Quantum mechanics for non-inertial observers
QQuantum mechanics for non-inertial observers
Marko Toroˇs,
1, 2, ∗ Andr´e Großardt,
1, 2, † and Angelo Bassi
1, 2, ‡ Department of Physics, University of Trieste,Strada Costiera 11, 34151 Miramare-Trieste, Italy Istituto Nazionale di Fisica Nucleare,Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy (Dated: January 17, 2017)
Abstract
A recent analysis by Pikovski et al. [Nat. Phys. 11, 668 (2015)] has triggered interest in thequestion of how to include relativistic corrections in the quantum dynamics governing many-particlesystems in a gravitational field. Here we show how the center-of-mass motion of a quantum systemsubject to gravity can be derived more rigorously, addressing the ambiguous definition of relativisticcenter-of-mass coordinates. We further demonstrate that, contrary to the prediction by Pikovski etal., external forces play a crucial role in the relativistic coupling of internal and external degrees offreedom, resulting in a complete cancellation of the alleged coupling in Earth-bound laboratories forsystems supported against gravity by an external force. We conclude that the proposed decoherenceeffect is an effect of relative acceleration between quantum system and measurement device, ratherthan a universal effect in gravitational fields. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ qu a n t - ph ] J a n ow does a quantum particle move in a gravitational field? Thus far, only the nonrel-ativistic case of a particle in a Newtonian, homogeneous gravitational potential has beenexperimentally tested [1]. The relativistic description of the dynamics of interacting quan-tum particles, on the other hand, is believed to require quantum field theory [2], sincerelativistic effects must include particle creation and annihilation. Strictly speaking, generalrelativistic corrections to the evolution of quantum matter should, therefore, be describedwithin the framework of quantum field theory in curved spacetime [3, 4].Nonetheless, it is instructive and of relevance for experiments [5–11] to discuss relativisticeffects in terms of higher order corrections to the nonrelativistic Schr¨odinger equation: amodel applicable to quantum systems for which the number of particles is conserved, to goodapproximation. These corrections can, for instance, be obtained as higher orders in 1 /c ofthe c → ∞ expansion for a classical Klein-Gordon field in curved spacetime [12, 13]. Thisresult, however, must be considered as a single particle evolution equation. A more difficultproblem, within this scheme, is how to analyze a system of many interacting particles. Thisis a necessary step towards correctly understanding how relativistic corrections affect themotion of atoms or molecules. The first problem one faces, is how to define a center-of-masswave-function and how to derive its dynamics.It is tempting to extend the one-particle dynamics to the center of mass (c.m.) of acomposite system, simply replacing the one-particle rest mass m by M + H (0)rel /c , M beingthe sum of all rest masses and H (0)rel the internal Hamiltonian to nonrelativistic order. Onethen obtains, up to order 1 /c , a coupling of internal and c.m. degrees of freedom proportionalto ( M c − T (0) + U (0) ) H (0)rel /c , where T (0) and U (0) are the nonrelativistic c.m. kinetic andgravitational potential energy, respectively. Based on this coupling, decoherence of the c.m.wave-function due to an external, homogeneous gravitational field was predicted by Pikovskiet al. [14], followed by a vivid debate [15–19]. One immediate question which, to date, hasnot been raised, is the compatibility of this scheme with the well-known problems in definingrelativistic c.m. variables [20]: no definition exists in which the c.m. is (i) uniformly movingin absence of external forces, (ii) positions and momenta obey the canonical commutationrelations, and (iii) the c.m. is frame independent [21]. A perturbative procedure to constructcanonical c.m. variables, in which the dynamics in flat spacetime take the single particle form,was presented by Krajcik and Foldy [22].In this letter, we show how to generalize this prescription to accelerated observers, or to2 xy Minkowski observer (a) no coupling = x ↑ g y Rindler observer (b) gravity coupling = x ↑ g y (c) potential coupling = x ↑ g y (d) grav. + pot. = no coup. FIG. 1. Overview of the different observer/particle constellations: (a) inertial (Minkowski) observerwith free particle; (b) accelerated (Rindler) observer with free particle; (c) inertial observer withaccelerated particle; (d) accelerated observer with accelerated particle. The equality signs denotethat, according to the equivalence principle, inertial and accelerated motion correspond to beingin free fall and at rest with respect to the Earth, respectively. In situation (b) a coupling of c.m.and internal degrees of freedom occurs due to acceleration. In (c) the opposite coupling occurs dueto the accelerating external potential. In (d) both effects cancel each other. a homogeneous gravitational field by virtue of the equivalence principle. Thereby, a precisedefinition of the coordinates in which the alleged coupling of c.m. and internal degrees offreedom holds can be given.Our analysis shows that, in this precise context, the c.m. motion couples to the internaldynamics, both for a particle held at constant height in the gravitational potential as seenby a free falling observer and for a free falling particle as seen by an accelerated (e. g. Earthbound) observer (see Fig. 1 b, c). Yet, no coupling of the c.m. to the internal Hamiltonianoccurs in situations where both the particle and the observer are accelerated (see Fig. 1 d).In this case, the decoherence effect for an accelerated observer, hovering at constant heightin the gravitational field, is exactly canceled by the potential that is keeping the particlefrom falling. The decoherence effect reported by Pikovski et al. [14], therefore, is an effectof relative acceleration of particle and observer (i. e. the experimental set-up), rather thana universal decoherence in the presence of a gravitational field.3 . EARTH BOUND OBSERVERS
