Is Einstein's equivalence principle valid for a quantum particle?

5 Pages·0.11 MB·English

by  Andrzej Herdegen

Is Einstein’s equivalence principle valid for a quantum particle?
∗
Andrzej Herdegen
Physics Department, University College Cork, Ireland
and
Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Krak´ow, Poland
†
Jaros law Wawrzycki
Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Krak´ow, Poland
stein (“strong”) equivalence principle is uniquely deter-
Einstein’sequivalenceprincipleinclassicalphysicsisarule
mined by one mass parameter. Thus the dynamics is
stating that the effect of gravitation is locally equivalent to
mass-dependent ((1) not true), but there is no room for
theacceleration ofan observer. Theprincipledeterminesthe
3 independent inertial and gravitational masses (in other
motion oftest particles uniquely(moduloverybroad general
0 words: the masses are equal, (2) true ).
assumptions). We show that the same principle applied to a
0 We turn now to this case. In the literature various
quantum particle described by a wave function on a Newto-
2
nian gravitational background determines its motion with a opinions onthe status of the equivalence principle in the
n similar degree of uniqueness. quantumworldareexpressed[3],andvarious,sometimes
a
rather far removed from the original geometrical notion
J
of equivalence, ideas are proposed [4] (but see also the
0
final discussion). We think, however, that the extension
3
of the Einstein equivalence principle in the form stated
In this note we address one of the conceptual issues
2 above to the quantum case experiences no logical diffi-
arisingfromtheeffortstoreconcilequantumtheorywith
v culty, at least in the setting in which it has often been
1 gravitation,the question of the status of the equivalence
considered. We feel, therefore, that it may be of interest
2 principle for quantum matter. In classical physics the
to see the simplicity of its action in this setting. The
0 Einstein equivalence principle is a rule making one half
0 setting referred to consists of a structureless particle de-
of the universal interdependence of geometry and mat-
1 scribedbyawavefunctiononagravitationalbackground
ter, namely the influence of geometry on matter, more
1 of the nonrelativistic spacetime (we use this term reluc-
0 specific. It states that the effect of gravitation is locally
tantly: it is deeply rooted in the physicists’ jargon, but
/ indistinguishable from the effects arising from the accel-
c misleading; see below). This setting has been adopted
q eration of the observer [1]. Put differently, gravitational
by several authors addressing the issue of covariance or
- effects may be locally “transformed away” by an appro-
r equivalence [4–6]. Within the path-integral formalism
g priate choice of the reference system. This is the princi-
conclusions similar to ours were reached earlier for the
: pleusedbyEinsteinhimself;someauthorscallit“strong
v Feynamn propagator of a structureless particle by De
equivalence principle” [2]. We adopt this version of the
i Bi`evre [6]. Our derivation, however, needs less assump-
X equivalence principle in this note as we believe that it
tions, refers directly to the wave function, and has the
r hits the heart of the matter, other “equivalence princi-
a advantage of great simplicity, both conceptual and tech-
ples” (cf. below) being more accidental or secondary.
nical (see also the closing discussion).
Within Einstein’s gravitation theory one shows that
Thereasonforchoosingthenonrelativisticratherthan
the above equivalence principle implies (modulo some
Einsteinian spacetime is that we want to avoidthe com-
natural assumptions on the general nature of the equa-
plications arising from creation and annihilation of par-
tions of motion) that all test particles placed in this
ticles and their quantum field-theoretical description,
spacetime move along geodesics. This fact is often ex-
which has to replace (“first-quantized”) quantum me-
pressed in one of two ways: (1) that the motion of the
chanicsinthiscase(thereexistingnoconsistentrelativis-
particleismass-independent,or(2)thattheinertialmass
tic quantum mechanics). The adopted setting is, how-
of the particle is equal to its gravitational mass. These
ever,nontrivial enough and, in fact, contraryto the cus-
two statements are sometimes used interchangeably as
tomaryname,possessesageometricalstructure(Newton-
“weak equivalence principle” in the literature [2]. This
Cartan) interpretable in physical terms as a relativity
use of terminology is rather confusing, as the two state-
theory, but with Galilean rather than Lorentzian local
ments are logically independent. They happen to co-
inertial observers [7].
