Classical TGD involves several key questions waiting for clearcut answers.
Answering questions is extremely useful activity. During last years Hamed has posed continually questions related to the basic TGD. At this time Hamed asked about the derivation of field equations of TGD. In "simple" field theories involving some polynomial non-linearities the deduction of field equations is of course totally trivial process but in the extremely non-linear geometric framework of TGD situation is quite different.
- The notion of preferred extremal emerges naturally in positive energy ontology, where Kähler metric assigns a unique (apart from gauge symmetries) preferred extremal to given 3-surface at M4 time= constant section of imbedding space H=M4× CP2. This would quantize the initial values of the time derivatives of imbedding coordinates and this could correspond to the Bohr orbitology in quantum mechanics.
- In zero energy ontology (ZEO) initial conditions are replaced by boundary conditions. One fixes only the 3-surfaces at the opposite boundaries of CD and in an ideal situation there would exist a unique space-time surface connecting them. One must however notice that the existence of light-like wormhole throat orbits at which the signature of the induced metric changes (det(g4)=0) its signature might change the situation. Does the attribute "preferred" become obsolete and does one
lose the beautiful Bohr orbitology which looks intuitively compelling and would realize quantum classical correspondence?
- Intuitively it has become clear that the generalization of super-conformal symmetries by replacing 2-D manifold with metrically 2-D but topologically 3-D light-like boundary of causal diamond makes sense. Generalized super-conformal symmetries should apply also to the wormhole throat orbits which are also metrically 2-D and for which conformal symmetries respect detg(g4)=0 condition. Quantum classical correspondence demands that the generalized super-confornal invariance has classical counterpart. How could this classical counterpart be realized?
- Holography is one key aspect of TGD and mean that 3-surfaces dictate everything. In positive energy ontology the content w of this statement would be rather obvious and reduce to Bohr orbitology but in ZEO situation is different. On the other hand, TGD strongly suggests strong form of holography based stating that partonic 2-surfaces (the ends of wormhole throat orbits at boundaries of CD) and tangent space data at them code for quantum physics of TGD. General coordinate invariance would be realied in strong sense: one could formulate the theory either in terms of space-like 3-surfaces at the ends of CD or in terms of light-like wormhole throat orbits. This would realize Bohr orbitology also in ZEO by reducing the boundary conditions to those at partonic 2-surfaces. How to realize this explicitly at the level of field equations? This has been the challenge.
While answering the questions I made what I immediately dare to call a breakthrough discovery in the mathematical understanding of TGD. To put it concisely: one can assume that the variations at the light-like boundaries of CD vanish for all conformal variations which are not isometries. For isometries the contributions from the ends of CD cancel each other so that the corresponding variations need not vanish separately at boundaries of CD! This is extremely simple and profound fact. This would be nothing but the realisation of the analogs of conformal symmetries classically and give precise content for the notion of preferred external, Bohr orbitology, and strong form of holography. And the condition makes sense only in ZEO!
I attach below the answers to the questions of Hamed almost as such apart from slight editing and little additions, re-organization, and correction of typos.
The physical interpretation of the canonical momentum current
Hamed asked about the physical meaning of Tnk== ∂ L/∂(∂n hk) - normal components of canonical momentum labelled by the label k of imbedding space coordinates - it is good to start from the physical meaning of a more general vector field
Tαk == ∂ L/∂(∂α hk)
with both imbedding space indices k and space-time indices α - canonical momentum currents. L refers to Kähler action.
- One can start from the analogy with Newton's equations derived from action
principle (Lagrangian). Now the analogs are the partial derivatives ∂ L/∂(dxk/dt). For a particle in potential one obtains just the momentum. Therefore the term canonical momentum current/density: one has kind of momentum current for each imbedding space coordinate.
- By contracting with generators of imbedding space isometries (Poincare and color) one indeed obtains conserved currents associated with isometries by Noether's theorem:
jA α= TαkjAk .
By field equations the divergences of these currents vanish and one obtains conserved charged- classical four-momentum and color charges:
Dα TA α=0 .
- The normal component of conserved current must vanish at space-like boundaries if one has such
if one has boundaries with Minkowskian signature of induced metric. Now one has wormhole throat orbits which are not genuine boundaries albeit analogous to them and one must be very careful. The quantity Tnk determines the values of normal components of currents and must vanish at possible space-like boundaries.
Note that in TGD field equations reduce to the conservation of isometry currents as in hydrodynamics where basic equations are just conservation laws.
The basic steps in the derivation of field equations
First a general recipe for deriving field equations from Kähler action - or any action as a matter of fact.
Expressing this summary more explicitly, the variation of the Kahler action with respect to the gradients of the imbedding space coordinates reduces to an integral of
- At the first step one writes an expression of the variation of the Kähler action as sum of variations with respect to the induced metric g and induced Kähler form J. The partial derivatives in question are energy momentum tensor and contravariant Kähler form.
- After this the variations of g and J are expressed in terms of variations of imbedding space coordinates, which are the primary dynamical variables.
