The equations of physics are derived from general theories. The odd situation of the moment is that the equations are quite well supported by experiment. One would logically conclude that the theories are as well supported, but this is not the case. The equations, or laws, themselves are very clear; their support by experiment is undeniable; it is in the interpretation of the equations that one finds difficulty.
This post arises from my reading a physics blog recently which mentioned that it seemed that in the 19th century, new physics ideas appeared in the form of “laws” while in the 20th century they were called “theories.” I think that the difference is not just a matter of terminology, but instead that theories and laws are not at all the same sort of thing.
When an experimenter takes data, the data can be arranged in various ways. If we are able to describe the data by fitting an equation to them, then I will call that a “law.” For example, Maxwell’s equations are laws. Given measurements for electric field, magnetic field, charge, velocity, etc., one can compute various things. This is more than curve fitting, but it is quite a bit less than theorizing. Theories are more general than laws. One theory can be used to define any number of laws. For example, the theory of quantum mechanics can be used to derive many different equations.
The power of equations is that a single equation can be used to fit the data of many experiments. As more and more experiments test a law, our belief in it can become more and more certain. Of course any experiment can only take a finite number of measurements each with limited accuracy, but so many experiments with so many varied conditions have been run over so many years with results quite compatible with each other that many laws are now assumed to be quite perfect.
The relationship between theories and laws is similar, in a way, to the relationship between equations and experiments. One equation explains a large number of experiments each of which tests the equation and, for the usual physics, produces evidence in favor of it. One theory produces a large number of equations. When each of the equations is verified by experiment, this is interpreted as evidence for both the law and the theory that produced the law.
As far as experimental support, the strongest physics theory is quantum mechanics. Despite the overwhelming evidence in favor of the accuracy of the equations derived from the theory, the theory itself is not at all well supported by the evidence. The original theory of quantum mechanics was a somewhat vague collection of techniques for moving from a classical theory of motion to a quantum theory. These were ironed out into a complete theory known as the Copenhagen interpretation.
The Copenhagen interpretation can be thought of as the “don’t ask, don’t tell” version of quantum mechanics. Unhappiness with this state of affairs led theorists to develop other theories that had the remarkable property of being completely different from the Copenhagen interpretation, but nevertheless produced equations that are identical. Thus these alternative interpretations cannot be distinguished from the standard interpretation by experiment. They are identical in terms of experimental evidence. The various theories differ in how they approach the problem of how one attaches probabilities to the wave function. The Copenhagen interpretation assumes wave function collapse is a probabilistic process which one cannot understand but simply uses to make calculations.
The most popular alternative interpretation is the “many worlds” interpretation. There are various, slightly different, versions of this theory. Basically, they amount to assuming that everytime a quantum object is measured, the classical universe splits into several classical universes according to the results of the measurement. This avoids the wave function collapse of the Copenhagen interpretation by instead assuming both situations continue on. Schroedinger’s cat is lives forever in one of these universes. I’d give a link to an article on the subject but given the attractive science fiction nature of the subject I’m sure my readers have already learned as much as the theory contains. I find it hard to believe that such a silly idea would be supported by as many physicists as it is.
My favorite well known alternative interpretation of quantum mechanics is Bohmian mechanics, or the Bohm interpretation. In this theory, quantum particles become classical, but with an extra “quantum force.” It’s a rather ugly theory in that one ends up with each particle’s motion determined by two objects, a wave function identical to that of the Copenhagen interpretation plus a classical particle position. For any given experiment, the particle position is unknown, but it can be shown that their overall statistics match that of quantum mechanics. In the Bohm interpretation, the wave functions never collapse; instead, the particle position selects one branch or another and the alternative branches simply do not contribute to the particle position. In this model, the universe contains huge amounts of unused wave function. In a sense, the Bohm interpretation is as extravagant with the wave function as the many worlds theory.
My own interpretation of quantum mechanics is similar to the Bohm interpretation. For me, the primary inadequacy of modern theoretical physics is in its inability to distinguish between the past and the future. That is, it is unable to pick out a moment in time as “now” but instead treats all time values equally. A result of this is that standard quantum mechanics exhibits an arrow of time only through collapse of wave functions.
If an experiment was run in the distant past, then it is clear that it has produced a result according to the probability laws. This would be a good place to represent the experiment with the classical part of the Bohm interpretation. Experiments in the future clearly need a wave function description as they must include all possibilities; thus the wave function. So for me, wave collapse is the sharpening of the wave function (as various possible Bohm paths are eliminated) until only a single path is left.
A consequence of this interpretation is that one must be able to deform a wave function into a classical particle path, as a topological process of mathematics. If the wave function were real, it is clear that one could do this by transforming the wave function into a delta function. But wave functions are not real, they are complex and this is a subtle problem. To get into this would take me too far afield on this post. So instead I will simply say “density matrix formalism” and let the readers try to work the consequences out for themselves.
