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Symmetry arguments in physics
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Stud. Hist. Phil. Sci., Vol. 30, No. 3, pp. 479–492, 1999 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0039–3681/99 $ - see front matter
Symmetry Arguments in Physics Peter Kosso* Physicists often appeal to the beauty of a theory as a way to judge its credibility, and the most prevalent component of this beauty is symmetry. This paper describes the role and structure of symmetry arguments in physics. It demonstrates that the epistemic authority of an appeal to symmetry is based on empirical evidence and is independent of any aesthetic judgment. Furthermore, symmetry in nature is not evidence of design. Just the opposite, symmetry indicates a lack of planning. It is about nature’s disregard for details. 1999 Elsevier Science Ltd. All rights reserved
It is more important to have beauty in one’s equations than to have them fit experiment. Paul Dirac
Dirac is an extremist on this issue of beauty and science, but respect for beauty is frequently and sincerely expressed by many physicists, sometimes in more specific terms such as symmetry, simplicity, or elegance. The purpose of this paper is to clarify the role of beauty, symmetry in particular, in physics. Is it beauty for its own sake that is, or should be, valued, or is beauty, like agreement with experiment, valued as a means to accepting theories that are more likely to be true? Does the beauty that Dirac values have a purely aesthetic role in physics, or does it have an epistemic purpose? Physics is essentially about simple objects and their interactions, that is, about particles and forces. Many of the most fundamental claims about these things are founded on principles of symmetry, the most predominant feature of what a physicist thinks of as beauty. Concerning forces, John Barrow (1988, p. 180) explains, ‘Today’s best working descriptions of all the known forces of nature . . . are predicated entirely upon symmetries.’ Concerning particles, the class of particles called vector bosons, things like photons and gluons that transmit the energy of interactions, show up in the theories primarily as ways of preserving symmetry. As Heinz Pagels (1986, p. 216) puts it, ‘gluons exist because of symmetry’. Symmetry clearly plays an authoritative role in theoretical physics. There are also contexts in which beauty, again in the form of symmetry, is
* Department of Philosophy, Box 6011, Northern Arizona University, Flagstaff, AZ 86011, U.S.A. Received 3 April 1998; in revised form 14 August 1998.
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suggested as a surrogate for experiment and empirical testing. In high energy physics, for example, hypotheses about the most basic constituents of matter and speculations about the beginning of time are beginning to exceed our ability to experiment. In place of experiment, beauty has become an authoritative criterion for accepting theories. A so-called Theory of Everything such as superstring theory describes events and objects at such high energy that experimental testing for the predicted particles is out of the question, perhaps forever. In a recent commentary in Physics Today, two particle physicists, Ginsparg and Glashow (1986, pp. 7–9), describe the situation this way. ‘We are stuck with a gap of 16 orders of magnitude between theoretical strings and observable particles, unbridgeable by any currently envisioned experiment.’ Without experiment, internal attributes of the theory rule the day, and ‘elegance, uniqueness and beauty define truth’ (Ginsparg and Glashow, 1986, p. 7). The description of superstring theory indicates an intended epistemic role for beauty. So does the account by Steven Weinberg (1992, p. 133) of state-of-the-art physics. In a chapter called ‘Beautiful Theories’, he describes ‘how the usefulness of our sense of beauty is a sign of our progress toward a final theory’. Given this epistemic intent for the beauty of a theory, we must ask the following important epistemological questions. Why should we think that beauty indicates truth? And how do we, that is, how do the physicists, evaluate beauty? Are there objective standards for telling a beautiful theory from an ugly one, or at least a more beautiful from a less? Is the assessment of beauty any more precise than a vague you-knowit-when-you-see-it? There have been interesting attempts to explicitly link beauty to truth, and thereby establish its epistemic credentials. They involve preliminary assumptions about the natural world. Anthony Zee (1986), for example, in Fearful Symmetry: the Search for Beauty in Modern Physics, assumes that the universe and all its parts were brought about by design. So when we describe the creation with elegance and symmetry, we capture the intent and the plan, and we describe it accurately. James Martin also advocates beauty as an indicator of truth. He too secures the link by making substantial metaphysical assumptions. Martin (1989, p. 363) presumes that the physical world is naturally harmonious and that we as humans have ‘a real but inevitably tacit access to reality’. Scientific knowledge, in this account, amounts to articulating our tacit understanding of things. Our sensibilities are innately connected to the harmony in nature, so when we manage to articulate a beautiful description, one that pleases the sensibilities, it is likely to be an accurate picture of the real harmony. Zee and Martin give epistemic value to beauty only by making these questionable metaphysical assumptions. It would be valuable to ask whether beauty can be used as a means to truth without the metaphysical commitment. Perhaps we can take some of the pressure off beauty by giving it somewhat less epistemic responsibility, less than the authority to grant justification and good rea-
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son to believe a theory. Beauty may be a good way to generate new ideas, an inspiration to creativity, but not a good way to test ideas or judge their truth. In this role, beauty is good reason to take a theory seriously, but not in itself good reason to believe the theory is true. And indeed, when we casually describe an experiment or some aspect of theorizing as being as much an art as a science, it is the setting up and getting things to work, or the initial inspiration of a hypothesis that requires finesse and creativity. The artistry is gone, though, from the standards used to prove that the experiment has worked or the hypothesis is worthy of belief. A good case for beauty in this suggestive, formative role is that of the subatomic particle ⍀−. Physicists knew of nine particles with properties that put them on a symmetric matrix, symmetric except for one gap, like an equilateral triangle with a missing corner. Under the influence of symmetry, physicists thought there must be a tenth particle in the set to fill the gap and complete the triangle. They noted the properties the particle would need to be in the empty place, and with this as a guide they looked for and found the tenth particle, ⍀−. Even in this proto-epistemic role, the influence of beauty is important and worth understanding. By telling us which experiments are worth doing, looking for ⍀−, for example, the initial criteria direct where huge investments of time and money are to be spent. Because of the large size and expense of experiments in high-energy physics, there is a lot of inertia in any theoretical program. Research programs are hard to get started and then hard to stop. Anything that influences the launching plays an important role in shaping the future of research. Furthermore, in light of Kuhnian arguments dissolving the distinction between contexts of discovery and justification, it would be naive to think that reasons to consider a theory are isolated from reasons to believe it. Once introduced, a theory might secure its own evidence. Besides, beauty is not restricted to a preliminary role of merely suggesting theories to test. Beauty is sometimes the decisive factor in justification. Weinberg (1992, p. 133) explicitly recognizes the distinction between suggesting and testing a theory, and just as explicitly puts beauty in a key role for both. He has the examples to show that ‘something as personal and subjective as our sense of beauty helps us not only to invent physical theories but even to judge the validity of theories’. This shows a high regard for beauty in physics, but endorsements and optimism alone cannot establish its epistemic authority. To find the source, if there is any, of this authority we have to see how beauty, in particular symmetry, is used in evaluating theories. 1. Beauty is Symmetry A snowflake is a thing of beauty. It is also a thing of symmetry. Each spoke displays a mirror reflection along its radial axis, and the whole crystal shows rotational symmetry by repeating the design exactly six times in one complete turn. Perhaps the beauty we feel is rooted in the geometric symmetry of structure.
