Philosophy of Adelic Physics

Abstract

The p-adic aspects of Topological Geometrodynamics (TGD) will be discussed. Introduction gives a short summary about classical and quantum TGD. This is needed since the p-adic ideas are inspired by TGD based view about physics.

p-Adic mass calculations relying on p-adic generalization of thermodynamics and super-symplectic and super-conformal symmetries are summarized. Number theoretical existence constrains lead to highly non-trivial and successful physical predictions. The notion of canonical identification mapping p-adic mass squared to real mass squared emerges, and is expected to be a key player of adelic physics allowing to map various invariants from p-adics to reals and vice versa.

A view about p-adicization and adelization of real number based physics is proposed. The proposal is a fusion of real physics and various p-adic physics to single coherent whole achieved by a generalization of number concept by fusing reals and extensions of p-adic numbers induced by given extension of rationals to a larger structure and having the extension of rationals as their intersection.

The existence of p-adic variants of definite integral, Fourier analysis, Hilbert space, and Riemann geometry is far from obvious and various constraints lead to the idea of number theoretic universality (NTU) and finite measurement resolution realized in terms of number theory. An attractive manner to overcome the problems in case of symmetric spaces relies on the replacement of angle variables and their hyperbolic analogs with their exponentials identified as roots of unity and roots of e existing in finite-dimensional algebraic extension of p-adic numbers. Only group invariants—typically squares of distances and norms—are mapped by canonical identification from p-adic to real realm and various phases are mapped to themselves as number theoretically universal entities.

Also the understanding of the correspondence between real and p-adic physics at various levels—space-time level, imbedding space level, and level of “world of classical worlds” (WCW)—is a challenge. The gigantic isometry group of WCW and the maximal isometry group of imbedding space give hopes about a resolution of the problems. Strong form of holography (SH) allows a non-local correspondence between real and p-adic space-time surfaces induced by algebraic continuation from common string world sheets and partonic 2-surfaces. Also local correspondence seems intuitively plausible and is based on number theoretic discretization as intersection of real and p-adic surfaces providing automatically finite “cognitive” resolution. The existence of p-adic variants of Kähler geometry of WCW is a challenge, and NTU might allow to realize it.

I will also sum up the role of p-adic physics in TGD inspired theory of consciousness. Negentropic entanglement (NE) characterized by number theoretical entanglement negentropy (NEN) plays a key role. Negentropy Maximization Principle (NMP) forces the generation of NE. The interpretation is in terms of evolution as increase of negentropy resources.

Appendices

Appendix: Super-Symplectic Conformal Weights and Zeros of Riemann Zeta

Since fermions are the only fundamental particles in TGD one could argue that the conformal weight of the generating elements of supersymplectic algebra could be negatives for the poles of fermionic zeta ζ _F . This demands n > 0 as does also the fractal hierarchy of super-symplectic symmetry breakings. NTU of Riemann zeta in some sense is strongly suggested if adelic physics is to make sense.

For ordinary conformal algebras there are only finite number of generating elements ( − 2 ≤ n ≤ 2). If the radial conformal weights for the generators of g consist of poles of ζ _F , the situation changes. ζ _F is suggested by the observation that fermions are the only fundamental particles in TGD.

• 1. Riemann Zeta ζ(s) = ∏ _p (1∕(1 − p ^−s) identifiable formally as a partition function ζ _B (s) of arithmetic boson gas with bosons with energy log(p) and temperature 1∕s = 1∕(1∕2 + iy) should be replaced with that of arithmetic fermionic gas given in the product representation by ζ _F (s) = ∏ _p (1 + p ^−s) so that the identity ζ _B (s))∕ζ _F (s) = ζ _B (2s) follows. This gives

  $$\displaystyle{ \frac{\zeta _{B}(s)} {\zeta _{B}(2s)}.}$$

  ζ _F (s) has zeros at zeros s _n of ζ(s) and at the pole s = 1∕2 of zeta(2s). ζ _F (s) has poles at zeros s _n ∕2 of ζ(2s) and at pole s = 1 of ζ(s).

  The spectrum of 1∕T would be for the generators of algebra {(−1∕2 + iy)∕2, n > 0, −1}. In p-adic thermodynamics the p-adic temperature is 1∕T = 1∕n and corresponds to “trivial” poles of ζ _F . Complex values of temperature do not make sense in ordinary thermodynamics. In ZEO quantum theory can be regarded as a square root of thermodynamics and complex temperature parameter makes sense.

• 2. If the spectrum of conformal weights of the generating elements of the algebra corresponds to poles serving as analogs of propagator poles, it consists of the “trivial” conformal h = n > 0- the standard spectrum with h = 0 assignable to massless particles excluded—and “non-trivial” h = −1∕4 + iy∕2. There is also a pole at h = −1.

