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Home , Identical particles, Invariant (physics), Symmetry (physics), Symmetry breaking

Proc. Natl. Acad. Sci. USA Vol. 93, pp. 14256–14259, December 1996 Colloquium Paper

This paper was presented at a colloquium entitled ‘‘ Throughout the Sciences,’’ organized by Ernest M. Henley, held May 11–12, 1996, at the National Academy of Sciences in Irvine, CA.

The role of in fundamental

DAVID J. GROSS

Department of Physics, Princeton University, Princeton, NJ 08544

ABSTRACT The role of symmetry in fundamental physics rize the regularities that are independent of the initial condi- is reviewed. tions. The laws are often difficult to discover, since they can be hidden by the irregular initial conditions or by the influence of Until the 20th century principles of symmetry played little uncontrollable factors such as friction or thermal conscious role in . The Greeks and others fluctuations. were fascinated by the symmetries of objects and believed that Symmetry principles play an important role with respect to these would be mirrored in the structure of nature. Even the laws of nature. They summarize the regularities of the laws Kepler attempted to impose his notions of symmetry on the that are independent of the specific dynamics. Thus invariance of the planets. Newton’s laws of embodied principles provide a structure and coherence to the laws of symmetry principles, notably the principle of equivalence of nature just as the laws of nature provide a structure and inertial frames, or . These symmetries coherence to the set of events. Indeed, it is hard to imagine that implied conservation laws. Although these conservation laws, much progress could have been made in deducing the laws of especially those of and , were regarded to be nature without the existence of certain symmetries. The ability of fundamental importance, these were regarded as conse- to repeat experiments at different places and at different quences of the dynamical laws of nature rather than as is based on the invariance of the laws of nature under consequences of the symmetries that underlay these laws. - translations. Without regularities embodied in the Maxwell’s equations, formulated in 1865, embodied both laws of physics we would be unable to make sense of physical events; without regularities in the laws of nature we would be Lorentz invariance and gauge invariance. But these symme- unable to discover the laws themselves. Today we realize that tries of electrodynamics were not fully appreciated for over 40 symmetry principles are even more powerful—they dictate the years or more. form of the laws of nature. This situation changed dramatically in the 20th century beginning with Einstein. Einstein’s great advance in 1905 was Classical Symmetries to put symmetry first, to regard the symmetry principle as the primary feature of nature that constrains the allowable dy- In classical dynamics the consequences of continuous symme- namical laws. Thus the transformation properties of the elec- tries are most evident using Hamilton’s action principle. tromagnetic were not to be derived from Maxwell’s According to this principle the classical motion is determined equations, as Lorentz did, but rather were consequences of by an extremum principle. Thus if we describe the system by a relativistic invariance, and indeed largely dictate the form of generalized coordinate x(t) (for example the position of a Maxwell’s equations. This is a profound change of attitude. point in space) then the actual motion of the system, Lorentz must have felt that Einstein cheated. Einstein recog- given the values of x(t)attϭt1and at t ϭ t2, is such that the nized the symmetry implicit in Maxwell’s equations and ele- action, S[x(t)], is extremal. The action is a local functional of vated it to a symmetry of space-time itself. This was the first x(t), namely it can be written as the integral over time of a instance of the geometrization of symmetry. Ten years later this function of x(t) and its time derivative—the Lagrangian, S ϭ point of view scored a spectacular success with Einstein’s t2 ͐t1 dtL[x(t), x(t)]. Hamilton’s principle means that if x¯(t)isthe construction of . The principle of equiva- actual motion then S[x¯(t) ϩ ␦x¯(t)] ϭ S[x¯(t)] for any infini- lence, a principle of local symmetry—the invariance of the laws tesimal variation, ␦x¯(t), of ¯(x t) that leaves its values at t ϭ t1 of nature under local changes of the space-time coordinates— and t ϭ t2 unchanged. This is often called the principle of least dictated the dynamics of gravity, of space-time itself. action, although the extremum can be either a minimum or a With the development of in the 1920s maximum of the action. The classical for symmetry principles came to play an even more fundamental x¯(t) follow from this principle. role. In the latter half of the 20th century symmetry has been A symmetry of a classical system is a transformation of the the most dominant concept in the exploration and formulation dynamical variable x¯(t), x¯(t) 3 ᏾[x¯(t)], that leaves the action of the fundamental laws of physics. Today it serves as a guiding unchanged. If follows that the classical equations of motion principle in the search for further unification and progress. are under the symmetry transformation, since if x¯(t) is an extremum of the action, and ᏾ generates a The Meaning of Symmetry symmetry of the action, then ᏾[x¯(t)] is also an extremum. The symmetry can then be used to derive new solutions. Progress in physics depends on the ability to separate the Thus, if the laws of motion are invariant under spatial analysis of a physical phenomenon into two parts. First, there , then if x(t) is a solution of the equations of motion, are the initial conditions that are arbitrary, complicated, and say an orbit of the earth around the sun, then the spatially unpredictable. Then there are the laws of nature that summa- rotated x(t), is also a solution. This is interesting and sometimes useful. The publication costs of this article were defrayed in part by page charge A more important implication of symmetry in physics is the payment. This article must therefore be hereby marked ‘‘advertisement’’ in existence of conservation laws. For every global continuous accordance with 18 U.S.C. §1734 solely to indicate this fact. symmetry—i.e., a transformation of a physical system that acts