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The link between beauty and symmetry does indeed exist in physics, and it is safe to say that beauty essentially means symmetry in that context. The theoretical description of particles that led to the discovery of the ⍀− is beautiful because of the symmetric matrix into which the particles fit. String theory, or in its most ambitious version, supersymmetric string theory, is most beautiful when it maximizes the symmetries in its description of nature. And this is hopeful news for beauty in physics, because symmetry is well understood and neatly codified in the precise mathematics of group theory. With beauty linked to symmetry, there is hope of an objective, group-theoretical way of deciding what is symmetric and what is not, or at least what is more or less symmetric. For this reason, and because it is true to the physicists’ concept of beauty, we will focus on symmetry and its role in physics. In understanding symmetry it is natural to think first of something visual that displays a geometric regularity. A snowflake is symmetric. The human body is roughly symmetric left-to-right, at least on the outside. An equilateral triangle is more symmetric than a scalene triangle. And so on. All of these things are symmetric because their appearance is unchanged after some distinct operation of physical change on the object. The equilateral triangle looks exactly the same even after being rotated by 120 degrees or after being reflected across a line that bisects one of the angles. The scalene triangle does not look the same after either of these operations. The human body looks the same after the mirroring operation of inverting left side for right. These geometric symmetries amount to an identity of appearance after some physical operation such as rotation, inversion, reflection, or translation. This is just the tip of the symmetry iceberg, but it leads in the right direction toward a general definition of symmetry. The most general concept of symmetry is this. Any operation that leaves some property unchanged is a symmetry. In the case of the snowflake, the property that is unchanged is its appearance, and the operation that preserves appearance is rotation by sixty degrees. There are other appearance-preserving operations on the snowflake, such as rotation by a full turn or reflection through an axis along any spoke, and these are all symmetries of the object. The concept of symmetry can apply to things, like a snowflake, or to phenomena, or to laws of nature. The phenomenon of gravity, for example, the attractive force between any massive objects, is independent of the color of the objects. Two blue balls exert a slight gravitational force on each other. If you switch one of the balls for a pink one (of the same mass and at the same position), the gravitational effect will be the same, exactly the same. When it comes to gravity, nature is indifferent as to the color of the objects. Gravity, in other words, is color symmetric. It is also flavor symmetric, smell symmetric, and a lot of other things symmetric. It is helpful to think of a symmetry as an ordered pair, an operation and an invariant of that operation. Thus, a snowflake has the symmetry of ⬍ 60° rotation, appearance > , meaning that the operation of rotating by sixty degrees, one-sixth of a full rotation, leaves the appearance unchanged. The snowflake has several
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other symmetries, and each of these could be characterized by an ordered pair. The phenomenon of gravity has the symmetry ⬍ change the color, force > , indicating that the force between two objects is unaffected by a change in the color of one or both of the objects. And again, gravity has other symmetries that could be expressed as ordered pairs. The color symmetry of gravity is by no means amazing. It’s not even worth noticing or mentioning except as an easy way to come to terms with the concept of symmetry. But having noticed color symmetry, we are now in a position to insist that any theory of gravity, any articulation of the laws of gravity, must be color symmetric. That is, the equations describing the phenomena of gravity must remain the same if the color variables are changed. Newton’s law of universal gravitation and the general theory of relativity both fulfill this requirement by default. Neither has any color variables at all in its equations. Nonetheless, color symmetry is a necessary condition of acceptability of a theory of gravity. It is not a very strict condition, since it is almost trivially easy to meet, but we are on the way to seeing how beauty in the form of symmetry plays an epistemic role in choosing theories. With the very general characterization of symmetry as invariance under some operation, the law-like characteristics of universality and uniformity are themselves symmetries. Uniformity over time, the claim that the laws of nature are the same now as they were in the past, is a symmetry of the form ⬍ different time, form of the law > . For example, the uniformity of Newtonian mechanics, in particular the second law, can be expressed as the symmetry ⬍ different time, F ⫽ ma > . The physical objects in nature lack this symmetry over time. People age, the Grand Canyon deepens, and so on. Observe these objects at a different time and things will be different. But the laws stay the same. In fact, it is the uniformity of laws, that is, their time symmetry, that allows us to tell just how old something like the Grand Canyon is. Laws are the same at all times; they are also the same at all places. The universal aspect of Newton’s law of gravity has its empirical origin in the discovery that the force between the earth and an apple can be described by the same law as describes the force between the earth and the moon. Gravity, in other words, is terrestrial–celestial symmetric. Similarly, if F ⫽ ma is true anywhere, it is true everywhere. This is a symmetry of the law. It means that you cannot use the laws by themselves to tell where you are in the universe. The contingent properties of the objects, their masses, for example, vary from object to object. Different things will be accelerating at different rates, depending on where you are, and they will be under the influence of various forces. But wherever you are, the different masses, forces, and accelerations will always be related in the same way, namely F ⫽ ma. The law is invariant over positions, and this is a symmetry. The great generality in the concept of symmetry makes it nothing very special. It also makes room for subjectivity, an element of personal choice, in the justifi-
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cation of theories. To see this, realize that any system whatsoever, and any theory or model or law of nature, will be symmetric in some way. All this requires is that some property be left unchanged under some operation. It may be a trivial symmetry in the sense that the property is not even specified by the theory and so changing the property is irrelevant to the theory. This was the case of the color symmetry of gravity. Some of the most important and informative symmetries in physics are trivial symmetries in this sense. The equation describing the acceleration of an object in response to a force has no variable for the absolute position of the event. It does not matter whether it happens in New York or San Francisco, the law is the same. This position symmetry is trivially fulfilled by the law, but it is onto something of fundamental importance, namely the nature of space itself. The position invariance of laws, their location symmetry, is related to the observation that we are unable to detect absolute position in space. We can only observe the position of something relative to something else. This observation leads to the requirement of position symmetry of all laws of nature, including the relation between force and acceleration, and we are on the way to the theory of relativity. Symmetry in all its generality is a very weak necessary condition for theorizing, too weak to be of any use. If symmetry considerations are to be of any epistemic value they must be specific. Since every theory is symmetric in some way or another, the only meaningfully strict symmetry requirements on theorizing will be those that specify a particular operation, a particular symmetry. We might demand translation or rotation symmetry of our theories, for example. So, the question to ask in evaluating a new theory is not simply, Is it symmetric? We have to ask, Is it symmetric under such-and-such specific operation? Then the epistemological question is focused to ask why symmetry under this particular operation is a necessary condition of a true description of nature. Why should a theory have this symmetry and not some other? The need to be specific shows the lurking possibility of subjectivity in symmetry arguments. If every theory is symmetric in some way, then symmetry is wherever we choose to find it. Why can’t we all just choose our own favorite symmetry operation and then use it, insist on it, to judge other ideas? Given the ubiquity of symmetry in general, how can symmetry discriminate between good theories and bad? Everything is beautiful, in its own way. Here is a graphic example of the flexibility in the concept of symmetry. A snowflake, a crystallized state of water, has been my paradigm for symmetry and beauty in nature. The contrast would be water in an amorphous state, a vapor of molecules in a nondescript cloud, a random distribution that has no distinguishable parts or patterns. Nobody uses the cloud of vapor as an example of symmetry, but they could. While the snowflake has obvious rotational symmetries, the random cloud of molecules has a translational symmetry. If the cloud of molecules is entirely homogeneous, such that there is no way to tell one point from another, then there
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is no distinguishing the pattern as it is from a pattern that is shifted a bit to the left or to the right. The crystal, the snowflake, lacks this symmetry of translation. With each molecule precisely located in the crystal, we can detect a change of position in this system.1 This is not to say that the laws of nature governing the crystal will lack the translation symmetry. They will certainly be the same in one spot as the next. It is the appearance of the system itself that is translation symmetric in one case (the cloud of vapor) and is not translation symmetric in the other (the snowflake). The point is that we cannot say simply that the snowflake is symmetric and the amorphous cloud is not. Each system has its own symmetry. I am resisting the even more counterintuitive conclusion from this comparison between snow and clouds, that the amorphous cloud is even more symmetric than the crystalline snowflake. Here is what tempts me. The cloud of vapor has the translation symmetry that the snowflake lacks, and the cloud has the rotational symmetry that drew us to the snowflake in the first place. In fact, the cloud has more rotational symmetry than the flake. Its utter lack of precision in the placement of molecules allows rotation by any angle whatsoever to preserve its appearance. The snowflake must be rotated by exactly sixty degrees, or some multiple of sixty. But, if the cloud is isotropic such that there is no distinguishing axis and no way to tell one orientation from any other, then any rotation leaves the cloud unchanged. The cloud has an unlimited variety of continuous symmetries, continuous operations that leave the system unchanged, while the snowflake has only a few discrete symmetries. But I am reluctant to declare the amorphous cloud the more symmetric system. There is no reason to think that there is an absolute count of symmetries, given the tremendous freedom in choosing the operation and the invariant in the ordered pair. Surely there will be symmetries in the snowflake that are missing in the cloud. The cosmological principle, an observational foundation of modern astronomy, is another example of a symmetry in randomness. The universe appears to be a roughly homogeneous distribution of galaxies. On the large scale, one place in the universe is essentially indistinguishable from another. In other words, matter distribution is location symmetric. If there were some pattern in the distribution, the universe would lack this symmetry. In the Big Bang model of the universe, there is a pattern in time, though not in space. Over time, galaxies are getting further apart. The distribution of matter is space symmetric in this model, but not time symmetric. In the Steady-State model, on the other hand, there is no beginning of the universe and matter is continuously created to maintain the same density in space. In this model, the universe is both space and time symmetric. The examples suggest that in some way the overall symmetry of a system increases as the system becomes more disordered. A system in total disorder is 1 G. Nicolis and I. Prigogine even describe the process of freezing as an event of “symmetry breaking”, that is, of losing the symmetry in the transition from liquid to solid crystal. See Nicolis and Prigogine (1989, p. 41).
