Paul Dirac, and the almost but not quite
I was thinking . . . and I just finished a book of four lectures on quantum mechanics yesterday. . .
Dirac was a physicist whose writing seems very concerned with a mathematical explication of quantum field theory. But, like all modern physicists, he was keenly aware of the problem, and tension between, quantum physics and the theory of relativity. So, he sets himself a task of examining the field, in terms of quanta, and how that would relate to a relativistic space.
So we have relativity. It is a pretty tried and tested theory of how motion, time, space and gravity are connected. It provides good predictions and corrects the errors of classical mechanics in terms of the orbits of planets. It accounts for the finite speed of the propagation of light. But, if one, like Dirac, wants to examine a quantum physics in terms of relativity, we have to make the mathematical descriptions of what happens to the "particle" agree with any other mathematical description regardless of the frame of reference. But this, as we know, will only make it agree with special relativity. General relativity goes on to show that the space-time continuum is "bent" due to the effect of massive bodies (gravitation). So, not only will our mathematical description have to be equally valid without regard to "absolute" velocity, or the time of one frame of reference, but it must be made to agree as described on a curved continuous surface.
Dirac's mathematics are difficult to grasp, but he proceeds in a very logical order. He begins with two elementary equations, one he calls an "action integral" which is a particular formal mathematical description that is an equation of motion. It makes it easier to think of a Cartesian plane, with the equation y= -gx^2 + b, describing a parabola. Following along the Galilean thought lines, this upside down parabola describe the motion of a falling body moving along the x-axis with some uniform velocity. The action integral that Dirac uses is the same sort of thing, only the Cartesian plane is eliminated. He follows this by deriving a mathematical way to proceed from this "classical" mechanics of particles to what is called the "Hamiltonian" giving way to a quantum mathematical description of the motions of particles. The first two of the four lectures follow along these lines, first picturing how we could mathematically go from classical mechanics and a description of motion, to a quantum description. The second lecture does just that, moves from classical mechanics into quantum mechanics.
But once he has passed from classical mechanics to quantum mechanics he wonders about how this would agree with relativity. The equations starting off his derivation, though not explicitly in agreement with relativity, they are made from relativistic assumptions. But the problem arises when he uses quantifiers to pass over from classical mechanics to quantum mechanics, there are certain assumptions that must be made. We (as mathematicians and physicists) have to choose the order of our factors, and in so choosing, we may introduce a quantifier which will make the result at odds with relativity. So Dirac starts over. He redoes the "action integral" with a different time variable. He then shows that with this derivative - going quickly through the same steps as the first two lectures, yields an equation that also is quantum, thus showing that there is not any absolute time variable. Then he has shown that there is not one observer in his descriptions, because there is not one time, and therefore at least in this way it agrees with relativity.
Dirac then tries to construct his equations on a curved surface. He shows that it can't work. Finally, he shows that he can make some simple cases in quantum mechanics work out on flat Euclidean space, which is good enough o get his mechanics to agree in simple cases with special relativity, but which cannot be in accord with general relativity. Close, but not quite the "Grand Unified Theory". He recapitulates by pointing out all of the difficulties in quantization of quantum mechanics, and how far mathematical physics is from being complete. He brings up again the idea of "degrees of freedom" in the motion. They have to restrict the degrees of freedom to a certain finite amount in order for his theory of quantization to work. But he points out that the actual fact is that the thing described has "infinite degrees of freedom". Specifically, he says:
But with field theory, we have an infinite number of degrees of freedom, and this infinity may lead to trouble. It usually does lead to trouble.
He then enumerates the problems of the infinite variables, mathematically inconsistent methods of dealing with divergent integrals, and so forth. He concludes, finally, on a grim note, saying the work is far from complete. Others are trying methods completely different from his. But Dirac maintains that even if the method he uses is not the answer, the answer will look something like his method.
This is getting quite long, but I have a couple preliminary reflections to make, to be elaborated later. First, compare relativity with Aristotelian motion and time (with the given empirical data superadded to Aristotle). Consider quantum mechanics then as the problem of the way too small, both in the breaking of substance, and the problem with the intellectual observer in the way too small. Consider, finally, the priority of the more known to the less known, and the relationship of relativity to quantum mechanics in that light. Perhaps these few questions will shed some light on a melding of natural philosophy, what is heard of nature, and physics, what is calculated about nature.
