Introduction to Groups, Invariants and Particles Frank W. K. Firk, Emeritus Professor of Physics, Yale University 2000 CONTENTS 2 1. INTRODUCTION 3 2. GALOIS GROUPS 8 3. SOME ALGEBRAIC INVARIANTS 20 4. SOME INVARIANTS IN PHYSICS 29 5. GROUPS CONCRETE AND ABSTRACT 43 6. LIES DIFFERENTIAL EQUATION, INFINITESIMAL ROTATIONS, AND ANGULAR MOMENTUM 57 7. LIES CONTINUOUS TRANSFORMATION GROUPS 68 8. PROPERTIES OF n-VARIABLE r-PARAMETER LIE GROUPS 9. MATRIX REPRESENTATIONS OF GROUPS 10. SOME LIE GROUPS OF TRANSFORMATIONS 11. THE GROUP STRUCTURE OF LORENTZ TRANSFORMATIONS 12. ISOSPIN 78 83 94 107 115 13. LIE GROUPS AND THE STRUCTURE OF MATTER 128 14. LIE GROUPS AND THE CONSERVATION LAWS OF THE PHYSICAL UNIVERSE 159 15. BIBLIOGRAPHY 163 1 3 INTRODUCTION The notion of geom etrical symmetr y in Art and in Natu re is a familiar one. In Modern Physics, this notion ha s evolved to include symmetries of an abstract kind. These new symmetries play an essential par t in the theories of the microstructure of matter. The basic symmetries found in Natu re seem to originate in t he ma thematical stru cture of t he laws themselves . laws that go vern the motions of the g alaxies on the one hand and the motions of quarks in nucleons on the oth er. In the N ewtonian era, t he laws of Natu re we re de duced from a sm all number of imperfect obs ervations by a small number of renowned scientists and mathematicians. It was not until the Einsteinian era , however, that the significance of the symmetries associated with the laws was fully app reciated. The discovery of space-time symmetries has led to the widely-held belief that the laws of Nature c an be derived from symmetry , or invariance, principles. Our incomplete knowledge of the fundamental interactions means th at we are not yet in a position to confirm this belief. We therefore use arguments based on emp irically established laws and rest ricted sym metry principles to guide us in our search for the fundamental symmetries. Fre quently, it is important to understand w hy the symm etry of a system is observed t o be broken. In Geometry, an object with a definite shape, size, location, and 4 orientation constitutes a st ate whose symmetry proper ties, or i nvariants, are to be studied. Any transformation that leaves the state un changed in form is called a s ymmetry transformation. The greater the number of symmetry transformations that a s tate can underg o, the higher its symmetry . If the number of conditions that define th e state is reduced then the symm etry of the state is increased. For exa mple, an object characterized by oblateness alone is symmet ric under all transformat ions except a change of shape. In describing the symmetry of a state of the most general kind (not simply geometric), the algebraic structure of the set of symmetry operators must be given; it is not sufficient to give the number of operations, and not hing else. The law of combination of the operators must be stated. It is the algebraic group that fully characterizes the symmetry of t he general state. The Theory of Groups came ab out unexpectedly. G alois showed that an equation of degree n , where n is an integer greater than or equal to five cannot, in general, be solved by algebraic means. In the course of this gre at wo rk, h e developed the ideas of Lagra nge, Ruffini, and Ab el and introduced th e concept of a group. Galois discussed the functional relationships amon g the roots of an equation, and showed t hat the relationships have symm etries associated with them un der permut ations of the roots. The operators that transform one functional relationship into 5 another are elements of a se t that is characteristic of the equation ; the set of operators is called the G alois group of the equati on. In the 1 850s, Cayley showed that ever y finite group is isomorp hic to a cer tain permu tation group. The geometrical symm etries of crystals are described in t erms of finite group s. These symmetries are discussed in many standard works (see bibliography) and therefore, they will not be discussed in this book. In the b rief period bet ween 1924 and 1 928, Quantum Mechanics was developed. Almost immediately, it was recognized by Weyl, and by Wigner, that certain parts o f Group Theory could be used as a p owerful analytical tool in Quan tum Physics. Their ideas have been developed over the decades in man y are as that range from the Theory of Solids to Particle Phy sics. The essential role played by groups th at are characterized by par ameters th at va ry continuously in a given range was first emphasized by Wigner. T hese group s are know n as Lie Groups. They have become increasingly important in many branches of contempora ry physics, par ticularly Nuclear an d Particle Phys ics. Fifty years after Galois had introduced th e concept of a group in t he Theory of Eq uations, Lie introduced th e concept of a continuous transformation group in the Theory of Differential Equations. Lies theo ry unified many of t he disconnected methods of solving differential equations t hat h ad evolved over a period of two hundred years. Infinitesimal unitary transformat ions play a key role in 6 discussions of the fundamental conservation laws of Physics. In Classical Dynamics, the invariance of the equations of motion of a particle, or system of particles, unde r the Galilean transformation is a basic par t of everyday relativity. The search for the transformation that leaves Maxwells equations of Electromag netism unc hanged in form (invariant) under a linear transformation of the space-time coordinates, led to the discovery of the Lorent z transformation. The fundamental importance of this tra nsformation, and its related invariants, cannot be overstated. This introduction to Group Theory , with its emph asis on Lie Groups and their application to the stud y of symmetries of t he fundamental constituents of matter, has its origin in a one-semester course that I ta ught at Yale University for more than ten y ears. The course was developed for Seniors, and advanced Juniors, majoring in the Physical Sciences. The students had generally completed the core c ourses for their majors, and had taken intermediate level courses in Linear A lgebra, Real and Com plex Analysis, Ord inary Linear Differential Equations, and some of t he Sp ecial Fun ctions of Physics. Group Theory was not a mathematical requirement for a degree in the Phy sical Sciences. The majority of existing undergra duate textbooks on Group Theory and its applications in Phys ics tend to be either highly qua litative or highly mathematical. The pu rpose of this introduction is to steer a middle course that provides the stud ent w ith a sound mathematical basis for st udying the symm etry prope rties of t he fundamental particles. It is not generally appreciated by Physicists that 7 continuous tr ansformation groups originated in t he Theory of Differential Equ ations. The infinitesimal generators of Lie Group s therefore have forms that involve differential operators and their commutators, and these operators and their algebraic properties have found, and continue to find, a natural place in the development of Quantum Physics. Guilford, CT. June, 2000 2 8 GALOIS GROUPS In the early 19th - century, Abel proved that it is not possible to solve the general polynomial equation of degree greater than four by algebraic means. He attempted to characterize all equations that can be solved by radicals. Abel did not solve this fundamental problem. The problem was taken up and solved by one of the greatest innovators in Mathematics, namely, Galois. 2.1. Solving cubic equations The main ideas of the Galois procedure in the Theory of Equations, and their relationship to later developments in Mathematics and Physics, can be introduced in a plausible way by considering the standard problem of solving a cubic equation. Consider solutions of the general cubic equation Ax3 + 3Bx2 + 3Cx + D = 0, where A - D are rational constants. If the substitution y = Ax + B is made, the equation becomes y3 + 3Hy + G = 0 where H = AC - B2 and G = A2D - 3ABC + 2B3. The cubic has three real roots if G2 + 4H3 < 0 and two imaginary roots if G2 9 + 4H3 > 0. (See any standard work on the Theory of Equations). If all the roots are real, a trigonometrical method can be used to obtain the solutions, as follows: the Fourier series of cos3u is cos3u = (3/4)cosu + (1/4)cos3u. Putting y = scosu in the equation y3 + 3Hy + G = 0 (s > 0), gives cos3u + (3H/s2)cosu + G/s3 = 0. Comparing the Fourier series with this equation leads to s = 2 v(-H) and cos3u = -4G/s3. If v is any value of u satisfying cos3u = -4G/s3, the general solution is 3u = 2np 3v, where n is an integer. Three different values of cosu are given by u = v, and 2p/3 v. The three solutions of the given cubic equation are then scosv, and scos(2p/3 v). Consider solutions of the equation x3- 3x + 1 = 0. 10 In this case, H = -1 and G2 + 4H3 = -3. All the roots are therefore real, and they are given by solving cos3u = -4G/s3 = -4(1/8) = -1/2 or, 3u = cos-1(-1/2). The values of u are therefore 2p/9, 4p/9, and 8p/9, and the roots are x1 = 2cos(2p/9), x2 = 2cos(4p/9), and x3 = 2cos(8p/9). 2.2. Symmetries of the roots The roots x1, x2, and x3 exhibit a simple pattern. Relationships among them can be readily found by writing them in the complex form 2cos. = ei. + e-i. where . = 2p/9 so that x1 = ei. + e-i. , x2 = e2i. + e-2i. , and x3 = e4i. + e-4i. . Squaring these values gives x12 = x2 + 2, x22 = x3 + 2, and x32 = x1+ 2. 11 The relationships among the roots have the functional form f(x1,x2,x3) = 0. Other relationships exist; for example, by considering the sum of the roots we find x1 + x22 + x2 - 2 = 0 x2 + x32 + x3 - 2 = 0, and x3 + x12 + x1 - 2 = 0. Transformations from one root to another can be made by doubling-the- angle, . The functional relationships among the roots have an algebraic symmetry associated with them under interchanges (substitutions) of the roots. If is the operator that changes f(x1,x2,x3) into f(x2,x3,x1) then f(x1,x2,x3) . f(x2,x3,x1), 2f(x1,x2,x3) . f(x3,x1,x2), and 3f(x1,x2,x3) . f(x1,x2,x3). 3 The operator = I, is the identity. In the present case, - 2) = (x22 - x3 - 2) = 0, (x12 - x2 and 2(x12 - x2 - 2) = (x32 - x1 - 2) = 0. 12 2.3. The Galois group of an equation. The set of operators {I, , 2} introduced above, is called the Galois group of the equation x3 - 3x + 1 = 0. (It will be shown later that it is isomorphic to the cyclic group, C3). The elements of a Galois group are operators that interchange the roots of an equation in such a way that the transformed functional relationships are true relationships. For example, if the equation x1 + x22 + x2 - 2 = 0 is valid, then so is (x1 + x22 + x2 - 2 ) = x2 + x32 + x3 - 2 = 0. True functional relationships are polynomials with rational coefficients. 2.4. Algebraic fields We now consider the Galois procedure in a more general way. An algebraic solution of the general nth - degree polynomial aoxn + a1xn-1 + ... an = 0 is given in terms of the coefficients ai using a finite number of operations (+,- ,,,v). The term "solution by radicals" is sometimes used because the operation of extracting a square root is included in the process. If an infinite number of operations is allowed, solutions of the general polynomial can be obtained using transcendental functions. The coefficients ai necessarily belong 13 to a field which is closed under the rational operations. If the field is the set of rational numbers, Q, we need to know whether or not the solutions of a given equation belong to Q. For example, if x2 - 3 = 0 we see that the coefficient -3 belongs to Q, whereas the roots of the equation, xi = v3, do not. It is therefore necessary to extend Q to Q', (say) by adjoining numbers of the form av3 to Q, where a is in Q. In discussing the cubic equation x3 - 3x + 1 = 0 in 2.2, we found certain functions of the roots f(x1,x2,x3) = 0 that are symmetric under permutations of the roots. The symmetry operators formed the Galois group of the equation. For a general polynomial: xn + a1xn-1 + a2xn-2 + .. an = 0, functional relations of the roots are given in terms of the coefficients in the standard way x1 + x2 + x3 .. .. + xn = -a1 x1x2 + x1x3 + .. x2x3 + x2x4 + ..+ xn-1xn = a2 x1x2x3 + x2x3x4 + .. .. + xn-2xn-1xn = -a3 . . x1x2x3 .. .. xn-1xn = an. Other symmetric functions of the roots can be written in terms of these 14 basic symmetric polynomials and, therefore, in terms of the coefficients. Rational symmetric functions also can be constructed that involve the roots and the coefficients of a given equation. For example, consider the quartic x4 + a2x2 + a4 = 0. The roots of this equation satisfy the equations x1 + x2 + x3 + x4 = 0 x1x2 + x1x3 + x1x4 + x2x3 + x2x4 + x3x4 = a2 x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 = 0 x1x2x3x4 = a4. We can form any rational symmetric expression from these basic equations (for example, (3a43 - 2a2)/2a42 = f(x1,x2,x3,x4)). In general, every rational symmetric function that belongs to the field F of the coefficients, ai, of a general polynomial equation can be written rationally in terms of the coefficients. The Galois group, Ga, of an equation associated with a field F therefore has the property that if a rational function of the roots of the equation is invariant under all permutations of Ga, then it is equal to a quantity in F. Whether or not an algebraic equation can be broken down into simpler equations is important in the theory of equations. Consider, for example, the equation x6 = 3. It can be solved by writing x3 = y, y2 = 3 or 15 x = (v3)1/3. To solve the equation, it is necessary to calculate square and cube roots . not sixth roots. The equation x6 = 3 is said to be compound (it can be broken down into simpler equations), whereas x2 = 3 is said to be atomic. The atomic properties of the Galois group of an equation reveal the atomic nature of the equation, itself. (In Chapter 5, it will be seen that a group is atomic ("simple") if it contains no proper invariant subgroups). The determination of the Galois groups associated with an arbitrary polynomial with unknown roots is far from straightforward. We can gain some insight into the Galois method, however, by studying the group structure of the quartic x4 + a2x2 + a4 = 0 with known roots x1 = ((-a2 + )/2)1/2 , x2 = -x1, x3 = ((-a2 - )/2)1/2 , x4 = -x3, where = (a 22 - 4a4)1/2. The field F of the quartic equation contains the rationals Q, and the rational expressions formed from the coefficients a2 and a4. The relations x1 + x2 = x3 + x4 = 0 are in the field F. 16 Only eight of the 4! possible permutations of the roots leave these relations invariant in F; they are x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4 { P1 = , P2 = , P3 = , x1 x2 x3 x4 x1 x2 x4 x3 x2 x1 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4 P4 = , P5 = , P6 = x2 x1 x4 x3 x3 x4 x1 x2 x3 x4 x2 x1 x1 x2 x3 x4 x1 x2 x3 x4 P7 = , P8 = }. x4 x3 x1 x2 x4 x3 x2 x1 The set {P1,...P8} is the Galois group of the quartic in F. It is a subgroup of the full set of twentyfour permutations. We can form an infinite number of true relations among the roots in F. If we extend the field F by adjoining irrational expressions of the coefficients, other true relations among the roots can be formed in the extended field, F'. Consider, for example, the extended field formed by adjoining (= (a22 - 4a4)) to F so that the relation x12 - x32 = is in F'. We have met the relations x1 = -x2 and x3 = -x4 so that x12 = x22 and x32 = x42. Another relation in F' is therefore x22 - x42 = . 17 The permutations that leave these relations true in F' are then {P1, P2, P3, P4}. This set is the Galois group of the quartic in F'. It is a subgroup of the set {P1,...P8}. If we extend the field F' by adjoining the irrational expression ((-a2 - )/2)1/2 to form the field F'' then the relation x3 - x4 = 2((-a2 - )/2)1/2 is in F''. This relation is invariant under the two permutations {P1, P3}. This set is the Galois group of the quartic in F''. It is a subgroup of the set {P1, P2, P3, P4}. If, finally, we extend the field F'' by adjoining the irrational ((-a2 + )/2)1/2 to form the field F''' then the relation x1 - x2 = 2((-a2 - )/2)1/2 is in F'''. This relation is invariant under the identity transformation, P1 , alone; it is the Galois group of the quartic in F''. The full group, and the subgroups, associated with the quartic equation are of order 24, 8, 4, 2, and 1. (The order of a group is the number of distinct elements that it contains). In 5.4, we shall prove that the order of a subgroup is always an integral divisor of the order of the full group. The order of the full group divided by the order of a subgroup is called the index 18 of the subgroup. Galois introduced the idea of a normal or invariant subgroup: if H is a normal subgroup of G then HG - GH = [H,G] = 0. (H commutes with every element of G, see 5.5). Normal subgroups are also called either invariant or self-conjugate subgroups. A normal subgroup H is maximal if no other subgroup of G contains H. 2.5. Solvability of polynomial equations Galois defined the group of a given polynomial equation to be either n, (see 5.6). The Galois method therefore involves the following steps: the symmetric group, Sn, or a subgroup of S 1. The determination of the Galois group, Ga, of the equation. 2. The choice of a maximal subgroup of H max(1). In the above case, {P 4. 1, ...P8} is a maximal subgroup of Ga = S 3. The choice of a maximal subgroup of Hmax(1) from step 2. In the above case, {P1,..P4} = Hmax(2) is a maximal subgroup of Hmax(1). The process is continued until Hmax = {P1} = {I}. The groups Ga, Hmax(1), ..,Hmax(k) = I, form a composition series. The composition indices are given by the ratios of the successive orders of the groups: gn/h(1), h(1)/h(2), ...h(k-1)/1. 19 The composition indices of the symmetric groups Sn for n = 2 to 7 are found to be: n Composition Indices 2 2 3 2, 3 4 2, 3, 2, 2 5 2, 60 6 2, 360 7 2, 2520 We shall state, without proof, Galois' theorem: A polynomial equation can be solved algebraically if and only if its group is solvable. Galois defined a solvable group as one in which the composition indices are all prime numbers. Furthermore, he showed that if n > 4, the sequence of maximal normal subgroups is Sn, An, I, where An is the Alternating Group with (n!)/2 elements. The composition indices are then 2 and (n!)/2. For n > 4, however, (n!)/2 is not prime, therefore the groups Sn are not solvable for n > 4. Using Galois' Theorem, we see that it is therefore not possible to solve, algebraically, a general polynomial equation of degree n > 4. 