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one with no specified positions, spacing, or trajectories of its parts. Change any of these parameters in any way and it is still a system in total disorder. The system is unchanged. The correlation between symmetry and disorder is imprecise without a concept and a measure of the degree of symmetry, but Joe Rosen (1995, p. 150) clearly endorses the idea that the extreme of disorder, if only a hypothetical state, corresponds to the extreme of symmetry: ‘Total disorder is maximal symmetry.’ This makes sense, given the general definition of symmetry, but it is the opposite of the original intuition that associated symmetry and beauty with order and precision. It is certainly opposite to Zee’s link between symmetry and design. Symmetry cannot be evidence of careful planning and meticulous placement of things or implementation of laws. It is just the contrary. Symmetry shows where there is a disregard for details, where things are at least in some way haphazard. Symmetries reveal what doesn’t matter in nature, what can be altered to no effect. Symmetries are about nature’s indifference to the details of change.2 2. Substantive Symmetry Arguments A symmetry is an invariance under some operation of change. How is a symmetry, a specific symmetry, used as an epistemic tool? That is, how can a criterion of symmetry be used to direct theorizing in the general direction of the truth? Of course, it will not be fool-proof, but perhaps it can better the odds. Bas van Fraassen (1989, p.242) distinguishes between substantive symmetry arguments and ‘symmetry arguments proper’. It is the former, the substantive arguments, that are of more epistemological interest. Symmetry arguments proper involve exploiting a symmetry of a theory, a model, or a physical system, as a way of facilitating solving problems. Proper arguments are analytic in the sense of unpacking the details and implications of a symmetry already known to be a feature of the system. These are indeed common in physics, and symmetry is an efficient way of revealing the features of a new or complicated theory. But these arguments make no epistemic progress. As van Fraassen (1989, pp. 260–261) puts it, ‘Symmetry arguments [proper] have that lovely air of the a priori, flattering what William James called the sentiment of rationality. And they are a priori, and powerful; but they carry us forward from an initial position of empirical risk, to a final point with still exactly the same risk.’ Advancement in science demands taking risks, and to find the epistemic value of symmetry in physics, if there is any, we have to deal with substantive symmetry arguments. These amount to the empirically risky claims that nature in fact has a particular symmetry, and the enforcement of that symmetry as a necessary condition for an acceptable theory. This is the initial risk that van Fraassen talks about, and it is a prerequisite to a symmetry argument proper. We have to know the 2 Joe Rosen (1990, p. 291) uses similar words, ‘The physical significance of symmetry of evolution is that nature is indifferent to certain aspects of physical systems.’
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symmetry is there before we use it as a tool of inference, and the claim that the symmetry is there is where the epistemic action takes place. It is certainly this substantive sort of argument that Ginsparg and Glashow (1986, p. 7) are talking about in their assessment of string theory, where ‘elegance, uniqueness and beauty define truth’. In other words, beauty in the form of a specific symmetry is thought to be an aspect of nature, so a true theory must have this symmetry. A good example of symmetry arguments in physics, both proper and substantive arguments, is Einstein’s principle of relativity as it functions in the special theory of relativity. The principle of relativity states simply that the laws of physics are the same in all inertial reference frames. In other words, at different places in the universe, and in reference frames going at different uniform speeds, the laws will be the same. This is clearly a symmetry in the laws of physics. The fact that the laws of physics are the same in an earth-bound laboratory as in a speeding airplane, the fact that you cannot use the laws of physics alone to detect which of those reference frames you are in, and the fact that we cannot detect any uniquely privileged reference frame that is stationary in an absolute space, are all expressions of the same symmetry. The principle of relativity was not a new idea with Einstein. His contribution was to stick to it as no one had before, forcing the form of the laws of nature to fit the symmetry. This is a substantive symmetry argument that we will look at in a moment. First there is Einstein’s symmetry argument proper. The proper argument uses the symmetry, the principle of relativity, together with relevant background knowledge, to deduce the details of the theory. The principle of relativity, together with Maxwell’s equations as the laws of electricity and magnetism, are enough to derive the Lorentz transformations and other consequences for the special theory of relativity. Length contraction, time dilation, and E ⫽ mc2 are all consequences of symmetry arguments proper, using the principle of relativity as the specific symmetry. These arguments are now standard exercises for students of physics, great pedagogical tools for using symmetry to unpack the theory, but they were amazing in Einstein’s time. They were amazing both for their surprising results and for their insightful technique for revealing the consequences of a theory. With the details of the theory articulated, in particular with the Lorentz transformations down on the page, the special theory of relativity and its foundational symmetry in the principle of relativity are used as a standard of acceptability for other theories. This is a substantive argument. The symmetry argument proper allows us to express inertial reference frame symmetry in terms of Lorentz invariance. Lorentz invariance is thus a necessary feature of all physical theories. This criterion becomes a way to both discover and test new theories. In Einstein’s words, ‘This [Lorentz transformation] is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature.’ (Einstein, 1995, p. 43). The case of the principle of relativity is characteristic of how substantive sym-
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metry arguments work generally in physics. Understanding the general form of these arguments is the key to answering the central epistemological question of how beauty, in this case symmetry, indicates truth. First note that there is no argument that relies on a link between symmetry in general and truth. The arguments are always about some specific symmetry, such as Lorentz invariance, or, as we will soon see, gauge symmetry. The arguments have to be specific in this way, since every theory and every system is symmetric in some way. So the good news is that we do not need to establish a connection between symmetry in general and truth. The bad news is that we will need a link to truth individually for any specific symmetry invoked. This (the good news) means we can avoid the grand and dubious metaphysical assumptions about design and natural harmony that were the basis of Zee’s and Martin’s arguments, respectively. The pattern of a specific symmetry argument in physics follows three basic steps. First, a symmetry is observed in nature. This is how relativity got started, with the observation that the laws of physics, as far as we can tell, are in fact the same in all inertial reference frames. This is not a guess or an assumption or the expression of some aesthetic predilection. It is the result of riding on a train and seeing that everything behaves while the train is moving exactly as it does while the train is stopped at the station. We observe this invariance under the operation of changing reference frames. The second step is the symmetry argument proper, the deduction of the consequences of this particular symmetry and the full nature of the invariance. The third step is the risk, the generalization that the observed symmetry applies to relevantly similar, unobserved systems as well. From the observation of the invariance of the laws of physics in the reference frames we have experienced, we infer that all laws of physics are invariant in all inertial reference frames. The third step, the substantive part of the argument, is an argument by analogy, extending what is observed to analogous systems that are not observed. It bears the standard strengths, risks, and limitations of arguments by analogy. It is fallible and subject to revision. Its strength is dependent on knowing which particular symmetries can be generalized, and to what sort of systems. Like all arguments by analogy, these substantive symmetry arguments are evaluated in terms of the amount and variety of evidence for the particular symmetry, and in terms of the similarities between the circumstances in observed and projected cases of the symmetry. The case of the principle of relativity fits this pattern of argument by analogy. We observe that the laws of mechanics are the same in all inertial reference frames. In accelerating reference frames, on the other hand, relations such as F ⫽ ma apparently need to be revised by the addition of inertial forces. On the basis of this evidence we endorse the analogy between inertial reference frames in which we have made measurements and inertial frames in which we have not, and the symmetry is extended to describe transformations between inertial reference frames
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in general. But we do not endorse the analogy to non-inertial reference frames, and the symmetry is explicitly limited in this way. This establishes the Lorentz symmetry for laws of mechanics, but Einstein’s principle of relativity is more general than this. It applies to all the laws of physics, and this generalization is the result of another argument by analogy. The laws of electricity and magnetism are relevantly similar to the laws of mechanics in that the reference frame invariance of the latter must be a feature of the former. The particular symmetries that are at the forefront of foundational physics these days fit into arguments with this same basic structure. Given the deeply entrenched status of the special theory of relativity, Lorentz invariance, the symmetry of the principle of relativity, is at least a prima facie necessary condition on any new theory. But we need lots of necessary conditions in order to weed out the false theories and zero in on the true. This means we need to discover additional specific symmetries and to demand not simply more symmetry in the new theories but specifically these symmetries. The discoveries of which particular symmetries to apply are rooted in observation, observing what does not change under particular operations. Gauge symmetry is particularly important in high energy, theory-of-everything physics. It has ‘taken command of elementary particle physics’ (Aitchison and Hey, 1982, p. 179). Gauge symmetry begins with the observation that electrostatic effects between two objects or points in space depend on the potential difference between the two but not on the absolute value of the potential. As an example, if a powerline has an electrical potential of a million volts, it’s a danger. It can spark or electrocute something on the ground with zero electrical potential, but it is harmless to something else that is at the same million-volt potential. The electrical effect requires a potential difference. And the electrical effect will be the same between a wire with two million volts and an object at one million, as it will be between the one million and something at zero. All that matters is the difference. It is analogous to mechanical laws that depend only on relative distance and speed between objects, not on any absolute position or speed in space. The point is that you can set your electrostatic potential to zero at any point and then measure the differences in values from that point. Call the one-million volt wire zero, if you like, and then the two-million volt line will be rescaled to one million, and the previous zero will be negative one million. With these values, all the laws will be unchanged. It is just like the freedom to put the origin of a coordinate system anywhere you like without affecting the mechanical laws of nature. You also have the freedom to choose the scale, the units of measurement. Shrink or expand the scale, the gauge, of measurement and the laws are unaffected. The rescaling of the electrostatic potential is a global transformation in the sense that the change is uniform throughout space. This global symmetry is apparent in the laws of electrostatics, and it is related to the conservation of electric charge.
