" 262 THE ELEMENTS OF PHYSICS not exclude the possibility that distant galaxies may consist entirely of antimatter. Since the visible light emitted by atoms in no way differs from the light emitted by antiatoms there is no optical way of finding this out. However, radioactive processes in antistars differ from the corre- sponding processes in stars in that they are accompanied by emission of anlineuirinos instead of neutrinos, and antineutrinos differ from neutrinos by their handedness. Thus, if it ever becomes feasible to develop neutrino telescopes, one may yet be able to decide whether distant galaxies do or do not contain antistars. Additional Suggested Reading Chew, G. F., M. Gfll-Mann, and A. H. Rosenfei.d. "Strongly Interacting Particles,'' Scientific American, February 1964, 74-93. Dyson, F. J. "Mathematics in the Physical Sciences," Scientific American, September 1964, 128-146. For d , KL. W . 1 ~h e World of ' Elemen tary Particles, W<ham, Mass, : B la i sde 1 1 , 1963. Frlsch, D. H., and A. M. Thorndikf. Elementary Panicles, D. Van Nostrand Co., Inc., Princeton, N.J., 1964. Fowler, W. B., and N. P. Samios. "The Omega-Minus Experiment," Scientific American, October 1964, 36-45. Graetz.fr, G. "Discovery of Nuclear Fission," American Journal of Physics *2(l%4), 9 15. Hill, R. D. "Resonance Particles, " : Scientific American, January 1963, 38-47. Kay, W. A. "Recollections of Rutherford," The Natural Philosopher I, 129-155., Waltham, Mass. : Blaisdcll 1963. Leach man, R. B. "Nuclear Fission," Scientific American, August 1965, 49-59. LEDERMAN, L. M. "The Two-Neutrino Experiment," Scientific American, March 1963, 60 70. O'Neill, G. R. "The Spark Chamber," Scientific American, August 1962, 36-43. Penman, S. "The Muon," Scientific American, July 1961, 46-55. Rossi, B. "High- Energy Cosmic Rays," Scientific American, November 1959, 134-149. Telegdi, V. L, "Hypernuclei," Scientific. American, January 1962, 50-56. YANG, C. N. Elementary Particles, Princeton University Press, Princeton, N.J., 1961. CHAP T E R 24 Conservation Laws and the Rano-e of Their Applicability The realm of Newtonian physics can be adumbrated by tracing the limits within which the laws of conservation of energy, momentum, and angular momentum are valid, and within which these arc the only conservation laws. Energy, momentum, and angular momentum are apparently sufficient to grasp the entire kinematics of the various forms matter is capable of assuming, and their conservation can be considered an experimental fact within the entire universe accessible to observation from our particular vantage point. The discovery of at least three types of interaction, beyond the universal gravitational interaction known to Newton, namely electromagnetism. and the "strong" and "weak" interactions assumed to account for transmutations among elementary particles, has led to recognition of additional attributes of matter not associated with its kinematic state (i.e., with rest or motion in space and time). Some of these attributes appear to be conserved universally, independent of the kind oT interaction responsible for the processes in which these attributes are passed on from one form of matter to another. Others appear to be conserved only in certain processes which are mediated by one, or more, but never all, of the supposedly basic interactions mentioned above. Although until this century the laws of conservation of energy, momentum, and angular momentum were tested only in processes governed by gravitation and electromagnetism, they have survived, unshaken by the discovery of strong and weak interactions; the belief in their universal validity was, in fact, instrumental in the discovery of new particles such as the neutrino (see Chapter 9). The proliferation of conserved attributes had a forerunner in the discovery of the law of conservation of electric charge, which was the subject of 263 264 THE ELEMENTS OF PHYSICS Chapter 17. In view or the indivisibility of the electronic charge e, one can formulate it in terms of a charge quantum number Q, which can have integer values only, so that Qe is the electric charge of the object characterized by the quantum number Q: the sum total of all quantum numbers Q in any dosed system remains constant. No exceptions to this law have been observed to date; charge Q seems to belong to the class of universally conserved attributes. For example, emission of an electron e~{Q — —1) by a nucleus requires an increase in Q of that nucleus by one unit, as in the reaction ™Co{Q = +27)^™Ni(0 = +28) + e (0 = -1) + *(G = 0). There exist at least three more quantum numbers (the baryon number B., the