3 20 SOME ALGEBRAIC INVARIANTS Although algebraic invariants first appeared in the works of Lagrange and Gauss in connection with the Theory of Numbers, the study of algebraic invariants as an independent branch of Mathematics did not begin until the work of Boole in 1841. Before discussing this work, it will be convenient to introduce matrix versions of real bilinear forms, B, defined by mn B = .. aijxiyj I=1 j=1 where x = [x1,x2,...xm], an m-vector, y = [y1,y2,...yn], an n-vector, and aij are real coefficients. The square brackets denote a column vector. In matrix notation, the bilinear form is B = xTAy where a11 . . . a1n A = am1. . . amn The scalar product of two n-vectors is seen to be a special case of a bilinear form in which A = I. If x = y, the bilinear form becomes a quadratic form, Q: 21 Q = xTAx. 3.1. Invariants of binary quadratic forms Boole began by considering the properties of the binary quadratic form Q(x,y) = ax2 + 2hxy + by2 under a linear transformation of the coordinates x' = Mx where x = [x,y], x' = [x',y'], and | i j| M = . | k l| The matrix M transforms an orthogonal coordinate system into an oblique coordinate system in which the new x'- axis has a slope (k/i), and the new y'- axis has a slope (l/j), as shown: y 22 y' [i+j,k+l] [j,l] x' [0,1] x' [i,k] [1,1] [0,0] [1,0] x Fig. 1. The transformation of a unit square under M. The transformation is linear, therefore the new function Q'(x',y') is a binary quadratic: Q'(x',y') = a'x'2 + 2h'x'y' + b'y'2. The original function can be written Q(x,y) = xTDx where | a h| D = , | h b| and the determinant of D is detD = ab - h2, called the discriminant of Q. The transformed function can be written Q'(x',y') = x'TD'x' 23 where a' h' D' = , h' b' and detD' = a'b' - h'2, the discriminant of Q'. Now, Q'(x',y') = (Mx)TD'Mx = xTMTD'Mx and this is equal to Q(x,y) if MTD'M = D. The invariance of the form Q(x,y) under the coordinate transformation M therefore leads to the relation (detM)2detD' = detD because detMT = detM. The explicit form of this equation involving determinants is (il - jk)2(a'b' - h'2) = (ab - h2). The discriminant (ab - h2) of Q is said to be an invariant of the transformation because it is equal to the discriminant (a'b' - h'2) of Q', apart from a factor (il - jk)2 that depends on the transformation itself, and not on the arguments a,b,h of the function Q. 3.2. General algebraic invariants 24 The study of general algebraic invariants is an important branch of Mathematics. A binary form in two variables is n f(x1,x2) = aox1n + a1x1n-1x2 + ...anx2 = . aix1n-ix2i If there are three or four variables, we speak of ternary forms or quaternary forms. A binary form is transformed under the linear transformation M as follows f(x1,x2) => f'(x1',x2') = ao'x1'n + a1'x1'n-1x2' + .. The coefficients ao, a1, a2,... ao', a1', a2' .. and the roots of the equation f(x1,x2) = 0 differ from the roots of the equation f'(x1',x2') = 0. Any function I(ao,a1,...an) of the coefficients of f that satisfies rwI(ao',a1',...an') = I(ao,a1,...an) is said to be an invariant of f if the quantity r depends only on the transformation matrix M, and not on the coefficients ai of the function being transformed. The degree of the invariant is the degree of the coefficients, and the exponent w is called the weight. In the example discussed above, the 25 degree is two, and the weight is two. Any function, C, of the coefficients and the variables of a form f that is invariant under the transformation M, except for a multiplicative factor that is a power of the discriminant of M, is said to be a covariant of f. For binary forms, C therefore satisfies rwC(ao',a1',...an'; x1',x2') = C(ao,a1,...an; x1,x2). It is found that the Jacobian of two binary quadratic forms, f(x1,x2) and g(x1,x2), namely the determinant .f/.x1 .f/.x2 .g/.x1 .g/.x2 where .f/.x1 is the partial derivative of f with respect to xsimultaneous covariant of weight one of the two forms. The determinant 1 etc., is a .2f/.x12 .2f/.x1.x2 , .2g/.x2.x1 .2g/.x22 called the Hessian of the binary form f, is found to be a covariant of weight two. A full discussion of the general problem of algebraic invariants is outside the scope of this book. The following example will, however, illustrate the 26 method of finding an invariant in a particular case. Example: To show that (aoa2 - a12)(a1a3 - a22) - (aoa3 - a1a2)2/4 is an invariant of the binary cubic f(x,y) = aox3 + 3a1x2y + 3a2xy2 + a3y3 under a linear transformation of the coordinates. The cubic may be written f(x,y) = (aox2+2a1xy+a2y2)x + (a1x2+2a2xy+a3y2)y = xTDx where x = [x,y], and aox + a1y a1x + a2y D = . a1x + a2y a2x + a3y Let x be transformed to x': x' = Mx, where i j M = k l then 27 f(x,y) = f'(x',y') if D = MTD'M. Taking determinants, we obtain detD = (detM)2detD', an invariant of f(x,y) under the transformation M. In this case, D is a function of x and y. To emphasize this point, put detD = f(x,y) and detD'= f'(x',y') so that f(x,y) = (detM)2f'(x',y' = (aox + a 1y)(a2x + a3y) - (a1x + a2y)2 = (aoa2 - a12)x2 + (aoa3 - a1a2)xy + (a1a3 - a22)y2 = xTEx, where E = Also, we have (aoa2 - a12 ) (aoa3 - a1a2)/2 (aoa3 - a1a2)/2 (a1a3 - a22 ) . f'(x',y') = x'TE'x' = xTMTE'Mx 28 therefore xTEx = (detM)2xTMTE'Mx so that E = (detM)2MTE'M. Taking determinants, we obtain detE = (detM)4detE' = (aoa2 - a12)(a1a3 - a22) - (aoa3 -a1a2)2/4 = invariant of the binary cubic f(x,y) under the transformation x' = Mx. 4 29 SOME INVARIANTS IN PHYSICS 4.1. Gal ilean invariance. Eve nts o f finite extension and du ration are part of t he ph ysical wor ld. It will be conv enient to introduce the n otion of ideal ev ents that have neither extension nor duration. Ideal events may be repre sented as mathematical points in a space-time ge ometr y. A particular eve nt, E, is described by the four components [t,x,y,z] where t is the time of the event, and x,y,z, are its three spatial coordinates. The time and space coordinates are referred to arbitrarily chosen origins. The spatial mesh need not be Cartesian. Let an event E[t,x], recorded by an observer O at the origin of an x- axis, be reco rded as the event E'[t ',x'] by a second ob server O', moving at constant speed V along the x -axis. We suppose that their clocks are synchron ized at t = t' = 0 when t hey coincide at a common origin, x = x' = 0. At time t, we write the plausible equations t' = t and x' = x - Vt, where Vt is the distance travelled by O' in a time t. The se equations can be written E' = GE where 30 1 0 G = . -V 1 G is the operator of the Galilean transformation. The inverse equations are t =t' and x = x ' + V t' or E = G-1E' where G-1 is the inverse Galilean ope rator. (It undoes the effect of G). If we mu ltiply t and t' by the constants k and k ', respectively, where k a nd k' have dimensions of velocity then all terms h ave dimensions of length. In space-space, we have the Pythagorean form x2 + y2 = r2, an invariant under rotations. We are the refore led to ask th e question: is (kt)2 + x2 invariant under the op erator G in space-time? C alculation gives (kt)2 + x2 = (k't ')2 + x'2 + 2Vx't' + V 2t'2 = (k't' )2 + x'2 o nly if V = 0. We see, therefore, that Galilean space-time is not Pythagorean in its algebraic form. We not e, however, the key role played by acceleration in Galilean-Newtonian physics: The velocities of the events according to O and O' are obtained by 31 differentiating the equation x' = -Vt + x with respect to time, giving v' = -V + v, a p lausible result, based up on ou r experience. Differentiating v' with resp ect t o time gives d v'/dt' = a ' = d v/dt = a where a and a' are the accelerations in the two frames of reference. The classical acceleration is invariant un der the Galilean transformation. If the relationship v' = v - V is used to describe the motion of a pulse of light, moving in emp ty space at v = c . 3 x 108 m/s, it does not fit the facts. All studies of very high speed particles that e mit electromagnetic radiation show tha t v' = c f or all va lues of the relative speed, V. 4.2. Lorentz invariance and Einstein's space-time symmetry . It was E instein, above all others, who adv anced our understanding of the true natu re of space-time and relative motion. We shall see tha t he made use of a symm etry argum ent to find the changes that must b e made to the G alilean transformation if it is to account for the relative motion of rapidly moving objects and o f beams of light. H e rec ognized an inconsistency in t he Galilean-Newtonian equ ations, based as they are , on everyday experience. H ere, we shall restrict the discussion to non- accelerating, or so called inertial, frames We have seen that the classical equations relating the events E an d 32 E' a re E' = GE, and the inverse E = G-1E' where 1 0 1 0 G = and G-1 = . -V 1 V 1 These equations ar e connected by the substitution V .-V; this is an algebraic statement of the N ewtonian principle of relativity. Einstein incorporated this principle in his theory. He a lso retained th e lin earity of the classical equations in t he absence of a ny ev idence to the c ontra ry. (Equispaced interv als of time and distance in one inertial frame remain equispaced in any other inertial frame). He the refore symmetrized the space-time eq uations as follows: t' 1 -V t = . x' -V 1 x Note, however, the inconsistency in the dimensions of the time-equation that has now been introduced: t' = t - Vx. The term Vx has dimensions of [L]2/[T], and not [T]. This can be cor rected by introducing the invariant speed of light, c . a postulate in Einstein's theory that is consistent w ith experiment: ct' = ct - Vx/c so that all terms now h ave dimensions of length. Einstein went further, and introd uced a dimensionless quantity . 33 instead of the scaling factor of unity that appears in the Galilean equations of space-time. This factor must be consistent w ith all observations. The equations then become ct ' = .ct - .x x ' = -.ct + .x, where =V/c. These can be written E' = LE, where . -. L = , and E = [ct,x] -. . L is the operator of the Lorentz transformat ion. The inverse equation is E = L-1E' where . . L-1 = . . . This is the inverse Lorentz transformation, obta ined from L by changing . - (or ,V .-V); it h as the effect o f undoing the transformat ion L. We can t herefore write LL-1 = I or . -. .. 1 0 34 = . -.. .. 0 1 Equ ating elements gives .2 -2.2 = 1 therefore, . = 1/v(1 -2) (taking the positive root). 4.3. The invariant interval. Pre viously, it was shown tha t the space-time of Galileo and Newton is not Pythagorean in form. We now ask the question: is Einsteinian space- time Pythagor ean in for m? Direct calculation leads to (ct)2 + (x)2 = .2(1 + 2)(ct')2 + 4.2x'c t' +.2(1 + 2)x'2 . (ct')2 + (x')2 if > 0. Note, however, that the dif ference of squares is an invariant under L: (ct)2 - (x)2 = (ct')2 - (x')2 because .2(1 -2) = 1. Space-time is said to be pseudo-Euclidean. The negative sign that characterizes Lorent z invariance can be included in t he th eory in a general way as follows. We introduce two k inds of 4-vectors x = [x0, x1, x2, x3], a contravariant vector, 35 and x = [x0, x1, x2, x3], a covariant ve ctor, where x = [x0,-x1,-x2,-x3]. The scalar product of t he vectors is defined as x Tx = (x0, x1, x2, x3)[x0,-x1,-x2,-x3] = (x0)2 - ((x1)2 + (x2)2 + (x3)2) The event 4-vector is E = [ct, x, y, z] and the covariant for m is E = [ct,-x,-y,-z] so that the Lorent z invariant scalar produc t is ETE = (ct)2 - (x2 + y2 + z2). The 4-vector x tr ansforms a s follows: x' = Lx where . -. 0 0 -. . 0 0 L = . 0 0 1 0 0 0 0 1 This is the o perator of the Lorentz transformation if the motion of O' is along the x-axis of O's frame of reference. Importan t con sequences of the Lorentz transformation are that intervals of time measured in two different inertial frames are not the same but are related by the equation .t' = ..t 36 where .t is an interval measured on a clock at res t in O's frame, and distances are given by .l' = .l/. where .l is a length measured on a ruler at rest in O's frame. 4.4. The energy-momentum invariant. A d ifferential time interval, dt, cannot be used in a Lorentz-invariant way in kinematics. We must use the proper time differential interval, dt, defined by ( cdt)2 -dx2 = (cdt')2 -dx'2 = (cdt)2. The Newt onian 3-velocity is vN = [dx/dt, dy/dt, dz/dt], and this must be replaced by the 4-velocity V = [d(ct)/dt, dx/dt, dy/dt, dz/dt] = [ d(ct)/dt, dx/dt, dy/dt, d z/dt]dt/dt = [.c,.vN] . The scalar product is then VV = (.c)2 -(.vN)2 = (.c)2(1 - (vN/c)2) = c2. (In form ing the scalar produ ct, the transpose is unde rstood). The magn itude of t he 4-velocity is .V . = c, the invariant speed of light. In Classical Mechanics, the concept of momentum is importa nt because 37 of its role as an invariant in an isolated system. We the refore introduce the concept of 4-momentum in Relativistic Mechanics in order to find possible Lorentz invariants involving this new q uantity. The contra variant 4-momentum is defined as: P = mV where m is the mass of the p article. (It is a Lorentz scalar, t he ma ss measured in t he frame in which th e par ticle is at res t). The scalar product is P P = Now, P = therefore, P P = Writing (mc)2. [m.c, m.vN] (m.c)2 - (m.vN)2. M = .m, the relativistic mass, we obta in PP = (Mc)2 - (MvN)2 = (mc)2. Multiplying throug hout by c2 gives M2c4 - M2vN2c2 = m2c4. The quan tity Mc2 ha s dimensions of ener gy; we the refore write E = Mc2 the total energy of a freely moving pa rticle. This leads to the fundamental invariant of dynamics 38 c2PP = E2 - (pc)2 = Eo2 where Eo = mc2 is the rest energy of the p article, and p is its relativistic 3-momentum. The total energy can be written: E = .Eo = Eo + T, where T = Eo(.- 1), the relativistic kinetic energy. The magn itude of t he 4-momentum is a Lorent z invariant .P . = mc. The 4- m omentum transforms a s follows: P' = LP. For relative motion along the x-axis, this equation is equivalent to the equations E' = .E - .cpx and cpx = -.E + .cpx . Using the Planck-Einstein equations E = h. an d E = pxc for photons, the ener gy eq uation becomes .' = .. - .. = ..(1 -) 39 = .(1 -)/(1 -2)1/2 = .[(1 -)/(1 + )]1/2 . This is the relativistic Dop pler shift for the frequency .', measured in an inertial frame (primed) in t erms of the frequency . me asured in another inertial frame (unprimed). 4.5. The frequency-wavenumber invarian t Par ticle-Wave duality, one o f the most prof ound discoveries in Physics, has its origins in Lorentz invariance. It w as pro posed by deBroglie in the early 192 0's. He used the following arg ument. The displacement o f a wave can be written y(t ,r) = Acos(.t -kr) where . =2p. (the angular frequency), .k. =2p/. (the wavenumber), and r = [x, y , z] (the position vector). The ph ase (.t - kr) can be written ((./c)ct - kr), and t his has the form of a Lorentz invariant obtained from the 4-vectors E[ct , r], and K[./c, k] where E is the event 4-vector, and K is the "frequency-wavenumber" 4- vector. deB roglie not ed th at the 4-momentum P is conn ected to the event 4 - vector E through the 4-velocity V, and the frequency-wavenumber 4- vector K is conn ected to the event 4 -vector E through the Lorentz invariant pha se of a wa ve ((./c)ct - kr). He therefore proposed that a 40 direct connectionmustexist betw een P an d K; it is illustrated in the following diagram: E[ct ,r] ( Einstein) P P =inv. P [E/c,p] E K =inv. (deBroglie) K[./c,k] (deBroglie) Fig. 2. The coup ling between P an d K vi a E. deB roglie proposed that the conn ection is the simplest possible, name ly, P an d K are proportional to ea ch ot her. He realized that there could be only one value for the c onstant of pro portionality if the Planck-Einstein resu lt for photons E = h ./2p is but a special case of a general result . it must be h/2p, where h is Plancks constant. Therefore, d eBroglie proposed th at P . K or P = (h/2p)K. Equ ating the elements of the 4-vectors gives E = (h/2p). and p = (h/2p)k . In these remarkable equations, our notions of particles and waves are 41 forever merged. The smallness of the value of Planck's co nstant prevents us from observing the d uality directly; however, it is clearly observed at the molecular, atomic, nuclear, and pa rticle level. 4.6. deBroglie's invariant. The invariant form ed from the frequency-wavenumber 4-vector is KK = (./c, k)[./c,-k] =(./c)2 - k2 = (.o/c)2, where .o is the proper angular frequency. This invariant is the w ave version of Einstein's energy-momentum invariant; it gives the dispersion relation .o2 = .2 - (kc)2. The ratio ./k is the phase velocity of the w ave, vf. For a wa ve-packet, the group velocity vG is d./dk; it can be obt ained by differentiating the dispersion eq uation as follows: .d.- kc 2dk = 0 therefore, vG = d./dk = kc2/.. The deBr oglie invariant involving the produ ct of the phase and group velocity is therefore vfvG = (./k).(kc2/.) = c2. This is the w ave-equivalent of Einstein's famous E = Mc2. 42 We see that v fvG = c2 = E/M or, v G = E/Mvf = Ek/M. = p/M = vN, the particle velocity. This result played an important p art in the development of Wave Mechanics. We shall find in later chapters, that Lorentz transformations form a gro up, and that invariance principles are related directly to symmetry tra nsformations and the ir associated g roups. 5 43 GROUPS CONCRETE AND ABSTRACT 5.1 Some concrete examples The elements of the set {1, i}, where i = v-1, are the roots of the equation x4 = 1, the fourth roots of unity. They have the following special properties: 1. The product of any two elements of the set (including the same two elements) is always an element of the set. (The elements obey closure). 2. The order of combining pairs in the triple product of any elements of the set does not matter. (The elements obey associativity). 3. A unique element of the set exists such that the product of any element of the set and the unique element (called the identity) is equal to the element itself. (An identity element exists). 4. For each element of the set, a corresponding element exists such that the product of the element and its corresponding element (called the inverse) is equal to the identity. (An inverse element exists). The set of elements {1, i} with these four properties is said to form a GROUP. In this example, the law of composition of the group is multiplication; this need not be the case. For example, the set of integers Z = {.., -2, -1, 0, 1, 2, ...} forms a group if the law of composition is addition. In this group, the identity element is zero, and the inverse of each integer is the integer with the same magnitude but with opposite sign. In a different vein, we consider the set of 44 matrices: 44 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 {M} = 0 1 0 0 , 1 0 0 0 , 0 0 0 1 , 0 0 1 0 . 