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When electric charges move, that is, when the phenomena become dynamic, electric and magnetic effects become interrelated. In the unified theory of electromagnetism, the gauge symmetry is local. The change of scale of the electromagnetic potential can differ from place to place. The rescaling is a function of position and time. The uneven changes in the electrostatic potential are just compensated by the corresponding changes in the magnetic vector potential so that there is no change at all in the combined electromagnetic laws. This symmetry under local gauge transformation is what is called gauge symmetry, and a theory that is gauge symmetric is a gauge theory. The field of interaction that compensates for the space–time dependent rescaling, in this case the electromagnetic potential, is called the gauge field. The gauge symmetry, that is, the local gauge symmetry, of electromagnetism is apparent in the theory itself. It is manifest in the Maxwell equations. It is not an imposition from any theoretical or aesthetic predilection, but a direct result of the discovery that electricity and magnetism are manifestation of a unified electromagnetic phenomenon. Though the gauge symmetry was discovered from the Maxwell equations, physicists like to point out that the logic can be reversed. From a requirement of gauge symmetry (and Lorentz symmetry) the Maxwell equations can be derived. In this way, the symmetry condition can be used to generate a theory of interactions, and the technique can be extended to find new theories of other interactions. To understand the nature of nuclear interactions, for example, ‘We now mimic as literally as possible, the discussion of electromagnetic gauge (phase) invariance.’ (Aitchison and Hey, 1982, p. 185) The theory of electromagnetism is a local gauge theory. Similarly, the general theory of relativity is based on a local symmetry.3 The Lorentz symmetry that characterizes the special theory of relativity is a global symmetry in that each inertial reference frame extends uniformly throughout spacetime. The effect of making the symmetry local, that is, making the coordinate systems variable in space and time and insisting that the laws of nature have the same form in all of these different systems, changes the theory into the general theory of relativity, and, as in a local gauge theory, there is a resulting field of interaction. This field is gravity. Two fundamental interactions, electromagnetic and gravitational, are seen to be locally symmetric. By analogy to electromagnetism, other fundamental interactions are thought to be locally gauge symmetric. Without knowing the laws of nuclear interactions, the condition of (local) gauge symmetry is imposed from the start to derive theories of interactions with new gauge fields. Gauge theory has ‘taken
3 For insight into the differences and similarities between the symmetries of electrodynamics, general relativity, and special relativity, see Wigner (1967) ch. 2.
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command’ of fundamental physics, but only on its empirical authority from electromagnetism. The power and elegance of the symmetry arguments sometimes allow us to lose sight of their empirical foundations. So do some of the claims of the theoreticians. At one point, John Mauldin (1986, p. 221) makes it sound too good to be true: ‘Physicists can now deduce a gauge theory such as Maxwell’s electromagnetism without knowing any experimental evidence about electricity and magnetism,’ and ‘working physical laws can be pulled out of a mathematical hat’. This certainly flatters the sentiment of rationality, but only by overlooking the empirical basis of the symmetry. We need to remember, as it says in an often-used textbook on subatomic physics, that ‘Experiment is the only judge as to whether a symmetry principle holds.’ (Frauenfelder and Henley, 1974, p. 154) And Mauldin (1986, p. 221) points out that the hat trick with gauge symmetry is licensed ‘with hindsight from using gauge symmetry’, that is, with evidence of gauge symmetry in more familiar systems and laws. This crucial link to observation is the first of the three steps in a substantive symmetry argument. We have observed repeatedly that electromagnetism is in fact gauge symmetric, and so there is empirical reason to believe that this particular symmetry is an aspect of nature. To continue with Mauldin’s metaphor, working physical laws can be pulled out of a mathematical hat only because gauge symmetry has been put into the hat already, and put there on the authority of observations. 3. Conclusion The case of gauge symmetry is similar to that of the principle of relativity. Together they point to general conclusions about substantive symmetry arguments in physics. These arguments make no assumptions or inferences about symmetry in general. They are always about some specific symmetry or other. Furthermore, these arguments do not simply assume that the specific symmetry is a fact of nature, and then use it as a standard for judging theories. They begin with observation. Sunny Auyang (1995, p. 102) puts this point about symmetry in Kantian terms. General ideas of symmetry and the group-theoretic way of thinking are ‘our construction’. Symmetry is a physicist’s way of categorizing nature. But knowing what specific symmetries are true of nature ‘must be determined empirically’ (Auyang, 1995, p. 93). Once we see how symmetry is understood and used in physics, it seems misleading to associate symmetry with beauty. This is not to say that it is wrong or counterproductive to appreciate the aesthetic appeal of something like gauge symmetry or the principle of relativity, and I do not mean to discourage finding beauty wherever we can. But aesthetic value is entirely gratuitous to the epistemic role. It is not that physicists find gauge symmetry to be beautiful and on the basis of that beauty they believe it to be a real feature of nature. It is rather that, for mundane empirical reasons, physicists believe this particular symmetry is an aspect of nature, and, by
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the way, they also find it to be a thing of beauty. A tidy accountant might find a neat, orderly ledger of numbers to be a thing of beauty, and that’s fine. But what really counts is that the arithmetic is done properly, and that is independent of the beauty. The specific symmetries used in substantive arguments in physics are discovered; they are not artifacts of our aesthetic judgment. They are invariant aspects of observed phenomena. The only subjective influence is in using the word ‘beauty’ to describe symmetry. Furthermore, symmetry arguments do not lead to claims about design or planning or harmony in nature. If the metaphor of a personified nature is appropriate at all, then symmetries reveal nature’s indifference rather than careful attention. The role of symmetry in physics amounts to discovering nature’s indifference. References Aitchison, I. and Hey, A. (1982) Gauge Theories in Particle Physics (Bristol: Adam Hilger). Auyang, S. (1995) How is Quantum Field Theory Possible? (New York: Oxford University Press). Barrow, J. (1988) The World within the World (New York: Oxford University Press). Einstein, A. (1995) (originally published 1920) Relativity (Amherst: Prometheus Books). Frauenfelder, H. and Henley, E. (1974) Subatomic Physics (Englewood Cliffs: Prentice Hall). Ginsparg, P. and Glashow, S. (1986) ‘Desperately Seeking Superstrings?’, Physics Today 39, 7–9. Martin, J. (1989) ‘Aesthetic Constraints in Theory Selection’, British Journal for the Philosophy of Science 40, 357–364. Mauldin, J. (1986) Particles in Nature (Blue Ridge Summit: Tab Books). Nicolis, G. and Prigogine, I. (1989) Exploring Complexity (New York: W. H. Freeman). Pagels, H. (1986) Perfect Symmetry: the Search for the Beginning of Time (New York: Simon and Schuster). Rosen, J. (1995) Symmetry in Science (New York: Springer Verlag). Rosen, J. (1990) ‘Fundamental Manifestations of Symmetry in Physics’, Foundations of Physics 20, 283–307. van Fraassen, B. (1989) Laws and Symmetry (New York: Oxford University Press). Weinberg, S. (1992) Dreams of a Final Theory (New York: Pantheon Books). Wigner, E. (1967) Symmetries and Reflections (Bloomington: Indiana University Press). Zee, A. (1986) Fearful Symmetry: the Search for Beauty in Modern Physics (New York: MacMillan).
