The problems of quantum theory

Update: should discuss the original work of de Broglie and the interesting work of Grete Hermann. Also the hydrodynamic analogues involving bouncing droplets on vibrating fluid surfaces.

Quantum physics represents a very imperfect, incomplete and highly unsatisfactory theory.  Its interpretations and approaches are multi-faceted and complex. Quantum physics took a drastically wrong turn with the Hilbert space operator  based Von Neumann axiomatization (and the subsequent more sophisticated, but likewise inadequate, $C^\ast$-algebra approach) - abandoning the much more interesting initial historical connections to the 'dualistic theory of radiation', statistical mechanics,  the photo-electic effect, geometric optics, the Hamilton-Jacobi equations, 'wave mechanics', 'matrix mechanics' - and above all the insights of Paul Dirac who was one of the giants of 20th-century physics (Penrose's Twistor Theory is in some sense a continuation of Dirac).  The investigations carried out during the initial development of quantum physics gave rise to meaningful and interesting mathematics and physics which did not depend in any way on probabilistic interpretations (or the collapse) of the wave function - for example the study of the Lorenz-invariant Klein-Gordon and Dirac equations. If the Hilbert-space and and $C^\star$-algebra based quantum theory was at least mathematically rigorous and interesting in its own right (it ultimately gave rise to Alain Connes' Non-Commutative Geometry) this is not the case of Feynman's and Schwinger's approach to  (perturbative) quantum field theory.  Cf.  G.B. Folland's Quantum Field Theory: A Tourist Guide for Mathematicians (2008) where it is stated that once we leave the free field (itself requiring a staggering amount of functional analysis and distribution theory) we have left the realm of a correct mathematical formulation of physics. The problem is that QFT is not only bad mathematics it is also bad experimental science.

Something is rotten in the state of QED (Oliver Consa)

Consa says the much-touted precision of QED is based on measurements of the electron g-factor, but that “this value was obtained using illegitimate mathematical traps, manipulations and tricks”.Theoreticians come up with a calculation that exactly matches an experiment. Then a later experiment shows that the earlier experiment wasn’t quite correct. Then the theoreticians change their calculation to match the new experiment. And so on (...) Consa quotes Dyson from 2006: “As one of the inventors of QED, I remember that we thought of QED in 1949 as a temporary and jerry-built structure, with mathematical inconsistencies and renormalized infinities swept under the rug. We did not expect it to last more than 10 years before some more solidly built theory would replace it. Now, 57 years have gone by and that ramshackle structure still stands”. It still stands because it’s been propped up by scientific fraud. Here we are fourteen years later, and it’s still the same, and physics is still going nowhere. How much longer can this carry on? Not much longer, because now we have the internet.

https://physicsdetective.com/something-is-rotten-in-the-state-of-qed/

Maybe a clue to improving this situation involves a critique and reform of distribution theory - for example along the lines of Sato's theory of hyperfunctions. This has of course already been suggested in the context of the divergent infinite sums of $\delta$-functions appearing in QFT.

