A N N A L E N D E R PHYSIK ~~ ~ 7. Folge. Band 37. 1980. Heft 2, S.81-160 Galilei Covariance Does Not Imply Minimal Electromagnetic Coupling1) By KARLKR-4US Pliysikalisches Institut der Universitat Wiirzburg Abstract. The postulate of Galilei covariance in one-particle classical and quantum mechanics is reinvestigated, with particular intent to correct some current misconceptions concerning the rdle of minimal electromagnetic coupling in Galilei covariant theories. Die elektromagnetisehe Minimalkopplung ist keine Folge der Ualilei-Kovarianz Inhal tsubersicht. Das Postulnt der Galilei-Kovarianz bei der klassischen und quantenniechanischen Beschreibung eines einzelnen Teilchem wird diskutiert. Insbesondere sol1 dabei ein verbrciktes MiDverstiindnis iiber die Rolle der elektromagnetischen Minimalkopplung in Galileikovariantcn Theorien korrigiert werden. 1. Motivation and Outlook A couple of years ago JAUCH [ l ] formulated “a suitably generalized form of the principle of Galilei invariance” ([l],p. 286), which restricts the interaction of a spinless quantum iiiechanical particle with external fields t o niinimal electromagnetic coupling, as described by the Hamiltonian 1 H = - 2m (P - e 4 Q t t)>2 e m , t ) . (1.1) + Since then, Jauch’s argument has been repeated and reformulated several times [2 -41, and some authors have tried t o generalize it t o particles with spin [5], or to apply siinilar argunients to classical mechanics [3,4,6]and t o the case of the Lorentz group [6]. We want t o show here that Jauch’s derivation of (1.1)is not convincing from a physical point of view. I n fact, the crucial invariance requirement of Ref. [l] is 90 restrictive that it is satisfied by the Hamiltonian (1.1)for a particular subgroup of the Galilei group only. As this subgroup is not distinguished physically, whereas the extension of Jauch’s invariance requirement t o the whole Galilei group would further reduce (1.1)t o the case A = 0, this requirement is not suitable t o establish a privileged position of the minimal coupling Hamiltonian (1.1)in quantum mechanics. On the contrary, Galilei covariance - which is the proper substitute for invariance if external fields are present - involves considerably weaker requirements only, which are satisfied by a large class of Hamiltonians of which (1.1) is but one particular cam. Similar objections apply t o the modifications and extensions of the original argument, and in particular to the corresponding reasoning for classical mechanics [3, 4, 61. 1) Dedicated to Professor G G Ann. PIiysik. 7. Folge, Bd. 37 ~ LUDWIG R on the occasion of his sixtieth birthday. K. &AUS 82 I n order t o prepare the ground for the proof of these statements, we first want t o present a short but self-conhined discussion of Galilei covariance in classical and quantum mechanics. For more detailed reviews of the r61e of the Galilei groiip in physics see, e.g., LEVY-LEBLOND [4]or STEINWEDEL [7]. 2. Cfaliei Covariance in Classical RIecliaiiics with External Fields The inhoaiogeneous Galilei group consists of the transformations x‘= R x f v t + o , t’=t+z (2.1) connecting the space and tinie coordinates s and t of a given event with the coordinates x’ and t’ of the milie event with respect t o a different inertial frame, with R, u, a and T denoting a rotation inatris, the relative ve1ocit.y of the frames of reference, and a shift of the space and time coordinates, respectively. I n t h i s and the following Section we will discuss transforniations (2.1) with z = 0 only. Position q and velocity Q of a point particle interacting with external fields transforiii iinder Galilei transforiiiations (2.1) with T = 0 according to q’W = and Q’(4= + vl + RQ(t)+ RqV) (2.2) 0 (2.3) u7 whereas t.lie external field components y V ,Y = 1 ... 11, transform according t o with wine finite-dimensional representation D,,,(R, u) of the homogeneous Galilei group. A Galilei covariant theory is obtained froni a Lagrangian (2.8) t?1 Lo(& = 2 QZ (2.6) and an interaction part2) L,(q, q: 1) = 4 4 . p:Y(q.4 ) with the invariance property3) (2.7) - Al(Q9v,Y(q.0 ) = 4(Q’, qC(q’*0) (2.8) I n the “priiiied” inertial frame, the Lagrangian is given by the same function of the particle velocity and the external fields, i.e.. Due to the different functional form of the external fields entering Al. the functional fornis of L, and 1;; (and thus of L and L’) in general are different. but by (2.8) we have L,(q, q , t ) = L;(q’.Q’,t ) . 2) The quantity relevant for particle iiiechanics is L = Lo + L, as a function of q, q, and 1. Eq. (2.7) says that L,(q, 4. I ) i R calculated from another function A , of q and the external fields F,.(!l, I ) . 3) Eq. ( 2 3 ) for pure space tmnslations escliicles any explicit q dependence of A,. Galilei Covariance Does Not Imply Minimal Electromagnetic Coupling 83 On the other hand, Lo(Q’)= L,(Q) + mRQ - v + 9, so that L and L’ are also nunierically different. However, since L’(q‘,Q’, t ) - L(q, q, t ) = L,(Q’)- Lo(*) is a total time derivative, Lagrange’s equations of motion (2.10) in the .,unprimed” frame are e q u i v a l e n t t o the corresponding equations in the ‘‘prinied’’ frame. Moreover, with (2.9) the equations (2.10) become f o r m i n v a r i a n t under Galilei transformations, in the s e w that the left hand sides of (2.10) and of the corresponding “primed” equations are given by the same universal (= frame-independent) fiinction of the argunients Equivalence and equal functional form of the equations of motion (2.10) in different frames of reference together constitute what usually is called Galilei c o v a r i a n c e . Both requirementa are consequences of the physical equivalence of all inertial frames: Fornr invariance means that identical external fields if realized in different frames will lead to identical particle trajectories, whereas equivalence of the equations of motion in different frames nieans that a given particle trajectory q(t), if transfornied via (2.2) into another inertial frame, solves the equations of motion (2.10) in the new frame. On the other hand, as L and L’ in general are different functions of time t and the particle variables q, Q resp. q’, q’, the description of the particle’s motion in given external fields obviously is n o t i n v a r i a n t under Galilei transformations, in the sense that a solution q(t) of (2.10) in general will not solve the corresponding “primed” equations. I n other words, the set of possible trajectories is different in different inertial frames. due to the breaking of Galilei invariance in the presence of external fields. The similarities between classical and quantum mechanics are manifest only if the former is put into canonical (Hamiltonian) form. One introduces the canonical moment u m p ( t ) with components (2.11) Under Galilei transformatione (2.1), p(t)is changed into