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Stud. Hist. Phil. Sci., Vol. 30, No. 3, pp. 479–492, 1999 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0039–3681/99 $ - see front matter
Symmetry Arguments in Physics Peter Kosso* Physicists often appeal to the beauty of a theory as a way to judge its credibility, and the most prevalent component of this beauty is symmetry. This paper describes the role and structure of symmetry arguments in physics. It demonstrates that the epistemic authority of an appeal to symmetry is based on empirical evidence and is independent of any aesthetic judgment. Furthermore, symmetry in nature is not evidence of design. Just the opposite, symmetry indicates a lack of planning. It is about nature’s disregard for details. 1999 Elsevier Science Ltd. All rights reserved
It is more important to have beauty in one’s equations than to have them fit experiment. Paul Dirac
Dirac is an extremist on this issue of beauty and science, but respect for beauty is frequently and sincerely expressed by many physicists, sometimes in more specific terms such as symmetry, simplicity, or elegance. The purpose of this paper is to clarify the role of beauty, symmetry in particular, in physics. Is it beauty for its own sake that is, or should be, valued, or is beauty, like agreement with experiment, valued as a means to accepting theories that are more likely to be true? Does the beauty that Dirac values have a purely aesthetic role in physics, or does it have an epistemic purpose? Physics is essentially about simple objects and their interactions, that is, about particles and forces. Many of the most fundamental claims about these things are founded on principles of symmetry, the most predominant feature of what a physicist thinks of as beauty. Concerning forces, John Barrow (1988, p. 180) explains, ‘Today’s best working descriptions of all the known forces of nature . . . are predicated entirely upon symmetries.’ Concerning particles, the class of particles called vector bosons, things like photons and gluons that transmit the energy of interactions, show up in the theories primarily as ways of preserving symmetry. As Heinz Pagels (1986, p. 216) puts it, ‘gluons exist because of symmetry’. Symmetry clearly plays an authoritative role in theoretical physics. There are also contexts in which beauty, again in the form of symmetry, is
* Department of Philosophy, Box 6011, Northern Arizona University, Flagstaff, AZ 86011, U.S.A. Received 3 April 1998; in revised form 14 August 1998.
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suggested as a surrogate for experiment and empirical testing. In high energy physics, for example, hypotheses about the most basic constituents of matter and speculations about the beginning of time are beginning to exceed our ability to experiment. In place of experiment, beauty has become an authoritative criterion for accepting theories. A so-called Theory of Everything such as superstring theory describes events and objects at such high energy that experimental testing for the predicted particles is out of the question, perhaps forever. In a recent commentary in Physics Today, two particle physicists, Ginsparg and Glashow (1986, pp. 7–9), describe the situation this way. ‘We are stuck with a gap of 16 orders of magnitude between theoretical strings and observable particles, unbridgeable by any currently envisioned experiment.’ Without experiment, internal attributes of the theory rule the day, and ‘elegance, uniqueness and beauty define truth’ (Ginsparg and Glashow, 1986, p. 7). The description of superstring theory indicates an intended epistemic role for beauty. So does the account by Steven Weinberg (1992, p. 133) of state-of-the-art physics. In a chapter called ‘Beautiful Theories’, he describes ‘how the usefulness of our sense of beauty is a sign of our progress toward a final theory’. Given this epistemic intent for the beauty of a theory, we must ask the following important epistemological questions. Why should we think that beauty indicates truth? And how do we, that is, how do the physicists, evaluate beauty? Are there objective standards for telling a beautiful theory from an ugly one, or at least a more beautiful from a less? Is the assessment of beauty any more precise than a vague you-knowit-when-you-see-it? There have been interesting attempts to explicitly link beauty to truth, and thereby establish its epistemic credentials. They involve preliminary assumptions about the natural world. Anthony Zee (1986), for example, in Fearful Symmetry: the Search for Beauty in Modern Physics, assumes that the universe and all its parts were brought about by design. So when we describe the creation with elegance and symmetry, we capture the intent and the plan, and we describe it accurately. James Martin also advocates beauty as an indicator of truth. He too secures the link by making substantial metaphysical assumptions. Martin (1989, p. 363) presumes that the physical world is naturally harmonious and that we as humans have ‘a real but inevitably tacit access to reality’. Scientific knowledge, in this account, amounts to articulating our tacit understanding of things. Our sensibilities are innately connected to the harmony in nature, so when we manage to articulate a beautiful description, one that pleases the sensibilities, it is likely to be an accurate picture of the real harmony. Zee and Martin give epistemic value to beauty only by making these questionable metaphysical assumptions. It would be valuable to ask whether beauty can be used as a means to truth without the metaphysical commitment. Perhaps we can take some of the pressure off beauty by giving it somewhat less epistemic responsibility, less than the authority to grant justification and good rea-
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son to believe a theory. Beauty may be a good way to generate new ideas, an inspiration to creativity, but not a good way to test ideas or judge their truth. In this role, beauty is good reason to take a theory seriously, but not in itself good reason to believe the theory is true. And indeed, when we casually describe an experiment or some aspect of theorizing as being as much an art as a science, it is the setting up and getting things to work, or the initial inspiration of a hypothesis that requires finesse and creativity. The artistry is gone, though, from the standards used to prove that the experiment has worked or the hypothesis is worthy of belief. A good case for beauty in this suggestive, formative role is that of the subatomic particle ⍀−. Physicists knew of nine particles with properties that put them on a symmetric matrix, symmetric except for one gap, like an equilateral triangle with a missing corner. Under the influence of symmetry, physicists thought there must be a tenth particle in the set to fill the gap and complete the triangle. They noted the properties the particle would need to be in the empty place, and with this as a guide they looked for and found the tenth particle, ⍀−. Even in this proto-epistemic role, the influence of beauty is important and worth understanding. By telling us which experiments are worth doing, looking for ⍀−, for example, the initial criteria direct where huge investments of time and money are to be spent. Because of the large size and expense of experiments in high-energy physics, there is a lot of inertia in any theoretical program. Research programs are hard to get started and then hard to stop. Anything that influences the launching plays an important role in shaping the future of research. Furthermore, in light of Kuhnian arguments dissolving the distinction between contexts of discovery and justification, it would be naive to think that reasons to consider a theory are isolated from reasons to believe it. Once introduced, a theory might secure its own evidence. Besides, beauty is not restricted to a preliminary role of merely suggesting theories to test. Beauty is sometimes the decisive factor in justification. Weinberg (1992, p. 133) explicitly recognizes the distinction between suggesting and testing a theory, and just as explicitly puts beauty in a key role for both. He has the examples to show that ‘something as personal and subjective as our sense of beauty helps us not only to invent physical theories but even to judge the validity of theories’. This shows a high regard for beauty in physics, but endorsements and optimism alone cannot establish its epistemic authority. To find the source, if there is any, of this authority we have to see how beauty, in particular symmetry, is used in evaluating theories. 1. Beauty is Symmetry A snowflake is a thing of beauty. It is also a thing of symmetry. Each spoke displays a mirror reflection along its radial axis, and the whole crystal shows rotational symmetry by repeating the design exactly six times in one complete turn. Perhaps the beauty we feel is rooted in the geometric symmetry of structure.
