From the problem of time in general relativity to the TGD counterpart of F= ma

This contribution was inspired by a posting of Lawrence B. Crowell related to one particular problem related to the notion of time in general relativity: the general coordinate invariance implies interpretational problems since its is difficult to identify any preferred time coordinates used by observers. This is however not the only problem.

The 3 problems related to the notion of time

It is good to beging with my first comment to the posting of Lawrence Crowell.

1. In the materialistic ontology subjective and geometric time are identified and this leads to deep problems.

In TGD zero energy ontology (see this and this) allows both times and solves both the measurement problem and the problem of free will. Subjective time corresponds to a sequence of “small” state function reductions (SSFRs) replacing the Zeno effect, in which nothing happens, with the notion of self.

Zero energy ontology replaces space-time surfaces as analogs of Bohr orbits for 3-surfaces having an interpretation as a geometric representation of particles. The classical dynamics is slightly non-deterministic although field equations are satisfied. This has crucial implications for the description of fundamental interactions (see this) and this). This non-determinism also makes possible the description of physical correlates of cognition.

The second problem relates to the geometric time. General coordinate invariance allows an endless number of identifications of the time coordinate. In TGD, space-times are 4-surfaces in H=M⁴× CP₂ and M⁴ provides linear Minkowskian time or light-cone proper time (cosmic time) as a preferred time coordinate for the space-time surfaces.

The loss of Poincare invariance in General Relativity is the third problem and led to TGD. In TGD one obtains classical conservation laws due to the Poincare invariance of M⁴ factor of H. What does F= ma mean in TGD?

Lawrence Crowell asked about how the counterpart of F= ma emerges in TGD. At the general level the answer is as follows.

1. F= ma states momentum conservation for a particle plus its environment by characterizing the momentum is exchanged between the particle and environment. In TGD, momentum conservation generalizes to field equations for the space-time surfaces as analogs of Bohr orbits of particles identified as 3-surfaces. The field equations state the conservation of Poincare and color charges classically.

Newton’s equations are not given up as in General Relativity and their generalization defines the dynamics of space-time surfaces. Formally TGD is therefore like hydrodynamics. Einstein’s equations follow naturally at the QFT limit as a remnant of Poincare invariance when the sheets of the many-sheeted space-time are replaced with a slightly curved region of M⁴ carrying the sum of induced gauge fields and gravitational fields defined as deviation from M⁴ metric.

This can be made more precise Holography = holomorphy (H-H) hypothesis ( (see this, (see this, and this) reduces the field equations to local algebraic equations in terms of generalized holomorphy irrespective of action as long as it general coordinate invariant and expressible in terms of the induced geometry. This generalizes the role of holomorphy in string models.

In conformal field theories holomorphy is a correlate for criticality. In TGD it would be a correlate for quantum criticality in the 4-D sense. This principle is extremely powerful since various dynamical parameters are analogous to a critical temperature.

Space-time surfaces as minimal surfaces become analogs of Bohr orbits. Minimal surface equations generalize massless field equations and the TGD counterparts of field equations of gauge theories follow automatically.

Holomorphy is violated as 3-D surfaces which are analog for the singularities of analytic functions. This can be seen as a generalization of the fact that analytic functions can be expressed in terms of the holographic data given at poles and cuts.

What is crucial is that there is a light failure of determinism (but not of field equations). This occurs also for soap films, modellable as 2-D minimal surfaces: the frames do not uniquely determine the soap film. In TGD, the identification of this non-determinism as a p-adic non-determinism is attractive and leads to a generalization of the notion of p-adic number field to a function field (this and this ).

H-H vision leads to a long-sought-for understanding of the origin of p-adic length scales hypothesis which for 30 years ago led to a surprisingly successful particle mass calculations based on p-adic thermodynamics (see this, this and this). The most recent article is about the application to the calculation of the mass spectrum of quarks and hadrons this). The p-adic non-determinism and p-adic length scale hypothesis would have origin in the iterations of polynomials defining dynamical symmetries in H-H vision and also giving a connection to the Mandelbrot fractals and Julia sets becomes possible. Geometric and fermionic counterparts of F= ma in TGD

The next question concerns the counterpart of F= ma at geometric and fermionic levels respectively.

F= ma is a simple model for interactions. How are the interactions of two space-time surfaces A and B as analogs of Bohr orbits described geometrically?

1. Geometric vision suggests that a generalization of a contact interaction is in question. The intersection A and B defines the contact points. Without additional assumptions the intersection of A and B would be a discrete set of points. One can argue that this is not enough.
2. The intuitive idea is that there must exist additional prerequisites for the formation of a quantum coherent structure, at least in the interaction region. The proposal is that A and B share a common generalized complex structure, which I call Hamilton-Jacobi structure (see this) so that the conformal moduli defining the H-J structure would identical.

The common H-J coordinates involve hypercomplex coordinate pair u,v with light-like coordinate curves, common complex M⁴ coordinate w and common complex CP₂ coordinates. The analytic functions defining A and B as their root must be generalized analytic functions of the same H-J coordinates.

The solution of field equations in Minkowskian space-time regions implies that either u or v is a passive coordinate since it cannot appear in the generalized analytic functions (the real hypercomplex coordinates u and v are analogous to z and z). The 2-D surfaces at which u and v vary, are generalizations of straight strings of M⁴ and dynamically very simple. Vibrational string degrees of freedom are frozen but the string ends at the partonic orbits are dynamical and can carry fermion numbers.

In this case, the intersection of A and B consists of 2-D string world sheets connecting light-like partonic orbits. This gives a connection with string model type description.