  Both the non-trivial pole with real part h _R = −1∕4 and the pole h = −1 correspond to tachyons. I have earlier proposed conformal confinement meaning that the total conformal weight for the state is real. If so, one obtains for a conformally confined two-particle states corresponding to conjugate non-trivial zeros in minimal situation h _R = −1∕2 assignable to N-S representation.

  In p-adic mass calculations ground state conformal weight must be − 5∕2 [34]. The negative fermion ground state weight could explain why the ground state conformal weight must be tachyonic − 5∕2. With the required five tensor factors one would indeed obtain this with minimal conformal confinement. In fact, arbitrarily large tachyonic conformal weight is possible but physical state should always have conformal weights h > 0.

• 3. h = 0 is not possible for generators, which reminds of Higgs mechanism for which the naive ground states correspond to tachyonic Higgs. h = 0 conformally confined massless states are necessarily composites obtained by applying the generators of Kac-Moody algebra or super-symplectic algebra to the ground state. This is the case according to p-adic mass calculations [34], and would suggest that the negative ground state conformal weight can be associated with super-symplectic algebra and the remaining contribution comes from ordinary super-conformal generators. Hadronic masses, whose origin is poorly understood, could come from super-symplectic degrees of freedom. There is no need for p-adic thermodynamics in super-symplectic degrees of freedom.

A General Formula for the Zeros of Zeta from NTU

Dyson’s comment about Fourier transform of Riemann Zeta [1] (http://tinyurl.com/hjbfsuv) is interesting from the point of NTU for Riemann zeta.

• 1. The numerical calculation of Fourier transform for the imaginary parts iy of zeros s = 1∕2 + iy of zeta shows that it is concentrated at discrete set of frequencies coming as log(p ⁿ), p prime. This translates to the statement that the zeros of zeta form a 1-dimensional quasicrystal, a discrete structure Fourier spectrum by definition is also discrete (this of course holds for ordinary crystals as a special case). Also the logarithms of powers of primes would form a quasicrystal, which is very interesting from the point of view of p-adic length scale hypothesis. Primes label the “energies” of elementary fermions and bosons in arithmetic number theory, whose repeated second quantization gives rise to the hierarchy of infinite primes [46]. The energies for general states are logarithms of integers.

• 2. Powers p ⁿ label the points of quasicrystal defined by points log(p ⁿ) and Riemann zeta has interpretation as partition function for boson case with this spectrum. Could p ⁿ label also the points of the dual lattice defined by iy.

• 3. The existence of Fourier transform for points log(p _i ⁿ) for any vector y _a in class C(p) of zeros labelled by p requires $p_{i}^{iy_{a}}$ to be a root of unity inside C(p). This could define the sense in which zeros of zeta are universal. This condition also guarantees that the factor n ^{−1∕2−iy} appearing in zeta at critical line are number theoretically universal (p ^1∕2 is problematic for Q _p : the problem might be solved by eliminating from p-adic analog of zeta the factor 1 − p ^−s).

  • (a) One obtains for the pair (p _i , s _a ) the condition log(p _i )y _a = q _ia 2π, where q _ia is a rational number. Dividing the conditions for (i, a) and (j, a) gives

    $$\displaystyle{p_{i} = p_{j}^{q_{ia}/q_{ja} }}$$

    for every zero s _a so that the ratios q _ia ∕q _ja do not depend on s _a . From this one easily deduce p _i ^M = p _j ^N, where M and N are integers so that one ends up with a contradiction.

  • (b) Dividing the conditions for (i, a) and (i, b) one obtains

    $$\displaystyle{\frac{y_{a}} {y_{b}} = \frac{q_{ia}} {q_{ib}}}$$

    so that the ratios q _ia ∕q _ib do not depend on p _i . The ratios of the imaginary parts of zeta would be therefore rational number which is very strong prediction and zeros could be mapped by scaling y _a ∕y ₁ where y ₁ is the zero which smallest imaginary part to rationals.

  • (c) The impossible consistency conditions for (i, a) and (j, a) can be avoided if each prime and its powers correspond to its own subset of zeros and these subsets of zeros are disjoint: one would have infinite union of sub-quasicrystals labelled by primes and each p-adic number field would correspond to its own subset of zeros: this might be seen as an abstract analog for the decomposition of rational to powers of primes. This decomposition would be natural if for ordinary complex numbers the contribution in the complement of this set to the Fourier transform vanishes. The conditions (i, a)and (i, b) require now that the ratios of zeros are rationals only in the subset associated with p _i .

