Does the presence of cosmological constant term make Kähler coupling strength a genuine coupling constant classically?
The addition of the volume term to Kähler action has very nice interpretation as a generalization of equations of motion for a world-line extended to a 4-D space-time surface. The field equations generalize in the same manner for 3-D light-like surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian, for 2-D string world sheets, and for their 1-D boundaries defining world lines at the light-like 3-surfaces. For 3-D light-like surfaces the volume term is absent. Either light-like 3-surface is freely choosable in which case one would have Kac-Moody symmetry as gauge symmetry or that the extremal property for Chern-Simons term fixes the gauge.
The known non-vacuum extremals are minimal surface extremals of Kähler action and it might well be that the preferred extremal property realizing SH quite generally demands this. The addition of the volume term could however make Kähler coupling strength a manifest coupling parameter also classically when the phases of Λ and αK are same. Therefore quantum criticality for Λ and αK would have a precise local meaning also classically in the interior of space-time surface. The equations of motion for a world line of U(1) charged particle would generalize to field equations for a "world line" of 3-D extended particle.
This is an attractive idea consistent with standard wisdom but one can invent strong objections against it in TGD framework.
- All known non-vacuum extremals of Kähler action are minimal surfaces and the minimal surface vacuum extremals of Kähler action become non-vacuum extremals. This suggest that preferred extremals are minimal surface extremals of Kähler action so that the two dynamics apparently decouple. Minimal surface extremals are analogs for geodesics in the case of point-like particles: one might say that one has only gravitational interaction. This conforms with SH stating that gauge interactions at boundaries (orbits of partonic 2-surfaces and 2-surfaces at the ends of CD) correspond classically to the gravitational dynamics in the space-time interior.
Note that at the boundaries of the string world sheets at light-like 3-surfaces the situation is different: one has equations of motion for geodesic line coupled to induce Kähler gauge potential and gauge coupling indeed appears classically as one might expect! For string world sheets one has only the topological magnetic flux term and minimal surface equation in string world sheet. Magnetic flux term gives the Kähler coupling at the boundary.
- Decoupling would allow to realize number theoretical universality since the field equations would not depend on coupling parameters at all. It is very difficult to imagine how the solutions could be expressible in terms of rational functions with coefficients in algebraic extension of rationals unless αK and Λ have very special relationship. If they have different phases, minimal surface extremals of Kähler action are automatically implied. If the values of αK correspond to complex zeros of Riemann ζ, also Λ should have same complex phase, in order to have genuine classical coupling. This looks somewhat un-natural but cannot be excluded.
The most natural option is that Λ is real and αK corresponds to zeros of zeta. For trivial zeros the phases are different and decoupling occurs. For trivial zeros Λ and αK differ by imaginary unit so that again decoupling occurs.
- One can argue that the decoupling makes it impossible to understand coupling constant evolution. This is not the case. The point is that the classical charges assignable to super-symplectic algebra are sums over contributions from Kähler action and volume term and therefore depend on the coupling parameters. Their vanishing conditions for sub-algebra and its commutator with entire algebra give boundary conditions on preferred extremals so that coupling constant evolution creeps in classically!
The condition that the eigenvalues of fermionic charge operators are equal to the classical charges brings in the dependence of quantum charges on coupling parameters. Since the elements of scattering matrix are expected to involve as building bricks the matrix elements of super-symplectic algebra and Kac-Moody algebra of isometry charges, one expectes that discrete coupling constant evolution creeps in also quantally via the boundary conditions for preferred extremals.
- For Option I the coupling between the two dynamics could be induced just by the condition that the space-time surface becomes an analog of geodesic line by arranging its interior so that the U(1) force vanishes! This would generalize Chladni mechanism! The interaction would be present but be based on going to the nodal surfaces! Also the dynamics of string world sheets is similar: if the string sheets carry vanishing W boson classical fields, em charge is well-defined and conserved. One would also avoid the problems produced by large coupling constant between the two-dynamics present already at the classical level. At quantum level the fixed point property of quantum critical couplings would be the counterparts for decoupling.
- For Option II the coupling is of conventional form. When cosmological constant is small as in the scale of the known Universe, the dynamics of Kähler action is perturbed only very slightly by the volume term. The alternative view is that minimal surface equation has a very large perturbation proportional to the inverse of Λ so that the dynamics of Kähler action could serve as a controller of the dynamics defined by the volume term providing a small push or pull now and then. Could this sensitivity relate to quantum criticality and to the view about morphogenesis relying on Chladni mechanism in which field patterns control the dynamics with charged flux tubes ending up to the nodal surfaces of (Kähler) electric field (see this)? Magnetic flux tubes containing dark matter would in turn control and serve as template for the dynamics of ordinary matter.
- Number theoretic vision suggests the existence of preferred extremals represented by rational functions with rational or algebraic coefficients in preferred coordinates. For Option I one has preferred extremals of Kähler action which are minimal surfaces so that there is no coupling and no constraints on the ratio of couplings emerges: even better, both dynamics are independent of the coupling. All known non-vacuum extremals of Kähler action are indeed also minimal surfaces. For Option II the ratio of the coefficients Λ/8π G and 1/4παK should be rational or at most algebraic number. One must be however very cautious here: the minimal option allowed by strong form of holography is that the rational functions of proposed kind emerge only at the level of partonic 2-surfaces and string world sheets.