Self-interactions of the space-time surface correspond to self-intersections consisting of string world sheets. For instance, the description of the internal dynamics of hadrons (see this and this) is realized in terms of self-intersections. In the fermionic sector modified/induced Dirac equation at the space-time level holds true for the induced spinor fields and can be solved exactly by the holomorphy just as in the case of string models. At the level of scattering amplitudes the dynamics reduces to the fermionic N-point functions.

Propagators are free propagators in H and the hard problem is to understand how fermion pair creation is possible when fermions are free in H. The notion of exotic smooth structure, possible only for 4-D space-time surfaces, solves the problem.

How to translate F=ma to a view about the transfer of isometry charges between initial and final state particles?

Let us return to the original questions. How can one understand the generalization of F= ma in terms of a transfer of isometry charges of A (momenta color charges) from the initial state particles A and B to the final state particles? The classical field equations state the local conservation of isometry currents. How can this give rise to a transfer of total charges?

1. In the interaction region the Hamilton-Jacobi structures for A and B must be identical. Intersection consists of string world sheets. The interacting state therefore differs from the non-interacting state. Intuitively it is clear that the incoming states in the distant geometric past approach disjoint Bohr orbits. This is true also in the remote future except that the scattering need not be elastic and the particles identifiable bremsstrahlung can be emitted in the interactions. The interaction can also induce the decay of A and B. What happens in hadronic reactions gives a good idea of what happens.
2. The key notion is the mild failure of classical determinism for the Bohr orbits, which also characterizes criticality. The minimal surfaces describing the space-time surfaces have 3-D loci of non-determinism at which the classical determinism fails. These loci are analogous to the 1-D frames spanning 2-D soap films, which are also slightly non-deterministic minimal surfaces. There are several soap films spanned by the same collection of frames. The non-determinism gives rise to a sequence of small state function reductions (SFRs) generalizing the Zeno effect of standard quantum mechanics.

The description of the scattering in space-time degrees of freedom

Consider first the situation for a single particle as a 4-D Bohr orbit.

1. The non-determinism gives rise to internal interactions assignable to the self intersection as string world sheets. In the TGD inspired theory of consciousness, they can be identified in terms of geometric correlates of cognition.
2. Thinking is however dangerous also at the elementary particle level! The non-determinism is associated with quantum criticality and can lead to the decay of a partonic orbit to two or even more pieces. It can also change the topology of the partonic 2-surface characterized by genus g (CKM mixing). The partonic decay can in turn induce the decay of the space-time surface itself. The outcome would be a particle decay.

This also leads to an emission of virtual particles as Bohr orbits, which appear as exchanges in 2-particle interactions. Massless extremals/topological light rays (see this) as counterparts of massless modes of gauge fields can be emitted. Closed 2-sheeted monopole flux tubes as geometric particles can be created by a splitting of a single monopole flux pair by a reconnection: this would be involved with a particle decay and emission of a virtual particle.

The scattering of two interacting particles A and B reduces to self interaction in the interaction region behaving like a single particle. The slight non-determinism makes possible a geometric generalization of the Feynman diagram type description. Now however all would be discrete and finite. There would be no path integral and therefore no divergences.

The outcome would be a classical description and the generalization of F= ma would code for the transfers of isometry charges from A and B to the final state particles generated in the scattering. The slight classical non-determinism would make this possible. The description of the scattering in fermionic degrees of freedom

The description of the scattering in fermionic degrees of freedom involves highly non-trivial aspects.

1. In the fermionic degrees of freedom, the fermionic propagators between points of the singular 3-surfaces defining the interaction vertices as ends of string world sheets of the intersection would describe the situation.
2. The crucial point is the possibility of fermion pair creation only for 4-D space-time surfaces due to the existence of the exotic smooth structures (see this, this). In other space-time dimensions the fermions would be free.
3. Fermion pair creation corresponds intuitively to the turning of a fermion line in time direction. At the 3-D holomorphic singularities X³ (analogous to cuts of analytic functions) the minimal surface equations fail and the additional pieces of the classical action, in particular 4-volume as the analog of cosmological constant term, become relevant. The twistor lift of TGD (see this and this) implies that the action is sum of K”ahler action and volume term.
4. X³ represents a defect of the standard smooth structure and the turning of a fermion line at X³ at which standard smoothness fails corresponds to the transformation of u-type coordinate curve to a v-type coordinate curve takes place in the pair creations. u and v are associated with the parallel Minkowskian space-time sheets of the 2-sheeted space-time region. The creation of fermion occurs at the boundaries of two string world sheets at different monopole flux tubes so that a decay of monopole flux tube to a pair of them occurs.

The pair creation would take place in the interaction regions and lead to the generation of final state fermions as the decay to 3-surfaces takes place. Closed 2-sheeted monopole flux tubes carrying fermion lines at their ends defined by Euclidean wormhole contacts connecting two sheets are generated.

At the 3-D defects minimal surface property fails and the trace of the second fundamental form, call it H^k, which vanishes almost everywhere, has a delta function singularity. By its group theoretical properties H^k has an interpretation as a generalized Higgs field. What is new is its M⁴ part, which has an interpretation as a local acceleration for a 4-D Bohr orbit. The same interpretation applies to the TGD counterpart of the ordinary Higgs.

The vertices at 3-D singularities are analogous to points at which the direction of the motion for a Brownian particle changes. Conformal invariance suggests that the 8-D Higgs vector H^k is light-like so that one has H_kH^k=0. Higgs has the dimension of ℏ/length and its vanishing gives rise to the analog of 8-D massless fixing M⁴ mass squared in terms of CP₂ mass squared. A reasonable guess is that the square of M⁴ part of H^k is proportional to particle mass squared. This would give quantum-classical correspondence.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/10/from-problem-of-time-in-general.html