incide in the context of classical general relativity, but
Wecannowstatethemainclaimofthisnote. Consider
may diverge in another setting. This is, as we shall see
a quantum particle described by a wave function ψ in a
below, what happens in the quantum case. The quan-
geometrical background with the Newton-Cartan struc-
tum dynamics of a test particle following from the Ein-
ture. Assume that the probability density ofthe particle
1
is a scalar field. Then the Einstein equivalence principle (t′,x′i) is also a Galilean system if and only if it is re-
determines the motion of the particle. This motion is lated to (t,xi) by a transformationof the form:
not mass-independent, but the inertial and gravitational
massesare necessarilyequal(whichis whatone observes t′ =t+b, ~x′ =R~x+~a(t), (1)
in experiment, see Ref. [8]; we shall return to the exper-
where R is an orthogonal transformation and ~a(t) is
imental aspect of the equivalence principle in the con-
an arbitrary time-dependent translation. Let us denote
cluding discussion). The choice of the mass parameter
(t,~x)≡X, (t′,~x′)≡X′ and let us write the transforma-
is the only freedom of the equation. The equation it-
tionasX′ =rX. Thetwofieldsφandφ′ correlatedwith
self, when written in an appropriate coordinate system,
the two systems are then related by the transformation
is nothing else but the usual Schr¨odinger equation with
φ′(X′) = φ(X)−~a¨(t)·~x′ + arbitrary function of time.
the Newtonian potential term. We move now on to the
We choose the function to be zero and write the trans-
details.
formation law in the form
We start by giving a brief account of the Newton-
Cartan geometry. We shall not discuss the underly-
φ′(X)=φ(r−1X)−~a¨(t−b)·~x. (2)
ing axioms and the logical structure of this geome-
try, referring the reader to the existing literature [7],
The field φ is no longer a true scalar field, as with each
but rather summarize the resulting structure in simple
of the two systems a different gauge has been fixed.
terms. The Newton-Cartan geometry is defined on a
The Newton-Cartan spacetime is flat if and only if
four-dimensional differential manifold. This manifold is
there existsa Galileansystemin whichφ is a functionof
equippedwithanabsolutetimetdefiningthefoliationof
time only (and may be chosen equal to zero). The same
the spacetime by simultaneity hypersurfaces, a positive-
holdstheninallthoseGalileansystemswhicharerelated
definite metric on each of these hypersurfaces, and a
by a transformation from the Galilean group (a¨(t) = 0)
covariant derivative (affine connection) compatible with
tothefirstone(and,consequently,toeachother). These
these structures. However, as a metric on a hypersur-
special Galilean systems are called global inertial sys-
face is a form, there is no unique way of embedding it in
tems. If the spacetime is curved there are no global in-
the four dimensional manifold without additional struc-
ertial systems, but, as is evident from the transforma-
tures. This is how a gauge freedom in the choice of a
tionlaw (2) andthe formof the connection,for any cho-
four-dimensional metric arises. Nevertheless, both the
sen point of spacetime there exist Galilean systems in
metric properties on the hypersurfaces of constant time
whichconnectionvanishesinthis point. Therestrictions
as well as the compatible connection are unique (gauge-
ofthesesystemstoasmallneighborhoodofthispointare
independent). With respect to the thus defined connec-
relatedtoeachotherbyGalileantransformationsandare
tiontheleavesofconstanttimeareflat. Thenon-flatness
called local inertial systems. Both global (when they do
ofthe geometry reflects only the wayin whichthe leaves
exist) and local inertial systems have exactly the same
fit together to form the four-dimensional spacetime, and
physicalinterpretationintermsofspecialobserversasin
is encoded in one single scalar field φ. This field, how-
Einstein’s theory.
ever, is again non-uniquely determined by the connec-
The Newton-Cartan spacetime is the spacetime of the
tion, being subject to a gauge freedom. All this sounds
Newtonian world with gravitation. The field equation
rather more complicated than for a Lorentzian manifold
of the form: “the Ricci tensor of the connection = 0”
of Einstein’s theory of gravity, but now a great simpli-
turns out to be identical in a Galilean system with the
fication comes. In the Newton-Cartan geometry there
Laplace equationfor φ, andthe geodesicequation has in
existsaclassofprivilegedglobalcoordinatesystems,the
this system the form of the Newton’s second law for a
so-called Galilean coordinates. One of the coordinates
particle in the gravitational potential field φ. In parallel
in each of these systems is always the time coordinate t
with the Einstein theory of gravity the geodesic law of
up to a translation by a constant. The other will be de-
motion of classicaltest particles may be obtained by the
notedbyxi(i=1,2,3). Thespacepartofthecoordinate
application of the Einstein equivalence principle. And
basis is a Cartesian system with respect to the metric:
here again the two minor equivalence principles are true
∂/∂xi · ∂/∂xj = δ . Moreover, vectors ∂/∂xi are
ij in the classical case.