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the same way everywhere and at all times—there exists an For example the lowest energy state, the ground state of the associated time independent quantity: a conserved charge. Hydrogen is such a rotational invariant singlet state. This connection went unnoticed until 1918, when Emmy Other superpositions of rotated states will yield other irreduc- Noether proved her famous theorem relating symmetry and ible representations of the symmetry . Indeed any state conservation laws. Thus due to the invariance of the laws of can be written as a sum of states transforming according to physics under spatial transformations momentum is con- irreducible representations of the . These served, due to time translational invariance energy is con- special states can be used to classify all the states of a system served and due to the invariance under a change in phase of possessing symmetries and play a fundamental role in the the wave functions of charged is analysis of such systems. Consequently the theory of repre- conserved. It is essential that the symmetry be continuous; sentations of continuous and discrete groups plays an impor- namely that it is specified by a set of parameters that can be tant role deducing the consequences of symmetry in quantum varied continuously, and that the symmetry transformation can mechanics. With the tools of many consequences be arbitrarily close to the identity transformation (which does of symmetry are revealed. For example, the selection rules that nothing to the system). The discrete symmetries of nature (all govern atomic spectra are simply the consequences of rota- of which are approximate symmetries), such as time reversal tional symmetry. invariance or mirror reflection, do not lead to new conserved Quantum mechanics also revealed a new kind of symmetry, quantities. that of exchange of . This lead to a classifi- cation of all elementary particles as either , whose wave One can give a simple geometrical argument that illustrates function is invariant under interchange of two identical par- the connection between symmetry and conservation laws. ticles, or , whose changes sign when two Consider the motion of a particle described by x(t), from x to i identical particles are interchanged. The quantum statistics of x f. Assume that the action is invariant under spatial transla- such particles is different, with profound implications for their tions. If so the action for the actual path, S[x(t)] will be equal behavior in aggregate. to the action for the displaced path—i.e., S[x(t)] ϭ S[x(t) ϩ In relativistic quantum mechanics the implications of sym- a]. Now consider the path of motion from xi to xi ϩ a to xf ϩ metry are greater. Here the symmetry group is the Poincare´ a to xf.Ifais very, very small (infinitesimal) then, according group, of space-time translations, rotations, and boosts to to Hamilton’s principle, the action along this path is the same moving frames. The analysis of the representations of this as the original action. Thus the difference of the two vanishes, group leads to a complete classification of physical irreducible and since the action is additive we have representations–elementary particles: (i) Massive representations: M Ͼ 0. These irreducible S͓x 3 x ϩ a 3 x ϩ a 3 x ͔ Ϫ S͓x 3 x ͔ ϭ S͓x 3 x ϩ a͔ i i f f i f i i representations are labeled by the and the J, which ϩ S͓x ϩ a 3 x ͔ ϭ 0. is quantized in half-integer units, J ϭ 0, 1/2, 1, . . . . f f (ii) Massless representations: M ϭ 0. In this case the only finite dimensional representations of this group are one di- The action along the infinitesimal path from xi to xi ϩ a must