For the general option the Fourier transform can be delta function for x = log(p ^k) and the set {y _a (p)} contains N _p zeros. The following argument inspires the conjecture that for each p there is an infinite number N _p of zeros y _a (p) in class C(p) satisfying

$$\displaystyle{p^{iy_{a}(p)} = u(p) = e^{ \frac{r(p)} {m(p)} i2\pi },}$$

where u(p) is a root of unity that is y _a (p) = 2π(m(a) + r(p))∕log(p) and forming a subset of a lattice with a lattice constant y ₀ = 2π∕log(p), which itself need not be a zero.

In terms of stationary phase approximation the zeros y _a (p) associated with p would have constant stationary phase whereas for y _a (p _i ≠ p) the phase $p^{iy_{a}(p_{i})}$ would fail to be stationary. The phase e ^ixy would be non-stationary also for x ≠ log(p ^k) as function of y.

• 1. Assume that for x = qlog(p), where q not a rational, the phases e ^ixy fail to be roots of unity and are random implying the vanishing/smallness of F(x).

• 2. Assume that for a given p all powers p ^iy for y ∉ {y _a (p)} fail to be roots of unity and are also random so that the contribution of the set y ∉ {y _a (p)} to F(p) vanishes/is small.

• 3. For x = log(p ^k∕m) the Fourier transform should vanish or be small for m ≠ 1 (rational roots of primes) and give a non-vanishing contribution for m = 1. One has

  $$\displaystyle{\begin{array}{l} F(x = log(p^{k/m}) =\sum _{1\leq a\leq N(p)}e^{k\frac{M(a,p)} {mN(p)} i2\pi }u(p), \\ u(p) = e^{ \frac{r(p)} {m(p)} i2\pi }.\end{array} }$$

  Obviously one can always choose N(a, p) = N(p).

• 4. For the simplest option N(p) = 1 one would obtain delta function distribution for x = log(p ^k). The sum of the phases associated with y _a (p) and − y _a (p) from the half axes of the critical line would give

  $$\displaystyle{F(x = log(p^{n})) \propto X(p^{n}) \equiv 2cos(n \frac{r(p)} {m(p)}2\pi ).}$$

  The sign of F would vary.

• 5. For x = log(p ^k∕m) the value of Fourier transform is expected to be small by interference effects if M(a, p) is random integer, and negligible as compared with the value at x = log(p ^k). This option is highly attractive. For N(p) > 1 and M(a, p) a random integer also F(x = log(p ^k) is small by interference effects. Hence it seems that this option is the most natural one.

• 6. The rational r(p)∕m(p) would characterize given prime (one can require that r(p) and m(p) have no common divisors). F(x) is non-vanishing for all powers x = log(p ⁿ) for m(p) odd. For p = 2, also m(2) = 2 allows to have | X(2ⁿ) | = 2. An interesting ad hoc ansatz is m(p) = p or p ^s(p). One has periodicity in n with period m(p) that is logarithmic wave. This periodicity serves as a test and in principle allows to deduce the value of r(p)∕m(p) from the Fourier transform.

What could one conclude from the data (http://tinyurl.com/hjbfsuv)?

• 1. The first graph gives | F(x = log(p ^k) | and second graph displays a zoomed up part of | F(x = log(p ^k) | for small powers of primes in the range [2, 19]. For the first graph the eighth peak (p = 11) is the largest one but in the zoomed graphs this is not the case. Hence something is wrong or the graphs correspond to different approximations suggesting that one should not take them too seriously.

  In any case, the modulus is not constant as function of p ^k. For small values of p ^k the envelope of the curve decreases and seems to approach constant for large values of p ^k (one has x < 15 (e ¹⁵ ≃ 3. 3 × 10⁶).

• 2. According to the first graph | F(x) | decreases for x = klog(p) < 8, is largest for small primes, and remains below a fixed maximum for 8 < x < 15. According to the second graph the amplitude decreases for powers of a given prime (say p = 2). Clearly, the small primes and their powers have much larger | F(x) | than large primes.

There are many possible reasons for this behavior. Most plausible reason is that the sums involved converge slowly and the approximation used is not good. The inclusion of only 10⁴ zeros would show the positions of peaks but would not allow reliable estimate for their intensities.

• 1. The distribution of zeros could be such that for small primes and their powers the number of zeros is large in the set of 10⁴ zeros considered. This would be the case if the distribution of zeros y _a (p) is fractal and gets “thinner” with p so that the number of contributing zeros scales down with p as a power of p, say 1∕p, as suggested by the envelope in the first figure.

