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Relativistic Unruh-DeWitt detectors with quantized center of mass

In this paper, we extend the Unruh-DeWitt (UDW) model to include a relativistic quantized center of mass (COM) for the detector, which traditionally has a classical COM and follows a classical trajectory. We develop a relativistic model of the detector following two different approaches, starting from either a first- or second-quantized treatment, which enables us to compare the fundamental differences between the two schemes. In particular, we find that the notion of localization is different between the two models, and leads to distinct predictions which we study by comparing the spontaneous emission rates for the UDW detector interacting with a massless scalar field. Furthermore, we consider the UDW system in both a vacuum and medium, and compare our results to existing models describing a classical or quantized COM at low energies. We find that the predictions of each model, including the two relativistic cases, can in principle be empirically distinguished, and our results can be further extended to find optimal detector states and processes to perform such experiments. This would clarify both the role of a quantized COM for interactions with an external field, and the differing localizations between the first- and second-quantized treatments.


I. INTRODUCTION
The UDW model, originally proposed by Unruh [1] and later simplified to its monopole form by DeWitt [2], has become the customary model for particle detectors in quantum physics. It was originally formulated for the investigation of quantum field theory (QFT) in both flat and curved spacetimes [3][4][5], is commonplace in the study of relativistic quantum information [6][7][8] and entanglement harvesting [9][10][11], and is used in quantum optics as an idealized model of the atom-light interaction [12][13][14].
Conventionally, the detector is modelled as a quantized two-level system following a classical trajectory, i.e., with a classical COM. The traditional UDW model is an idealization which does not account for fully quantized dynamics, but can be extended to include quantized external degrees of freedom. Such a detector model with a quantized COM was previously studied in the non-relativistic regime by Stritzelberger and Kempf [15] for inertial and uniformly accelerating detectors (but see also [16] for an earlier, atom-light treatment). This model was proposed in order to study the dynamics of a quantized COM, and the effects of the COM coherence on the detector-field interaction.
However, a non-relativistic treatment of a quantized COM does not come without issues, and raises the question of the consistency of the resulting model. These subtleties are especially relevant in the case of quantum electrodynamics (QED) and quantum optics, where a quantized COM extension of the detector model is often considered, see e.g., [14] and references within. In an early paper studying the effects of a COM in QED, Wilkens [17] found spurious velocity-dependent * e.gale@uq.edu.au † m.zych@uq.edu.au effects due to the omission of requisite relativistic corrections, namely the Röntgen interaction term [18], which were resolved once this correction was introduced [19]. However, the inclusion of the Röntgen term alone is not enough to deal with all spurious effects; the mass defect of the detector must also be considered in order to avoid anomalous friction forces [20]. These non-physical results are ultimately due to the mixing of the Galilean and Lorentz groups, which respectively govern the dynamics of the detector and field. The problems of the model can be partially remedied by introducing the Röntgen term and modelling the mass defect, but as Wilkens noted, "the ultimate cure of this deficiency may be expected from a relativistically covariant description of the atomic motion . . . this program is, however, highly nontrivial" [17].
There are two possible approaches to developing a relativistic covariant formalism, one starting from a relativistic second-quantized description and the other by developing a relativistic first-quantized description. While the detector in the UDW model is most commonly described as a first-quantized system, a second-quantized model was also proposed in Unruh's original paper [1], and has been considered in some more recent analyses [21,22].
In this paper, we extend the UDW detector model to incorporate a relativistic quantized COM for both the first-and second-quantized approaches. We find that these two models lead to distinct notions of localization, the consequences of which can be studied for a given physical process, and specifically consider in the case of spontaneous emission. We compare the first-and secondquantized models by studying their predictions on the rate of spontaneous emission, for which we obtain analytical expression as a functional of the detector's initial state. An advantage of working within the UDW model is that spin is neglected, which is particularly convenient for the comparison of the different localizations. In addition to questions of localizability, we also consider a detector interacting with a field in both a vacuum and medium, from which we are able to obtain previously proposed models for the non-relativistic [15] and "semirelativistic" [23] regimes as limiting cases.