- I have proposed that that the inverse of Kähler coupling strength has spectrum coming as zeros of zeta or their imaginary parts (see this). The phases of complexified 1/αK and Λ/2G must be same in order to avoid the decoupling of Kähler action and minimal surface term implying minimal surface extremals of Kähler action.
This conjecture is consistent with the rational function property only if αK and vacuum energy density ρvac appearing as the coefficient of volume term are proportional to the same possibly transcendental number with proportionality coefficient being an algebraic or rational number.
If the phases are not identical (say Λ is real and one allows complex zeros) one has Option I and effective decoupling occurs. The coupling (Option2)) can occur for the trivial zeros of zeta if the volume term has coefficient iΛ/8πG rather than Λ/8π G to guarantee same phase as for 1/4παK. The coefficient iΛ/8πG would give in Minkowskian regions large real exponent of volume and this looks strange. In this case also number theoretical universality might make sense but SH would be broken in the sense that the space-time surfaces would not be analogous to geodesic lines.
- At quantum level number theoretical universality requires that the exponent of the total action defining vacuum functional reduces to the product of roots of unity and exponent of integer existing in finite-dimensional extension of p-adic numbers. This would suggest that total action reduces to a number of form q1+iq2π, qi rational number, so that its exponent is of the required form. Whether this can conform with the properties of zeros of zeta and properties of extremals is not clear.
- Option I: Decoupling of the dynamics of Kähler action and volume term in space-time interior for all values of coupling parameters.
- Option II: Coupling of dynamics for trivial zeros of zeta and Λ→ iΛ.
Consider a typical particle physics experiment. There are incoming and outgoing free particles moving along geodesics, these particles interact, and emanate as free particles from the interaction volume. This phenomenological picture does not follow from quantum field theory but is put in by hand, in particular the idea about interaction couplings becoming non-zero is involved. Also the role of the observer remains poorly understood.
The motion of incoming and outgoing particles is analogous to free motion along geodesic lines with particles generalized to 3-D extended objects. For both options these would correspond to the preferred extremals in the complement of CD within larger CD representing observer or measurement instrument. Decoupling would take place. In the interaction volume interactions are "coupled on" and particles interact inside the volume characterized by causal diamond (CD). What could be the TGD view translation of this picture?
- For Option I one would still have decoupling and the interpretation would be in terms of twistor picture in which
one always has also in the internal lines on mass shell particles but with complex four-momenta. In TGD framework the momenta would be always complex due to the contribution of Euclidian regions defining the lines of generalized scattering diagrams. As explained coupling constant evolution can be understood also in this case and also classical dynamics depends on coupling parameters via the boundary conditions. The transitory period (control action) leading to the decoupled situation would be replaced by state function reduction, possibly to the opposite boundary.
- For Option II the transitory period would correspond to the coupling between the two classical dynamics and would take place inside CD after a phase transition identifiable as "big state function reduction" to time reversed mode. The problem is that in the interacting phase αK would not have a value approximately equal to the U(1) coupling strength of weak interactions (see this) so that the physical picture breaks down.
- For Option I state preparation and state function reduction would be in symmetric role. Also now there would be inherent asymmetry between zero energy states and their time reversals. With respect to observer the time reversed period would be invisible.
- For Option II state preparation for CD would correspond to a phase transition to a time reversed phase labelled by a trivial zero of zeta and Λ→ iΛ. In state function reduction to the original boundary of CD a phase transition to a phase labelled by non-trivial zero of zeta would occur and final state of free particles would emerge. The phase transitions would thus mean hopping from the critical line of zeta to the real axis and back and change the values of αK and possibly Λ. There would be strong breaking in time reversal symmetry.
One cannot of course take this large asymmetry as an adhoc assumption: it should be induced by the presence of larger CD, which could also affect quite generally the values of αK and Λ (having also a spectrum of values).
What happens within sub-CD could be fundamental for the understanding of directed attention and sensory-motor cycle.
- The target of directed attention would correspond to the volume of CD - call it c - within larger CD - call it C representing the observer - attendee having c as part of its perceptive field. c would serve as a target of directed attention of C and thus define part of the perceptive field of c. c would correspond also to sub-self giving rise to a mental image of C. This would also allow to understand why the attention is directed rather than being completely symmetric with respect to C and c. For both options directed attention would correspond to sub-self c interpreted as mental image. There would be no difference.
- Quite generally, the self and time-reversed self could be seen as sensory input and motor response (Libet's findings). Directed attention would define the sensory input and sub-self could react to it by dying and re-incarnating as time-reversed subself. The two selves would correspond to sensory input and motor action following it as a reaction. Motor reaction would be sensory mental image in reversed time direction experienced by time reversed self. Only the description for the reaction would differ for the two options.
The motor action would be time-reversed sensory perception for Option I. For Option II motor action would correspond
to a different phase in which Kähler action and volume term couple classically.
The free geodesic line dynamics with vanishing U(1) Kähler force indeed brings in mind the proposed generalization of Chladni mechanism generating nodal surfaces at which charged magnetic flux tubes are driven (see this).
- For Option I the interiors of all space-time surfaces would be analogous to nodal surfaces and state function reductions would correspond to transition periods between different nodal surfaces. The decoupling would be dynamics of avoidance and could highly analogous to Chladni mechanism.
- For Option II the phase labelled by trivial zeros of zeta would correspond to period during which nodal surfaces are formed. This view about state function reduction and preparation as phase transitions in ZEO would provide classical description for the transition to the phase without direct interactions.
For background see the new chapter How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD? of "Towards M-matrix" or the article with the same title.
For a summary of earlier postings see Latest progress in TGD.