Feynman's way out (of the classical arena)

Physics has inherited the notions of Classical Mechanics. The physical explanation of the problem of motion is essentially deterministic. So, we have seen that is necessary to dig deeply in order to find the roots of the problem. And, above all, it is imperative to cut these roots in order to understand Quantum Theory. This is not an easy enterprise due to the fact that mechanic philosophy is ubiquitous in our language and has installed in our minds. The most difficult point to interpret is understanding the reason of movement. Notice that, contrary to what happens in the mechanical Universe, at quantum scale, the interactions do not cause changes; they only open the door so as the transitions can take place. The quantum processes occur spontaneously by chance. 

The interpretations of Quantum Theory that try to conserve the ontology of Classical Mechanics exacerbate the role of Schroedinger’s equation in the theory. It is claimed that the state of the system evolves continuously, as if after an event the system followed by “inertia” the continuous evolution of probabilities determined by this equation. But this path takes us to a dead end: explaining the discontinuity through continuity, and chance by determinism. 

So, the conceptual revolution of Quantum Theory will only be completed after the new conception of the problem of motion will be generally accepted. On the whole, quantum phenomena show us that aleatory processes are spontaneous and chance plays the role of the “inertia principle” in the theory (the spontaneous persistence of motion). Any other attempt would be another way of returning to the theory of “hidden variables”. 

At present, the big question is the opposite one. How to explain the apparent determinism we observe at classical level from the aleatory behavior quantum phenomena. The first answer to this question was outlined by Dirac [...3] then Feynman [19, 20] completed this elegant idea developing his path integral formalism. In this frame work, all paths, continuous or discontinuous, are allowed and the classical trajectory is just the most probable sequence in the network of potential events. 

In conclusion, the paradigm of a legal Universe is absent in Quantum Theory. Physical “laws” are just symmetries or reduced to the “law of big numbers”. Chance is the true physical principle of the theory. The second principle, symmetry, only plays the role of a descriptive one. In fact, it proposes a framework for the theory from which the logic and probabilistic distributions for the physical processes can be deduced. In fact, as the processes are random, we are just limited (as in the case of the die) to find the symmetries in the system which allow us to determine the probability for these processes. 

As symmetries have the group mathematical structure, we can establish an homomorphism with the non singular square matrix. That is a linear representation of the group of symmetries of the system. In particular, the mathematical framework of Quantum Theory is the projective representation in a complex ray space. This is to say that the representation of rays and matrices are defined up to a phase factor, because, at the end of the day, probabilities only depend on the square of the amplitudes. This phase factor is a non trivial element. This one is responsible for the non commutativity of translations in phase space and, in general, for the interference effects characteristic of quantum processes. Moreover, it also explains the classical limit of the theory. 

To sum up, we have seen that the geometry of the ray space determines the logic of the quantum processes and, as a consequence, the quantum probabilistic rules such as Born’s and von Neumann’s ones, historically postulated ad hoc. This logic and probabilistic rules a priori sound enigmatic, since we have to leave aside the familiar Boolean logic and Kolmogorov’s notion of probabilities. However, if we  finally accept that the ray space is the true scenery where the events occur, the new logic is inherently derived from geometry. The propositions of this logic are chains of sequential paths of events (histories) that have two equivalent representations. One uses of projector operators, and the other one works in terms of complex amplitudes which follow Feynman rules. In this rules, which describe the basic experiments of polarization and interference, is encapsulated the mysterious logic of the quantum processes.

The Logic behind Feynman's Paths

Edgardo T. Garcia Alvarez, November 22nd 2010