Often, one develops relativistic quantum mechanics (RQM) within the framework of a relativistic QFT, i.e., second-quantized approach; the status of a relativistic non-field-theoretic quantum mechanics is controversial, and is often objected to on both formal and ontological grounds, e.g., regarding the non-localizability and observer-dependence of particles (for some discussion on this and related topics, see e.g., [24][25][26][27][28]). While a second-quantized approach is required for a fully relativistically covariant treatment of standard quantum mechanics, our intention here is to present and explore consequences of both of these approaches. Furthermore, analogous to how non-relativistic quantum mechanics (NRQM) exists as a limiting case of relativistic QFT, a first-quantized RQM is also "embedded" within such a theory, provided that the system remains in a low-energy regime with respect to the Compton scale, as for example evidenced by experiments such as those studying the electron's magnetic moment [29,30].
Throughout this paper, we refer to the different formal descriptions of quantum mechanics as either "first-" or "second-quantized." Although this terminology was originally coined based on a historical misunderstanding, we adopt this terminology to avoid potential confusion when referring to the detector and field constituents. That is, we refrain from referring to the fully covariant approach as a "field" theory, and instead refer to "first-" or "second-quantized" approaches, either of which may be treated relativistically.
In Sec. II, we review the conventional UDW model, its history and various extensions, paying close attention to how the different models are related. We review the standard point-like and smeared detector models, the quantized COM extension, and second-quantized treatments of the detector. In Sec. III, we present our relativistic first-quantized model of the UDW detector and compare to the second-quantized formulation, primarily with regards to the differences in localization. Additionally, we consider the detector-field system in both a vacuum and medium, and compare the relativistic first-and secondquantized models to earlier results featuring a classical, or semi-/non-relativistic COM. In Sec. IV, we derive the spontaneous emission rate of the detector as a functional of the initial COM wavefunction, which we take to have a Gaussian profile, and again compare our relativistic treatment to past models. Finally, we discuss our results in Sec. V, where we systematically address the issues of localizability, time, and mass-energy in RQM, before concluding in Sec. VI. We consider (3 + 1)-dimensional Minkowski spacetime with metric signature (+, −, −, −), and abbreviate (x, t) by x. We work in natural units = c = 1, although we restore units when needed for clarity.

II. UNRUH-DEWITT DETECTOR MODELS
Unruh [1], in their seminal paper, originally proposed two detector models: a "particle in a box" detector model and a second-quantized detector model. The latter is generally not studied, while the former was simplified to the well-known point-like "monopole" description by De-Witt [2], now known simply as the UDW model. While the UDW detector has become the standard model, and the one most often treated in the literature, various iterations have since been made. In this section, we review the monopole detector model and some common extensions made by past authors, and conclude by re-introducing and evaluating the second-quantized model, which has been the subject of some renewed interest [21,22].
In its most general form, the UDW model features a detector coupled to a field, with total Hamiltonian For the moment, we leave the Hamiltonians for the detec-torĤ D and interactionĤ I unspecified, but subsequently treat in detail. In the UDW model, a real-valued scalar field is most commonly considered, with a free Hamiltonian given byĤ where ω(k) denotes the dispersion relation for the field, andâ † (k) andâ(k) are respectively the creation and annihilation operators, which satisfy the commutation re- The scalar field, coupled to the detector in the interaction Hamiltonian, is characterized by the field operator φ(x). Expandingφ(x) into its positive and negative frequency modes, we choose the planewave basis such that where ν is the propagation speed of the field, and we have restricted our analysis to Minkowski spacetime. A full derivation ofφ(x), and summary of conventions employed, can be found in the appendix. We distinguish the propagation speed of the field from the speed of light in our analysis for two reasons: (i) to separate the relativistic nature of the detector from the field, which allows for easier determination of the non-relativistic limit; (ii) for the study of detectors in media (which has been previously treated in [15,31,32]), and for possible connections to analog models of relativity [33].
The standard treatment of a scalar field in a vacuum can be re-obtained by taking ν = c, which in natural units is ν = 1. The explicit form of the interaction Hamiltonian depends on the model of the detector and its coupling with the field. In the simplest case, one has a pointwise coupling between the field and monopole detector, but this is often extended to include a spatial smearing in the detector-field coupling, which we discuss in the following section.
A. Point-like and smeared monopole detectors The point-like detector model is generally treated as an idealization of a particle confined to a box, or alternatively as an atom, ion or molecule, which can be modelled as a two-level system with ground state |g and excited state |e , and energies E g and E e respectively. Furthermore, a "particle" is said to be detected when the detector transitions from its ground state to its excited state. While the detector has quantized internal states, it is assumed to have a classical COM and to follow a corresponding classical worldline (t(τ ), x(τ )), which is conventionally parameterized by its proper time τ .