We postulate axiomatically that the 'position', 'momentum', even 'energy' of a 'particle' are given by distributions over space-time (we still have our standard PDEs for distributions).  Thus a 'particle' does not necessarily have a definite position at a given moment of time. Nor is a 'particle' a wave or field defined aver space-time. It is a completely different kind of entity which subsumes as particular cases or approximations the aspect of wave or particle (for the wave-like aspect we have regular distributions, for the particle aspect $\delta$-functions, or something similar). There must be an "interactive" (we must carefully re-evaluate the controversies surrounding the interactive interpretation of the collapse of the wave-function as well as the hidden-variable approaches) or alternative way of explaining the collapse of the wave function and the probabilistic aspect based on this perspective (cf. the work of A. Hobson (2012)).  Since experimentally we can only prepare 'test functions' with a limited degree of precision it is not surprising that the output of the distribution should also exhibit a corresponding degree of uncertainty. But of course we need to ask what is the physical nature of the test functions? Do not they have to be (regular) distributions as well?  In practice the test functions will not be exactly regular distributions but only approximately so (determined by some boundary conditions). Thus observations - which correspond to evaluating the test functions - or interactions of localized distributions along a boundary - will have uncertainty corresponding to the non-regular components of the approximate test function.  Note that a distribution is essentially non-local (cf. Hobson's analogy to a bursting balloon) although they can be restricted. It would be interesting to explore how this approach looks like from the point of view of Sato's hyperfunction theory - and sheaf cohomology (Penrose would endorse this !). And maybe Penrose's Twistor theory has an even greater significance in a completely different philosophical context than the one adopted by Penrose himself (who still adheres to the 'collapse of the wave-function' dogma). Consider the double-slit experiment. We need a concept of boundary and interaction for distributions. And to be able to deduce probabilistic information from distributions, boundaries and completely deterministic equations. But we must not forget that the plate used in the double-slit experiment is only approximately a plane - in reality it has a highly irregular surface and there will always be one local region which is the first to "touch" the wave-front proceeding from the slits.

Note that a regular distribution may be localized according to the support of its associated function $f$ in $L_{loc}(\Omega)$.  That is its value for  is equal to the value of its restriction to an open set containing the support of $f$ . Or the support can be disconnected, so we have two disjoint localized centers. A distribution may be regular or localized according to its restriction to a certain time interval but evolve into a different situation - this can be used for a solution to paradoxes similar to the EPR paradox.  The photon ceases to be a localized wave-packet (a regular distribution) becoming a non-regular one (i.e. having a non-local holistic character) (we have a continuous path in the space of Radon measures for instance) thus explaining why an observation (i.e. an interaction) at location A can determine the outcome of an observation at a distant location B.

The whole proposal above is obviously highly sketchy and unsatisfactory.  We need not only the non-locality (which transcends both the field and particle approach) using distribution theory but also the fundamental postulate that the linearity of the theories and equations is only an approximation of the fundamentally non-linear or even chaotic (but deterministic) physics at a finer scale (cf. the Casimir effect which QFT interprets as 'fluctuations of the vacuum'). It is this framework that could explain that in reality observations are interactions with a non-linear component - in general expressing what happens when a non-localized (non-regular) distribution interacts with a regular localized one (recall that there is no satisfactory definition of a product of distributions in general). Maybe we must extend physics to account for an equivalence between energy and information (in observations, the measurement process) perhaps embodied in Psi-phenomena. The wave-function is like the continuous holistic coordinated movement of juggling (or swimming).  If the mind stops and focuses on a localized part (i.e. local interaction energy is exchanged) the system implodes and its non-linear dynamics leads to only apparently random final outcomes or crashes.

In another place we have proposed that the fundamental issue at stake regarding quantum theory is that it is necessary to abandon the postulate of differentiability and continuity in our mathematical models of nature - but not necessary computability and determinism. This entails immediately that we have no longer in general any criterion or concept of 'identity' or 'individuality' which in classical (and relativistic) physics is entailed by the temporal continuity of (particle, field) solutions of differential equations. It is has not been proven that $\pi$ is a normal number. Its expansion is computable and deterministic yet it is conjectured that this sequence  is probabilistically indistinguishable from a random sequence (i.e. any finite subsequence occurs with equal probability). We also raised questions about the ordinal (order type) of time. The fundamental problem is studying how this measurable framework can approximate the continuous and differential framework (and this was Boltzmann's problem) - were we need to go beyond the clumsy chimera of the continuous wave-function codifying discrete random information.

It is not because nature is random or non-deterministic that we are driven to use smooth probability distribution equations but rather because nature is not-continuous or smooth and this is the most convenient and logical way of doing modelling bearing in mind the history of physics. However discrete dynamical systems and computability theory may open up new possibilities. And it may be that the non-smooth dynamics are generated by an underlying smooth structure and we return to the beginning.

