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Dažas domas, lasot Mario Livio grāmatu «Vai Dievs ir matemātiķis?»
Get PDFGalvenais grāmatas jautājums ir:
Vai matemātika ir atklājums vai izgudrojums
Trying to fuse the basic mathematical ideas of quantum TGD to a single coherent whole
Get PDFThe theoretical framework behind TGD involves several different strands and the goal is to unify them to a single coherent whole. TGD involves number theoretic and geometric visions about physics and duality, analogous to Langlands duality, is proposed to unify them. Also quantum classical correspondence (QCC) is a central aspect of TGD. One should understand both the duality and QCC at the level of detail.
The following mathematical notions are expected to be of relevance for this goal.
\begin{enumerate}
\item Von Neumann algebras, call them , in particular hyperfinite factors of type (HFFs), are in a central role. Both the geometric and number theoretic side, QCC could mathematically correspond to the relationship between and its commutant .
For instance, symplectic transformations leave induced K\"ahler form invariant and various fluxes of K\"ahler form are symplectic invariants and correspond to classical physics commuting with quantum physics coded by the super symplectic algebra (SSA). On the number theoretic side, the Galois invariants assignable to the polynomials determining space-time surfaces are analogous classical invariants.
\item The generalization of ordinary arithmetics to quantum arithmetics obtained by replacing and with and allows us to replace the notions of finite and p-adic number fields with their quantum variants. The same applies to various algebras.
\item Number theoretic vision leads to adelic physics involving a fusion of various p-adic physics and real physics and to hierarchies of extensions of rationals involving hierarchies of Galois groups involving inclusions of normal subgroups. The notion of adele can be generalized by replacing various p-adic number fields with the p-adic representations of various algebras.
\item The physical interpretation of the notion of infinite prime has remained elusive although a formal interpretation in terms of a repeated quantization of a supersymmetric arithmetic QFT is highly suggestive. One can also generalize infinite primes to their quantum variants. The proposal is that the hierarchy of infinite primes generalizes the notion of adele.
\end{enumerate}
The formulation of physics as K\"ahler geometry of the "world of classical worlds" (WCW) involves f 3 kinds of algebras ; supersymplectic isometries acting on , affine algebras acting on light-like partonic orbits, and isometries of light-cone boundary , allowing hierarchies .
The braided Galois group algebras at the number theory side and algebras at the geometric side define excellent candidates for inclusion hierarchies of HFFs. duality suggests that corresponds to the degree of the polynomial defining space-time surface and that the roots of correspond to braid strands at side. Braided Galois group would act in and hierarchies of Galois groups would induce hierarchies of inclusions of HFFs. The ramified primes of would correspond to physically preferred p-adic primes in the adelic structure formed by p-adic variants of with and replaced with and
Some New Ideas Related to Langlands Program viz. TGD
Get PDFLanglands' program seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Langlands program is described by Edward Frenkel as a kind of grand unified theory of mathematics.
In the TGD framework, duality assigns to a rational polynomial a set of mass shells in and by associativity condition a 4-D surface in , and its it to . means that number theoretic vision and geometric vision of physics are dual or at least complementary. This vision could extend to a trinity of number theoretic, geometric and topological views since geometric invariants defined by the space-time surfaces as Bohr orbit-like preferred extremals could serve as topological invariants.
Concerning the concretization of the basic ideas of Langlands program in TGD, the basic principle would be quantum classical correspondence (QCC), which is formulated as a correspondence between the quantum states in the "world of classical worlds" (WCW) characterized by analogs of partition functions as modular forms and classical representations realized as space-time surfaces. L-function as a counter part of the partition function would define as its roots space-time surfaces and these in turn would define via Galois group representation partition function. QCC would define a kind of closed loop giving rise to a hierarchy.
If Riemann hypothesis (RH) is true and the roots of L-functions are algebraic numbers, L-functions are in many aspects like rational polynomials and motivate the idea that, besides rationals polynomials, also L-functions could define space-time surfaces as kinds of higher level classical representations of physics.
One concretization of Langlands program would be the extension of the representations of the Galois group to the polynomials to the representations of reductive groups appearing naturally in the TGD framework. Elementary particle vacuum functionals are defined as modular invariant forms of Teichm\"uller parameters. Multiple residue integral is proposed as a manner to obtain L-functions defining space-time surfaces.