From these assumptions, the free Hamiltonian for the detector is simply given bŷ where E = E e − E g denotes the energy gap, and without loss of generality we have taken E g = 0.
The interaction between the detector and scalar field is most often treated as a linear coupling, taken as analogous to the atom-light interaction between a dipole and quantum electromagnetic field −d ·Ê. The Hamiltonian for the detector-field interaction in the interaction picture is described byĤ where λ is the coupling strength,φ(x) is the aforementioned quantized scalar field, andμ(t) is the monopole operator for the detector, defined bŷ enabling the field to excite and de-excite the detector. The simple coupling between the detector and field in Eq. 5 is commonly considered in the literature, where the interaction is generally assumed to be time-independent. However, if the interaction is confined to a finite time interval, then one encounters ultraviolet divergences in the response of the detector [34]. These divergences can be removed by allowing the coupling to vary in time, such that λ → λ(t), and requiring the interaction to be smoothly switched on and off [35][36][37].
Further problems arise from the point-like nature of the detector model. Ironically, while the point-like approximation originally introduced by DeWitt [2] was intended to simplify the model, it leads to divergences that must in turn be regularized by modelling the finite extension of the detector. As treated in [3,[38][39][40], these problems are resolved by modelling the spatial profile of the detector, which can be introduced by modifying the interaction Hamiltonian to include a smeared coupling between the detector and field, i.e., λ → λ(x).
To ensure that the response of the detector is safe from the above-mentioned problems, both the switching of the interaction and the finite extension of the detector must be included in the model. Conventionally, the time-dependent and smeared coupling is decomposed as where χ(t) is the switching function, and F (x − x D ) is the smearing function centered around the stationary detector's COM at x = x D . The interaction Hamiltonian is obtained by integrating over the spatial extent of the detector The switching and smearing functions are smooth functions, commonly chosen to have compact support, or to have a Gaussian or Lorentzian profile. While one largely has freedom to choose the switching function for the interaction, the introduction of a smearing profile is physically dubious if done in an ad hoc manner. Instead, one can derive the smearing profile from firstprinciples by considering a specific physical description for the detector. For example, the smearing associated with the discrete energy levels of a particle trapped in a box [41], or the smearing of a two-level atom given by its atomic wavefunction [13].
While the introduction of a spatial smearing is sufficient to the problems caused from the point-like idealization, the model still assumes a detector following a classical trajectory. A fully quantum treatment would include the quantized degrees of freedom associated with the detector's COM. Moreover, by quantizing the COM degrees of freedom, the motivations for introducing a smearing profile do not arise; thus, smearing is not considered in the subsequent models.

B. Detectors with quantized center of mass
To provide a more physical account of the detector and its smearing, one can take the detector to be localized inside a box, rather than assume a point-like monopole. Such a model was given by Unruh and Wald [41], who considered a scalar field coupled to a detector with a quantized COM, i.e., x D →x D describes the position operator of the detector. The interaction between the box-detector and field in the Schrödinger picture iŝ and by introducing a completeness relation over position states, can equivalently be written aŝ Because the detector is confined in a box, we may express the above in terms of the detector's energy eigenstates |j . Further insertion of completeness relations giveŝ where ψ j (x) ≡ x|j denotes the position-space wavefunction for the jth energy level. Following [41], one may restrict the detector to two energy levels j = {g, e} and drop diagonal terms in the interaction, i.e., |g g| and |e e|. One thereby obtains an interaction with a position-dependent monopole operator wherê which may be compared to the earlier introduction of smearing for the point-like detector (8).
A different extension of the UDW model, which idealizes the atom-light interaction, was considered by Stritzelberger and Kempf [15]. Their detector model extends the traditional UDW model to include the quantized COM, in addition to the usual quantized internal degrees of freedom for the two-level system. In their original paper, Stritzelberger and Kempf consider an inertial detector whose evolution is described by the free nonrelativistic Hamiltonian wherep is the COM momentum operator and M the detector's mass. As opposed to the traditional UDW model, which features a detector following a classical trajectory, a COM rest frame (where p = 0) cannot be freely chosen, since this would violate the Heisenberg uncertainty relation. Therefore, the external COM degrees of freedom must be included in the free evolution, where the detector states now live in the product Hilbert space H D = H ext ⊗ H int . The free Hamiltonian has eigenstates of the form where E j is the jth energy level, corresponding to either the ground or excited state of the detector j = {g, e}.