p(cid:0)arallel(cid:1)pr(cid:0)opagat(cid:1)ed by the connection, so th(cid:0)e cova(cid:1)riant The existence of the globalGalileansystems simplifies
derivative of (∂/∂t) gives the only nontrivial character-
greatly the investigation of the covariance of equations.
istic of the connection, and must be expressible in terms
To see whether an equation has a geometrical, indepen-
of φ. In fact, with each choice of a Galilean system a
dentofthe choiceofcoordinates,meaning,itissufficient
natural gauge of the field φ is chosen by the formulas:
tocheckwhetherithasthesameforminallGalileansys-
∇ (∂/∂t)ν = φν∇ t, where in the coordinate basis the
µ µ tems. If it has, writing its coordinate independent form
vector field φν is given by φ0 = 0, φi = ∂φ/∂xi . This
may pose some technical difficulties, but is possible. In
fixes the choice of φ up to an addition(cid:0)of an ar(cid:1)bitrary what follows we use the Galilean systems only.
function of time. If (t,xi) is a Galilean system, then
2
We are now prepared to place a quantum particle in withoutlossofgeneralitythata,b andc areconstant.
i ik
this Newtonian geometry. We assume that it is de- Consider now general Galilean transformations. For the
scribed by a wave function ψ(X), such that the corre- first two coefficients the covariance now demands that
lated “probability density” ρ(X) ≡ ψ¯ψ(X) is a scalar a = λa, R~b+2a~v = λ~b (where λ may be a function of
field: ρ′(X′) = ρ(X). We use this quantum mechanical the transformation). These conditions can be satisfied
language, but the argument is purely field theoretic in for all R and ~v only if a = 0 and~b = 0. The transfor-
spirit, and no a priori assumptions on the integration of mation of the third coefficient is now R R c = λc .
ij kl jl ik
probability need to be made. The scalar transformation This may be satisfied only if c = cδ . Rescaling the
ik ik
law of ρ does not determine the transformation law of equation by an appropriatephase factor one can assume
ψ, as it says nothing about the phase of ψ. Therefore, that c ≥ 0. However, if c = 0 the equation is at most
the problem to be solved is this: Can we ascribe in each ofthe firstorder,andthenconsiderationssimilartoours
Galilean system a phase to the ψ in such a way that a show that d = 0 and f~ = 0. This case is trivial. Thus
consistent transformation law of the form c > 0 and λ = 1. Now one writes the transformation of
the equation in full. From the invariance of the ∂ -term
ψ′(X)=e−iθ(r,X)ψ(r−1X) (3) one finds that d is a constant. The condition fortf~(X)
takes the form
wouldholdandψwouldsatisfyaforminvariantequation
in all those systems? f~(X)=R−1 f~(X′)−d~v−2ic∂~′θ(r,X′) .
Following standard assumptions we restrict consid-
h i
erations to the class of linear equations of second or-
Considering this condition for translations and rotations
der at most. Wishing to make use of the equivalence
one finds that Ref~(X) is an invariant vector field, thus
principle we first have to answer this question for ψ
Ref~(X)=0,andthend=ik,withrealk. Applying ∂~×
in flat space, with the restriction of coordinate sys-
totheimaginarypartoftheconditiononefindsthatalso
tems to inertial ones. One could make use of the stan-
∂~×Imf~(X) is an invariant field, so Imf~(X) = ∂~h(X).
dard quantum-mechanical arguments, which would pro-
At this point looking back to the equation we realize
duce the free particle Schr¨odinger equation with the
that what remains of this term may be got rid off by
well known transformation law of ψ consistent in the
a redefinition of the phase of ψ (by −(i/2c)h(X)). We
quantum-mechanical sense, as a projective unitary rep-
can assume then that f~(X)=0 and find that θ(r,X) =
resentation of the Galilean group. However, we think
−(k/2c)~v ·~x +θ˜(r,t). Finally, the condition for g(X)
that it is instructive to see how the same result follows
reads now
by purely geometrical arguments, without any use of a
Hilbertspace. Wesketchtheargumentbriefly. Thecoor-
dinatetransformationsarenowrestrictedtotheGalilean g(X)=g(X′)−(k2/4c)~v2+k∂t′θ˜(r,t′).