mensional. These are labeled by a single helicity, ␭, that is be proportional to a, i.e., S[xi 3 xi ϩ a] ϵ pi⅐a, which defines the momentum p. Similarly the action along the infinitesimal half-integer. An example of such a representation is the left-handed , which only has one helicity state with ␭ path from xf ϩ a to xf is given by S[xf ϩ a 3 xf] ϵϪpf⅐a(the minus sign is because the path runs in the opposite direction). ϭ 1/2. (If we include then irreducible representations Consequently the momentum p is conserved, namely it is a time contain both positive and negative helicities, Ϯ␭.) This group independent constant along the path of motion. theoretic analysis makes it clear that massless spinning parti- cles are fundamentally different from massive particles. This Symmetry in Quantum Mechanics difference has profound implications for dynamics; indeed, it requires that massless spinning particles be described by gauge In quantum theory, invariance principles permit even further theories. reaching conclusions than in classical mechanics. In quantum Symmetry Breaking mechanics the state of a physical system is described by a ray in a Hilbert space, ⌿͘. A symmetry transformation gives rise ͉ The secret of nature is symmetry, but much of the texture of to a linear , R, that acts on these states and the world is due to mechanisms of symmetry breaking. The are transforms them to new states. Just as in classical physics the a variety of mechanisms wherein the symmetry of nature can symmetry can be used to generate new allowed states of the be hidden or broken. The first is explicit symmetry breaking system. However, in quantum mechanics there is a new and where the dynamics is only approximately symmetric, but the powerful twist due to the linearity of the symmetry trans- magnitude of the symmetry breaking is small, so that formation and the superposition principle. Thus if ͉⌿͘ is an one can treat the symmetry violation as a small correction. allowed state than so is R͉⌿͘, where R is the operator in the Such approximate symmetries lead to approximate conserva- Hilbert space corresponding to the symmetry transformation tion laws. Many of the symmetries observed in nature are of ᏾. So far this is similar to classical mechanics. However, we this sort, not really symmetries of the laws of physics at all, can now superpose these states—i.e., construct a new al- but—for what appears sometimes to be accidental reasons— lowed state: ͉⌿͘ ϩ R͉⌿͘. (There is no classical analogue for approximate symmetries for a certain class of phenomena. The such a superposition; of say the superposition of two orbits isotopic symmetry of the nuclear is an example of an of the earth.) approximate symmetry; good due to the small values of the up The superposition principle means that we can construct and down and the weakness of the electromag- linear combinations of states that transform simply under the netic force. symmetry transformations. Thus superimposing all states that A more profound way of hiding symmetry is the phenom- are related by rotations we obtain a state ͉⌽͘ ϭ •R R͉⌿͘ that enon of spontaneous symmetry breaking. Here the laws of is rotationally invariant, the singlet representation of the physics are symmetric but the state of the system is not. This group. Namely situation is common in classical physics. The earth’s orbit is an example of a solution of Newton’s equations that is not R͉⌽͘ ϭ ͸RRЈ͉⌿͘ ϭ ͸RЉ͉⌿͘ ϭ ͉⌽͘. rotationally invariant, although the equations are. Conse- RЈ RЉ quently, for an observer of the solar system, the rotational Downloaded by guest on September 26, 2021 14258 Colloquium Paper: Gross Proc. Natl. Acad. Sci. USA 93 (1996)