The detector-field interaction has the same form as the box detector model of Unruh and Wald [41], but now features a monopole operator separate from the COM degrees of freedom, which couples to the field aŝ where the detector's monopole and COM are evolved with respect to the detector's free Hamiltonian (14) Given the non-relativistic form of Eq. (14), the Hamiltonian is no longer Lorentz invariant and cannot describe relativistic trajectories. Moreover, as discussed in the introduction, the mixing of the Galilean and Lorentz groups between the detector and field leads to nonphysical effects, particularly in the full atom-light interaction, although it can be partially remedied by inclusion of relativistic corrections, e.g., the Röntgen term.
In the case of the UDW model proposed by Stritzelberger and Kempf [15], an incorporation of the detector's mass defect was recently studied in [23], whereby one models the change in the detector's mass-energy resulting from emission and absorption. Such a "semirelativistic" approach is advantageous insofar that one wants to avoid a relativistic treatment, although the aforementioned problems can be completely resolved only in a relativistic second-quantized model.

C. Second-quantized detector models
In addition to the box detector model, Unruh [1] proposed a fully relativistic second-quantized model, where the detector was represented by a composite scalar field. Despite its initial proposal almost a half-century ago, the second-quantized model has seen comparatively little study. A likely reason for this unpopularity is that one loses the convenient localization, and simplicity, of the point-like UDW model. Despite these disadvantages, a full consideration of relativistic effects in a detector model is certainly of interest and has been considered in a number of previous analyses, most recently in [22].
In the past treatments, the second-quantized detector has generally been assumed to stay confined to some region, such as in a box-detector by Unruh or the field in a cavity considered by Grove and Ottewill [42]. A disadvantage of these models is that additional care must be taken to define the rigidity of the container, namely how one should formalize the detectors' walls, but is generally resolved by taking the walls to remain fixed relative to some inertial observer.
However, one might well ask whether a full treatment of the detector's container is really necessary, particularly given the simplicity of the UDW model where no mention or treatment of a box is required. Indeed, due to the complexities of the second-quantized model, it is much easier to simplify the model to a monopole detector by restricting to the low-energy regime [42]. Likewise, in a separate analysis by Colosi and Rovelli [43], they found that for a box-detector sufficiently large with respect to its Compton wavelength λ c ≡ /mc, the correlation functions of the localized states exponentially converge to the global states defined on Minkowski spacetime [44]. Consequently, one can justify neglecting the detector's container on the basis that one remains in the low-energy regime for large wavelengths with respect to the size of the detector, and where boundary effects are neglected.
In the analysis by Giacomini and Kempf [22], who studied the second-quantized model in the context of quantum reference frames, the detector is modelled as a composite real-valued scalar field with a continuum normalization. Like previous treatments, the detector has two energy levels j = {g, e} corresponding respectively to the ground and excited states. In the Schrödinger picture, the detector is defined bŷ with E j (p) = p 2 + M 2 j , and total mass-energy M j = m + E j given by the sum of the detector's rest mass and internal energy. The creation and annihilation operators satisfy the non-covariant commutation relations The free Hamiltonian for the detector is given by the sum of two fields describing the internal energieŝ while the Hamiltonian for the detector-field interaction is given bŷ In [22], to simplify the model's structure, the interaction Hamiltonian was restricted to the single-particle sector, resulting in which is close, but not identical to, the first-quantized interaction (16) introduced in [15]. Before we discuss the above interaction, let us observe that since (20) preserves the particle number of the detector fieldψ, so restricting it to the one-particle sector is redundant in the sense that applying this interaction to an initial state containing, say, one detector yields final states with also just one detector, possibly with a different internal energy.
In addition to different dispersion relations for the COM of the detector between (16) and (21), the second-quantized model has non-orthogonal "position states" defined by which are equivalently given by According to the textbook interpretation, the above states are position states in the second-quantized theory [45]; however, the presence of the integration measure 1/ √ 2E obfuscates such a straightforward interpretation. Clearly, while these "position states" transform covariantly, they do not correspond to eigenstates of a position operator, or any observable for that matter. This is significant given that the second-quantized position states do not correspond to an observable, i.e., Eq. (22) are not eigenstates of a Hermitian operator, a fact which was previously pointed out in [46].
The above-mentioned non-orthogonality of these second-quantized "position states" is also due to the presence of the integration measure. Indeed, these states do not correspond to a Fourier transform in momentum space, as is the case for the first-quantized position states [47]. This non-orthogonality is generally interpreted as a non-localizability in RQM, which has been the subject of extensive discussion in the literature, and has recently been studied in several papers analyzing the transition from relativistic QFT to NRQM [46,48,49].