- The integral defining the variation can be decomposed to a total divergence plus a term vanishing for extremals for all variations: this gives the field equations. Total divergence term gives a boundary term and it vanishes by boundary conditions if the boundaries in question have time-like direction.
If the boundary is space-like, the situation is more delicate in TGD framework: this will be considered in the sequel. In TGD situation is also delicate also because the light-like 3-surfaces which are common boundaries of regions with Minkowskian or Euclidian signature of the induced metric are not ordinary topological boundaries. Therefore a careful treatment of both cases is required in order to not to miss important physics.
Tαk ∂αδ hk + (∂ L/∂ hk) δ hk .
The latter term comes only from the dependence of the imbedding space metric
and Kähler form on imbedding space coordinates. One can use a simple trick. Assume
that they do not depend at all on imbedding space coordinates, derive field equations,
and replaced partial derivatives by covariant derivatives at the end. Covariant
derivative means covariance with respect to both space-time and imbedding space vector
indices for the tensorial quantities involved. The trick works because imbedding space
metric and Kähler form are covariantly constant quantities.
The integral of Tαk ∂αδ hk decomposes to two parts.
- The first term, whose vanishing gives rise to field equations, is integral of
Dα Tαk δ hk .
- The second term is integral of
∂α (Tαk δ hk) .
This term reduces as a total divergence to a 3-D surface integral over the boundary of the region of fixed signature of the induced metric consisting of the ends of CD and wormhole throat orbits (boundary of region with fixed signature of induced metric). This term vanishes if the normal components Tnk of canonical momentum currents vanishes at the boundary like region.
In the sequel the boundary terms are discussed explicitly and it will be found that
their treatment indeed involves highly non-trivial physics.
Boundary conditions at boundaries of CD
In positive energy ontology one would formulate boundary conditions as initial conditions by fixing both the 3-surface and associated canonical momentum densities at either end of CD (positions and momenta of particles in mechanics). This would bring asymmetry between boundaries of CD.
In TGD framework one must carefully consider the boundary conditions at the boundaries of CDs. What is clear that the time-like boundary contributions from the boundaries of CD to the variation must vanish.
While writing this a completely new idea popped to my mind. What if one poses the vanishing of the boundary terms at boundaries of CDs as additional boundary conditions for all variations except isometries? Or perhaps for all conformal variations (conformal in TGD sense)? This would not imply vanishing of isometry charges since the variations coming from the opposite ends of CD cancel each other! It soon became clear that this would allow to meet all the challenges listed in the beginning!
- This is true if the variations are assumed to vanish at the ends of CD. This might be however too strong a condition.
- One cannot demand vanishing of Ttk (t refers to time coordinate as normal coordinate) since this would give only vacuum extremals. One could however require quantum classical correspondence for any Cartan sub-algebra of isometries, whose elements define maximal set of isometry generators. The eigenvalues of quantal variants of isometry charge assignable to second quantized induced spinors at the ends of space-time surface are equal to the classical charges. Is this actually formulation of Equivalence Principle, is not quite clear to me.
- These conditions would realize Bohr orbitology also to ZEO approach and define what "preferred extremal" means.
- The conditions would be very much like super-Virasoro conditions stating that super conformal generators with non-vanishing conformal weight annihilate states or create zero norm states but no conditions are posed on generators with vanishing conformal weight (now isometries). One could indeed assume only deformations, which are local isometries assignable to the generalised conformal algebra of the δ
M4+/-× CP2. For arbitrary variations one would not require the vanishing. This could be the long sought for precise formulation of super-conformal invariance at the level of classical field equations!
It is enough co consider the weaker conditions that the conformal charges defined as integrals of corresponding Noether currents vanish. These conditions would be direct equivalents of quantal conditions.
- The natural interpretation would be as a fixing of conformal gauge. This fixing would be motivated by the fact that WCW Kähler metric must possess isometries associated with the conformal algebra and can depend only on the tangent data at partonic 2-surfaces as became clear already for more than two decades ago. An alternative, non-practical option would be to allow all 3-surfaces at the ends of CD: this would lead to the problem of eliminating the analog of the volume of gauge
group from the functional integral.
- The conditions would also define precisely the notion of holography and its reduction to strong form of holography in which partonic 2-surfaces and their tangent space data code for the dynamics.
Needless to say, the modification of this approach could make sense also at partonic orbits.
Isometry charges are complex
One must be careful also at the light-like 3-surfaces (orbits of wormhole throats) at which the induced metric changes its signature.
- Should one assume that det(g4)1/2 is imaginary in Minkowskian and real in Euclidian region? For Kähler action this is sensible and Euclidian region would give a real negative contribution giving rise to exponent of Kähler function of WCW ("world of classical worlds") making the functional integral convergent. Minkowskian regions would give imaginary contribution to the exponent causing interference effects absolutely essential in quantum field theory. This contribution would correspond to Morse function for WCW.
The implication would be that the classical four-momenta in Euclidian/Minkowskian regions are imaginary/real. What could the interpretation be? Should one accept as a fact that four-momenta are complex.