p’(0 = Bp(t) mu, as follows from (2.3) and (2.8) by a simple calculation. With the Hamilton function6) + X ( Q , Pt,) = P Q - L(q, Q,t ) , (2.12) (2.13) from which Q has to be eliminated by means of (2.11)6), the equations of iiiotion are now given by Hamilton’s equations (2.14) 4) 5) 6) 8‘ Note that form invariance as defined her refers to particle and field variables together. The term “Hamiltonian”is reserved here for the componding quantity in quantum mechanics. This requires that (2.11) may be solved for 4 as a function of q, p , and t. K. &tans 84 I n the “primed” inertial frame, corresponding equations hold true with 4‘ - L’(q’,4’, t ) . JE“’(q’,p’, t ) = p’ * (2.15) Since the definitions of &‘ and &” involve the same functions of particle velocity, momentum, and external fields in both frames, the Hamilton function 2 is given by a frame-independent function of momentum p a n d t h e f i e l d s v,,(q,t ) , and Eqs. (2.14) are form invariant in a similar sense as explained above. Again this does not mean that &‘ is a frame-independent function of time t and the p a r t i c l e variables q and p;on the contrary, the functional form of JE“’(q’,p’,t ) differs from that of H ( q ,p, t ) since different field functions y:(q’, t ) enter JE“’. Likewise, Eq. (2.11) yields a functional connection between p , q a n d the fields q,,(q,t ) which is form invariant, whereas the corresponding functional connection between the particle variables p, q, and q is frame-dependent. I n other words, if (2.11) is written in the form P = mz,4 2 0 , or solved for q in the form 4 = w q ,P , t ) the corresponding functions IT and Y obtained in the “primed” frame will be different from lI and Y. The only exception is the case of a velocity-independent interaction, 9 (2.10) in which case (2.8) implies L, = L,(q, t ) = - U q , t ) with a scalar (potential) field V ( q ,t). The resulting functional connection p=mq (2.17) between q and p is indeed frame-independent. Vice versa, assume the function 17 defined above t o be frame-independent. The same, then, holds true for d(q, q, t ) = n(q,q, t ) - mq = p - mq. df. From the transformations equations (2.2), (2.3) and (2.12) we obtain d(Rq ~t Q, R q U ,t ) = d(q’, q’, t ) = p’ - mq’ = R ( p - m q) = w q , q, t ) , which is easily shown t o imply d = 0. (For instance, with R = 1, u = 0 we find that d is independent of q ; etc.). We ale thus led back t o (2.17) which, by (2.11), is equivalent to (2.16). Note that even in this case the function 1 (2.18) *(Q, P,t ) = 2m P2 V q . ,t ) + + + + is frame-dependent unless V V = 0, since otherwise V’(q’,t ) = V ( q ,t ) have different functional form, V and V‘ defined by Galilei Covariance Does Not Imply Minimal Electromagnetic Coupling 83 Returning to the general caw, we obtain from (2.3), (2.12), and the tramformation behaviour of L,the formula X‘(q’,P’, t ) = Z ( q , P , t ) + Rp + * (2.19) v2. 11 The equivalence of Hamilton’s equations (2.14) in different inertial frames follows either from the equivalence of (2.10) and (2.14) in each frame, or more directly from the fact that the transformation formulae (2.2), (2.12) and (2.19) correspond to a canonical transformation with t.lie generating function F ( q , p ’ ,t ) = ( R q + ut + a ) .p‘ - mRq u m - - v2t. 2 To give a n example, we consider [7] a particle interacting with a symmetric contrnvnriant Galilean four-tensor field p, Y = 1 ... 4 , T = ( T f i ” (t~) ),, with TI“’ = T’fi.This field may be decomposed, according t o ”) T = ( ”A T* - 2 v ’ into R R (2.20) syinnietric three-tensor field i, k = 1, 2, 3, 8 = (Oir), vector field and its transpose AT = A2,4), and n scalar field v. Under a Galilei transformation (2.1), these fields are transformed according t o [71’) 8’(~ t’)’ = , Re(%, t)RT, (2.21) A’(%’,t’) = RA(x, t ) - R e ( % ,t)RT * U, 1 F’(X’, t ’ ) = Y(X, t ) RA(x, t ) * u - - u * R e ( $ , t)RT - U. 2 In particular, for Euclidian transformations (v = 0, t = 0), 8, A , and p thus transforiu like a tensor, vector, and scalar field, respectively. Moreover, tensor fields (2.20) with the particular properties 0 = 0, or 0 = 0 and A = 0, retain these properties + 7) Xotation: With a three-tensor8 and vectors u and v , u . Q and 8 . v are the vectors with i-th components 2 t&k@ki and 2 @irvk, respectively, andu 8 v is the scalar 2 ui@&. RU has k as 8’-tli component . . k Rikuk, and the ik-component of R 8 R T is 2 R&,,Rh. ik For a symmetric c Im tensor 0 the vectors u 8 and 8 u coincide unless, a8 happens in quantum mechanics, the components of u and 8 are non-commuting operators rather than c-numbers. - . K. KRans 86 under arbitrary Galilei transformations, and lead to the transformation formulae A' = RA, v' = v f RA v , or v' = y , of a contravariant Galilean four-vector field [7] (2.22) or a scalar field v, respectively. With the covariant Galilean four-velocity of the particle, an invariant interaction Lagrangian is then [7] A - _e l - p v x 2 x P ' e = - 4 . 8 ( q ,t ) 2 (2.23) - 4 + eci. A(q, t ) - ev(q,0 , the invariance of which is easily checked by means of (2.21). In particular, the cases 8 = 0, or 8 = 0 and A = 0, yield the usual theories of a particle of charge e interacting with an external electromagnetic four-potential (2.22) or a pure electric potential y , respectively. The canonical momentum p is given explicitly by P = @(q,t ) * with #(x, t ) = m l df. 4 + eA(q, (2.24) t), + e@(x, t ) , 1 denoting the unit tensor. From this, @ is obtained as a function of q, p, and t in the form (2.25) Q = @-'(q, t ) * (P - eA(q,0 ) . Here we have assumede) that the tensor @(x,t) has a n inverse for all x and t. The Hamilton function (2.13) of this theory, by (2.23) and (2.25), becomes W q , P,t ) = 1 (P - eA(q, t ) ) * @-l(!?, t ) ) * (P- eA(q, 0) + ev(q, t), (2.26) and Hamilton's equations (2.14) read 4 = 0 - 1 . (p - eA) (coinciding with (2.25)) and 1 2#-1 Iji = - - ( p - eA) 2 8% .