Dynamics of Particles with Internal ,,Spin" By H . R u n d Summary A generalised relativistic Iagrangian is suggested, which is such t h a t the momenturn vector possesses a component- normal to thc direction of the velocity, while the conservation of total angular momentum requires the existence of a non-vanishing internal angular momentum tensor (spin). The resulting mechanical system is very similar to the spin models of clementary particlesdiscussed by H o n l , P a p a p e t r o u , B o p p and others, a s these emerge as special cases. Although the latter arc usually derived from variational principles for which the Lagrangian depends on the acceleration, the present theory is based on a Lagrangian depending solely on position and velocity. This method avoids certain mathematical diffieulties,and permits theimmediate application of the Hamiltonian theory associated with such simple variational principles. Although no radiation cffects arc considered, the A b r a h a m vector plays an important role in this purely mechanical theory. $j1. Introduction In his monograph H. H o n l l ) expresses the vicw t h a t the mechanics of his pole-dipole particle (and its generalisations) represents a simple extension of the relativistic mechanics of a particle. The close affinity bctwcen the dynamics of particles of this type and electrodynamic field theory on the one hand, and the striking transition t o the corresponding quantum-mechanical systems with spin properties on the other, certainly seem to confirm this view. In the present note we shall cndcavour to present an alternative and more general mathematical theory of the relativistic mechanics of a particle, cxtended in a similar sense. While the original discussion of the pole-dipole particle of Htinl and P a p a p ~ t r o uwas ~ ) bascd on a method of approximation suggested by the gravitational theory of general relativity i t was later found that the equations of motion of such models could be derived from a variational principle, in which, however, the Lagrangian function depends on second derivatives of the position vector, this mcthod making free use of undctermined multipliers. Such procedures are not only subject to techniques which are somewhat questionable from a purely mathematical point of view, but they also tcnd t o obscure the situation when a dcepcr physical insight and further mathematical developments are bcirig sought. -~ l) H. Hon l , Feldmechanik des Elektroris urd der Elementarteilchen. Ergebn. exak. Naturw., Bd. 26, (Springer, 1962), p. 326. 2) H. H o n l u. A. Papape tr ou, Z. Physili 112, 512 (1939); 114, 478 (1939). 2 Ann. Physik. 7. Folge, Bd. 7 18 Annalen der Physik. 7. Yoke. Band 7.1961 I t will be shown t h a t one ma.y carry out a similar programme on the basis of a variational principle whose Lagrangian involves no derivatives higher than the first. of the position vector. No part.icular physical model is postulated, so t h a t a very general form of the Lagrangian may be presupposed : it is subjected only t o one condition, namely, t h a t the total angular momentum (to be defined in a special manner) be conserved in the absence of external fields. This determines the generaliscd momentum vector - which has, i n gcneral, a component normal to the direction of the velocity - t o within a scalar function. The above-mentioned theories emerge as special cases when t.his function is chosen in various ways. The introduction of the Hamiltonian associated with our variational principle immediately leads t o the Hamiltonian cquations, which form the basis of the t,ransition t,o the corresponding quantummechanical systems. A significant feature of the method is t.he fact that t.he A b r a h a m vector, which plays a fundamental role in the classical theory of the radiating electron, appews with a certain inevitability, although no radiation cbffects are considered. We restrict our