Critique of Quantum Mechanics and Quantum Field Theory

Here we will present some short logical and philosophical critiques and questions concerning quantum mechanics and quantum field theory with the ultimate goal of going beyond these theories and constructing more satisfactory ones. We will also attempt to bridge the gap between some of our previous speculations and the actual mathematical structure of quantum theory.

We focus on the collapse of the wave-function. But the root of the problem can already be found in the singular situation of observables corresponding to Hermitian operators while the temporal evolution of the state of a system corresponds to unitary operators. The collapse is essentially a projection onto an eigenvector. In classical physics measurement and observations are external to the system. In quantum mechanics they become internalized. The observation process, the applications of hermitian operators, suggest a highly sensitive non-linear response of a system whose nature is still not understood.

What if we viewed space as discrete (like the lattice structures used to explain free fields in QFT)? Then the wave function becomes just a finite vector, the space points mere indexes - just like time is a mere index. We loose the topology.  It is curious that  position can be promoted to an operator but time cannot. A time operator in quantum theory seems to be a notion derived from momentum and as such recalls the classical Aristotelian notion of time being a "measure of change".  

Understanding what the vacuum state $|0\rangle$ is in QFT is not easy - and many different accounts are given in the textbook of why the Green function takes on a particular interpretation of a particle being created at point $x$ and destroyed a point $y$.  Understanding what the spatial-temporally indexed operators are - and what measurements they correspond to - is also not easy. The creation and annihilation operators are not Hermitian and do not correspond to observations and measurements and yet Hermitian operators (or operator-valued distributions) can be written in terms of them.  These supposedly correspond to measurements.  We must first of all understand the Vacuum Expectancy Value, what $\langle 0 | \phi(y) \phi(x) | 0 \rangle$ means (which is obviously similar to the expected value in quantum mechanics $\langle \phi | A | \phi \rangle$). And understand it experimentally. The first $\phi(x)$ corresponds to the high-energy localized injection of a particle (perhaps the product of decay) while the second $\phi(y)$ could correspond to collision with a detector (which thus destroys the particle).

We think that the statistical mechanics motivations of second quantization and QFT is particularly interesting. Our "wave function" gives a probabilistic distribution over occupation numbers $(N_1,...,N_n)$. The interpretation of the creation and annihilation operators in this context is subtle. How are we really to understand

$a_i \Phi (N_1,...,N_i,...,N_k,...) = \sqrt{N_i} \Phi(N_1,...,N_i -1,...,N_k,...)$?

Too often we see physical metaphors applied rather vaguely to mathematical constructs. To us these operators only really make sense in the context of the rigorous formulation of Fock spaces (tensor products, symmetric and anti-symmetric products of Hilbert space $H$). Given an orthonormal basis $\epsilon_i$ of $H$ then the operator $\Sigma_{i=1}^\infty a^\dagger(\epsilon_i)a (\epsilon_i)$ has very clear properties ($Sv = nv$ for $v$ in the $n$th component of the Fock space), far more illuminating than speaking of "particle number". But of course Fock space pertains only to the theory of free fields. The general case, the perturbative case, seems intimately connected to the general concept of formal deformation of an object by formal power series (cf. quantum groups or Hopf algebras in general), but there is no rigorous mathematical structure for perturbative QFT.

 A central flaw in the theoretical treatment of QFT is that it systematically ignores the central unspoken role of human agency, the experimenter and experimental setup, behind the mathematical formalism. Operators (operator valued distributions) are treated as independent active entities in their own right which somehow decide to "act" on fields. The glaring shortcoming is that there is no clear criteria of demarcation between the human observer the what is observed - nature.  Can the measurement process itself be considered a natural process when considered and observed externally by a third observer ? Are Hermitian operators really then descriptions of something objective happening to the system, the external observed observer "zapping" or collapsing the system? And should not this whole process itself be capable of superposition?

This most curious situation: observables must correspond to Hermitian operators - this can be deduced by the requirements of measurement in a probabilistic framework. Also, the unperturbed evolution of a system corresponds to unitary operators (or in the Heisenberg picture conjugation of observables by unitary operators). But what a coincidence that - for a finite discrete approximation - the lie algebra of the unitary group is the space of anti-hermitian operators - itself isomorphic (via multiplication by $i$) to that of hermitian operators. That is, every unitary matrix is can be written $e^J$ for $J$ anti-hermitian or $e^{iH}$ for $H$ hermitian. So measurements correspond to objects in the tangent space of the group of unitary operators - but they are also applied in their own right to the wave function.

Why could not the total energy of the vacuum be infinite? Why could not the total energy of the universe be infinite and yet there be still, in some sense, conservation of energy?  For instance we could have a countable infinite collection of non-communicating finite universes with finite energy and in which the conservation of energy held. Then the total set of universes would be a universe with infinite energy in which there was still conservation of energy in a meaningful sense.  We could even allow communication between the finite components satisfying a conservation law.

It is curious how the discreteness of the energy states is connected to the finitude of the system. 

The vacuum state in QFT is similar to consciousness (to alayavijñana) - and we can argue that classical field theory could never be an adequate representation.

We have discussed before the interest of considering material and energetic constraints in information - and this is precisely the idea behind linear logic (Girard's first papers explicitly mention chemistry), as well as partially already present in the memory management aspect of programming languages. QFT scattering is a lot like communication with material constraints.  The "answer" of the system has to be materially destroyed to be received.

This is certainly a bizarre idea, but could it be that the experiments with LHCs are not teaching us about nature but rather a kind of training process analogous to AI? Nature is changing her habits in conformity to what physicists want to observe? And finally consider how we attempted previously to characterize the massive datasets which allowed LLMs to be trained and perform as they do.  The qualities of the massive data sets in question could they not correspond to fundamental properties of the vacuum or quantum fields in general? Of course the analogy is limited because of the apparent inescapable indeterminism in QFT while LLMs are just deterministic automata.

Type theory / QTF analogy.   Type = Operator, Term inhabiting a type = Vector and application of operator to the vector,  Simply Typed Lambda-Calculus = Quantum Mechanics, Dependent Type Theory = Quantum Field Theory, operators depend on values. Loops, non-termination = infinities. 

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.