Praman.a, Vol. 23, No. 4, October 1984, pp. 445-457. ~() Printed in India.

Symmetry groups of mathematical physics

P RUDRA Department of Physics, University of Kalyani, Kalyani 741235, India

Abstract. Recent work on Lie's method ofextended groups to obtain symmetry groups and invariants of differential equations of mathematical physics is surveyed. As an essentially new contribution one-parameter Lie groups admitted by three-dimensional harmonic oscillator, three-dimensional wave equation, Klein-Gordon equation, two-component Weyl's equation for neutrino and four-'component Dirac equation for Fermions are obtained. Keywords. Mathematical physics; symmetry groups; differential equations; Lie groups; harmonic oscillator; wave equation; Klein-Gordon equation; Weyl equation; Dirac equation.

PACS No. 02.20+b

I. Introduction

Application of the Lie groups theory to the characterization of atomic spectra by Wigner (1931) followed Weyl's work (1925, 1926) on the representation theory of continuous groups. The symmetry group used in this characterization was the three- dimensional orthogonal group leaving the tbrm Y? ~ x 2 invariant. The/-degeneracy for the hydrogen atom in a potential V(r) = l~/r could not be explained on this basis; this gave rise to the concept of accidental degeneracy (Mclntosh 1971; Elliott and Dawber 1979), until Fock (1935) showed that the full symmetry group of the non-relativistic hydrogen problem was O (4), the orthogonal group in four-dimension. Full degeneracy of the problem was thus accounted for. Bargmann (1936) showed that besides the angular momentum operator L an extra operator, the Runge-Lenz vector A = L x p - ltr/r, was required for the complete description of the orbit. The experience with non-relativistic hydrogen problem has led physicists to search for complete symmetry groups of other classical and quantum mechanical systems. Maxwell's electromagnetic equations form one such system of interest of the former class. Bateman (1910) showed that Maxwell's equations are invariant under the 15- parameter conformal group consisting of the 10-parameter inhomogeneous Lorentz group, scale transformation, inversion and 4-parameter acceleration transformations. Use ofconformal symmetry in quantum field theory and other branches of physics has been investigated in some detail (Wess 1960: Fulton et al 1962). The other physical system that has drawn the attention of physicists is the harmonic oscillator, both isotropic and anisotropic. Jauch and Hill (1940) investigated in detail the isolropic harmonic oscillator and showed that its symmetry group is the unitary unimodular group. Cisneros and Mclntosh (1970) made a similar study of the anisotropic harmonic oscillator. It will be appropriate to mention here that a connected problem is to obtain the Casimir invariants (Racah 1951) of the symmetry group. By Noether's theorem (1918) the symmetry groups are characterized by their Casimir invariants, and 445 446 P Rudra obtaining the Casimir invariants of the problem was equivalent to obtaining the symmetry group. Methods of solving both the problems, finding the symmetry group or obtaining the Casimir invariants, remained an art which only the ingeneous could utilize for different physical systems. It somehow escaped most of the physicists that Lie and Scheffers (1891) and Dickson (1924) had long ago given very detailed algorithm of extended group for finding out generators of Lie groups that keep the form of a differential equation invariant. Only recently this method (Olver 1976; Sattinger 1979; Hamermesh 1983) has been used extensively to find symmetry groups of different equations of physics. Wulfman and Wybourne 0976) considered the classical one-dimensional harmonic oscillator and found that its symmetry group is the 8-parameter non- compact Caftan group A2. Lutzky (1978) showed that a 5-parameter subgroup of this 8-parameter group leaves the action integral invariant, thus having 5 conserved quantities of which only 2 are functionally independent. In a series of papers Leach and coworkers (Leach 198 la; Prince 1983a; Prince and Eliezer 1980, 1981; Prince and Leach 1980) solved the classical problems of N-dimensional time-dependent harmonic oscillator and Kepler motion. Boyer et a! (1976), and Harnard and Winternitz 0980) studied both linear and non-linear one-dimensional Schr6dinger equation and found the corresponding symmetry groups. Vinet 0980) considered the linear hyperbolic equation in two variables. Kalinis and Miller 0974) and Boyer et a! (1975) applied the method of Lie to time-dependent Schr6dinger equation of free particles. As mentioned before, a connected problem is obtaining the Casimir invariants. The usual method of Noetber keeping the action integral invariant has the disadvantage that it does not give the Runge-Lenz vector for Kepler motion. In this method Runge- Lenz vector is obtained if one goes beyond point transformations and involve the velocity as an independent variable (L~vy-Leblond 1971). In the Lie's method of differential equation all the invariants appear as a result of point transformations. The method has b~n successfully applied to the Kepler problem, time-dependent harmonic oscillator and quadratic hamiltonians (Leach 1978, 1980; Guenther and Leach 1977; Prince 1983b). Makarov et al (1967) obtained linear and quadratic invariants of motion for non-relativistic Schr6dinger equation. Patera et al (1976a, b) used Lie's method to obtain the invariants of continuous subgroups of Poincar~ group, for all real algebras of dimension up to 5 and for all real nilpotent algebras of dimension 6. These authors made a distinction between Casimir operators (polynomials in the generators), rational invariants (rational functions of generators) and general invariants (irrational and transcendental functions of the generators). Gonz~lez-Gascon (1977), and Gonz~lez- Gascon and Gon~lez-L6pez 0983) have applied the theory of partial differential equations to ordinary differential systems of classical mechanics to obtain first integrals and the upper bounds for the number of independent point-like symmetry vectors of differential equations. In §2 Lie's theory of extended group is described and the symmetry groups for some of the important differential equations of mathematical physics are obtained. Section 3 shows how the invariants are obtained from Lie's theory and in ~ the method is applied to Korteweg-deVries nonlinear equation of soliton physics. Finally, another aspect of Lie's theory is considered (Eisenhart 1961). This is the concept of one-parameter Lie groups admitted by a complete system of differential equations. If ~ is a solution of the system then all the functions X~ for a set of linear operators X~(~ = 1,..., r) are also solutions when the X~'s satisfy a particular Symmetry groups of mathematical physics 447

