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arXiv:1906.03725v1 [quant-ph] 9 Jun 2019 srpeetdb ntr operators unitary by represented is element ( oa translations poral .Glligroup Galilei try. [ rule 22 mass-superselection the for ment [ variables dynamical conjugate are 11 time proper mass where and developed been has formalism [ consistent fully or- experimentally MSR in test setups to optical der quantum of with dynamics simulate [ mass-superpositions to forward physical put all were questioned proposals at nally, been are has it mass-superpositions frames [ whether reference Newtonian quantum of not the- text quantum is non-relativistic the ory in that proposed been ok tdigrltvsi ffcsi o-nryquantum of [ low-energy body systems in growing effects the relativistic to studying of relevant works are degrees results internal Our low-energy dynamical of freedom. with terms of particles in theory masses relativistic Newtonian result dynamical the The with of particles dynamics. and interpretation its system an defi- of offers composite correct limit also a the non-relativistic for is the parameter resolution of mass the a of to non-relativistic nition key the The in do we ? mass-superpositions Why question: observe the not answering MSR, the of content are MSR of validity nor understood. meaning fully physical neither that h nlss otayt oeagmns[ arguments some to contrary analyses, the fsae soitdwt ieetmse [ masses different some with that superpositions associated considered suppressing states been be vari- of has could a It process from dynamical discussed perspectives. been of have ety (MSR) rule selection R~ x asspreeto rule.– Mass-superselection , physical the of explanation natural a out point we Here shows and approaches above the challenges This ]. h ttsadpyia enn ftems super- mass the of meaning physical and status The 23 + ,aie o unu hoywt aiensymme- Galilean with theory quantum a for arises ], ~wt g 12 + ,~w,~a,bR, ~a, t – 2 19 al o experiments. particles, cla top composite and table relativistic physics describing non-standard any formalism invoking a r particl without a composite MSR of the offer dynamics ma of we ho relativistic of Here of issues: notion limit two relativistic tonian of physics. dynamical understanding fundamentally Newtonian the the in is from present result the be supersele for a to Crucial yield seem not not does orig approach do fea Its consistent MSR a debated. 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( n boosts and m m − to ue e,sproiin fmasses of superpositions Yet, rule. ction h oino asprmtrarises parameter mass a of notion the w seeg,adwa stecretNew- correct the is what and ss-energy, nldrvto skont einconsis- be to known is derivation inal U 2 stems aaee.Tecu fteargu- the of crux The parameter. mass the is m iitcqatmter superpositions theory quantum tivistic a ietpyia nepeaini terms in interpretation physical direct a has ) K ( 26 ′ ~ e ok 03,NwYr,USA York, New 10031, York, New f (i) ) g ue vni etok,isvalidity, its textbooks, in even tures ie h eainbtenMRand MSR between relation the rifies 1 − ewe h tts n hsadffrn state different a thus and states, the between ?( rae ymtysol edt more to should symmetry greater (A ]? = ersnsiett in identity represents ) ~a nater ihms en parameter a being mass with theory a In ) m~ x U g slto fteMRpuzzle. MSR the of esolution ( m ± g 10 − g − ~w − , ~ ~w and 21 ilsa identity an yields sadPhysics, and cs ~a 2 t~p ) 22 g U , − where , ,wihbcmseietwhen evident becomes which ], ( 23 ~w g stralia 24 m ~a g – ~a ) m ′ ,bthwcudagreater a could how but ], 25 U g ~w rdcsarltv phase relative a predicts ( .A eut h follow- the result, a As ]. = 2 g 23 ~ x oasproiinstate superposition a to ) = ~w m stepsto operator position the is ]; = ) id G ′ oee,snethe since However, . (ii) G tcno le phys- alter cannot it , e − id aga’ argu- Bargman’s im~a ~w etna the- Newtonian G in I G int : 22 ~p . ]: n of and ~ x ′ (1) (2) = 2

