Fundamental Constants in Physics

Everybody makes use of constants in everyday life, quite often without even being aware of it; we use constants as a set of references in which we base our relationship and communication with other people. Let me clarify the previous sentence with two examples.

When we return home we expect to find it always in the same place, that is, we expect its location to be constant. Because the location does not change from day to day it is a temporal constant but actually it is also a constant relative to the different people who may wish to seek us, because we express our house's location through a door number and we give that number to all those who want to visit us; can we imagine the confusion if the door number was different
for each visitor? Our door number is thus a constant which we use as reference, allowing us to convey the information of a constant location.

Let us examine a different example; suppose someone states that a worker's monthly pay is proportional to the time he worked during that month. That person will be implying a mathematical proportionality relation between the number that expresses the worker's pay, in pounds, say, and the number of work hours in the corresponding month. Implicitly she will be saying that the result of dividing pay by work time is always the same, whatever the month;
that number is a proportionality constant. Everyday language is less precise than mathematical language and it is acceptable to say that two numbers are proportional even when the result of their quotient fluctuates. In this essay I want to look at constants used in physics, trying to identify those that are truly fundamental, that is, those that we have to accept without justification independently of the advancement of science. Some of the ideas I express further along are highly controversial and I warn the reader to take them with caution, making a confrontation with the mainstream posture in the scientific community.

If we use physics to solve real everyday problems, the question of which and how many constants are fundamental becomes irrelevant. For instance, we know exactly what calculations are needed in order to find the energy for heating a room and those calculations are independent of the fact that some constants may be fundamental and others just convenient. Even in relativistic or quantum mechanical problems it does not matter if the constants are fundamental or not. The question becomes meaningful only for the unification of physics, that is, when we try to place all branches of physics under the umbrella of a single theory. Someone wrote that physics will be mature when it can be condensed in a single formula to be warn on the front of a T-shirt, something like E = m c^2.

It is important to fully understand this objective because it does not mean that the unifying formula will spell the end of physics and certainly not the end of other sciences. I usually make an analogy to clarify the unification objective. We all make use of word processors, it is indifferent whether we work with Microsoft Word or LaTeX, and we associate to those programs an identity independent of the particular computer where they are running. Computers differ from each other but they are all essentially based on electrical currents which are turned on or off; these can also be assigned to 0s or 1s of a binary algebra. Certainly we are not close to understanding the how the word processor works just because we know the binary algebra or the currents flowing in the integrated circuits; this is because those programs exist on a different level, which becomes manifest if we realize that they can exist in different computers without loosing their identity. In a similar way the different branches o physics and the various sciences exist on a higher level of reality and their development can and must be carried on, independently of what may become the language of physics at a fundamental level.

All physics' theories are based on principles, fundamental truths that we don't question and whose validity is established when the theories predictions are confronted with experimental results or observations of the world. Fundamental constants' values are also accepted without justification; they are part of the theory's essence. When several theories compete for the explanation of a given set of observations, it always happens that one of them becomes prominent because it makes better predictions or is applicable to a wider set of phenomena. Einstein's relativity dethroned Newton's dynamics because while both theories predictions are equally effective when low energies are involved, relativity produces far more accurate predictions for high energy experiments. One always expects any new theory to include or revert to the existing ones in those cases where these produce good predictions.

At present physicists have to reach for different theories depending on whether they are dealing with cosmology and astrophysics phenomena, interaction between atoms or elementary particles. The three main theories, general relativity, quantum mechanics and the standard model of particle physics coexist, not always peacefully. Actually it is only on everyday physics, what we will call human scale physics, that there are no uncertainties about which theory must be used. Here Newton's dynamics and Maxwell's electromagnetism are the rulers and one can be certain that any new theory will have to incorporate them. One should say, however, that quantum physicists are perfectly pleased with quantum mechanics and astrophysicists are equally happy with general relativity; things get more difficult when the two theories have to come together in order to build models of primordial Universe. A unified theory of physics, when it exists, will have to be applicable to the whole Universe, from elementary particles to the cosmos, and to all times, from the beginning to the present time, allowing predictions for its evolution. Physicists are not only concerned with the validity of their theories but also with their elegance, a vague concept which has to do with having a small number of founding principles; this means that a theory is all the more elegant the smaller the number of its principles. It is in the scope of a search for elegance that determining which and how many fundamental constants there are finds its reason.

First of all I would like to remove from the list of fundamental constants a large set of constants relative to laws of approximate proportionality. For example, I know that the force needed for pushing a chair is approximately proportional to the velocity of the chair. Putting it differently, if I divide my force by the chair's velocity I get always approximately the same number, which I
designate by friction constant. However everybody understands that there are so many variables in a friction problem that reducing it to a simple proportionality law can only be a rough approximation. This type of constants, constants of convenience, are typically known with low precision, are usually written with no more than one decimal place and are usually dependent on the particular units' system that is used. Those constants are the result of our inability to deeply understand the phenomenon in question or else the need to simplify a complex set of relations among all the variables involved. They are a practical application of Taylor's theorem, familiar to all physicists and mathematicians, and could theoretically be eliminated by the use of exact formulas. Those constants cannot qualify as fundamental.

Some constants in physics are associated with relations of exact proportionality, unlike the precedent case; one example is the proportionality law between the mean energy of a gas' molecules and its temperature; this law involves the Boltzman constant. For those less familiar with the subject it is sufficient to say the the molecules in a gas are in constant motion, which can be measured either by the gas temperature or by the molecules' mean energy. Actually temperature and mean energy in this case measure exactly the same thing and one wanders why one is expressed in Kelvin and the other in Joules. The main reason for this state of affairs is tradition; statistical mechanics had to be invented for us to understand that measuring temperature was indeed measuring energy and by then there were two different units for the same thing. It means that Boltzman's constant is essentially a units conversion factor. Contrary to those we called constants of convenience it is known with high accuracy; we can however do without it, for that sufficing to express temperatures in Joules. We don't do it ordinarily because we might no be understood by a large fraction of the population, but we do it frequently when writing equations. Since they are units conversion factors, all constants like this have their value dependent on the units' systems that are used; Boltzman's constant has different values when expressed in Joules per Kelvin or in Joules per degree Fahrnheit.

In human scale physics, that is leaving aside sub-atomic world and the cosmos, one usually considers four fundamental constants; if we consider also those areas the number grows to over twenty. The four fundamental constants are: speed of light in vacuum, universal gravitational constant, which appears when evaluating the attraction between two masses, Planck's constant, relating a photon's frequency to its energy, and one constant chosen among electrical entities, for which we can elect the proton's electric charge. All these constants are known with great accuracy and their values are units' system dependent; the main question is: are they really fundamental or, just like Botzman's constant, they are the result of parallel developments of our knowledge, which led us to define different units for things that are not essentially different?

Let us look first at vacuum speed of light, that every one recalls being 300,000 Km/s. This constants allows us to convert a time interval into the distance light travels during that time and so, if needed, we can measure length in seconds or years. One usually writes light-seconds or light-years to stress that distance is meant but these are not different seconds or years; they are exactly the same thing. A light-second is a second but we add the light qualifier to mean we are referring to a 300,000 Km distance. With Einstein and Minkowski distance and time measurements became undistinguishable; distance and time concepts are intimately related and don't allow a distinction in terms of units. In relativity work it is common practice to measure time and distance with the same units, eliminating the vacuum light speed constant. I am convinced that since the formulation of special relativity theory that is no longer a fundamental constant.

Can we do a similar thing with the other fundamental constants? As we said we have to consider four fundamental constants and significantly we have also four fundamental dimensions in any units' system. The choice of these may fall, for instance, in length, time, mass and electric charge. Operating the four fundamental constants among themselves we can build standards for each of those dimensions; in doing so we make our measurements independent standard meters or atomic clocks, because our standards are all built with constants that we consider fundamental and immutable. This procedure allows, in principle, the use of a single standard for all measures, time, length, mass, etc., since we can convert among them through the fundamental constants. The single standard purpose is the establishment of a universal scale factor, so that we all use the same number to express the same measurement. Measurements made in this way don't need units because they are all expressed in the same unit; they can be called non-dimensional; at the same time all fundamental constants are eliminated because they become unity. Such units' system is usually designated by Planck or natural units but most of all I would like to stress that we can apparently do away with all the fundamental constants.

