Preferred extremals of Kähler action as manifolds with constant Ricci scalar whose geometric invariants are topological invariants

The recent progress in the understanding of the preferred extremals led to a reduction of the field equations to conditions stating for Euclidian signature the existence of Kähler metric. The resulting conditions are a direct generalization of corresponding conditions emerging for the string world sheet and stating that the 2-metric has only non-diagonal components in complex/hypercomplex coordinates. Also energy momentum of Kähler action and has this characteristic (1,1) tensor structure. In Minkowskian signature one obtains the analog of 4-D complex structure combining hyper-complex structure and 2-D complex structure.

The construction lead also to the understanding of how Einstein's equations with cosmological term follow as a consistency condition guaranteeing that the covariant divergence of the Maxwell's energy momentum tensor assignable to Kähler action vanishes. This gives T= kG+Λ g. By taking trace a further condition follows from the vanishing trace of T:

R = 4Λ/k .

That any preferred extremal should have a constant Ricci scalar proportional to cosmological constant is very strong prediction. Note however that both Λ and k∝ 1/G are both parameters characterizing one particular preferred extremal. One could of course argue that the dynamics allowing only constant curvature space-times is too simple. The point is however that particle can topologically condense on several space-time sheets meaning effective superposition of various classical fields defined by induced metric and spinor connection.

The following considerations demonstrate that preferred extremals can be seen as canonical representatives for the constant curvature manifolds playing central role inThurston's geometrization theorem known also as hyperbolization theorem implying that geometric invariants of space-time surfaces transform to topological invariants. The generalization of the notion of Ricci flow to Maxwell flow in the space of metrics and further to Kähler flow for preferred extremals in turn gives a rather detailed vision about how preferred extremals organize to one-parameter orbits. It is quite possible that Kähler flow is actually discrete. The natural interpretation is in terms of dissipation and self organization.

A. The geometrical invariants of space-time surfaces as topological invariants

An old conjecture inspired by the preferred extremal property is that the geometric invariants of the space-time surface serve as topological invariants. The reduction ofKähler action to 3-D Chern-Simons terms gives support for this conjecture as a classical counterpart for the view about TGD as almost topological QFT. The following arguments give a more precise content to this conjecture in terms of existing mathematics.

  1. It is not possible to represent the scaling of the induced metric as a deformation of the space-time surface preserving the preferred extremal property since the scale of CP2 breaks scale invariance. Therefore the curvature scalar cannot be chosen to be equal to one numerically. Therefore also the parameter R=4Λ/k and also Λ and k separately characterize the equivalence class of preferred extremals as is also physically clear.

    Also the volume of the space-time sheet closed inside causal diamond CD remains constant along the orbits of the flow and thus characterizes the space-time surface. Λ and even k∝ 1/G can indeed depend on space-time sheet and p-adic length scale hypothesis suggests a discrete spectrum for Λ/k expressible in terms of p-adic length scales: Λ/k ∝ 1/Lp2 with p≈ 2k favored by p-adic length scale hypothesis. During cosmic evolution the p-adic length scale would increase gradually. This would resolve the problem posed by cosmological constant in GRT based theories.

  2. One could also see the preferred extremals as 4-D counterparts of constant curvature 3-manifolds in the topology of 3-manifolds. An interesting possibility raised by the observed negative value of Λ is that most 4-surfaces are constant negative curvature 4-manifolds. By a general theorem coset spaces H4/Γ, where H4= SO(1,4)/SO(4) is hyperboloid of M5 and Γ a torsion free discrete subgroup of SO(1,4). Geometric invariants are therefore topological invariants. It is not clear to me, whether the constant value of Ricci scalar implies constant sectional curvatures and therefore hyperbolic space property. It could happen that the space of spaces with constant Ricci curvature contain a hyperbolic manifold as an especially symmetric representative. In any case, the geometric invariants of hyperbolic metric are topological invariants.