~x − ξ~(t), t = t′ with arbitrary ξ~(t). The non-relativistic with m replaced by M. Instead of the pure phase e−im~a~w Schr¨odinger equation (~p 2/2m)ψ(~x, t) = i~ψ˙(~x, t), where we thus have an operator e−iM~a~w implementing the same ψ˙ := dψ/dt, in the above primed coordinates becomes shift along q as found in eq. (4). ′2 ¨ ′ i~ϕ˙ = ~p + mξ~ · ~x′ ϕ, where ψ(~x, t) = eif(~x ,t)ϕ(~x′,t) We can now appreciate the essence of the MSR im- 2m passe: mass superpositions are not constrained by any ˙ and f(~x′,t) = m(ξ~ · ~x′ + dt ξ˙2/2)/~. Generalising the superselection rule in a theory which actually includes sequence of boosts and shifts to an arbitrary closed path, different mass states that could be superposed. But if R ξ~(0) = ξ~(T ) = 0, the state in the primed frame at the end indeed there are no constraints on mass-superpositions, ′ ~ T ˙2 why aren’t they ubiquitous in Newtonian physics? On acquires a phase ψ (T ) = exp{(im/ ) 0 dt ξ /2}ψ(T ). Proper time elapsing during this round-trip as measured the other hand, superposing masses unavoidably yields ′ TR ˙2 2 1/2 proper time effects, which have no interpretation in a in the primed coordinates is T = 0 dt(1 − ξ /c ) and the above phase factor is given by time dila- non-relativistic theory. Below we present our resolu- tion ∆τ = T − T ′ between the twoR frames mc2∆τ = tion to the problem, structured along the lines suggested T ˙2 −4 in [23]: find the system for which the states with dif- m 0 dt ξ /2+ O(c ). These observations will be cru- cial for our resolution. ferent masses can actually be prepared; and address the NewtonianR particles with a dynamical mass.– To in- question what is then measurable. vestigate the problem in a consistent manner, Newtonian Relativistic composite particles.– The extended theory has been extended to include mass as a dynamical Galilei transformation hints that the resolution of the operator M. The simplest extension yields a Hamilto- MSR puzzle must include the notion of proper time. nian [4, 7–9, 11, 23] The failure of the dynamical-mass extension of the New- tonian framework to yield MSR hints that dynamical ~p 2 mass should be taken seriously. We therefore begin H = + MΦ(x), (3) dm 2M with a fully relativistic theory of particles with internal degrees of freedom (DOFs) [12, 13, 27, 28]. ¨ where we have included the non-inertial term ξ~ · ~x′ as a The square of the relativistic four momentum pµ, µ = gravitational potential Φ(x), in of the equivalence 0, .., 3 is an invariant quantity describing particle’s energy 2 µ ν principle. Since the Newtonian limit of relativistic dy- in the rest frame [29] Hrc = − p gµν p , where gµν namics contains the term mc2, the corresponding term is the with signature (−, +, +, +), and c P Mc2 is often included in (3). The argument we outline is the . For a composite system, e.g. an below works in either case. , Hr comprises not only the sum of the masses of the The surprising finding is that no MSR arises in this constituents but also their binding and kinetic energies dynamical-mass extension of the Newtonian dynamics, – it is a fundamentally dynamical quantity describing see ref. [23] for details. The reason is that the symme- internal DOFs. In an arbitrary reference frame, the total try of eq. (3) is not G but its central extension G˜, with energy is H ≡ cp0 and for a static symmetric metric reads M as the central element1. The new group elements 2 j 2 g˜α,R, ~w,~a,b ∈ G˜ implement the action of the Galilei group H = −g00(c pj p + Hr ), (5) on the spacetime coordinates, as well as shifts along an j q i j additional coordinate q associated with the mass. One where pj p ≡ i,j=1,2,3 p gij p ; For a field-theory deriva- tion see [18, 27, 30, 31], for derivation as a limit of an can thus writeg ˜ ≡ (α, g) whose action on all the coordi- P 1 2 N-particle bound system see [28]. Locally, the symmetry nates reads [23]g ˜(q, ~x, t) = (q + α − ~wR~x − 2 ~w t, R~x + ~wt +~a, t + b). Crucially, the chain of transformations (1) of eq. (5) is the central extension of the Poincar´egroup, in G˜ does not result in an identity but in a shift of the with Hr being the central element. Hr is a generator of internal coordinate by ~w~a internal dynamics and commutes with all other genera- tors. The