To summarize, one can easily define a second-quantized model of the detector dynamics, which in the singleparticle sector has an interaction that resembles, but is not identical to, the first-quantized treatment due to the different localization that one obtains. However, if this resulting second-quantized model is adopted, then the question arises whether one can start directly from a relativistic first-quantized treatment of the detector? Such a model would already be constrained to the single-particle sector and would necessarily treat the detector as a localized system. In the following section, we therefore construct and analyze such a first-quantized model from the standpoint of RQM, and compare its predictions to both the second-quantized model, as well as semi-and non-relativistic ones.

III. RELATIVISTIC DETECTOR WITH FIRST-QUANTIZED CENTER OF MASS
In our review of past detector models, the physical description of the detector is often abstracted in order to focus exclusively on the field. Generally, in the context of quantum field theory, particle detectors are introduced in order to give an operational meaning to "particles," i.e., "particles are what particle detectors detect." There are a number of remarks to be made here, the first being that the idealization of a detector by the monopole model leads to problems which must be resolved by reintroducing the very structure that was originally abstracted away. Second, the operational definition of a particle detector risks circularity; just as a "particle" has some effective ontology, so too does a particle detector. Care must be taken to adequately define and formalize what precisely one means by a "detector." Here, we consider a "detector" to be a suitably localized system such that one can employ a first-quantized formulation, but where the free dynamics may not be abstracted. In formulating a relativistic first-quantized model, we take the interaction to be as given in the nonrelativistic case (16), which can equivalently be expressed in a form closer to the relativistic second-quantized model (21) (24) where we have combined the detector's external and internal states with the first-quantized position states. Note that we distinguish between the first-and secondquantized coupling constants, which have different dimensions between the two models. We take the firstquantized position states to be equivalent to the nonrelativistic theory, defined by These first-quantized position states, unlike those derived from the second-quantized theory, do not transform covariantly [50], but are orthogonal and correspond to eigenstates of a respective position operator. More specifically, they correspond to the Newton-Wigner position operator [51], which was derived by invariance conditions that a suitable position operator should reasonably satisfy. While we discuss the two localizations in more detail in Sec. V, at this stage we consider both the firstand second-quantized models and subsequently compare their predictions. Working in the interaction picture, we evolve the position states with respect to the coordinate time t (26) with the time-evolutions for the first-and secondquantized position states respectively defined by For convenience, we introduce the general position state where f j (p) is the integration measure for the second-(f j (p) = 1/ 2E j (p)) and first-quantized (f j (p) = 1) position states, cf. Eqs. (22) and (25) respectively. For the free dynamics of the detector, in RQM one conventionally considers a Hamiltonian of the form where the mass is treated as a c-number. However, this is inappropriate for the model we consider since the above Hamiltonian is unable to characterize the detector's internal degrees of freedom. Instead, we reformulate this expression to include a quantized mass-energŷ where we have introduced the mass-energy operator whose eigenvalues are given by the action on a corresponding mass eigenstatê For a detector described by a simple two-level system, one may express the mass-energy operator bŷ with M j = m + E j corresponding to the total massenergy of the detector, as in the second-quantized model. There are various motivations to quantize the massenergy, which we outline and discuss in Sec. V.

A. Calculation of spontaneous emission rate
For simplicity, we consider only a single physical process, namely spontaneous emission, which is primarily done to more easily compare with past analyses [15,23], and because it is a frequent subject of study in the UDW model, particularly from the context of quantum optics. One can also consider alternate physical processes, such as absorption and vacuum excitation, or in principle obtain more general results by formalizing in terms of correlations functions, which we defer to future work.
In the case of spontaneous emission, one starts with an excited detector in the vacuum with an arbitrary center-of-mass wavefunction ψ i of the detector, which can be represented in the momentum basis as For the final state, the detector is in its ground state with a spontaneously emitted field quantum Expanding the evolution operator perturbatively up to first order, we obtain The resulting transition amplitude evaluates to It is most interesting to find the total transition rate, which can be obtained by first deriving the total probability of the detector's final internal state, given by where we have traced over the final momenta for the field and detector.