group,~a(t)=~vt+~ainEq.(1). Weassumethattheequa- Applying∂~tothisconditionandconsideringtranslations
tionforψ hasthe followingform(the sameinallinertial
androtationswefindthat∂~g vanishes,sog(X)=g(t)≡
systems):
G′(t). Here again we realize that redefining the phase of
a∂2+b ∂ ∂ +c ∂ ∂ +d∂ +f ∂ +g ψ =0, ψ (by (i/k)G(t)) we remove the remaining freedom of
t i i t ik i k t i i
the phase in ψ and the g term from the equation. (If
(cid:2) (cid:3)
where a(X), ..., g(X) are the same functions in each k = 0 then g is invariant, so g = const. We get the
coordinate system, and ψ transforms according to a law Helmholtz equationandscalartransformationlawfor ψ.
of the form (3) with θ to be determined. In the “un- This, being not a dynamic equation, we discard.) We
primed” version of this equation we substitute ψ(X) = finally get, with the standard notation of constants,
eiθ(r,X′)ψ′(X′) in accordance with Eq. (3), express the
differentiations in terms of the new variables X′ and di- i~∂ + ~2 ∂~2 ψ =0, θ(r,X)=−m~v·~x+ m~v2t,
vide the resulting equation by the phase factor function (cid:20) t 2m (cid:21) ~ 2~
eiθ(r,X′). In the resulting equationthe ratios of the coef-
which, of course, is the standard free particle theory.
ficient functions standing at the distinct differential op-
Einstein’s equivalence principle implies now that if
erators must be equal to the ratios of a(X′), ..., g(X′).
in the flat space we transform the Schr¨odinger equa-
It is easy to see that the transformations of the coeffi-
tion to all arbitrary Galilean systems (noninertial), then
cient functions at the secondorder operators remainun-
we can identify all local modifications to the equation
affectedbythephasefunctionθ. Consideringfirstspace-
which can appear in an arbitrary Galilean system in
time translations X′ = X +Y, Y = (b,~a), one finds, in
curvedspacetime. Weassumethetransformationlaw(3)
particular,thattheratiosofa(X),b (X),c (X)mustbe
i ik
and find that the transformed equation differs from the
equaltothoseofa(X+Y), b (X+Y), c (X+Y)forall
i ik
Schr¨odingerequationatmostbyadditionaltermsonthe
X and Y. It follows that rescaling the original equation
left-handsideoftheformiχ~·∂~ψ+Λψ,whereχ~ isreal. In
by an appropriate factor function of X one can assume
3
curvedspacetimetheEinsteinequivalenceprinciplegives equivalent to the connection, so the meaning is exactly
then the equation the same as in the classical case. The covariance condi-
~2 tion now simplifies to −(m/2)~a˙2(t)+~∂t′θ˜(r,t′)=0. In
i~∂ + ∂~2+iχ~(X)·∂~+Λ(X) ψ(X)=0, this way we finally obtain the equation
t
(cid:20) 2m (cid:21)
~2
i~∂ + ∂~2−mφ(t,~x) ψ(t,~x)=0, (4)
where χ~(X) and Λ(X) are now fields characterizing ge- (cid:20) t 2m (cid:21)
ometry. We assume that these fields are determined lo-
and the transformation exponents
cally by the geometry. Assuming the transformationlaw
oftheform(3)anddemandingthecovarianceoftheequa- θ(r,X)=−m~a˙(t−b)·~x+ m t~a˙2(τ −b)dτ. (5)
tion we find the condition ~ 2~Z
0
~2 Until now we have considered the relation between
χ~(X)=R−1 χ~′(X′)−~~a˙(t)− ∂~′θ(r,X′) .