Our approach will involve considering a measurable function or field (an atomized field) over space-time (or a generalized space-time).  For instance having nowhere dense support or a condition of being discrete up to the Planck scale. A postulate: for every line we have one point (or Planck region) of non-empty intersection. Think of sand on the surface of a drum.

We postulated that a truly free particle (atomized field or light-dust) can be with equal probability anywhere in space at a give moment. This is the analogue of Boltzmann's postulate that for a molecule of a perfect gas each velocity vector has equal probability.

A real 'theory of everything' would be a theory which allows one to solve all (differential) equations. 

TPC, self and temporality

We have described TPC as being involved with the transcendental awareness of the total continuum or process of thought considered purely as such, as merely a process of thought.  But TPC has an important aspect: transcendental philosophical consciousness involves the transcendental awareness of those preconditions and structures upon which ordinary consciousness entirely depends and at the same time of which ordinary consciousness as a rule is oblivious of. It is almost as if ordinary consciousness only exists, can only exist, under the condition of it carrying a forgetfulness of its own transcendental preconditions and predetermining structures. Transcendental consciousness is thus also the consciousness of what non-transcendental consciousness must necessary forget.   If TPC in turns does not have full knowledge of such transcendental preconditions and structures, it falls back into a false dualism, limitation and illusory reification.  Two of the most important of such preconditions are selfhood and temporality.  That which claims to be a being which we carry along as a huge burden - all possible transcendental questioning, untangling and analysis being forgotten - and that which we must forget in the first place in order for the very being-ness of being to arise.  Only then we arise to the selfless liberated insight into the pure universal sphere and flux of pure thought processes: only thus is further progress of TPP possible. The ordinary self is the protoype of illusory unquestioned, posited, composite being. The original prototype of being in transcendental temporal oblivion. One task is to study the formative tendency which constructs this prototype employing among other things certain classes of conscious phenomena.This prototype is the the scaffold upon which world-positing and world-directedness and the mutual feedback of the identity construct takes place. Being-in-the world is not a transcendental condition or principle but a non-transcendental illusion conditioned by transcendental ignorance and forgetfulness. Following Plato a philosophical consciousness must include knowledge concerning the phenomenology of love and beauty.  This is a transcendental illusion which yet in its purest form  participates of the some of the modalities of TPP-liberated consciousness.  It paradoxically offers a glimpse of the bliss of beinglessness falsely conditioned and limited by being, by the illusory directed network of the self-prototype and its world. When we are born we are forced to build our identity and learn the world; when 'born into the spiritual life' our duty is to unbuild this identity and learn to see through the illusion and construct of the world. It is as if the first phase were like being plunged into the water, a downwards journey into illusion and forgetfulness. The second phase is when we begin to rise again and make our way back towards the surface and the air of light and truth. TPP is not about an 'individual' detaching itself from the 'world' and still less about an 'individual' transferring its feelings, desires, volitions and tendencies to alternative 'imaginary friends' or any kind of ontological construct - rather it is about seeing through both the 'individual' and the 'world' and their dependent interplay and mutual constitution. There is no contradiction between Yogacara and Madhyamaka. Two of the main tasks of TPC-based philosophy are: i) the untangling of all concepts and categories and the showing forth that they all exhibit the mark of the alleged being of the prototype (self), ii) exhibing the absoluteness of the moral law,  that the absolute can be characterized as morality (dharmakaya). The profound affinity between the consciousness and act of TPC and TPP and that of the essence of morality.  In all of the above the great interest of Kant is patent.  A model of the universal  flux of pure consciousness (Indra's net): neural nets, cellular automata, music theory, formal grammar, tessellations, Wang tiles, knots (and our own theory related to this) - unfolding of the concept of computation (paticcasamupada) and its concurrent parallel interconnected forms. And most importantly the theory of (meta) reflection and representation.  And the logical and mathematical theory of programs, specially the hierarchy of function definitions. The way forward in physics, the way to transcend and rectify quantum field theory, seems to us to involve a radical new foundation for physics based not on the formalism of functional analysis and operator algebras but rather on finitary combinatorics, theoretical computer science, graph theory and algebra. Maybe something following Birkhoff's lattice approach. 