- -e-. - ( p - eA) + e - . @aA -l. 8P.i ( p - eA) (2.27) av &i T h e Galilei transformation formulae for the corresponding field strengths E and B = V X A are discussed, e.g., in ";I. B, 9) See footnote 6. This assumption is Galilei invariant by (a.21). = -vrp aA -at Gdilei Cowrinnce Does Sot Iniply Minimal Electromagnetic Coupling 87 LIhY-r,EBLoND [4,61 and PIRON [3] claiiiied t o have shown that the only inter&.tioii of a classical particle consistent with the requirements of Galilei covariance is given by iiiininial electromagnetic coupling, i.e., by the particular case 8 = 0 of tlie internction Lagrangian (2.23). However, on the one hand, the covariance requireiiirnts of, e.g., Ref. [GI iinrnediately follow fi*omour Eqs. (2.2), (2.3) and (2.12) for tlie paitir.ular case R=l,a=-ut,t=O (”28) of (Ll), called “instantaneous Galilei transforination” i n Ref. [G]’o). Since, 011 the other Iiand, Eqs. (2.2), (2.3) and (2.12) obviously are satisfied by the esaiiiple considered also if 8 0 (and not only for “instantnneous” but for a r b i t r a r y Gdilei [ i ) )that somet,ransforinations!), it is obvious (as remarked already by STEINWEDEL thing must be wrong with the proofs mentioned. An explicit discuwion of this point will be postponed until after we have discussed the corresponding probleni in quant uin inechanics. + 3. Galilei Covariance in One-Particle Quantum Mechanics Quantuiii mechanics becomes very similar in form t o classical niechanics if formiilnted in the Heisenberg picture, which therefore is adopted here. (For a short discussion of the Schrodinger picture see Section 5 . ) Basic variables are now, in a given inertial frame, two self-adjoint operator faniilies Q ( 1 ) and P ( t ) which replace the classic.al variables q ( t ) and p ( t ) and provide, for each t, a n irreducible representation of the ceanonical coniinutation relations i[P&), Q k ( t ) ] = &.l, [ Q i ( t ) ,Qn ( t ) ] = [P,(t),P,(t)]= 0. i , E The equations of motion are given by Q ( t ) = i [ H ( t ) ,Q(t)l, = W(Q P(t)l , with the (Heisenberg picture) Hainiltonian = 1, 2. 3. (3.1) (5.2) H ( t ) = 2 , m ( Q ( t ) , V), t), (3.3) obtained from the classical Hamilton function Yi’ by substituting tlie operators Q and P for their classical counterparts. The suffix sym nieans that &‘ has t o be suitably syninietrized in order to make H (formally) self-adjoint. In virtue of the commutation relations (3.1), Eqs. (3.2) will then becoine properly symnietrized analoga in terms of operators of the classical Eqs. (2.14). Since we are discussing the kineiiiatical aspects of quantum mechanics only, we will assume that the “dynamical problem” posed by (3.1) and (3.2) may he solved in the usual way, thus taking for granted that the following assuniptions are satisfied: A s s u m p t i o n a. For given irreducible initial values (3.4 ) O(0) = Qo, P ( 0 ) = Po mtisfying the coniniutation relations (3.1), there exists a unique one-parameter faiiiil y of unitary time evolution operators V ( t ) satisfying”) V ( t )= iV(t)H&), V ( 0 )= 1 (3.5) lo) P m o ~ [3], using a different parametrization of the Galilei group, calls then1 “Galilei transformations”. 11) Eqs. (3.5) are equivalent to the Schrdinger eqiiation -i;fs(t) = ~S(t)fS@) for the Schrodinger state veztor fs(t) = V * ( l ) f , with f denoting the Heisenberg state vector, and the initial condition fS(0) = f. with the Schrodinger picture Haiiiiltonian &At) = X s y n , ( Q 0 , Po, 0. (3.d) -4ssuiiiption b. The solution of Eqs. (3.1) and (3.2) with initial values (3.4) is unique, and is given by Q ( t ) = V ( t ) QoV*(t), P ( t ) = V ( t )P,V*(t). With (3.7). the first equation of (3.5) beconies equivalent to T’(t) = i V ( t ) H&) V * ( l ) V ( t ) = i3fS,,,,(Q(t). P ( t ) ,t ) V ( t ) = i H ( t ) V ( t ) . (3.7) (3.8) Using this, (3.1) indeed follows a t least fornially by differentiating (3.7). Actually Assuiiiptions a and b should be proved rather than postulated, but as far as we know. no rigorous proof exists even for the case which is most interesting physically, i.e., for a particle in a given electromagnetic field. However, the success of conventional quantum iiieclianics suggests that the above assumptions are basically correct. Now consider a new (“prinied”) inertial franie, related t o the original (“unpriined”) one by a Galilei transformation (1.1) with T = 0. The canonical variables for the new frniiie are then postulated to be, as in tlie classical case (Eqs. (2.2) and (2.12)), Q’(t) = R Q ( t ) + ut + a. (3.9) P’(t) = RP(1) $- 1)t.W wliic.11 also implies ~ @ ( t ) = RO(t) +u. (3.10) P ’ ( t ) = RP(I). The first. equations of botli (3.9) and (3.10) are direct consequences of the physical interpretation of Q ,Q‘ and Q , 0’ in terms of particle positions and velocities. On the other hand, tlie second equation of (3.9) need not always have an immediate physical interpret.ation. e.g., due to gauge-noninvariance (arid thus non-nieasurability) of the canonical iiioiiientiini of a particle in an electromagnetic field. Eqs. (3.9) iiiiinediately iniply that Q ’ ( t ) and P’(t) satisfy the commutation relations (3.1) and are irreducible for all t . I n the new franie: the external fields are given by (2.4). The new Haiiiiltonian is t.akeii to be (3.11) H ’ ( t ) = Yf;,,,)(Q‘(t): P ’ ( t ) .1 ) with the transfornied Haiiiilton function (2.15), and yields equations of iiiotion of the foriii (3.2) for the “prinied” variables Q’ and P’. This procedure guarantees foriii i n v a r i a n c e of the equations of motion (3.2): like 2 ,the right hand sides of Eqs. (3.2) liecoiiie frame-independent functions of P and the external fields q,,(Q,t ) . Since H is given (up t o syninietrization) by the classical function JV of Q , P and t.: arid the transformation foriiiulae for Q and P are tlie wine as in the classical case, we also expect, in analogy to (2.19): t.lie transforination foriiiula H ’ ( t ) = H ( t ) f R P ( ! )* +7 1% W 17’ (3.12) for the Hamiltonians to hold true. This equation guarantees the e q u i v a l e n c e of the *‘unpriiiied‘’and “prinied” equations of niotion. To show this, it is sufficient t o prove froni (3.2) and (9.12) that Q‘ and P‘ as given by (3.9) satisfy the “priiiied” versioii 89 Galilei Cowtriance Does Kot Imply Minimal Electromagnetic Coupling of the equations of inotion (3.2). Now, indeed, (3.1), (3.9), (3.10) and (3.12) imply i[H’,Qi] = i [ ( H + 2 RJ‘Lvk), C &jQi] Ll =2 &jQj + vi = QI, j j Forin invariance and equivalence together again constitute Galilei c o v a r i a n c e , the physical ineaning of which is essentially the same as described above for the case of classical mechanics. Tinie evolution operators in the “primed” frame niay be defined explicitly by V ‘ ( t )= V ( t )e ((RP,* Y f? 2 v1 I ) . (3.13) Inscrting the initial values of Q’ and P’, + + Q’(0)= Qh = RQ, a, P’(0) = Pi = RP, mu, as following froin (3.9), and using the first one (for t = 0) of the equations eiP(f)*b Q ( t ) e-iP(0.b = Q ( t ) + 6 , e i Q ( 0 . C p ( t )e - i Q ( W = p ( t ) - c which follow froin (3.1), we easily find V ‘ ( t )Q;V’*(t) = HQ(t) ~t u = Q’(1) and V’(t)Pi V’*(t) = RP(t) mu = P‘(t). Moreover, V’(0)= 1, and differentiation of (3.13) yields (3.14) (3.15) + + + P(t)= iH’(t) V’(1)= iV’(t) H i ( t ) , by ( S . 7 ) and (3.12) and since, as in the “unprimed” frame, H’(t) V ’ ( t )= V’(1)H $ ( t ) . Starting froin (3.13) we have thus derived, a t least formally, Assumptions a and b for the “primed” frame from their “unpriined” counterparts. It is clear, besides this, that due t o form invariance any rigorous proof of these assumptions will apply anyhow to both frames. A concrete example is provided by the “quantized” version of the theory considered in Section 2. Froin (2.26) we obtain the Hainiltonian 1 (3.16) H ( t ) = (P - W Q , t ) ) @ - l ( Q , t ) (P - eA(Q, t ) ) q ( Q , 1 ) . - - + No additional syinnietrization is needed here, due to the symmetric appearance of Eq. (2.