G en eral Relativ ity an d G ravi tation , Vol. 29, No. 2 , 1997

Testing the Equivalence Princip le in the Quantum

Regime

Catalina Alvarez1 ,2 and Robert Mann1 ,3

Rece ived May 28, 199 6

We consider possible test s of the Einstein Equivalence P rinciple for physi-

cal syst ems in which quantum -mechanical vacu um energies cannot be ne-

glect ed. Speci ® c t ests include a search for the m anifestat ion of non-metric

eŒect s in Lamb-shift t ransit ions of hydrogen ic atoms and in anomalous

m agnet ic m om ent s of massive leptons. We discuss how current experi-

m ent s already set bounds on the violat ion of the equivalence principle in

this sector and how new (high-precision) measurem ent s of these quant i-

t ies could provide further informat ion to this end.

KEY WORDS : Variatns of the equivalence principle

Our understanding of gravitation is built upon the foundat ions of the

Equivalence P rinciple. Originally regarded as the cornerstone of mechan-

ics by Newton, and used later by Einstein in the development of general

relat ivity, it has come to be understood as the basis for the not ion that

spacet ime has a unique operat ional geometry. It consequent ly ensures that

the eŒects of gravity on matter can be described in a purely geometric fash-

ion.

Of the several exist ing variant s of the equivalence principle, it is the

Einstein Equivalence Principle (eep ) which plays a pivotal role in this

regard. This principle has three component s as follows. The ® rst is that

1 Departm ent of P hysics, University of Wat erloo, Wat erloo, Ontario N2L 3G1, Canada2 E-mail: calvarez@avat ar.uwaterloo.ca3 E-mail: m ann@avat ar.uwaterloo.ca

2 4 5

0001-7701/ 97/ 0200-0245$09.50/ 0 1997 P lenum Publishing Corporation

2 4 6 A lva re z an d M an n

all freely falling bodies (i.e. bodies which are not acted upon by non-

gravitational forces such as electromagnet ism and which are small enough

so that tidal eŒects are negligible) move independent ly of their composit ion

(the Weak Equivalence Principle, or wep ). The second component is the

assertion that the result s of any non-gravit ational experiment (such as the

measurement of an electromagnet ic current in a wire) are independent of

where and when in the universe it is carried out (Local Posit ion Invariance,

or lp i) , and the third component is the assertion that such results are

independent of the velocity of the freely falling reference frame in which the

experiment is performed (Local Lorentz Invariance, lli). Metric theories

(such as general relat ivity and Brans± Dicke Theory) realize the eep by

endowing spacet ime with a symmetric, second-rank tensor ® eld gmu that

couples universally to all non-gravit ational ® elds [1], so that that in a

local freely falling frame the three postulates are satis® ed. By de® nit ion,

non-metric theories do not have this feature: they violat e universalit y and

so permit observers performing local experiments to detect eŒects due to

their posit ion and/ or velocity in an external gravitational environment.

Each of the three components of eep have been subjected to severe

experimental scrut iny. Empirical limits on wep violat ion are typically set

by torsion balance experiments, whereas limits on lp i and lli violat ion are

set by gravitational red-shift [2] and atomic physics experiments [3] respec-

tively, all to varying degrees of precision. The universalit y of gravit ational

redshift has been veri® ed to 1 part in 5000 [2], wep to 1 part in 1012 [4]

and lli to 1 part in 1021 [3]. Signi® cant ly improved levels of precision are

anticipated in future experiments [5].

Impressive as these limits are, the dominant form of mass-energy gov-

erning the systems these experiments study is nuclear electrostatic energy,

although violat ions of the eep due to other forms of energy (virtually all

of which are associat ed with baryonic matter) have also been estimated

[6]. However there are many physical systems dominat ed by other forms of

mass energy for which the validity of the equivalence principle has yet to

be empirically checked, including matter/ ant imatter systems [7], (hypoth-

esized) dark matter, photons of diŒering polarizat ion [8], massive leptons

[9], neutrinos [10], second and third generat ion matter [7], and quantum

vacuum ¯ uctuat ions [11]. Comparat ively lit t le is known about empirical

limits on eep -violat ion in these other sectors [12].