condition. The one-parameter Lie groups generated by the X.'s are said to be admitted by the given system ofdifferential equations. A key theorem states that either the system admits a r-parameter Lie group or solutions of the system are obtained by direct process. In §5 the generators of the one-parameter Lie groups admitted by three- dimensional isotropic harmonic oscillator, three-dimensional wave equation, Klein- Gordon equation, Weyl's equation for neutrino and Dirac equation for spin-l/2 particles are obtained. It should be pointed out that in the last two cases the differential equations are not scalar equations, but are 2- and 4-component equations respectively.

2. Symmetry groups of differential equations

In this section we describe Lie's theory of extended group and obtain the symmetry group that keep the form of the classical Lagrangian equation invariant. We consider a point transformation t' = t +t$~t~(q,t), q~ = qi +tSat~li(q,t), i= 1..... N, (1) in the space t, q = (ql). The generator of the transformation is g = ~(q, t)t~/t3t + ~th(q, t)~/t~ql. (2) i The generator for the nth extension group is given by

X ~"> =- ~t~/~t + ~ [q,~/t~q, + t/[')~/eql 1' +... + t/l")~/~ql"'] where t/~k)(q,q(l) ..... q(k),t)= ~d t/[,-l) --~ti.(k) -~,d (k = 1...... n)

d dt = t~/t3t + ~ [ q~l)a/gqi +... + q~g)t~/t~q~k- 1)] i and q~k) = (d/dt)kqo (3) The finite transformation of the extended group can be expressed as t'= (exp~tX)t, q'= (expatX)q, qtk),= (expctX(k))qtk). (4) For an nth order differential equation ql,,) + g,(q, q(l) .... , q(.- 1), t) = 0, (5) we have to find the unknown functions ~ and t/is from the condition X(.)[q~.) +g~(q, q(~)..... q(.- l), t)] = 0, i = 1..... N. (6) By equating the coefficients of powers ofq~k), k = 1.... , (n - 1), to zero, after replacing q~") by - g~ in the left side of (6), we get partial differential equations for ~ and t/,.'s, whose solutions give the unknown functions. For the second order Newtonian equation of the form (Leach 1981b) q,+g,(q,t)=O, i= 1..... N (7) the generator for the extended group is 448 P Rudra