It is important to point out the significance of the observer for a quantum mechanicaldescription. In contrast to classical relativity, where the observer is simply an expressionof the set of coordinates used to describe the physics, in quantum mechanics the bipartitenature of the theoretical description of experiments must be taken into account: in additionto the dynamical evolution of a quantum state, this description must also contain the reduc-tion of the state (either objectively or effectively, regardless of the preferred interpretation ofquantum mechanics) to the eigenstate of some operator upon preparation and measurement.The most convenient choice for the observer frame is, therefore, generally the rest-frame ofthe experimental set-up. Any other observer frame will require an appropriate transforma-tion, not only of the dynamics but also of the measuring procedure. In this context, thediscussion of decoherence due to a moving detector [19] is insightful.The frame attached to an observer moving along a timelike curve γ in curved spacetimeis usually described in terms of Fermi normal coordinates. For nearly-local experiments, i. e.observations performed in the vicinity of γ , Riemann tensor corrections can be neglected,and the analysis of a system in an external gravitational field reduces to the description ofan accelerated observer in flat spacetime [23]. Specifically, for a radially infalling observerin the Earth’s gravitational field the Fermi normal coordinates ( ct, x, y, z ) near a timelikecurve γ yield the flat spacetime Minkowski metric, as required by the equivalence principle.For a shell observer, hovering at constant altitude, the metric expressed in Fermi normalcoordinates reduces to the Rindler metric [24, 25]d s = − (cid:16) gxc (cid:17) c d t + d x + d y + d z + O (∆ ) , (1)where ∆ denotes the spatial distance from γ , x corresponds to vertical displacements and g is the acceleration on γ .In other words, we have Einstein’s presentation of the equivalence principle [26], accordingto which the local physics in the gravitational field can be described entirely as the physics ofan accelerated observer in flat spacetime. The rest of this letter, therefore, will be concernedonly with c.m. variables for Minkowski and Rindler observers.4 I. RELATIVISTIC C.M. FOR INERTIAL OBSERVERS
Early works have dealt with the extension of nonrelativistic quantum mechanics torelativistic single-particle situations where particle creation and annihilation can be sup-pressed [27, 28]. Newton and Wigner [29] show the existence of a unique position operatorfor elementary systems whose components satisfy the usual commutation relations. Thisposition operator is frame dependent already in the single-particle case [30]; a fact that islittle surprising since measurements are attached to a spacelike hypersurface in a specific frame.This framework can be generalized to many particles. Relativistic c.m. variables whichsatisfy the canonical commutation relations can then be constructed, starting from thePoincar´e algebra [22] (cf. Refs. [31, 32] for earlier works). The ten generators of the Poincar´egroup (translations P , rotations J , boosts K , and time translation H ) satisfy the Lie algebra[ P i , P j ] = [ P i , H ]= [ J i , H ] = 0 , ( i, j = 1 , , J i , J j ] = i (cid:15) ijk J k , [ J i , P j ] = i (cid:15) ijk P k , [ J i , K j ] = i (cid:15) ijk K k , [ K i , H ] = i P i , [ K i , P j ] = i δ ij H /c , [ K i , K j ] = − i (cid:15) ijk J k /c . (2)For a system of N spinless particles with masses m µ , the generators can be expressed interms of the particle coordinates r µ and momenta p µ ( µ = 1 , . . . , N ): P = N (cid:88) µ =1 p µ , J = N (cid:88) µ =1 r µ × p µ , (3a) H = N (cid:88) µ =1 T µ + U , (3b) K = N (cid:88) µ =1 (cid:18) c { r µ , T µ } − t p µ (cid:19) + V , (3c)where { · , · } denotes the anti-commutator, U is the interaction potential between the par-ticles, V the “interaction boost” [33], and the single-particle kinetic energy is T µ = (cid:113) p µ c + m µ c . (4)The Galilean group algebra can be straightforwardly obtained as the group contraction c → ∞ [34], where only the last line of Eq. (2) changes to[ K i , P j ] = i δ ij M , [ K i , K j ] = 0 , (5)5here M = (cid:80) m µ is the total mass.In the nonrelativistic case, it is natural to introduce the c.m. coordinates P = (cid:80) p µ and R = (cid:80) m µ r µ /M ; if we also require that the total internal angular momentum be zero, thenthe generators of the Galilean group assume a single particle form P = P , J = R × P H = P M + H rel , K = M R − t P . (6)The internal Hamiltonian H rel contains internal kinetic energies and the interaction poten-tial U . The