invariance of the law of gravitation is not manifest. The are invariant under the transformation. Thus global rotations particular orbit is picked out by the asymmetric initial condi- rotate the laboratory, including the observer and the physical tions of the planet. Thus this mechanism for hiding symmetries apparatus, and all observations will remain unchanged. Gauge of physics is related to the induced by asymmetric symmetry is of a totally different nature. Gauge symmetries initial values. In quantum mechanics the situation is different. are formulated only in terms of the laws of nature; the Quantum mechanical systems with a finite number of degrees application of the symmetry transformation merely changes of freedom (an adequate description of atomic physics, for our description of the same physical situation, does not lead to example) always have a symmetric ground state. Classically a a different physical situation. particle under the influence of gravity can sit anywhere on a Gauge symmetry first appeared in Maxwell’s electrodynam- flat surface; yet the quantum mechanical state of lowest energy ics. Here the physical observables are the electric and magnetic (the ground state) is a superposition of all these classical fields, E and B. It was discovered early on that one could allowed states. Thus the quantum mechanical particle is ev- simplify the equations by introducing a vector potential A␮,in erywhere at one—a state that exhibits the translational invari- terms of which both the electric and magnetic fields could be ance of the laws of motion. expressed. Yet this description was not unique, one could In systems with an infinite number of degrees of freedom, perform a gauge transformation, A␮(x) 3 A␮(x) ϩ Ѩ␮␾(x), however, global symmetries may be realized in two different without changing the values of E and B. This symmetry was ways. The first way is standard: the laws of physics are invariant regarded for decades as rather artificial. As Wigner, one the and the ground state of the theory is unique and symmetric, as pioneers of symmetry in this century put it, is the case for quantum mechanical systems with a finite number of degrees of freedom. However in systems with an This gauge invariance is, of course, an artificial one, similar to infinite number of degrees of freedom a second mode is that which we could obtain by introducing into our equations possible, in which the ground state is asymmetric. Such spon- the location of a . The equations must then be invariant taneous symmetry breaking is responsible for the existence of with respect to changes of coordinates of that ghost. One does crystals (that break translational invariance), magnetism (in not see, in fact, what good the introduction of the coordinate which is broken), superconductivity (in of the ghost does. which the phase invariance of charged particles is broken), and This attitude toward gauge invariance has changed dramat- the structure of the unified electro-weak theory and more. ically in the last two decades. Gauge theories have assumed a Indeed the spontaneous symmetry breaking of global and local central position in the fundamental theories of nature. They gauge symmetries is a recurrent theme in modern theoretical provide the basis for the extremely successful , physics. The search for new symmetries of nature is based on a theory of the fundamental, nongravitational forces of na- this possibility, for a new symmetry that we discover must be ture—the electromagnetic, weak, and strong interactions. To somehow broken otherwise it would have been apparent long be sure gauge invariance is a symmetry of our description of ago; it would have been an old symmetry. nature, yet it underlies dynamics. Gauge invariance forces the Spontaneous broken symmetries have consequences. Al- existence of special particles, gauge bosons. These are massless though the symmetry is not manifest it has implications. Thus spin one particles that are associated with the vector potential for every broken global symmetry there exist fluctuations with and mediate the forces. Thus the SU(3) ϫ SU(2) ϫ U(1) very low energy. These appear as massless particles. Examples gauge symmetry of the standard model implies the existence are sound waves in solids, spin waves in magnetics and pions of eight that mediate the , three gauge in nuclear physics. bosons, the WϮ and the Z bosons, that mediate the weak Associated with spontaneous symmetry breaking is the interactions and the of light. As Yang has stated: phenomenon of symmetry restoration. If one heats a system Symmetry dictates interaction. The first example of this was that possesses a broken symmetry it tends to be restored at high general relativity where Einstein employed the symmetry of . Thus a ferromagnetic material can be magne- space-time under local changes of coordinate to determine the tized at low temperature (or even at room temperature) with laws of gravity. all the little atomic magnets aligned in the same direction. This Furthermore, the realization that gauge symmetry is based is a state of broken . As the temperature on the fiber bundle, a sophisticated geometrical concept, has increases the vibrate more and more. Finally when the provided a deep and beautiful geometrical foundation for temperature is greater than a certain critical value the fluc- gauge symmetry. The fiber bundle is a beautiful mathematical tuations win out over the forces that tend to align the atomic magnets and the average magnetization vanishes. Above the construction that combines an internal space together with critical temperature the system exhibits rotational symmetry. space-time, to form a unified geometrical object which exhibits Such a transition from a state of broken symmetry to one the gauge symmetry. It has also been understood, a century where the symmetry is restored is a phase transition. We after Maxwell discovered his equations, that the vector poten- believe that the same phenomenon occurs in the case of the tial is not just an artificial construct, but has direct observable symmetries of the fundamental forces of nature. Many of these meaning. This is most evident in the Bohm–Aharonov effect, are broken at low . Very early in the history of the wherein an beam propagates in a region where there universe, when the temperature was very high, all of these is no electromagnetic field, yet due to the geometry the vector symmetries of nature were presumably restored. The resulting potential is nonvanishing. Due to the change of phase that phase transitions, as the universe expanded and cooled, from accompanies a charged particle moving in a background vector symmetric states to those of broken symmetry have important potential one can observe interference effects directly. Thus cosmological implications. the vector potential is primary. Indeed today we believe that global symmetries are unnat- Gauge Symmetry ural. They smell of action at a distance. We now suspect that all fundamental symmetries are local gauge symmetries. The traditional symmetries discovered in nature were global Global symmetries are either all broken (such as parity, time symmetries, transformations of a physical system in a way that reversal invariance, and charge symmetry) or approximate is the same everywhere in space. Global symmetries are (such as isotopic spin invariance) or they are the remnants of regularities of the laws of motion but are formulated in terms spontaneously broken local symmetries. Thus, Poincare´invari- of physical events; the application of the symmetry transfor- ance can be regarded as the residual symmetry of the mation yields a different physical situation, but all observations Minkowski vacuum under changes of the coordinates. Downloaded by guest on September 26, 2021 Colloquium Paper: Gross Proc. Natl. Acad. Sci. USA 93 (1996) 14259