Poincar´ealgebra is otherwise unchanged com- g˜−~ag˜− ~wg˜~ag˜~w = ( ~w~a, idG). (4) pared to the case for a structureless particle where m is a parameter labelling the representation. Such a central The unitary operators implementing G˜ are the same as in extension is ‘trivial’, it is a product of the Poincar´eand the Galilei group, with the replacement of m → M, where of the internal symmetry group. M is a generator of translations along q. Therefore, the We now seek the non-relativistic limit of eq. (5). The unitary representation of the loop remains as in eq. (2) usual approach is to take a low-energy limit (small veloc- ities and weak gravity)

~p 2c2 Φ(x) 1 Hle = Hr + + Hr 2 , (6) Central extension of a group is a group whose quotient by the 2Hr c one-parameter subgroup, here generated by M, is isomorphic to the original group, and where this subgroup commutes with all where ~p = (p1,p2,p3) is the three-momentum. Crucially other group generators. the dynamics of the centre of mass (CM) given by Hle 3 and by Hdm is fully equivalent – as can directly seen by first observation above) gives the correct non-relativistic replacing the internal energy in eq. (6) with the dynam- Hamiltonian of a composite system 2 ical mass in appropriate units: Hr → Mc . Below we 2 2 ~p show that Hle, eq (6), and therefore also Hdm, (3) do mc + H0 + + mΦ(x). (9) not define a non-relativistic theory. 2m Time of an internal observable a under Hle Note that the dynamical mass-energy is not entirely sup- ~ is described bya ˙ = i[Hle,a]/ =: ω where pressed but survives in the rest energy term, resulting in the familiar expression for a total energy of a non- 2 ~v Φ(x) relativistic composite system – where internal energy H0 ω = ω0 1 − 2 + 2 , (7) 2c c ! simply adds to the CM kinetic and potential energies. Composite particles and MSR.– We can now anal- ~ where ~v = ∂Hle/∂~p and ω0 := i[Hr,a]/ . Note that ω0 is yse what happens in the correct Newtonian limit (9) the rest frame speed of internal dynamics and describes for a superposition state of two mass-energies | M1i + time evolution with respect to proper time τ, i.e. ω0 = 2 | M2i, where | Mii, i = 1, 2 is the eigenstate of M = da ~v 2 2 dτ . To lowest post-Newtonian order dτ = (1 − 2c2 + Hr/c ≡ mIint + H0/c with the eigenvalue Mi. We Φ(x) 2 2 )dt and eq. (6) thus includes the lowest order time have M(| M1i+| M2i)= m(| M1i+| M2i)+E1/c | M1i+ c 2 dilation effects: the velocity-dependent term describes E2/c | M2i, where Ei is the internal energy H0| Mii = the special relativistic and the potential- Ei| Mii. The state is a superposition of mass-energies, dependent term describes the gravitational time dilation, but it has a well-defined mass, since by definition the see also [19, 32]. Newtonian mass is an operator proportional to iden- The first essential observation is that for composite tity, mIint. Mass-energy superpositions are therefore nei- particles, one needs to better define when a theory is non- ther forbidden nor lead to superpositions of Newtonian relativistic. We propose the following: a non-relativistic masses: In the correct Newtonian limit the dynamical limit should give rise to Euclidean notion of spacetime, part of the mass-energy H0 is negligible in the inertial with global time. This is the case in eq. (7) when ω ≈ ω0. and gravitational potential energy terms, where only the To understand under what conditions this happens it is static part mIint contributes. It is this static part which instructive to split the rest energy Hr into a static part we recognise as “the mass” in the non-relativistic physics. E0 ·Iint, where Iint is the identity operator on the inter- Allthough in the Newtonian limit the state | M1i + nal DOFs (which we skip hereafter) and the remaining | M2i has a fixed value m of inertia and weight, it is in dynamical part H0 := Hr − E0, so that Hr ≡ E0 + H0. a superposition of rest mass-energies, as the dynamical We can now take the limit H0 ≪ E0 of eq. (6), which to part of the mass-energy survives as the additive internal lowest order in 1/c2 yields energy H0 in eq (9). How does G emerge from G˜? As anticipated [23], un- ~p 2c2 Φ(x) derstanding MSR also explains how G emerges from G˜ as H ≈ E + + E + H le 0 2E 0 c2 0 a symmetry in non-relativistic physics. Note first that 0 (8) ~p 2c2 Φ(x) non-relativistic limit of the Poincar´egroup (its In¨on¨u- +H − + . 