    By Mostow rigidity theorem finite-volume hyperbolic manifold is unique for D>2 and determined by the fundamental group of the manifold. Since the orbits under the Kähler flow preserve the curvature scalar the manifolds at the orbit must represent different imbeddings of one and hyperbolic 4-manifold. In 2-D case the moduli space for hyperbolic metric for a given genus g>0 is defined by Teichmueller parameters and has dimension 6(g-1). Obviously the exceptional character of D=2 case relates to conformal invariance. Note that the moduli space in question plays a key role in p-adic mass calculations \cite{allb}{elvafu}.

    In the recent case Mostow rigidity theorem could hold true for the Euclidian regions and maybe generalize also to Minkowskian regions. If so then both "topological" and "geometro" in "Topological GeometroDynamics" would be fully justified. The fact that geometric invariants become topological invariants also conforms with "TGD as almost topological QFT" and allows the notion of scale to find its place in topology. Also the dream about exact solvability of the theory would be realized in rather convincing manner.

These conjectures are the main result of this posting independent of whether the generalization of the Ricci flow discussed in the sequel exists as a continuous flow or possibly discrete sequence of iterates in the space of preferred extremals of Kähler action. My sincere hope is that the reader could grasp how far reaching these result really are.

B. Generalizing Ricci flow to Maxwell flow for 4-geometries and K\"ahler flow for space-time surfaces

The notion of Ricci flow has played a key part in the geometrization of topological invariants of Riemann manifolds. I certainly did not have this in mind when I choose to call my unification attempt "Topological Geometrodynamics" but this title strongly suggests that a suitable generalization of Ricci flow could play a key role in the understanding of also TGD.

B.1. Ricci flow and Maxwell flow for 4-geometries

The observation about constancy of 4-D curvature scalar for preferred extremals inspires a generalization of the well-known volume preserving Ricci flow introduced by Richard Hamilton and defined in the space of Riemann metrics as

dgαβ/dt= -2Rαβ+ (2/D)Ravggαβ .

Here Ravg denotes the average of the scalar curvature, and D is the dimension of the Riemann manifold. The flow is volume preserving in average sense as one easily checks (<gαβdgαβ/dt> =0). The volume preserving property of this flow allows to intuitively understand that the volume of a 3-manifold in the asymptotic metric defined by the Ricci flow is topological invariant. The fixed points of the flow serve as canonical representatives for the topological equivalence classes of 3-manifolds. These 3-manifolds (for instance hyperbolic 3-manifolds with constant sectional curvatures) are highly symmetric. This is easy to understand since the flow is dissipative and destroys all details from the metric.

What happens in the recent case? The first thing to do is to consider what might be called Maxwell flow in the space of all 4-D Riemann manifolds allowing Maxwell field.

  1. First of all, the vanishing of the trace of Maxwell's energy momentum tensor codes for the volume preserving character of the flow defined as

    dgαβ/dt= Tαβ .

    Taking covariant divergence on both sides and assuming that d/dt and Dα commute, one obtains that Tαβ is divergenceless.

    This is true if one assumes Einstein Maxwell equations with cosmological term. This gives

    dgαβ/dt= kGαβ+ Λ gαβ =k Rαβ + (-kR/2+Λ)gαβ .

    The trace of this equation gives that the curvature scalar is constant. Note that the value of the Kähler coupling strength plays a highly non-trivial role in these equations and it is quite possible that solutions exist only for some critical values of αK. Quantum criticality should fix the allow value triplets (G,Λ,αK) apart from overall scaling

    (G,Λ,αK)→ (xG,Λ/x, xαK) .

    Fixing the value of G fixes the values remaining parameters at critical points. The rescaling of the parameter t induces a scaling by x.

  2. By taking trace one obtains the already mentioned condition fixing the curvature to be constant, and one can write

    dgαβ/dt= kRαβ -Λ gαβ .

    Note that in the recent case Ravg=R holds true since curvature scalar is constant. The fixed points of the flow would be Einstein manifolds satisfying

    Rαβ= (Λ/k) gαβ .