One challenge is to construct Riemann zeta and the associated function and the Hadamard product leads to a proposal for the Taylor coefficients of as a function of . One would have , . is the hyperbolic analogy for a root of unity and defines a finite-D transcendental extension of p-adic numbers and together with th roots of unity powers of define a discrete tessellation of the hyperbolic space .
This construction leads to the question whether also finite fields could play a fundamental role in the number theoretic vision. Prime polynomial with prime order and integer coefficients smaller than can be regarded as a polynomial in a finite field. If it satisfies the condition that the integer coefficients have no common prime factors, it defines an infinite prime. The proposal is that all physically allowed polynomials are constructible as functional composites of these
Some comments of the physical interpretation of Riemann zeta in TGD
Get PDFThe Riemann zeta function and its generalizations are very interesting from the point of view of the TGD inspired physics. duality assumes that rational polynomials define cognitive representations as unique discretizations of space-time regions interpreted in terms of a finite measurement resolution. One implication is that virtual momenta for fermions are algebraic integers in an extension of rationals defined by a rational polynomial and by Galois confinement integers for the physical states.
In principle, also real analytic functions, with possibly rational coefficients, make sense. The notion of conformal confinement with zeros of interpreted as mass squared values and conformal weights, makes and L-functions as its generalizations physically unique real analytic functions.
If the conjecture stating that the roots of are algebraic numbers is true, the virtual momenta of fermions could be algebraic integers for virtual fermions and integers for the physical states also for . This makes sense if the notions of Galois group and Galois confinement are sensible notions for .
In this article, the properties of and its symmetric variant and their multi-valued inverses are studied. In particular, the question whether might have no finite critical points is raised
TGD Inspired Model for Freezing in Nano Scales
Get PDFFreezing is a phase transition, which challenges the existing view of condensed matter in nanoscales. In the TGD framework, quantum coherence is possible in all scales and gravitational quantum coherence should characterize hydrodynamics in astrophysical and even shorter scales. The hydrodynamics at the surface of the planet such as Earth the mass of the planet and even that of the Sun should characterize gravitational Planck constant assignable to gravitational flux tubes mediating gravitational interactions. In this framework, quantum criticality involving phases of ordinary matter located at the magnetic body (MB) and possibly controlling ordinary matter, could be behind the criticality of also ordinary phase transitions.
In this article, a model inspired by the finding that the water-air boundary involves an ice-like layer. The proposal is that also at criticality for the freezing a similar layer exists and makes possible fluctuations of the size and shape of the ice blob. At criticality the change of the Gibbs free energy for water would be opposite that for ice and the Gibbs free energy liberated in the formation of ice layer would transform to the energy of surface tension at water-ice layer.
This leads to a geometric model for the freezing phase transition involving only the surface energy proportional to the area of the water-ice boundary and the constraint term fixing the volume of water. The partial differential equations for the boundary surface are derived and discussed.
If at the critical for the two phases at the boundary layer, the boundary consists of portions, which are minimal surfaces analogous to soap films and conformal invariance characterizing 2-D critical systems is obtained. Clearly, 3-D criticality reduces to rather well-understood 2-D criticality. For , conformal invariance is lost and analogs of soap bubbles are obtained.
In the TGD framework, the generalization of the model to describe freezing as a dynamical time evolution of the solid-liquid boundary is suggestive. An interesting question is whether this boundary could be a light-like 3-surface in and thus have a vanishing 3-volume. A huge extension of ordinary conformal symmetries would emerge
Hen and egg problems of biology from TGD point of view
Get PDFBiology has several hen and egg problems. What came first: DNA,RNA, amino-acids or proto-cell membrane? Did metabolism precede genetic code or vice versa? The stimulus leading to this article could have been the finding that organic molecules are formed in interstellar space at ultralow temperatures of few Kelvin in which chemistry should freeze completely. Therefore the formation of glycine peptides, which has been demonstrated in the laboratory, should be impossible.