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 If the law of composition is matrix multiplication , then {M} is found to obey: 1 --closure and 2 --associativity, and to contain: 3 --an identity, diag(1, 1, 1, 1), and 4 --inverses. The set {M} forms a group under matrix multilication. 5.2. Abstract groups The examples given above illustrate the generality of the group concept. In the first example, the group elements are real and imaginary numbers, in the second, they are positive and negative integers, and in the third, they are matrices that represent linear operators (see later discussion). Cayley, in the mid-19th century, first emphasized this generality, and he introduced the concept of an abstract group, Gn which is a collection of n distinct elements (...gi...) for which a law of composition is given. If n is finite, the group is said to be a group of order n. The collection of elements must obey the four rules: 1. If gi, gj . G then gn = gj gi . G . gi, gj . G (closure) 45 2. gk(gjgi) = (gkgj)gi [leaving out the composition symbol] (associativity) 3. . e . G such that gie = egi = gi . gi . G (an identity exists) 4. If gi . G then . gi-1 . G such that gi-1gi = gigi-1 = e (an inverse exists). For finite groups, the group structure is given by listing all compositions of pairs of elements in a group table, as follows: e . gi gj . .(1st symbol, or operation, in pair) e gi . . gigi gigj . gj . gjgi gjgj . gk . gkgi gkgj . . . If gjgi = gigj . gi, gj . G, then G is said to be a commutative or abelian group. The group table of an abelian group is symmetric under reflection in the diagonal. A group of elements that has the same structure as an abstract group is a realization of the group. 5.3 The dihedral group, D3 The set of operations that leaves an equilateral triangle invariant under rotations in the plane about its center, and under reflections in the three planes through the vertices, perpendicular to the opposite sides, forms a group of six elements. A study of the structure of this group (called the 46 dihedral group, D3) illustrates the typical group-theoretical approach. The geometric operations that leave the triangle invariant are: Rotations about the z-axis (anticlockwise rotations are positive) Rz(0) (123) . (123) = e, the identity Rz(2p/3)(123) . (312) = a Rz(4p/3)(123) . (231) = a2 and reflections in the planes I, II, and III: RI (123) . (123) = b RII (123) . (321) = c RIII (123) . (213) = d This set of operators is D3 = {e, a, a2, b, c, d}. Positive rotations are in an anticlockwise sense and the inverse rotations are in a clockwise sense., so that the inverse of e, a, a2 are e-1 = e, a-1 = a2, and (a2)-1 = a. The inverses of the reflection operators are the operators themselves: b-1 = b, c-1 = c, and d-1 = d. We therefore see that the set D3 forms a group. The group multiplication table is: e 2 b e 2 b a 2 e a2 a2 e b 2 c 2 e d 2 e aa d c aa e d c aa c b d b d c a aa e d c b ab d c a aa c b d In reading the table, we follow the rule that the first operation is written on 47 the right: for example, ca2 = b. A feature of the group D3 is that it can be subdivided into sets of either rotations involving {e, a, a2} or reflections involving {b, c, d}. The set {e, a, a2} forms a group called the cyclic group of order three, C3. A group is cyclic if all the elements of the group are powers of a single element. The cyclic group of order n, Cn, is Cn = {e, a, a2, a3, .....,an-1}, where n is the smallest integer such that an = e, the identity. Since k n-k = a n = e, a a an inverse an-k exists. All cyclic groups are abelian. The group D3 can be broken down into a part that is a group C3, and a part that is the product of one of the remaining elements and the elements of C3. For example, we can write D3 = C3 + bC3 , b . C3 = {e, a, a2} + {b, ba, ba2} = {e, a, a2} + {b, c, d} = cC3 = dC3. This decomposition is a special case of an important theorem known as Lagranges theorem. (Lagrange had considered permutations of roots of equations before Cauchy and Galois). 5.4 Lagranges theorem The order m of a subgroup H m of a finite group Gfactor (an integral divisor) of n. n of order n is a 48 Let Gn = {g1=e, g2, g3, ...gn} be a group of order n, and let Hm = {h1=e, h2, h3, ...hm} be a subgroup of Gn of order m. If we take any element gk of Gn which is not in Hm, we can form the set of elements {gkh1, gkh2, gkh3, ...gkhm} = gkHm. This is called the left coset of Hm with respect to gk. We note the important facts that all the elements of gkhj, j=1 to m are distinct, and that none of the elements gkhj belongs to Hm. Every element gk that belongs to Gn but does not belong to Hm belongs to some coset gkHm so that Gn forms the union of Hm and a number of distinct (non-overlapping) cosets. (There are (n - m) such distinct cosets). Each coset has m different elements and therefore the order n of Gn is divisible by m, hence n = Km, where the integer K is called the index of the subgroup Hm under the group Gn. We therefore write Gn = g1Hm + gj2Hm + gk3Hm + ....g.KHm where gj2 . Gn . Hm, 49 gk3 . Gn . Hm, gj2Hm . gnK . Gn . Hm, gj2Hm, gk3Hm, ...gn-1, K-1Hm. The subscripts 2, 3, 4, ..K are the indices of the group. As an example, consider the permutations of three objects 1, 2, 3 ( the group S3 ) and let Hm = C3 = {123, 312, 231}, the cyclic group of order three. The elements of S3 that are not in H3 are {132, 213, 321}. Choosing gk = 132, we obtain gkH3 = {132, 321, 213}, and therefore S3 = C3 + gk2C3 ,K = 2. This is the result obtained in the decomposition of the group D3 , if we make the substitutions e = 123, a = 312, a2 = 231, b = 132, c = 321, and d = 213. The groups D3 and S3 are said to be isomorphic. Isomorphic groups have the same group multiplication table. Isomorphism is a special case of homomorphism that involves a many-to-one correspondence. 5.5 Conjugate classes and invariant subgroups If there exists an element v . Gn such that two elements a, b . Gn are 50 related by vav-1 = b, then b is said to be conjugate to a. A finite group can be separated into sets that are conjugate to each other. The class of Gn is defined as the set of conjugates of an element a . Gn. The element itself belongs to this set. If a is conjugate to b, the class conjugate to a and the class conjugate to b are the same. If a is not conjugate to b, these classes have no common elements. Gn can be decomposed into classes because each element of Gn belongs to a class. An element of Gn that commutes with all elements of Gn forms a class by itself. The elements of an abelian group are such that bab-1 = a for all a, b . Gn, so that ba = ab. If Hm is a subgroup of Gn, we can form the set {aea-1, ah2a-1, ....ahma-1} = aHma-1 where a . Gn . Now, aHma-1 is another subgroup of Hm in Gn. Different subgroups may be found by choosing different elements a of G n. If, for all values of a . Gn aHma-1 = Hm (all conjugate subgroups of Hm in Gn are identical to Hm), 51 then Hm is said to be an invariant subgroup in Gn. Alternatively, Hm is an invariant in Gn if the left- and right-cosets formed with any a . Gn are equal, i. e. ahi = hka. An invariant subgroup Hm of Gn commutes with all elements of Gn. Furthermore, if hi . Hm then all elements ahia-1 . Hm so that Hm is an invariant subgroup of Gn if it contains elements of Gn in complete classes. Every group Gn contains two trivial invariant subgroups, Hm = Gn and Hm = e. A group with no proper (non-trivail) invariant subgroups is said to be simple (atomic). If none of the proper invariant subgroups of a group is abelian, the group is said to be semisimple. An invariant subgroup Hm and its cosets form a group under multiplication called the factor group (written Gn/Hm) of Hm in Gn. These formal aspects of Group Theory can be illustrated by considering the following example: The group D3 = {e, a, a2, b, c, d} ~ S3 = {123, 312, 231, 132, 321, 213}. C3 is a subgroup of S3: C3 = H3 = {e, a, a2} = {123, 312, 231}. Now, bH3 = {132, 321, 213} = H3b cH3 = {321, 213, 132} = H3c 52 and dH3 = {213,132, 321} = H3d. Since H3 is a proper invariant subgroup of S3, we see that S3 is not simple. H3 is abelian therefore S3 is not semisimple. The decomposition of S3 is S3 = H3 + bH3 = H3 + H3b. and, in this case we have H3b = H3c = H3d. (Since the index of H3 is 2, H3 must be invariant). The conjugate classes are e = e eae-1 = ea = a aaa-1 = ae = a a2a(a2)-1 = a2a2 = a bab-1 = bab = a2 cac-1 = cac = a2 dad-1 = dad = a2 The class conjugate to a is therefore {a, a2}. The class conjugate to b is found to be {b, c, d}. The group S3 can be decomposed by classes: S3 = {e} + {a, a2} + {b, c, d}. S3 contains three conjugate classes. If we now consider Hm = {e, b} an abelian subgroup, we find 53 aHm = {a,d}, Hma = {a.c}, a2Hm = {a2,c}, Hma2 = {a2, d}, etc. All left and right cosets are not equal: Hm = {e, b} is therefore not an invariant subgroup of S3. We can therefore write S3 = {e, b} + {a, d} + {a2, c} = Hm + aHm + a2Hm. Applying Lagranges theorem to S3 gives the orders of the possible subgroups: they are order 1: {e} order 2: {e, d}; {e, c}: {e, d} order 3: {e, a, a2} (abelian and invariant) order 6: S3. 5.6 Permutations A permutation of the set {1, 2, 3, ....,n} of n distinct elements is an ordered arrangement of the n elements. If the order is changed then the permutation is changed. the number of permutations of n distinct elements is n! We begin with a familiar example: the permutations of three distinct objects labelled 1, 2, 3. There are six possible arrangements; they are 123, 312, 231, 132, 321, 213. These arrangements can be written conveniently in matrix form: 54 1 2 3 1 2 3 1 2 3 p1 = , p2 = , p3 = , 1 2 3 3 1 2 2 3 1 1 2 3 1 2 3 1 2 3 p4 = , p5 = , p6 = . 1 3 2 3 2 1 2 1 3 The product of two permutations is the result of performing one arrangement after another. We then find p2p3 = p1 and p3p2 = p1 whereas p4p5 = p3 and p5p4 = p2. The permutations p1, p2, p3 commute in pairs (they correspond to the rotations of the dihedral group) whereas the permutations do not commute (they correspond to the reflections). A general product of permutations can be written s1 s2 . . .sn 1 2 . . n 1 2 . . n = . t1 t2 . . .tn s1 s2 . . sn t1 t2 . . tn The permutations are found to have the following properties: 1. The product of two permutations of the set {1, 2, 3, ...} is itself a permutation of the set. (Closure) 2. The product obeys associativity: 55 (pkpj)pi = pk(pjpi), (not generally commutative). 3. An identity permutation exists. 4. An inverse permutation exists: s1 s2 . . . sn p-1 = | | 1 2 . . . n such that pp-1 = p-1p = identity permutation. The set of permutations therefore forms a group 5.7 Cayleys theorem: Every finite group is isomorphic to a certain permutation group. Let Gn ={g1, g2, g3, . . .gn} be a finite group of order n. We choose any element gi in Gn, and we form the products that belong to Gn: gig1, gig2, gig3, . . . gign. These products are the n-elements of Gn rearranged. The permutation pi, associated with gi is therefore g1 g2 . . gn pi = . gig1 gig2 . . gign If the permutation pj associated with gj is g1 g2 . . gn pj = , gjg1 gjg2 . . gjgn where gi . gj, then g1 g2 . . gn pjpi = . (gjgi)gi (gjgi)g2 . . (gjgi)gn This is the permutation that corresponds to the element gjgi of Gn. 56 There is a direct correspondence between the elements of Gn and the n- permutations {p1, p2, . . .pn}. The group of permutations is a subgroup of the full symmetric group of order n! that contains all the permutations of the elements g1, g2, . . gn. Cayleys theorem is important not only in the theory of finite groups but also in those quantum systems in which the indistinguishability of the fundamental particles means that certain quantities must be invariant under the exchange or permutation of the particles. 6 57 LIES DIFFERENTIAL EQUATION, INFINITESIMAL ROTATIONS AND ANGULAR MOMENTUM OPERATORS Although the field of continuous transformation groups (Lie groups) has its origin in the theory of differential equations, we shall introduce the subject using geometrical ideas. 6.1 Coordinate and vector rotations A 3-vector v = [vx, vy, vz] transforms into v = [vx, vy, vzgeneral coordinate rotation R about the origin of an orthogonal coordinate ] under a system as follows: v = R v, where i.i j.i k.i R =.i.j j.j k.j. i.k j.k k.k cos.ii . . = .cos.ij . . . cos.ik . cos.kk where i, j, k, i, j, k are orthogonal unit vectors, along the axes, before and after the transformation, and the cos.iis are direction cosines. The simplest case involves rotations in the x-y plane: . vx . = . cos.ii cos.ji ..vx . vy cos.ij cos.jj vy = . cosf sinf. = Rc(f)v 58 -sinf cosf where Rc(f) is the coordinate rotation operator. If the vector is rotated in a fixed coordinate system, we have f . -f so that v = Rv(f)v, where Rv(f) = . cosf -sinf. . sinf cosf 6.2 Lies differential equation The main features of Lies Theory of Continuous Transformation Groups can best be introduced by discussing the properties of the rotation operator Rv(f) when the angle of rotation is an infinitesimal. In general, Rv(f) transforms a point P[x, y] in the plane into a new point P[x, y]: P = Rv(f)P. Let the angle of rotation be sufficiently small for us to put cos(f) . 1 and sin(f) . df, in which case, we have Rv(df) = . 1 -df. df 1 and x = x.1 - ydf = x - ydf y = xdf + y.1 = xdf + y Let the corresponding changes x . x and y . y be written 59 so that We note that where x = x + dx and y = y +dy dx = -ydf and dy = xdf. Rv(df) = .1 0. + .0 -1. df 0 1 1 0 = I + idf i = .0 -1. = Rv(p/2). 1 0 Lie introduced another important way to interpret the operator i = Rv(p/2), that involves the derivative of Rv(f) evaluated at the identity value of the parameter, f = 0: dRv(f)/df. = .-sinf-cosf.. = .0 -1. = i f =0 cosf-sinf 1 0 f = 0 so that Rv(df) = I + dRv(f)/df. .df, f = 0 a quantity that differs from the identity I by a term that involves the infinitesimal, df: this is an infinitesimal transformation. Lie was concerned with Differential Equations and not Geometry. He was therefore motivated to discover the key equation dRv(f)/df = . 0 -1..cosf -sinf. 60 1 0 sinf cosf = iRv(f) . This is Lies differential equation. Integrating between f = 0 and f = f, we obtain Rv(f) f .dRv(f)/Rv(f) = i .df I 0 so that ln(Rv(f)/I) = if, or Rv(f) = Ie if , the solution of Lies equation. Previously, we obtained Rv(f) = Icosf + isinf. We have, therefore Ie if = Icosf + isinf . This is an independent proof of the famous Cotes-Euler equation. We introduce an operator of the form O = g(x, y, ./.x, ./.y), and ask the question: does dx = Of(x, y; df) ? Lie answered the question in the affirmative; he found dx = O(xdf) = (x./.y - y./.x)xdf = -ydf and dy = O(ydf) = (x./.y - y./.x)y.f = xdf . 61 Putting x = x1 and y = x2, we obtain dxi = Xxidf , i = 1, 2 where X = O = (x1./.x2 - x2./.x1), the generator of rotations in the plane. 6.3 Exponentiation of infinitesimal rotations We have seen that Rv(f) = eif, and therefore Rv(df) = I + idf, for an infinitesimal rotation, df Performing two infinitesimal rotations in succession, we have Rv2(df) = (I + idf)2 = I + 2idf to first order, = Rv(2df). Applying Rv(df) n-times gives Rvn(df) = Rv(ndf) = eindf = e if = Rv(f) (as n .8 and df . 0, the product ndf . f). This result agrees, as it should, with the exact solution of Lies differential equation. A finite rotation can be built up by exponentiation of infinitesimal 62 rotations, each one being close to the identity. In general, this approach has the advantage that the infinitesimal form of a transformation can often be found in a straightforward way, whereas the finite form is often intractable. 6.4 Infinitesimal rotations and angular momentum operators In Classical Mechanics, the angular momentum of a mass m, moving in the plane about the origin of a cartesian reference frame with a momentum p is Lz = r p = rpsinfnz where nz is a unit vector normal to the plane, and f is the angle between r and p. In component form, we have Lzcl = xpy - ypx, where px and py are the cartesian components of p. The transition between Classical and Quantum Mechanics is made by replacing px by -i(h/2p)./.x (a differential operator) and py by -i(h/2p)./.y (a differential operator),where h is Plancks constant. We can therefore write the quantum operator as LzQ = -i(h/2p)(x./.y - y./.x) = -i(h/2p)X and therefore X = iLzQ/(h/2p), 63 and dxi = Xxi df = (2piLzQ/h)xi df, i = 1,2. Let an arbitrary, continuous, differentiable function f(x, y) be transformed under the infinitesimal changes x = x - ydf and y = y + xdf . Using Taylors theorem, we can write f(x, y) = f(x + dx, y + dy) = f(x - ydf, y + xdf) = f(x, y) + ((.f/.x)dx + ((.f/.y)dy) = f(x, y) + df(-y(./.x) + x(./.y))f(x, y) = I + 2pidfLz/h)f(x, y) = e2pidfLz/h f(x, y) = Rv(2pLzdf/h) f(x, y). The invatriance of length under rotations follows at once from this result: If f(x, y) = x2 + y2 then .f/.x = 2x and .f/.y = 2y, and therefore f(x, y) = f(x, y) + 2xdx + 2ydy = f(x, y) - 2x(ydf) + 2y(xdf) = f(x, y) = x2 + y2 = invariant. This is the only form that leads to the invariance of length under rotations. 64 6.5 3-dimensional rotations Consider three successive counterclockwise rotations about the x, y, and z axes through angles , ., and f, respectively: z z' y y' y about x x x, x' z' y' z'' y', y'' . about y x' x'' x' y'' y''' z'' z''' f about z '' '' '''xxxThe total transformation is Rc(, ., f) = Rc(f)Rc(.)Rc() cosfcos. cosfsin.sin + sinfcos-cosfsin.cos + sinfsin = .-sinfcos.-sinfsin.sin + cosfcos sinfsin.cos + sinfsin. sin.-cos.sin cos.cos For infinitesimal rotations, the total rotation matrix is, to 1st-order in the ds: 1 df -d. Rc(d, d., df) = .-df 1 d. . d. -d 1 The infinitesimal form can be written as follows: 65 1 df 0 1 0 -d. 1 0 0 Rc(d, d., df) = .-df 1 0.. 0 1 0 ..0 1 d. 0 0 1 d. 0 1 0 -d 1 = . I + Y3df .. I + Y2d. .. I + Y1d. where 0 0 0 0 0 -1 0 1 0 Y1 = . 0 0 1. , Y2 = . 0 0 0. , Y3 = . -1 0 0. . 0 -1 0 1 0 0 0 0 0 To 1st-order in the ds, we have Rc(d, d., df) = I + Y1d + Y2d. + Y3df . 