Weak equivalence principle violation with a quantum particle in a gravitational wave

James Q. Quach∗

Institute for Photonics and Advanced Sensing and School of Chemistry and Physics",

University of Adelaide, South Australia 5005, Australia

I. INTRODUCTION

The weak equivalence principle (WEP) states that

point particles in free-fall will follow trajectories that are

independent of their mass. This principle underpins clas-

sical gravitational theory. In the context of classical the-

ory the WEP is well defined; in quantum theory however",

the WEP is ill-defined. This is because under Heisen-

berg’s uncertainty principle, point particles and trajec-

tories are ambiguous concepts. The problem is further

highlighted when one compares the classical action of a

particle with mass m in a gravitational field with the

quantum action of a scalar particle φ. The classical ac-

tion is

SC = −mc

∫

ds , (1)

where ds2 = gµνdx

µdxν . As m appears simply as a mul-

tiplicative factor, it does not feature in the equations of

motion. This is consistent with the WEP. In comparison",

the quantum action is

SQ =

~2

2m

∫

d4x

√

−g() . (2)

The corresponding equation of motion

1√

−g

∂µ(

√

−ggµν) , (3)

is dependent on m, in apparent violation of the WEP.

Given the difficulties with interpreting the WEP in a

quantum context, an alternative formulation has been of-

fered by Seveso et al. [seveso17]. In their formulation",

they encode an object’s trajectories in the Fisher infor-

mation. The WEP’s notion that free-falling trajectories

should be independent of mass, is reformulated as the

statement that the Fisher information of a free-falling ob-

ject is invariant with mass. In this information-theoretic

framework, violation of the WEP means that one may

extract information about an object’s mass in free-fall.

This information-theoretic formulation of the WEP has

the advantage that it is extensible to quantum objects in

an unambiguous manner.

Under this formulation Seveso et al. showed that quan-

tum objects in a static uniform gravitation field does not

violate the WEP, whereas in a non-uniform gravitational

field they do. The classical analogue of this is the fact ex-

tended classical objects in non-uniform gravitational field

do not follow geodesics [geroch75]. In this paper we show

that quantum objects in a uniform but time-dependent

gravitational field violates the WEP; in particular, we

look at gravitational waves. This does not have a classi-

cal analogue, as extended classical objects in uniform but

time-dependent gravitational fields do follow geodesics.

As such, this work reinforces the idea that quantum ob-

jects violates the WEP in fundamental ways. In Sec.

II. FISHER INFORMATION [MOVE TO

INTRO?]

The Fisher information gives the amount of informa-

tion that an observable random variable provides about

an unknown parameter. In our case, the random vari-

able is the position of the particle x, and the unknown

parameter is it mass m. For a particle with wave function

ψ(x, t), the Fisher information is

Fx(m) =

∫

dx|ψ(x, t)|2[∂m log |ψ(x, t)|2]2 . (4)

In the absence of gravity, observation of the position

of the particle can betray information about its mass.

For example, a free Gaussian wave packet spreads with

variance σ2(t) = σ2(0) + ~t/2m; one may extract infor-

mation about its mass by monitoring its position. Formu-

lation of the WEP in terms of Fisher information states

that the presence of a gravitational field should not pro-

duce more information about the mass of a particle, i.e.

Fx(m) = Fx(m)|free, where F freex (m) is the Fisher infor-

mation in the absence of a gravitational field.

III. SCHWARZSCHILD FIELD

Prior work took the non-relativistic limit of the Klein-

Gordon equation in a curved space-time background, to

derive the Hamiltonian in a weak gravitational field [1].

As an alternative derivation, we begin with the Dirac

equation in curved space-time, which describes a spin-

1/2 particle of rest mass m in a gravitational field (SI

units)",

i~γaeaµ(∂µ − Γµ)ψ = mcψ . (5)

The spacetime metric gµν can be related at every point

to a tangent Minkowski space ηab via tetrads e

a

µ, gµν =

eaµe

b

νηab. The tetrads obey the orthogonality conditions

2

eaµe

ν

a = δ

ν

µ, e

a

µe

µ

b = δ

a

b . We use the convention that Latin

indices represent components in the tetrad frame. The

spinorial affine connection Γµ =

i

4e

a

ν(∂µe

νb+Γνµσe

σb)σab",

where Γνµσ is the affine connection and σab ≡ i2 [γa, γb]

are the generators of the Lorentz group. γa are gamma

matrices defining the Clifford algebra {γa, γb} = −2ηab",

with spacetime metric signature (−",+",+",+). We use the

Einstein summation convention where repeated indices

(µ, ν, σ, a, b = {0, 1, 2, 3}) are summed.

