This book tries to provide a view about the applications of TGD to condensed matter physics. Quantum TGD in its recent form. Quantum TGD relies on two different views about physics: physics as an infinite-dimensional spinor geometry based on the notion of "World of Classical Worlds" (WCW) and physics as a generalized number theory.
WCW picture generalizes Einstein's geometrization program to a geometrization of the entire quantum physics. Number theoretic vision states that so-called adelic physics provides a dual view about physics.
M8-H duality realizes these dual views in terms of space-time surfaces X4⊂ H and X4⊂ M8 mapped to each other byM8-H duality. This duality turns out to be a generalization of momentum-position duality of wave mechanics. Also the duality of number theory and geometry suggested by Langlands correspondence pops up into mind.
The view about physics at the level of H
An important guiding principle in the development of TGD has been quantum classical correspondence (QCC), whose most profound implications follow almost trivially from the basic structure of the classical theory forming an exact part of quantum theory (here TGD differs from quantum field theories (QFTs)).
- 4-D General Coordinate Invariance (GCI) forces holography: the space-time surface associated with a given 3-surface is almost unique as an analog of Bohr orbit. X4 is therefore a preferred extremal of an action principle. This realizes QCC at space-time level and leads to zero energy ontology (ZEO) generalizing the ontology of standard physics.
- The new view about space-time as 4-surface X4⊂ H= M4× CP2 is central for applications. One manner to formulate this is that X4 is simultaneously minimal surface and extremal of Kähler action SK analogous to Maxwell action. The twistor lift of TGD forces the presence of both SK and the volume term in the action.
- Especially important minimal surfaces are CP2 type extremals representing building bricks of elementary particles, cosmic strings and magnetic flux tubes as their deformations so that their M4 and CP2 projections have dimension larger than 2, and so called massless extremals (MEs). Magnetic flux tubes appear as two variants depending on whether they carry monopole flux or not. Monopole flux tubes require no current to create the magnetic field and are not possible in Maxwellian theory. Both are in a crucial role also in condensed matter applications.
- The new view about space-time differs dramatically from that of GRT. The space-time surface is topologically non-trivial in all scales and many-sheeted in the sense that CP2 coordinates as function of M4 coordinates and vice versa are many-valued. The space-time of GRT is obtained from the many-sheeted one in long length scale limit by replacing the sheets with a single region of M4 and by deforming its metric. The gauge potentials are defined as sums of induced gauge potentials for sheets. The deviation of the metric is the sum of the deviations of the induced metric from the M4 metric.
The new physics related to the many-sheetedness is not describable in terms of the QFT approach.
- The classical field equations reduce to conservation laws for the conserved charges defined by the isometries of H. Therefore they are essentially hydrodynamical and this together with QCC is essential for TGD inspired quantum hydrodynamics (QHD). The conjecture that the extremals allow generalized Beltrami property, which implies the existence of a global coordinate varying along the flow lines of flow. For instance, Beltrami property provides purely classical geometric correlates for supra flows and supracurrents. Global coordinates allow identification of order parameters having interpretation in terms of quantum coherence.
- The requirement that modified Dirac operator at the level of space-time surface is in a well-defined sense a projection of the Dirac operator of H implies that for preferred extremals the isometry currents are proportional to projections of the corresponding Killing vectors with proportionality factor constant along the projections of their flow lines.
This implies as generalization of the energy conservation along flow lines of hydrodynamical flow (ρ v2/2+p=constant).
This also leads to a braiding type representations for isometry flows of H in theirs of their projections to the space-time surface and it seems that quantum groups emerge from these representations. Physical intuition suggests that only the Cartan algebra corresponding to commuting observables allows this representation so that the selection of quantization axes would select also space-time surface as a higher level state function reduction.
One also ends up to a generalization of Equivalence Principle stating that the charges assignable to "inertial" or "objective" representations of H isometries in WCW affecting space-time surfaces as analogs of particles are identical with the charges of "gravitational" or subjective representations which act inside space-time surfaces. This has also implications for M8-H duality.
The view about physics at the level of M8
Over the years, the number theoretical vision has evolved to what I call adelic physics. M8-H duality as analog of momentum-position duality and of Complementarity Principle crystallizes number theoretical vision.
- Complexified octonions M8c have interpretation as an analog of momentum space. There are 4-surfaces in both M8 and H and they are related by M8-H duality. 4-surfaces X4 ⊂ M8 have associative normal spaces and are are defined as "roots" of an octonionic polynomial defined as an algebraic continuation of a real polynomial P with rational coefficients.