6.6 Algebra of the angular momentum operators The algebraic properties of the Ys are important. For example, we find that their commutators are: 0 0 0 0 0 -1 0 0 -1 0 0 0 [Y1, Y2] = | 0 0 1|| 0 0 0| - | 0 0 0|| 0 0 1| 0 -1 0 1 0 0 1 0 0 0 -1 0 = -Y3 , [Y1, Y3] = Y2 , and [Y2, Y3] = -Y1 . These relations define the algebra of the Ys. In general, we have [Yj, Yk] = Yl = ejkl Yl , where ejkl is the anti-symmetric Levi-Civita symbol. It is equal to +1 if jkl is 66 an even permutation, -1 if jkl is an odd permutation, and it is equal to zero if two indices are the same. Motivated by the relationship between Lz and X in 2-dimensions, we introduce the operators Jk = -i(2p/h)Yk , k = 1, 2, 3. Their commutators are obtained from those of the Ys, for example [Y1, Y2] = -Y3 . [2piJ1/h, 2piJ2/h] = -2piJ3/h or -[J1, J2](2p/h)2 = -2piJ3/h and therefore [J1, J2] = ihJ3/2p . These operators obey the general commutation relation [Jj, Jk] = ihejkl Jl /2p . The angular momentum operators form a Lie Algebra. The basic algebraic properties of the angular momentum operators in Quantum Mechanics stem directly from this relation. Another approach involves the use of the differential operators in 3- dimensions. A point P[x, y, z] transforms under an infinitesimal rotation of the coordinates as follows P[x, y, z] = Rc(d, d., df]P[x, y, z] Substituting the infinitesimal form of Rc in this equation gives dx = x - x = ydf -zd. 67 dy = y - y = -xdf + zd dz = z - z = xd. - yd . Introducing the classical angular momentum operators: Licl, we find that these small changes can be written 3 dxi = . dak Xkxi k = 1 For example, if i = 1 dx1 = dx = d(z./.y -y./.z)x + d.(-z./.x + x./.z)x + df(y./.x -x./.y)x = -zd. + ydf . Extending Lies method to three dimensions, the infinitesimal form of the rotation operator is readily shown to be 3 Rc(d, d., df) = I + . (.Rc/.ai)| dai . i=1 All ais = 0 7 68 LIES CONTINUOUS T RANSF ORMATION G ROUPS In the p revious chapter, we discussed the p roper ties of infinitesimal rot ations in 2- and 3-dimensions, and we found that they are related directly to the angular momentum operators of Quantum Mechanics. Importan t algebraic propert ies of the matrix representations of the operators also were introduc ed. In this chapter, we shall consider the subject in general terms. Let xi, i = 1 to n be a set of n v ariables. They may be considered to be the coordinates of a point in an n-dimensional vector s pace, Vn. A set of equations involving the x is is obtained by the transformat ions xi = fi(x1, x2, ...xn: a1, a2, ....ar), i = 1 to n in which the set a1, a2, ...ar co ntains r-independent para meters. The set Ta, of transformations maps x . x. We shall write x = f(x;a) or x = for the set of functions. Tax It is assumed that the functions fi are differentiable with res pect to the xs and t he a s to any required order. These functions necessarily depend o n the essential para meters, a. This means th at no two tra nsformations with different numbers of p arameters are the same. r is the smallest number required to characterize the transformation, completely. The set of functions fi forms a finite co ntinu ous group if: 1. The result of two successive transformations x . x . x is equivalent 69 to a single transformat ion x . x: x = f(x; b) = f(f(x; a); b) = f(x; c) = f(x; .(a; b)) where c is the set of p arameters c. = .. (a; b) , . = 1 to r, and 2. To e very transformation there corresponds a unique inverse that belongs to the set: . a such that x = f (x; a) = f(x; a) We have -1 TaTa = Ta-1Ta = I, the identity. We shall see that 1) is a highly restrictive req uirement. The transformation x = f(x; a0) is the identity. Without loss of generality, we can take a0 = 0. The essential point of Lies th eory of continuous tr ansformation groups is to consider that part of the group that is close to the identity, and not to consider the group as a whole. Successive infinitesimal changes can be used to build up the finite change. 7.1 One-param eter group s Consider the transformation x . x unde r a f inite change in a single par ameter a, and t hen a change x + dx. There are two pa ths from x . x + dx; they are as shown: x 70 an infinitesimal da a , a finite para meter change x + dx a + da a differential x (a = 0) We have x + d x = f(x; a + da) = f(f(x; a); da) = f(x; da) The 1st-order Taylor expansion is dx = .f(x; a)/.a The lie group cond itions then dem and a + da = .(a; da). But .(a; 0) = a, (b =0) therefore da = u(x) da a = 0 a + da = a + ..(a; b)/. b da b = 0 so that d a = ..(a; b)/. b da b = 0 or da = A(a)da. Therefore dx = u (x)A(a)da, leading to dx/u(x) so that x .dx/u(x) x We therefore obtain = = A(a)da 71 a . A(a)da = s, (s = 0 . the identity). 0 U (x) - U(x) = s. A t ransformat ion of coordinates (new v ariables) therefore transfers all elements of t he group b y the same transformation: a o ne-pa rameter group is equivalent to a group of translatio ns. Two cont inuous tra nsformation groups are said to be s imilar when they can be obtained from one another by a change of variable. For example, consider the g roup defined by x1 a 0 x1 x2 = 0 a2 x2 The identity coprr esponds to a = 1. The infinitesimal transformation is therefore x1 ( 1 + da) 0 x1 x2 = 0 (1+ da)2 x2 . To 1st-order in da we have x 1 = x1 +x1da and x 2 = x2 + 2x2da or dx1 = x1da and dx2 = 2x2da. 72 In the limit, these equations give dx1/x1 = dx2/2x2 = da. These are the differential equations t hat c orrespond to the infinitesimal equations above. Integrating, we have x1 a x2 a .dx 1/x1 = .da and .dx 2/2x2 = da , x10x2 0 so that lnx1 -lnx1 = a = ln (x1/x1) and ln(x2/x2) = 2a = 2ln(x1/x1) or U = (x2/x12) = U = (x2/x12) . Put ting V = lnx1, we obt ain V = V + a and U = U, the translation group. 7.2 Determin ation of t he finite equations from the infinitesim al for ms Let the finite equations of a one-parameter group G(1) be x1 = f(x1, x2 ; a) and x2 = .(x1, x2 ; a), and let the identity correspond to a = 0. 73 We consider the transformation of f(x1, x2) to f(x1, x2). We expand f(x1, x2) in a Maclaurin series in the p arameter a (at definite values of x 1 and x2): f(x1, x2) = f(0) + f(0)a + f(0)a2/2! + ... where f(0) = f(x1, x2)| a=0 = f(x1, x2), and f(0) = (df(x1, x2)/da| a=0 = {(.f/.x1)(dx1/da) + (.f/.x2)(dx2/da)}| a= 0 = {(.f/.x1)u(x1, x2) + (.f/.x2)v(x1, x2)}|a=0 therefore f (0) = {(u(./. x1) + v( ./.x2))f}|a=0 = Xf(x1, x2). Continuing in this way, we have f(0) = {d2f(x1, x2)/da2}|a=0 = X2f(x1, x2), etc.... The function f(x1, x2) can be expanded in t he series f(x1, x2) = f(0) + af (0) + (a2/2!)f(0) + ... = f(x1, x2) + aXf + (a 2/2!)X2f + ... Xnf is the symb ol for operating n-times in succession of f with X. The finite eq uations of the group are therefore x1 = x1 + aXx1 + (a2/2!)X2x1 + ... and x2 = x2 + aXx2 + (a2/2!)X2x2 + = ... If x1 and x2 are definite values to which x1and x2 reduce for the identity 74 a=0, then the se equations ar e the series solutions of the differential equations dx1/u(x1, x2) = d x2/v(x1, x2) = d a. The group is referred to as the g roup Xf. For exam ple, let Xf = (x1./.x1 + x2./.x2)f then x1 = x1 + aXx1 + (a2/2!)X2f ... = x1 + a(x1./.x1 + x2./.x2)x1 + ... = x1 + ax1 + (a2/2!)(x1./.x1 + x2./.x2)x1 + = x1 + ax1 + (a2/2!)x1 + ... = x1(1 + a + a2/2! + ...) = x1ea. Also, we find x2 = x2ea. Put ting b = ea, we have x1 = bx1, and x2 = bx2. The finite group is the group of magnifications. If X = (x./.y -y./.x) we find, for example, that the finite group is the gro up of 2-dimensional rotat ions. 7.3 Invariant function s of a group 75 Let Xf = (u./.x1 + v./.x2)f define a one-parameter gro up, and let a=0 give the identity. A fu nction F(x1, x2) is ter med a n invariant un der the transformation group G (1) if F(x1, x2) = F (x1, x2) for all values of the p arameter, a. The function F(x1, x2) can be expanded as a series in a: F(x1, x2) = F(x 1, x2) + aXF + (a 2/2!)X(XF) + ... If F(x1,x2) = F (x1, x2) = in variant for all values of a, it is necessary fo r XF = 0, and this mean s that {u(x1, x2)./.x1 + Consequently, v(x 1, x2)./.x2}F = 0 . F(x1, x2) = constant is a solution of dx1/u(x1, x2) = dx2/v(x1, x2) . This equation has one solution that depends on one ar bitrary constant, and therefore G(1) ha s only one basic invariant, and all othe r pos sible invariants can be given in terms o f the basic invariant. For exam ple, we now rec onsider the the invariants of rotat ions: The infinitesimal transformations are given by 76 Xf = (x 1./.x2 -x2./.x1), and the differential equation tha t gives the invariant function F of the gro up is obta ined by solving the characteristic differential equations dx1/x2 = so that dx1/x2 + dx2/x1 = The solution of this equation is 22 x1 + x2 = df, and dx 2/x1 = -df, 0. constant, and therefore the invariant function is 22 F (x1, x2) = x1 + x2. 2 All functions of x 1 + x22 are the refore invariants of the 2-dimensional rot ation group. This method c an be generalized. A group G(1) in n-variables defined by the e quation xi = f(x1, x2, x3, ...xn; a), i = 1 to n, is equivalent to a unique infinitesimal transformation Xf = u1(x1, x2, x3, ...xn).f/.x1 + ...un(x1, x2, x3, ...xn).f/.xn . If a is the g roup parameter then the infinitesimal transformation is x i = xi + ui(x1, x2, ...xn)da (i = 1 t o n), then, if E(x1, x2, ...xn) is a function that can be differentiatedn-times with respect to its arg uments, we have E (x1, x2, ...xn) = E(x1, x2, ...xn) + aXE + (a 2/2!)X2E + . 77 Let (x1, x2, ...xn) be the coordinates of a point in n-space and let a be a par ameter, independent of the xis. As a var ies, the p oint (x1, x2, ...xn) will describe a trajectory, starting from the initial point (x1, x2, ...xn). A necessary and sufficient con dition tha t F(x 1, x2, ...xn) be an invariant function is that XF = 0. A curve F = 0 i s a t rajectory and t herefore a n invariant curve if XF(x1, x2, x3, ...xn) = 0. 8 78 PROPERTIES OF n-VARIABLE, r-PARAMETER LIE GROUPS The change of an n-variable function F(x) produced by the infinitesimal transformations associated with r-essential parameters is: n dF = . (.F/.xi)dxi i = 1 where r dxi = . ui.(x)da. , the Lie form. . = 1 The parameters are independent of the xis therefore we can write rn dF = .da.{. ui.(x)(./.xi)F} . = 1 i = 1 r = .da. X. F . = 1 where the infinitesimal generators of the group are n X.= . ui.(x)(./.xi) , .= 1 to r. i = 1 The operator r I + . X.da.. = 1 differs infinitesimally from the identity. The generators X. have algebraic properties of basic importance in the Theory of Lie Groups. The X.s are differential operators. The problem is therefore one of obtaining the algebraic structure of differential operators. This problem has its origin in the work of Poisson (1807); he introduced the following ideas: The two expressions X1f = (u11./.x1 + u12./.x2)f and X2f = (u21./.x1 + u22./.x2)f 79 where the coefficients ui. are functions of the variables x1, x2, and f(x1, x2) is an arbitrary differentiable function of the two variables, are termed linear differential operators. The product in the order X2 followed by X1 is defined as X1X2f = (u11./.x1 + u12./.x2)(u21.f/.x1 + u22.f/.x2) The product in the reverse order is defined as X2X1f = (u21./.x1 + u22./.x2)(u11.f/.x1 + u12.f/.x2). The difference is X1X2f -X2X1f = X1u21.f/.x1 + X1u22.f/.x2 -X2u11.f/.x1 -X2u12.f/.x2. = (X1u21 -X2u11).f/.x1 + (X1u22 -X2u12).f/.x2 = [X1, X2]f. This quantity is called the Poisson operator or the commutator of the operators X1f and X2f. The method can be generalized to include . = 1 to r essential parameters and i = 1 to n variables. The ath-linear operator is then Xa = uia.f/.xi n = . uia.f/.xi , ( a sum over repeated indices). i = 1 Lies differential equations have the form .xi/.a. = uik(x)Ak.(a) , i = 1 to n, . = 1 to r. Lie showed that (.ckts/.a.)uik = 0 in which 80 ujs.uit/.xj -ujt.uis/.xj = ckts (a)uik(x), so that the c ktss are constants. Furthermore, the commutators can be written [X., Xs] = ( ck.sujk)./.xj = ck.sXk. The commutators are linear combinations of the Xks. (Recall the earlier discussion of the angular momentum operators and their commutators). The ck.ss are called the structure constants of the group. They have the properties ck.s = -cks. , c.sc.t + cstc.. + ct.c.s = 0. Lie made the remarkable discovery that, given these structure constants, the functions that satisfy .xi/.a. = uikAk.(a) can be found. (Proofs of all the above important statements, together with proofs of Lies three fundamental theorems, are given in Eisenharts standard work Continuous Groups of Transformations, Dover Publications, 1961). 8.1 The rank of a group Let A be an operator that is a linear combination of the generators of a group, Xi: A = aiXi (sum over i), and let 81 X = xjXj . The rank of the group is defined as the minimum number of commuting, linearly independent operators of the form A. We therefore require all solutions of [A, X] = 0. For example, consider the orthogonal group, O+(3); here A = aiXi i = 1 to 3, and X = xjXj j = 1 to 3 so that [A, X] = aixj[Xi, Xj] i, j = 1 to 3 = aixjeijkXk . The elements of the sets of generators are linearly independent, therefore aixjeijk = 0 (sum over i, j,, k = 1, 2, 3) This equation represents the equations -a2 a1 0 x1 0 .a3 0 -a2..x2. = .0. . 0 -a3 a2 x3 0 The determinant of is zero, therefore a non-trivial solution of the xjs exists. The solution is given by xj = aj (j = 1, 2, 3) so that A = X . O+(3) is a group of rank one. 8.2 The Casimir operator of O+(3) The generators of the rotation group O+(3) are the operators. Yks, 82 discussed previously. They are directly related to the angular momentum operators, Jk: Jk = -i(h/2p)Yk (k = 1, 2, 3). The matrix representations of the Yks are 0 0 0 0 0 -1 0 1 0 Y1 = . 0 0 1., Y2 = . 0 0 0. , Y3 = . -1 0 0. . 0 -1 0 1 0 0 0 0 0 The square of the total angular momentum, J is 3 J2 = . Ji2 1 = (h/2p)2 (Y12 + Y22 + Y32) = (h/2p)2(-2I). Schurs lemma states that an operator that is a constant multiple of I commutes with all matrix irreps of a group, so that [Jk, J2] = 0 , k = 1,2 ,3. The operator J2 with this property is called the Casimir operator of the group O+(3). In general, the set of operators {Ci} in which the elements commute with the elements of the set of irreps of a given group, forms the set of Casimir operators of the group. All Casimir operators are constant multiples of the unit matrix: Ci = aiI. The constants ai are characteristic of a particular representation of a group. 9 83 MATRIX REPRESENTATIONS OF GROUPS Matrix r epresentations of linear operators are important in Linear Algebra; we s hall see that they are eq ually important in Group Theory. Ifa group of m m matrices Dn(m) = {D1(m) (g1),...Dk(m) (gk), ...Dn(m) (gn)} can be f ound in which each element is associated with the corre sponding element gk of a group o f ord er n Gn = {g1,...gk,....gn}, and the matrices obey Dj(m) (gj)Di(m) (gi) = Dji(m) (gjgi), and D1(m) (g1) = I, the identity, then the matrices Dk(m) (gk) are said to form an m-dimensional rep resentation of Gn. If the association is one-to-one we have an isomorph ism and the rep resentation is said to be fai thful . The subject o f Group Representations forms a ver y large bra nch o f Gro up Theory. The re are man y standard works on this topic (see the bibliography), each one cont aining numerous definitions, lemmas and theorems. He re, a rather brief account is given of some o f the more importan t res ults. The read er sh ould delve into the deeper asp ects of the subject as the need arises. The subject will be introduced by considering representations of the rotat ion groups, and their corresponding cyclic 84 gro ups. 9.1 The 3-di mensional representation of rotations in the plane The rotation of a vector through an angle f in the plane is characterized by the 2 x 2 matrix c osf -sinf Rv(f) = . sinf cos f The group of symmetry transformat ions that leaves an equilateral triangle invariant unde r rotations in the p lane is of order three, and each element of the group is of d imension two Gn ~ R3(2) = {R(0), R(2p /3), R(4p /3)} = 1 0 , -1/2 -v3/2 , -1/2 v3/2 . 0 1 v3/2 -1/2 , -v3/2 -1/2 { 123, 312, 231} = C3. These matrices form a 2-dimensional representation of C3 . A 3 -dimensional representation of C3 can be obtained as follows: Consider an equilateral triangle located in the plane and let the coordinates of the three ver tices P1[x, y], P2[x, y], and P3[x, y] be written as a 3-vector P13 = [P1, P2, P3], in normal order . We introduce 3 3 matrix ope rators Di(3) that ch ange the o rder of the elements of P13, cyclically. The identity is P13 = D1(3) P13, where D1(3) = diag(1, 1, 1). The rearrangement 85 P13 . P23[P3, P1, P2] is given by P23 = D2(3) P13, where 0 0 1 D2(3) = 1 0 0 , 0 1 0 and the rearrangement P13 . P33[P2, P3, P1] is given by P33 = D3(3) P13 where 0 1 0 D3(3) = 0 0 1 . 1 0 0 The set of matrices {Di(3) } = {D1(3) , D2(3) , D3(3) } is said to form a 3- dimensional r epresentation of the original 2-dimensional representation {R3(2) }. The elements Di(3) ha ve th e same group multiplication ta ble as that associated with C3. 9.2 The m-di mensional representation of sy mmetry tra nsformations in d-di mensions Consider the case in which a group of order n Gn = {g1, g2, ...gk, ...gn} is repre sented by Rn(m) = {R1(m) , R2(m) , .....Rn(m) 86 where Rn(m) ~ Gn, and Rk(m) is an m m matrix representation of gk. Let P1d be a vector in d-dimensional space, written in normal order: P1d = [P1, P2, ...Pd], and let P1m = [P1d, P2d, ....Pmd] be an m-vector, written in normal order, in which the comp onents are each d-vectors. Introd uce t he m m matrix ope rator Dk(m) (gk) such t hat P1m = D1(m) (g1)P1m P2m = D2(m) (g2)P1m . . Pkm = Dk(m) (gk)P1m , k = 1 to m, the number of symmetry operations, where Pkm is the kth (cyclic) permuta tion of P1m , and Dk(m) (gk) is called the m-d imensional representation of g k. Infinitely many representations of a given repre sentation can be 87 found, for, if S is a matrix repre sentation, and M is any definite matrix with an inverse, we can form T(x) = MS(x)M-1, . x . G. Sin ce T(xy) = MS(xy)M-1 = MS(x)S(y)M-1 = MS(x)M-1MS(y)M-1 = T(x)T(y), T is a representation of G. The new re presentation simply involves a change o f variable in t he corresponding substitutions. Representations related in the man ner of S an d T are equivalent , and are not regarded as different representations. All representations that are e quivalent to S are equivalent to each othe r, and the y for m an infinite class. Two equivalent rep resentations will be written S ~ T. 9.3 Direct sums If S is a representation of dimension s, and T is a representation of dimension t of a group G, the mat rix S(g) 0 P = , (g . G) 0T(g) of dimension s + t is called the direct sum of the matrices S(g) and T(g), written P = S . T. Therefore, given two representations (they can be the same), we can obta in a third by adding them directly. Alternatively, let P be a rep resentation of dimension s + t; we suppose that, for all x . G, the matrix P(x) is of the form A(x) 0 0B(x) where A(x) and B(x) are s s and t t matrices, respectively. (The 0s 88 are s t and t s zero matrices). Define the matrices S an d T as follows: S(x) = A(x) and T(x) = B(x), . x . G. Since, by the group property , P(xy) = P(x)P(y), A(xy) 0A(x) 0A(y) 0 = 0B(xy) 0B(x) 0B(y) A(x)A(y) 0 = . 