0 2E2 c2 Wigner contraction [24, 33]) for structureless particles is  0  the central extension of the Galilei group and not the The first three terms do not contribute to internal dy- Galilei group [24]. Consequently, the low-energy limit namics and the term H0 alone gives universal internal of the central extension of the Poincar´e group, with 2 evolution, independent of the CM. The notion of a global mc → Hr, is the central extension of the Galilei group 2 time is therefore recovered when the remaining terms, the with Hr/c as the centre [27]. This can be seen as a phys- second line of eq. (8), are negligible. When these terms ical reason why G˜ arises as a symmetry of “Newtonian” are absent, the rate of internal dynamics is independent particles with dynamical masses, found in ref. [23]. of the velocity or gravitational potential difference be- We have just shown that a consistent non-relativistic tween the rest frame of the particle and the frame with limit for composite particles is obtained when the inertial time coordinate t. and gravitational mass-energies are effectively a param- The second essential observation is that we do not au- eter. In the unitary representation of the boost we are tomatically have a notion of a mass-parameter for a rel- then left with a parameter m instead of the operator ativistic composite system – we only have a dynamical M, since it is the inertial mass which is relevant here. quantity Hr, the energy in the rest frame of the sys- The commutator between the boost and the translation tem [10], which in the low-energy limit defines all the generators thus becomes pK − Kp → m. The resulting mass-energies: the rest mass-energy, inertia and weight, symmetry group is a product of the central extension of see eq. (6). Eq. (8) offers a natural definition of the mass the Galilei group with a parameter m at the centre and 2 parameter as the static part of Hr, i.e. m := E0/c . In- a one parameter group of internal symmetries generated 2 corporating this into eq. (8) and taking H0/mc ≪ I (the by H0. However, transformations resulting from such a 4 central extension of the Galilei group (with a parameter non-relativistic limit (9) rest mass-energy is essentially m in the centre) are indistinguishable from those origi- unchanged (H0 and Hr only differ by an unobservable nating from the Galilei group itself. The two symmetries constant E0), whereas inertia and weigh are given by 2 are empirically indistinguishable [21, 23–25], both in the E0/c . This preserves the validity of the remaining part quantum and in the classical case 2 of the EEP, the Weak Equivalence Principle. For dis- What is measurable in the Newtonian framework? cussion and experimental tests of the EEP for composite For a non-relativistic limit of any theory to be meaning- quantum particles see refs [18, 31, 38–40]. ful, we have to assume that measurements can only have a finite precision. If we could measure internal states ar- Conclusion.– Lack of mass-superpositions in Newto- bitrarily precisely, time dilation of their evolution would nian physics is a consequence of the operational defini- never be negligible. Experiments with atomic clocks have tion of the mass-parameter and of the non-relativistic already measured time dilation between clocks with rel- limit of dynamics of composite particles. It does not re- ative speed of ∼ 10m/s, and at a height difference of quire any restriction on kinematics. The problem with ∼ 30cm [34]. State of the art clocks could even mea- MSR has been rooted in a presumption that promoting sure time dilation due to relative velocities ∼30cm/s and mass to a dynamical variable in a Newtonian Hamilto- height difference ∼2cm [35]. This further illustrates that nian still yields a non-relativistic theory [1–5, 11, 23]. for systems with internal DOFs one cannot