%)l2). With this Hainiltonian, R simple calculation, using (3.1) and the identity (3.17) 12) However, there exist other symmetrized versions of the same function X , e.g., 1 Z= { ( p- eA) ( ( p - eA) ( @ - I - (p - eA)) (p eA)} +- ep. 4 - em-’) + . - Insert,iiipQ and P for q and p we now arrive, after some calciilat,ion with (3.1’7),at another Hamiltonisn 1 P@Z1(Q,t ) H(t) = H ( t ) -9 4 $ aQiOQk Substituting fi for H in (3.?) leaves unchanged the equation for Q but changes the equat,ion for P. Different symmetrizations of 2 may thus lead to different “quantized” versions of one and the same classical theory. - + - --. 90 K. KRAUS following from (3.1), yields the equations of motion (3.2) in the foriii 1 (@-l(Q,t ) . P P . @-l(Q,t ) ) , Q = -e@-l(Q, t ) . A ( Q , t ) + 1 Pi= - T ( P - e A ) . - . a#-1 2Qi + (3.18) (P - eA) @ - l . (P - eA) + (P - eA) .@-I. - (3.19) which correspond to syiniiietrized versions of the classical equations (2.25) and (2.27). (Eq. (3.18), by the way, may be solved for P in the foriii 1 Q @) e A 2 (@. Q which is a symmetrized analog of (2.24)). The Galilei transformation formula (3.12) is easily checked for the Haniiltonian (3.16) (or also for cf. footnote 12) by using Eqs. (2.21) and (3.9). The model considered is thus perfectly covariant under Galilei transformations, whereas JAUCH [l,21 and others [3, 41 arrived a t the conclusion that Galilei covariance forces H to be of the form (l.l),corresponding to 0 = 0 in our niodel (9.16). This apparent contradiction is related t o the unitary iiiipleiiientability of Galilei transformationsj whic*h will be discussed in the subsequent Section. + p=- + s(t), 4. Unitary Implementability of Galilei Transformations in Quantum Mechanics A celebrated theorem of von Neuniann (see, e.g., [2]) states that any two irreducible representations of the canonical commutation relations (3.1) are unitarily equivalent *3). I n our case this theoreiii iniplies the existence of unitary operatorn U , ( R , v . o )for each time t and each Galilei transformation (2.1) with t = 0 which implenient, for the given time t , the Galilei transforiiiations (3.9) of position and nioiiientiini : ut(R,V , a) Q ( t ) t’t(R, V , a) = Q’(t) = R Q ( t ) + vt + a, (4.1) U:(R, v , a) P ( t ) U,(R,v , a) = P’(t)= R P ( t ) + mu. Irreducibility implies that the operators U,(R,v , a) are unique up t o phase factors, whereas successive application of (4.1) for two Galilei transforniations shows that U , ( R ,v , a ) U,(S, w , b) and U,(RS,v Rw, u Rb) must be equal up t o a. lihasc factor. The operators Ut(R,v , a) thus provide, for each fixed t , a unitary ray representation of the inhomogeneous Galilei group (without tiiiie translations). For the particular cases of space translations, pure Galilei transforniations, and space rotations, Eqs. (4.1) are satisfied by the following explicit choices for Ut(R ,v , a): + + (4.2) with denoting equality up t o a phase factor. The first two definitions are iniiiiediately suggested by (3.15). I n the third equation, r = r ( R ) is a vector in tlie dircction of the rotation axis with length equal t o the angle of rotation, whereas L = L(t) = Q ( t ) x P ( t ) 13) Von Neumann’s theorem actoally applies to tlie canonical commutation relations in Weyl’s formulation which, although being more restrictive than (3.1)[l],are satisfied in the case considered here (cf. Eq. (3.15)). Gal ilei Covariance Does Not Imply Minimal Electromagnetic Coupling 91 is the angular momentum operator. For a general Galilei transforination we may exploit the ray representation property of U,(R, u, a) to define im Ut(R,v , a) = e T ’ O U t ( l , 0, a) U t ( l ,v, 0) Ut(R,0, 0). (4.3) With (4.2) and (4.3) the phases of U t ( R ,v , a) are fixed by convention, and a straightforward calculation (see, e.g., [4]) yields the inultiplication law im -(vRb-Rw.a) U,(R,v , a) U t ( S ,w , b ) = e 2 U,(RS,v + Rw, u + R b ) . (1.4) Eqs. (1.2) and the irreducibility of Q and P imply that the ray representation U,(R. v , a) is irreducible for each t. Moreover, the ray representations U,(R, v , a) for different values of t are unitarily equivalent14). I n fact, as L may also be written as Q- - Pt xP,(4.2) inqdies ( m l )1 that U J R ,u, a) results from Uo(R,u, a) by substituting Q = Q - - Pt for Qo and m i P for Po. Since the unitary operator V ( t )e - I p ” this also implies U t ( R ,v , a) _- i = z transforms Qo into Q and Po into P , i V ( t )e mpotUo(R,v , a) e Gap” V * ( t ) A closely related formula is Ut(R,V , a) = V ( t ) U,(IZ, V , a) V’*(t),. with V’(t)given by (3.13). Using the relation which follows from (4.1) for Uo(R,v , a), (4.G) is easily obtained froiii (4.5) and (3.13). For a free particle with i t 1 H=p2, V ( t ) = eSPO1 (4.8) 2m the ray representation U,(R, v , a) becomes time-independent, Ut(R, ~3 0) Uo(R,v , a), (4.9) as follows either from the explicit form of U,(R,u, a) (Eqs. (4.2) and (4.3)) or, even simpler, from (4.5) and (4.8). Vice versa, t o require that (4.1) can be satisfied with time-independent operators U,(R,v , a) leaves the free-particle case (4.8) as the only possibility. I n fact, since (4.1) fixes U t ( R ,u, a) up t o phase factors, U,(l, 0, a), U t ( l ,u. 0) and U t ( R ,0, 0) inust coincide up t o phase factors eia(OJ), e@(”J),and ev’(r*f), respectively, with the operators defined by (4.2). They may thus be made time-independent only if it is possible t o choose the phases a , and y so that P * a - &(a,1 ) = 0 (mQ - P - Pt) - u L .r - j ( r ,t ) = 0 for all a, + @(u,t ) = o for all u, and 1’) for all r . It suffices, therefore, to verify (4.4) for the particularly simple case t = 0. K. K R A U S 92 These conditions can be satisfied only if with suitable c-number vectors c(t), d(t), and e(t). If inserted into i=QxP +QxP, this leads to d x P - C X Q= e , which is inconsistent with (3.1) unless c = d = e = 0. Thus, indeed, (4.9) forces the equations of motion t o be a 1 (4.10) P=O, Q=-P, rn as for a free particlels). Another typical difference between a free particle and a particle interacting with external fields is found by extending the previous discussion to include the time translations x-+ x, = x, t - t t , = 1 T. (4.11) Denoting a pure time translation (4.11) by {T} and a Galilei transformation (2.1) with T = 0 by { R , u, a}, we obtain from (2.1) the multiplication law (4.12) {R,u, a>{TI = {TI {R,u, a 4- v.1- + The transforinations of Q and P corresponding t o (4.11) are given by (4.13) Q ( 1 ) + Q,(t) = Q(l - T), P ( t )-+ P,(t) = P(t - 7 ) . By von Neumann’s theorem, the transforinations (4.13) are also unitarily implementable, Q2@) = UTW QW Ut(r), P2(t)= U f ( t )P ( t ) ut (T)2 and (3.7) implies that U , ( t ) = V ( t ) V*(t - T) is n