0B(x)B(y) Therefore, S(xy) = S(x)S(y) and T(xy) = T(x)T(y), so that S an d T are rep resentations. The representation P is said to be decomposable, with components S an d T. A representation is indecomposable if it cannot be decomposed. If a com ponent of a decomposable repre sentation is itself decomposable, we c an continue in this manner to decompose any rep resentation into a finite number of indecomposable comp onents. (It should be not ed th at the property of indecomposablity depends on the field of the representation; the real field must sometimes be extended to the complex field to check for indecomposability). A w eaker form of d ecomposability arises when we consider a matrix of the form A(x) 0 P(x) = E(x) B(x) where A(x), and B(x) are matrices of dimensions s s and t t 89 respectively and E(x) is a matrix that de pends on x, and 0 is the s t zero matrix. The matrix P, and an y equ ivalent form, is said to be reducible. An irreducible representation is one t hat cannot be reduced. Every decomposable matrix is reducible (E(x) = 0), whereas a reducible rep resentation need not be decomposable. If S an d T are red ucible, we can continue in this way to obtain a se t of irreducible com ponents. The components are d eterm ined uniquely, up to an equivalence. The set of distinct irr educible representations of a finite gro up is (in a given field) an invariant of the group . The com ponents form the building bloc ks of a representation of a group. In Physics, decomposable representations are ge nerally referred to as reducible representatio ns (reps). 9.4 Sim ilarity and unitary transforma tions and matrix dia gonal ization Before discussing the q uestion of the possibility of reducing the dimension of a given re presentation, it will be useful to consider s ome importan t res ults in theTheory of Matr ices. The proofs of these statements are given in the standard works o n Matrix Theory. (See bibliography). If there exists a matrix Q such th at Q-1AQ = B , then the matrices A an d B are related by a similarity transformation. If Q is unitary (QQ = I: Q =(Q*)T , the hermitian conjugate) 90 then A an d B are related by a unitary transformation. If A = Q-1AQ; B = Q-1BQ; C = Q-1CQ..then any algebraic relation amon g A, B, C...is also satisfied by A, B, C ... If a similarity transformation produces a d iagonal matrix then the pro cess is called dia gonal ization. If A an d B can be diagonalized by the same matrix the n A an d B commute. If V is form ed from the eigenvectors of A then th e similarity tra nsformation V-1AV will produce a diagonal matrix whose elements are the eigenvalues of A. If A is hermitian then V will be unitary a nd therefore an hermitian matrix can always be diagonalized by a unitary transformat ion. A real symmetric mat rix can always be diagonalized by an orthogonal tra nsformation. 9.5 The Schur-Auerbach theorem This theorem states Every matrix representation of a finit e group is equi valent to a uni tary matrix representatio n Let Gn = {D1, D2, ....Dn} b e a matrix group, and let D be the matrix formed b y tak ing the sum of pairs of elements n D = . DiDi i = 1 where Di is the hermitian conjugate of Di. Since Di is non-singular, each term in the sum is positive definite. 91 Therefore D itself is positive definite. Let Ld be a diagonal matrix that is equivalent to D, and let Ld1/2 be the positive definite ma trix formed by replacing the elements of Ld by their positive square roots. Let U be a unitary matrix with the property that Ld = UDU-1. Introdu ce th e mat rix S = Ld-1/ 2U, then SDiS-1 is unitary. (This property can be dem onstrated by co nsidering (SDiS-1)(SDiS-1), and showing that it is equal to the identity.). S will tra nsform the original matrix representation Gn into diagonal form. Every unitary matrix is diagonalizable, and therefore every matrix in ever y finite matrix r epresentation can be diagonalized. 9.6 Schurs lemmas A m atrix representation is reducible if every element of t he representation can be put in block-diagonal form by a single similarity tra nsformation. Invoking the result of the previous section, we need only discuss unitary representations. IfGn = {D(.)(R)} is an irreducible rep resentation of dimension . of a g roup Gn, and {D()(R)} is an irreducible representation of dimension of the same g roup, Gn, and if there exists a matrix A such th at D(.)(R)A = AD()(R) . R . Gn then either i) A = 0 92 or ii) A is a sq uare non-singular matrix (so that . = ) Let the co lumns of A be written c1, c2, ...c, then, for any ma trices D(.) an d D() we have D(.)A = an AD() = therefore D(.)cj = (D(.)c1, D(.)c2, ...D(.)cn) ( . D() k1ck, . D() k2ck, ....D() kck). k = 1 k = 1 k = 1 . D() kjck k = 1 and therefore the c-vectors span a sp ace that is invariant under the irreducible set of .-dimensional matrices {D(.)}. The c-vectors are therefore the null-vector or they span a .-dimensional vector spa ce. The first case corresponds to A = 0, and the second to = . an d A . 0. In the second case, the hermitian conjugates D(.) 1, ...D(.) n an d D() 1, ...D() n also ar e irreducible . Furthermo re, since D(.) i(R)A = AD() i(R) D() iA = AD(.) i , and therefore, following the method ab ove, we find that .= . We must therefore have . = , so tha t A is square.. Since the .-columns of A span a .-dimensional space, the matrix A is necessarily non-singular. As a cor ollary, a matrix D that commutes with an irr educible set of matrices must be a scalar ma trix. 9.7 Characters 93 If D(.)(R) and D()(R) are related by a similarity transformat ion then D(.)(R) gives a representation of G t hat is equivalent to D()(R). These two sets of matrices are ge nerally different, whereas their structu re is the same. We wish, therefore, to answer the question: what intrinsic properties of the matrix r epresentations are invariant u nder coord inate transformations? Consider . [ CD(R)C-1]ii = . CikDkl(R)Cli-1 i ikl = .dklDkl(R) kl = . Dkk(R) , the trace of D(R). k We see that the trace, or character, is an invariant under a change of coordinate ax es. We write the character as .(R) = . Dii(R) i Equ ivalent representations have t he same set of characters. The ch aracter of R in the repre sentation is written .()(R) or [ ; R]. Now, the conjugate elements of G have the form S = URU-1, and then D(R) = D(U)D(R)[D(R)]-1 therefore .(S) = .(R). We can describe G by giving its characters in a particular representation; all elements in a class have the same .. 10 94 SOME LIE GROU PS OF TRANSFORM ATIONS We shall consider those Lie group s that can be described b y a finite set of continuously varying essential parameters a1,...ar: xi = fi(x1,...xn; a1,...ar) = f(x; a) . A set of para meters a e xists that is associated with the inverse tra nsformations: x = f(x; a). These equations mu st be solvable to give th e xis in terms o f the xis. 10.1 Linear group s The general linear group GL (n) in n-dimensions is given b y the set of equations n xi = . aijxj , i = 1 to n, j = 1 in which det |aij| . 0. The group con tains n2 pa rameters t hat h ave values covering an infinite ran ge. The group GL(n) is said to be not closed. All linear groups with n > 1 are non-abelian. The group GL(n ) is isomorph ic to the group of n n matrices; the law of comp osition is therefore mat rix multiplication. The special linear group of transformations SL(n) in n-dimensions is obt ained from GL(n ) by imposing the condition de t|aij| = 1. A functional relation therefore exists among the n2- parameters so th at the num ber of required para meter s is reduced to (n2 -1). 10.2 Orthogonal groups 95 If the transformat ions of the general linear group GL(n) are such that n . xi2 . invariant , i = 1 then the rest ricted group is called th e orthogonal gr oup, O(n), in n- dimensions. There are [n + n(n -1)/2] conditions imposed on the n2 par ameters of GL(n ), and the refore the re are n(n - 1)/2 essential par ameters of O(n). For exam ple, in three d imensions x = Ox ; O = { O33: OOT = I, detO = 1, aij . R} where a11 a 12 a 13 O = a21 a 22 a 23 . a 31 a32 a 33 We have x12 +x22 + x32 = x12 +x22 +x32 . invariant under O(3). This invariance imposes six conditions on the original nine par ameters, and therefore O(3) is a three-parameter group. 10.3 Unitary grou ps If the x is and t he aijs of the general linear group GL(n) are complex, and the transformat ions are required to leave xx invariant in the complex space, then we obtain the unitary group U(n) in n-dimensions: U(n) = { Unn: UU = I, detU . 0, uij . C}. There are 2n2 independent real parameters ( th e real and imaginary parts of the aijs), and the unitary condition imposes n + n(n -1) c onditions on them so the group has n2 real paramet ers. The unitary condition means 96 that .j |a ij|2 = 1, and therefore |aij|2 = 1 fo r all i, j. The para meters are limited to a f inite range of values, and the refore the gro up U(n) is said to be closed. 10.4 Sp ecial unit ary g roups If we impose the restriction detU = +1 on the unitary group U (n) then we obtain the special unita ry group SU(n) in n-dimensions. We have SU(n) = { Unn: UU = I, detU = +1, uij . C}. The dete rminantal condition reduces the num ber of required real par ameters to (n2 -1). We shall see th at the groups S U(2) and S U(3) play an important part in Modern Physics. 10.5 The group SU(2), the infinitesimal form of SU(2), and the Pauli sp in matrices The special unitary group in 2-dimensions, SU(2), is defined as SU (2) = { U22: UU = I, detU = +1, uij . C}. It is a three-parameter group. The defining conditions can be used to obta in the mat rix rep resentation in its simplest form; let a b U = c d where a, b, c, d . C. 97 The hermitian conjugate is a* c* U = , b* d* and therefore | a|2 + |b|2 ac* + bd* UU = . a*c + b*d |c |2 + |d|2 The unitary c ondition gives |a|2 + |b|2 = |c|2 + |d|2 = 1, and the determinantal condition gives ad -bc = 1 . Solving these equations , we obta in c = -b*, and d = a*. The general form of SU(2) is therefore a b U = . -b* a* We now study the infinitesimal form of SU(2); it must have the structur e 1 0 da db 1 + da db Uinf = + = . 0 1 -db* da* -db* 1 + da* The dete rminantal condition therefore gives d etUinf = (1+ da)(1 + da*) +dbdb* = 1 . To first order in the ds, we obtain 1 + da* + da = 1, or 98 da = -da*. so that 1+ da db Uinf = . -db* 1-da The matrix elements can be written in their comp lex forms: da = ida/2 , db = d/2 + id./2. (The factor of two has been introduced for later conv enience). 1 + ida/2 d/2 + id./2 Uinf = . -d/2 + id./2 1 - ida/2 Now, any 22 matrix can be written as a linear co mbination of the matrices 1 0 0 1 0 -i 1 0 , , , . 0 1 1 0 i 0 0 -1 as follows a b 1 0 0 1 0 -i 1 0 = A + B + C + D , c d 0 1 1 0 i 0 0 -1 where a = A + D, b= B -iC, c = B + iC, and d = A - D. We then have a b (a + d) 1 0 (b+ c) 0 1 i(b -c) 0 -i (a- d) 1 0 = + + + . c d 2 0 1 2 1 0 2 i 0 2 0 -1 The infinitesimal form of SU(2) can th erefore be written Uinf = I + (id./2) 1 + (id/2) 2 + (ida/2) 3 , 99 or Uinf = I + (i/2). dtjThis is the Lie form. j . j = 1 to 3. The js are the Pau li sp in-ma trices:; they are the generators o f the group SU(2): 0 1 0 -i 1 0 = = = ,2 ,3 . 1 1 0 i 0 0 -1 They play a f undamental role in t he description of spin-1/2 particles in Quantum mechanics. (See later discussions). 10.6 Commutators of th e spi n mat rices and structure constants We have previously introduced the commutators of the infinitesimal generators of a Li e group in conn ection with the ir Lie Algebra. In this section, we c onsider the com mutat ors o f the generators of SU(2); they are found to have the symmetric forms [ 1, 2] = i 3, 2 [ 2, 1] = -2i [ 1, 3] = i 2, [ -2 3, 1] = [ 2, 3] = i 1, 2 [3, 2] = 3, i 2, 2 i 1. -2 We see that the commutator of any pair of t he th ree matrices gives a constant multiplied by the v alue of the remaining matrix, thus [ j, k] = ejkl2i l . where the quantity ejkl = 1, depending on the permutations of the indices. (e(xy )z = +1, e(yx )z = -1 ..etc...). The quan tities 2iejkl are the structure constants associated with the group. 100 Other propert ies of the spin matrices are found to be 2 = 2 = 2 = I; 1 2 = i 3, 2 3 = i 1, 3 1 = i 2. 1 2 3 10.7 Homomor phism of S U(2) and O+(3) We can form the ma trix P = xT = xj j, j = 1, 2, 3 from the matrices x = [x1, x2, x3] and = [ 1, 2, 3] : therefore x 3 x1 -ix2 P = . x1 + ix2 -x3 We see that x3 x1 -ix2 P = (P*)T = = P, x1 + ix2 -x3 so that P is hermitian. Furtherm ore, TrP = 0, and detP = -(x12 + x22 + x32). Another matrix, P, can be formed b y car rying out a similarity tra nsformation, thus P = UPU , (U . SU(2)). A similarity transformation leaves bot h the trace and the determinant unchanged, therefore 101 T rP = TrP, and d etP = detP. However, the condition d etP = detP means that xxT = xxT, or x 12 + x22 + x32 = x12 + x22 + x32 . The transformation P = UPU is therefore equivalent to a three- dimensional or thogonal transformat ion that leaves xxT invariant. 10.8 Ir reducible representations of S U(2) We have seen that the b asic form of the 22 matrix representation of the group SU(2) is a b U = , a,b . C; |a|2 + |b|2 =1. -b* a* Le t the basis vectors of this space be 1 0 x1 = and x2 = . 0 1 We then have a x1 = Ux1 = = ax1 -b*x2 , -b* and b x2 = Ux2 = = bx1 + a*x2 , a* and therefore x = Utx. If we wr ite a 2-dimensional vector in this complex space as c = [u, v ] then the components transform under SU(2) as 102 u = au + bv and v = -b*u + a*v , and therefore c = Uc . We see that the components o f the vect or c tr ansform dif ferently from those of the basis vect or x the transformat ion matrices are the tra nsposes of each othe r. T he vector c = [u, v ] in this complex space is called a spinor (Cartan, 1913). To find an irreducible repre sentation of SU(2) in a 3-dimensional space, we need a s et of three linearly independent ba sis functions. Following Wigner (see bibliograph y), we can choo se the polynomials u2, uv, and v2, and introduce the polynomials defined by 1 + m 1 -m j = 1 u v f = m v { (1 + m)! (1 + m)!} where j = n/2 (the dimension of the space is n + 1) . and m = j, j- 1,...-j . Inthe p resent case, n= 2,j = 1 , and m = 0, 1. (The factor 1/v{(1 + m)! (1 - m)!} is chosen to make the representative matrix unitary). 103 We have, therefore f11 = u2/v2 , f01 = uv, and f-11 = v2/v2. A 3 3 r epresentation of an element U . SU(2) in this space can be found by defining the transformation Ufm1(u, v) = fm1(u, v). We then obtain m Ufm1(u, v) = (au + bv)1 + m(-b*u +a*v)1 -,m = 0, 1, v{(1 + m)!(1 -m)!} so that Uf11(u, v) = (au + bv)2/v 2 = (a2u2 + 2abuv + b2v2)/v 2 , Uf01(u, v) = (au + bv)(-b*u + a*v) = -ab*u2 + (|a|2 -|b|2)uv + a*bv2 , and Uf-11(u, v) = (-b*u + a*v)2/v 2 = (b*2u2 -2a*b*uv + a*2v2)/v 2 . We then have 1 a2 v 2ab b2 f1 f11 -v 2ab* |a|2 -|b|2 v2a*b f01 = f01 1 b*2 -v2a*b* a*2 f-1 f-11 or UF = F. We find that UU = I an d the refore U is, indeed, unitary. This pro cedure can be generalized to an (n + 1)-dimensional space as follows Let fmj(u, v) = uj + mvj -m , m = j, j - 1, ...-j. 104 v{(j + m)!(j -m)!} (Note that j = n/2 = 1/ 2, 1/1, 3/2, 2/1, ..). For a given v alue of j, there are 2j + 1 linearly independent p olynomials, and therefore we c an form a (2j + 1) (2j + 1 ) representative mat rix of an element U of SU(2): Ufmj(u, v) = fmj(u, v). The deta ils of this general case are g iven in Wigner s classic text. He demonstrates the irredu cibility of the (2j + 1)-dimensional representation by showing that an y matrix M wh ich commut es with Uj for all a, b such that |a|2 + |b|2 = 1 mus t nec essarily be a constant ma trix, and therefore, by Schurs lemma, Uj is an irredu cible representation. 10.9 Representations o f rotations and the concept of tens ors We have discussed 2- a nd 3-dimensional representations of the ort hogonal group O (3) and their connection to an gular momentum operators. Higher-dimensional representations of the orthogonal group can be obtained b y con sidering a 2-index q uantity , Tij a tensor that consists of a set of 9 elements t hat transform under a rotation of t he coordinates as follows: Tij . Tij = RilRjmTlm (sum over repeated indices 1, 2, 3). If Tij = Tji (Tij is symm etric), then th is symmetr y is an invariant un der rot ations; we have Tji = RjlRimTlm = RjmRilTml = RilRjmTlm = Tij . 105 If TrTij = 0, then so is TrTij, for Tii = RilRimTlm = (RTR)lmTlm = dlmTlm = Tll = 0. The comp onents of a symmetric traceless 2-index tensor contains 5 members so th at the transformation Tij . Tij = RilRjmTlm de fines a new rep resentation of them of dimension 5. Any tensor Tij can be written Tij = (Tij + Tji)/2 + (Tij -Tji)/2 , and we have Tij = (Tij + Tji)/2 = (Tij -(dijTll)/3) + (dijTll)/3 . The decomposition of the ten sor Tij gives any 2-index tensor in terms of a sum of a single component, proportional to the identity, a set of 3 independent q uantities combined in an anti-symmetric tensor (Tij -Tji)/2, and a se t of 5 independent c ompon ents of a symmetric traceless tensor. We write the dimensional eq uation 9 = 1 . 3 . 5. This is as far as it is possible to go in t he process of d ecomposition: no oth er su bsets of 2 -index ten sors can be found that preserve the ir identities under the defining transformation of t he coordinates. Rep resentatio ns with no subsets of tensors that preserve th eir identities under the defining rotation s of tensors are irreducible representations. We shall see that the decomposition of tensor products into 106 symmetric and anti-symmetric parts is important in the Quark Mo del of elementary particles. The representations of the o rthog onal group O(3) are found to be importan t in defining the int rinsic spin of a particle. The dy namics of a par ticle of finite mass can always be descibed in its rest frame (all inertial frames are eq uivalent!), and therefore the particle can be char acterized by rot ations. All known p articles have dynamical states that can be described in terms of t he te nsors of some irredu cible representation of O (3). If the dimension of the irrep is (2j + 1 ) then the particle spin is found to be pro portional to j. In Particle Physics, irreps with values of j = 0 , 1, 2,... and with j = 1/2, 3/2, ... are found that corre spond to the fundamental bosons and fermions, resp ectively. The three dimensional ortho gonal group SO(3) (det = +1) and the two dimensional group SU(2) have the same Lie algebra. In the case of the group SU(2), the (2j + 1 )-dimensional representations are a llowed for bot h integer and h alf -integer va lues of j, whereas, the representations of the group SO(3) are limited to integer values of j. Since all the rep resentations ar e allowed in SU(2), it is called th e covering gr oup. We not e tha t rotations thr ough f a nd f +2p ha ve different effects on the 1/2- integer repre sentations, and therefore they are (spinor) t ransfomations associated with SU(2). 