meaningfully However, promoting mass to a dynamical quantity is op- define the Newtonian limit only in terms of the CM. erationally equivalent to incorporating relativistic mass- The lack of signatures of dynamical masses in Newto- energy equivalence, which brings in the time dilation ef- nian physics is therefore not due to the lack of observ- fects [7, 8, 10, 12, 18]. In fact, mass-energy equivalence ables that can measure mass-energy in superposition, but has been used by Einstein to derive the gravitational due to the fact that the Newtonian limit is defined as time dilation [41]. The symmetry of the dynamical-mass the limit where dynamical mass-energy contributions to framework comprises Galilean transformations for space- inertia and weight are negligible. Consider Rabi oscil- time coordinates and translations along a new coordinate lations between internal states of an atom [36]. They associated with the dynamical mass [4, 11, 23]. The low- demonstrate between different eigenstates of energy relativistic theory is instead invariant under cen- H0, and thus of M. But unless their frequency is very tral extension of the Galilei group with mass-energy as high, the physical effects arising from the dynamical part the central element, which is the Lorentz symmetry up to of the atom’s inertia and weight are negligible. On the 1/c2 [27], and proper time takes the role of the additional other hand, the above mentioned time dilation measured coordinate [7, 8, 10]. with atomic clocks directly verifies the dynamical na- ture of inertia and weight as described by Hle (these efffects are fully explained by eq. (7)). We usually do Independently of its interpretation, the regime of low- not think of time dilation effects as demonstrating mass- energy particles with dynamical mass-energy has its own superpositions – nor as effects violating MSR – although symmetry and phenomenology, and can be studied fully 2 Hle with Hr → Mc is operationally the same as Hdm, in its own right. It has already allowed exploring proper with the added rest mass term, as in refs [7–11]. time effects in unstable [9] and interfering quantum par- Einstein Equivalence Principle.– The notion of a ticles [12, 14–16, 42], it was used to assess the limits to mass parameter emerges in the non-relativistic limit the notion of an ideal clock [19, 43] and to the notion of by splitting the mass-energy into two separate quanti- time [30], and to study the role of mass-energy equiva- ties: mass and internal energy.3 The Newtonian limit lence in atom-light interactions [17, 44]. The approach thus breaks two of the three constituents of the Ein- has further enabled a quantum formulation of the EEP stein Equivalence Principle [37]: Local Lorentz Invari- for composite particles [18] and can shed light on the role ance and Local Position Invariance, where the former re- of proper time in quantum-to-classical transition [13, 45– quires equality of rest mass-energy and inertia and the 47]. latter – of rest mass-energy and weight. Indeed, in the Acknowledgment.– M.Z acknowledges Australian Re- search Council (ARC) DECRA grant DE180101443 and 2 ARC Centre EQuS CE170100009. This publication was Te extended Galilei group already appears as a symmetry in clas- sical : commutators of group elements in the quan- made possible through the support of a grant from the tum case and their Poisson brackets in the classical case are the John Templeton Foundation. The opinions expressed in same up to the imaginary unit. Consequently, central extension this publication are those of the authors and do not nec- of the Galilei group also appears as a symmetry in a classical essarily reflect the views of the John Templeton Founda- theory where the mass is taken to be dynamical [23]. 3 Even more broadly, the masses of most standard model particles tion. The authors acknowledge the traditional owners of arise as a static energy term – where the energy is that of the the land on which the University of Queensland is situ- Higgs field. ated, the Turrbal and Jagera people. 5

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