X TM = ¢O/Ot+ ~ [q, O/Oq, + (ih - ~q,)~/Oil, i + (i~, - ~'q,- 2~ q,) i~/c~4,]. (8) Separating out the terms of the second and the third power of the///s, we get ~z ~/c3qic3qj = O, (9) t~2th/t')qiC~qj -- ¢~jlt~2¢/~qit3t -- 3ilt'~z~/t~qjct = O. 00) Equations (9) and (10) give

= a(t) + ~bk(t)qk k

q, = ~ [bkqkq, + C,k (t)qk] + di(t). (1 I) k Thus the generators of the symmetry group are of the form

X = [a + Zbkqk] O/¢'~l + ~ [di + Z (bkqkq, -~ cikqk)] "/?q,, (12) k i k where a, bk, Cik, di are determined from (7) and (8). For 2-dimensional Kepler motion (Prince and Eliezer 1981) +pr/r 3 = 0, (t31 we get the 3 generators for the symmetry group

Xl~t~/?l, Xz~-x(~/t~y-yc"/?x, X3~-lg/f'l"+2x[~/'~.;.'~d'-2.Vg/'~')' (14) with the commulation relations [x,, x2] -- 0, Ix,, x3] = x,, Ix2, x3] = 0. (15) For l-dimensional harmonic oscillator (Wulfman and Wybourne 1976) .#+x = 0, (16) we get 8 generators for the symmetry group X 1 = (1 + x 2) sin t ?/i~x - x cos t ~/gt,

X 2 = (1 -- x 2 ) sin t (/(x + x cos t ~'/?t, X3 = (1 + x 2) cos t ('/?x + x sin t ?/?t, X4 = (1 -x2)cost ~/?x-x sint ?/?t. X5 =?/?t, Xo =x?/?x, X7 =xcos2tg.,i'x+sin2tP/?t, X8 = - x sin 2t ?/?x + cos 2t ?/~t, (17) forming the non-compact Cartan's algebra A2.

3. lnvariants of symmetry groups

In this section we first explain Lie's method of obtaining the invariant functions of tile generators for a Lie group. The problem of finding the invariants reduces 1o solving a Symmetry oroups of mathematical physics 449

certain set of linear first order partial differential equations. These equations may have polynomial solutions giving rise to Casimir invariants, rational solutions giving rise to rational invariants (i.e. ratio of two polynomials), or more general solutions including transcendental functions leading to general invariants (Patera et al 1976a, b). If X~, i = 1,..., r are generators of the symmetry group with structure constants Ci~

Ix,, xj] = xk (18) k then the invariants are those functions F(X) for which [Xi, F(X)]=0, i= 1.... ,r. (19) A differential representation of Xi's are

X, --, X, = ~. C~j Xk ~/~Xj. (20) j,k The commutation relation (19) is then replaced by the partial differential equation ;~F=0, i=1 .... ,r. (21) For a single equation of the form of (21) we are led to the total differential equation

~, Izj,. dxff~ xkC~j #j,~ -- independent of m (22) j jk with the subsidiary condition C0/z~.,k =/~./Zk., (23) J i.e. #~,. is an eigenvector of the matrix Cik belonging to the eigenvalue 2., (suppressing the dependence of/~jm and ).m on i). Then