A real polynomial (its roots) therefore defines the entire 4-surface, which means holography taken to an extreme. This also motivates the preferred extremal property at the H side, where one has partial differential equations and variational principle instead of algebraic equations. The analog of Bohr orbit property is one manner to formulate the restrictions.
- The notion of cognitive representation is motivated by adelic physics, and corresponds to a subset of points of X4 ⊂ M8 in M8c such that the coordinates of the points are in the extension of rationals defined by P. They define a unique discretization of the 4-surface. Cognitive representation contains common points of the real 4-surface and its p-adic variants and makes possible the number-theoretical universality of adelic physics. It is not completely clear whether the cognitive representations are needed only on the M8 side or at both sides of the duality.
- The interpretation of the points of M8 as momenta leads to the proposal that the points of cognitive representation are algebraic integers. Contrary to the naive expectations, all algebraic numbers of extension belong to the cognitive representation for the roots of octonionic polynomials. A natural restriction is that 4-momenta correspond to algebraic integers. A further restriction is that the "active" points of the cognitive representation are occupied by quarks (in TGD leptons can correspond to bound states of quarks). Therefore the 4-surface in H is analogous to the Fermi ball containing discrete quark momenta as an analog of cognitive representation. Condensed matter physics relies strongly on the use of momentum space so that M8 picture is central for the applications in condensed matter.
- Galois group of the polynomial defining X4 ⊂ M8 acts as symmetries for the roots of the polynomials. The order of Galois group of P is identified as effective Planck constant heff/h0=n, where the ordinary Planck constant is a multiple of h0. n is in general not the same as the dimension of extension defined by the degree of P. The original identification was as the dimension of extension counting the number of roots. One must keep an open mind here.
The identification of as heff as the order of the Galois group (rather than dimension of extension) finds support from the following. The order gives the number of regions at the orbit of Galois group (in the case that the isotropy group of the point is trivial!) and at the level of H, the action is sum of identical contributions over these regions so that Planck constant would be nh0.
The phases of matter with different values of heff behave like dark matter relative to each other (this does not of course imply that galactic dark matter and energy correspond to heff>h phases). Various quantum scales, typically proportional to heff, can be arbitrarily long and quantum coherence is possible in all scales and assigned with the magnetic body (MB).
Galois confinement generalizes the notion of periodic boundary conditions and could serve as a universal mechanism for the formation of bound states. One implication is that the total momentum of a bound state formed by quarks has total momentum, which is a rational integer. This serves as an extremely powerful constraint.
The evolutionary hierarchies formed by the extensions of rationals defined by functional composition of polynomials are characterized by root inheritance if the condition P(0)=0 is satisfied. This gives rise to an evolutionary hierarchy of bound states such that the new level contains all bound states of the previous level (conserved genes serve as an analogy). What is nice is that the states of Galois representations which are not Galois singlets can serve as composites of Galois singlets at the next level.
For these reasons Galois confinement is expected to play a fundamental role in TGD and in the TGD based view about condensed matter.
- Nottale introduced the notion of gravitational Planck constant ℏeff=ℏgr = GMm/v0, which allows us to understand the planetary planets as Bohr orbits. The form of ℏgr is dictated by the Equivalence Principle. In the TGD framework, the hierarchy of Planck constants, the infinite range of gravitational interaction, and the absence of screening motivate this notion.
The gravitational Compton length for a particle with mass m is Λgr = GM/v0=rsc/2v0 and thus expressible in terms of Schwarzschild radius and has rs/2 as lower bound. For Earth of order TGD view about living matter involves ℏgr in an essential manner.
Amazingly, there are indications that Λgr for Earth could play a key role in super-conductivity, superfluidity, and in quantum analogies of hydrodynamical systems. Also the role of Λgr for Sun is suggestive. Note that the original motivation for the large value of heff was the TGD based model for effects of em radiation at ELF frequencies on the vertebrate brain.
The proposal is that the description of physics in terms of geometry and number theory are dual to each other. There are several observations motivating M8-H duality.
Could M8 and H= M4× CP2 provide alternative dual descriptions of physics?
- There are four classical number fields: reals, complex numbers, quaternions, and octonions with dimensions 1,2,4,8. The dimension of the embedding space is D(H)= 8, the dimension of octonions. Spacetime surface has dimension D(X4)=4 of quaternions. String world sheet and partonic 2-surface have dimension D(X2) =2 of: complex numbers. The dimension D(string)=1 of string is that of reals.
- Isometry group of octonions is a subgroup of automorphism group G2 of octonions containing SU(3) as a subgroup. CP2=SU(3)/U(2) parametrizes quaternionic 4-surfaces containing a fixed complex plane.
See the book TGD and Condensed Matter.