interaction boost V is constrained by the shape of the potential U , and itsnonrelativistic contribution can be set equal to zero without loss of generality [35].Krajcik and Foldy [22] notice that this single particle form of the generators can, alter-natively, be understood as a definition of c.m. variables. This is the starting point of therelativistic extension of c.m. variables. One requires that the Poincar´e generators take a relativistic single particle form: P = P , J = R × P H = (cid:113) P c + H , K = 12 c { R , H} − t P . (7)The c.m. variables can then be constructed in a perturbative way order by order in 1 /c . Inthis construction, the Hamiltonian describing the c.m. motion in Minkowski spacetime—orthe motion of a free falling particle as seen by a free falling observer—follows immediatelyfrom the time translation Poincar´e group generator: H Mink.c.m. = (cid:113) P c + H . (8)It is then straightforward to obtain the lowest order relativistic correction by expanding toorder 1 /c : H Mink.c.m. ≈ M c + P M + H (0)rel + 1 c (cid:32) − P M + H (1)rel − P H (0)rel M (cid:33) . (9)In the first line, this Hamiltonian consists of the rest mass energy and the nonrelativistickinetic and internal energies. The second line contains the relativistic corrections, both forthe kinetic energy and the internal interactions. The last term couples the (nonrelativistic)c.m. kinetic energy to the (nonrelativistic) internal energy. The explicit form of the c.m.coordinates and of the internal Hamiltonian can be found in Ref. [22].6 ' = -c /g x't t 't 't 't ' spacetime regionof interest Minkowski observerRindler observer ∂ t' ∂ t' ∂ t ∂ t FIG. 2. Transformation from free falling Minkowski observer to accelerated Rindler observer. Inthe spacetime diagram on the left hand side, the simultaneity hypersurfaces for the Rindler observerare depicted by the red lines, with the dotted line representing the Rindler horizon. We also includethe trajectories given by hyperbolic rotations. The spacetime region of interest is highlighted ingreen and the magnified drawing on the right hand side compares the simultaneity hypersurfaces forthe Rindler and Minkowski observers and depicts the corresponding time-evolution Killing vectors.
III. RELATIVISTIC C.M. FOR ACCELERATED OBSERVERS
So far, only inertial systems have been considered. Let us now discuss the dynamics ofthe same free falling composite particle, but from the perspective of an accelerated observer.In general, this can be considered as a noninertial transformation from Minkowski to Rindlercoordinates (see Fig. 2). We can construct the Hamiltonian operator for the Rindler ob-server by considering a family of inertial frames instantaneously at rest with the Rindlerobserver. We express the Rindler time evolution Killing vector ∂ t (cid:48) in the coordinates of theinstantaneous inertial frame, and use the correspondence between Killing vectors and thegenerators of the Poincar´e group in inertial frames. In this construction, the Hamiltoniandescribing the c.m. motion in Rindler spacetime—or the motion of a free falling particleas seen by a suspended observer on Earth—is given by (cf. appendix A for a step by stepconstruction): H Rindlerc.m. = H Mink.c.m. + g c { X, H
Mink.c.m. } , (10)7here these operators are defined on the Rindler equal-time hypersurfaces. Note that thisis the well-known Rindler Hamiltonian [36, 37] rewritten within Foldy’s formalism. Again,the lowest order relativistic correction is obtained by expanding to order 1 /c : H Rindlerc.m. = M c + H (0) + 1 c H (1) (11a) H (0) = P M + H (0)rel + M g X + U (0)ext (11b) H (1) = − P M + H (1)rel − P H (0)rel M + g M { X, P } + H (0)rel g X + U (1)ext , (11c)In addition to the terms present in the Minkowski Hamiltonian (9), at the nonrelativisticlevel one obtains the homogeneous potential M gX , as expected. The relativistic correctionsto this gravitational term consist of a pure c.m. term, coupling the c.m. position to the c.m.kinetic energy, as well as the coupling of c.m. position to the internal Hamiltonian. Thelatter is the term of interest when discussing potential decoherence effects [14]. We alsoincluded an external interaction U ext .In Refs. [13, 14] the Hamiltonian of a particle in the gravitational field of the Earth wasderived as the limit of Schwarzschild spacetime for radii large compared to the Schwarzschildradius. The result differs in two respects from ours: an additional term M g X , expressingcurvature corrections which were neglected in our limit from Fermi normal coordinates toRindler coordinates, as well as a pre-factor of 3 in the term proportional to { X, P } , which iseasily understood comparing the Schwarzschild metric to the Rindler metric [38]. However,recalling the above discussion of the role of the observer, it is important to keep in mind thatthe description in Schwarzschild coordinates is that of an observer at infinite radial distance.This is not an issue in any classical situation, where the choice of coordinates is only a matterof convenience, but in the quantum scenario direct predictions for experimental outcomesmust be made in a frame attached to the laboratory. IV. SUPPORTING EXTERNAL POTENTIAL