(cid:20) m (cid:21) two coordinate systems only. Is the resulting structure,
the equation (4) and the transformation laws (2) and
Applying∂~×tothis equationwefindthat∂~×χ~ isavec- (5), consistent with the composition of transformations?
tor field, in particular it is a vector field with respect to That is, do we get the same result if we choose to break
rotations. Butφ,theonlycharacteristicofthegeometry, the transformation X 7→ X′ into two steps with an in-
is a scalar field with respect to rotations, so there is no termediate system on the way: X 7→ X′′ 7→ X′? The
localwayinwhichaχ~ givingrisetoanonzerovectorfield answer is that the two final gravitational potentials dif-
∂~×χ~ can be formed with it. Hence χ~(X)=∂~ξ(X). We fer in general by a time-dependent (~x-independent) ad-
observenow,thatthislongitudinalfieldmaybeabsorbed ditive term, while the two final wave functions differ by
intothephaseofψ (withtheappropriatemodificationof a time-dependent phase factor. This, however, poses no
Λ), so one can assume χ=0. The transformationcondi- difficulty. The difference in the potentials is consistent
tionthensimplifiestoθ(r,X)=−(m/~)~a˙(t−b)·~x+θ˜(r,t). with the freedom in their definition, while a time depen-
The covariance condition for the Λ term now takes the dent phase factor in the wave function does not change
form the state vector (in L2(R3,d3x)) at any time, and in the
equation induces only another change of φ by an ad-
Λ(X)=Λ′(X′)−m~a¨(t)·~x′−(m/2)~a˙2(t)+~∂t′θ˜(r,t′). dition of a function of time. We mention as an aside
that one could begin the whole analysis by classifying
Atthispointletus lookbackoncemoretothe flatspace
on group-theoreticalbasis all exponents θ(r,X) fulfilling
case and assume that X is an inertial system. Then
thisconsistencycondition. Suchaclassificationhasbeen
Λ(X)=0andwefindthattheadditionaltermsintheop-
achieved by one of us (J.W.) and will be published else-
eratoractingonψ′ arisingfromthenon-inertialityofthe
where.
systemX′arem~a¨(t′−b)·~x′+(m/2)~a˙2(t′−b)−~∂t′θ˜(r,t′). We have thus shown that Eq. (4) is uniquely deter-
We learn two things. First, the terms are real, so by the mined by Einstein’s equivalence principle. In particular,
equivalence principle Λ(X) is real in general. Second, a we have shown that the principle implies equality of in-
changeofcoordinatesproducesdefinitetermsuptolinear ertialand gravitationalmasses. The equation, of course,
order in ~x. The equivalence principle then implies that is standard, andhas been discussedmany times, but the
in the general case it should be possible by a change of derivation of its geometrical uniqueness within standard
coordinates to eliminate in the neighborhood of a given quantummechanicsisnew. Withinthepath-integralfor-
pointX0 termsindependentof,andlinearin~x−~x0. Put malism De Bi`evreobtained earlier similar results for the
differently, it should be possible to transform away the Feynman propagator of a particle in gravitational field.
valueandthefirstderivative∂~ ofΛ(X)atthispoint. Let His derivationis basedon anadditionalgeometricstruc-
us introduce Λ˜(X) by Λ(X)=−mφ(X)+Λ˜(X). Using tures (the Bargmann bundle over spacetime and an as-
Eq. (2) we find that the covariance condition now takes sociated vector bundle). The propagator is assumed to
the form havegeometricpropertieswithrespecttothisstructures.
Λ˜(X)=Λ˜′(X′)− m2~a˙2(t)+~∂t′θ˜(r,t′), Aqusatnhtuemcomnneecchtainonicsoifstnhoist efoxrpmliaciltismat wthiitshstthageesittanisdnarodt
quite obvious what are the correspondingrestrictions on
whichimplies∂~′Λ˜′(X′)=R∂~Λ˜(X). Itisnowclearthatif the wave function. However, they must amount at least
∂~Λ˜(X)6=0 at some point then it cannot be transformed to some restrictions on the transformation properties of
away. ThereforeΛ˜(X)=Λ˜(t), andmay be removedby a the wave function. On the other hand in our derivation
change of phase of ψ. Thus the unique solution for Λ is the phase of the wave function is completely free at the
Λ = −mφ. We now see the geometrical meaning of the start. The equivalence principle and the adjustment of
condition that the first derivative ∂~Λ may be removed thephaseyieldboththedynamicequationandthetrans-
at a point by a change of coordinates: this derivative is formation law in an extremely simple way.