We describe in this paper the results of an approach for examining

potent ial violat ions of the eep due to eŒects which are peculiarly quantum-

mechanical in origin (i.e. are due solely to radiat ive corrections) . EŒects

of this type include Lamb-shift transit ion energies in hydrogenic atoms

and anomalous magnet ic moments of massive leptons. Tests of the eep in

Te s t in g t h e E q u iva le n c e P r in c ip le in t h e Q u a nt u m R e g im e 2 4 7

this sector push the confrontation between quantum mechanics and gravity

ever closer, providing us with qualit atively new empirical windows on the

foundat ions of gravit ational theory.

The action appropriat e for Quantum Electrodynamics in a back-

ground gravitational ® eld (gqed) is

S = s d4x Ö ± g[ w ( i /Ñ + e /A ± m ) w ±

1

4Fmu F

mu], (1)

where Ñ is the covariant derivat ive, Fmu º Au,m ± Am ,u and /A = eam c a Am ,

eam being the tetrad associat ed with the metric. Our approach is to ex-

tend this action to the wide class of non-metric theories described by the

TH em formalism [13]. This formalism (which has as its limit ing case all

metric theories) assumes that the external gravitational environment of a

given physical system is desribed by a static, spherically symmetric metric

which does not necessarily couple universally to all forms of matter. More

concretely,

gmu = diag ( ± T, H, H, H ) and Fmu Fmu

= 2(eE2 ± B

2 / m) ,

where®E º ±

®

Ñ A0 ± ¶®A / ¶ t and

®B º

®

Ñ £®A. e and m are arbit rary functions

of the Newtonian background potential U = GM / r (which approaches

unity as U ® 0) as are T and H , which in general will depend upon the

species of part icles within the system, which we shall take to be massive

leptons.

The action (1) will in general depend upon the velocity®u of a given

(sub)atomic system relat ive to the preferred frame as well as the TH em

parameters. This dependence can be obtained using a Lorentz transfor-

mation to transform ® elds and coordinat es from the preferred frame to the

rest frame of the (sub)atomic system, whose small spacet ime size permits

us to ignore spat ial variat ions in the THem parameters. Analysis shows

that , upon local rescaling of physical parameters, it is only the electromag-

netic sector of the action that depends explict ly on®u and the dimensionless

parameter jF º 1 ± c2F = 1 ± H0 / T0e0 m0 , ª 0º denot ing evaluat ion at the

system’s center of mass, with c F the ratio of the limiting speed of lepton F̀ ’

to the speed of light [11]. It is straight forward to check that all divergences

in gqed can be renormalized by proper rede® nit ions of the parameters of

the theory, (which now include the TH em funct ions) and that the Ward

ident ities are satis® ed. This is a consequence of gauge invariance. In ex-

tracting predict ions from gqed, we note that the natural scale for j is

set by the magnitude of U, which empirically is much smaller than unity,

permitting a perturbat ive analysis in j.

2 4 8 A lva re z an d M an n

The Lamb shift is an energy shift caused by quantum vacuum ¯ uc-

tuat ions between normally degenerate states in a hydrogenic atom [11].

To extract the predict ion for this eŒect in gqed, it is necessary to solve

the ® eld equat ions for the electromagnet ic vector potential produced by a

point like nucleus of charge Ze at rest in the moving frame. Employing pre-

viously established techniques [14] yields the result that this degeneracy is

lifted before radiat ive correct ions are introduced. This non-metric energy

shift is isotropic in®u and vanishes when

®u = 0. Evaluat ing the relevant

radiat ive corrections to the required accuracy O(jF )O(®u2 )O(a(Za)4 ) en-

tails a very lengthy and tedious calculat ion, leading to an expression for a

gravitationally modi® ed Lamb shift energy D EL (jF ,®u ) which is no longer

isotropic in®u .

