emerging also now (for decades already, Grothendienck) but now in p-adic sense. It will have powerful implications also for real number based physics.


The conjecture represented in the posting is just one example about how completely new mathematical continents become visible as mathematical consciousness evolves.

*By number theoretic universality p-adic integrals are algebraic continuations as functions of the parameters characterizing the variety and real function integrated. This is just the TGD based manner to formulate the basic idea of motivic integration.

*The integrals should be also obtainable as sums for the numbers of p-adic points with given value of function weighted by the value of function. This should be the practical manner to calculate the integrals if such method exists. aI do not know whether mathematicians have proposed something like this.

These two definitions should be equivalent in some sense. At his moment the truth of this conjecture looks like a complete mathematical miracle. Why real volume integrals could be integrated as numbers of p-adic points of p-adic variant of variety (recall that rational m/n corresponds to infinite p-adic integer finite in real sense). Can one device any argument making this plausible?

Something extremely deep about which I - and perhaps even the mathematicians- do not have a slightest idea yet, must be involved if the conjecture is true. The conjecture could of course fail as most mathematical conjectures do.

 
At 7:00 PM, Blogger Matti Pitkanen said...

Sorry for typo: in the paragraph before the last one
"infinite p-adic integer finite in real sense" should read "finite p-adic integer infinite in real sense".

 
At 8:40 AM, Blogger ThePeSla said...

Matti,

Most interesting and developing notions of which terminology aside the intensity of enquiry may leave us a little lost, but anyone should be able to grasp the excitement and intensity.

I think I might adopt your term phantom DNA for some recent ideas. It seems that the transcendental issue you raise is one of my concerns to as to how it affect the mathematics- after all it seems the circle-square is only true around 8 dimensions or so.

I am also topologically interested in these ideas of flux tubes and wormholes, certainly the triality is involved at the base level and even within that (after all in your next post are ideas of which I find it harder to question some effect not there as it is part of the natural code- can life itself and the DNA forms arise just from the water? Hmmmmm...) What are the limits and structures of flux tube wormholes etc? Can we exceed that?

Keep up the deep thinking. I wish we understood each others visions a little better. Oh did you catch the info that Plancks constant may be smaller than what we now think?

The PeSla

 
At 4:09 AM, Anonymous riemann sum calculator said...

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How riemann sum calculator will be used and where .Please tell me it's benefit ..

 

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