Speculations on Physics V

Chris Cole

This is the fifth installment of my speculations about physics, which occur sporadically in Noesis. For the record, the previous installments have been:

        Noesis 43 (Nov 89): Schrodinger’s Cat and Bell’s Theorem
        Noesis 75 (Nov 92): Causality (with Dean Inada and Mike Price)
        Noesis 137 (Mar 98): Duality between Space, Objects, and Interactions
        Noesis 138 (Sep 98): Octonions (with Robert Low)

Einstein once said that the laws of mathematics were certain but that their application to the real world was uncertain. I interpret this to mean that it is unclear that any particular mathematical model represents the physical system to which it is applied. For example, the first thing any text on the mathematical basis of quantum mechanics will tell you is that the state of the system is an infinite dimensional vector in Hilbert space. What I want to focus on here is not the esoteric mathematics of infinite dimensional function spaces, but rather the mundane and rarely questioned mathematics of the integers.

Underlying Hilbert space, and pretty much any mathematical model of the real world that has been entertained by the bulk of the physics community in the last few hundred years, is the idea that space is continuous and infinitely divisible. The natural model that has been used for this space for the last hundred years or so has been the real numbers. The name alone tells the story of their importance. This is because the real number field is closed under the process of taking the limit. In other words, the limit of a series of real numbers is a real number (or does not exist at all). The real numbers form the smallest field with this property that includes the nontrivial mathematical operations, such as addition, multiplication, and division.

One problem with the real numbers is that they are not complete, in the sense that it is possible to write down an equation that has only real coefficients but does not have a real solution. One such equation is x² = -1. This of course led to the discovery of the complex numbers. It is worth noting that quantum mechanics cannot be formulated without the complex numbers. In fact, Feynman once claimed that all the mysterious consequences of quantum mechanics could be traced back to the mathematics of the square root. I take this to mean that since the wave function is a complex function, and the probability is the absolute value of the wave function, effects such as interference are inevitable.

There are objects such as vectors, spins, and rotations that cannot be modeled with complex numbers, but since the invention of linear algebra (i.e., matrix algebra), this difficulty has been ignored. Most physicists are only faintly aware that the quaternions can be used to model these physical quantities without the need for matrices, and that some quantities cannot be modeled with matrices since matrix multiplication is always associative. Non-associative algebras are considered beyond the pale by most physicists.

However, I do not want to move in that direction. I only cite the existence of these other number fields to point out that the real
numbers are not etched in granite. In fact, the real numbers have many peculiar properties, but I don’t want to focus even on that. What evidence do we have that these are a good model for physics?

The main property of the world that real numbers seem well suited to model is the property of continuity. There are many quantities that appear to be continuous in nature, such as position, velocity, acceleration, mass, temperature, etc. In other words, there is always a value of these quantities that lies between any two other values; if you have a temperature of 98.6 at one time and a temperature of 98.7 a little later, you must have had a temperature of 98.65 at some intermediate time.

However, the real numbers are overkill for this property. This version of continuity can be handled with the rational numbers alone. This is because there is a rational number between any two rational numbers. And most real numbers are irrational; in fact, the rationals can be counted whereas the reals cannot. This means that there are only as many rationals as there are counting numbers.

So why use the extra power of the reals? Mainly for convenience; it is not convenient that series of rationals do not always converge to a rational. But this convenience is not evidence; we have no evidence that there are more points in space than there are rational numbers, for example.

Once we start thinking in this direction, it suggests to me even a further question. If there may be only countably many points in space, then maybe there are only finitely many points in space.

Many philosophers think that Zeno showed that space is continuous. To recap: an arrow cannot fly because in order to move one inch, it must move one half inch; in order to move one half inch, it must move one quarter inch, and so on. The conclusion: you need to take the limit of an infinite number of motions in an infinite number of instants to give you a finite speed. You need to take limits to get the arrow to move. Ergo, space is continuous.

But actually, another logical possibility exists: namely, the arrow jumps discontinuously from point to point. Since there are only finitely many points, the arrow moves.

So, what about indirect evidence for continuous space? One kind of indirect evidence would be if the laws of nature are scale invariant, meaning that they are the same on any scale. And this is true, to some extent. However, there are certain constants of nature that have units like length, time, and mass. One of these is the speed of light. These constants provide a natural scale for the laws of nature. In fact, many physicists now believe that the Planck constant is related to the smallest unit of space, time, and mass.

Another kind of very indirect evidence would be if the axioms of set theory imply that there must be an infinite set. In other words, what if some very basic assumptions about the way the world works (i.e., the axioms of set theory) logically imply that there must be at least one set that contains infinitely many objects? But this is not the case. In fact, the standard Zermelo-Fraenkel axioms of set theory include a separate axiom called the Axiom of Infinity that explicitly states that
there is at least one infinite set. This axiom is independent of the other axioms.

What evidence do we have that there are infinitely many points in space? For that matter, what evidence could we ever have that there are infinitely many of anything? The most you can ever show is that you haven’t yet found the end of something. To show that something is actually infinite would require all of space and time to complete. Any finite number, no matter how large, is tiny compared to infinity.

So, the assumption that space is continuous is a matter of mathematical convenience. This assumption is perfectly innocuous as long as the phenomena we are investigating are large compared to the smallest unit of space, time, and mass. Or maybe not. If we get used to the mathematical convenience of continuity, we may not explore the mathematics of “approximate” continuity.

What kind of objects exhibit “approximate” continuity? I am not aware of any number systems with this property. There are a lot of unusual and unexplained phenomena in physics that occur at a very low level: CP violation, for example. Wouldn’t it be interesting if “approximately” continuous numbers exhibited “approximate” symmetries that mirrored some of these odd phenomena?