possible choice. This leads t o (4.14) (4.15) + U,(t,) ul-71(7.2) = L’&l T.). Moreover, a straightforward calculation with (4.4),(4.5) and (1.7) yields (4.16) UdR, u, 4 VdT) = UdT) Ut-,(R, u, a VT). For a free particle V ( t )is a one-parameter group, so that (4.17) + U&) = V ( T ) lwcoiiies time-independent, and provides a representation of the Abelian group of time translations (4.11). In this case Eq. (4.17) reads U @ , u, 0 ) V ( T ) = V ( T ) U J R , u, a + VT), 1 Eqs. (4.10) do no lead directly to (4.8) but only to H = -Pz + U(t), with an arbitrary 2m c-number function U ( t ) . However, as U ( t ) does not affect physical statements like (4.1@),it may be dropped altogether, or removed by a gauge transformation since (3.12) impliefi that U ( t )behaves uiidrr Galilei transformations like an electric potential v. Is) Galilei Covariance Does Xot Imply Minimal Electromagnetic Coupling 93 which corresponds to the group multiplication law (4.12), thus indicating that Uo(R,v . a ) and V ( t ) generate a unitary ray representation of the inhomogeneous Galilei group including time translations. Although still properly reflecting the structure of this group, the formulae (4.16) and (4.17) cannot lead t o a ray representation in the presence of external fields since this would require, among other things, that (4.9) could be satisfied by a suitable choice of phase factors of U , ( R ,u, a ) . This, clearly, indicates the breaking of Galilei invariance by the external fields. Moreover, Eqs. (4.4) and (4.lf.i) show that this symmetry breaking affects in a very different way and to a very different degree the “kinematic” (t = 0) and the “dynamic” (= time translation) subgroup of the inhoniogeneous Galilei group, respectively. The reason for this is simple. Whereas “kinematic” Galilei transformations of the basic variables Q and P for fixed tare always of the form (3.9) independent of the presence and the particular kind of external fields, the latter are of decisive importance for the time translations, since the functional connections between variables Q and P at different times are determined by the equations of motion (3.2). 5. SchrSdinger Picture Although the Heisenberg picture used so far has considerable conceptual and formal advantages, we shall also give a short account a t least of the Schrijdinger picture, since the latter has been used in previous discussions of Galilei covariance [l,21. The formal simplicity of the Heisenberg picture and its close correspondence t o classical mechanics arise froin the convention of describing a given (pure) state of the particle by a Hilbert space vector f which is t h e s a m e in all frames of reference. Accordingly, Galilei transformation formulae for Heisenberg picture operators like, e.g., (3.9) or the first equation of (3.10), are of exactly the same form as the corresponding classical equations (2.2), (2.3), or (2.12), and in fact follow immediately from them since expectation values in any given state are required t o transform like the corresponding classical quantities. This simplicity of the formalism and, in particular, of Galilei transformation formulae, is lost in the Schrodinger picture. Assume the state of the particle, i.e., the Heisenberg state vector f , t o be kept fixed. I n a given ((‘unprimed’’)frame of reference, this state is now described by the time-dependent family of Schradinger state vectors fs@)= Y*(t)f , (5.1) whereas the Schrodinger picture operators Os(t) are related to the corresponding Heisenberg operators O(t) by O,(t) = V*(t) O(t) V ( t ) . I n particular, by (3.7), the position and momentum operators Qdt) Qo, P s ( ~ ) Po (3.2) (5.3) become time-independent whereas, e.g., the Hamiltonian H&) (cf. Eq. (3.G)), or the velocity operator Qdt)= ; [ H , s ( ~ 0) ,0 1 , (5.4) in general are time-dependent also in the Schrodinger picture. The transition t o another (“primed”) inertial frame by a Galilei transformation (2.1) with z = 0 implies, in case u 0, also the transition t o another Schrodinger picture (indicated by the suffix S’),with vectors f s 4 ) = V’*(t)f (5.5) + K. G A U S 94 describing the given state, and operators O&t) = V’*(t)O’(t) V’(t) (5.6) replacing the Heisenberg operators O‘(t) referring to the “primed” frame. From (5.6) we obtain, e.g., Q$,(t)= V’*(t)Q’(t) V’(t)3 QA, Pi.*(t)G Pb. (5.7) H i 4 ) = JKym(Qb, Pb. t ) (5.8) (analogous t o ( X G ) ) , and o$*(t) = i [ H k ( t ) ,Q;]. (5.9) The use of different Schrodinger picture in fraiiies with non-vanishing relative velocity v is unavoidable, since form invariance of the theory requires that Heisenberg and Schrodinger picture are connected by (5.1) and (5.2) in all reference frames whereas, by (3.13), the time evolution operators V ( t ) and V’(t) are different unless v = 0. The d i f f e r e n t Schrodinger state vectors (5.1) and (5.5) describing thesamestatein frames nioring relative to each other are connected, in virtue of (3.13)) by the forinulal6) -i 21 RP,~v+--v’ 1 (5.10) I&). fs4) =e ( This frame-dependence of the Schrodinger state vectors also leads to transformation forinulae for Schrodinger picture operators which are more complicated and less intuitively appealing than the corresponding formulae in the Heisenberg picture. To illustrate this by examples, consider first Eq. (3.12) for the Heisenberg picture Hamiltonians. If combined with (3.13), this formula implies m ~ $ , ( t= ) e--iR“,.vlH s(t ) eiRPo.vt RP, v - v2, (5.11) 2 since + . + V’*(t)P ( t ) V’(t) = V*(t)P ( t ) V ( t )= P o . Hiniilarly, the first equation of (3.9) implies &(t) = e--iRPo.vl ds(t)eiRPn.vt + V (5.12) for the Schrodinger picture velocity operators. Finally, one obtains from the first equation of (3.8) with the help of (3.15) the expected formula (cf. (3.14)) Q’0 -- e-iRP,wt ( R Q , vt a) eiRP*.”l = R Q , a (5.13) + + + for position operators. Clearly the lack of correspondence between Eqs. (5.11), (5.12), and (5.13) and their classical counterparts (2.19), (2.3)) and (2.2) is due t o the fact that. the former lead to the classical transformation formulae for expectation values only if combined with Eq. (5.10) for the state vectors. This lack of correspondence is likely t o lead to mistakes - except in simple cases like (6.13) where the absence of ut on the r.h.s. is almost obvious - if transformation formulae for Schrodinger operators are guessed rather than derived. (An example for this is Eq. (6.11) in Section 6.) The Galilei transformations (with z = 0) of position and momentum operators are unitarily implemented also in the Schrodinger picture. I n fact, Eqs. (4.1) for t = 0 lead to 0; = U,*(R,v , a) QoUo(R, v , a) , (5.14) PA = U,*(R,v , a) P,U,(R, v , a). Alternatively, (5.14: also follows from Eqs. (3.7), the corresponding ‘‘primed” equations, and (4.6). le) An even more coniplicated fonnula is obtained if r + 