dx'm/2,.x',. = independent of m, where x~, = ~xj#j.. (24) J If the eigenv¢ctors are complete, we get the solution F (x) = O(x'2a'/x'l ~2, x,aa,/x,l a,, ..... x,~,/x,l a.). (25) To obtain the first integrals of motion l(q, LI, t) of (5) we have to solve the partial differential equation Xll)l(q,q,t)=O, dl(q, il, t)/dt=O, i=1 ..... r. (26) Here Xll's are the first extensions of the generators Xi's. The first member of (26) has the associated Lagrangian system dt/¢ = dqJr h = dqi/r/~~). (27) If u/s are constants of integrals for the first pair of (27) and vi's for any other pair, then I is a function of u and v: l(u, v) and dvi/du J = ~(u, v). From these sets of equations the first integral is obtained by quadrature. In the two-dimensional Kepler motion the generators X~ and X 2 yield the integrals

I~ = ½r2+ lz/r = energy, 12 = x~-y~/= z-component of angular momentum. pro2 450 P Rudra

For the generator X 3 the associated Lagrangian system is dt/t = dx/Zax = dy/]y = d~/(-]~) = dj,/(- ~) (28) yielding ul(x,t) = x2/t 3, u2(y,t) = y2/t3, Vx(X,~,t) = t~/s, v2(Y,~, t) = t~ 3. (29) The corresponding integrals are the x and y components of Runge-Lenz vector x~ 2 - ygy~ - #x/r = constant, y~2 _ xJi~ - #y/r = constant. (30) It must be admitted that Lie's method for obtaining the invariants is not as straightforward as Noether's. Nevertheless, it has the advantage that it gives us all the invariants of the problem.

4. Symmetry groups of non-linear differential equations

Not all differential equations of mathematical physics are linear. There are a whole lot of nonlinear equations that have come to the attention of physicists (Eilenberger 1981). These are the partial differential equations having soliton-like solutions. A particularly important equation in this set is the Korteweg-deVries equation au/& + uau/t3x + c~Su/(dx) 3 = 0. (31) We now describe the method (Hamermesh 1983) for obtaining the largest local symmetry group of the set of partial differential equations in q dependent variables u ~, l = 1..... q and n independent variables xi, i = 1..... n. For this we first construct a space encompassing all the derivatives that appear in the partial differential equations. The number of different kth order partial derivatives is

/1 k ~ k and the partial deriyatives are denoted by

0, = 8Pl/dx~J,... ax j. with J = (j, ..... J.), [JI = ,~J,- i Here the j[s are non-negative integers. If X is a generator of the product space (x, u) then the kth extension X *kl of X is given by

X = ~'. ~l (x, u) d/dx, + ~" tp s (x, u)c3/aul. (32) i 1

X ~k) = X + ~ ~., tpf (x, utk})c~/c~u~ (33) Here ¢/= D~(~, - y.4¢i)+ X'u z ,i /- J,~'* , (34) i i

where [,l Ii ~ Ou~/Oxi, J,i ~ (Jl,. . . ,Ji-l,Ji + l,Ji+l .... ,J.) (35) Symmetry groups of mathematical physics 451

and D J = DIJID2J2...D j', Dj = ~/tgx~-b E (36) I tJI ~ k A system of partial differential equations A'(x, u tk)) = O, ot = 1..... p (37) has the symmetry group G if for every generator X ~ G

Xlk)A~(x, u (k)) = 0, ~t --- 1 ..... p. (38)

Detailed calculation shows that for the Korteweg-deVries equation (equation (31) ) G is a four-parameter Lie group with generators X t =d/dx, X 2=d/dt, X 3=td/Ox+d/du, X, = xa/Ox + 3td/at + 2uO/tgu (39) with the non-vanishing commutators Ix1, x,] = x,, x3] = xl, IX2, x,] = 3x2, IX 3, X,] = - 2X3. (40)

5. One-parameter Lie groups admitted by differential equations

In this section we come to another aspect of Lie's theory (Eisenhart 1961). If we know the solution ~ of a complete system of linear partial differential equations A'~k=0, ct=l ..... p, (41) then the functions X~k, where

