Invitation to a Contemporary Physics (2004)

he starved to death. But, of course, we know that the ass will eat — the slightest bias, even if an autosuggestion, or merely thinking about it, may make him turn to one or the other of the two stacks of hay!).

Now, we turn to the physically interesting example of a system of many interacting particles in thermal equilibrium. The branch of physics that deals with such systems is called statistical mechanics (see Appendix C). Most inanimate systems are of this kind. A good example is that of a ferromagnet. Take a piece of iron. For our purpose, we may regard the atoms of iron as tiny magnets, compass needles if you like. The origin of these tiny magnets, or the magnetic moments as they are called, lies in the spinning electrons. But this detail is not relevant for our discussion. These tiny magnets, shown as arrows in Fig. 1.14, interact with each other. The interaction is due to the quantum-mechanical ‘exchange’ of electrons because of their indistinguishability. But this is again a detail not important for our discussion. What is really important is that the interaction energy depends on the angle between these magnetic moments. Thus if we turn all the atomic magnetic moments around by the same angle about the same axis, the energy of the system will remain the same. We say that the law governing the system is spherically symmetric. Furthermore, for a ferromagnet the energy is minimum when the magnetic moments are all parallel to each other. It is clear, therefore, that these atomic moments will tend to align parallel to each other. At high temperatures, however, the thermal agitation will make these moments point in di erent directions at random so that there is no net magnetization. The state of the system will be spherically symmetric

— it has the same symmetry as the law of interaction. We call this disordered, hightemperature symmetric phase the paramagnetic phase. As the sample is su ciently cooled, however, the interaction energy favoring parallel alignment of the magnetic moments wins over the disrupting tendency of the thermal agitation. When this happens the magnetic moments align parallel to each other on average and thus the

→

f(M)

Mx

→

M

Figure 1.14: Spontaneous symmetry breaking in a ferromagnet at Tc. f(M) denotes thermodynamic potential.

system develops a net magnetization M, which grows in magnitude with decreasing temperature. This low-temperature ordered state is called the ferromagnetic phase. The temperature Tc at which the system makes the continuous transition from the high-temperature disordered phase to the low-temperature ordered phase is called the critical temperature, or the Curie temperature. The magnetization which is a measure of order is referred to as the order parameter. (The physics of continuous phase transition, often called the second-order phase transition, at and about the critical temperature has been an extremely active area of research of our times. It is determined almost entirely by the symmetry and the dimensionality of the order parameter and is quite independent of the microscopic details of chemical composition, etc. This ‘universality’ of the ‘critical behavior’ is most fascinating but we must let it pass). The question now is what should be the direction of this net magnetization M. Inasmuch as the energy depends only on the relative orientation of the atomic moments, all directions of M are equally probable statistically. And yet in a given realization some direction of M must get selected. This state is then symmetric only for rotations about this direction of M — it has only an axial symmetry which is a subgroup of the full spherical symmetry of the interaction law. The symmetry is thus spontaneously broken! For a large, in principle infinite, system an arbitrarily small magnetic field or anisotropy will fix the direction of M. The connection with our mechanical example should be obvious. The order parameter (magnetization M) plays the role of displacement x. Instead of mechanical potential energy V (x) which was to be minimized, here we have a thermodynamic potential (free energy) f(M), which is to be minimized. The only detail that di ers is that whereas the position x in our mechanical case was a scalar (one-dimensional), the order parameter M is a vector. Thus, in the mechanical case the two equivalent minima were separated by a potential barrier in the broken symmetry phase, while in the ferromagnetic case all the equivalent minima (di ering only by the direction of M) are degenerate (i.e., have the same free energy) and M can in principle freely gyrate among them. Perhaps a better mechanical analogue would have been the marble in a punted wine bottle. (Incidentally, this freedom leads to the possibility of certain waves propagating in the broken symmetry phase whose frequency tends to zero as the wavelength tends to infinity (i.e., they are massless). We call these Goldstone modes. For the (antiferro-)magnetic case, these are the spinwaves). All this plays an important role in the physics of phase transition in condensed matter. We emphasize that this is a highly cooperative phenomenon resulting from interaction among large number, infinite in principle, of particles, their spins in this case.

Now, how can all this possibly bring about unification of the fundamental forces of Nature? This is a magnificent and highly technical obsession of contemporary physics. We will try to give just the flavor of it in plain words. Any symmetry can be broken spontaneously. In particular and most importantly, it can be the gauge symmetry. It is the combination of gauge symmetry and spontaneous symmetry breaking that is central to unification. Consider the simplest case when the