0. Galilei Covariance Does Kot Iniply ithima1 Electromagnetic Coupling 6. Derivations of Minimal Electromagnetic Coupling in Quantum Mechanics : A Critical Review Quantum niechanics of a particle interacting with external fields is Galilei covariant, as shown in Section 3, provided only the interaction Lagrangian satiefies the invariance requirenient (2.8). This still leaves considerable freedom for the construction of such lagrangians; e.g., (2.33) niay be generalized t o tensor fields of higher rank. Additional postulates, going beyond mere Galilei covariance, are thus necessary if one wants t o single out iiiore specific interactions. A requirement of this type, suggested by Refs. [l] t o [4), is the assuinption that, in addition t o and simultaneously with the Galilei transformations (3.9) of position and momentum, the corresponding transforniation (3.10) of velocity is also unitarily implementable, so that there are unitary operators U,(R,u, a) which, besides (Al), also satisfy + U:(R, u, a) o(t)U 1 ( Ru, , a) = @ ( t ) = R d ( t ) u. (6.1) (We now return t o the Heisenberg picture.) h’ote that (6.1) is n o t implied by Galilei covariance. and in particular is not necessary to guarantee the transforniation formula (3.10) for the velocity, which in fact holds true in arbitrary theories of the type discussed in Section 3. (A corresponding remark applies to Eq. (6.12) below.) Since U , ( R ,u, a) is already fixed by (4.1) up to a phase factor, Eqs. (4.2) may still be assumed to hold true. For a free particle we have that (6.1) follows trivially from (4.1). Vice versa, (4.1) and (6.1) toget,her imply (6.2). Kainely, by (4.1) and (6.1), BO A @ )= P ( t ) - m&t) df. satisfies U:(R, U ,a) A ( t ) U,(R,U, U ) = R d ( t ) . (G.3) Using (4.2). we conclude from (6.3) for R = 1, u = 0 that A commutes with P ;then (ti.3) for R = 1, a = 0 implies that A also ~oiiiiiiiiteswith Q. Hence A iiiust be a c-nuiiiber. This, however, is inconipatible with (6.3) for R =i= 1 unless A G 0. Together with (3.1) and (3.2). Eq. (6.2) yields 1 Zm so that H - - P2 comniutes with Q, and therefore must be a function of Q and 1. T h u s (6.2) also implies 1 H =2na P2 + V ( Q ,t ) , (6.4) as in classical mechanics (cf. Eq. (2.18)). The close correspondence between this reasoning and the discussion in Section 2 is realized by noticing that, provided (4.1) is true, the additional requirement (6.1) is satisfied if and only if the functional fornt of the first half of Eqs. (3.2), Q = WQ,P,t ) , (6.5) is frame-independent. I n fact, (6.5), (6.1) and (4.1) imply that the “primed” variables K. m A U S 96 satisfy Q’= U:(R,V, U ) Y ( QP, , 1 ) U , ( R , v ,U ) = Y ( Q ’ , P’, t ) with the same function Y; vice versa, (4.1) and frame-independence of Y yield (6.1). Because, therefore, condition (6.1) for arbitrary Galilei transforniations (with 7 = 0) excludes any velocity-dependent interaction, a Haiiiiltonian like (1.l) can be consistent with, or derived froin, Eq. (6.1) only if the validity of the latter is restrict,ed t o particular Galilei transformation^^^). Indeed, (6.1) is postulated in Refs. [3, 41 for instantaneous Galilei transforinations only, as given by (2.28), with the same value o f t in (2.28) which also occurs in (6.1). For such transforinations, Eq. (0.3) reads UfAU,=A, with eiQ’mu U , = ~ ~ v( , 1-vt) , by (4.2) and (4.3). Thus A coniinutes with Q , and therefore must be a function e A ( Q , t) of Q and t only. Hence we obtain * 1 Q =(P - eA(Q, t ) ) (6.6) m which, if combined with (3.2), inmediately leads to the Hainiltonian (1.1).(See the above derivation of (6.4) from (6.2), or Refs. [l]t o [4]). Originally this argument wag formulated by JAUCH [l, 21 in the Schrodinger picture. If translated by means of (5.2), (5.6) and (4.6) into that picture, (6.1) reads U,*(R,V , 4 dsv, Uo(& v , 4 = d i w (6.i) For the particular transformations (2.28), Eq. (6.7) coiiibined with (5.12) leads t o u,*&(t) U , = O&t) = e- iP,.vt ds(qeiPo.ut + V ? (6.8) where u0 - uo( 1, v , -vt) = e i Q o meiP& by (4.1) and (4.3) for t = 0. On the other hand, (5.14) implies U,*PoUo= Pb = Po mv. (6.9) Multiplication of (6.8) and (6.9) by eiPo’mvfroin the left and by e-ipo’av froiii the right and subtraction yields + e--iQo,mu (p0 - mQs(t)) eiQo.mv = P, - m o s ( t ) . This implies 1 Qs(Q= ; (Po - eA(Qo9 0) (6.10) which, a s excepted, is just the Schrodinger picture version of (G.6). Actually, however, the argument presented in Refs. [l,21 is somewhat different. Eq. (6.7) is postulated there t o hold true for p u r e Galilei transforinations, and is 17) By the way, the Hamiltonian (1.1) is already excluded if (6.1) is postulated for space translatiom only. For (1.1) implies A = eA(Q,t ) , which contradicts Eq. (6.3) for space translations unless A = A ( t ) ;such vector potentials, however, can be removed by a gauge transformation. This fact was overlooked by JAUCH;actually, in [2], the second of Eqs. (13-51) (for u = 0) implies that a,. as defined by Eq. (13-57) is independent of Q . Onlilei Covariance Does Xot Imply 3Linimnl Electromagnetic Coupling 97 coinbined with the velocity transforination foriiiula @&) = + (6.11) which, although apparently obvious, is definitely wrong (cf. Eq. (5.12)). One thus obtains U,*(1,u>0) Odl) Uo(L u, 0) = dsco + ¶J with U0(1,u, 0) = eiQ0.m” from (4.2). Together with the iiioinentuiii transforination foriiiula (Ll-l), this yields (6.10) as above. I n Refs. [l,21 one is thus led t o the wrong impression that the Hamiltonian (1.1) is obtained, via (6.10), from postulating (6.7) for pure rather than for “instantaneous” Galilei transformations. As mistakes like (6.11) are easily avoided in the Heisenberg picture, this illustrates once more the advantages of the latter. The given derivation of the particular Hainiltonian (l.l), however, is not very convincing froni a physical point of view. The principle of Galilean relativity states that all inertial frames are physically equivalent. Thus, a priori, all Galilei transformations (2.1) leading from a given fraine t o other frames are also equivalent. This doex not mean that different types of Galilei transformations cannot enter a Galilei covariant theory in a formally different way. I n fact, the “dynaniic” subgroup of tiine translations and the “kinematic” subgroup of transformations (2.1) with z = 0 are described quite differently in quantum mechanics, as indicated and explained in Section 4. (The situation in classical iiiechanics is similar.) However, nobody a s yet has given a convincing argument, according to which the particular subgroup (2.28) of the ”kinematic” Galilei group should play a distinguished r61e in quantum or classical mechanics. To require that Eq. (6.1) is satisfied just for this subgroup must thus be considered a purely formal rather than a physically motivated postulate. Or, t o express i t