X = ~.. ¢,~ (x) t3/dxm, (42) m will also be solutions of the complete set if [X, a ~] = ~2}(x)AP. (43) P If moreover Xi and X~ are two operators satisfying (43), then their commutator [Xi, Xi] also satisfies (43). These X's form the different one-parameter Lie groups admitted by the complete set of linear partial differential equations. The key theorem is that either the operators X, i = 1.... , r, satisfying (43) form an r-parameter Lie group or the equations are solvable by direct process. The importance of these one- parameter groups lies in the fact that they give the complete set of solutions, if one of the solutions is known. In second order partial differential equations

A" =- a~(x) + ~,aT (x)d/t~x, + ~, aTj(x)O2/dxidxj, = 1 ..... p, (44) i ij we get a set of partial differential equations for the ~'s

¢i ¢z ¢K aat m # 452 P Rudra

[¢,, + ~] .OaTlOx. - - a ,.O = Z 2~a£ rn m mn

nl

,n p ~t, fl = 1 ..... p; i,j,m,k = 1 ..... n. (45) We now apply this method to obtain the generators of the one-parameter Lie groups for the three-dimensional isotropie harmonic oscillator, three-dimensional wave equation, Klein-Gordon equation, two-component Weyl equation for neutrino and four-component Dirac equation for Fermions.

5.1 Three-dimensional isotropic harmonic oscillator There is a single equation ihA~b = 0, with A =- O/Ot - (ih/2m)V 2 + (imto~/2h) (x 2 + y2 + z2). (46)

Taking X = ~ 10/Ox + ~2 O/Oy + ~30/Oz + ~40/Ot, (47) equation (45) takes the form

~,X "at"~2 Y + ~3 Z : ,~ (X 2 "1" y2 "t- Z 2)/2;

(ih/2m)V2~i + O¢~/Ot = 0, i = 1, 2, 3; (ih/2m)V2~4 + O~4/Ot = - 2;

a~,/Ox = O~2/Oy = O~3/Oz = -2/2;

O¢,lOx = Ù~4/Oy = O~41Oz = 0;

O~21Ox + O~,lOy = O~310x + O~,lOz = O~31OY + O~210z = O. (48) Since the ~{s are analytic functions of the real variables x, = x, x2 = y, x3 = z, x4 = t, we use the necessary and sufficient condition O2~/OxmOx. = O2~jOx.Ox,., and get the 4 generators

X 1 = - i(yO/Oz - zO/Oy), X 2 = - i(zO/Ox - xO/Oz),

X 3 = -i(xO/Oy- yO/Ox), X4 = -iO/Ot, (49) with the non-vanishing commutators

[X,,X2]=iX3, [X2, Xa]=iX,, IX 3,X,]=iX 2. (50) These generators form the unitary group U(2) with X4 forming an exceptional subgroup.

5.2 Three-dimensional wave equation There is again a single equation A~ = 0, with A - V 2 -02/(0"r) 2, where • = ct. (51)

With X = ~,O/Ox+~20/ay+~30/Oz+~,O/&, (52) Symmetry 9roups ofmathematical physics 453 we get the following partial differential equations for the ~{s:

O¢l/Ox = 0~2/0y = ~9~3/~z = ~¢4/0~ = 2/2; V2¢i-O2{J(d,) 2 = 0, i = 1, 2, 3, 4;

O~4/dz - O~a/~gT = O~,,/Oy - O~2/O~ = O¢,,/Ox - 0~/0~ = O; O~/Oy + O~2/Ox = O~a/Oz + O~a/Ox = O~2/Oz + O~3/Oy = O. (53) If we again impose the analyticity condition on the ~{s, we get 11 generators Xt=-iO/Ox, X2=-iO/Oy, Xa=-iO/c3z, X4=-iO/c3z, X s = -i(yO/Oz - zO/Oy), X 6 = -i(zO/Ox -xO/Oz),

X 7 = - i(xO/Oy- yO/Ox), X s = xO/Oz + xO/Ox,

X 9 = yd/O~ -I- '~O/Oy, Xl0 = zO/O'r, -~ "cO/Oz, X1 ~ = xO/Ox + yO/Oy + zO/Oz + ~O/O~ (54) with the non-vanishing commutators