  3. It is by no means obvious that continuous flow is possible. The condition that Einstein-Maxwell equations are satisfied might pick up from a completely general Maxwell flow a discrete subset as solutions of Einstein-Maxwell equations with a cosmological term. If so, one could assign to this subset a sequence of values tn of the flow parameter t.
  4. I do not know whether 3-dimensionality is somehow absolutely essential for getting the classification of closed 3-manifolds using Ricci flow. This ignorance allows me to pose some innocent questions. Could one have a canonical representation of 4-geometries as spaces with constant Ricci scalar? Could one select one particular Einstein space in the class four-metrics and could the ratio Λ/k represent topological invariant if one normalizes metric or curvature scalar suitably. In the 3-dimensional case curvature scalar is normalized to unity. In the recent case this normalization would give k= 4Λ in turn giving Rαβ= gαβ/4. Does this mean that there is only single fixed point in local sense, analogous to black hole toward which all geometries are driven by the Maxwell flow? Does this imply that only the 4-volume of the original space would serve as a topological invariant?

B.2. Maxwell flow for space-time surfaces

One can consider Maxwell flow for space-time surfaces too. In this case Kähler flow would be the appropriate term and provides families of preferred extremals. Since space-time surfaces inside CD are the basic physical objects are in TGD framework, a possible interpretation of these families would be as flows describing physical dissipation as a four-dimensional phenomenon polishing details from the space-time surface interpreted as an analog of Bohr orbit.

  1. The flow is now induced by a vector field jk(x,t) of the space-time surface having values in the tangent bundle of imbedding space M4× CP2. In the most general case one has Kähler flow without the Einstein equations. This flow would be defined in the space of all space-time surfaces or possibly in the space of all extremals. The flow equations reduce to

    hkl Dα jk(x,t) Dβhl= (1/2)Tαβ .

    The left hand side is the projection of the covariant gradient Dαjk(x,t) of the flow vector field jk(x,t) to the tangent space of the space-time surface. D α is covariant derivative taking into account that jk is imbedding space vector field. For a fixed point space-time surface this projection must vanish assuming that this space-time surface reachable. A good guess for the asymptotia is that the divergence of Maxwell energy momentum tensor vanishes and that Einstein's equations with cosmological constant are well-defined.

    Asymptotes corresponds to vacuum extremals. In Euclidian regions CP2 type vacuum extremals and in Minkowskian regions to any space-time surface in any 6-D sub-manifold M4× Y2, where Y2 is Lagrangian sub-manifold of CP2 having therefore vanishing induced Kähler form. Symplectic transformations of CP2 combined with diffeomorphisms of M4 give new Lagrangian manifolds. One would expect that vacuum extremals are approached but never reached at second extreme for the flow.

    If one assumes Einstein's equations with a cosmological term, allowed vacuum extremals must be Einstein manifolds. For CP2 type vacuum extremals this is the case. It is quite possible that these fixed points do not actually exist in Minkowskian sector, and could be replaced with more complex asymptotic behavior such as limit, chaos, or strange attractor.

  2. The flow could be also restricted to the space of preferred extremals. Assuming that Einstein Maxwell equations indeed hold true, the flow equations reduce to

    hklDα jk(x,t) ∂βhl= 1/2(kRαβ -Λ gαβ) .

    Preferred extremals would correspond to a fixed sub-manifold of the general flow in the space of all 4-surfaces.

  3. One can also consider a situation in which jk(x,t) is replaced with jk(h,t) defining a flow in the entire imbedding space. This assumption is probably too restrictive. In this case the equations reduce to

    (Dr jl(x,t)+Dljr)∂αhrβhl= kRαβ -Λ gαβ .

    Here Dr denotes covariant derivative. Asymptotia is achieved if the tensor Dkjl+Dkjl becomes orthogonal to the space-time surface. Note for that Killing vector fields of H the left hand side vanishes identically. Killing vector fields are indeed symmetries of also asymptotic states.

It must be made clear that the existence of a continuous flow in the space of preferred extremals might be too strong a condition. Already the restriction of the general Maxwell flow in the space of metrics to solutions of Einstein-Maxwell equations with cosmological term might lead to discretization, and the assumption about reprentability as 4-surface in M4 × CP2 would give a further condition reducing the number of solutions. On the other hand, one might consiser a possibility of a continuous flow in the space of constant Ricci scalar metrics with a fixed 4-volume and having hyperbolic spaces as the most symmetric representative.

