How would Quantum Field Theory look like if we stopped for a while developing it further as if it were the draft of a yet-to-be-discovered Theory of Everything, and just started to reformulate the Standard Model as a mathematically and conceptually coherent physical theory? And what would such a theory tell us about the world and about ourselves, which remains hidden in the ill-defined formulations we’ve grown up with through the last decades? As I started back in 2010 to reflect on these questions, I didn’t have yet a clear vision of what this work would lead me to.

I just had the feeling that these very basic questions hadn’t been interesting anyone any more for a far too long time, and that we should actually have the means by now, with our understanding of Renormalization, of writing down a well-defined Quantum Field Theory reasonably accounting for all known experimental data (excepted General Relativity phenomena) – which means essentially that it has to be compatible with the Standard Model at known energy scales. I was quite confident that I could find a physically sound regularization of the Standard Model, which I simply wouldn’t consider as an approximation, but take as the exact theory itself, the Standard Model being an ill-defined idealization of it. The models used in computer simulations of lattice Quantum Chromodynamics, for instance, would show me the way. Of course, I knew that I wouldn’t be able to derive the theory from the usual first principles any more, but given that all the attempts of axiomatic Quantum Field Theory to construct well-defined interacting fields upon these first principles had failed miserably, I thought that maybe they could be misleading in the end. Anyhow, I had never been very fond of the heuristical construction of Quantum Field Theory based on Gauge and Poincaré invariance. Developing the whole mathematical apparatus of Representation Theory to simply derive the expression of spin 1 and spin 1/2 spinors as irreducible unitary representations of the Poincaré group had always seemed far too expensive to me, and Gauge transformations mixing particle fields far too artificial to make up a fundamental symmetry of Nature.So I felt free to redefine the Hilbert space of the quantum states without paying much attention to these first principles and focused instead on the mathematical well-definedness of the theory, and in particular of the Schrödinger equation.

The most evident way of insuring a well-defined solution at all times is to make the Hilbert space finite dimensional, which has two major physical implications. The most important one is that the physical space itself, too, has to be finite,From a cosmological perspective, the finiteness of space is also a very interesting aspect.

It addresses the old question of knowing whether there is something like a frontier of the universe or if the universe is infinite, and it offers a very original answer. According to this theory, the universe is both finite and boundless; it actually has a toroidal structure, which is not of topological nature, but reveals itself at the level of the field dynamics: Wave packets will transit smoothly from one side of the finite lattice to the opposite one without experiencing any discontinuity. So the light we emit, for instance, could come back to us from the opposite direction after having traveled through the whole universe. Yes, if the universe were smaller, maybe you could see the Earth looking at the stars... and the position of the closest images of the Earth in the night sky would give you the direction of the lattice axes, by the way.The second physical implication of the finite dimension of the Hilbert space is the existence of a maximum occupation number for boson fields.

I wondered if there were any good theoretical reason to assume an unbounded number of bosons per field mode, and I actually didn’t find any. Of course, the commutation relations usually considered as essential properties of the creation operators would break down when the maximum number of particles is being reached, but these relations, relicts of a heuristical construction of Quantum Field Theory based on the harmonic oscillator model of Quantum Mechanics, are not really necessary to define creation operators. In fact, it is quite straight-forward to define a basis of the Hilbert space on a finite lattice, you just have to take as basis vectors field configurations defined as functions giving the number of particles of each kind at each lattice site. And it isn’t more complicated either to define creation operators as adding one particle of a given kind at a given lattice site, as long as a given maximum occupation number hasn’t been reached. The normalization factors implied by the commutation relations can then be moved to the spinors, where they actually belong. The situation is quite similar for fermions: If you don’t construct the Hilbert space heuristically as a Fock space over the one-particle Hilbert space of Quantum Mechanics, the sign factors implied by the anticommutation of the creation operators can be moved to the spinors too. So in the end, there isn’t any qualitative distinction to be made between bosons and fermions; the same creation operators can be used in both cases, differing only in their maximum occupation numbers. In fact, if you don’t construct the Hilbert space as a Fock space, but define it directly (or use a Fock space modulo particle labels permutations), there is no Spin-Statistics Theorem classifying particles into bosons and fermions according to their spin any more. This famous theorem relates the integer or half-integer character of the spin to the possible sign change happening to the quantum state when the labels of two particles of the same type are being exchanged. But the notion of exchanging the labels of two particles doesn’t actually have any physical meaning, it only makes sense in the Fock space formalism, and is a mere mathematical artifact. I think it is important to realize that the Spin-Statistics Theorem, traditionally considered as one of the greatest insights provided by Special Relativity into Quantum Field Theory, actually doesn’t have any profound physical meaning, and doesn’t establish, as it is often being stated, a connexion between the geometry of space-time and the collective behavior of particles. It only expresses a property of the “unphysical” Fock space formalism, and becomes meaningless as soon as you consider the “physical” quantum states modulo particle labels permutations. So the categories of ‘bosons’ and ‘fermions’ are not implied by Special Relativity, as far as their collective behavior is concerned; only the form of the spinors is. Determining experimentally the maximum occupation number for each boson field is still an open question: For “heavy” bosons like the Z boson, for instance, I don’t think that a lower bound much greater that one can already be established with current experimental data...Once I had constructed this well-defined framework for Quantum Field Theory and made a first proof-of-concept by integrating Quantum Electrodynamics, I left the paper draft I had written by that time rest for a while, took care of my new-born son and started reading a book from the Philosophy library of my wife that had been intriguing me for a while: A French translation of Spinoza’s *Ethics*.

As soon as I had developed this Spinozist model of the mental world (which builds up, together with the material world of quantum fields, the physical world as a whole), I got confronted with the old question of the status of time in Quantum Physics.

The controversies on this subject have been summarized very concisely by Wolfgang Pauli in his statement that there cannot be any time observable in Quantum Physics. In the Copenhagen interpretation, indeed, time isn’t a property of the quantum system under observation; it isn’t being measured quantum physically, butIn the end, the model I’m proposing can be roughly described in very simple terms: A mental state is being experienced while the quantum state is undergoing an elementary unitary evolution, then a new mental state is being randomly moved to as the quantum state gets projected to the corresponding subspace, an so on.

In the meanwhile, this almost sounds trivial to me, so I guess I’m eventually understanding Quantum Physics, at least in this form. This alone would be a revolution in this field of science. But I’m not interested in pretending to have discovered deep truths about “the inmost force which binds the world”, to speak with Goethe; I just wanted to show that it is possible, and actually quite easy, to give Quantum Field Theory a form and an interpretation which make it a formally and conceptually closed theory, capable of giving a well-defined answer to any question we can ask it – even if we may eventually find out that it wasn’t the right one. This interpretation challenges all existing ones insofar as it is the first time that this degree of conceptual precision and formal well-definedness has been reached, and I hope this will be motivation enough for others to work out alternative interpretations and achieve the same level of quality – so that we can finally know what Quantum Theory is actually about...Sébastien Fauvel, born 1983, graduated from the Ecole Normale Supérieure of Paris in Physics and Comparative Literature. He has been working as a Consultant, Software and Web Developer in Lyon, Freiburg and Basel.