It is not really like that. What can we say about numbers pi or Neper? The former pops up unavoidable every time we relate the perimeter of a circle with its diameter and the latter is similarly unavoidable although in less common situations. There is no units' system manipulation capable of changing them, quite simply because they are the result of mathematical relations and thus they are non-dimensional. These constants must not be called fundamental because their value can be obtained mathematically and does not need to be known a priori. There is at least one non-dimensional constant in physics, which is thus units' system independent; this is the fine structure constant, which we find in the interaction between two electrons. This is truly a fundamental constant of physics, unless someone someday discovers a mathematical series with this point of convergence; then its value will cease to be a postulate and this will become another number like pi.

What can we say about the pléiade of constants in elementary particle physics? The present state of knowledge in this area is so incipient that we can reasonably hope the fall off for several of them in the years to come, as the theory evolves. Something similar can probably happen in cosmology, where constants such as the critical density of the Universe and the cosmological constant have all the looks of temporary answers to the insufficiencies of present science. We hope that on its coming of age physics will dispense with all the constants, exception made for the non-dimensional ones; these may eventually become mathematical, leaving physics completely devoid of fundamental constants. If and when this vision becomes real, physics will become a set of mathematical theorems and relations, derived from a small number of axioms; these will be the first principles of physics. Among the principles that will remain, one must certainly establish the number of space dimensions where physics is developed; will the four dimensions of general relativity theory be enough or will we need eleven dimensions, as super-string theorists advocate? Most scientists today think that we will need eleven dimensions but a small group of people aims to prove that five dimensions will suffice; it seems definitely clear that we will need to go beyond four dimensions.

What do physicists mean when they speak of multidimensional universes? Our senses and our instruments tell us about three dimensions to which we have grown used to. Everybody knows that three numbers are needed in order to place a point relative to the Earth: latitude, longitude and altitude; this is what we mean when we say the physical universe is three-dimensional. Einstein add time to these three dimensions; this dimension is notably different from the other three, at least in the way we perceive it. In fact time is different even for the relativity theory; we can never go back in time, for instance. Apparently there is no place for new dimensions because we have no direct perception for them nor do our instruments show evidence of their existence. If there are more than four dimensions, the additional ones must have distinguishing
characteristics to separate them from the three physical world dimensions and the time, so that we can be perfectly unaware of them. In super-string type theories the new dimensions are curled on themselves in such way that in the macroscopic world we live in they are impossible to detect. The concept is by no means intuitive and it is difficult to grasp but it provides motivation for a large number of scientists, just because it holds the promise of one day unifying the different areas of physics. For the small community exploiting the five dimensional possibility, the new dimension is detectable only at the cosmic level. This dimension is undistinguishable from the time in ordinary laboratory experiments but separates from it at the galaxy level and in experiments involving velocities approaching the speed of light.

It is generally accepted that the unification of physics cannot be achieved without a change of paradigm; the problem is that everyone knows a change is needed but nobody can say for sure which change will work. The revolution brought by Einstein is a change of paradigm in itself because prior to that no one could conceive an upper bound to velocity; the fact that the speed of light is an absolute upper limit to all velocities means that even if it were possible for someone to travel in a train close to the speed of light and fire a bullet inside the train, also close to the speed of light, the "sum" of the two velocities would still be below that limit. Einstein was responsible for other important concepts which together made a radical change to the way we see the Universe. In a parallel development the quantum revolution initiated by Planck produced another paradigm change, this time with relevance mainly for the microscopic world. Here one says that energy cannot change continuously but just by quantum leaps. One says also that there are no certainties but just probabilities in the quantum world. It is hard to imagine a paradigm change similar to any of those and nevertheless we will probably have to wait for a revolution of comparable caliber in order to place all physics under a single theory.

What is the purpose of a unified theory? Why are we concerned with having to change theories when we look at different problems, if the different theories are reliable within their domain? This is first of all an aesthetics question because an elegant theory of universal application will be a work of art capable of immortalizing its author. But there is more. By looking at an object from different viewpoints one gains perspective over that object; similarly, by approaching a problem in different ways one usually perceives details that are hidden in a single approach. Furthermore we are far from happy with the reliability of existing theories at the cosmic and elementary particle scales and so it is legitimate to hope that the forthcoming revolution will bring answers to some of the remaining problems.

Where do we stand then in respect of what is fundamental in Physics and in the Universe through it? There is no doubt that some concepts we use and associate to different manifestations and properties of reality are in such way related that they could be merged into a single one, removing several constants in the process. If this can one day be extended to particle physics and the cosmos is a question of faith; it is equally legitimate to believe or not believe. Because faith moves mountains, those who believe find in it enough motivation to proceed with their efforts.