The paradox disappears in the TGD framework as do also the hen and egg problems. Magnetic body carrying dark matter as phases allows a universal realization of genetic code and of the analogs of basic bio-molecules in terms of dark nucleon and dark photon triplets. Chemical realization emerged later and the question is whether they emerged simultaneously or whether there was some natural order for the chemical steps. The prebiotic form of metabolic machinery based on hydrogen bonds and dark protons emerged at the same time. A metabolism with metabolic energy quantum assignable electrons which corresponds to average energy of a photon of microwave background is predicted and shows itself via miniature potentials of the neuronal membrane
About the role of Galois groups in TGD framework
Get PDFThis article was inspired by the inverse problem of Galois theory. Galois groups are realized
as number theoretic symmetry groups realized physically in TGD a symmetries of space-time
surfaces. Galois confinement is as analog of color confinement is proposed in TGD inspired
quantum biology.
Two instances of the inverse Galois problem, which are especially interesting in TGD, are
following:
Q1: Can a given finite group appear as Galois group over Q? The answer is not known.
Q2: Can a given finite group G appear as a Galois group over some EQ? Answer to Q2 is
positive as will be found and the extensions for a given G can be explicitly constructed
Sobre_tres_límites_intrínsecos_de_la_física_y_su_tratamiento_filosófico
Get PDFThe item is a handout for a seminar session held on 21.09.2021 at the Dpt. of Philosophy of the University of Navarre, Pamplona, Spain
The presentation approaches the question "Why is there Mathematics in Physics" from the starting point of reductionisms in Physics. If these reductionisms are omitted, the consequences drawn from "prescientific" experience turn out to be the beginning of answering the said question
What could 2-D minimal surfaces teach about TGD?
Get PDFIn the TGD Universe space-time surfaces within causal diamonds (CDs) are fundamental
objects.
1. M8 − H duality means that one can interpret the space-time surfaces in two manners:
either as an algebraic surface in complexified M8
or as minimal surfaces in H = M4×CP2.
M8 − H duality maps these surfaces to each other.
2. Minimal surface property holds true outside the frame spanning minimal surface as 4-D
soap film and since also extremal of K¨ahler action is in question, the surface is analog
of complex surface. The frame is fixed at the boundaries of the CD and dynamically
generated in its interior. At frame the isometry currents of volume term and K¨ahler
action have infinite divergences which however cancel so that conservation laws coded
by field equations are true. The frames serve as seats of non-determinism.
3. At the level of M8
the frames correspond to singularities of the space-time surface. The
quaternionic normal space is not unique at the points of a d-dimensional singularity and
their union defines a surface of CP2 of dimension dc = 4 − D < d defining in H a blow
up of dimension dc.
In this article, the inspiration provided by 2-D minimal surfaces is used to deepen the TGD
view about space-time as a minimal surface and also about M8 − H duality and TGD itself.
1. The properties of 2-D minimal surfaces encourage the inclusion of the phase with a
vanishing cosmological constant Λ phase. This forces the extension of the category of
real polynomials determining the space-time surface at the level of M8
to that of real
analytic functions. The interpretation in the framework of consciousness theory would
be as a kind of mathematical enlightenment, transcendence also in the mathematical
sense.
2. Λ > 0 phases associated with real polynomials as approximations of real analytic functions
would correspond to a hierarchy of inclusions of hyperfinite-factors of type II1
realized as physical systems and giving rise to finite cognition based on finite-D extensions
of rationals and corresponding extensions of p-adic number fields.
3. The construction of 2-D periodic minimal surfaces inspires a construction of minimal
surfaces with a temporal periodicity. For Λ > 0 this happens by gluing copies of minimal
surface and its mirror image together and for Λ = 0 by using a periodic frame.
A more general engineering construction using different basic pieces fitting together like
legos gives rise to a model of logical thinking with thoughts as legos. This also allows
an improved understanding of how M8 − H duality manages to be consistent with the
Uncertainty Principle (UP).
4. At the physical level, one gains a deeper understanding of the space-time correlates
of particle massivation and of the TGD counterparts of twistor diagrams. Twistor lift
predicts M4 K¨ahler action and its Chern-Simons implying CP breaking. This part is
necessary in order to have particles with non-vanishing momentum in the Λ = 0 phase