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The link between beauty and symmetry does indeed exist in physics, and it is safe to say that beauty essentially means symmetry in that context. The theoretical description of particles that led to the discovery of the ⍀− is beautiful because of the symmetric matrix into which the particles fit. String theory, or in its most ambitious version, supersymmetric string theory, is most beautiful when it maximizes the symmetries in its description of nature. And this is hopeful news for beauty in physics, because symmetry is well understood and neatly codified in the precise mathematics of group theory. With beauty linked to symmetry, there is hope of an objective, group-theoretical way of deciding what is symmetric and what is not, or at least what is more or less symmetric. For this reason, and because it is true to the physicists’ concept of beauty, we will focus on symmetry and its role in physics. In understanding symmetry it is natural to think first of something visual that displays a geometric regularity. A snowflake is symmetric. The human body is roughly symmetric left-to-right, at least on the outside. An equilateral triangle is more symmetric than a scalene triangle. And so on. All of these things are symmetric because their appearance is unchanged after some distinct operation of physical change on the object. The equilateral triangle looks exactly the same even after being rotated by 120 degrees or after being reflected across a line that bisects one of the angles. The scalene triangle does not look the same after either of these operations. The human body looks the same after the mirroring operation of inverting left side for right. These geometric symmetries amount to an identity of appearance after some physical operation such as rotation, inversion, reflection, or translation. This is just the tip of the symmetry iceberg, but it leads in the right direction toward a general definition of symmetry. The most general concept of symmetry is this. Any operation that leaves some property unchanged is a symmetry. In the case of the snowflake, the property that is unchanged is its appearance, and the operation that preserves appearance is rotation by sixty degrees. There are other appearance-preserving operations on the snowflake, such as rotation by a full turn or reflection through an axis along any spoke, and these are all symmetries of the object. The concept of symmetry can apply to things, like a snowflake, or to phenomena, or to laws of nature. The phenomenon of gravity, for example, the attractive force between any massive objects, is independent of the color of the objects. Two blue balls exert a slight gravitational force on each other. If you switch one of the balls for a pink one (of the same mass and at the same position), the gravitational effect will be the same, exactly the same. When it comes to gravity, nature is indifferent as to the color of the objects. Gravity, in other words, is color symmetric. It is also flavor symmetric, smell symmetric, and a lot of other things symmetric. It is helpful to think of a symmetry as an ordered pair, an operation and an invariant of that operation. Thus, a snowflake has the symmetry of ⬍ 60° rotation, appearance > , meaning that the operation of rotating by sixty degrees, one-sixth of a full rotation, leaves the appearance unchanged. The snowflake has several
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other symmetries, and each of these could be characterized by an ordered pair. The phenomenon of gravity has the symmetry ⬍ change the color, force > , indicating that the force between two objects is unaffected by a change in the color of one or both of the objects. And again, gravity has other symmetries that could be expressed as ordered pairs. The color symmetry of gravity is by no means amazing. It’s not even worth noticing or mentioning except as an easy way to come to terms with the concept of symmetry. But having noticed color symmetry, we are now in a position to insist that any theory of gravity, any articulation of the laws of gravity, must be color symmetric. That is, the equations describing the phenomena of gravity must remain the same if the color variables are changed. Newton’s law of universal gravitation and the general theory of relativity both fulfill this requirement by default. Neither has any color variables at all in its equations. Nonetheless, color symmetry is a necessary condition of acceptability of a theory of gravity. It is not a very strict condition, since it is almost trivially easy to meet, but we are on the way to seeing how beauty in the form of symmetry plays an epistemic role in choosing theories. With the very general characterization of symmetry as invariance under some operation, the law-like characteristics of universality and uniformity are themselves symmetries. Uniformity over time, the claim that the laws of nature are the same now as they were in the past, is a symmetry of the form ⬍ different time, form of the law > . For example, the uniformity of Newtonian mechanics, in particular the second law, can be expressed as the symmetry ⬍ different time, F ⫽ ma > . The physical objects in nature lack this symmetry over time. People age, the Grand Canyon deepens, and so on. Observe these objects at a different time and things will be different. But the laws stay the same. In fact, it is the uniformity of laws, that is, their time symmetry, that allows us to tell just how old something like the Grand Canyon is. Laws are the same at all times; they are also the same at all places. The universal aspect of Newton’s law of gravity has its empirical origin in the discovery that the force between the earth and an apple can be described by the same law as describes the force between the earth and the moon. Gravity, in other words, is terrestrial–celestial symmetric. Similarly, if F ⫽ ma is true anywhere, it is true everywhere. This is a symmetry of the law. It means that you cannot use the laws by themselves to tell where you are in the universe. The contingent properties of the objects, their masses, for example, vary from object to object. Different things will be accelerating at different rates, depending on where you are, and they will be under the influence of various forces. But wherever you are, the different masses, forces, and accelerations will always be related in the same way, namely F ⫽ ma. The law is invariant over positions, and this is a symmetry. The great generality in the concept of symmetry makes it nothing very special. It also makes room for subjectivity, an element of personal choice, in the justifi-
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cation of theories. To see this, realize that any system whatsoever, and any theory or model or law of nature, will be symmetric in some way. All this requires is that some property be left unchanged under some operation. It may be a trivial symmetry in the sense that the property is not even specified by the theory and so changing the property is irrelevant to the theory. This was the case of the color symmetry of gravity. Some of the most important and informative symmetries in physics are trivial symmetries in this sense. The equation describing the acceleration of an object in response to a force has no variable for the absolute position of the event. It does not matter whether it happens in New York or San Francisco, the law is the same. This position symmetry is trivially fulfilled by the law, but it is onto something of fundamental importance, namely the nature of space itself. The position invariance of laws, their location symmetry, is related to the observation that we are unable to detect absolute position in space. We can only observe the position of something relative to something else. This observation leads to the requirement of position symmetry of all laws of nature, including the relation between force and acceleration, and we are on the way to the theory of relativity. Symmetry in all its generality is a very weak necessary condition for theorizing, too weak to be of any use. If symmetry considerations are to be of any epistemic value they must be specific. Since every theory is symmetric in some way or another, the only meaningfully strict symmetry requirements on theorizing will be those that specify a particular operation, a particular symmetry. We might demand translation or rotation symmetry of our theories, for example. So, the question to ask in evaluating a new theory is not simply, Is it symmetric? We have to ask, Is it symmetric under such-and-such specific operation? Then the epistemological question is focused to ask why symmetry under this particular operation is a necessary condition of a true description of nature. Why should a theory have this symmetry and not some other? The need to be specific shows the lurking possibility of subjectivity in symmetry arguments. If every theory is symmetric in some way, then symmetry is wherever we choose to find it. Why can’t we all just choose our own favorite symmetry operation and then use it, insist on it, to judge other ideas? Given the ubiquity of symmetry in general, how can symmetry discriminate between good theories and bad? Everything is beautiful, in its own way. Here is a graphic example of the flexibility in the concept of symmetry. A snowflake, a crystallized state of water, has been my paradigm for symmetry and beauty in nature. The contrast would be water in an amorphous state, a vapor of molecules in a nondescript cloud, a random distribution that has no distinguishable parts or patterns. Nobody uses the cloud of vapor as an example of symmetry, but they could. While the snowflake has obvious rotational symmetries, the random cloud of molecules has a translational symmetry. If the cloud of molecules is entirely homogeneous, such that there is no way to tell one point from another, then there
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is no distinguishing the pattern as it is from a pattern that is shifted a bit to the left or to the right. The crystal, the snowflake, lacks this symmetry of translation. With each molecule precisely located in the crystal, we can detect a change of position in this system.1 This is not to say that the laws of nature governing the crystal will lack the translation symmetry. They will certainly be the same in one spot as the next. It is the appearance of the system itself that is translation symmetric in one case (the cloud of vapor) and is not translation symmetric in the other (the snowflake). The point is that we cannot say simply that the snowflake is symmetric and the amorphous cloud is not. Each system has its own symmetry. I am resisting the even more counterintuitive conclusion from this comparison between snow and clouds, that the amorphous cloud is even more symmetric than the crystalline snowflake. Here is what tempts me. The cloud of vapor has the translation symmetry that the snowflake lacks, and the cloud has the rotational symmetry that drew us to the snowflake in the first place. In fact, the cloud has more rotational symmetry than the flake. Its utter lack of precision in the placement of molecules allows rotation by any angle whatsoever to preserve its appearance. The snowflake must be rotated by exactly sixty degrees, or some multiple of sixty. But, if the cloud is isotropic such that there is no distinguishing axis and no way to tell one orientation from any other, then any rotation leaves the cloud unchanged. The cloud has an unlimited variety of continuous symmetries, continuous operations that leave the system unchanged, while the snowflake has only a few discrete symmetries. But I am reluctant to declare the amorphous cloud the more symmetric system. There is no reason to think that there is an absolute count of symmetries, given the tremendous freedom in choosing the operation and the invariant in the ordered pair. Surely there will be symmetries in the snowflake that are missing in the cloud. The cosmological principle, an observational foundation of modern astronomy, is another example of a symmetry in randomness. The universe appears to be a roughly homogeneous distribution of galaxies. On the large scale, one place in the universe is essentially indistinguishable from another. In other words, matter distribution is location symmetric. If there were some pattern in the distribution, the universe would lack this symmetry. In the Big Bang model of the universe, there is a pattern in time, though not in space. Over time, galaxies are getting further apart. The distribution of matter is space symmetric in this model, but not time symmetric. In the Steady-State model, on the other hand, there is no beginning of the universe and matter is continuously created to maintain the same density in space. In this model, the universe is both space and time symmetric. The examples suggest that in some way the overall symmetry of a system increases as the system becomes more disordered. A system in total disorder is 1 G. Nicolis and I. Prigogine even describe the process of freezing as an event of “symmetry breaking”, that is, of losing the symmetry in the transition from liquid to solid crystal. See Nicolis and Prigogine (1989, p. 41).