In general, the external interaction U ext can of course be of any form, depending on theexperimental situation. Notably, it can also depend on time, c.m. momentum, or internaldegrees of freedom of the particle. We now want to discuss the requirements on this potential8n order to support the particle and keep it from falling (i. e. be at rest with respect to theobserver). A more detailed discussion is provided in the appendix B.Note that a potential U ext ( X ), which would have no effect on the coupling of c.m. andinternal dynamics [39, 40], cannot keep the particle from falling in a relativistic situation.For a classical system the argument is quite straightforward. In order to keep the particleat rest, we need to require a static momentum. With Hamilton’s equations of motion,˙ P = − ∂H Rindlerc.m. /∂ R = 0, this implies that the external interaction must exactly cancelall g -dependent terms, including the coupling of c.m. position to internal Hamiltonian and,therefore, the potential cannot depend on the c.m. coordinate X only.The necessary condition for a quantum particle in order to be at rest in the observer’sframe can be expressed as the condition that the position expectation value is not acceler-ated: d (cid:104) X (cid:105) / d t = 0. This only requires the classical behavior to mimic the evolution inMinkowski spacetime. It does not reinforce any further restrictions on the quantum dynam-ics. Using the Ehrenfest theorem, at the nonrelativistic level this yields the expected result: U (0)ext = − M gX .At the order 1 /c the condition for the acceleration of (cid:104) X (cid:105) to vanish is (cid:104) [ P x , U (1)ext ] (cid:105) − i2¯ h (cid:104) [[ X, U (1)ext ] , P ] (cid:105) = i¯ hg (cid:18) (cid:104) H (0)rel (cid:105) + (cid:104) P (cid:105) − (cid:104) P x (cid:105) M (cid:19) (12)Although this condition could be satisfied by a time-dependent potential which is tunedto the respective expectation values, a more realistic model for physical interactions is anoperator-valued U (1)ext , depending on both position and momentum [41]. In this case, thecondition (12) yields the interaction U (1)ext = − H (0)rel gX − g M { X, P } . (13)Unsurprisingly, this interaction exactly cancels all acceleration terms in the Hamilto-nian (11). V. CONCLUSION
The term c − H (0)rel gX in the Hamiltonian (11) which couples the internal energy of aquantum system to its position through acceleration has been the source for the prediction9f decoherence [14]. Our discussion from the perspective of the Poincar´e group and theinclusion of the external interaction clarifies the meaning of this result and answers some ofthe concerns about whether it is in contradiction to the equivalence principle.The predicted decoherence effect clearly exists for a free falling particle observed in anaccelerated laboratory (or a laboratory on Earth) as depicted in Fig. 1 (b). Experimentally,this is the situation one finds for example in interferometry experiments with atoms [5]and molecules [6, 7]. From the perspective of the equivalence principle, it is important topoint out that it is the detector and not the particle that is accelerated in this situation.In this regard, Bonder et al. [16] rightfully remark that the situation analysed by Pikovskiet al. [14] is equivalent to that of an accelerated observer studying free, isolated systemswithout gravity; however, their conclusion that “such scenarios cannot lead to decoherenceas, without gravity, there is nothing to cause it” is incorrect. (Of course, the equivalenceprinciple allows to describe the particle evolution both from the perspective of the acceler-ated Rindler observer moving with the detector and from the perspective of the free fallingMinkowski frame moving with the particle.) Although there is no gravity in the latter case,decoherence still results from the accelerated detector [19]. On the other hand, a free fallingdetector measuring the very same particle will, of course, not see any decoherence.For a particle which is held at constant position in the laboratory frame, as in Fig. 1 (d),we learn that the interaction required to keep the particle from falling cannot be ignoredin the discussion. Quite to the contrary, this interaction will generally cancel the couplingterms that are supposed to lead to decoherence.The requirement of canonical commutation relations for the c.m. variables—which is theonly choice that can easily be reconciled with the principles of Quantum Mechanics—leads tothe definition adopted from Krajcik and Foldy [22]. The resulting Hamiltonians (9) and (11)should be understood as generators of the dynamics of the c.m. coordinate in this particularchoice . It must be stressed that the c.m. coordinate found in this way is frame dependent ,and a distinguished role is given to the instantaneous rest frame of the detector. Althoughthis seems to be the most plausible extension of the principles of Quantum Mechanics torelativistic situations, the ultimate decision about whether or not it is correct must be madeby experiment.Note that acceleration actually couples the c.m. position to the full Hamiltonian, as isevident from Eq. (10). Even in absence of acceleration, or when an external potential cancels10ll contributions of the observer’s acceleration, the remaining special relativistic correctionproportional to P H (0)rel still couples the c.m. dynamics to the internal Hamiltonian. ACKNOWLEDGMENTS
The authors thank D. Giulini, D. Sudarsky, and S. Bacchi for insightful discussions, andgratefully acknowledge funding and support from
INFN and the University of Trieste (FRA2016). A.G. acknowledges funding from the German Research Foundation (DFG).