considerations t o t,he (flat) M i n k o w s k i world. The coordinates (2,y. z, t ) of the particlc are t o be reprcsent.ed by xi (i = 1, . . ., 4), where we have put. d == i c t. The summation convention is used t.hroughoiit1, so that the line element is given b y ds2 = c2 at2 - ax2 - dy2 - dz2 = - axi axi. Denoting derivatives with respect t o s by dots, we put in particular rl.ri/ds, so that we have the identities 7ci = x i == while where w is t.he velocit,y of thc particle. In the sequel t will dcnote a n arbitrary parameter, and differentiation with respect t o t will be denoted by a primc: c . g. xli = dxi/dt. Q 2. The relativistic Lagrangian We denote by Lo ( x i , x t i ) the Lagrangian corresponding t o a particle i n the absence of elect,romagnetic fields. It will be shown t h a t it is n e c e s s a r y t o i n s i s t t h a t Lo be h o m o g e n e o u s of t h e first d e g r e e i n t h e di. Xot only does this ensure th a t t h e action integral [ Lo ( x i , 3 9 ) a t taken along any path is independent of the parametrisation of t h a t path (so that the resulting E u l e r - L a g r a n g e equations arc also independent of the choice of z),but it will also be evident t h a t this condition is necessary for consistency if it is assumed that the terms representing the external electromagnetic field in the complete Lagrangian L are given in their usual form : El. Rund: Dynamic,? of 19 Particles with Internal ,,Spii~" where the Ai denote the well-known retarded potentials. This is easily seen as follows. The E u l e r - L a g r n n g e equations corresponding to (2.1) are simply where F i j is the skew-symmetric field tensor p 1 3. aAj aA' ax' as' _=--- (2.3) Thus the equations (2.2) imply t h a t the condition [&(aLo)-5],ti dt as" = 0 (2.4) must hold identically. However, this is possible if, and only if, Lo (xi,x'i) is homogeneous of the first degree in di: i. e. if Lo (xi,2 :di) = 3. Lo (xi, XI;), (I, =+ 0); (2.5) for clearly we have - so that condition (2.ha) m y constant of integration being absorbed in Lo. On replacing x'i by 1, x'f, (3, =+ O), in (2.5a) we obtain from which the required relation (2.5) follows directly after integration with respect t o i. A most significant advantage of a homogeneous Lagrangian is the fact that it avoids undetermined multipliers; this is in direct contrast to the methods adopted by many authors, e. g. by H o n l or I n f e l d 3 ) , who regard the condition xi zi = - 1 as a constraint and use the multiplier method accordingly. From a purely mathematical point of view this approach appears to be somewhat questionable, as t h i s condition is really one which merely dctcrmines the parameter t,while the use of multipliers mag, in general, lead to severe theoretical difficulties4). Furthermore, the condition of homogerieity removes all ambiguity from the Lagrangian, in the sense t h a t it allows u s to determine whether or not a term should or should not be multiplied by the unit factor (- 2i &i)'/* when the proper time s is used. In passing we remark t h a t similar observations apply to the method associated with Lagrangians involving higher order derivatives. Here the hgrangian - H. H o n l , Footnote 1, p. 320 e t seq; L. Tnfcld, Bull. Acad. Polon. Sci.,CI. III, G, -. . . 8) 491 (1967). 4) H. Rund, Math. Nachr. 11, 61 (1954). o* 20 Annalen der Physik. 7. Folge. B a d 7. 