[X,,X6] = 'X3,[X~,X d = -iX2,[X,,X,] = X,,[X2,X~] = -iXa,

IX2, XT] = iX1, IX2, Xg] = X,, [Xa, Xs] = iX 2, IX3, X6] = -iXl, x 0] = x,, Ix,, = x,, [x,, x+] = Ix,, x,0] = [Xs,X6] = iXT, [Xs, X7] = --iX 6, [Xs,X9] = iSlo, [Xs,XIo ] = X 9,

[X6,X7] = iXs,[X6,X8] = -iXlo,[X6,X,o ] = iX8,[XT,Xs] .~. iX9, [Xv,X9] = -iXa, [Xa,X9]=iXT, [Xs,X,o] = -iX6, [Xg,X,o] = iX5, [Xk,X,,] = Xk, k = 1, 2, 3, 4. (55)

The first 4 generators are those for translations along the x, y, z, z axes; X 5, X 6 and X7 are those for rotations in the coordinate space of x, y and z; Xs, X9 and X~o are those for the 3 Lorentz boosts; finally X1 ~ is the generator for the scale transformation x '~ = sx ~. This I 1-parameter Lie group has no exceptional subgroup. The generators X 1..... X ~o are those for the inhomogeneous Lorentz group I t4 and form a subgroup. The group obtained here does not contain the inversion generator and those for the four-parameter abelian group of acceleration transformations (Wess 1960) of the conformal group. The reason is that the vectors for those generators are not analytic functions of the coordinates and we have obtained here only those that are analytic functions.

5.3 Klein-Gordon equation Here also we have a single equation h2c2A~ = 0, with A - V z -02/(~'1s) 2 --(mc/h) 2 (56)

Taking the coordinates x~ = x, x 2 = y, x a = z, x4 = z = ct, and

x = (57) i 454 P Rudra

Equations (45) give

2(mc/h) ~ = 0; [V ~ - di/(a~) 2 ] ~ = 0, i = 1, 2, 3, 4;

dC1/~xi = dC~/dx2 = ~gC~/t3x3 = dC~/dx4 = 0; d~ldxl - dCll~x~ = dC,~ldx 2 - d¢21dx,~ = ~¢~/dx 3 - ~Caldx~ = 0;

~/ax3 + dC3/dx2 = aC3/ax, + aC~/ax3 = aC~/ax2 + aC~/ax~ = o. (58)

Imposing the analyticity restriction we get the ten generators X~ ..... X~0 of the inhomogeneous Lorentz group I,~ of (54) with the commutation relations of (55). The scaling generator X~ t is absent because of the presence of the non-zero mass term in A.

5.4 Two-component Weyl equation for neutrino This is again a single equation

iticA~b = 0, with A -- ~" V + I~/~. (59) Since the coefficients appearing in A are 2 x 2 matrices, we write the Ci's as 2 x 2 matrices and expand them in terms of the complete set of bases, the identity matrix I and the 3 components of Pauli matrices ¢ri's.

Ct=C°l+~CTtr,, 2=2oI+~,2,a,, ao=0, ao=0,

a4 = l, al = a~ (i = 1, 2, 3). (60) The condition [Ct, aj] + [C~, all = 0 (61) gives C, = C° I, C, = C° I + err, (i = 1, 2, 3) (62)

The partial differential equations connecting ¢°'s, C, 2o, 2~'s are

~ (C,° - C) - aC°/a~ = - ia~°/az = iq°/ay = - 2, - a~/ax

a I C° - C)/ ay = ia~° / az = aC°/aT = - ~aC° / ax = - 22 - aC/ ~y a(q ° - C)/az = - ~aC°/ay = ~a~°/ax = a¢°/a~ = - 23 - a¢/az ~( C° - ~)/ ~.c = ~O / ~x = ~CO l ~y = ~C° / ~z = - 20 - c3~/c3z. (63) The admissible generators are