lepton number L, and the muon number LV) which are universally conserved, in the following sense. If one assigns to any baryon (see the definition in the preceding chapter, and Figure 23.6) the number B = + 1 , to any antibaryon the number B = ■— X, and the number B = to any particle not classified as a baryon, then the sum total of all baryon numbers in any closed system remains constant in time. This means, for example, that one cannot transform any boson into two baryons, even though energy, momentum, angular momentum, and electric charge were conserved in such a process. One can, however, transform a boson (B — 0) into a baryon-antibaryon pair (B = +1 — 1 = 0), provided ail the other conservation laws are also satisfied. In other words, creation or annihilation of a baryon requires simultaneous creation or annihilation of an antibaryon. Examples are the production of an antineutron n in the reaction p(B = +1) + p(B = +1) -> p(B = +1) + p(B = +1) + n {B = +4 + k£2?= — 1), and the annihilation or an antiprolon p in the reaction p{B = -1) + p(B = +1) -* vr 1 {B = 0) + 7t~{B = 0) + 7r\B = 0). A similar situation prevails, independent of baryon number conservation, among the leptous. By assigning the lepton number L — + 1 to neutrino v, electron e , and muon pr, L = - 1 to the respective antilcptons v, e+, p~ (see Figure 23.3), and £ = to all particles that are not leptons, one can summarize the outcome of a large number of experiments resulting in oc- currence or nonoccurrence of processes involving leptons as a conservation law; the sum total of all lepton numbers in any closed system remains constant in time. For example, the massless fermion emitted together with the electron in the decay of the neutron is definitely an antincutrino, n -> p + e~ + v, and not a neutrino, as this would violate the conservation of lepton number in the process. Another example is the decay of the "positronium," an "atom" consisting of an electron e (L= +\) and a positron eft = -1) into photons y(L = 0) as in the reaction e~ + e~ ->■ y + y. CONSERVATION LAWS AND THEIR APPLICABILITY 265 Finally, the discovery of the second type of neutrino v u , emitted in the decay of the muon, for example in the reaction pT -> e~ + v + v p , can be viewed as one of the many known consequences of a law of conservation of muon number. If one assigns the muon number L p = +1 to u~, v p , e~, v, and L = — 1 to the respective antiparticles (see Figure 23.3), one can summarize a large number or known facts by saying that the sum total of muon numbers in any closed system remains constant in time. For example, the decay of the muon u~ into an electron e~ and a photon y has never been observed, since it violates conservation of muon number, even though it would be consistent with the laws of conservation of energy, momentum, angular momentum, electric charge, and lepton number. Examples of reactions for which muon number L fi is conserved are the capture reactions v + %—p-p -f fi~ and v + p -" n + p~, which are used in the detection of muon neutrinos. All the conservation laws mentioned thus far hold without exception. Until 1956 it was widely believed that another conservation law of universal validity existed — the law of conservation of parity P (see the definition off in Figure 23.4). However, the discovery of processes, mediated by weak interactions, which violate inversion symmetry (as described in Chapter 16) has played havoc with that notion, and the parity P has become recognized as an attribute that is conserved only in processes mediated by strong and electromagnetic interactions. Thus the electromagnetic decay of the pion ir" into two photons tt° -*■ 2y conserves parity P, but the weak decay of the neutron n -> p + e~ + v does not conserve P. Example of an attribute whose conservation has an even smaller range of validity is the isospin T (see definition in Chapter 23). Any object, whether elementary or composite, engaging in strong interactions can be characterized by two quantum numbers T and 7 3 , where IT + 1 is the number of possible charge states of the object, and T s tells which of these charge states is realized. These isospin quantum numbers are analogous to the quantum numbers J and7 3 devised to characterize the angular momentum state of an object, if the angular momentum J is capable of 2/ + 1 orientations, and /«, tells which of these orientations is realized. For example, an object consisting of (a) a nucleon, T N i.e., 27\r + 1 =2 possible charge states exist (namely, proton T 3 = +£ and neutron T 3 = -r— f) and (b) a" pion, F, = 1, i.e., 2T r + 1 = 3 possible charge states exist (namely, •n- with 