A mathematical model of transcendental pure consciousness (which can only be a limited exterior projection, a partial mirroring...) will involve a spatialization of concepts (structures) wherein their interconnection and mutual (meta) reflection takes place - in a temporal dimension. A general theory of networks which generalizes neural nets, cellular automata, tessellations, crystallography, (higher)category theory and proof theory. This will include the spatial, topological aspects of music, and the architecture of a semantic universe (semantic space) or lexicology. But we can take also an atemporal space-time approach. Has the effects of special relativity on the physical realization of Turing machines been studied? Could special relativity be the key to how physical systems could compute beyond the Turing limit? Perhaps the quantum effects on physical computational systems can be studied in a way different from the current work in 'quantum computing'. In Hegel's  Logic we can discern a marked spatial, geometric way of thinking of concepts, their unfolding, mirroring and interaction.

The knot, the tangle, the projection, the multiplicity is not real, what is real is the formative energy which ties and then unties, tangles and then untangles, projects and then dissolves, multiplies and then unites. Is this not the ultimate meaning of the Hegelian Idea, the holomovement of skepsis which though necessarily dealing in formalism is not itself entirely formalizable ? Each thought (unit of consciousness) containing implicitly all other thoughts and each thought only existing through its relationship to them? That by trying to isolate a thought 'in itself' it inevitably looses the essence of what it really is.

Another approach is to view space-time as characterizing non-transcendental consciousness and being characterized by its algebraic and logical properties. Thus pure transcendental consciousness should be represented by a structure in which some of these logical or algebraic properties are different, the most famous being Heyting rather than Boolean (Brouwer), non-commutativity (Connes) or non-distributivity (Birkhoff).  Birkhoff's explaining away of the uncertainty principle as simply an expression of non-distributivity. However this approach does not go beyond the previous proposal, it is still a priori and phenomenologically a spatialized or space-time based 'collection' of components, parts, connected to each other by some kind of relation or network. And the essence of representation theory is bringing this spatial substrate further into the foreground.

The philosophical elephant in the living room is that we have not explored something which should occupy a fundamental place in TCP. That of symmetry, quasi-symmetry and analogy. Like temporality and the prototype of the self, it is so pervasive as to be habitually consigned to an implicit oblivion. Symmetry and its allied modes pervade biology, psychology, linguistics, chemistry, physics, mathematics, logic, computer science, engineering, the structure of philosophical systems, poetry, music,...so deeply, in such a far-reaching manner, that like temporality or the construction of self, we simply forget and cannot fathom its central, universal transcendental a priori role (symmetries abound in the architecture of Kant's and Hegel's systems, also in the Pali suttas, and linear logic brought forth the implicit symmetries and modalities in classical logic). We end with the following preliminary (Hegelian) descriptions: symmetry is the quality of a being which allows it to change and yet remain the same. Symmetry is also that by which a being may produce itself (notice than in symmetric tiling we can transport any tile to any other tile while preserving the identity of the whole). Symmetry is the foundation for reflection-into-another and reflection-into-self. Recall how spinors capture the 'memory' (homotopy class) of a rotation in SO(3), much like the dialectic in Hegel. An object may be brought back to itself and yet have gained something permanent.  Symmetry is that by which an inherently dynamic process can produce the illusion of stability and being. Symmetry allows us to fathom essentially interrelated holistic systems (illustration of 'maps of consciousness': dictionaries, grammars, thesauri, manuals of stylistics and translation, encyclopedias, biographies, novels).