11 107 THE GROUP STRUCTURE OF LORENTZ TRANSFORMATIONS The square of the invariant interval s, between the o rigin [0, 0, 0, 0] of a spacetime coordinate system and an arbitrary event x = [x0, x1, x2, x3] is, in index notation s2 = xx = xx , (sum over = 0, 1, 2, 3 ). The lower indices can be raised using the metric tensor .. = diag(1, 1, 1, 1), so that s2 = ..xx. = ..xxv , (sum over an d .). The vect ors n ow ha ve contrav ariant for ms. In matrix notation, the invariant is s2 = xT x = xT x . (The transpose must be written explicitly). The primed and unp rimed column matrices (contrav ariant vec tors) are related by the Lorentz matrix ope rator, L x = Lx . We therefore have xT x =(Lx)T (Lx) = xTLT Lx . The xs are arbitrary, therefore LT L =. This is the defining propert y of the Lorent z transformations. 108 The set of all Lor entz transformations is the set L of all 4 4 matrices that satisfies the defining proper ty L ={L: LT L =; L: all 4 4 real matrices; = diag(1, 1, 1, 1}. (Note that each L ha s 16 (independent) r eal matrix elements, and therefore belongs to the 16-dimensional space, R16). 11.1 The group structure of L Consider the result of two successive Lorentz transformations L1 and L2 that transform a 4-vector x as follows x . x . x where x = L1x , and x = L2x. The resu ltant vector x is given b y x = L2(L1x) = L2L1x = Lcx where Lc = L2L1 (L1 followed by L2). If the c ombined op eration Lc is always a Lorentz transformation then it must satisfy T Lc Lc = . 109 We must therefore have (L2L1)T (L2L1) = or L1T(L2T L2)L1 = so that T L1 L1 =, (L1, L2 . L) therefore Lc = L2L1 . L . Any number of successive Lorentz transformations may be carried out to give a resultant that is itself a Lorentz transformat ion. If we take th e det erminant o f the defining equation of L, d et(LT L) = det we obtain (detL)2 = 1 (d etL = detLT) so that detL = 1. Since t he determinant of L is not zero, an inverse transformation L1 exists, and t he equation L1L = I, the identity, is always valid. Consider the inverse of the defining equation (LT L)1 = 1 , or L1 1(LT)1 = 1 . 110 Using = 1, and rearran ging, gives L1 (L1)T = . This result shows that the inverse L1 is always a member of the set L. We therefore see that 1. If L1 an d L2 . L , then L2 L1 . L 2. If L . L , then L1 . L 3. The identity I = diag(1, 1, 1, 1 ) . L and 4. The matrix operators L ob ey associativity. The set of all Lor entz transformations therefore form s a group. 11.2 The rotation grou p, revisited Spatial rotat ions in two and three dimensions are Lorentz tra nsformations in which the time-component remains unchanged. Let R be a real 33 matrix that is part of a Lorent z transformation with a constant time-component. In this case, the defining property of t he Lorentz transformations leads to RTR = I , the identity matrix, diag(1,1,1). This is the d efining propert y of a three-dimensional ortho gonal matrix If x = [x1, x2, x3] is a t hree-vector that is transformed und er R to give x then xTx = xTRTRx 111 = xTx = x12 + x22 + x32 = invariant unde r R. The action of R on any three-vector preserves length. The set of all 33 ort hogonal matrices is denoted by O(3), O(3) = {R: RTR = I, r ij . R}. The elements of this set satisfy the four group axioms. The group O(3) can be split into two parts that are said to be disconnected:: one w ith detR = +1 an d the othe r with det R = -1. The two parts are written O+(3) = {R: detR = +1} and O-(3) = {R: detR = -1} . If we define the parity operator , P, t o be the o perator that reflects all points in a 3-dimensional cartesian system throug h the origin then -1 0 0 P = 0 -1 0 . 0 0 -1 The two parts of O(3) are related by the operator P: if R . O+(3) then PR . O-(3), and if R . O-(3) then PR . O+(3). 112 We can t herefore c onsider only that pa rt of O(3) that is a group, namely O+(3), together with the operator P. 11.3 Connected and disconnected parts of the Lorentz grou p We have shown, previously, that e very Lorent z transformation, L, has a determinant equal to 1. The matrix elements o f L ch ange continuously as the relative velocity changes continuously. It is not possible, however, to move continuously in such a way that we c an go from the set of tr ansformations with detL = +1 to thos e with det L = -1; we say that the set { L: detL = +1} is disconnected fr om the set {L: detL =- 1}. If we wr ite the Lorentz transformation in its component form L . L . where = 0,1,2,3 labels the rows, and . = 0,1,2,3 labels the c olumns then the time comp onent L00 ha s the values L00 = +1 or L00 = -1. The set of tr ansformations can th erefore be split into four disconnected parts, labelled as follows: { L. +} = {L: detL = +1, L00 = +1} {L. -} = {L: detL = -1, L00 = +1} {L. +} = {L: detL = +1, L00 = -1}, and {L. -} = {L: detL = -1, L00 = -1}. 113 The identity is in {L. +}. 11.4 Parity, time-reversal and orthochronous transformations Two discrete Lorentz transformations are i) the p arity transformation P ={P: r . -r, t . t} = di ag(1, -1, -1, -1), and ii) the time-rever sal transfprmat ion T ={T: r . r, t . -t} = di ag(-1, 1, 1, 1} . The disconnected p arts of {L} are related by the transformations that involve P, T, and PT, as shown: PT L. + L. - P T L. -L. - Fig. 3 Connections b etween the disconnected p arts of the Lorentz transformations The proper orthochronou s tra nsformations ar e in the g roup L. +. We 114 see that it is not necessary to consider the complete set {L} o f Lorentz tra nsformations we need consider only tha t sub set { L. +} t hat forms a gro up by itself, and either P, T, or PT co mbined. Experiments have shown clear v iolations under the parity transformation, P an d violations under T ha ve been inferred fro m exp eriment an d the ory, combined. However, not a single experiment has been carried out that shows a violation of the p roper orthochro nous transformations, {L. +}. 12 115 ISOSPIN Par ticles can be distinguished from one another by their intrinsic pro perties: mass, charge, spin, parity, and their electric and magnetic mom ents. In o ur on-going que st for an under standing of the true natu re of the fundamental particles, and their interactions, other intrinsic proper ties, with nam es such as isospin and stra ngeness, have been discovered. The intrinsic properties are defined b y qua ntum numbers; for example, the quantum number a is defined b y the eigenvalue equation Af = a.f where A is a linear operator, f is the wavefunction of the system in the zero-momentum frame, and a is an eigenvalue of A. In this chapter, we shall discuss the first of t hese new p roper ties to be introduced, namely, isospin. The building blocks of n uclei are proto ns (positively charged) and neutrons (neutral). Nu merou s experiments on the scattering of proto ns by pro tons, and proto ns by neut rons, have shown tha t the nuclear forces between pairs have the same strength, provided t he angular momentum and spin states are the same. These observations form the basis of an importan t con cept the charge-independence of.the n ucleon-nuc leon. force. (Corrections for the coulomb e ffects in proto n-proton s cattering must be made). The origin of this concept is found in a new symmetry principle. In 1932, Chadwick no t only identified th e neutron in studying the interaction of alpha-particles on beryllium nuclei but also showed that 116 its mass is almost equal to the mass of the proton. (Recent measurements give m ass of proton = 93827231(28) MeV/c2 and ma ss of neut ron = 93956563(28) MeV/c2) Within a few month s of Chadwicks discovery , Heisenberg introdu ced a theory of nuclear forces in which he considered the n eutro n and the proto n to be tw o st ates of t he same ob ject the nucleon. He introd uced an intrinsic var iable, later ca lled isospin, that p ermits the char ge states (+, 0) of the nucleons to be distinguished. This new variable is needed (in addition to the traditional space-spin variables) in the description of nucleon- nucleon scattering. In nuclei, protons and neutr ons behave in a remarkably sym metrical way: the binding energy of a nucleus is closely propo rtional to the number of neutr ons and protons, and in light nuclei (mass number <40), the number of neutrons can be equal to the number of protons. Before discussing the isospin of particles and n uclei, it is necessary to introduce an extended Pa uli Exclusion Principle. In its or iginal form, th e Pauli Exclusion Principle was introduc ed to account for features in the observed spectra of atoms th at co uld not be unde rstood using the the n cur rent models of atomic str uctur e: no.two electrons in an.atom.can exist.in the same quantum.state defined.117 by.the q uantum num bers.n, , m , m s wh ere n is the principal quantum number, is the.orbital angular momentum.quantum nu mber, m is the. magnetic quan tum n umber, and ms is the.spin.quantum nu mber. For a sy stem of N particles, the complete wavefunction is written as a p roduct of single-particle wavefunctions .(1, 2, ...N) = .(1).(2)....(N). Consider this form in t he simplest case for two identical particles. L et one be in a s tate labelled .a an d the othe r in a state .b. For i dentical particles, it makes no difference to the probability density |.|2 of the 2- particle system if the particles are e xchanged: |.(1, 2)|2 = |.(2, 1)|2 , (the .s are n ot measurable) so that, either .(2, 1) = .(1, 2) (symmetric) or .(2, 1) = -.(1, 2) (anti-symmetric). Let .I = .a(1).b(2) (1an a, 2 in b) and .II = .a(2).(1) (2in a, 1 in b). The two particles are indistinguishable, therefore we have no way of 118 knowing whether .I or .II de scribes the system; we po stulate that the system spends 50% of its time in .I an d 50% of its time in .II. The two- particle system is considered to be a lin ear combination of .I an d .II: We have, therefore, either .sym m = (1 /v2){.a(1).b(2) + .a(2).b(1)} (BOS ONS) or .ant isymm = (1/v2){.a(1).b(2) -.a(2).b(1)} (FER MIONS) . (The coefficient (1/v2) normalizes the sum of the squares to be 1). Exc hanging 1. 2 leaves .sym m un changed, whereas exchanging pa rticles 1. 2 rever ses the sign of .ant isymm . If two p articles are in .S, both particles can exist in the same state with a = b. If two particles are in .AS , and a = b, we have .AS = 0 they cannot e xist in the same quantum state. Electrons (fermions, spin = (1/2)h) are described by anti-symmetric wavefunctions. We can now introdu ce a more general Pauli Exclusion Principle. Write the nucleon wavefunction as a product: .(., q) = .(.)fN(q) , where . = .(r, s) in which r is the space vect or, s is the spin, and q is a charge or isospin label. For two nucleons, we write 119 .(.1, q1; .2, q2), for two proto ns: .2p = .1(.1, .2)fN(p1)fN(p2), for two neutr ons: .2n = .2(.1, .2)fN(n1)fN(n2), and for an n-p pair: .np = .3(.1, .2)fN(p1)fN(n2) or = .4(.1, .2)fN(n1)fN(p2). If we re gard the p roton and neutr on as different states of the same object, labelled by the c harge or i sospin coo rdinate, q, we must extend the Pau li principle to cover the new coordinate: the total wavefunction is then .(.1, q1; .2, q2) = -.(.2, q2; .1, q1) . It.must.be an ti-symmetric un der the fu ll ex change.. For a 2p- ora 2n-pair, the exchange q 1. q2 is symm etrical, and therefore the space-spin part mus t be anti-symmetrical. For an n-p pair, the symmetr ic (S) and anti-symmetric (AS) is ospin wav efunctions are I) FS = (1/v2){fN(p1)fN(n2) + fN(n1)fN(p2)} (symmetr ic under q1 . q2), and therefore the space-spin part is anti-symmetrical, II) FAS = (1/v2){fN(p1)fN(n2) -fN(n1)fN(p2)} 120 (anti-symmetr ic under q1 . q2), and therefore the space-spin part is symmet rical. We shall need these results in later discussions of t he symmetr ic and ant i- symmetric properties of quark systems. 12.1 Nuclear -decay Nuclei are bound states of n eutro ns and protons. If the n umber of pro tons in a nucleus is Z and the number of neut rons is N then the mass number of the nucleus is A = N + Z. Some n uclei are natur ally unstable. A p ossible mode of decay is by the emission of a n electron ( th is is -decay a process that typifies the fundamental wea k interaction) . We write the decay as A ZXN . AZ+1 XN-1 + e1 + .e ( ߖ-decay) or, we c an have AA ZXN . Z-1 XN-1 + e+ + .e ( + -decay). A r elated process is that of electron capture of an orbital electron that is sufficiently close to the po sitively charged nuc leus: e A + AZXN . Z+1 XN+1 + .e. Other related processes are .e + AZXN . AZ-1 XN-1 + e+ and .e + AZXN . AZ+1 XN-1 + e . The decay of the free.pr oton has not been observed at the present time. 121 The experimental limit on the half-life of the p roton is > 1031 ye ars! Many cur rent theor ies of the microstructure of m atter predict that the proton decays. If, however, t he life-time is > 1032 - 1033 ye ars t hen t here is no realistic possibility of observing the decay directly (The limit is set by Avogadro s nu mber and t he finite number of proto ns th at can be assembled in a suitable experimental apparatus). The fundamental -decay is that of the free n eutro n, first o bserved in 1946. The process is n0 . p + +e + .e0 ,t1/2 = 1037 + 019 minutes. This measured life-time is of fundamental importance in Particle Phy sics and in Cosmology. Let us set up an algebraic description of t he -decay process, recognizing that we have a 2-state system in which the transformation p . n occurs: In the ߖ-decay of a free n eutro n n . p + +e + .e, and in t he +-decay of a p roton , bound in a nucleus, p . n + e+ + .e . 12.2 Isospin of t he nucleon The spontaneous tr ansformations p. n observed in -decay lead us to introduce the o perators = fp , +fp = 0, (eliminates a p roton ) that transform p . n: +fn and 122 -fp = fn , -fn = 0, (eliminates a n eutro n). Since we are dealing with a two-state system, we choo se the isospin par ts of the proto n and neut ron wavefunctions to be 1 0 f(p) = a nd f(n) = , 0 1 in which case the operators must have the forms: 0 1 0 0 = and - = . + 0 0 1 0 They are singular and n on-hermitian. We have, for example 0 1 0 1 +fn = = , fn .fp, 0 0 1 0 and 0 1 1 0 = = 0 0 0 0 +fp ( + removes a proton). To make the p resent algebraic description analogous to the two-state system of the intrinsic spin of the e lectron, we introduce linear combinations of the : 0 1 = = 1 0 1, a Pauli matrix, + + - = 1 and 0 -i 123 -+) = = 2 = i( - 2, a Pauli matrix. i 0 A t hird operator that is diagonal is, as expected 1 0 = = 3 3, a Pauli matrix. 0 1 The three operators { 1, 3} t herefore obey t he commutation 2, relations [ j/2, k/2] = iejkl l/2 , where the factor of(1/2) is introduced because of the 2:1 homom orphism between SU(2) and O+(3): the vect or operator t = is called the isospin operator o f the nucl eon. /2 To classify the isospin states of the nucleon we may use the pro jection of t on the 3rd axis, t3. The eigenvalues, t3, of t3 co rrespond to the proton (t3 = +1/2) and neutr on (t3 = -1/2) states. The nucleon is said to be an isospin doublet with isospin quan tum number t = 1/2. (The number of states in the multiplet is 2t + 1 = 2 fo r t = 1/2). The char ge, QN of the nucleon can be written in terms of the isospin quantum numbers: QN = q(t 3 +(1/2)) = q or 0, where q is the proton c harge. (It is one o f the grea t unsolved problems of Par ticle Phys ics to und erstand wh y the char ge on the proto n is equal to the charge on the electron). 12.3 Isospin in nuclei. 124 The concept o f isospin, and of ro tations in isospin space, associated with individual nucleons can be applied to nuclei systems of many nucleons in a boun d state. Let the isospin of the ith-nucleon be ti, and let ti = i /2. The operator of a system of A nucleons is defined as T = .Ai=1 ti = .Ai=1 i/2 . The eigenvalue of T3 of the isospin operator T3 is the sum o f the individual components T3 = .Ai=1 t3i = .Ai=1 t3i/2 = (Z N)/2. The char ge, QN of a nucleus can be written QN = q.Ai=1 (t3i + 1)/2 = q( T3 + A/2) . For a given eigenvalue T of the o perator T, the state is (2T + 1) -fold degenerate. The eigenvalues T3 of T3 are T3 = -T, -T + 1,...0,...T + 1, T . If the H amiltonian H of the nucleus is char ge-independent then [H, T] = 0. and T is said to be a good q uantum number. In light nuclei, where the isospin-violating coulomb interaction between pairs of pro tons is a small effect, the c oncept of isospin is particularly useful. The study of isospin effects in nuclei was first applied to the observed p roper ties of the lowest- lying states in the three nuclei with mass number A = 14: 14C, 14N, and 14O. 125 The relative energies of the states are shown in the following diagram: Ene rgy ( MeV) 6 0+ T = 1, T3 =1 4 0+ T = 1 , T3 =0 2 0+ T = 1 , T3 =-1 1+ T = 0 , T3 =0 0 Fig 4. An isospin si nglet (T = 0) a nd an isospin triplet (T = 1) in the A = 14 system. In the absence of the coulomb in teraction, the three T = 1 states woul d be degenerate. The spin and parity of the g round state of 14C, the first excited state o f 14N and the groun d state of 14O a re measured to be 0+; these three states are characterized by T = 1. The ground st ate o f 14N h as spin and parity 1+; it is an isospin singlet (T = 0). 12.4 Isospin and mesons We have seen that it is possible to classify the char ge states of nucleons and nuclear isobars using the concept o f isospin, and the a lgebra of SU(2). It will be useful to classify other particles, including field par ticles (quanta) in t erms of their isospin. Yukawa (1935), first pr oposed tha t the stro ng nu clear force bet ween a p air of nucleons is carried by massive field particles called mesons. Yukawas method was a masterful development of the theory of the 126 electrom agnetic field to include the c ase of a m assive field particle. If .p is the mes on wavefunction the n the Yukawa differential equation for t he meson is .. .p + (E0/hc)2.p = 0. where .. = (1/c2).2/.t2 -.2 . The r-dependent (spatial) form of .2 is .2 . (1/r2)d/dr(r2d/dr) The static (time-independent) solution of t his equation is read ily checked to be .(r) = (-g2/r)exp(-r/rN) where rN = h/mpc = hc/mpc2 = hc/Ep 0, so that 2 1 /rN = (Ep 0/hc)2 The ran ge of the nuclear force is defined by the condition r = rN = h/mpc 2 10-13 cm . This gives the mass of the meson to be close to the measured value. It is importan t to note that the range of t he force . 