matter consists of charged particles (electrons). As we have already seen, the global symmetry (namely, the irrelevance of absolute phase) when gauged locally, generates the electromagnetic field automatically, which in the quantum version is the photon. (It is inherent in this mechanism that the photon — the gauge field — be massless). Now in quantum theory, the interactions between particles are mediated by the exchange of quanta of some field (much the same way as the exchange of a handball between two players will exert an e ective force of repulsion between them. For attraction, let them exchange boomerangs!). Thus, the photon mediates interaction between charges. It also follows from general quantum principles that the range of interaction be inversely proportional to the rest mass of the quanta exchanged. This is why the range of the electromagnetic interaction is infinite. This intimate ‘genetic’ connection between matter (electrons) and the gauge field (photons) leads us to expect an induced change in the character of the gauge field when the matter undergoes a phase transition in which the very global symmetry, whose local gauging generated the gauge field, undergoes spontaneous breakdown. In point of fact, it would be very surprising if it were otherwise. We already see this e ect in a superconductor in the laboratory (see Chapter 3 on Superconductivity). Here electrons undergo a transition in which the phase of this collective (macroscopic) wave function takes on a definite value, breaking thus the global symmetry spontaneously. This induces the photon to acquire a non-zero mass, making it impossible for it to propagate very far into the superconductor. This explains the famous Meissner e ect, namely that the magnetic field is expelled from the bulk of a superconductor.

This was the simplest, but a most striking demonstration, of the change of character of a gauge field induced by the SSB of the matter (field). All we have to do now is to generalize to more complicated internal symmetries that can be imagined and indeed have been postulated. Thus, there may be several gauge fields generated by local gauging. They may be all symmetry related and thus of the same character. Now, if the matter undergoes SSB, the group of these gauge fields may be split into subgroups, and di erent subgroups may acquire di erent masses. Successive phase transitions (and the associated SSB’s) may generate thus a gamut of fields with di erent characters. This generation of di erent gauge fields (fundamental forces) by the descent of symmetry due to SSB, from the single most symmetric initial entity is the dream of the grand unified theory (GUT). It has already been partially realized in the unification of the electromagnetic and the weak interaction by Glashow, Salam and Weinberg for which they won the 1979 Nobel Prize in Physics.

One has a plausible scenario in mind that the universe began totally symmetric with a Big Bang some 15 billion years ago. As it expanded it cooled and underwent successive phase transitions. The associated SSB’s led to the diversity of fields that survive at the present epoch. And what a diversity — if the strong interaction measures unity on a certain scale, the electromagnetic interaction will measure 10−2, the weak interaction 10−5 and the gravitational interaction 10−34!

The strong interaction acts on hadrons (protons, neutrons, pions, etc., or their postulated building blocks, the quarks) but not on leptons (electrons, neutrinos, etc.) and is short-ranged. The weak interaction involves neutrinos and has a still shorterrange. The electromagnetic interaction acts on all charged particles and has infinite range. The gravitational interaction is the weakest of all, but acts universally on everything and has infinite range. And yet all may have a common origin. This reminds one of the concept of the Nirguna Brahma of the ancient Hindus — formless, featureless, totally symmetric pure existence, from which all diversity originated, shall we say, by spontaneous symmetry breaking!

This brings us to the end of our exploration of symmetry. We have seen its power. Obviously, symmetry cannot answer all the why’s and how’s, but it does reduce them to fewer why’s and how’s. To the philosophical question of why Nature is so symmetric, we can perhaps answer thus. Symmetry is, in the ultimate analysis, absence of bias. It is an expression of justice. There is a principle of insu cient reason against asymmetry. A sphere is admitted. But a deviation from sphericity must bide our question.

Galileo had spoken of the great Book of Nature. We should perhaps add that the first and the last Chapters of this Book are on symmetry and its spontaneous breakdown, respectively.

1.6 Summary

Symmetry means invariance of an object with respect to a set of operations called symmetry operations to be performed on it. The object may be the geometrical form of body such as a crystal of common salt, and the set of operations may the geometrical operations of translation along a direction, rotation about an axis, or reflection in a plane. The symmetry operations may be continuous or discrete, physically performable or non-performable. More importantly, the object may be a law of nature itself expressed mathematically by a certain equation. Symmetry then means the invariance, or rather covariance, of the form of the equation under the mathematical transformations corresponding to the symmetry operations, that may not be geometrical in nature. Symmetry is a powerful physical principle that helps us not only simplify calculations and classify and unify diverse objects, it also restricts the possibilities in the absence of complete knowledge of the physical world. It creates new physics when a symmetry is requested on intuitive grounds. There is a branch of mathematics called group theory that provides the proper and powerful language for dealing with symmetry. Symmetry has played a fundamental role in quantum physics, particularly in the domain of high-energy physics, where its predictive power has been fully vindicated. The rather abstract idea of gauge symmetry is one of the profoundest concepts produced by the human mind. Any symmetry, however compelling aesthetically it may be, must be established experimentally.

Thus, parity signifying the left-right symmetry between an object or a process and its mirror image turned out to be false in certain fundamental processes involving neutrinos. Symmetry can also be broken spontaneously when there is a phase transition. The idea of spontaneous symmetry breaking has played a decisive role in our understanding of phase transitions in general, and in the context of grand unified theories and the early universe in particular. The search for deeper, hidden symmetries of Nature continues.

