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.