Appendix A: Step by step construction of Rindler Hamiltonian
Let us first review how to obtain the Rindler space Killing vectors in the coordinates ofthe instantaneous inertial frame [42]. Consider a family of inertial frames instantaneouslyat rest with the Rindler observer: specifically, consider the reference frame S (¯ t (cid:48) ) at theRindler time ¯ t (cid:48) . The transformation from the Rindler coordinates x µ (cid:48) = ( ct (cid:48) , x (cid:48) ) to the S (¯ t )coordinates x µ = ( cT, X ) is given by: cT = ( x (cid:48) + c g ) sinh( g ( t (cid:48) − ¯ t (cid:48) ) c ) , (A1) X = ( x (cid:48) + c g ) cosh( g ( t (cid:48) − ¯ t (cid:48) ) c ) − c g . (A2)We omit the other two coordinates Y = y (cid:48) and Z = z (cid:48) in the discussion. The inversetransformation is given by: ct (cid:48) = c g tanh − ( cTX + c g ) + c ¯ t (cid:48) , (A3) x (cid:48) = (cid:115) ( X + c g ) − c T − c g . (A4)We now consider the Killing vectors c ∂ t (cid:48) , c ∂ T , ∂ X : the Minkowski metric, expressed interms of the Rindler coordinates or the S (¯ t (cid:48) ) coordinates, does not depend on t (cid:48) or T , X ,respectively. In particular, we will rewrite ( c ∂ t (cid:48) ) µ (cid:48) = (1 ,
0) in terms of ( c ∂ T ) µ = (1 ,
0) and( ∂ X ) µ = (0 ,
1) at the time t (cid:48) = ¯ t (cid:48) : 11 µ = ∂x µ ∂x µ (cid:48) | t (cid:48) =¯ t (cid:48) (cid:18) c ∂ t (cid:48) (cid:19) µ (cid:48) (A5)To this end we calculate the transformation matrix ∂x µ (cid:48) ∂x µ | t (cid:48) =¯ t (cid:48) from Eqs. (A3), (A4), whichcan be easily inverted to give: ∂x µ ∂x µ (cid:48) | t (cid:48) =¯ t (cid:48) = gX/c
00 sign(1 + gX/c ) (A6)where sign( a ) = a/ | a | is the sign function.It is then straightforward, using Eqs. (A5), (A6), to obtain v µ = (1 + gX/c , ∂ t (cid:48) = (1 + g/c X ) ∂ T . (A7)
1. From Killing vectors to Hilbert space operators
Now let us construct the Hamiltonian operator ˆ H that generates the time evolution onthe Hilbert space H for inertial and non-inertial observers. As we will discuss in detail, theoperator ˆ H depends on the type of motion, i. e. inertial or non-inertial, in general can changeover time, and is also hypersurface dependent. This is not surprising, but rather expectedwithin the chosen framework [22]: the experimental apparatus, modeled by operators onHilbert space, performs measurements on equal-time hypersurfaces.In inertial reference frames, the scheme of the derivation is to map the time evolutionKilling vector (corresponding to the Poincar´e algebra time evolution generator) on an equal-time hypersurface to an operator on Hilbert space: this operator generates the infinitesimaltime evolution associated to the specific motion and to the specific hypersurface. To con-struct this map, i.e. the representation of the Poincar´e algebra on the Hilbert space H , weconsider an inertial reference frame with coordinates ( cT, X , X , X ). The isomorphism ξ between elements of Poincar´e algebra A and the Killing vectors V (at time T = 0) is given12y H (cid:55)→ ∂ T , (A8) K i (cid:55)→ T ∂∂ X i (cid:12)(cid:12) T =0 + X i ∂c ∂T (cid:12)(cid:12) T =0 = X i ∂c ∂T (cid:12)(cid:12) T =0 , (A9) P i (cid:55)→ ∂ X i , (A10) J i (cid:55)→ (cid:15) ijk X j ∂ X k (A11)and the Lie algebra representation γ on the Hilbert space H (at time T = 0) is given by: H (cid:55)→ ˆ H, (A12) K i (cid:55)→ ˆ K i (cid:12)(cid:12) T =0 , (A13) P i (cid:55)→ ∂ X i , (A14) J i (cid:55)→ (cid:15) ijk X j ∂ X k . (A15)This establishes the linear map (homomorphism) φ = γ ◦ ξ − between Killing vectors V andHilbert space operators A ( H ) in an inertial frame (at time T = 0). We have the followingpicture A ξ (cid:47) (cid:47) γ (cid:15) (cid:15) V φ (cid:124) (cid:124) A ( H ) , where A , V and A ( H ) denote the Poincar´e algebra, the Killing vectors and the algebra ofHilbert space operators, respectively.We first discuss the situation for the Minkowski observer in inertial motion, where thecoordinates are denoted by ( ct, x , x , x ). Specifically, at time t = ¯ t we consider a new iner-tial frame with coordinates ( cT, X , X , X ) such that the T = 0 hypersurface correspondsto the t = ¯ t hypersurface, and we set x i = X i . Using this new inertial frame one thenestablishes that the generator of time evolution on the equal-time hypersurface T = 0 is theoperator ˆ H , i.e. we use the map between Killing vectors and Hilbert space operators: φ : ∂ T (cid:55)→ ˆ H. (A16)We define the Minkowski Hamiltonian ˆ H Mink. on the t = ¯ t hypersurface to be given by theoperator ˆ H . We repeat this construction on each equal-time hypersurface of the inertial13Minkowski) observer by varying ¯ t : in this way we define ˆ H Mink. for all t . Specifically, theMinkowski Hamiltonian in c.m. coordinates is given byˆ H Mink.c.m. = (cid:113) ˆ P c + ˆ H , (A17)where we have inserted the time-evolution generator expressed in c.m. coordinates (seeEq. (7) from the main text).We next discuss time evolution for the non-inertial (Rindler) observer, where the coordi-nates are denoted by ( ct (cid:48) , x (cid:48) , x (cid:48) , x (cid:48) ). We will define the Hamiltonian ˆ H Rindler for the Rindlerobserver exploiting the map between Killing vectors and Hilbert space operators in inertialframes (see Eqs.(A8) -(A15)). Specifically, at time t (cid:48) = ¯ t (cid:48) we consider an inertial frame withcoordinates ( cT, X , X , X ) such that the T = 0 hypersurface corresponds to the t (cid:48) = ¯ t (cid:48) hypersurface, and we set x (cid:48) i = X i . We rewrite the time evolution Killing vector ∂ t (cid:48) in thecoordinates of the inertial frame instantaneously at rest with the Rindler observer on the T = 0 hypersurface (see Eq. (A7)), i.e. ∂ t (cid:48) = ∂ T + g Xc ∂ T . From this point onwards, thederivation mirrors the derivation of the previous paragraph for the Minkowski observer. Us-ing the linearity of the map φ from Killing vectors to Hilbert space operators (see Eqs.(A8)-(A15)) we obtain the following operator (at time t (cid:48) = ¯ t (cid:48) ): φ : ∂ T + g X i c ∂ T (cid:12)(cid:12) T =0 (cid:55)→ ˆ H + g ˆ K i (cid:12)(cid:12) T =0 . (A18)We define the Rindler Hamiltonian ˆ H Rindler at time t = ¯ t (cid:48) ( T = 0) to be given by theoperator: ˆ H Rindler = ˆ H + g ˆ K i (cid:12)(cid:12) T =0 . (A19)We repeat this construction on each equal-time hypersurface of the non-inertial (Rindler)observer by varying ¯ t (cid:48) : in this way we define ˆ H Rindler for all t (cid:48) . We now consider acceler-ated motion along the x (cid:48) axis. From Eq. (A19) it is straightforward to write the RindlerHamiltonian in c.m. coordinates:ˆ H Rindlerc.m. = ˆ H Mink.c.m. + g c { ˆ X, ˆ H Mink.c.m. } , (A20)where we have inserted the time-evolution and boost generators expressed in c.m. coordinates(see Eq. (7) from the main text) and ˆ H Mink.c.m. = (cid:113) ˆ P c + ˆ H .14 ppendix B: Restrictions on supporting potentials For the c.m. motion in the local coordinates of an accelerated observer we found theHamiltonian H Rindler c.m. = H Mink. + g c { X, H