1/(mass of t he field quantum). In the case of the electrom agnetic field, the mass of the field quantum (the photon) is zero , and therefore the force has an infinite ran ge. The mesons come in three charge states: +, -, and 0. The mesons 127 have intrinsic spins equal to zer o (they ar e field particles and the refore they are bosons), and t heir rest energies are measured to be Ep 0 = 1395 MeV, and Ep00 = 1356 MeV. They are therefore considered to be me mbers of an isospin triplet: t = 1,t3 = 1,0. In Particle Physics, it is the custom to de signate the isospin quantum number by I, we shall follow this conv ention fro m now on. The third com ponent of the isospin is an additive quantum number. The comb ined values of the isospin projections of the two particles, one with isospin projection I3(1) , and t he ot her with I3(2) , is I3(1+ 2) = I3(1) + I3(2) . Their isospins combine to give states with different numbers in each multiplet. For example, in pion (meson)-nucleon scattering p + N . st ates with I3(1 + 2) = (3/2) or (1/2). These values are o btained by noting that Ip (1) = 1, and IN(2) = 1/2, so that I3p (1) + I3N(2) = (1, 0) + (1/2) = (3/2), an isospin quartet, or (1/2), an isospin doublet. Sym bolically , we write 3 . 2 = 4 . 2. (This is the rule for forming the product (2I3(1) + 1).(2I3(2) + 1). 13 128 GROUPS AND THE STRUCTURE OF MATTER 13.1 St rangeness In the e arly 1950 s, our understanding of t he ultimate str uctur e of matter s eemed to be com plete. We required neutr ons, proto ns, electrons and neut rinos, and mesons and pho tons. Our optimism was short-lived. By 1953, excited states of t he nu cleons, and more massive mesons, had been discovered. Some of the new particles had completely unex pected pro perties; for example, in the interaction betw een proton s and p-mesons (pions) the following decay mode was observed: Pr oton (p+) . Pion (p +) Sig ma (.+) Pion (p0 ) Kaon (K+) . Pion (p +) .. Initial interaction Final decay lasts ~10-23 seconds ta kes ~ 10-10 seconds ( Strong force a cting) (Wea k force a cting) Gell-Mann, and independently Nishijima, pro posed that the kaons (heavy mesons) were endowed with a new intrinsic proper ty not affected by the strong force. Gell-Mann called this proper ty s trang eness. S trang eness is conserved in t he strong interactions but cha nges in the weak interactions. The Gell-Mann - Nishijima interpretation of the strangeness-129 changing involved in the proton-pion interaction is p + (S = 0) .+ (S = 1) p0 (S = 0) p+ (S = 0) . K + (S = +1) . p+ (S = 0) .. . S = 0 . S = 1 In the strong part of t he interaction, there is no ch ange in the num ber defining the strangeness, whereas in t he weak part, the strangeness changes by one u nit. Having de fined the values of S for the particles in this interaction, they are d efined for ever. All subsequent exp eriments involving these objects have been consistent with the original assignments. 13.2 Pa rticl e patterns In 1961, Gell-Mann, and independently Neeman, introd uced a scheme t hat c lassified the strong ly interacting particles into family groups (Mendeleev again?). They were not, at this point in the d evelopment of t he theory, concerned with the n ature of p ossible building blocks of the nucleons and mesons. They w ere c oncerned, however, with the inclusion of stra ngeness i n the theo ry, and therefore they were led to study the arr angements of particles in an abstract space de fined by their electric charge a nd st rangeness. The common feature of each family was chosen to be their intrinsic spin; the family of spin-1/2 baryons (strongly 130 interacting particles) has eight members: n0, p+ ,. ,.0 ,. ,.0 , and .0 . Their strangeness quantum numbers are: S = 0: n0, p+ ; S = 1: . ,.0 , and .0 ;and S = 2: .0, . (No negatively- charged proto n exists, and n o positively-charged . ex ists). If the positions of these eight particles are given in charge-strangeness space, a remarkable pattern emerges: Str angen ess n0 p+ . 0 . 0 .0 . .+ .0 +1 .0 Charge Fig. 5. charge-strangeness space 1 The family po rtrait of spin-1/2 baryons in There are two particles at the center, each with zero char ge and zer o strangeness; they are the .0 an d the .0. ( They have different rest masses). 1 2 They studied the struct ure of other families. A particularly 131 importan t set of p articles consists of all baryo ns with sp in 3/2. A t the time, there we re nine kn own p articles in this category: .0, . 1, . +2, .*0, .*1, .0, and .-1 . They have the following pattern in char ge-strangeness space: Charge: 1 0 +1 +2 Str angen ess . 0 .- .0 .+ . ++ .*0 .+ .* .*0 1 .* 2 O The symm etry pattern of the family of spin-3/2 baryon s, shown b y the known nine ob jects was sufficiently compelling for Gell-Mann, in 196 2, to suggest that a ten th member of the family should exist. Furthe rmore, if the symm etry has a physical basis, the tenth member s hould have spin-3/2, charge 1, strangeness 3, and its mass should be about 150MeV greater than the mass of t he .0 pa rticle. Two ye ars after this suggestion, the tenth member of the family was identified in high ener gy pa rticle collisions; it decayed via weak interactions, and possessed the predicted properties. This could not hav e been by chance. The discovery of the O pa rticle was 3 crucial in helping to establish the concept of the Gell-Mann Neeman 132 symmetry mode l. In addition to the symm etries of baryo ns, groupe d by their spins, the model was used to obtain sym metries of mesons, also grouped by their spins. 13.3 The special unitary group SU(3) and particle st ructu re Several years before the work of Gell-Mann and Neeman, Sakata had atte mpted to build-up the kno wn pa rticles from {neutro n- proton- lambda0} t riplets. The lambda particle was required to car ry the strangeness. Alt hough the model was shown not to be valid, Ikeda et al. (1959) introd uced an importa nt mathematical analysis of the three-state system that involved th e group SU(3). The notion tha t an under lying gro up structu re of elementary particles might ex ist was popular in t he early 1960s. (Special Unitary Groups were used by J. P. Elliott in the late1950s to describe symmetry proper ties of light nuclei). The problem facing Particle Physicists, at the time, was to find the appropriate g roup and its fundamental repre sentation, and to co nstruct higher-dimensional representations that wou ld account for the w ide variety of symmetries observed in charge-strangeness space. We have seen th at the char ge of a particle can be written in terms of its isospin, a concept that has its origin in the c harge-independence of the nucleon-nucleon for ce. When app ropriate, we shall discuss the symm etry properties of p articles in isospin-strangeness space. Pre viously, we discussed the properties of the Lie group SU(2). It is 133 a g roup characterized by its three generators, the Pa uli spin matrices. Two-state systems, such as t he electron with its quan tized spin-up and spin- down, and the isospin states of n ucleons and nuc lei, can be treated quantitatively using this group. The symmetries of n ucleon and meson families discovered by Gell-Mann and N eeman, implied an underlying structur e of nucleons and me sons. It could not be a structure simply associated with a two-state system bec ause the o bserved particles were endowed not only with positive, negative, and zero charge but a lso with strangeness. A three-state system was therefore considered necessary, at the very least; the most promising candidate was the group SU(3). We shall discuss the infinitesimal form of this gro up, and we shall find a suitable set of generators. 13.3.1 The a lgebra of SU(3) The group of special unitary transformations in a 3-dimensional complex space is defined as SU( 3) = {U33 : UU = I, detU = +1, uij .C}. The infinitesimal form of SU(3) is SU(3)inf = I +idaj j/2, j = 1 to 8. (There a re n2 -1 = 8 generators). The quan tities daj are rea l and infinitesimal, and the 3 3 matrices j are the linearly independent gen erators of the group . The rep eated index, j, means that a sum over j is taken. The defining properties of the group restrict the form of the 134 generators. For example, the unitary condition is UU = (I +idaj j/2)(I idaj j/2) = I idaj j/2 + idaj j/2 to 1s t-order, = I if j = j. The generators must be hermitian. The dete rminantal condition is det = +1; and the refore Tr j = 0. The generators must be traceless. The finite form of U is obta ined by ex ponentiation: U = exp{iaj j/2}. We can find a suitable set of 8 g enerators by ex tending the met hod used in our discussion of isospin, thus: Let three fundamental states of t he system be chosen in the simplest way, namely: 1 0 0 u = 0 , v = 1 , an d w = 0 . 0 0 1 If we wish to transform v . u, we can do so by de fining the operator A+: 0 1 0 0 1 A+ v = u, 0 0 0 1 = 0 . 0 0 0 0 0 We can introd uce other operators that transform the states in pairs, thus 135 0 0 0 A =1 0 0 , 0 0 0 0 0 0 0 0 0 B+ = 0 0 1 , B = 0 0 0 , 0 0 0 0 1 0 0 0 0 0 0 1 C+ = 0 0 0 , C = 0 0 0 . 1 0 0 0 0 0 These matrices are singular and n on-hermitian. In the discussion of isospin and the group SU(2), the non -singular, traceless, hermitian matrices 1, and 2 are formed from the ra ising and lowering operators ma trices by introducing the complex linear co mbinations = + + = 1 an d = i( 1 2) = 1 2 The generators of SU(3) are formed fro m the operators A, B, C by constructing complex linear combinations. For example: the isospin operator 1 = 1 = + + 0 0 = A+ + A = 1, a generator of SU(3). 2. , a generator of SU(2) becomes 1 0 0 0 Continuing in this way, we obtain A+ = 1/2 + i 2/2, 136 where 0 2 2 = 0 , 0 0 0 and C+ + C = 4, C+ C = i 5, B+ + B = 6 a nd B + B = i 7 . The remaining generators, 3 an d 8 are traceless, diagonal, 33 matrices: 0 1 0 0 3 0 , 8 = 0 1 0 . = 3 0 0 0 0 0 -2 The set of matrices { 1, ..... 8} are ca lled the G ell-Mann m atrices, introduced in 1961 . They ar e normalized so that Tr ( j k) = 2djk. The normalized for m of 8 is therefore 1 0 0 8 = (1/v3) 0 1 0 . 0 0 2 If we pu t Fi = i/2. we find A = F1 iF2 , B = F6 iF7, and 137 C= F4 + iF5 . Let A3 = F3, B3 = F3/2 + (v3/4)F8 , and C3 = (1/2)F3 -(v3/4)F8., so th at A3 + B3 + C3 = 0. The last cond ition mean s that only eight of the nine operators are independent. The generators of the g roup are readily shown to obey the Lie commutation relations [Fi, Fj] = ifijkFk , i,j,k = 1 to 8. where the quantities fijk are the non-zero structure constants of the group; they are found to obey fijk = fjik, an d the Jacobi identity. The commutation relations [Fi, Fj] can be written in terms of the operators A, ...Some typ ical results ar e [ A+, A-] = 2A3, [ A+, A3] = -A+, [ A-, A3] = +A-, [A3, B3] = 0, [A3, C3] = 0, [B3, C3] = 0 [B+, B-] = 2B3, [ B+, B3] = -B-, [B-, B3] = +B-, etc. The two diagonal operators c ommut e: [F3, F8] = 0 . Now, F1, F2, and F3 co ntain the 22 isospin ope rators (Pauli matrices), each with zer os in the third row and column; they obe y the commutation relations of isospin. We therefore make th e identifications F1 = I1, F2 = I2, and F3 = I3 138 where the Ijs are the components o f the isospin. Par ticles that exp erience th e str ong n uclear interaction are ca lled hadrons; they are separated into two sets: the baryons, with half-integer spins, and the mesons with zero or integer spins. Pa rticles that do not experience th e str ong interaction are called leptons. In or der to qua ntify the difference bet ween baryo ns and leptons, it h as been found n ecessary to introduce the baryon nu mber B = +1 to denote a baryon, B = 1 to denote a n ant i-baryon a nd B = 0 f or all other pa rticles. Lepto ns ar e characterized by the lepton number L = +1, anti-leptons ar e assigned L = 1, and all other particles are a ssigned L = 0. It is a p resent-day fact, based up on nu merou s observations, that the total baryon an d lepton number in any interaction is conserved. For exa mple, in t he decay of the free neutron we find n0 =p+ +e + .e0 B = +1 = +1 + 0 + 0 L = 0 = 0 + 1 +(1). The fundamental symmetr ies in Nat ure responsible for these conservation laws are not known at this time. These conservation laws may, in all likelihood, be broken. In discussing the patterns o f baryon families in char ge-strangeness space, we wish to incorporat e the fact that we a re de aling with baryons that interact via the strong nuclear force in which isospin an d strangeness are conserved. We therefore choose to describe their patterns in isospin-139 hypercharge s pace, where the hype rchar ge Y is defined to i nclude bot h the strangeness and the bar yon a ttribute o f the particle in an additive way: Y = B + S. The diagonal operator F8 is therefore assumed to be directly associated with the hype rchar ge op erator, F8 = (v3/2)Y. Because I3 an d Y co mmute, states can be chosen th at are simultaneous eigenstates of the o perators F3 an d F8. Since no other SU(3) operators commute with I3 an d Y, no other ad ditive quantum numbers are associated with the SU(3) symmetr y. The op erators F4,...F8 are considered to be new con stants-of-the-motion of t he strong interaction ham iltonian. 13.4 Irreducible r epresentations of SU(3) In an earlier discussion of the irredu cible representations of SU(2), we found that the commutation relations of the g enerators of the group were satisfied not only by the fundamental 22 matrices but also by matrices of h igher dimension [(2J + 1) . (2J + 1)], where J can have the values 1/2, 1, 3/2, 2, ....The J-values cor respond to the spin of the par ticle whose state is given by a sp inor (a column vector with special tra nsformation properties). In t he 22 r epresentation, both covariant an d contravariant spinors are allowed: i) covariant spinors (with lower indices) are written as 2-component 140 columns that transform under U . SU(2) as i = Uij j , where a1 = , a2 and ii) contravariant spinors (with uppe r indices) are written as 2-compon ent rows that transform as: j = i Uij , where = (b1, b2). The co-and contra -variant spinors are transformed with the aid of t he anti- symmetric ten sors ij an d ij. For example, j = i ij tra nsforms as a covariant sp inor with the form b2 i = . -b1 The higher-dimensional repre sentations are built up f rom the fundamental form by taking ten sor p roduc ts of the fundamental spinors i , j , or i and by symmetrizing and anti-symmetrizing the re sult. We state, without pro of, the th eorem that is used in this method: when a tensor product o f spi nors has been broken down into its symmetric and anti-symmetric parts, it has been decomposed into irreducible 141 representatio ns of the SU(n). (See Wigner s standard work for the original discussion of the method, and de Swart in Rev. Mod. Ph ys. 35, (1963) for a detailed discussion of tensor analysis in the stud y of the irreps of SU(n)) As an example, we wr ite the te nsor produ ct of two covariant spinors i an d j in the following way i. j = i j =( i j + j i)/2 + ( i j -j i)/2 There are four elements associated with the product (i,j can have values 1 and 2). The symm etric part of t he product has three independent elements, and transforms as an ob ject that has spin J=1. (There are 2J + 1 members of the symmetric set). The anti-symmetric part has one element, and therefore transforms as an object with spin J = 0. This result is familiar in the theo ry of angu lar momentum in Quantum M echanics. The explicit forms of the four elements are: J3 =+1: 1 1 J = 1 J3 =0 :(1/v2)(12 + 2 1) J 3 =1 : 2 1 and J = 0 J3 = 0 : (1/v2)(12 21) . Higher-dimensional representations are built up from the tensor products 142 of covariant and contra variant 3-spinors, an d respectively. The pro ducts are then written in terms of their symm etric and anti-symmetric par ts in order to obtain the irreducible representations. For example, the pro duct i j, i,j = 1,2,3, can be written i j = ( i j -(1/3)dij k k) + (1/3)dij k k , in which the trace has been separated out. The trace is a zero -rank tensor with a single co mponent. The other tensor is a traceless, symmetr ic tensor with eight independent components. The decomposition is written symbolically as: 3 . 3 = 8 . 1. We can form the te nsor produ ct of two covariant 3-spinors, i j as follows: = (1/2)( i j + j i) + (1/2)( i j j i), i,j = 1,2,3. ij Sym bolically, we have 3 . 3 = 6 . 3 , in which the symmetric tensor has six components and the a nti-symmetric tensor h as three c omponents. Other tensor produ cts that w ill be of interest are 3 . 3 . 3 = 10 . 8 . 8 . 1 , and 8 . 8 = 27 . 10 . 10 . 8 . 8 . 1 . The appearance of the octet 8 in the 3 . 3 decomposition (recall 143 the observed octet of spin-1/2 baryons), and the decuplet 10 in the triple pro duct 3 . 3 . 3 de composition (recall the o bserved decuplet of spin-3/2 bar yons), was of p rime importance in t he development of the group theory of elementary pa rticles. 13.4.1 Weight diagrams Two of t he Gell-Mann ma trices, 3 an d 8, are diagonal. We can write the eigenvalue equations: 3u = auu, 3v = avv, and 3w = aww, an d 8u = uu, 8v = vv, and 8w = ww , wh ere ai an d i are the eigenvalues. Let a and b be normalization factors associated with the operators 3 and 8, r epectively, so that a 0 0 b 0 0 N = 0 a 0 , an d 8 N= 0 b 0 . 3 0 0 0 0 0 2b If u = [1, 0 , 0], v = [0, 1 , 0], and w = [0, 0 , 1] (columns), we find 3Nu = au , 8Nu =bu, 3Nv = av , 8Nv =bv , and 3Nw =0w , 8 Nw = 2bw. The weight v ectors are formed from the pairs of eigenvalues: [au, u] = [a, b], 144 [av, v] = [-a, b], and [aw, w] = [0, -2b]. A w eight diagram is obt ained by plotting these vectors in the a space, thus: 2b b 2a a a 2a -b 2 b a This weight diagram for the fundamental 3 representation of SU(3) was well-known to Mathematicians at the time of the first use of SU(3) symmetry in P article Physics. It was to play a key role in the development of the q uark model. 