differently : If Hamiltonians like, e.g., (3.16) were t o be rejected becanse they violate (6.1) for “instantaneous” Galilei transformations, one should as well reject the Hamiltonian (1.1) because it violates (6.1) for space translations (cf. footnote 17). Accepting this, one still seems t o have the choice of either postulating (6.1) for the whole inhoniogeneous Galilei groupie) or dropping (6.1) altogether. However, the second alternative is so much more attractive that, t o our opinion, there is no choice a t all. First, a postulate like (6.1) is quite contrary t o the spirit of quantum mechanics with external fieldsl9). The unitary implementability (cf. Eqs. (4.1) and (4.14)) of arhitrary Galilei transformations of Q and P follows from the fraine-independence of the canonical coniniutation relations (3.1). The variables Q and P, on the other hand, nre specified as functions of Q,P and t by the equations of motion ( 3 4 , which are expected t o be - and in fact inust be - frame-dependent in the presence of external fields. From this point of view it does not appear very natural, t o say the least, if (4.1) is extended t o hold true, in the foriii (G.l), also for velocity operators. This is especially obvious if (6.1) is translated into the equivalent statement (cf. (6.5))that the first one of the equations of motion (3.2) takes a frame-independent form. Secondly, if (6.1) is postulated t o be true, thenthe intrinsic symmetry of the equations of motion (3.2) strongly suggest t o require U,*(R,u, a) k(t)U,(R,u, a) = Py)= RP(t) (6.12) 18) AS (6.1) already implies (G.‘?), the extension of (6.1) to time translations imposes no arlditional restrictions. 18) This criticism also applies if (6.1)i8 postulated for particular Galilei transfwmations only. K.m A U S 9s as well. Together with (Ll), however, this excludes anything except tlie free particle case (4.10)?since coiiipnrison with (6.3) shows that (G.12) implies P = 0, wheieas (G.l) leads to ( 0 . 2 ) 2 0 ) . After all. we feel that a condition like (G.l), which so severely restricts the ”natural” franie-dependence of dynamics in external fields, should not be imposed unless this is absolutely necessary, e.g.. for covariance reasons. However, snch niotivation of (6.1) is inissing. That, nevertheless, (6.1) iiiay be satisfied in particular cases for all or at least for soiiie Galilei transformations, does not mean that the corresponding Haniiltonians (G.4) or (1.1) are iiiore satisfactory theoretically, but rather that they are just simpler than others. For, obviously, inore coinplicated Haniiltonians yield iiiore coniplicated - and thus more strongly fraine-dependent - equations of motion (3.2). Last hnt not least, the strongest argument against postulating (6.1) in full generality is that this postulate would exclude the Haiiiiltonian (1.1)which is realized in nature. The nioiiientiiiii P is not always an observable, since it is not gauge-invariant in external elcctromagnetic fields, or in tlie particular theory discussed in Section 3 which also admits the usual gauge transformations of A and 9. I n such cases it might thus he interesting to ask whether there are unitary operators TVt(R.u, a) which implement the Galilei transformations of the o b s e r v a b l e s Q and Q a t time t but whose action on P is not specified a priori21). It is sufficient, in order t o answer this equation in the negative, to consider the particular Hainiltonian (1.1).I n this case Q is given hy (6.(;), and using (3.17) wc obtain for the coinmutntors of velocity coniponeiits e %Ai P A , (6.13) Bi,(Q, t ) = i[&t), & t ) ] = - (??a2 \SQ, df. -). i3Qi Assuniing that unitary operators W t ( R ,u, a) with the above-mentioned properties esiht. at least for u = 0, we arrive a t Bip(RQ + U.t) = WF(R,0, a) &(Q, t ) IVt(R, 0, a) 41 = s RilBkn8i[&Qm] = qQi. = 1in 2 Ri8tmBrnt(Q, t ) . lm Froin this we conclnde, setting R = 1, that Eu is independent of Q. Then. setting R x 1and taking into account the antisynimetrj- of Bik, we arrive a t Bi, = 0 which, by (6.13), ineaiis ‘ i x A E 0. (0.14) Tlius A is a pure gradient, which can bc innde t o vanish by a suitable gauge transforination. Sontrivial electroinagiietic interactions are thus incompatible with the siniultnneoiis validity of (6.1)and (4.1) even if the latter is restricted t o position operators only. Note. uioreover, that in order to derive (6.14) i t waq sufficient to consider Euclidean transformations (u = 0) only. 20) Eqs. (Ll) and (G.12) are automatically satisfied, and thus the particle is free, if the eqnntiom of niotion (3.2) are invariant under Galilei transformations. Moreover, (G.1) and (G.11) follow trivially by differentiating (4.1) with respect to t if U,(R,Y . a) does not depend on 1. This proves once more that time-independent operators U t ( R .a’, a) satisfying (4.1) exist for a free particle only. 21) At first sight it appears as if JAUCH [l] XVRS postulating the existence of such operators H’, rather than of tlie U’s discussed before, since Eq. (1.9) of [l] - which corresponds to the second one of Eqs. (4.1) here - is derived rather than postulated there. Actually. however, this “derivation” of Eq. (9.9) in [l] just aniounts to replacing it by mother, equally strong but more coiiiplicnted nrsuinption. Galeilei Covariance Does Kot Imply Minimal Electromagnetic Coupling 99 Suiiiiiiarizing the present discussion, we feel it is not unfair to the authors of Refs. [I] to [4] to state that their attempt of deriving the minimal coupling Hainiltonian (1.1) froin sufficiently motivated invariance postulates has failed. Consequently we need not discuss in detail the recent generalization [5] of this atteiiipt t o particles with spin. Actually the comparison of JAUCH’S work [l,21 with later papers [3, 41 reveals a slightly but noticeably different attitude towards the problem in question. Jauch, obviously, was quite aware of the fact 6hat the simultaneous validity of the implementability conditions (4.1) and (6.1) is a more or less plausible additional postulate rather than a self-evident covariance requirement. A success like the derivation of the Hamiltonian (1.1) from this postulate would indeed have provided a strong argument in favour of the latter. However, Jauch apparently did not notice that, actually, this postulate is satisfied by (1.1) for particular Galilei transformations only, so that the same postulate if also applied, e.g., to translations would further reduce (1.1) to the form (6.4). PIRON [3] and I,~VT-LEBLOND [4], in contrast t o this, seem t o consider condition (6.1) as selfevident. This condition would indeed follow immediately froni the statement that the “new Haniiltonian ... is just the old one with the new variables”, which in [3] is claimed to hold true in classical mechanics, and thus most likely is taken for granted also in quantum mechanics. L~w-LEBLOND, finally, claims that the condition (6.1) for instantaneous Galilei transformations is implied by the velocity transformation law (3.10) or. in his words, by the requirement that “the Galilean law of velocity addition ieitiains valid in the presence of external forces” ([4], p. 284). 