Mink. } + U ext (B1a) H Mink. = M c + H (0) + 1 c H (1) (B1b) H (0) = P M + H (0)rel (B1c) H (1) = − P M + H (1)rel − P H (0)rel M (B1d)We will also need the commutation relations[ X, H (0) ] = i¯ hM P x (B2a)[ X, { X, H (0) } ] = i¯ hM { X, P x } (B2b)[[ X, H (0) ] , { X, H (0) } ] = 2¯ h M H (0) (B2c)[[ X, { X, H (0) } ] , H (0) ] = − h M P x (B2d)[[ X, { X, H (0) } ] , X ] = 2¯ h M X (B2e)With the Ehrenfest theorem, dd T (cid:104) A (cid:105) = − i¯ h (cid:104) [ A, H ] (cid:105) + (cid:28) ∂A∂T (cid:29) , (B3)one obtains the equations of motion for the c.m. expectation value:d d T (cid:104) X (cid:105) = − h (cid:104) [[ X, H ] , H ] (cid:105) − i¯ h (cid:104) [ X, ∂∂T U ext ] (cid:105) , (B4)where, for now, we allow for the external potential to be time dependent. Inserting theRindler Hamiltonian (B1a), we obtain:d (cid:104) X (cid:105) d T = − g h c (cid:10) [[ X, H
Mink. ] , { X, H
Mink. } ] (cid:11) − h (cid:10) [[ X, H
Mink. ] , U ext ] (cid:11) − g h c (cid:10) [[ X, { X, H
Mink. } ] , H Mink. ] (cid:11) − g h c (cid:10) [[ X, { X, H
Mink. } ] , { X, H
Mink. } ] (cid:11) − g h c (cid:10) [[ X, { X, H
Mink. } ] , U ext ] (cid:11) − h (cid:10) [[ X, U ext ] , H Mink. ] (cid:11) − g h c (cid:10) [[ X, U ext ] , { X, H
Mink. } ] (cid:11) − h (cid:104) [[ X, U ext ] , U ext ] (cid:105) − i¯ h (cid:104) [ X, ∂∂T U ext ] (cid:105) (B5)15 . Classical, time dependent potential Let us first discuss how the particle can be kept from falling, i. e. be co-accelerated withthe Rindler observer, by a classical potential which is assumed to be a function of X and T : U ext ( T, X ) = U (0)ext ( T, X ) + 1 c U (1)ext ( T, X ) . (B6)Hence, all commutators of U ext with X are zero and at lowest order (1 /c ) we getd (cid:104) X (cid:105) d T (cid:12)(cid:12)(cid:12)(cid:12) O ( c ) = − gM ¯ h (cid:10) [[ X, H (0) ] , X ] (cid:11) − h (cid:68) [[ X, H (0) ] , U (0)ext ] (cid:69) = − i g ¯ h (cid:104) [ P x , X ] (cid:105) − i¯ hM (cid:68) [ P x , U (0)ext ] (cid:69) = − i¯ hM (cid:68) [ P x , M gX + U (0)ext ] (cid:69) , (B7)which vanishes for the choice U (0)ext = − M gX , as expected. With this result, the first ordercorrection becomes c d (cid:104) X (cid:105) d T (cid:12)(cid:12)(cid:12)(cid:12) O ( c − ) = − g h (cid:10) [[ X, H (0) ] , { X, H (0) } ] (cid:11) − h (cid:68) [[ X, H (0) ] , U (1)ext ] (cid:69) − g h (cid:10) [[ X, { X, H (0) } ] , H (0) ] (cid:11) (B8a)= − gM (cid:10) H (0) (cid:11) − i¯ hM (cid:68) [ P x , U (1)ext ] (cid:69) + gM (cid:10) P x (cid:11) . (B8b)Requiring this acceleration of the particle to vanish (for the Rindler observer), using thedefinition (B1c) of H (0) , we obtain the condition (cid:104) [ P x , U (1)ext ] (cid:105) = i¯ hg (cid:18) (cid:104) H (0)rel (cid:105) + (cid:104) P (cid:105) − (cid:104) P x (cid:105) M (cid:19) , (B9)which is satisfied for the potential U (1)ext ( T ) = − (cid:18) (cid:104) H (0)rel (cid:105) + (cid:104) P (cid:105) − (cid:104) P x (cid:105) M (cid:19) gX . (B10)Therefore, the potential keeping the particle from falling to order 1 /c will generally be timedependent , and must be tuned to match the expectation values of momentum squared andinternal energy.This potential can be a reasonable choice in order to provide an effective description ofthe classical interventions of the experimenter, e. g. externally tuning a trapping potential insuch a way that a self-consistent behavior is achieved. However, as a fundamental equation,describing external interactions of the quantum system, the resulting nonlinear Schr¨odingerequation would lead to superluminal signaling [43]. To this end, the Schr¨odinger equationwith the potential (B10) is not a general evolution equation, but only applies to a specificchoice of initial c.m. state and external potential.16 . Quantum interaction with external systems
