About TGD counterparts of classical field configurations in Maxwell's theory
Classical physics is an exact part of TGD so that the study of extremals of dimensionally reduces 6-D Kähler action can provide a lot of intuition about quantum TGD and see how quantum-classical correspondence is realized.
In the following the goal is to develop further understanding about TGD counterparts of the simplest field configurations in Maxwell's theory.
About differences between Maxwell's ED and TGD
TGD differs from Maxwell's theory in several important aspects.
In this article CP2 type extremals will be considered from the point of view of quantum criticality and the view about string world sheets, their lightlike boundaries as carriers of fermion number, and the ends as point like particles as singularities acting as sources for minimal surfaces satisfying non-linear generalization of d'Alembert equation.
- The TGD counterparts of classical electroweak gauge potentials are induced from component of spinor connection of CP2. Classical color gauge potentials corresponds to the projections of Killing vector fields of color isometries.
- Also M4 has Kähler potential, which is induced to space-time surface and gives rise to an additional U(1) force. The couplings of M4 gauge potential to quarks and leptons are of same sign whereas the couplings of CP2 Kähler potential to B and L are of opposite sign so that the contributions to 6-D Kähler action reduce to separate terms without interference term.
Coupling to induced M4 Kähler potential implies CP breaking. This could explain the small CP breaking in hadronic systems and also matter antimatter asymmetry in which there are opposite matter-antimatter asymmetries inside cosmic strings and their exteriors respectively. A priori it is however not obvious that the CP breaking is small.
- General coordinate invariance implies that there are only 4 local field like degrees of freedom so that for extremals with 4-D M4 projection corresponding to GRT space-time both metric, electroweak and color gauge potentials can be expressed in terms four CP2 coordinates and their gradients. Preferred extremal property realized as minimal surface condition means that field equations are satisfied separately for the 4-D Kähler and volume action reduces the degrees of freedom further.
If the CP2 part of Kähler form is non-vanishing, minimal surface conditions can be guaranteed by a generalization of holomorphy realizing quantum criticality (satisfied by known extremals). One can say that there is no dependence on coupling parameters. If CP2 part of Kähler form vanishes identically, the minimal surface condition need not be guaranteed by holomorphy. It is not at all clear whether quantum criticality and preferred extremal property allow this kind of extremals.
- Supersymplectic symmetries act as isometries of "world of classical worlds" (WCW). In a well-defined sense supersymplectic symmetry generalizes 2-D conformal invariance to 4-D context. The key observation here is that light-like 3-surfaces are metrically 2-D and therefore allow extended conformal invariance.
Preferred extremal property realizing quantum criticality boils down to a condition that sub-algebra of SSA and its commutator with SSA annihilate physical states and that corresponding Noether charges vanish. These conditions could be equivalent with minimal surface property. This implies that the set of possible field patterns is extremely restricted and one might talk about "archetypal" field patterns analogous to partial waves or plane waves in Maxwell's theory.
- Linear superposition of the archetypal field patterns is not possible. TGD however implies the notion of
many-sheeted space-time and each sheet can carry its own field pattern. A test particle which is space-time surface itself touches all these sheets and experiences the sum of the effects caused by fields at various sheets. Effects are superposed rather than fields and this is enough. This means weakening of the superposition principle of Maxwell's theory and the linear superposition of fields at same space-time sheet is replaced with set theoretic union of space-time sheets carrying the field patterns whose effects superpose.
This observation is also essential in the construction of QFT limit of TGD. The gauge potentials in standard model and gravitational field in general relativity are superpositions of those associated with space-time sheets idealized with slightly curved piece of Minkowski space M4.
- An important implication is that each system has field identity - field body or magnetic body (MB). In Maxwell's theory superposition of fields coming from different sources leads to a loss of information since one does not anymore now which part of field came from particular source. In TGD this information loss does not happen and this is essential for TGD inspired quantum biology.
Remark: An interesting algebraic analog is the notion of co-algebra. Co-product is analogous to reversal of product AB= C in the sense that it assigns to C and a linear combination of products ∑ Ai⊗ Bi such that AiBi=C. Quantum groups and co-algebras are indeed important in TGD and it might be that there is a relationship. In TGD inspired quantum biology magnetic body plays a key role as an intentional agent receiving sensory data from biological body and using it as motor instrument.
- I have already earlier considered a space-time correlate for second quantization in terms of sheets of covering for heff=nh0. I have proposed that n factorizes as n=n1n2 such that n1 (n2) is the number sheets for space-time surface as covering of CP2 (M4). One could have quantum mechanical linear superposition of space-time sheets, each with a particular field pattern. This kind state would correspond to single particle state created by quantum field in QFT limit. For instance, one could have spherical harmonic for orientations of magnetic flux tube or electric flux tube.
One could also have superposition of configurations containing several space-time sheets simultaneously as analogs of many-boson states. Many-sheeted space-time would correspond to this kind many-boson states. Second quantization in quantum field theory (QFT) could be seen as an algebraic description of many-sheetedness having no obvious classical correlate in classical QFT.
- Flux tubes should be somehow different for gravitational fields, em fields, and also weak and color gauge fields. The value of n=n1n2 for gravitational flux tubes is very large by Nottale formula hbareff= hbargr= GMm/v0. The value of n2 for gravitational flux tubes is n2∼ 107 if one accepts the formula G= R2/n2hbar. For em fields much smaller values of n and therefore of n2 are suggestive. There the value of n measuring in adelic physics algebraic complexity and evolutionary level would distinguish between gravitational and em flux tubes.
Large value of n would mean quantum coherence in long scales. For gravitation this makes sense since screening is absent unlike for gauge interactions. Note that the large value of heff=hgr implies that αem= e2/4πℏeff is extremely small for gravitational flux tubes so that they would indeed be gravitational in an excellent approximation.
n would be the dimension of extension of rationals involved and n2 would be the number space-time sheets as covering of M4. If this picture is correct, gravitation would correspond to much larger algebraic complexity and much larger value of Planck constant. This conforms with the intuition that gravitation plays essential role in the quantum physics of living matter.
There are also other number theoretic characteristics such as ramified primes of the extension identifiable as preferred p-adic primes in turn characterizing elementary particle. Also flux tubes mediating weak and strong interactions should allow characterization in terms of number theoretic parameters. There are arguments that in atomic physics one has h=6h0. Since the quantum coherence scale of hadrons is smaller than atomic scale, one can ask whether one could have heff<h.
I will also discuss the delicacies associated with M4 Kähler structure and its connection with what I call Hamilton-Jacobi structure and with M8 approach based on classical number fields. I will argue that the breaking of CP symmetry associated with M4 Kähler structure is small without any additional assumptions: this is in contrast with the earlier view.
The difference between TGD and Maxwell's theory and consider the TGD counterparts of simple em field configurations will be also discussed. Topological field quantization provides a geometric view about formation of atoms as bound states based on flux tubes as correlates for binding, and allows to identify space-time correlates for second quantization. These considerations force to take seriously the possibility that preferred extremals besides being minimal surfaces also possess generalized holomorphy reducing field equations to purely algebraic conditions and that minimal surfaces without this property are not preferred extremals. If so, at microscopic level only CP2 type extremals, massless extremals, and string like objects and their deformations would exist as preferred extremals and serve as building bricks for the counterparts of Maxwellian field configurations and the counterparts of Maxwellian field configurations such as Coulomb potential would emerge only at the QFT limit.
See the chapter About the Identification of the Preferred Extremals of Kähler Action or the article About TGD counterparts of classical field configurations in Maxwell's theory.