• 2. The infinite sum, which should vanish, converges only very slowly to zero. Consider the contribution $\Delta F(p^{k},p_{1})$ of zeros not belonging to the class p ₁ ≠ p to $F(x = log(p^{k})) =\sum _{p_{i}}\Delta F(p^{k},p_{i})$, which includes also p _i = p. $\Delta F(p^{k},p_{i})$, p ≠ p ₁ should vanish in exact calculation.

  • (a) By the proposed hypothesis this contribution reads as

    $$\displaystyle{\begin{array}{l} \Delta F(p,p_{1}) =\sum _{a}cos\left [X(p^{k},p_{1})(M(a,p_{1}) + \frac{r(p_{1})} {m(p_{1})})2\pi )\right ]. \\ X(p^{k},p_{1}) = \frac{log(p^{k})} {log(p_{1})}.\\ \end{array} }$$

    Here a labels the zeros associated with p ₁. If p ^k is “approximately divisible” by p ¹ in other words, p ^k ≃ np ₁, the sum over finite number of terms gives a large contribution since interference effects are small, and a large number of terms are needed to give a nearly vanishing contribution suggested by the non-stationarity of the phase. This happens in several situations.

  • (b) The number π(x) of primes smaller than x goes asymptotically like π(x) ≃ x∕log(x) and prime density approximately like 1∕log(x) − 1∕log(x)² so that the problem is worst for the small primes. The problematic situation is encountered most often for powers p ^k of small primes p near larger prime and primes p (also large) near a power of small prime (the envelope of | F(x) | seems to become constant above x ∼ 10³).

  • (c) The worst situation is encountered for p = 2 and p ₁ = 2^k − 1—a Mersenne prime and $p_{1} = 2^{2^{k} } + 1$, k ≤ 4—Fermat prime. For (p, p ₁) = (2^k, M _k ) one encounters X(2^k, M _k ) = (log(2^k)∕log(2^k − 1) factor very near to unity for large Mersennes primes. For (p, p ₁) = (M _k , 2) one encounters X(M _k , 2) = (log(2^k − 1)∕log(2) ≃ k. Examples of Mersennes and Fermats are (3, 2), (5, 2), (7, 2), (17, 2), (31, 2), (127, 2), (257, 2), . . . Powers 2^k, k = 2, 3, 4, 5, 7, 8, . . are also problematic.

  • (d) Also twin primes are problematic since in this case one has factor $X(p = p_{1} + 2,p_{1}) = \frac{log(p_{1}+2)} {log(p_{1})}$. The region of small primes contains many twin prime pairs: (3,5), (5,7), (11,13), (17,19), (29,31),….

  These observations suggest that the problems might be understood as resulting from including too small number of zeros.

• 3. The predicted periodicity of the distribution with respect to the exponent k of p ^k is not consistent with the graph for small values of prime unless the periodic m(p) for small primes is large enough. The above-mentioned effects can quite well mask the periodicity. If the first graph is taken at face value for small primes, r(p)∕m(p) is near zero, and m(p) is so large that the periodicity does not become manifest for small primes. For p = 2 this would require m(2) > 21 since the largest power 2ⁿ ≃ e ¹⁵ corresponds to n ∼ 21.

To summarize, the prediction is that for zeros of zeta should divide into disjoint classes {y _a (p)} labelled by primes such that within the class labelled by p one has $p^{iy_{a}(p)} = e^{(r(p)/m(p))i2\pi }$ so that has y _a (p) = [M(a, p) + r(p)∕m(p))]2π∕log(p).

More Precise View About Zeros of Zeta

There is a very interesting blog post by Mumford (http://tinyurl.com/zemw27o), which leads to much more precise formulation of the idea and improved view about the Fourier transform hypothesis: the Fourier transform or its generalization must be defined for all zeros, not only the non-trivial ones and trivial zeros give a background term allowing to understand better the properties of the Fourier transform.

Mumford essentially begins from Riemann’s “explicit formula” in von Mangoldt’s form.

$$\displaystyle{\sum _{p}\sum _{n\geq 1}log(p)\delta _{p^{n}}(x) = 1 -\sum _{k}x^{s_{k}-1} - \frac{1} {x(x^{2} - 1)},}$$

where p denotes prime and s _k a non-trivial zero of zeta. The left-hand side represents the distribution associated with powers of primes. The right-hand side contains sum over cosines

$$\displaystyle{\sum _{k}x^{s_{k}-1} = 2\frac{\sum _{k}cos(log(x)y_{k})} {x^{1/2}},}$$

where y _k is the imaginary part of non-trivial zero. Apart from the factor x ^−1∕2 this is just the Fourier transform over the distribution of zeros.