- Twistor approach to TGD is now in quite good shape. M4× CP2 is the unique choice is one requires that the Cartesian factors allow twistor space with Kähler structure and classical TGD allows twistor formulation.
In the recent formulation the fundamental fermions are assumed to propagate with light-like momenta along wormhole throats. At gauge theory limit particles must have massless or massive four-momenta. One can however also consider the possibility of complex massless momenta and in the standard twistor approach on mass shell massless particles appearing in graphs indeed have complex momenta. These complex momenta should by quantum classical correspondence correspond directly to classical complex momenta.
- A funny question popping in mind is whether the massivation of particles could be such that the momenta remain massless in complex sense! The complex variant of light-likeness condition would be
p2Re= p2Im , pRe• pIm=0 .
Could one interpret p2Im as the mass squared of the particle? Or could p2Im code for the decay width of an unstable particle?
Boundary conditions at the wormhole throat orbits and connection with quantum criticality and hierarchy of Planck constants defining dark matter hierarchy
The contributions from the orbits of wormhole throats are singular since the contravariant form of the induced metric develops components which are infinite (det(g4)=0). The contributions are real at Euclidian side of throat orbit and imaginary at the Minkowskian side so that they must be treated as independently.
- One can consider the possibility that under rather general conditions the normal components Tnkdet(g4) 1/2 approach to zero at partonic orbits since det(g4) is vanishing. Note however the appearance of contravariant appearing twice as index raising operator in Kähler action. If so, the vanishing of Tnkdet(g4) 1/2 need not fix completely the "boundary" conditions. In fact, I assign to the wormhole throat orbits conformal gauge symmetries so that just this is expected on physical grounds.
- Generalized conformal invariance would suggest that the variations defined as integrals of Tnkdet(g4) 1/2δ hk vanish in a non-trivial manner for the conformal algebra associated with the light-like wormhole throats with deformations respecting det(g4)=0 condition. Also the variations defined by infinitesimal isometries (zero conformal weight sector) should vanish since otherwise one would lose the conservation laws for isometry charges. The conditions for isometries might reduce to Tnkdet(g4) 1/2→ 0 at partonic orbits. Also now the interpretaton would be in terms of fixing of conformal gauge.
- Even Tnkdet(g4) 1/2=0 condition need not fix the partonic orbit completely. The Gribov ambiguity meaning that gauge conditions do not fix uniquely the gauge potential could have counterpart in TGD framework. It could be that there are several conformally non-equivalent space-time surfaces connecting 3-surfaces at the opposite ends of CD.
If so, the boundary values at wormhole throats orbits could matter to some degree: very natural in boundary value problem thinking but new in initial value thinking. This would conform with the non-determinism of Kähler action implying criticality and the possibility that the 3-surfaces at the ends of CD are connected by several space-time surfaces which are physically non-equivalent.
The hierarchy of Planck constants assigned to dark matter, quantum criticality and even criticality indeed relies on the assumption that heff=n× h corresponds to n-fold coverings having n space-time sheets which coincide at the ends of CD and that conformal symmetries act on the sheets as gauge symmetries. One would have as Gribov copies n conformal equivalence classes of wormhole throat orbits and corresponding space-time surfaces. Depending on whether one fixes the conformal gauge one has n equivalence classes of space-time surfaces or just one representative from each conformal equivalent class.
- There is also the question about the correspondence with the weak form of electric magnetic duality. This duality plus the condition that jαAα=0 in the interior of space-time surface imply the reduction of Kähler action to Chern-Simons terms. This would suggest that the boundary variation of the Kähler action reduces to that for Chern-Simons action which is indeed well-defined for light-like 3-surfaces.
If so, the gauge fixing would reduce to variational equations for Chern-Simons action! A weaker condition is that classical conformal charges vanish. This would give a nice connection to the vision about TGD as almost topological QFT. In TGD framework these conditions do not imply the vanishing of
Kähler form at boundaries. The conditions are satisfied if the CP2 projection of the partonic orbit is 2-D: the reason is that Chern-Simons term vanishes identically in this case.
- A further intuitively natural hypothesis is that there is a breaking of conformal symmetry: only the generators of conformal sub-algebra with conformal weight multiple of n act as gauge symmetries. This would give infinite hierarchies of breakings of conformal symmetry interpreted in terms of criticality: in the hierarchy the integers ni would satisfy ni divides ni+1.
Similar degeneracy would be associated with the space-like ends at CD boundaries and I have considered the possibility that the integer n appearing in heff has decomposition n=n1n2 corresponding to the degeneracies associated with the two kinds of boundaries. Alternatively, one could have just n=n1=n2 from the condition that the two conformal symmetries are 3-dimensional manifestations of single 4-D analog of conformal symmetry.
As should have become clear, the derivation of field equations in TGD framework is not just an application of a formal recipe as in field theories and a lot of non-trivial physics is involved!
See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article
The vanishing of conformal charges as a gauge conditions selecting
preferred extremals of Kähler action.