B.3. Dissipation, self organization, transition to chaos, and coupling constant evolution

A beautiful connection with concepts like dissipation, self-organization, transition to chaos, and coupling constant evolution suggests itself.

  1. It is not at all clear whether the vacuum extremal limits of the preferred extremals can correspond to Einstein spaces except in special cases such as CP2 type vacuum extremals isometric with CP2. The imbeddability condition defines a constraint force which might well force asymptotically more complex situations such as limit cycles and strange attractors. In ordinary dissipative dynamics an external energy feed is essential prerequisite for this kind of non-trivial self-organization patterns. As a matter fact, the fact that the Kähler action equals to

    In the recent case the external energy feed could be replaced by the constraint forces due to the imbeddability condition. It is not too difficult to imagine that the flow (if it exists!) could define something analogous to a transition to chaos taking place in a stepwise manner for critical values of the parameter t. Alternatively, these discrete values could correspond to those values of t for which the preferred extremal property holds true for a general Maxwell flow in the space of 4-metrics. Therefore the preferred extremals of Kähler action could emerge as one-parameter (possibly discrete) families describing dissipation and self-organization at the level of space-time dynamics.

  2. For instance, one can consider the possibility that in some situations Einstein's equations split into two mutually consistent equations of which only the first one is independent

    xJανJνβ = Rαβ , LK= xJανJνβ= 4Λ ,

    x=1/16παK .

    Note that the first equation indeed gives the second one by tracing. This happens for CP2 type vacuum extremals.

    Kähler action density would reduce to cosmological constant which should have a continuous spectrum if this happens always. A more plausible alternative is that this holds true only asymptotically. In this case the flow equation could not lead arbitrary near to vacuum extremal, and one can think of situation in which LK= 4Λ defines an analog of limiting cycle or perhaps even strange attractor. In any case, the assumption would allow to deduce the asymptotic value of the action density which is of utmost importance from calculational point of view: action would be simply SK= 4Λ V4 and one could also say that one has minimal surface with Λ taking the role of string tension.

  3. One of the key ideas of TGD is quantum criticality implying that Kähler coupling strength is analogous to critical temperature. Second key idea is that p-adic coupling constant evolution represents discretized version of continuous coupling constant evolution so that each p-adic prime would correspond a fixed point of ordinary coupling constant evolution in the sense that the 4-volume characterized by the p-adic length scale remains constant. The invariance of the geometric and thus geometric parameters of hyperbolic 4-manifold under the Kähler flow would conform with the interpretation as a flow preserving scale assignable to a given p-adic prime. The continuous evolution in question (if possible at all!) might correspond to a fixed p-adic prime. Also the hierarchy of Planck constants relates to this picture naturally. Planck constant hbareff=nhbar corresponds to a multi-furcation generating n-sheeted structure and certainly affecting the fundamental group.
  4. One can of course question the assumption that a continuous flow exists. The property of being a solution of Einstein-Maxwell equations, imbeddability property, and preferred extremal property might allow allow only discrete sequences of space-time surfaces perhaps interpretable as orbit of an iterated map leading gradually to a fractal limit. This kind of discrete sequence might be also be selected as preferred extremals from the orbit of Maxwell flow without assuming Einstein-Maxwell equations. Perhaps the discrete p-adic coupling constant evolution could be seen in this manner and be regarded as an iteration so that the connection with fractality would become obvious too.

B.4 Does a 4-D counterpart of thermodynamics make sense?

The interpretation of the Kähler flow in terms of dissipation, the constancy of R, and almost constancy of LK suggest an interpretation in terms of 4-D variant of thermodynamics natural in zero energy ontology (ZEO), where physical states are analogs for pairs of initial and final states of quantum event are quantum superpositions of classical time evolutions. Quantum theory becomes a "square root" of thermodynamics so that 4-D analog of thermodynamics might even replace ordinary thermodynamics as a fundamental description. If so this 4-D thermodynamics should be qualitatively consistent with the ordinary 3-D thermodynamics.