There is also a slowly varying term $1 - \frac{1} {x(x^{2}-1)}$, which has interpretation as the analog of the Fourier transform term but sum over trivial zeros of zeta at s = −2n, n > 0. The entire expression is analogous to a “Fourier transform” over the distribution of all zeros. Quasicrystal is replaced with union on 1-D quasicrystals.

Therefore the distribution for powers of primes is expressible as “Fourier transform” over the distribution of both trivial and non-trivial zeros rather than only non-trivial zeros as suggested by numerical data to which Dyson [1] referred to (http://tinyurl.com/hjbfsuv). Trivial zeros give a slowly varying background term large for small values of argument x (poles at x = 0 and x = 1—note that also p = 0 and p = 1 appear effectively as primes) so that the peaks of the distribution are higher for small primes.

The question was how can one obtain this kind of delta function distribution concentrated on powers of primes from a sum over terms cos(log(x)y _k ) appearing in the Fourier transform of the distribution of zeros.

Consider x = p ⁿ. One must get a constructive interference. Stationary phase approximation is in terms of which physicist thinks. The argument was that a destructive interference occurs for given x = p ⁿ for those zeros for which the cosine does not correspond to a real part of root of unity as one sums over such y _k : random phase approximation gives more or less zero. To get something non-trivial y _k must be proportional to 2π × n(y _k )∕log(p) in class C(p) to which y _k belongs. If the number of these y _k :s in C(p) is infinite, one obtains delta function in good approximation by destructive interference for other values of argument x.

The guess that the number of zeros in C(p) is infinite is encouraged by the behaviors of the densities of primes one hand and zeros of zeta on the other hand. The number of primes smaller than real number x goes like

$$\displaystyle{\pi (x) = \#(primes <x) \sim \frac{x} {log(x)}}$$

in the sense of distribution. The number of zeros along critical line goes like

$$\displaystyle{\#(zeros <t) = (t/2\pi ) \times log( \frac{t} {2\pi })}$$

in the same sense. If the real axis and critical line have same metric measure, then one can say that the number of zeros in interval T per number of primes in interval T behaves roughly like

$$\displaystyle{ \frac{\#(zeros <T)} {\#(primes <T)} = log(\frac{T} {2\pi } ) \times \frac{log(T)} {2\pi } }$$

so that at the limit of T → ∞ the number of zeros associated with given prime is infinite. This assumption of course makes the argument a poor man’s argument only.

Possible Relevance for TGD

What this speculative picture from the point of view of TGD?

• 1. A possible formulation for NTU for the poles of fermionic Riemann zeta ζ _F = ζ(s)∕ζ(2s) could be as a condition that is that the exponents $p^{ks_{a}(p)/2} = p^{k/4}p^{iky_{a}(p)/2}$ exist in a number theoretically universal manner for the zeros s _a (p) for given p-adic prime p and for some subset of integers k. If the proposed conditions hold true, exponent reduces p ^k∕4 e ^{k(r(p∕m(p)i2π} requiring that k is a multiple of 4. The number of the non-trivial generating elements of super-symplectic algebra in the monomial creating physical state would be a multiple of 4. These monomials would have real part of conformal weight − 1. Conformal confinement suggests that these monomials are products of pairs of generators for which imaginary parts cancel.

• 2. Quasicrystal property might have an application to TGD. The functions of light-like radial coordinate appearing in the generators of super-symplectic algebra could be of form r ^s, s zero of zeta or rather, its imaginary part. The eigenstate property with respect to the radial scaling rd∕dr is natural by radial conformal invariance.

  The idea that arithmetic QFT assignable to infinite primes is behind the scenes in turn suggests light-like momenta assignable to the radial coordinate have energies with the dual spectrum log(p ⁿ). This is also suggested by the interpretation of ζ as square root of thermodynamical partition function for boson gas with momentum log(p) and analogous interpretation of ζ _F .

  The two spectra would be associated with radial scalings and with light-like translations of light-cone boundary respecting the direction and light-likeness of the light-like radial vector. log(p ⁿ) spectrum would be associated with light-like momenta whereas p-adic mass scales would characterize states with thermal mass. Note that generalization of p-adic length scale hypothesis raises the scales defined by p ⁿ to a special physical position: this might relate to ideal structure of adeles.

• 3. Finite measurement resolution suggests that the approximations of Fourier transforms over the distribution of zeros taking into account only a finite number of zeros might have a physical meaning. This might provide additional understand about the origins of generalized p-adic length scale hypothesis stating that primes p ≃ p ₁ ^k, p ₁ small prime—say Mersenne primes—have a special physical role.