1961 function must be of the form Lo (9, x'k, Q ) , homogeneous of thc first degrrc i n the second set of variables x'k, while Q is given by where it may bc rioted t h a t Q'In represents the invariant first curvature of t h e world-line of the particle. We ahall not, however, concern ourselves wit ti variational principles of this type. Our considcrations will bc based on a Lagrangian function (in tho absence of electromagnetic fields) of the type Lo = - [k, + x."19(s'k)](- x'i x'i)lIt, (2.6) where ko and kl are simply constants, while g ( x t k )is a function homogeneous of dcgrec zero in x'kin accordance with our requirement above. Explicit forms of g ( x ' ~will ) emerge from the exigencies of our applications of (2.6). We remark that whcn kl = 0 the function (2.6) reduces t o the Lagrangian commonly used in the special theory of relativity, with ko = moc (m, = restmass). If we put (23) we obtain a vcctor whose component,s are homogeneous of dcgree - 1 in z'~. while (;i ( a ' k ) " i 0. (2.8) = . I Gi is always normal to the world-linc of our particle. We shall define thc first momentum vector as usual by so that Obviously this vector is homogeneous of degree zero in the xti; we may, therefore, replace the latter by the vclocty vector uc => dxilds, which permits us to simplify (2.!9) according t o thc identity (1.1). Wc then obtain Pi (u)= [ k , + k, g (?&)IMi - kl ci (u). (2.1 0) It follows, therefore, that thc momentum vector possesses a component not only in the direction of the velocity, b u t also one in the direction of Gi, which is normal to the vclocity vector. In the presence of a n electromagnetic field the equations of motion (2.2) now become dP' _ ds - (k, + k1 g) iii + k, (@ ?ci - 5) Uk - -F , , (2.1 1) u j . We remark that it is easily verified t h a t the rclation (dPi/ds)zci identically; for from (2.7) it follows t h a t = 0 holds ("12) H . Rund: Dynamics of Particles with Internal ,,&inrc 21 and hence, by homogeneity, we have (2.13) At this stage nothing further can be said about the equations of motion until we have at our disposal more information concerning the - so far unspecified - function g and its derivatives Gi. For this reason we shall first discuss the angular momentum tensor. 6 3. Orbital and Internal Angular Momentum In analogy with the usual relativistic definition of angular momentum we shall define the orbital angular momentum of our system about the origin of coordinates by means of the skew-symmetric tensor pk - xk P<. (3.1) In the absence of an external field ( P i = const.) the rate of change of angular Mi,= or, substituting from (2.10), by Thus, in general, orbital angular momentum is not conserved, unless Gk is of the form Gk = x uk, a contingency which we may well discard, since this would simply imply that x = 0 or g = const. in view of (2.8). We conclude, therefore, that the tensor (3.1) cannot represent the total angular momentum of the particle: in fact, we shall suppose that a skew-symmetric tensor mik has to be adjoined to Mi,such that the sum Mi, mikis conserved. Clearly, the precise form of mi, cannot be determined on the basis of our previous analysis alone. One would be inclined to think, however, that the commonly used momentum pk = k,, u k might give rise to the compensating terms. We thus introduce the following a d h o e a s s u m p t i o n , namely, that the ,,internal" angular momentum tensor mik may be represented as a vector + (3.3) where, without loss of generality, we may assume that zi is normal to ui, i. e. that ziui = 0, (3.4) since a component of zi in the direction of uicould not possibly contribute to the value (3.3) of miL' We shall see that this assumption will enable us to specify the Gi t o within a scalar factor. From (3.2) and (3.3) we now have zi u k - zk ui - ui ( i k + k, Gk) + U* (& + kl Gi) = 0, (3.5) 22 Annulen der Physik. 7 . Folge. Band 7. 1961 from which we deduce by multiplication with ui and application of (LI), (2.8) and (3.4) that ik kl Gk = - u k ( i i u$), (3.6) + which, when substituted in (3.5), yields zi c k - z k ki = 0. (3.7) It therefore follows that there exists a scalar function @ such that Thus, apart from the scalar function @, the vector zi is determined by our conditions. The same applies also t o the functions Gi;for from (3.4) we now have & uii = - zi zii = @ (u' a u') I , (3.10) which, when substituted together with (3.9) in (3.6), gives +d tik. kl Gk = @ [iik- (zijzij) uk] (3.11) Amost remarkable