X~ = I(-id/dx), X 2 = I(-id/dy), X 3 = I(-it~/dz), X 4 = I(-id/dT),

X s = l[ - i(yd/~z - z~/dy) + (xd/dz + z~/ax)],

X 6 = I [ - i(z~/~x - xa/dz) + (yd/~T + zd/~y) ],

X7 = I[ - i(xa/dy - yd/dx) + (zd/t~z + Tt3/cOz)],

X a = I(xd/~x + y~/dy + zd/~z + zd/dz),

X 9 = - ia" V (64) Symmetry oroups of mathematical physics 455 with the non-vanishing commutators

[Xl,Xs] = X,,, IX,, X63 = iX3, [X,, Xv] = - iX2, IX2, Xs] = -iX3, [x~,x~] = x4, [x~,xd = ~x,,[x3,xd = ix~, [x~,x~] = -ix~, [x,,xd = x4, [x4,xd = x,, [x,,x~] = x~, [x,,xd = x3,

[Xs,X6] = 2iX,, [Xs, XT] = -2iX6, [X6,X7 ] = 2iX 5, Ix j, x~] = X i(j= 1,2,3,4) [Xs, X9] = i(0-2X3 -- 0-3X2) - °'1X4, IX6', X9] = i(0-3X1 - 0-1X3)

-- o- 2 X4, [X7, X9 ] i(0"1X 2 -0-2X1) -0-3X4, IX8, X9] -~- --X 4 -X 9. (65)

Here X~ to X4 are the generators for translations along x, y, z, z axes; Xs, X6, X7 are generators for screw transformations along x, y and z directions; X a is that for scale transformation and X 9 for helicity transformation. It should be noted that commu- tators with X 9 have matrices rather than scalars as the structure constants. Thus X~ ..... Xs form the group of the eigenfunctions.

5.5 Four-component Dirac equation for fermions We have here a four-component equation

ihcA~ = 0, with A -- ~t. V + lO/t~ + (imc/h)fl. (66)

The functions ao, ai's, airs and ~{s are now 4 x 4 matrices and we express them in the bases formed by the Kronecker direct products (I ®I), (I t~i), (at ®I), (at ®o-j). On this basis Gt = (0-x (~a) and fl = (0-3(~)I). Solving the set of partial differential equations in ~'s embodied in (45) we get the following 12 generators:

Xa=-i(I(~I)t~/ax, X2=-i(l(~I)a/t~y, Xa=-i(I(~)I)O/Oz,

X 4 = - i(I(~I) 0/0~, X 5 = - ~'V- (mc/h) (0-2(~)l)0t'(r x V)

X 6 = -i(2mc/h)(I(~)l)(yt'~/Oz-zO/t~y)+ (0-a(~)l) ( aty - 0y,]

X 7 = -i(2mc/h)(l@I)(zO/Ox -xO/Oz)+ (0-3QI) (:=e. - e~O-zzo)

X a = -i(2mc/h)(l(x~l)(xO/Oy-yO/Ox)+ (0-3~)I) ot~ - ¢ty~x

X9 ~- (0-2(~I)(otyO/t~z-~tzS/OY), Xlo = (0-2QI) ~tz 0zJ

X11--(0-2QI)(o~x~-~,~-~), X,2= -i(0-,(~I)ot'V. 456 P Rudra

6. Discussion

We have surveyed recent activities in determining the symmetry groups for different classical and quantum mechanical systems. The problem reduces to solving a set of coupled partial differential equations. It is true that this is no mean proposition. However, there is a vast literature of classical mathematics on this branch and with a little trial and error symmetry groups for a large number of physical systems have been obtained. On the other hand the classic works of Bateman (1910) and Fock (1935) were only possible as results of individual ingenuity. It has been shown that Lie's method gives an algorithm for the systematic study of the problem.

References

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Note added in proof:

In equations (33) and (36), xi, u I and u~'s are to be considered as independent variables. The term [J [ = 0 is excluded from the summation over [JI in (33), while in (36) it is included.