4
Within standard quantum mechanics Kuchaˇr [5] has in the way. Measurements are done locally, which is the
derived the equation (in general covariant form) by operational foundation of the Einstein’s principle.
canonical quantization of the geodesic motion. (Where
in the process is the mass independence lost? It is, of
course,whenaftergoingovertotheHamiltonianformal-
ism, in which mass appears, the momentum looses any
memory of the mass upon replacement by −i~∂~.) How-
ever, canonical quantization is a heuristic procedure (it
∗
isratherclassicalmechanics,whichisbelieved,inprinci- Electronic address: [email protected]
†
ple, to be derivable fromthe quantum mechanics)and it Electronic address: [email protected]
is unable to decide the uniqueness question or to clarify [1] C. M. Will, Theory and Experiment in Gravitational
the intrinsic structure at play. On the other hand Du- Physics (Cambridge University Press, Cambridge, 1993).
val and Ku¨nzle [5] work from start with a wave function [2] F. Hehl, J. Lemke and E. W. Mielke, in Geometry and
of a particle. They show how the equation obtained by Theoretical Physics, editedbyJ.DebrusandA.C.Hirsh-
feld (Springer, Berlin 1991), p. 56.
Kuchaˇrmaybederivedbytheminimalcouplingprinciple
[3] D. Greenberger, Ann. Physics (NY) 47, 116 (1968); D.
if the wave function is assumed to have certain geomet-
GreenbergerandA.W.Overhauser,Rev.Mod.Phys.51,
rical properties (is a section of a vector bundle associ-
43 (1979); H.-J. Treder, Ann. Phys. (Leipzig) 39, 265
ated with the Bargmann bundle over spacetime). The
(1982).
geometricstructures introducedby them havebeen then [4] C. L¨ammerzahl, Class. Quantum Grav. 17, 13 (1998);
adoptedbyDeBi`evreinthepapermentionedabove,and Gen. Rel. Grav. 28, 1043 (1996); Acta Phys. Pol. B29,
alsobyChristian[5],whomakesitabasisforaconstruc- 1057 (1998).
tionofaNewton-Cartanquantumfieldtheoryofparticles [5] K.Kuchaˇr,Phys.Rev.D22,1285(1980);C.DuvalandH.
and gravitational field. While the structures introduced P.Ku¨nzle, Gen.Rel.Grav.16,333 (1984); D.Canarutto,
byDuvalandKu¨nzleilluminatethegeometryofthegen- A. Jadczyk and M. Modugno, Rep. Math. Phys. 36, 95
eralcovariantSchr¨odingerequation,theyincorporateas- (1995); J. Christian, Phys. Rev. D56, 4844 (1997); R.
Vitolo,Ann.Inst.H.Poinc.70,239(1999);J.Wawrzycki,
sumptions on the transformation properties of the wave
Int.J. Theor. Phys. 40, 1595 (2001).
functionandontheformoftheequationwhichwederive
[6] S. DeBi`evre, Class. Quantum Grav. 6, 731 (1989).
here from scratch.
[7] E´. Cartan, Ann. E´c. Norm. Sup. 40, 325 (1923); 41, 1
Another approach to the question of the validity of (1924); A. Trautman, C. R. Acad. Sci. Paris 257, 617
equivalenceprincipleinthequantumworldhasbeenpro- (1963); C.DuvalandH.P.Ku¨nzle,ibid.285,813(1977).
posed by La¨mmerzahl [4]. He gives arguments to the [8] R. Colella, A. W. Overhauser and S. A. Werner, Phys.
effect that Eq. (4) is favored by a principle which he in- Rev.Lett.34,1472(1975); M.KasevichandS.Chu,ibid.
troduces and calls quantum equivalence principle. This 67, 181 (1991).
principle formulates a condition for a possibility of the
extraction of mass-independent characteristics from ex-
perimental results. However, there is no obvious con-
nection of this principle with Einstein’s geometricalidea
and its compelling persuasiveness (in fact, La¨mmerzahl
avoids the covariance question completely). We do be-
lieve La¨mmerzahl’s results are important and interest-
ing, but see their role on the experimental side rather
than as a theoretical paradigm. What we mean, more
precisely, is this. Einstein’s principle is a local princi-
ple. For a classical test particle, which is a local object,
its content translates itself rather directly into experi-
mentalpredictions. Thequantummechanicalwavefunc-
tion,ontheotherhand,isanonlocalobject,andthereis
no simple analogoustranslation - in general gravitycan-
not be eliminated on any hypersurface of constant time.
La¨mmerzahl’s papers show how to extract experimental
consequencesofEinstein’sequivalenceprinciple fromex-
perimentaldata. Havingsaidthis,however,wealsowant
to express disagreement with the opinion that nonlocal-
ityofthewavefunctionprecludesoperationalmeaningof
Einstein’sprinciple. Itmaybe notobvioushowtoreveal
such meaning, but we can see no fundamental obstacle
5