A Mess called Quantum Mechanics

After 4 years of contemplation I can now officially declare that I no longer consider Quantum Mechanics in any seriousness with regard to ontology (reality and truth). 

Just to show its contingency I have been trying to publish a work that I first [here] considered of so little value that did not have the motivation to write it explicitly. 

My main reason is that the theory does not even have clear definitions to begin with! There are at least three different ways to define the quantum-mechanical momentum, first being the orthodox definition using linear self-adjoint operators $$\boxed{p_\mu=i\hbar\partial_\mu}$$ This definition has occupied theoretical physics and 99% of physicists working on (foundations of) quantum mechanics since 1926, when Schrödinger, led by the strong fashion of the time to explain discrete atomic spectra, thought that the natural mathematical implementation of these discrete radiation packets is done via eigenvalues, and since then the academic sheep could only follow suit.

From the conservation of probability $$\frac{\partial |\psi|^2}{\partial t}+\nabla\cdot\left(|\psi|^2 \frac{\nabla S}{m}\right)=0,$$ (that can be derived from the Schrödinger equation) one is led to another expression for the quantum-mechanical momentum, i.e. $$\textbf{p}=\nabla S,$$ which in terms of the wavefunction is $$\textbf{p}=\hbar\Im\frac{\nabla\psi}{\psi}.$$ Theoretically there is no reason why one should neglect the real part and ruin the structure of complex numbers by only considering the imaginary part. This has been my little contribution: I do not throw the real part away and define $$\boxed{p_\mu=i\hbar\frac{\partial_\mu\psi}{\psi}}$$ This allows to arrive at nonlinear generalizations of Schrödinger and Klein-Gordon equations that unlike orthodox quantum mechanics (and Bohmian mechanics) also allow for compressible Madelung fluids. 

In 1952 Schrödinger proposed a gauge transformation, without realizing that it is in fact using another definition for momentum!, viz. $$\boxed{p_\mu=\frac{i\hbar}{2}\partial_\mu\log\frac{\psi}{\psi^*}}$$ I realized this only yesterday from Vedral's note. 

You can check for yourself that using a simple plane wave $$\psi=e^{ik_\nu x^\nu}$$ all these expressions give the four-wavevector (hence momentum using de Broglie's hypothesis). But which one is the definition? How can we find out when theoretical physics has never considered these possibilities? 

This is why I cannot anymore continue thinking about anything that involves quantum mechanics and have the least sympathy for any theory (including all the current attempts of quantum gravity) that takes it seriously. They might achieve some results --like QFT-- but they will never have any significant consequence for our understanding of reality. 

So, what is to be done? Both aesthetically and according to our experience in history of physics, when something becomes so ambiguous, the solution is to leave it altogether and look for a totally new ground, one that does not borrow any of the key constituents of the messy theory. That is what I have tried to do here and here. 

Tension between Ontology and (Kantian a priori) Epistemology in Theoretical Physics

This is an elaboration of what I presented here. As far as I am concerned the idea of tension between apparent ontology of physical theories and a priori-Kantian aspects of epistemology as a driving force of theoretical physics has not been discussed so far in philosophy of physics. 

To be clear by epistemology I mean study of epistemic objects, i.e. study of objects in our mind as long as we think about them. 

Accordingly by ontology I mean study of ontic objects, i.e. study of objects in the external world. 

I take it as evident here that we (humans) are either objects or subjects (transcendental idealism) (but not both; in this context at least), i.e. we are either a knower (subject), one who wants to know the world of objects, or a being-known (object); hence the categorisation of objects into subjective (epistemology) and objective ones (ontology) is justified. 

The starting point of philosophy is the will to seek truth, i.e. becoming a subject (sujet) is the action by which one becomes a philosopher, hence the world philosopher=lover of wisdom. The will, as long as it is present, creates the reign of subjectivity in which the power of subjectivity rules (Fichte). The power of subjectivity is the driving force, the motivation, the subjectivity's vis viva that drives the philosopher to know the world, so a subject by merely being a subject is driven to the knowledge of objects. 

Ontologies are the focus of physics, and epistemologies of metaphysics à la Kant. An important part of Kant's epistemology which has been the subject of enormous controversy and silly readings is the notion of a priori aesthetics, a priori intuitions that exist prior to experience. Three main examples are res extensa, space and time. Kant barely enters ontology in his Critique of Pure Reason; but where he entered physics he created disasters (Metaphysical Foundations of Natural Science) --judging from the influence of Kant's physics on Schelling's, in which Schelling tries to explain action at a distance of gravity instead of trying to solve it, and he takes Galvani processes as fundamental mediator of electricity and magnetism; quite mislead indeed from the viewpoint of our knowledge today (this idea though was not neither new nor unfamiliar in his time and it helped Faraday to discover induction). I agree that Kant crashed at physics (at least he did not live up to my expectations) but he got epistemology quite right. This is an honest modern reading of (part of) Kant's epistemology expressed informally: 

• I do not care whether you are Witten, you cannot think about a manifold without embedding it into a background. This is what people usually mistake with Newtonian absolute space. 
• I do not give a damn whether you are a `genius', you cannot think about a mathematical point. 

But Kant is definitely not saying you cannot have any of these notions in ontology. He has restricted himself entirely to epistemology --objects in our mind. This is the root of all the controversy that `Kant based his epistemology on Newtonian physics. We already have relative space (in relativity) and discreteness (in quantum mechanics).' and such. As restrictions on thinking about objects in our mind, I challenge all of mankind to think of a manifold without embedding it into a higher-dimensional background. 

My main point is that as long as one is a subject, one is bound to conform to such a priori notions. The rulebook of the executive power of subjectivity is Kant's a priori intuitions. 

This means that as long as we want to know the objects in the external world we are driven by the power of our subjectivity to make ontology conform to epistemology; the more the better. 

This is a powerful tension that drives theoretical physics today and has driven it for most of its lifetime. 

I have three examples for the notion of res extensa; physics trying to make itself conform more and more with the notion of extension:

1. An example which actually (ontologically) worked: Weak interaction à la Fermi was a point interaction hence not conforming to res extensa. Standard Model extended it and it worked.
2. String theory, extending points to spheres and one-dimensional lines in Feynman diagrams to cylinders. 
3. Penrose's CCC, extending the singular point of big-bang to a sphere via conformal transformation.

This brings me to the vital question: what do we count in theoretical physics as requiring explanation?

For example, is it the discreteness of spacetime that requires explanation or its continuity? 

Development of recent serious attempts which are partially philosophically-aware like Loop Quantum Gravity has been along the latter and I think that is wrong. Our thirst of subjectivity is only quenched when we arrive at a theory that conforms maximally to a priori intuitions. Based on this, as extension is an important a priori intuition, it is the emergence of discreteness from continuity (not the other way around) that requires explanation.