13.5 The 3-quark model of m atter Although the octet and decuplet patterns of hadrons of a given spin and parity emerge as irreducible repre sentations of t he group SU(3), major problems rem ained that resu lted in a great deal of scepticism concerning the validity of t he SU (3) model of fundamental particles. The most pressing problem was: why are there no known particles associated 145 with the fundamental triplets 3, 3 of SU(3) that exh ibit the symmetry of the weight diagram discussed in t he last section? In 1964 , Gell-Mann, and independently, Zweig, proposed th at three fundamental entities do ex ist that correspond to the base states of SU(3), and that they form boun d states of the hadrons. That such entities have not been observed in the free state is simply related to their enorm ous binding energy. The three entities were called quarks by Gell-Mann, and aces by Zweig. The Gell- Mann term has surv ived. The anti-quarks ar e associated with the conjugate 3 representation. The three quarks, denoted by u, d, and s (u and d fo r the up-and d own-isospin states, and s for strangeness) have highly unusual properties; they are Lab el B Y I I3 Q=I3 +Y/2 S = Y - B u 1/3 1/3 1/2 +1/2 +2 /3 0 d 1/3 1/3 1/2 1/2 1 /3 0 s 1/3 2/3 0 0 1/3 1 s 1 /3 2/3 0 0 + 1/3 + 1 d 1 /3 1/3 1 /2 +1 /2 + 1/3 0 u 1 /3 1/3 1/2 1/2 2/3 0 The quarks occupy the following positions in I3 - Y space Y Y 146 d u I3 I3 s u d s These diagrams have the same relative forms as t he 3 an d 3 we ight diagrams of SU(3). The baryons are made up of q uark triplets, and t he me sons are made up of the simplest possible structures, namely quarkanti-quark pairs. The covariant and cont ravar iant 3-spinors introduced in t he previous section are now given physical significance: = [u, d , s], a covariant co lumn 3-spinor, and = (u, d, s), a contrav ariant row 3-spinor. where u = [1, 0, 0 ], d = [0, 1, 0 ], an d s = [0, 0, 1] represent the unitary symmetry part of t he to tal wavefunctions of the three quarks. The form al operators A, B, and C, introd uced in section 13.3.1, are now viewed as operators that transform one fla vor (type)of quark into another flavor (th ey ar e shift op erators): A = I(I3) . I3 1 , B = U(U3) . U3 1, called the U -spin operator, and C = V(V3) . V3 1, called the V-spin operator. 147 Exp licitly, we have I+(1/2) . 1 /2 : d . u I(+1/2) . 1 /2 : u . d U+(1/2) . 1/2 : s . d U(+1/2) . 1 /2 : d . s V+(1/2) . 1/2 : u . s and V(+1/2) . -1/2 : s . u. The quarks can be characterized by the three quantum numbers I3, U3, V3. Their positions in the I3-U3-V3 -space again show the unde rlying symmetry : d (-1/2, 1/2, 0) -I3 U3 -V3 +1/2 u(1/2, 0, -1/2) -1/2 + 1/2 I3 +1/ 2 s( 0, -1/2, 1/2) V3 -Y -U3 The members of the octet of mesons with JP = 0 are formed from qq-pairs 148 that belong to the fundamental 3, 3 representation of the quark s. The p0 and .0 mesons are linear combinations of the q q(bar) states, thus K0 d s Y K+ us s d u p d u p 0 p+ ud -1 .0 +1 u d s K su K0 sd I3 The none t for med from the tensor produ ct 3 . 3 is split into an octe t that is even under the label exchange of two particles, and a s inglet that is odd unde r label exchange : 3 . 3 = 8 . 1 where the 1 is .0 = (1/v 3)(uu + dd + ss ), an d the two members of the octet at the center are: p0 = (1/v2)(uu dd) and .0 = (1/v6)(uu + dd - 2ss). The action of I on p+ is to transform it into a p 0. This operation has the following mea ning in terms o f I acting on the ten sor p roduc t, u . d: I(u . d) = (Iu) . d + u . (Id) (c.f. derivative rule) 149 .. . I ( p+) = d . d+ u u .p 0 Omitting the tensor product sign, normalizing the amplitudes, and ch oosing the phases in the generally accepted w ay, we have: p0 = (1/v2)(uu dd). The singlet .0 is said to be orthogonal to p0 an d .0 at the origin. If the symmetry of the octet were exact, th e eight members of t he octet wo uld have t he same ma ss. This is not quite the case; the symmetry is broken by the d ifference in effective ma ss between the u- and d-quark (essentially the same effective masses: ~ 300 MeV/c2) and the s-quark (effective ma ss ~ 500 MeV/c2). (It should be noted that the effective masses of the quarks, derived from the mass differences of hadron-pairs, is not the same as the cu rrent-quark ma sses that appear in the fundamental theory . The discrepancy b etween the effective masses and the fundamental masses is not fully understood at this time). The decomposition of 3 . 3 . 3 is 3 . 3 . 3 =(6 . 3) . 3 = 10 . 8 . 8 . 1 in which the states of the 10 are symmetric, the 1 is antisymmetric, and the 8, 8 states are of mixed symmetry. The decuplet that appears in t his decomposition is associated with the o bserved decuplet of spin-3/2 baryon s. In terms of t he th ree fundamental quarks u, d, and s, the mak e -up of the individual members of the decuplet is shown schematically in the 150 following diagram: d dd ~ du d ~ u ud uuu ~ dds ~ dus ~ uus ~ sds ~ sus sss The prec ise make-up of each state, labelled by ( Y, I, I3,) is given in the following tab le: (1, 3/2, +3/2) (1, 3/2, +1/2) (1, 3/2, 1/2) (1, 3/2, 3/2) (0, 1, +1) (0, 1, 0) (0, 1, 1) (1, 1/2, +1/2) (1, 1/2, 1/2) (-2, 0, 0) = = = = = = = = = = uuu(++ ) (1/v3)(udu + duu + uud) (1/v3)(ddu + udd + dud) ddd() (1/v3)(usu + suu + uus) (1/v6)(uds + dsu + sud + dus + sdu + usd) (1/v3)(dsd + sdd + dds ) (1/v3)(ssu + uss + s us) (1/v3)(ssd + dss + s ds) sss() The general theory of t he permuta tion group of n entities, and its rep resentations, is out side the scope of this introdu ction. The use of t he Young ta bleaux in obtaining the mixed symmetry s tates is treated in Hamermesh (1962). The char ges of the .++ of the q uarks, namely , ., and O pa rticles fix the fractional values quark flavor charge (in units of the electron charge) 151 u +2/3 d 1/3 s 1/3 The char ges of the anti-quarks ar e opp osite in sign to the se values. Extensive rev iews of the 3-quark model and its application to the physics of the low-energy pa rt of the hadron spectrum can be found in Gasiorowicz (1966) and Gibson and Pollard (1976). 13.6 The need for a new quantum number: hidden color Immediately after the introduction of the 3-quark model by Gell-Mann and Zweig, it was recognized that the model was not c onsistent with the exte nded Pauli principle when applied to bou nd states of three quarks. For example, the structu re of the spin-3/2 .+ st ate is such tha t, if each qua rk is assigned a spin sq = 1/2, the three spins must be aligned ... to give a net spin of 3 /2. (It is assumed that the relative orbital angu lar mom entum of t he qu arks in the .+ is zero (a symmetric s-state) a reasonable assumption to mak e, as it corresponds to minimum kinetic energy, and t herefore to a s tate of lowest total ener gy). The quark s are fermions, and therefore they must obey the generalized Pau li Principle; they cannot e xist in a completely aligned spin state when they are in an s- state that is symm etric unde r particle (quark) exchange. The unitary spin component of the total wavefunction must be anti-symmetric. Greenberg (1964) proposed th at a new d egree of freedom mus t be assigned t o the quarks if the Pauli Principle is not to be violated. The new proper ty was 152 later ca lled color, a property with profound c onsequences. A quark with a certain flavor possesses color (red, blue, gre en, say) that cor responds to the triplet representation of another form of SU(3) namely SU(3)C, where the subscript C differentiates the group from that introduced by Gell-Mann and Zweig the flavor group SU(3)F. The anti- quarks (that possess anti-color) have a triplet repre sentation in SU(3)C that is the c onjugate representation (the 3). Although the SU(3)F symmetr y is known no t to be exact, we have evidence tha t the SU( 3)C symmetry is an exact symmetry of Nature. Baryons and mesons are found to be colorless; the color singlet of a baryo n occurs in the decomposition S U(3)C = 3 . 3 . 3 = 10 + 8 + 8 + 1 . The meson singlets consist of linear c ombinations of the form 1 = (RR + BB + GG)/v3 . Although the hadro ns ar e colorless, certain observable quantities are directly related to the number of colors in the model. For exa mple, the pur ely electromagnetic decay of t he neutral pion, p0, into two ph otons p0 = . + ., has a lifetime tha t is found to be closely propo rtionl to the square of the number of colors. (Adler (1970) gives G = h/t = 1(eV) (no.of colors)2 . The measurements o f the lifetime give a value of G ~8 eV, consistent with Ncol s = 3. Since these early mea surements, refined experiments have demonstrated that there are three, and only three, colors associated with 153 the quarks. In studies of electron-positron interactions in the G eV-region, the rat io of cros s sections: R= s(e+e . ha drons)/s(e+e .+ ) is found to depend linearly on the num ber of colors. Good agre ement between the theoretical mode l and the measured value of R, over a wide ran ge of ener gy, is obt ained for three co lors. The color attribute of the q uarks has been responsible for the development o f a t heory of t he strongly interacting particles, called quantum chrom odynamics. It is a field theo ry in which the quarks ar e generators of a new typ e of field the color field. The mediators of the field are called gluons; they p ossess color, the at tribute of the source of the field. Consequently, they can interact with each other through the color field. This is a field quite unlike the electrodynamic field of classical electrom agnetism, in which the field quanta do not ca rry the at tribute of the sour ce of the field, namely electric charge. The photons, therefore, do not interact with each other . The gluons tr ansform a quark of a particular color into a quark of a different color. For example, in the interaction bet ween a red quark and a blue quark, t he colors are e xchanged. This requires that the e xchanged gluon carry color and anti-color, as shown: qb qr 154 the color lines are continuo us. q r gluon, g rb(bar)-carries red and anti-blue: qb Three different co lors permit nine different way s of coupling quarks and gluons. Three of t hese are red-red, blue-blue, and green-green that do not change th e colors. A linear combination ~(R.R + B.B + G.G) is symmetric in the c olor labels, and this combination is the singlet state of the group SU(3)C. Eight gluons, each with two co lor indices, are the refore req uired in t he 3-color theo ry of quarks. 13.7 More ma ssive quarks In 1974, the results of two independent ex periments, one a study of the reac tion p + Be . e+ + e .. (Ting et al.) and t he ot her a stud y of e+ + e . ha drons ..(Richter et al) showed th e pre sence of a shar p resonance at a center-of-mass energy of 3.1 GeV. The lifetime of the resonant state was found to be ~1020 seconds more than 10 3 seconds longer than expected for a state formed in the strong interaction. The resonant state is called the J/.. It was quickly realized t hat the state cor responds t o the ground st ate o f a n ew qu arkanti-quark system, a bound st ate cc, where c is a four th, m assive, quark e ndowed with one unit of a new quan tum number c, called cha rm. The quantum numbers assigned to the c-quark are JP=1/2+, c = 1, Q/e= +2/3, and B = 1/3. 155 Sou nd theoretical argum ents for a four th quark, carry ing a new quantum number, had been put forward s everal years be fore the experimental observation of the J/. st ate. Since 197 4, a complex set of states of the cha rmonium s ystem has been observed, and t heir decay pro perties studied. De tailed com parisons have been m ade with sophisticated theo retical models of the system. The inclusion of a char med q uark in the set of q uarks mean s that the gro up SU(4)F must be used in place of the original Gell-Mann-Zweig group SU(3)F. Although the SU (4)F symmetr y is badly broken because the effective mass of the c harmed qua rk is ~ 1.8 GeV/c2, some u seful applications have been made using the model. The fundamental rep resentations ar e [ u, d, s, c], a covariant column spinor, an d ( u, d, s, c), a contr avariant row spinor. The irreps ar e constructed in a way th at is analogous to that u sed in SU(3)F, namely, by finding the symmetric and anti-symmetric decompositions of the v arious tensor p roducts. The most u seful are: 4 . 4 = 15 . 1, 4 . 4 = 10 . 6, 4 . 4 . 4 = 20sym . 20mix . 20mix . 4ant i, an d 15 . 15 = 1 . 15sym . 15ant i . 20sym . 45 . 45 . 84. 156 The 15 includesthe n on-charmed (JP = 0 ) mesons and the following charmed mesons: D0 = cu, D0 = cu, mass = 1863 MeV/c2 , D+ = cd, D = cd, mass = 1868 MeV/c2, F + = cs, F = cs, mass = 2.04 MeV/c2. In order to discuss the baryons, it is necessary to i nclude the quark spin, and therefore the group must be extended to SU(8)F. Relatively few bar yons have been studied in deta il in this extended framework. In 1977, well-defined resonant st ates were observed at ene rgies of 9.4, 10.01, and 10 .4 GeV, and were interpre ted a s bound st ates of another quark, t he b ottom qua rk, b , and its anti-partn er, t he b. Mesons can be formed that include the b-quark, thus 00 Bu = bu, Bd = bd, Bs = bs, and Bc = bc . The stud y of the w eak decay modes of t hese states is curre ntly fashionable. In 1994, definitive evidence was obtained for the existence of a sixth quark, c alled the top quark, t. It is a massive entity with a mass almost 200 times the mass of t he proton! We have seen that the q uarks interact strongly via gluon e xchange. They also tak e par t in the w eak interaction. In an earlier discussion of isospin, the group generators were introduc ed by considering the -decay of the free n eutro n: n0 . p+ +e + .0 . We now know that, at the microscopic level, this process involves the 157 tra nsformation of a d-quark into a u-quark, and the p roduc tion of the carrier of the weak force, the ma ssive W pa rticle. The W bo son (spin 1) decays instantly into a n electron anti-neutrino pair, as shown: .0 W1 e neutron, n0 d( 1/3) . u(+2/3) d u proton, p+ u u d d The carriers of the Weak Force, W , Z0, were first identified in p-p collisions at high cent er-of-mass ener gy. The processes involve quarkanti-quark interactions, and the dete ction of t he decay electrons and positrons. e+ e Z0 u(+2/3) u (2/3) .0 W+ e+ u(+2/3) d (+1/3) .0 W e d(-1/3) u (-2/3) The char ge is conserved at each vertex. The carriers have very large measured masses: 158 mass W ~ 8 1 GeV/c2, and ma ss Z0 ~ 93 Ge V/c2. (Recall that the range of a force . 1/(mass of carrier); the W and Z masses cor respond to a very sh ort range,~10-18 m, for the Weak F orce). Any qua ntitative discussion of curren t work using Group Theory to tackle Grand Unified Theories, requires a k nowledge o f Quantum Field Theory that is not expected of readers of t his introd uctor y boo k. 14 159 LIE GROU PS AND THE CONSERVATION LAWS OF THE PHY SICAL UNIVERSE 14.1 Po isson and Dirac Brackets The Poisson Bracket of two differentiable functions A(p1, p2, ...pn, q1, q2, ...qn) and B(p1, p2, ...pn, q1, q2, ...qn) of two sets of variables (p1, p2, ...pn) and (q1, q2, ...qn) is defined as {A, B} =.1n (.A/.qi)(. B/.pi) (.A/.pi)(. B/.qi) . If A = (pi, qi), a dyn amical variable, and B = H(pi, qi), the h amiltonian of a dyna mical system, where pi is the (canonical) momentum and qi is a (g eneralized) coordinate, then { , H} = .1n (. /.qi)(. H/.pi) (. /.pi)(.H/.qi) . (n is thenum ber of degrees of freedom of the system). Hamiltons eq uations ar e .H/.pi = dqi/dt and .H/.qi = dpi/dt , and therefore {, H} = .1n (. /.qi)(dqi/dt) + (. /.qi)(dpi/dt) . The total differential of (pi, qi) is d = .1n (. /.qi)dqi + (. /.pi)dpi. 160 and its time derivative is ( d /dt) = .1n (. /.qi)(dqi/dt) + (. /.pi)(dpi/dt) = { , H} = . If the Poisson Bra cket is zero, the ph ysical quantity is a constant of the m otion . In Quantum Mechanics, the re lation is replaced by (d (d /dt) = { , H} /dt) = -(i/h))[ , H], Heisenbergs equation of motion. It is the custom to refer to the commutator [, H] as the Dirac Bracket. If the D irac Bracket is zero, the quan tum m echanical quantity is a c onstant of the motion.. (Dirac p roved that the classical Poisson Br acket { , H} can be identified with the Heisenberg commuta tor (i/h)[ , H] b y mak ing a suitable choice of the order of the qs a nd ps in the Poisson Bra cket). 14.2 Infinitesimal unitary transforma tions in Q uantum Mec hanic s The Lie form of an infinitesimal unitary transformation is U = I +idaX/h , where da ia real infinitesimal parameter, and X is an hermitian operator. (It is straightforward to sh ow th at this form of U is, indeed, unitary). Let a dynamical operator change under an infinitesimal unitary 161 tra nsformation: . = UU1 = (I +idaX/h) (I idaX/h) = ida X/h +idaX /h to 1st-order = +i(daX daX)/h = +i(F F)/h. where F = daX. The infinitesimal change in is therefore d = = i[F, ]/h If we identify F with Hdt (the c lassical form f or a purely temporal change in the system) then d = i[-Hdt, ]/h, or d = i[H, ]dt/h , so that d /dt = i[H, ]/h. For a tempora l change in the system, d /dt = d /dt. The fundamental Heisenberg equati on of motion 162 d /dt = i[ , ]/h is therefore deduced from th e uni tary infin itesimal transformation of the operator . This approach was taken by Schwinger in his form ulation of Quan tum Mechanics. |F| = Hdt is directly related to the generator, X, of a Quantum Mechanical infinitesimal transformation, and the refore we can associate with every sy mmetr y transformation of the system an hermitian operator F that is a constant of the motion - its eigenvalues do not change with time. This is an example of Noethers Theorem: A c onservation law is associated with every symmetry of the equation s of motio n. I f the equations of m otion are unchanged by the tra nsformations of a Group then a property of the system w ill remain constant as t he system evolves with time. As a well-known exam ple, if the equations of motion of an ob ject are invariant u nder translations is space, the linear momentum of the system is conserved. 15 163 BIBLIOGRAPHY The following book s are typical of those that a re suitable for Undergra duates: Arm strong, M. A., Groups and Sy mmetry, S pringer-Verlag, New York, 1988. Burns, Gerald, Int roduction to Group Theory, Academic Press, New York, 1977. Fritzsch, Harald, Quarks: the S tuff of Matter, Basic Books, New York, 1983. Jones, H. F., Groups, Representations and Physics, Adam H ilger, Bristol, 1990. The following book s are of a specialized nature; they are typical of what lies beyond the present introduction. Carter, Roger; Segal, Graeme; and Macdonald, Ian, Lectures on Lie Groups and Lie Algebras, Cambridge University Press, Cambridge, 1995. Commins, E. D., and Bucksbaum, P. H., Weak Interactions of Leptons and Quarks, Cambridge U niversity Press, Cambridge, 1983 Dickson, L. H., Lin ear Groups, Dover, New York, 1960 . Eisenhart, L. P., Continuo us Groups of Tr ansformations, Dover, New York, 1961. Elliott, J. P ., and Dawber, P. G., Symmetry in P hysics, Vol. 1, Oxford University Press, New York, 1979. Gell-Mann, Murray, and Neeman, Yuval, The Eightfold Way, 164 Benjamin, New York, 1964. Gibson, W. M., and Pollard, B. R., Symmetry Principles in Elementary Particle Phys ics, Cambridge U niversity Press, Cambridge, 1976. Hamermesh, Morton, Group Theory and its Applications to P hysical Problems, Dover, New York, 1989 . Lichtenberg, D. B., Unitary Symmetry and Elementary P articles, Academic Pres s, New York, 1978. Lipkin, Harry J., Lie Groups fo r Ped estrians, North-Holland, Amsterdam, 1966. Lomont, J. S., Applications of Finite Groups, Dover, New York, 1993. Racah, G., Group Theory and S pectroscopy, Reprinted in CER N(61-68), 1961. Wigner, E. P. , Group Theory and its Applications to the Qu antum Mechanics of Atomic Spectra, Academic Press, New York, 1959.