1.7 Further Reading

Books

•M. Gardner, The New Ambidextrous Universe, 3rd Ed. (W. H. Freeman and Company, New York, 1995).

•H. Weyl, Symmetry (Princeton University Press, Princeton, 1952).

•I. Hargittai, Symmetry 2 : Unifying Human Understanding (Pergamon Press, Oxford, 1989).

•A. Zee, Fearful Symmetry: The Search for Beauty in Modern Physics

(Macmillan Publishing Company, New York, 1986).

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we will begin by considering light as it is normally found and discuss its characteristic features, so as to contrast them with the distinctive properties of laser light. There exist now many types of lasers, which use di erent substances as active media, achieve atomic excitations through di erent techniques and generate light at di erent wavelengths. They all share, however, the same basic principles. We will describe many applications of lasers, not only in technology, but also in basic research in physics where the use of this tool has led us to new directions, taken us to new frontiers.

2.2 Conventional Light Sources

Light is ordinarily produced in hot matter. In 1704, in his second great work, Opticks, Isaac Newton wrote:

“Do not all fixed Bodies, when heated beyond a certain degree, emit Light and shine; and is not this Emission performed by the vibrating motions of their parts?”

2.2.1 Light and Electromagnetic Radiation

Light can be described as a perturbation of space of the kind one may observe in the vicinity of an electric conductor or near the path of a rapidly moving charged particle. Once we have observed how iron filings on a glass plate through which a current-carrying wire passes point to form closed circles about the central wire, or watched a small probe charge attached to the end of a very thin thread respond to the particle’s motion, we are left in no doubt that space nearby is pervaded with a certain distribution of force that is called electromagnetic field. The precise way in which this field varies in space and time is described concisely by a set of di erential equations, due to James Clerk Maxwell (1873), which replaced and generalized all previous empirical laws of electricity and magnetism. According to the theory encapsulated in these equations and later confirmed by experiments, the electromagnetic field is a disturbance that propagates from point to point in all accessible directions and behaves at large distances from the source as a wave. For this reason we may refer to it as an electromagnetic wave. One of its basic properties is that it can convey energy through empty space without transferring matter, always moving at the same high speed, the speed of light (c = 300 000 km/s in empty space), and when it stops moving, it ceases to exist. We implicitly refer to this kind of energy transfer whenever we use the term ‘electromagnetic radiation.’ An electromagnetic wave has all the characteristics, except visibility, of light: it can be reflected, refracted or di racted. Light, in e ect, is the visible form of electromagnetic radiation.

A light wave, like any other electromagnetic wave, is described by the variations in space and time of two vectorial quantities, namely, an electric field E and a

E

B

Figure 2.1: Representation of an electromagnetic wave as space and time variations of electric field E and magnetic field B. In (a) E stays at all times in a plane passing by the propagation line, and the polarization is said to be planar. In (b) E changes direction as it evolves, and the polarization is nonplanar.

magnetic field B. These vectors remain at all times perpendicular to each other and perpendicular to the direction of propagation. So the wave in question is a traveling transverse wave, much like the ripples on a disturbed water surface. We refer to the direction of the electric field vector as the wave polarization. Over a period of time, the electric field defines a vibration pattern which may be projected in the E–B plane as a line segment, a circle or an ellipse (Fig. 2.1). You can easily discover in which way a light wave is polarized: let it pass through a polarizer — a tourmaline crystal or a Polaroid filter — and observe the output as you slowly rotate the polarizer. A typical polarizer is a substance composed of long straight molecules aligned perfectly parallel to one another which strongly absorb the electric component parallel to the molecules, but let the perpendicular component pass on through with almost no absorption.

Electromagnetic waves di er from one another in their characteristic wavelengths, λ, the distances from one peak to the next. The whole range of electromagnetic wavelengths, called the electromagnetic spectrum (Fig. 2.2), covers values from the very small (for high-energy gamma rays) to the very large (for low-energy radio waves). A small portion of it forms the optical, or visible spectrum, extending from about 400 nm (violet) to about 700 nm (deep red). (1 nm is 1 nanometer, or one billionth of a meter.) Instead of wavelength,1 we may equivalently speak of wave frequency, ν, which refers to the rate of vibration and is measured in cycles per second, or Hertz (Hz); or also angular frequency, ω, which is just the ordinary frequency multiplied by 2π, and so is given in radians per second (rad/s). In practice, we can characterize a given radiation by its wavelength, λ, or by its frequency, ν or ω. For example, orange light has wavelength λ = 600 nm, or frequency ν = 5 ×1014 Hz, or ω = 3 × 1015 rad/s. It is the same information said in di erent ways (Problem 2.1).

In many respects, light behaves as a stream of quantum particles, which we call photons. The production of an electric current by a sheet of copper when irradiated by intense light (a process known as the photoelectric e ect) and the absorption

1λ = c/ν, where λ and ν are the wavelength and ordinary frequency of the wave; the angular frequency ω is defined as ω = 2πν. A brief discussion of the concepts of waves and fields can be found in Appendix A.