feature of this equation is the fact that t he c o e f f i c i e n t of @ i s t h e w e l l - k n o w n A b r a h a m v e c t o r : r k = Ck - (Gii ~ i i Uk. ) (3.12) We recall that this vector plays a prominent role in the classical theory of radiating electronss), the radiation term which appears in the equations of motion being simply of the form 2 e 2 r k / 3c. Furthermore, the condition Fk = 0 characterises uniformly accelerated motion, i. e. a motion for which the time rate of change of the acceleration (3-vector), as measured in an instantaneous rest-system, vanishes identicallye). We shall return to these remarks in the next section. Meanwhile, we may uae (3.11) t o derive various expressions for the function g (u). By definition (2.7) and (3.11) we have (3.13) so that klg= ~(zikUk)di+tj(Ukuk)ddds, (3.14) or, alternatively, 1 d k,g= (UkUk)@--$@-((ick&k) 2 as ds, (3.15) where we have disregarded constants of integration which may be absorbed in the term - (k,, i& 9 ) in the Lagrangian. Substituting from (3.14) and -+ 5, 8, P .A. M. Dirac, Proc. Roy. Soc. Ser. A, 167, 148 (1938). E.L. Hill, Physic. Rev. 72,143 (1947). 23 H . Recnd: Dynamics of Particles with Internal ,,#pin" (3.11) in (2.10) we find that the most general expression for the momentup assumes the form Pi = [k, + 3 (uk uk)@ + $s (Uk uk)6 d s1ui-&tii-@ iii. (3.16) Here @ is an arbitrary scalar function. We shall see in $ 5that by suitable choice of @ one may arrive a t the various forms of the momentum vector which have been used in previous descriptions of spin models. I n conclusion, we note that the ,,internal" angular momentum tensor may now, by virtue of (3.3) and (3.8), be written in the form (3.17) m i k=@(zi,kui-ziiuk). Q 4. The Equations of Motion We may now substitute (3.16) in the equations of motion (2.2); and after some simplification we find that these equations assume the form: In the absence of an electromagnetic field ( F i j = 0) these equations are satisfied by 2ii = 0. However, we discard this solution, since by (3.14) the term kl g = 0 when ui = 0, which would simply mean a return to the classical case. Furthermore, since u s is normal t o ui,we deduce from (3.17) that m i % can vanish only if zii = 0 or @ = 0. Again, the second of these alternatives also implies merely a return t o the classicalcase, so t h a t w e h a v e t o c o n c l u d e t h a t in general m i k =k O. (4.2) The equations (4.1) may be written in a more concise form if we introduce the A b r a h a m vector (3.12) and its derivative: dri .... -= uz-as a as (u3. . ~3 ui - ( ~ &j)j Ui. (4.3) After substituting (3.12) and (4.3) in (4.1), we find on simplification: " _' -- ui (iifd )@ + 2ii (k, + kl g - 8; ) - 2 8, T i- @ ari ds = ,Fij uj. as (4.4) We have observed above that the condition = 0 characterises uniformly accelerated motion, i. e. motion devoid of radiation damping. It is natural, then, t o enquire whether or not we can find a suitable class of functions @ such that the equations of motion (4.4) would permit a solution for which Ti = 0 (in the absence of an electromagnetic field). From a purely physical point of view such a state of affairs seems improbable ; and we shall now show t h a t t h e m o t i o n s c h a r a c t e r i s e d b y t h e e q u a t i o n s Ti= 0 a n d dPi/ds = 0 a r e i r r e c o n c i l a b l e , e x c e p t f o r a p a r t i c u l a r choice of @ u n d e r s p e c i a l i n i t i a l conditions. ri 24 Annalen der Phyib. 7 ' . Folge. Band 7. 