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one with no specified positions, spacing, or trajectories of its parts. Change any of these parameters in any way and it is still a system in total disorder. The system is unchanged. The correlation between symmetry and disorder is imprecise without a concept and a measure of the degree of symmetry, but Joe Rosen (1995, p. 150) clearly endorses the idea that the extreme of disorder, if only a hypothetical state, corresponds to the extreme of symmetry: ‘Total disorder is maximal symmetry.’ This makes sense, given the general definition of symmetry, but it is the opposite of the original intuition that associated symmetry and beauty with order and precision. It is certainly opposite to Zee’s link between symmetry and design. Symmetry cannot be evidence of careful planning and meticulous placement of things or implementation of laws. It is just the contrary. Symmetry shows where there is a disregard for details, where things are at least in some way haphazard. Symmetries reveal what doesn’t matter in nature, what can be altered to no effect. Symmetries are about nature’s indifference to the details of change.2 2. Substantive Symmetry Arguments A symmetry is an invariance under some operation of change. How is a symmetry, a specific symmetry, used as an epistemic tool? That is, how can a criterion of symmetry be used to direct theorizing in the general direction of the truth? Of course, it will not be fool-proof, but perhaps it can better the odds. Bas van Fraassen (1989, p.242) distinguishes between substantive symmetry arguments and ‘symmetry arguments proper’. It is the former, the substantive arguments, that are of more epistemological interest. Symmetry arguments proper involve exploiting a symmetry of a theory, a model, or a physical system, as a way of facilitating solving problems. Proper arguments are analytic in the sense of unpacking the details and implications of a symmetry already known to be a feature of the system. These are indeed common in physics, and symmetry is an efficient way of revealing the features of a new or complicated theory. But these arguments make no epistemic progress. As van Fraassen (1989, pp. 260–261) puts it, ‘Symmetry arguments [proper] have that lovely air of the a priori, flattering what William James called the sentiment of rationality. And they are a priori, and powerful; but they carry us forward from an initial position of empirical risk, to a final point with still exactly the same risk.’ Advancement in science demands taking risks, and to find the epistemic value of symmetry in physics, if there is any, we have to deal with substantive symmetry arguments. These amount to the empirically risky claims that nature in fact has a particular symmetry, and the enforcement of that symmetry as a necessary condition for an acceptable theory. This is the initial risk that van Fraassen talks about, and it is a prerequisite to a symmetry argument proper. We have to know the 2 Joe Rosen (1990, p. 291) uses similar words, ‘The physical significance of symmetry of evolution is that nature is indifferent to certain aspects of physical systems.’
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symmetry is there before we use it as a tool of inference, and the claim that the symmetry is there is where the epistemic action takes place. It is certainly this substantive sort of argument that Ginsparg and Glashow (1986, p. 7) are talking about in their assessment of string theory, where ‘elegance, uniqueness and beauty define truth’. In other words, beauty in the form of a specific symmetry is thought to be an aspect of nature, so a true theory must have this symmetry. A good example of symmetry arguments in physics, both proper and substantive arguments, is Einstein’s principle of relativity as it functions in the special theory of relativity. The principle of relativity states simply that the laws of physics are the same in all inertial reference frames. In other words, at different places in the universe, and in reference frames going at different uniform speeds, the laws will be the same. This is clearly a symmetry in the laws of physics. The fact that the laws of physics are the same in an earth-bound laboratory as in a speeding airplane, the fact that you cannot use the laws of physics alone to detect which of those reference frames you are in, and the fact that we cannot detect any uniquely privileged reference frame that is stationary in an absolute space, are all expressions of the same symmetry. The principle of relativity was not a new idea with Einstein. His contribution was to stick to it as no one had before, forcing the form of the laws of nature to fit the symmetry. This is a substantive symmetry argument that we will look at in a moment. First there is Einstein’s symmetry argument proper. The proper argument uses the symmetry, the principle of relativity, together with relevant background knowledge, to deduce the details of the theory. The principle of relativity, together with Maxwell’s equations as the laws of electricity and magnetism, are enough to derive the Lorentz transformations and other consequences for the special theory of relativity. Length contraction, time dilation, and E ⫽ mc2 are all consequences of symmetry arguments proper, using the principle of relativity as the specific symmetry. These arguments are now standard exercises for students of physics, great pedagogical tools for using symmetry to unpack the theory, but they were amazing in Einstein’s time. They were amazing both for their surprising results and for their insightful technique for revealing the consequences of a theory. With the details of the theory articulated, in particular with the Lorentz transformations down on the page, the special theory of relativity and its foundational symmetry in the principle of relativity are used as a standard of acceptability for other theories. This is a substantive argument. The symmetry argument proper allows us to express inertial reference frame symmetry in terms of Lorentz invariance. Lorentz invariance is thus a necessary feature of all physical theories. This criterion becomes a way to both discover and test new theories. In Einstein’s words, ‘This [Lorentz transformation] is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature.’ (Einstein, 1995, p. 43). The case of the principle of relativity is characteristic of how substantive sym-
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metry arguments work generally in physics. Understanding the general form of these arguments is the key to answering the central epistemological question of how beauty, in this case symmetry, indicates truth. First note that there is no argument that relies on a link between symmetry in general and truth. The arguments are always about some specific symmetry, such as Lorentz invariance, or, as we will soon see, gauge symmetry. The arguments have to be specific in this way, since every theory and every system is symmetric in some way. So the good news is that we do not need to establish a connection between symmetry in general and truth. The bad news is that we will need a link to truth individually for any specific symmetry invoked. This (the good news) means we can avoid the grand and dubious metaphysical assumptions about design and natural harmony that were the basis of Zee’s and Martin’s arguments, respectively. The pattern of a specific symmetry argument in physics follows three basic steps. First, a symmetry is observed in nature. This is how relativity got started, with the observation that the laws of physics, as far as we can tell, are in fact the same in all inertial reference frames. This is not a guess or an assumption or the expression of some aesthetic predilection. It is the result of riding on a train and seeing that everything behaves while the train is moving exactly as it does while the train is stopped at the station. We observe this invariance under the operation of changing reference frames. The second step is the symmetry argument proper, the deduction of the consequences of this particular symmetry and the full nature of the invariance. The third step is the risk, the generalization that the observed symmetry applies to relevantly similar, unobserved systems as well. From the observation of the invariance of the laws of physics in the reference frames we have experienced, we infer that all laws of physics are invariant in all inertial reference frames. The third step, the substantive part of the argument, is an argument by analogy, extending what is observed to analogous systems that are not observed. It bears the standard strengths, risks, and limitations of arguments by analogy. It is fallible and subject to revision. Its strength is dependent on knowing which particular symmetries can be generalized, and to what sort of systems. Like all arguments by analogy, these substantive symmetry arguments are evaluated in terms of the amount and variety of evidence for the particular symmetry, and in terms of the similarities between the circumstances in observed and projected cases of the symmetry. The case of the principle of relativity fits this pattern of argument by analogy. We observe that the laws of mechanics are the same in all inertial reference frames. In accelerating reference frames, on the other hand, relations such as F ⫽ ma apparently need to be revised by the addition of inertial forces. On the basis of this evidence we endorse the analogy between inertial reference frames in which we have made measurements and inertial frames in which we have not, and the symmetry is extended to describe transformations between inertial reference frames
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in general. But we do not endorse the analogy to non-inertial reference frames, and the symmetry is explicitly limited in this way. This establishes the Lorentz symmetry for laws of mechanics, but Einstein’s principle of relativity is more general than this. It applies to all the laws of physics, and this generalization is the result of another argument by analogy. The laws of electricity and magnetism are relevantly similar to the laws of mechanics in that the reference frame invariance of the latter must be a feature of the former. The particular symmetries that are at the forefront of foundational physics these days fit into arguments with this same basic structure. Given the deeply entrenched status of the special theory of relativity, Lorentz invariance, the symmetry of the principle of relativity, is at least a prima facie necessary condition on any new theory. But we need lots of necessary conditions in order to weed out the false theories and zero in on the true. This means we need to discover additional specific symmetries and to demand not simply more symmetry in the new theories but specifically these symmetries. The discoveries of which particular symmetries to apply are rooted in observation, observing what does not change under particular operations. Gauge symmetry is particularly important in high energy, theory-of-everything physics. It has ‘taken command of elementary particle physics’ (Aitchison and Hey, 1982, p. 179). Gauge symmetry begins with the observation that electrostatic effects between two objects or points in space depend on the potential difference between the two but not on the absolute value of the potential. As an example, if a powerline has an electrical potential of a million volts, it’s a danger. It can spark or electrocute something on the ground with zero electrical potential, but it is harmless to something else that is at the same million-volt potential. The electrical effect requires a potential difference. And the electrical effect will be the same between a wire with two million volts and an object at one million, as it will be between the one million and something at zero. All that matters is the difference. It is analogous to mechanical laws that depend only on relative distance and speed between objects, not on any absolute position or speed in space. The point is that you can set your electrostatic potential to zero at any point and then measure the differences in values from that point. Call the one-million volt wire zero, if you like, and then the two-million volt line will be rescaled to one million, and the previous zero will be negative one million. With these values, all the laws will be unchanged. It is just like the freedom to put the origin of a coordinate system anywhere you like without affecting the mechanical laws of nature. You also have the freedom to choose the scale, the units of measurement. Shrink or expand the scale, the gauge, of measurement and the laws are unaffected. The rescaling of the electrostatic potential is a global transformation in the sense that the change is uniform throughout space. This global symmetry is apparent in the laws of electrostatics, and it is related to the conservation of electric charge.
