tag:blogger.com,1999:blog-10614348.post8465806063323759613..comments2024-01-22T11:26:37.599-08:00Comments on TGD diary: Does Riemann Zeta Code for Generic Coupling Constant Evolution?Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-10614348.post-26841701890480286522015-11-26T22:31:58.834-08:002015-11-26T22:31:58.834-08:00
The answer contained a stupid little euro related...<br />The answer contained a stupid little euro related to the measure dr/r. I wrote a posting where it is done correctly.<br /><br />I noticed also that the existence of supersymplectic representations free of pathologies in turn essential for the existence of WCW strongly suggests that zeros are at critical line: assuming that the conformal weights are forced by strong form of holography to be zeros of zeta. <br /><br />Scaling operators for plane waves at light-cone boundary corresponds to the dilation operator H=xp. <br /><br />Eigenvalues must however contain strange looking real part 1/2 which would spoil unitarity at real axis. But since the inner product defined by the integral from 0 to infty now ordinary plane waves would be pathological. Presumably so because probability could leak out at origin. Real part 1/2 removes the pathology by reducing the inner product to standard inner product at real line. <br />Matpitka6@gmail.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-90910355783251395922015-11-26T21:38:21.126-08:002015-11-26T21:38:21.126-08:00Very nice, Thanks for the comment. For some reaso...Very nice, Thanks for the comment. For some reason I had the idea this relates to yang-mills mass gap , if , from what I understand that it's the difference between the lowest and next lowest eigenvalue.. in the H=xp paper they say that the RiemannSiegel vartheta function has a symmetry that indicates the position and momentum eigenfunctions are time-reverses of each other and that H=xp "generates dilations" Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-90620924427534446582015-11-26T18:59:38.653-08:002015-11-26T18:59:38.653-08:00My own "Strategy for Proving Riemann Hypothes...<br />My own "Strategy for Proving Riemann Hypothesis" relies on coherent states instead of eigenstates of Hamiltonian. The above approach in turn absorbs the problematic 1/2 to the integration measure at light cone boundary and conformal invariance is also now central. <br /><br /> Quite generally, I believe that conformal invariance in extended form applying at metrically 2-D light-cone boundary (light-like orbits of partonic 2-surfaces) might be central for understanding why physics requires RH. <br /><br />For instance, generating elements of extended supersymplectic algebra are labelled by generators having zeros of zeta as conformal weights. The number of generators is infinite. If some generator(s). s=1/2+iy guarantees that the real parts of conformal weights for all states are half integers. By conformal confinement the sum of y:s vanish for physical states. If some weight is not at critical line the situation changes. s= x+iy gives all multiples of x shifted by all half odd integer values. And of course, the realisation as plane waves at boundary fails.<br />Matpitka6@gmail.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-63361353412610065052015-11-26T18:49:27.985-08:002015-11-26T18:49:27.985-08:00Concerning H=xp and 1/2 in the non-trivial zeros o...Concerning H=xp and 1/2 in the non-trivial zeros of zeta. This can be understood also physically in zero energy ontology.<br /><br /> When one constructs tangent space basis in the "world of classical worlds", one has by holography analogs of plane waves at light-cone boundary propagating in radial light-like direction with coordinate r. They have the form r^{q +iy) and preferred extremal condition gives extremely powerful additional constraints quantisation q+iy: the first guess is that non-trivial zeros of zeta are obtained q=1/2.<br /><br />How to understand q=1/2. The natural scaling invariant integration measure defining inner product for "plane" waves is dr/r =dlog(r). The inner product must be same as for ordinary plane waves and indeed is for plane waves psi= r^(1/2+iy) since in inner product r from Int psi_1*psi_2 dr/r is cancelled by 1/r from integration measure.<br /><br />One has analogs of ordinary plane waves with delta function normalisation. The identification of 1/2+iy is zero of zeta is natural by generalised conformal invariance. <br /><br />If one assumes that p-adic primes correspond to zeros of zeta in 1-1 manner in the sense that p^iy(p) is root of unity existing in all number fields (algebraic extension of p-adics) one obtains that the plane wave exists for p at points r= p^n: powers of p one obtains a delta function distribution concentrated on powers of p: logarithmic lattice. This can be seen as space-time correlate for p-adicity. Something very similar is obtained from the Fourier transform of distribution of zeros at critical line (Dyson).<br />Matpitka6@gmail.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-19183673619797618222015-11-26T13:04:56.405-08:002015-11-26T13:04:56.405-08:00Yeah.. mathematicians are annoying like that. Sorr...Yeah.. mathematicians are annoying like that. Sorry to go off topic.. I'm trying to understand H=xp .. here is an idea related to coherent states rather than eigenstates. I get what you are saying about the complexification .. it's a nice idea..<br /><br />https://scholar.google.com/scholar?cluster=16925851478872994728&hl=en&as_sdt=0,44&sciodt=0,44Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-45995874804166101872015-11-25T20:47:07.987-08:002015-11-25T20:47:07.987-08:00Not a slightest idea;-). My source of frustrati...<br />Not a slightest idea;-). My source of frustrations as a physicist is that mathematicians see so totally different things as relevant. They see only the technical challenges related to proving theorems - many of which are "obvious" for physicist and mere technical details. Mathematicians would have ideal skills for challenging the assumptions behind the basic notions- say the notion of state of von Neumann algebra but they are happy to just prove theorems. <br /><br />Kadison-Singer problem relates to extension of pure states in Abelian algebra of diagonal bounded operators -maximal commuting set of observable for physicist- to the entire algebra. State in the sense used is the one used by C*-algebra people trying to reduce quantum field theory to density matrices. <br /><br />The profound problem of algebraic quantum field theory -AQFT - is that it only produces the statistical aspects of quantum theory - "thermodynamics". States in this sense indeed are counterparts for density matrices. One can identify pure states as counterparts of quantum states but the description of interference effects etc become difficult and one loses the physical picture so relevant for quantum theory.