For static gravitational field we will consider the

Schwarzschild metric in isotopic coordinates (x0 ≡ ct)",

ds2 = V 2(dx0)2 −W 2(dx · dx) , (6)

where (r ≡

√

x · x)

V =

(

1− GM

2c2r

)(

1− GM

2c2r

)−1

, (7)

W =

(

1 +

GM

2c2r

)2

. (8)

Under this metric Eq. (5) can be written in the fa-

miliar Schrödinger picture i~∂tψ = Hψ, where (α ≡

γ0γ, β ≡ γ0",p ≡ −i~∇, F ≡ V/W , and indices i, .., n =

{1, 2, 3} [2]

H = βmc2V +

c

2

[(α · p)F − F (α · p)] . (9)

A means by which to write down the non-relativistic

limit of the Dirac Hamiltonian with relativistic correction

terms is provided by the Foldy-Wouthuysen (FW) trans-

formation [3]. The FW transformation is a unitary trans-

formation which separates the upper and lower spinor

components. In the FW representation, the Hamilto-

nian and all operators are block-diagonal (diagonal in

two spinors). There are two variants of the FW transfor-

mation known as the standard FW (SFW) [3] and exact

FW (EFW) [2, 4–6] transformations. We will use here

the EFW transformation.

Central to the EFW transformation is the property

that when H anti-commutes with J ≡ iγ5β, {H",J} = 0",

under the unitary transformation U = U2U1, where (Λ ≡

H/

√

H2)

U1 =

1√

2

(1 + JΛ), U2 =

1√

2

(1 + βJ) , (10)

the transformed Hamiltonian is even (even terms do not

mix the upper and lower spinor components, odd terms

do)",

UHU+ =

1

2

β(

√

H2 + β

√

H2β) +

1

2

(

√

H2 − β

√

H2β)J

={

√

H2}evenβ + {

√

H2}oddJ .

(11)

Note that as β is an even operator and J is an odd oper-

ator, Eq. (11) is an even expression which does not mix

the positive and negative energy states.

Our Hamiltonian satisfies the EFW anti-commutation

property. Using the identity αiαj = i�ijkσkI2+δ

ijI4, the

perturbative expansion of

√

H2 yields to first order",

H ≈ mc2V + 1

4m

(W−1p2F + Fp2W−1) . (12)

Note that

√

H2 = {

√

H2}even = HI2 contains only

even terms, and therefore {

√

H2}odd = 0 in Eq. (11).

Taking the weak-limit gravitational field limit so that",

V ≈ 1− GM

c2r

, W ≈ 1 + GM

c2r

, (13)

we get (g ≡ −GMr/r3)

H = mc2 +

p2

2m

+mg · x . (14)

In Eq. (14) we arrive at a Hamiltonian of particle in

a static gravitational well, which one may have simply

written down by intuition. However, we have chosen to

follow the formal derivation of taking the FW transfor-

mation of the Dirac equation in curved spacetime, as we

will use this formalism to derive Hamiltonian of a par-

ticle in gravitational wave background in Sec. [], which

can not be simply written down by intuition.

The evolution of a quantum particle is governed by the

time-evolution operator U = e−iHt. Taking the Baker-

Campbell-Hausdorff expansion of U to second-order, the

time-evolution operator in a Schwarzschild field is (~ =

1) [1]

U ≈ exp

( imt3

3

g2

)

exp

( it3

6m

∇g · ∇∇ − gt

2

2

· ∇

)

exp

(

− imtg · x

)

Ufree .

(15)

where Ufree = exp(−imc2t) exp(−it∆/2m) is the free

time-evolution operator in the absence of any gravita-

tional field. Note that the exp(−imc2t) term only acts

as a constant phase factor in the non-relativistic limit",

and therefore can be ignored.

A. Uniform gravitational field

As our gravitational field is spherically symmetric, we

can reduce our problem to one-dimension in the radial

direction. We consider a Gaussian wave packet [change

everything to 1D r]",

ψ(x, 0) =

( 2

π

)1/4

e−r

2

, (16)

as this is most amenable to comparison with a classi-

cal particle; our results however are generalisable to any

wave function. For probe particles travelling over small

3

distances, it is usual to take the terrestrial gravitational

field as uniform. In the uniform gravitational case",

U = exp

( imt3

3

g2

)

exp

(

− gt

2

2

· ∇

)

exp

(

− imtg · x

)

Ufree .