Special Theory of Relativity
A postulate of the special theory of relativity is that there is no preferred reference frame, and that the speed of light is the same no matter what reference frame it is measured in. This results in a set of laws that show how one can convert times and distances measured in one reference frame into another reference frame. It’s obvious that one this set of equations is equivalent, except for interpretation, to the equations one would obtain by instead assuming that there is a preferred reference frame, but that no experiment with matter and light can distinguish it. In this alternative interpretation (which sometimes goes by the name “Lorentzian relativity”), spacetime is again separated into space and time, but we do not have an experiment that does the separation. There is no problems with causality because there is only one true indication of time (we just don’t know what it is exactly, except that it ticks no faster than the time we’re measuring in our laboratory’s, undoubtedly moving, reference frame). Matter is (at least classically) built from stuff that satisfies Maxwell’s equations, and since Maxwell’s equations are Lorentz invariant, so is matter. But the central idea of Lorentzian relativity is that the absence of evidence of a preferred reference frame is not necessarily evidence of the absence of a preferred reference frame. As with the alternative interpretations of quantum mechanics, Lorentzian relativity is experimentally indistinguishable from the standard theory but is philosophically completely at odds.
Another, and closely related alternative to special relativity is Euclidean relativity. In special relativity, time durations and spatial measurements depend on choice of reference frame but the time experienced by an object moving along a path, the proper time, does not depend on reference frame.
There are a lot of attributes that one might define for a classical object in the setting of special relativity. The attributes that correspond to things we assume are direct manifestations of reality are typically agreed upon. For instance, if the object is cube shaped, different observers will agree as to how many faces the cube has, or if it’s composite, the observers will agree on the number of subsidiary objects. And different observers agree on the proper time.
If the other attributes of an object are elements of physical reality, why not proper time? This is the essence of Euclidan relativity. The usual mixed metric -+++ of spacetime becomes purely positive, ++++. This makes certain classical ideas simpler. Maybe the best theoretical evidence for this kind of idea is the unnatural mathematical convenience in of Wick rotations in quantum field theory.
The problem with Euclidean relativity is in defining what happens to the proper time dimension. In my own version, the proper time dimension is a hidden dimension, that is, it is curled up as the string theorists do with their hidden dimensions. Unlike the string theory case however, the assumption of a single hidden dimension does not introduce topological complications. There is only one natural compactification, U(1). An advantage of this interpretation is that it explains the superluminal phase velocities of de Broglie’s matter waves. De Broglie’s relationship was the first law later subsumed into quantum mechanics and as such is at the center of the theory. Eventually I’ll write a blog post devoted to the phase and group velocities of quantum waves and explain this in more complete detail than the paper.
My favorite quantum mechanics and special relativity heresies, the modified Bohm interpretation and Euclidean relativity with a U(1) hidden dimension are synergistic. Part of the problem with my Bohm interpretation is that I need to explain where quantum phase comes from. Quantum phase is a U(1) symmetry. And the problem with my special relativity version is that I’ve got an extra U(1) symmetry; the curled up proper time dimension. These two problems conveniently cancel one another. There are many other nice consequences from these assumptions.
There is also an alternative theory for gravitation that is distinct from the usual General Relativity in that it naturally uses a flat (Minkowski) background space rather than an arbitrary set of coordinates. It was written down by Lasenby, Doran and Gull some years ago and has been used steadily by various researchers to make quantum gravity calculations. I wrote up a short blog post with references a few months ago: General Relativity Painleve and QFT. There are several attractive features to this “gauge gravity” theory. First, it avoids all the science fiction aspects of the traditional GR. Second, it’s written in the mathematics of Dirac’s gamma matrices and so it is natural to do elementary particles in. Third, it uses a gauge principle, as does particle physics. All these things imply that it will be needed in a unification of the forces (though it will still need to be modified at Planck scale).
Quantum Field Theory
In the standard model, the elementary particles are described using quantum field theory. In a certain way, this is a generalization of quantum mechanics and so one might look for a modified interpretation from the above discussed heretical quantum mechanics theories. The Bohm interpretation for quantum mechanics had some difficulty being converted into an interpretation of quantum field theory. I believe that if the readers will search on arXiv, they will find that some researchers, mostly German, have been working on this problem and that they report having some success with it.
But let’s take a quick look at the problem of alternative theories for quantum field theory more generally. If the new theory gives only the same equations as the old theory we really haven’t gained anything. To make real progress, rather than just plowing over the Copenhagen interpretation once again, we need to be able to produce equations that are not already predicted.
In my view, the most important unsolved problem in elementary particles is the origin of particle mass. For a generalization of QFT to also solve the particle mass problem, the particle masses have to be written in a form that is compatible with the methods of QFT. And this is why I’ve worked on generalizing the Koide mass formula.