1961 Multiplying (3.12) by implies Uk, we deduce from (1.1)that the condition T k = 0 . . ~ c qi i = 0 q? 6)= C , (4.5) where C is a non-vanishing constant (noting that we exclude the case zii = 0, while zij, being normal t o u!, cannot be a null-vector). Hence the condition Ti = 0 could be compatible with dPg/ds = 0 in (4.4) only if @ were to satisfy the differential equation ko k1g -6 = 0, (4.6) where we recall that k, g is completely determined by (3.14) in terms of @. Let us then, for the moment, assume that di is derived according to (4.6), in which case the equations of motion (4.4) (with Fij= 0) reduce t o + ui (cici)@ - 2 $ , p dri ds = 0, -@ and, if we note that by (1.1)and (3.12) we have rj kj = cj ~i (4.7) these equations can be written linearly in Ti and its derivative : If, for some value s = so, the motion is such that Ti = 0, it follows t h a t dI'ilds = 0 a t s = so; and, since the pth derivative of Ti,resulting from (4.8) by ( p - 1)-fold differentiation, will be given in terms of Pi up t o the ( p - l ) t h derivative, it follows that all derivatives of Tivanish a t s = so, so that Ti = 0 for subsequent values of s. We conclude, therefore, that the condition Ti= 0 is compatible with the equations of motion only if di is given by (4.6) and if I'i = 0 a t s = so. We remark that for the special physically significant cases to be treated below, equation (4.6) cannot be satisfied. We shall now investigate the conditions under which a n a c c e p t a b l e s o l u t i o n of (4.1) m a y r e p r e s e n t a p l a n e c i r c u l a r m o t i o n in the absence of an electromagnetic field. Such solutions have previously been studied in connection with spin phenomena'). If we suppose that this motion is defined by the equations x1 = a cos ( n t ) , 9 = a sin ( n t ) , x3 = 0, x4 = i c t , so that the (constant) velocity of the particle is given by w 2 = a2 n2, we may evaluate ui, tii, ui,u i and substitute in the equations of motion (4.1). It is found that dPilds vanishes if and only if the following two conditions are satisfied : (4.9a) (4l + k, g - (4.9b) From (4.9a) we have to deduce that @ = 0 (since c =/= O), so that (4.9b) may be simplified considerably, particularly as g is now simply given by Q @ (uizij) ') H. H o d , Footnote 1, p. 305; L. I n fe ld , Bull. Acad. Polon. Sci., CI. 111, 5, 979 (1967). H.Rund: Dynamics of 25 Particles with Internal ,,SpinLL in virtue of (3.14). Equation (4.9b) may be solved for @: it is found t h a t circular motion with a n g u l a r f r e q u e n c y n a n d c o n s t a n t velocity tv is p o s s i b l e i f a n d o n l y i f @ is t h e c o n s t a n t (4.10) in t h e a b s e n c e of a n e x t e r n a l e l e c t r o m a g n e t i c field. In conclusion we remark that if we define the work done on a particle by + -f - a force P during a displacement d 7 as usual by the scalar product P d 4 , and regard this expression as the subsequent increase of energy of the particle, it is found (as for the classical case) that the energy E of the particle is of the form i E = c P4, apart from a constant of integration. But from (2.10) we have, quite generally, (4.11) pid = - (ko 4 9 ) + since Gi is normal to ui. Thus for a system in which PI = P2= P3 = 0, (4.12) We note that in the classical case the condition P1= P2 = P3 = 0 implies w = 0, so that (4.12) reduces to the usual expression when ko = mo c, kI = 0. 8 5. Special Cases A number of interesting special cases occur if we choose specific functional representations for the function @. The cases to be mentioned here are those which have been studied in detail in the past. (a) Firstly, we note that from a purely analytical point of view the work is simplified considerably if we assume that @ is a function of the variable ( u j u j ) only. For in that case we can evaluate the integrals which appear in (3.15) and (3.16), and therefore, for the rest of this section, we shall suppose that @ is such afunction. T h i s w o u l d c o r r e s p o n d t o t h e g e n e r a l c a s e t r e a t e d b y H o n l a n d Bopp. Secondly, it will be seen that it is sufficient t o assume that di may be represented in the form @ = a (&j&i)W, (5.1) (where a and m are constants, rn - 1)in order to obtain some of the special cases referred to above. When di is given by (5.1), we have =+ (n. 2 ) so that by (3.15) and (5.1) (5.3) where we may neglect any constant of integration which may have occurred in (5.2), as such a constant could have been absorbed in the total expression ho 9- +s Annalen der Physik. 