7 3 = +1, tt° with 7~ 3 = 0, and tr with T$ = —1), can have either the isopsin T = 71 + T. v = § (corresponding to 27" + I =4 values of charge, ranging from Q — +2 for/>7r+ with T K — +f, io Q — —1 for nit- with T A = — !]), or the isospin 7 = 71 — T$ = I (corresponding to 266 THE ELEMENTS OF PHYSICS TT + 1 = 2 values of charge, namely, cither Q = + 1 for p-n* and im + or Q = for/jTT and nir : ''). This addition of isospins is analogous Lo the addition of two angular momenta J L = 1 and J s — |, which may give rise either to a quartet 6f angular momentum states belonging to J = ,/, + ./ a = I + s = ; J' or to a doublet of states with / — 1 — | = ■■§ depending on whether the angular momentum vector belonging to ./ = | aligns itself parallel or anti- parallel to the vector belonging to J — 1. Now the evidence of strong interactions allows one to conclude that isospin is conserved in these, and only in these, interactions: the quantum numbers T and T% in any closed system remain constant in time, as iong as only strong interactions between tiie constituents of the system are taken into account. For example, the electromagnetic decay tt° -> 2y of the neutral pion (7' =1, r a = 0) into two photons {T = 0, 7\, = 0) is a transition in which the quantum number T changes by one unit. The weak interactions violate the conservation of both r 3 and T. For example, the weak decay X~-yn + n- of the baryon X" (7' = 1, 7T a = - 1 ) into a neutron n (7' = 4, t%~ —1.) mid a pion tr (f = 1, T 3 = - 1) is a transition in which both T and T z change by -J- unit. Finally, there is a law of conservation of hypercharge 7 (see definition in Chapter 23), valid for processes mediated by strong or electromagnetic interactions, but violated by the weak interactions. For example, the weak decay A° -^ p + tt~ of the baryon A°( Y = 0) into a proton p(Y = 1) and a pion tt "( Y — 0) is a transition in which Y changes by one unit. In Table 24.1 are summarized the ranges of validity of the various conser- vation laws discussed thus far. In view or this baffling evidence the question inevitably arises: Does one have to accept all these conservation laws as primary facts of nature, or can one find some underlying principles, fewer in number, from which one can derive these laws, and give reasons for the peculiar limitations encountered by some of them ? This question remains essentially unanswered today, even though for several decades now it has been at the center of attention among many workers in theoretical physics. The modern attack on this problem was initiated between 1910 and 1920 by Emmy Noether, who discovered a remarkable link between symmetry properties of physical objects and conservation laws. Noether found that if one requires the action (see Chapter 17) of any field to be unchanged under a continuous symmetry operation, then there exists a field quantity, characteristic of that operation, which is conserved. A continuous operation is one which can be obtained by a continuous succession of small steps from the identity operation. Examples of continuous operations are translations in time, translation in space, and rotations in space. A translation operation, for example, acquires the rank of a symmetry operation if upon application to the action no change in the action results, meaning no change in the CONSERVATION LAWS AND THEIR APPLICABILITY 267 "laws" following from the corresponding variational principle that is the subject of Chapter 18. It is quite basic to the very feasibility of physics that the so-called laws of nature be invariant with respect to translations in time. If an experiment is carried out. now under given conditions and has a certain result, the same experiment carried out. at a later time under the same conditions must give the same result. If this were not true it would make no sense lo write books Taule 24.1 Range of validity of conservation laws governing various quantum numbers attributable to physical objects. Conserved in strong electromagnetic weak Attribute interactions Energy E yes yes yes Momentum p yes yes yes Angular momentum / yes yes yes h yes yes yes Baryon number B yes yes yes Lepton number L yes yes yes Nuon number L fl yes yes yes Charge number Q yes yes yes Pari I v P yes yes no 1 lypercharge Y yes yes no Isospin T- d yes yes no T yes no no on physics whose empirical contents is intended to be valid permanently. By requiring this kind of in variance, for example, Noether's theorem enables one to see the law of conservation of energy as a consequence of the symmetry with respect to translations in time. In a similar vein the three conservation laws following