“I want to emphasize the necessity for a sound mathematical basis for any fundamental physical theory. Any philosophical ideas that one may have play only a subordinate role. Unless such ideas have a mathematical basis they will be ineffective.
As an example of a philosophical idea without a precise mathematical basis I would like to mention Mach's principle. Einstein has stated that he was indebted to this principle in his line of thought which led him to general relativity. But I do not see how this could be. I do not see how the principle can be formulated in a sufficiently definite way to be of any value in the search for a precise physical theory.
One should keep the need for a sound mathematical basis dominating one's search for a new theory. Any physical or philosophical ideas that one has must be adjusted to fit the mathematics. Not the other way around.
The need for putting the mathematics first comes from its more rigid nature. One can tinker with one's physical or philosophical ideas to adapt them to fit the mathematics. But the mathematics cannot be tinkered with. It is subject to completely rigid rules and is harshly restricted by strict logic.
The reason I feel so strongly about the views expressed above is because of the success I have had with them in the past. My early research work, in the early 1920's, was based on Bohr orbits, and was completely unsuccessful. I was taking the Bohr orbits as physically real and trying to build up a mathematics for them. I worked hard on this problem…
One sees now how futile such work was. Heisenberg showed that one needed a completely new mathematics, involving non-commutative algebra. The Bohr orbits were an unsound physical concept and should not be used as the basis for a theory.
I learnt my lesson then. I learnt to distrust all physical concepts as the basis for a theory. Instead one should put one's trust in a mathematical scheme, even if the scheme does not appear at first sight to be connected with physics. One should concentrate on getting an interesting mathematics.”
Paul Dirac Mathematical Foundations of Quantum Theory (1978).
Bohr orbit was a brilliant physical idea but was not of course mathematically sound. The mistake was to give up this notion instead of trying to develop it further. The price paid was the basic paradox of quantum measurement theory which is still with us. Wave functions in the space of Bohr orbits generalize the standard notion of wave function, give precise connection with the classical theory, and lead to zero energy ontology. The classical determinism is consistent with the quantum jumps since they occur between these wave functions. The experiments of Minev et al provide direct experimental evidence for the zero energy ontology. Quantum jumps correspond to smooth classical developments leading to the final state of quantum jump. Second big mistake was the neglect of the fact that the conservation laws are lost in general relativity.
The notion of Bohr orbit, in the sense that I use it in TGD, realizes holography. In TGD holography is forced by 4-D general coordinate invariance. Otherwise one would have path integral over all space-time surfaces, plagued by horrible infinities. This is certainly consistent with the idea of sound mathematics as a basis of theoretical physics.
TGD provides a concrete realization of holography as a 4-D generalization of holomorphy in terms of combination of 2-D complex and hypercomplex structures as analogs of complex structures. I call these structures Hamilton-Jacobi structures. An explicit general solution of field equations is in question so that TGD is an exactly solvable theory. See this and this .
Irrespective of the action principle, Bohr orbits are minimal surfaces locally if the action is a general coordinate invariant constructed in terms of induced geometry. Only the singularities such as light-like boundaries and interfaces of Minkowskian and Euclidean regions, and string world sheets depend on action. This reflects the universality of quantum criticality. Kahler action plus volume term is the action implied by the twistor lift of TGD.
The holomorphic ansatz works also in the case of string models based on area action. In TGD the conformal and Kac-Moody symmetries of string model are replaced by an infinite algebra of a 4-D generalization of conformal symmetries in terms of Noether currents and act as isometries of ther "world of classical worlds". Remarkably, the action is *not* invariant under these symmetries as the naive expectation would be. Also symplectic symmetries assignable to the orbits of partonic symmetries and satisfying field equations for Chern-Simons-Kahler action are symmetries of the theory and act as isometries of WCW.
TGD also leads to a very detailed new geometric and topological view of atoms, nuclei and hadrons and the relationship between strong and electroweak interactions relying strongly on the topology of the space-time surfaces representing the system and making strong and testable predictions. In particular, a problem in the standard atomic theory is discovered: the atoms with many electrons are not classically stable and the TGD view provides stability. See this and this .
So sum up, theoreticians must always start from the requirement that the theory is free of logical contradictions. Quantum measurement theory gave up this requirement and this stopped the progress.
TGD involves of course extremely beautiful mathematical structures: WCW as a mathematical construct based on Bohr orbitology and making physics unique from its mere mathematical existence, physics as geometry and physics as number theory as the basic approaches dual to each other, M^8-H duality generalizing momentum position duality and strongly suggesting a connection with Langlands duality, number theoretic vision predicting standard model symmetries and giving rise to possible correlates of cognition, hyperfinite factors and their inclusions,...
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.