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When electric charges move, that is, when the phenomena become dynamic, electric and magnetic effects become interrelated. In the unified theory of electromagnetism, the gauge symmetry is local. The change of scale of the electromagnetic potential can differ from place to place. The rescaling is a function of position and time. The uneven changes in the electrostatic potential are just compensated by the corresponding changes in the magnetic vector potential so that there is no change at all in the combined electromagnetic laws. This symmetry under local gauge transformation is what is called gauge symmetry, and a theory that is gauge symmetric is a gauge theory. The field of interaction that compensates for the space–time dependent rescaling, in this case the electromagnetic potential, is called the gauge field. The gauge symmetry, that is, the local gauge symmetry, of electromagnetism is apparent in the theory itself. It is manifest in the Maxwell equations. It is not an imposition from any theoretical or aesthetic predilection, but a direct result of the discovery that electricity and magnetism are manifestation of a unified electromagnetic phenomenon. Though the gauge symmetry was discovered from the Maxwell equations, physicists like to point out that the logic can be reversed. From a requirement of gauge symmetry (and Lorentz symmetry) the Maxwell equations can be derived. In this way, the symmetry condition can be used to generate a theory of interactions, and the technique can be extended to find new theories of other interactions. To understand the nature of nuclear interactions, for example, ‘We now mimic as literally as possible, the discussion of electromagnetic gauge (phase) invariance.’ (Aitchison and Hey, 1982, p. 185) The theory of electromagnetism is a local gauge theory. Similarly, the general theory of relativity is based on a local symmetry.3 The Lorentz symmetry that characterizes the special theory of relativity is a global symmetry in that each inertial reference frame extends uniformly throughout spacetime. The effect of making the symmetry local, that is, making the coordinate systems variable in space and time and insisting that the laws of nature have the same form in all of these different systems, changes the theory into the general theory of relativity, and, as in a local gauge theory, there is a resulting field of interaction. This field is gravity. Two fundamental interactions, electromagnetic and gravitational, are seen to be locally symmetric. By analogy to electromagnetism, other fundamental interactions are thought to be locally gauge symmetric. Without knowing the laws of nuclear interactions, the condition of (local) gauge symmetry is imposed from the start to derive theories of interactions with new gauge fields. Gauge theory has ‘taken
3 For insight into the differences and similarities between the symmetries of electrodynamics, general relativity, and special relativity, see Wigner (1967) ch. 2.
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command’ of fundamental physics, but only on its empirical authority from electromagnetism. The power and elegance of the symmetry arguments sometimes allow us to lose sight of their empirical foundations. So do some of the claims of the theoreticians. At one point, John Mauldin (1986, p. 221) makes it sound too good to be true: ‘Physicists can now deduce a gauge theory such as Maxwell’s electromagnetism without knowing any experimental evidence about electricity and magnetism,’ and ‘working physical laws can be pulled out of a mathematical hat’. This certainly flatters the sentiment of rationality, but only by overlooking the empirical basis of the symmetry. We need to remember, as it says in an often-used textbook on subatomic physics, that ‘Experiment is the only judge as to whether a symmetry principle holds.’ (Frauenfelder and Henley, 1974, p. 154) And Mauldin (1986, p. 221) points out that the hat trick with gauge symmetry is licensed ‘with hindsight from using gauge symmetry’, that is, with evidence of gauge symmetry in more familiar systems and laws. This crucial link to observation is the first of the three steps in a substantive symmetry argument. We have observed repeatedly that electromagnetism is in fact gauge symmetric, and so there is empirical reason to believe that this particular symmetry is an aspect of nature. To continue with Mauldin’s metaphor, working physical laws can be pulled out of a mathematical hat only because gauge symmetry has been put into the hat already, and put there on the authority of observations. 3. Conclusion The case of gauge symmetry is similar to that of the principle of relativity. Together they point to general conclusions about substantive symmetry arguments in physics. These arguments make no assumptions or inferences about symmetry in general. They are always about some specific symmetry or other. Furthermore, these arguments do not simply assume that the specific symmetry is a fact of nature, and then use it as a standard for judging theories. They begin with observation. Sunny Auyang (1995, p. 102) puts this point about symmetry in Kantian terms. General ideas of symmetry and the group-theoretic way of thinking are ‘our construction’. Symmetry is a physicist’s way of categorizing nature. But knowing what specific symmetries are true of nature ‘must be determined empirically’ (Auyang, 1995, p. 93). Once we see how symmetry is understood and used in physics, it seems misleading to associate symmetry with beauty. This is not to say that it is wrong or counterproductive to appreciate the aesthetic appeal of something like gauge symmetry or the principle of relativity, and I do not mean to discourage finding beauty wherever we can. But aesthetic value is entirely gratuitous to the epistemic role. It is not that physicists find gauge symmetry to be beautiful and on the basis of that beauty they believe it to be a real feature of nature. It is rather that, for mundane empirical reasons, physicists believe this particular symmetry is an aspect of nature, and, by
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the way, they also find it to be a thing of beauty. A tidy accountant might find a neat, orderly ledger of numbers to be a thing of beauty, and that’s fine. But what really counts is that the arithmetic is done properly, and that is independent of the beauty. The specific symmetries used in substantive arguments in physics are discovered; they are not artifacts of our aesthetic judgment. They are invariant aspects of observed phenomena. The only subjective influence is in using the word ‘beauty’ to describe symmetry. Furthermore, symmetry arguments do not lead to claims about design or planning or harmony in nature. If the metaphor of a personified nature is appropriate at all, then symmetries reveal nature’s indifference rather than careful attention. The role of symmetry in physics amounts to discovering nature’s indifference. References Aitchison, I. and Hey, A. (1982) Gauge Theories in Particle Physics (Bristol: Adam Hilger). Auyang, S. (1995) How is Quantum Field Theory Possible? (New York: Oxford University Press). Barrow, J. (1988) The World within the World (New York: Oxford University Press). Einstein, A. (1995) (originally published 1920) Relativity (Amherst: Prometheus Books). Frauenfelder, H. and Henley, E. (1974) Subatomic Physics (Englewood Cliffs: Prentice Hall). Ginsparg, P. and Glashow, S. (1986) ‘Desperately Seeking Superstrings?’, Physics Today 39, 7–9. Martin, J. (1989) ‘Aesthetic Constraints in Theory Selection’, British Journal for the Philosophy of Science 40, 357–364. Mauldin, J. (1986) Particles in Nature (Blue Ridge Summit: Tab Books). Nicolis, G. and Prigogine, I. (1989) Exploring Complexity (New York: W. H. Freeman). Pagels, H. (1986) Perfect Symmetry: the Search for the Beginning of Time (New York: Simon and Schuster). Rosen, J. (1995) Symmetry in Science (New York: Springer Verlag). Rosen, J. (1990) ‘Fundamental Manifestations of Symmetry in Physics’, Foundations of Physics 20, 283–307. van Fraassen, B. (1989) Laws and Symmetry (New York: Oxford University Press). Weinberg, S. (1992) Dreams of a Final Theory (New York: Pantheon Books). Wigner, E. (1967) Symmetries and Reflections (Bloomington: Indiana University Press). Zee, A. (1986) Fearful Symmetry: the Search for Beauty in Modern Physics (New York: MacMillan).