Upper bounds on jF from current experiments can be obtained by

assuming that eep -violat ing contribut ions to D E L are bounded by the

current level of precision for the Lamb shift [15]. Using the accepted

upper limit j ®u j < 10- 3 for the preferred frame velocity [4] we ® nd [11]

the dominant non-metric contribut ion to D E L is due to purely radiat ive

corrections, yielding the bound j je j < 10- 5 . If we assume that posit rons

and electrons do not have equivalent couplings to the gravitational ® eld

[16], we ® nd that there is an addit ional radiat ive contribut ion to D E L due

to vacuum polarizat ion. Making the same comparisons as above, we ® nd

the most stringent bound on this quant ity to be j je + j < 10- 3 from present

Lamb shift experiments.

In the case of anomalous magnet ic moments, we ® nd that an evalua-

tion of the Feynman amplit ude relat ed to the elast ic scat tering of a lepton

by a static external ® eld yields an eŒective interaction term in the gqed

Hamiltonian,

Hs = ±e

2mf g

®S . ®

B + g*

®S . ®

u®B . ®

u g º ± Ci j

S i B j , (2)

with g º 2 + (a/ p)[1 + jF (1 + (c 2 / 6)(1 + 7®u2 ))], g* º ± (a/ p)jF (4/ 3)c 2 ,

and c 2 = (1 ± j ®u j 2 )- 1 , where

®S º ®

s/ 2 is the spin operator. The presence

of preferred frame eŒects induces a new type of tensorial coupling between

the magnet ic ® eld and the spin described by C i j .

From this we ® nd that the precession frequency of the longit udinal

spin polarizat ion®S . ®

b is ’ (eB / m )a, where

a =a

2p {1 + j[1 +c 2

6(1 + 7(V

2+ b2

) ± 8V2

cos2 Q ) ]} (3)

and where®b is the lepton velocity with respect to the laboratory system

which moves in the preferred frame with velocity®V , whose angle with the

Te s t in g t h e E q u iva le n c e P r in c ip le in t h e Q u a nt u m R e g im e 2 4 9

magnet ic ® eld is Q . Assuming the eep -violat ing cont ribut ions to a are

bounded by the current level of precision for gyromagnet ic anomalies (and

that j®V j < 10- 3 ) then the discrepancy between the best empirical and

theoretical values for the electron yields the bounds [9] j je - j < 10- 8 and

j je - ± je + j < 10- 9 , the latter following from a comparison of posit ron and

electron magnet ic moments. For muons, a similar analysis yields j jm - j <

10- 8 and j jm - ± jm + j < 10- 8 . To our knowledge these limits are the most

stringent yet noted for these parameters.

Non-metric eŒects also induce oscillat ions in the spin polarizat ion

component (SB ) parallel to B . The ratio between the temporal average of

this quant ity and the init ial polarizat ion of the beam can be estimated to

be [9] d F = ( h SB i / S ) ~ jF Vb cos Q c 2 . In highly relat ivist ic situat ions this

eŒect is enhanced, and can be estimated by considering a typical experi-

ment with V ~ 10- 3 , where for electrons (b ~ 0.5), and so d e ~ 10- 11 ;

and for muons (b = 0.9994) , d m ~ 10- 8 . In both cases the corresponding

present const raints for jF were employed. Improved measurements of this

quant ity for diŒerent values of Q should aŒord the opportunity of putting

tighter constrains on the jF parameters. The rotation of the Earth will have

the eŒect of convert ing this orientation dependence into a t ime-dependence

with a period relat ed to that of the sidereal day.

Addit ional empirical informat ion can also be extracted from D E L

and C i j by evaluat ing their associat ed gravit at ional redshift parameters.

Analysis [11] shows that these parameters are two linearly independent

combinat ions of

C0 ºT0

T 90ln [Te2

H ]9 ||||0

and L0 ºT0

T 90ln [Tm2

H ]9 ||||0

.

