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Galilei Covariance Does Not Imply Minimal Electromagnetic Coupling.

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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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