A realistic model for a relativistic interaction keeping the particle from falling—the forcesdue to the electromagnetic interaction with the atoms of a table, for instance, on which theparticle is at rest—will not be of the form (B10), as relativistic interactions generally donot depend on position only but also on momentum. As an example, take the DarwinLagrangian [31], or a Klein-Gordon field minimally coupled to an electric potential φ , whererelativistic corrections to the Schr¨odinger equation resulting from a 1 /c expansion [12] willcontain momentum dependent terms proportional to φ P .Hence, for a more physical model for the interaction, we ask for a ˆ U ext which is an operatorvalued function, not only of the position operator but also of momentum. Appealing to thesituation where an equilibrium of forces is achieved, e. g. for the electromagnetic interactionwith the atoms of, say, a table, we assume that this function has no explicit time dependence,and make the ansatz ˆ U ext = ˆ U (0)ext + 1 c ˆ U (1)ext , (B11)where we require that ˆ U (1)ext can depend on P and, therefore, does not commute with X ,while ˆ U (0)ext does commute with X . At lowest order (1 /c ) we then get exactly the sameexpression (B7) as in subsection B 1, yielding again U (0)ext = − M gX . The first order correctionnow becomes c d (cid:104) X (cid:105) d T (cid:12)(cid:12)(cid:12)(cid:12) O ( c − ) = − g h (cid:10) [[ X, H (0) ] , { X, H (0) } ] (cid:11) − h (cid:68) [[ X, H (0) ] , ˆ U (1)ext ] (cid:69) − g h (cid:10) [[ X, { X, H (0) } ] , H (0) ] (cid:11) − h (cid:68) [[ X, ˆ U (1)ext ] , H (0) ] (cid:69) (B12a)= − gM (cid:10) H (0) (cid:11) − i¯ hM (cid:68) [ P x , ˆ U (1)ext ] (cid:69) + gM (cid:10) P x (cid:11) − h (cid:68) [[ X, ˆ U (1)ext ] , H (0) ] (cid:69) . (B12b)Rearranging this expression to collect the terms that depend and do not depend on theexternal potential, respectively, and requiring the acceleration to vanish (for the Rindlerobserver), as before, then yields (cid:104) [ P x , ˆ U (1)ext ] (cid:105) − i M ¯ h (cid:104) [[ X, ˆ U (1)ext ] , H (0) ] (cid:105) = i¯ hg (cid:18) (cid:104) H (0) (cid:105) − (cid:104) P x (cid:105) M (cid:19) . (B13)Using the definition (B1c) of H (0) , this further simplifies: (cid:104) [ P x , ˆ U (1)ext ] (cid:105) − i2¯ h (cid:104) [[ X, ˆ U (1)ext ] , P ] (cid:105) = i¯ hg (cid:18) (cid:104) H (0)rel (cid:105) + (cid:104) P (cid:105) − (cid:104) P x (cid:105) M (cid:19) . (B14)17n order to satisfy this relation, ˆ U (1)ext can contain arbitrary position independent terms ( P , P , P , ...), but on top of that only terms linear in X not higher than second order in P arepossible. Considering the commutators[ P x , X ] = − i¯ h , [[ X, X ] , P ] = 0 , (B15)[ P x , XP x ] = − i¯ hP x , [[ X, XP x ] , P ] = − ¯ h P x , (B16)[ P x , X P ] = − i¯ h P , [[ X, X P ] , P ] = − h P x , (B17)[ P x , P X ] = − i¯ h P , [[ X, P X ] , P ] = − h P x , (B18)we see that the no-acceleration condition is satisfied forˆ U (1)ext = − H (0)rel gX − g M ( αX P + β P X ) , (B19)with α + β = 1. Demanding that the X P and P X terms enter symmetrically, we obtainˆ U (1)ext = − H (0)rel gX − g M { X, P } . (B20) [1] R. Colella, A. W. Overhauser, and S. A. Werner, “Observation of gravitationally inducedquantum interference,” Phys. Rev. Lett. , 1472–1474 (1975).[2] D. G. Currie, T. F. Jordan, and E. C. G. Sudarshan, “Relativistic invariance and hamiltoniantheories of interacting particles,” Rev. Mod. Phys. , 350–375 (1963).[3] Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynam-ics , edited by Robert M. Wald, Henry J. Frisch, Gene F. Mazenko, and Sidney R. Nagel,Chicago Lectures in Physics (The University of Chicago Press, Chicago, 1994).[4] Romeo Brunetti, Claudio Dappiaggi, Klaus Fredenhagen, and Jakob Yngvason, eds.,
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