Redshift experiments can therefore set bounds on independent regions of

(C0 , L0 ) parameter space in the lepton sector. However this will be a

challenge to experimentalist s because of the small redshift due to earth’ s

gravity (< 10- 9 ) and the intrinsic uncertaint ies of excited states of hydro-

genic atoms. One would at least need to perform these experiments in a

stronger gravitational ® eld (such as on a satellite in close solar orbit ) with

1± 2 orders-of-magnitude improvement in precision.

To summarize, violat ion of gravit ational universalit y modi® es radia-

tive correct ions to lepton-photon interact ions in a rather complicat ed way,

giving rise to several novel eŒects. Re® ned measurements of atomic vac-

uum transit ions and anomalous magnet ic moments can provide an interest-

ing new arena for invest igat ing the validity of the eep in physical regimes

where quant um ® eld theory cannot be neglected. It will be a challenge to

2 5 0 A lva re z an d M an n

set new empirical bounds on such eŒects in the next generat ion of experi-

ments.

ACKNOWLEDGEMENTS

This work received an Honourable Mention in the 1996 Gravity Re-

search Foundat ion Essay Contest.

This work was support ed in part by the Natural Sciences and Engi-

neering Research Council of Canada. We are grateful to C. W. F. Everitt,

M. Haugan, E. Hessels and C. M. Will for interesting discussions at various

stages of this work.

REFERENCES

1. Thorne, K. S., Lightman, A. P., and Lee, D. L. (1973) . Phys. Rev . D7 , 3563.

2. Vessot, R. F. C., and Lev ine, M. W . (1979) . G en . Rel. G rav. 1 0 , 181.

3. P rest age, J . D., Bollinger, J . J ., Itano, W . M., and W ineland, D. J . (1985) . Phys .

Rev . Lett. 5 4 , 2387; Lamoreaux, S. K., et al. (1986) . ibid . 5 7 , 3125; Chupp, T . E.,

et al. (1989) . ibid. 6 3 , 1541.

4. W ill, C . M. (1992) . Theory an d Exper im en t in G ravi tation a l Physi cs (2nd ed., Cam -

bridge University P ress, Cambridge) .

5. B laser, J . P. et al. (1994) . ST EP Assessm en t Study Repo rt , ESA Docum ent SCI

(94) 5, May 1994.

6. Haugan, M. P., and W ill, C. M. (1976) . Phys. Rev . Lett. 3 7 , 1; (1977) . Phys. Rev .

D1 5 , 2711.

7. Holzscheit er, M. H., Goldman, T ., Nieto, M. M. (1995) . P reprint LA-UR-95-2776,

hep-ph/ 9509336 .

8. Gabriel, M., Haugan , M., Mann, R. B ., and Palmer, J . (1991) . Phys . Rev. Lett. 6 7 ,

2123.

9. Alvarez , C., and Mann, R. B . (1996) . Phys. Rev . D , to appear.

10. Mann, R. B ., and Sarkar, U. (1996) . Phys. Rev . Lett. 7 6 , 865; Gasp erini, M. (1989) .

Phys. Rev . D3 9 , 3606.

11. Alvarez , C., and Mann, R. B . (1996) . Mod. Phys. Lett. A 1 1 , 1757.

12. Hughes, R. J . (1993) . Con tem porar y Phys ics 3 4 , 177.

13. Lightman, A. P., and Lee, D. L. (1973) . Phys . Rev . D8 , 364.

14. Gabriel, M. D., and Haugan , M. P. (1990) . Phys . Rev. D4 1 , 2943.

15. Eides, M. I., and Shelyuto, V. A. (1995) . Phys . Rev. A5 2 , 954.

16. SchiŒ, L. (1959) . Proc. Nat. Acad . Sc i. 4 5 , 69.

17. van Dyck, R. S., Schwinberg, P. B ., and Dehm elt , H. G. (1987) . Phys. Rev. Lett.

5 9 , 26.