About Huygens Principle and TGD
Stephen made an interesting question about the relationship of Huygens principle to TGD
The answer to the question became too long to serve as a comment so that I decided to add it as a blog posting.
1. Huygens Principle
Huygens principle can be assigned most naturally with classical linear wave equations with a source term. It applies also in perturbation theory involving small non-linearities.
One can solve the d'Alembert equation Box Φ= J with a source term J by inverting the d'Alembertian operator to get a bifocal function G(x,y) what one call's Green function.
Green function is bi-local function G(x,y) and the solution generated by point infinitely strong source J localised at single space-time point y - delta function is the technical term. This description allows to think that every point space-time point acts as a source for a spherical wave described by Green function. Green function is Lorentz invariant satisfies causality: on selects the boundary conditions so that the signal is with future light cone.
There are many kind of Green functions and also Feynman propagator satisfies same equation. Now however causality in the naive sense is not exact for fermions. The distance between points x and y can be also space-like but the breaking of causality is small. Feynman propagators
form the basics of QFT description but now the situation is changing after what Nima et al have done to the theoretical physics;-). Twistors are the tool also in TGD too but generalised to 8-D case and this generalisation has been one of the big steps of progress in TGD shows that M4×CP2 is twistorially completely unique.
2. What about Huygens principle and Green functions at the level of TGD space-time?
In TGD classical field equations are extremely non-linear. Hence perturbation theory based Green function around a solution defined by canonically imbedded Minkowski space M4 in M4×CP2 fails. Even worse: the Green function would vanish identically because Kahler action is non-vanishing only in fourth order for the perturbations of canonically imbedded M4! This total breakdown of perturbation theory forces to forget standard ways to quantise TGD and I ended up with the world of classical worlds: geometrization of the space of 3-surfaces. Later zero energy ontology emerged and 3-surfaces were replaced by pairs of 3-surfaces at opposite boundaries of causal diamond CD defining the portion of imbedding space which can be perceived by conscious entity in given scale. Scale hierarchy is explicitly present.
Preferred externals in space-time regions with Minkowskian signature of induced metric decompose to topological light-rays which behave like quantum of massless radiation field. Massless externals for instance are space-time tubes carrying superposition of waves in same light-like direction proceeding. Restricted superposition replaces superposition for single space-time sheet whereas unlimited superposition holds only for the effects caused by space-time sheets to at test particle touching them simultaneously.
The shape of the radiation pulse is preserved which means soliton like behaviour: form of pulse is preserved, velocity of propagation is maximal, and the pulse is precisely targeted. Classical wave equation is "already quantized". This has very strong implications for communications and control in living matter . The GRT approximation of many-sheetedness of course masks tall these beauty as it masked also dark matter, and we see only some anomalies such as several light velocities for signals from SN1987A.
In geometric optics rays are a key notion. In TGD they correspond to light-like orbits of partonic 2-surfaces. The light-like orbit of partonic 2-surface is a highly non-curved analog of light-one boundary - the signature of the induced metric changes at it from Minkowskian to Eucldian at it. Partonic 2-surface need not expand like sphere for ordinary light-cone. Strong gravitational effects make the signature of the induced metric 3-metric (0,-1,-1) at partonic 2-surfaces. There is a strong analogy with Schwartscild horizon but also differences: for Scwartschild blackhole the interior has
3. What about fermonic variant of Huygens principle?
In fermionic sector spinors are localised at string world sheets and obey Kähler-Dirac equation which by conformal invariance is just what spinors obey in super string models. Holomorphy in hypercomplex coordinate gives the solutions in universal form, which depends on the conformal equivalence class of the effective metric defined by the anti-commutators of Kähler-Dirac gamma matrices at string world sheet. Strings are associated with magnetic flux tubes carrying monopole flux and it would seem that the cosmic web of these flux tubes defines the wiring along which fermions propagate.
The behavior of spinors at the 1-D light-like boundaries of string world sheets carrying fermion number has been a long lasting head ache. Should one introduce a Dirac type action these lines?. Twistor approach and Feynman diagrammatics suggest that fundamental fermionic propagator should emerge from this action.
I finally t turned out that one must assign 1-D massless Dirac action in induced metric and also its 1-D super counterpart as line length which however vanishes for solutions. The solutions of Dirac equation have 8-D light-like momentum assignable to the 1-D curves, which are 8-D light-like geodesics of M4×CP2. The 4-momentum of fermion line is time-like or light-like so that the propagation is inside future light-cone rather than only along future light-cone as in Huygens principle.
The propagation of fundamental fermions and elementary particles obtained as the composites is inside the future light-one, not only along light-cone boundary with light-velocity. This reflects the presence of CP2 degrees of freedom directly and leads to massivation.
To sum up, quantized form of Huygens principle but formulated statistically for partonic fermionic lines at partonic 2-surfaces, for partonic 2-surfaces, or for the masses quantum like regions of space-time regions - could hold true. Transition from TGD to GRT limit by approximating many-sheeted space-time with region of M4 should give Huygens principle. Especially interesting is 8-D generalisation of Huygens principle implying that boundary of 4-D future light-cone is replaced by its interior. 8-D notion of twistor should be relevant here.