- Actually a complexification M8c== E8c by adding an imaginary unit i commuting with octonion units is needed in order to obtain sub-spaces with real number theoretic norm squared. M8c fails to be a field since 1/o does not exist if the complex valued octonionic norm squared ∑ oi2 vanishes.
- The four-surfaces X4 ⊂ M8 are identified as "real" parts of 8-D complexified 4-surfaces X4c by requiring that M4⊂ M8 coordinates are either imaginary or real so that the number theoretic metric defined by octonionic norm is real. Note that the imaginary unit defining the complexification commutes with octonionic imaginary units and number theoretical norm squared is given by ∑i zi2 which in the general case is complex.
- The space H would provide a geometric description, classical physics based on Riemann metric, differential geometric structures and partial differential equations deduced from an action principle. M8c would provide a number theoretic description: no partial differential equations, no Riemannian metric, no connections...
M8c has only the number theoretic norm squared and bilinear form, which are real only if M8c coordinates are real or imaginary. This would define "physicality". One open question is whether all signatures for the number theoretic metric of X4 should be allowed? Similar problem is encountered in the twistor Grassmannian approach.
- The basic objection is that the number of algebraic surfaces is very small and they are extremely simple as compared to extremals of action principle. Second problem is that there are no coupling constants at the level of M8 defined by action.
Preferred extremal property realizes quantum criticality with universal dynamics with no dependence on coupling constants. This conforms with the disappearance of the coupling constants from the field equations for preferred extremals in H except at singularities, with the Bohr orbitology, holography and ZEO. X4⊂ H is analogous to a soap film spanned by frame representing singularities and implying a failure of complete universality.
- In M8, the dynamics determined by an action principle is replaced with the condition that the normal space of X4 in M8 is associative/quaternionic. The distribution of normal spaces is always integrable to a 4-surface.
One cannot exclude the possibility that the normal space is complex 2-space, this would give a 6-D surface. Also this kind of surfaces are obtained and even 7-D with a real normal space. They are interpreted as analogs of branes and are in central role in TGD inspired biology.
Could the twistor space of the space-time surface at the level of H have this kind of 6-surface as M8 counterpart? Could M8-H duality relate these spaces in 16-D M8c to the twistor spaces of the space-time surface as 6-surfaces in 12-D T(M4)× T(CP2)?
- Symmetries in M8 number theoretic: octonionic automorphism group G2 which is complexified and contains SO(1,3). G2 contains SU(3) as M8 counterpart of color SU(3) in H. Contains also SO(3) as automorphisms of quaternionic subspaces. Could this group appear as an (approximate) dynamical gauge group?
M8=M4× E4 as SO(4) as a subgroup. It is not an automorphism group of octonions but leaves the octonion norm squared invariant. Could it be analogous to the holonomy group U(2) of CP2, which is not an isometry group and indeed is a spontaneously broken symmetry.
A connection with hadron physics is highly suggestive. SO(4)=SU(2)L× SU(2)R acts as the symmetry group of skyrmions identified as maps from a ball of M4 to the sphere S3⊂ E4. Could hadron physics ↔ quark physics duality correspond to M8-H duality. The radius of S3 is proton mass: this would suggest that M8 has an interpretation as an analog of momentum space.
One implication of M8-H duality is that the image of a generic point of X4⊂ M8 is a single point of CP2. There is complete localization in color degrees of freedom in Einsteinian regions of space-time having 4-D M4 projection and one cannot even speak of color. This would solve in a trivial manner the problem of color confinement.
CP2 type extremals however correspond to singularities for which a single point of line has ill-defined normal space and normal spaces correspond to a 3-D surface in CP2. In this case, one can assign representations of color group SU(3) to the image of the line which is essentially CP2. Color therefore makes sense inside the Euclidean wormhole contacts.
For the string-like entities, the singularity at a given point of string world sheets corresponds to a 2-D surface of CP2, which is a complex manifold or Lagrangian manifold. For the two geodesic spheres the color group reduces to U(2) or SO(3), and one can speak about spontaneous breaking of the color symmetry. Also S1 singularity is possible for objects with 3-D M4 projection. In this case the color symmetry reduces to U(1).
- What is the interpretation of M8? Massless Dirac equation in M8 for the octonionic spinors must be algebraic. This would be analogous to the momentum space Dirac equation. Solutions would be discrete points having interpretation as quark momenta! Quarks pick up discrete points of X4⊂ M8.
States turn out to be massive in the M4 sense: this solves the basic problem of 4-D twistor approach (it works y for massless states only). Fermi ball is replaced with a region of a mass shell (hyperbolic space H3).
M8 duality would generalize the momentum-position duality of the wave mechanics. QFT does not generalize this duality since momenta and position are not anymore operators.