according to Noether's theorem from imposition of the symmetries with respect to the three spatial translations can be interpreted as the laws of conservation of the three components of momentum, and the invariance under spatial rotations can similarly be looked upon as the origin of the law of conservation of angular momentum. Although it may be esthetically very satisfying to have thus accounted for the dynamic conservation laws in terms of symmetries in space and time, from a strictly logical point of view Noether's theorem up to this point amounts to a tautology, because the number of symmetries that must be 26S THE ELEMENTS OF PHYSICS postulated is equal to the number of resulting conservation laws, and one might then as well postulate these laws directly. Nevertheless, Noether's theorem provided the impetus for a large number or ingenious attempts to link the other conservative laws with other symmetry operations, not necessarily confined to operations in space and time. For example, in electromagnetic theory one can perform gauge transformations of the potentials which leave the electromagnetic field invariant; a number of authors have attempted to understand the law of conservation of electrie charge as consequence of this peculiar symmetry of the electromagnetic fields. In recent years considerable attention has been paid to discontinuous symmetry operations, such as reflections or inversions of coordinates, which cannot, in principle, evolve continuously from the identity operation. They are, therefore, not within the domain of Noether's theorem. Some of these, such as the symmetry under inversion of coordinates, which is violated by weak interactions (see Chapter 16), can be understood as origin of conser- vation laws, such as the law of conservation of parity, which is violated hy weak interactions. Others, such as the symmetry under reversal of motion (see Chapter 6), do not lead to introduction of a conserved quantum number, such as parity, because of formal reasons whose physical significance is ill understood at present. Thus there arc symmetries that do not lead directly to any conservation law, and there are universally valid conservation laws, such as the conservation of B, L, £,„, whose link with symmetries, if there is any, remains obscure. Additional Suggested Reading Feinbf.ro, C, and M, Goldhabfr, "Conservation Laws," Scientific American, October 1963, 36-45. McCauley, G. P. "Parity," American Journal of Physics 29 (1961), 173-181. Wicner, E. 1\ "Violations of Symmetry in Physics," Scientific American, December 1965,28-37. APPENDIX O N F The Probability Concept in Physics One of the principal aims of theoretical physics is the development of mathematical techniques enabling one to predict the outcome of experiments. A notable example is Newtonian mechanics, which consists of a set of "laws" enabling one to predict the future position and velocity of an object under the influence of some given force, provided one knows the initial conditions, i.e., has certain knowledge about both position and velocity of the body at some initial instant of time. Thus Newton's laws of motion together with Newton's law of gravitation have enabled theoretical astronomers to compute tables of the future positions of all planets revolving around the sun on the basis of past observations of position and velocity of these objects. However, in many branches of physics one encounters objects and situations whose initial conditions are partially or wholly unknown. This ignorance about the initial conditions may have various origins. In most macroscopic cases of this kind it stems from the complexity of the object, making an exhaustive determination of the initial conditions of all us parts a practical impossibility. For example, application or Newtonian mechanics to the atmosphere for the purpose of weather prediction suffers from the impossibility of obtaining complete information about the meteorological state of the atmosphere at any given initial instant of time. Fortunately, the meteorologist is not con- demned to complete frustration by his limited knowledge of present weather conditions, became he has records, and on the basis of past records he can resort to probability statements about the future weather. This possibility of weather prediction in probabilistic terms extends to similar predictions of a very primitive kind. For example, when an old resident in a district looks at the sky, sniffs the air, and then pronounces, "It is probably going to rain tomorrow," what he really means to say, if he is at all reliable in his observations, is that, whenever in the past sky and 269