7. Criticism of Derivations of Minimal Electromagnetic Coupling in Classical Mechanics The proposed translations [3, 4, 61 of Jauch’s derivation of the minimal coupling Hamiltonian (1.1) into the formalism of classical mechanics are even less convincing than the original argument. The loopholes in these reasonings are easily discovered. As shown in Section 2, the functional connection (7.1) P = II(q, 4, t ) between q, Q, and p implied by (2.11) is frame-dependent unless &‘ is of the very particular form (2.18). Accordingly the proper behaviour (cf. (2.12)) + p = H(q, Q,t ) -+ p‘ = ZI’(q‘, q’, t ) = Rp mu (7.2) of niotnentum under arbitrary Galilei transforniations (2.1) with z = 0 results, in of the functional form of (7.1) and a general, from the interplay of a change II+ change (cf. (2.2), (2.3)) n‘ q - f q‘ = Rq of the arguments entering -+ vt + a, + q-+ 4‘ = Rq v II and II’.If solved for Q, Eq. (7.1) leads to Q = W q , P,4 (7.3) (7.4) which, as easily shown, is just the first half of Hamilton’s equations of motion (2.14). LI~Y-LEBLOND [4, 61 uses, for the particular case of an “instantaneous” Galilei transformation (2.28) for which (7.3) reads q’ = q, Q’ = 4‘ instead of (7.2) the formula +v , + (7.5) + (7.6) p’ = II(q’,q’, t ) = p mu = n(q,q, t ) mu, with t h e s a m e function II in both frames. (See Eq. (5.31) in [GI and the equation following Eqs. (18a) and (18b) in [4]. The function lT is called p , and the argument t is omitted in Refs. [1,61.) Eqs. (7.5) and (7.6) immediately lead t o p = n ( q ,4,t) = mq 7. + eA(q, t ) . (7.7) K. &taus 100 Therefore (7.4) becomes 1 Q = (p - eA(q, 0 ) m from which one easily obtains [6] the desired resultsa) 1 W q , P,t ) = 2m (P - eA(q, t,>% q ( q , t ) . + (7.8) However, the uae of Eq. (7.6) is justified only if one arbitrarily assumes that the function Lf in (7.1) is either totally frame-independent (which, however, would lead to (2.18) instead of (7.8))) or at least invariant under "instantaneous" Galilei transformations (as (7.7) indeed is). Since the function Y in (7.4), i.e., the first half of Hamilton’s equations (2.14), has the same invariance properties=) as l7,the close correspondence between LBvy-Leblond’s implicit assumption, hidden in Eq. (7.6), and Jauch’s condition (6.1) becomes obvious now. But whereas in quantum mechanics Eq. (Ll), although hardly justified physically, is at least a formally well-defined requirement, the uncommented use of the formula (7.6) in place of the correct momentum transformation law (7.2) in Refs. [4] and [6] appears much more like a plain mistake than like a deliberate additional assumption. In any case, however, the criticism of Eq. (6.1) in Section 6 applies equally well to any assumption which translates (6.1), more or less literally, into classical mechanics. The flaw in €’IRON’S argument [3] is even more conspicuous. This author, as already quoted, asserts that after an arbitrary Galilei transformation the new Hamilton function “is just the old one with the new variables” ([3], p. 306). Such complete frame-independence of the function % . ? would clearly imply frame-independence of the functions Y in (7.4) and Il in (7.1), so that it is not surprising that Piron is also able to derive the desired result (7.8). For a theory ofthe type considered here, however, Piron’e assertion is simply wrong unless there are no interactions with external fields at all. Namely, according to Section 2, frame-independence of Il already implies that &‘ is of the form (2.18), whereas the latter is frame-independent only if V V = 0, i.e., for a free particle. Actually the Hamilton function (7.8), in contrast to the corresponding function ZZ as given by (7.7), is not even invariant under “instantaneous” Galilei transformat.ions, since in this case Eqs. (2.21) (with 8 = 0) and (7.5) yield A‘(q’, t ) = 4% t ) = A&‘, t ) but + + y’(q‘, 4 = v(q,t ) 4% 4 = d q ’ , t ) AM’, t ) 9 dq‘, 0 . Because form invariance of 2 under “instantaneous” Galilei transformations is used explicitly in the derivation of (7.8) in Ref. [3], the reasoning in that paper is thus seen to be contradictory. L~VY-LEBLOND [6] has also tried to derive minimal electromagnetic coupling from “a suitably formulated principle of Lorentz invariance”. Because this attempt is so similar to the one just described, and in particular is based on eimilar hidden and physically unmotivated aesumptions, we need not discuss it in detail here. Such discussion, anyhow, would go beyond the scope of the present paper. The last remark *a) By the way, the assertion in [4] (p. 263) that the Hamilton function (7.8) does not have the expected transformation behaviour (2.19) is erroneous. m) Compare Eq. (6) in [S]. Note also that the change of 4 under an infinitesimal Galilei transformation is given by the Poisson bracket of Y with the corresponding generator (cf., e.g., Eqs. (9) and (16) in [6]) only if the functional form of Y is not changed by the tramformation. Gal ilei Covariance Does Not Imply Minimal Electromagnetic Coupling 101 alm applies to the recent work of HOOGLAND [8], who discusses quantum mechanics with constant external fields and the corresponding invariance (rather t.han covariance) under suitable subgroups of Galilei transformations. The idea of this paper originated from a conversation with Prof. H. STEINWEDEL. I want to thank him and Dr. G. REENTSfor useful discussions. A kind letter from Prof. J.-M. MVY-LEBLOND, although expressing partial disagreement with the present paper. is also gratefully acknowledged. The reaction of Prof. C. €’IRON was leas encouraging but, in a certain senge, very informative also. References [l] .J. 31. JAUCH, Helv. Phys. Acta 37, 284-292 (1964). [‘?I .J. 31. JAUCH,Foundations of Quantum Mechanics. Addison-Wesley Publ. a m p . , Reading, Massachusetts 1968. [3] C. &ON, Found. Phys. 2, 287-314 (1972). [4] J.-$1.L~vY-LEBLOND, Galilei Group and Galilean Invariance. In: E. M. LOEBL(Ed.), Group Theory and Its Applications. Vol. 11, p. 221-299. Academic Press, New York, London 1971. [5] E.CELEQE~TL, L. LUSANNA, and E. SORACE, Nuovo Cimento 81 A, 89-99 (1976). [6] J.-11. L~vY-LEBLOND, Ann. Phys. N.Y. 57, 481-495 (1970). [7] H. STEINWEDEL,Fortachr. P h p . 24, 211-236 (1976). J. Phys. Lond. A 11, 797-804 (1978). [8] H. HOOOLAND, Bei der Redaktion eingegangen am 16.Juni 1978, revidiertm Manuskript eingegengen am 6. April 1979. Anachr. d. Verf.: Prof. Dr. K. KRAUS Physikalisches Institut der Universitat Wiirzburg Am Hubland D-8700 Wiirzburg

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