<br /><br />Zero energy ontology leading to "complex square root" of thermodynamics is required and this brings in complex hermitian square roots of density matrices and also that of partition function. One obtains generalisation of quantum state and also of AQFT. Basic theorems like Tomita's theorem (I hope I remember correctly) should be generalised. Many things to do but ...Matpitka6@gmail.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-27733066351723206112015-11-25T20:00:50.789-08:002015-11-25T20:00:50.789-08:00a paper on the hot topic of interlacing polynomial...a paper on the hot topic of interlacing polynomials is at http://arxiv.org/abs/1306.3969v4<br /><br /><br />do you know if this applies to only polynomials? does it generalize to infinite dimensional case?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-86446858753065520272015-11-21T19:06:05.815-08:002015-11-21T19:06:05.815-08:00Fejer kernel seems to be average of approximations...Fejer kernel seems to be average of approximations to delta function at zero. Easier to remember. Matpitka6@gmail.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-77261017575158342642015-11-21T19:03:55.330-08:002015-11-21T19:03:55.330-08:00
About almost-Hermitian operators. The problem of...<br />About almost-Hermitian operators. The problem of standard approach is that zeros s= 1/2+iy are not eigenvalues of a unitary operator. In Zero Energy Ontology wave function is replaced with a couple square root of density matrix and vacuum functional with a complex square root of partition function. This interpretational problem disappears. It is a pity that a too restricted view about quantum theory leads to misguided attempts to understand RH in terms of physical analogies. But this is not my problem;-). Matpitka6@gmail.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-34737003970036836342015-11-21T18:58:40.555-08:002015-11-21T18:58:40.555-08:00Ordering by size is essential for obtaining realis...Ordering by size is essential for obtaining realistic coupling constant evolution.<br />Matti Pitkänenhttps://www.blogger.com/profile/13512912323574611883noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-24983008652634498802015-11-21T14:02:05.276-08:002015-11-21T14:02:05.276-08:00corrected
𝒥^(2,+)x(t)={(p,X)|x(t+z)⩽x(t)+p⋅z+(X:...corrected<br /><br />𝒥^(2,+)x(t)={(p,X)|x(t+z)⩽x(t)+p⋅z+(X:z⊗z)/2+o(|z|^2) as z→0}<br />𝒥^(2,-)x(t)={(p,X)|x(t+z)⩾x(t)+p⋅z+(X:z⊗z)/2+o(|z|^2) as z→0}<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-44451414488983077562015-11-21T13:57:24.421-08:002015-11-21T13:57:24.421-08:00This comment has been removed by the author.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-14965172863591636302015-11-21T13:55:05.101-08:002015-11-21T13:55:05.101-08:00https://en.wikipedia.org/wiki/Fej%C3%A9r_kernel
U...https://en.wikipedia.org/wiki/Fej%C3%A9r_kernel<br /><br />Unitary Correlations and the Fiejer kernel<br /><br />https://statistics.stanford.edu/sites/default/files/2001-01.pdf<br /><br />you might be on to something here, from what I can tell with my mathematical understanding...<br /><br />wikipedia has something about "almost-Hermitian operators" I think this might be found in the last section where I briefly mention the possibility<br /><br />http://vixra.org/pdf/1510.0475v6.pdf on the last page in section 2.3<br /><br />𝒥^(2,+)x(t)={(p,X)|x(t+z)⩽x(t)+p⋅z+(X:z⊗z)/2+o(|z|^2) as z→0}<br />𝒥^(2,-)x(t)={(p,X)|x(t+z)⩾u(x)+p⋅z+(X:z⊗z)/2+o(|z|^2) as z→0}<br /><br />what I think is so cool is that the error term just so happens to be small-o |z|^2 hapybe the 'approximation error' is also a (randomly.. at what level?) complex wavefunction?<br /><br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-39992117190759448162015-11-20T23:35:57.926-08:002015-11-20T23:35:57.926-08:00So the relation is the ordering in which they appe...So the relation is the ordering in which they appear and not some other permutation? Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-51808272272912036112015-11-20T19:19:12.213-08:002015-11-20T19:19:12.213-08:00
I do not see as a question of whether to believe ...<br />I do not see as a question of whether to believe or not.<br /><br />Number theoretical universality - one of the basic principles of quantum TGD - states that for given prime p p^iy exists for some set C(p) of zeros y. The strong form - supported now by the stunning success of the identification of zeros as inverses of U(1) coupling constant strength - states that correspondence is 1-1: C(p) contains only one zero.<br /><br />Another support for the hypothesis is that it works and predicts U(1) coupling at electron scale with accuracy of .7 per cent without any further assumptions and that it leads to a parametrisation of generic coupling constant evolution in terms of rational or integer parameter real Mobius transformation. This is incredibly powerful prediction: number theoretical universality would provide highly detailed overall view about physics in all length scales. No one has dared even to dream of anything like this.<br /><br />Dyson speculated that zeros and primes and their powers form quasicrystals. Ordinary crystal is such and zeros and primes would be analogous to lattice and reciprocal lattice and therefore in 1-1 correspondence naturally.<br /><br />Matpitka6@gmail.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-39118342360286504782015-11-20T14:50:46.943-08:002015-11-20T14:50:46.943-08:00The part I have doubts about is the validity of as...The part I have doubts about is the validity of assigning a prime to each zero. I really doubt there is a one to one correspondence.. unless I missed something<br />--StephenAnonymousnoreply@blogger.com