7. Folge. Band 7. 1961 26 (b) When a = - kl, m = 0, the relation (5 3) reduces to g =- while @ =- 4, and 4 (Giu )> (5.4) (3.16) becomes (5.5) and (4.12) reads (5.6) E q u a t i o n (5.5) r e p r e s e n t s t h e e x a c t f o r m of t h e m o m e n t u m of t h e p o l e - d i p o l e m o d e l a s d e v e l o p e d b y H o n l a n d P a p a p e t r o u a ) , who. however, based their analysis on a method of approximation due t o L u b a n s kis), this method being suggested by the gravitational theory of general relativity. The momentum vector (5.5) can also be obtained from a variational principle involving a Lagrangian depending on the acceleration vector uiby means of undetermined multiplierslO). We remark that in this system the circular motion discussed in Q 4 is well possible. (c) When a = - 4,m = - 2, the relation (5.3) reduces t o (5.7) with (5.8) T h i s is t h e c a s e t r e a t e d b y Weyssenhoffll). The physical significance of the various forms of E, as given by (5.6) and (5.8) has been described in detail by the authors concerned, and we need not, therefore, enter into a similar discussion here. None of the cases (a), (b) or (c) are compatible with condition (4.6). g 6. The Hamiltonian Function When considering the transition to quantum mechanics of the general system described by our Lagrangian (2.1), we make use of the fact that there exists an invariant function H , which, although it does not represent the total energy of the system, satisfies all the identities and canonical equations which one normally attributes to a Hamiltonian function 12). We define a new invariant parameter c by putting do = L ,j.( dxi) =- (k,, + ,tl 9 ) 1/- ( . so that L 8) 9, lo) 11) l2) z') 23,-- = aai ax<+ ~~ + ~i ax{, (6.1) 1. See Footnote 2. J. Lubanski. Acta Phvs. Polon. 6. 356 11937). " Footnote 1, p. 322. J. Weyssenhoff, Acta Phys. Polon. 9, 46 (1947). H. R u n d , Differential Geometry of Finsler Spaces. (Springer, 1959), Ch. I, ~I 8 5. H . Rund: Dynamics of Particles with Internal ,,Spin" 27 By solving the equations p*i = Lfor the xriin terms of the P*i and substituting in (6.2)) we obtain an equation of the form H (xi,P*i) = 1, (6.4) where H is a uniquely determined function. Since we have identically we regard (6.4) as the Hamiltonian equation. Furthermore, as H is homogeneous of the first degree in P*i, equation (6.4) may be written as aH +P*i = 1, or, by (6.5)) x'i p*i = 1. But by (6.1) and (6.2) we have do L (x3,d)= ds ' so that the Hamiltonian equation is equivalent to u j P*i = L (Xk)Uk), or, by (2.1), to " >+ ui P*j - - Af ( c (Lo + L, g (uk)) = 0. I t is t h i s f o r m of t h e H a m i l t o n i a n e q u a t i o n which s h o u l d b e used i n t h e t r a n s i t i o n t o t h e c o r r e s p o n d i n g q u a n t u m - m e c h a n i c a l s y s t e m . Its similarity with the Dirac wave equation is striking. In fact, for the special case considered by H o n l , s u b s t i t u t i o n f r o m (5.4) i n (6.7) y i e l d s p r e c i s e l y t h e e q u a t i o n s u g g e s t e d b y h i m f o r t h i s purpose13). For a discussion of this process - even for more general cases - we refer to the monograph of H o n l and the literature cited therel4). In a later publication we hope t o deal with the Hamilton-Jacobi theory based on (6.7) as well as with an alternative process of quantisation. See Footnote 1, p. 324. See Footnote 1, p. 368 et seq.; also F. Bopp u. F. L. B a u e r , Z. Naturforsch. 4a, 611 (1949); H. Honl u. H. Boerner, Z. Naturforsch. 5a, 353 (1950). 13) 14) D u r b a n (South Africa)) Department of Applied Mathematics of the University of Natal. Bei der Redaktion eingegangen am 30. Januar 1960.

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