  1. The first naive guess would be the interpretation of the action density LK as an analog of energy density e=E/V3 and that of R as the analog to entropy density s=S/V3. The asymptotic states would be analogs of thermodynamical equilibria having constant values of LK and R.
  2. Apart from an overall sign factor ε to be discussed, the analog of the first law de= Tds-pdV/V would be

    dLK = kdR +Λ dV4/V4 .

    One would have the correspondences S→ ε RV4, e→ ε LK and k→ T, p→ -Λ. k∝ 1/G indeed appears formally in the role of temperature in Einstein's action defining a formal partition function via its exponent. The analog of second law would state the increase of the magnitude of ε RV4 during the Kähler flow.

  3. One must be very careful with the signs and discuss Euclidian and Minkowskian regions separately. Concerning purely thermodynamic aspects at the level of vacuum functional Euclidian regions are those which matter.
    1. For CP2 type vacuum extremals LK ∝ E2+B2 , R=Λ/k, and Λ are positive. In thermodynamical analogy for ε=1 this would mean that pressure is negative.
    2. In Minkowskian regions the value of R=Λ/k is negative for Λ<0 suggested by the large abundance of 4-manifolds allowing hyperbolic metric and also by cosmological considerations. The asymptotic formula LK= 4Λ considered above suggests that also Kähler action is negative in Minkowskian regions for magnetic flux tubes dominating in TGD inspired cosmology: the reason is that the magnetic contribution to the action density LK∝ E2-B2 dominates.
Consider now in more detail the 4-D thermodynamics interpretation in Euclidian and Minkowskian regions assuming that the the evolution by quantum jumps has Kähler flow as a space-time correlate.
  1. In Euclidian regions the choice ε=1 seems to be more reasonable one. In Euclidian regions -Λ as the analog of pressure would be negative, and asymptotically (that is for CP2 type vacuum extremals) its value would be proportional to Λ ∝ 1/GR2, where R denotes CP2 radius defined by the length of its geodesic circle.

    A possible interpretation for negative pressure is in terms of string tension effectively inducing negative pressure (note that the solutions of the modified Dirac equation indeed assign a string to the wormhole contact). The analog of the second law would require the increase of RV4 in quantum jumps. The magnitudes of LK, R, V4 and Λ would be reduced and approach their asymptotic values. In particular, V4 would approach asymptotically the volume of CP2.

  2. In Minkowskian regions Kähler action contributes to the vacuum functional a phase factor analogous to an imaginary exponent of action serving in the role of Morse function so that thermodynamics interpretation can be questioned. Despite this one can check whether thermodynamic interpretation can be considered. The choice ε=-1 seems to be the correct choice now. -Λ would be analogous to a negative pressure whose gradually decreases. In 3-D thermodynamics it is natural to assign negative pressure to the magnetic flux tube like structures as their effective string tension defined by the density of magnetic energy per unit length. -R≥ 0 would entropy and -LK≥ 0 would be the analog of energy density.

    R=Λ/k and the reduction of Λ during cosmic evolution by quantum jumps suggests that the larger the volume of CD and thus of (at least) Minkowskian space-time sheet the smaller the negative value of Λ.

    Assume the recent view about state function reduction explaining how the arrow of geometric time is induced by the quantum jump sequence defining experienced time. According to this view zero energy states are quantum superpositions over CDs of various size scales but with common tip, which can correspond to either the upper or lower light-like boundary of CD. The sequence of quantum jumps the gradual increase of the average size of CD in the quantum superposition and therefore that of average value of V4. On the other hand, a gradual decrease of both -LK and -R looks physically very natural. If Kähler flow describes the effect of dissipation by quantum jumps in ZEO then the space-time surfaces would gradually approach nearly vacuum extremals with constant value of entropy density -R but gradually increasing 4-volume so that the analog of second law stating the increase of -RV4 would hold true.

  3. The interpretation of -R>0 as negentropy density assignable to entanglement is also possible and is consistent with the interpretation in terms of second law. This interpretation would only change the sign factor ε in the proposed formula. Otherwise the above arguments would remain as such.

For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the title "Preferred extremals of Kähler action as manifolds with constant Ricci scalar whose geometric invariants are topological invariants".