(17)

Using the fact that momentum operator is a transla-

tion operator in the conjugate position space, the time

evolution a wave function [ψ(x, t) = Uψ(x, 0)] is

ψ(x, t) = exp

( imt3

3

g2

)

exp

(

− imtg · x

)

ψfree(x−

gt2

2

, t) ",

(18)

where ψ(x, t)free = Ufreeψ(x, 0) is the free wave function

in the absence of a gravitational field.

Substituting Eq. (16) into Eq. (18), the expected po-

sition of the wave packet in a uniform gravitational field

is

〈x〉 =

∫ ∞

−∞

ψ(x, t)∗xψ(x, t)dx =

π

2

gt2

2

. (19)

This is the geodesic of a freely-falling classical particle

with no initial momentum in a uniform gravitational field

g. As with the classical case, the expected trajectory

of the quantum particle is independent of its mass, in

alignment with the WEP.

Along with translating the wave function by gt2/2",

the uniform gravitational field induces a mass-dependent

phase factor in Eq. (18). This mass-dependent phase

factor however, is not present in the probability distribu-

tion, |ψ(x, t)|2 = |ψfree(x − gt2/2, t)|2. Therefore, by a

change of variable (u = x− gt2/2), we see that the uni-

form gravitational field does not produce any extra mass

information, i.e.

Fx(m) =

∫

du|ψfree(u, t)|2[∂m log |ψfree(u, t)|2]2

= Fx(m)|free .

(20)

B. Non-uniform gravitational field

If we do not make the approximation that g is uni-

form, than we must use the time-evolutation operator of

Eq. (15). In this case the wave function is [1]

ψ(x, t) ≈ exp

( imt3

6

g2

)

exp

( imt

2

∇g2 · g

)

exp

(

− imtg · x

)

ψfree(x−

gt2

2

+ d, t) ",

(21)

where

d ≡ t

2

2

(x · ∇g − g) + t

3

3m

p · ∇g + 5t

4

48

∇g2 . (22)

-expected value is mass dep -fisher -one notes that if

p is replaced/apprxoximated by classical p=mgt, mass

dependence is removed -write in english these things

IV. GRAVITATIONAL WAVE

The metric for a generally polarised linear plane GW

is

ds2 = −c2dt2 +dz2 +(1−2v)dx2 +(1+2v)dy2−2udxdy ",

(23)

where u = u(t− z) and v = v(t− z) are functions which

describe a wave propagating in the z-direction. We will

consider the case of a circularly polarised GW travelling

along the z-direction, i.e. v = f = f0 cos(kz − ωt) and

u = if . Under this metric Eq. (5) can be written in the

familiar Schrödinger picture [7]

H = βmc2 + cαj(δij + T

i

j )pi , (24)

with

T =

v −u 0−u −v 0

## 0 0 0

. (25)

Applying the EFW (or SFW) transformation and ig-

noring higher-order terms one arrives at [8]

HGW =

1

2m

(δij + 2T ij)pipj +mc

2. (26)

We would like to know how a Gaussian wave packet

behaves in a GW background. We will consider the wave

packet in one dimension without loss of generality;

ψ(x, 0) =

( 2

π

)1/4

e−(x−x0)

2

. (27)

We apply the unitary transformation operator U =

e−iHGWt to Eq. (27) to get the time-evolution of a wave

packet in a GW background (see Appendix A for deriva-

tion)",

ψ(x, t) = 2

3

4π

1

4

e−(x−x0)

2/4b

√

b

(28)

where

b ≡ 1 + 2i~t

m

(1 + f0 cosωt) (29)

The expected position of the wavepacket in a GW

background is

〈x〉 = x0 . (30)

In other words, the particle remains at rest in co-ordinate

system of Eq. (23). This actually is not surprising as this

is also what happens in the classical case. In the classical

case, the presence of a GW is measured in the change of

the proper distance between two particles.

Let us now calculate the mass Fisher information of the

particle in a the GW, and compare it to the free case.

-expected x, same as classical [unlike the static gravita-

tional field case, the wave fucntion correction is indepent

of x, and therefore the expectation value is same as clas-

sical, but fisher information is non-zero, therefore can

extract mass info] -fisher information different from free

case, and is dependent on m

4

V. ACKNOWLEDGEMENTS

This work was financially supported by the Ramsay

Fellowship.

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