The Origin of Symmetry is a profound and beautiful extension of the geometric symmetries of space-time to include symmetries Why is nature symmetric? There are at least two views. The generated by fermionic (anticommuting) charges. We can first is based on the paradigm of condensed systems describe supersymmetry by saying that space-time is to be where unexpected and new symmetries often occur, although replaced by super-space-time, which has new coordinates in they are not present in the fundamental laws. The prime addition to the usual coordinates of space and time, that we example is the appearance of symmetry in the behavior of denote by ␪i, i ϭ 1, 2, . . . . The new feature of these long-range fluctuations of a system undergoing a second-order coordinates is that they are anticommuting numbers, i.e., ␪1␪2 phase transition. Here one has the phenomenon that at the ϭϪ␪2␪1. Supersymmetric physics is formulated in this su- fundamental, short distance or high energy, level there is no perspace. Thus all fields are function of x and t and the ␪i symmetry. Rather the symmetry emerges dynamically at large values. Super-symmetry is then a set of continuous transfor- distances. mation, rotations, of all of the coordinates of superspace. Could this be the reason for the ‘‘fundamental symmetries’’ These symmetries contain the usual relativistic symmetries of that we observe in nature? Could they be dynamical conse- , but in addition new symmetries, with new conse- quences of an asymmetric physics? I believe not. The lesson of quences. The most important consequence is that for every the in this century points to the opposite particle with spin J there must be another with spin J Ϯ 1/2. conclusion. As we explore physics at higher and higher energy, If the symmetry were exact these would be degenerate in mass. revealing its structure at shorter and shorter distances, we This is not what is observed in nature—thus supersymmetry discover more and more symmetry. This symmetry is usually must be broken. But, as we have learned, this is no problem; broken or hidden at low energy. I like to think of the first most symmetries are broke. If the scale of breaking is the scale paradigm as Garbage in—Beauty out, and the second as Beauty of the standard model then this symmetry could explain the in—Garbage out. At the fundamental level nature, for what- hierarchy problem. Furthermore it would then be visible at ever reason, prefers beauty and is marvelously inventive in which are just now becoming accessible. We eagerly inventing new forms of beauty. If this is the case then it await the experimental discovery of the signs of this (broken) provides us with an important tool for the exploration of symmetry at the next generation of particle accelerators—a nature. When searching for new and more fundamental laws discovery of new dimensions of space-time. of nature we should search for new symmetries. Finally, in recent years we have begun to seriously explore a new kind of theory based on a radical extension of the New Symmetries conceptual framework of local : . String theory is the most ambitious candidate for a Current theoretical exploration in the search for further unified theory of all the interactions that naturally embodies in unification of the forces of nature, including gravity, is largely a consistent fashion quantum gravity. It contains within it all based on the search for new symmetries of nature. Theorists the familiar symmetries which we have discovered play a role speculate on larger and larger local symmetries and more in nature. It indeed appears to have all the ingredients we need intricate patterns of symmetry breaking in order to further to derive or explain the standard model. In addition, there are unify the separate interactions. Most exciting is the speculation hints within the theory that it embodies new and strange concerning new kinds of symmetry, which could explain some symmetries that we are now trying to understand. of the most mysterious features of nature. Foremost among Thus, once again we are embarked on a new stage of these is supersymmetry that has the ability to unify bosons and exploration of fundamental laws of nature, a voyage guided fermions into a single pattern, to unify matter and force, and largely by the search for and the discovery of new symmetries. to help explain the mysterious fact that the mass scale of atomic and nuclear physics is so much smaller than the scale deter- Supported in part by the National Science Foundation under Grant mined by gravity (the hierarchy problem). PHY90-21984. Downloaded by guest on September 26, 2021