In conventional analyses of the UDW model, one separates the detector's selectivity from the response function of the field and only considers the latter [4]. However, due to the modified form of both the interaction (26) and free Hamiltonian (29), the internal and external degrees of freedom of the detector are coupled, and as a result its dynamics can no longer be excluded. Therefore, we consider the full transition rate rather than just the response rate of the field. Substituting the transition amplitude (36), one finds and recognizing that the above integrals over time depend only on the interval s = t − t , but not on the total time s = t + t , we may find the transition rateṖ Since switching effects are neglected, we restrict to asymptotic times ∆t → ∞, and re-introducing the integration over the initial momentum p, the above expression becomeṡ At this point, we consider the different localizations for the first-and second-quantized models separately. For the case of a massless scalar field, the dispersion relation is ω(k) = ν|k|, and the transition rate evaluates tȯ where T denotes the "template function," originally coined and defined for the non-relativistic case in [15]. Considering each localization separately, the relativistic template functions for the first-and second-quantized models are given by where is an auxiliary function of the form [52] (a, b, c) The above expressions for an interaction in media simplify dramatically in the vacuum case, i.e., when taking ν = c = 1. While it may initially appear that the tem-plate functions (38) diverge, the results are well-defined in the limit ν → 1. The vacuum template function corresponding to the first-quantized localization is given by while for the second-quantized localization one finds where notably the second-quantized model features the reciprocal of the relativistic dispersion relation, in contrast to the first-quantized case; see also Fig. 1 for a comparison of these two localizations.
The different behaviour between the two models results from the distinct integration measures, which enter in the first-and second-quantized position states, and is ultimately due to the different notions of "local interaction" between the field and detector in the two cases. Moreover, the different dimensions of the two template functions is offset by the effective difference in dimensions between the coupling constants of each model, which give the correct dimension for the transition rate.
Given that the coupling constants between the firstand second-quantized localizations are distinct, it is necessary to find a means to equate them so that the two cases can be meaningfully compared. Requiring that the first-and second-quantized template functions agree for |p| = 0, one obtains a mass-dependent factor relating the respective coupling constants by Employing this relation between coupling constants, one obtains corrected expressions for Eqs. (38b) and (39b) which agree in dimension with the first-quantized case.
In all instances where we compare the first-and secondquantized localizations, such as for the template functions given in Figs. 1 and 2, we use the above results and in general employ Eq. (40).

B. Re-derivation of transition rates for semi-and non-relativistic detectors
While the template functions are particularly simple in the vacuum case, one is unable to recover exact analytical expressions found in the Galilean limit. This is due to the fact that only the dynamics of the detector are governed by the Galilean group, while the field's dynamics are still described by the Lorentz group. Therefore, it is necessary to distinguish between the speed of light present in the detector's relativistic dispersion relation and the field's propagation speed, so that one has respectively separated the group structure governing the detector and field.
Given this, one may contract the Lorentz group for the detector by first subtracting the rest energy and taking |p| M j . Recall from Eq. (31) that the mass-energy for the two levels is defined by M j = m + E j , where m ≡ M g is the rest mass-energy of the detector, taking E g = 0.
Contracting the free Hamiltonian (29), one obtains where E is the detector's internal energy gap (equivalently taking E = E e ). Notably, Eq. (42) still features a quantized mass-energy, as one has contracted to the centrally extended Galilean group in this limit. Regarding the template functions, the first-quantized case (38a) reduces to which was previously obtained in [23] directly using the "semi-relativistic" Hamiltonian (42) for the detector's COM. In this semi-relativistic template function (43), the parameter ν has been previously interpreted as the speed of light, but should more precisely be understood as the propagation speed of the field, i.e., a parameter formally independent of the speed of light. One can fully contract to the non-relativistic limit by takingM → m1 with M g = M e = m, whereby the template function reduces to and one obtains the non-relativistic model analyzed in [15]. Comparing the semi-and non-relativistic models alongside the relativistic cases, as seen in Fig. 1, the difference between the semi-and non-relativistic cases is small, but becomes particularly notable when comparing to the relativistic cases. As can be most easily seen in Fig. 1a, the relativistic first-quantized template function increases without bound, while the semi-and nonrelativistic template functions asymptote to the propagation speed of the field, which for a vacuum is unity. In the classical COM limit, i.e., for sufficiently massive detectors |p| m, one re-obtains the results for the traditional UDW model. In this regime, the dynamics are independent of the detector's center of mass and simply proportional to the energy gap. Moreover, in the regime where the detector mass is sufficiently large and dominates all other dimensional quantities, such as E m, then all models coincide as seen in Fig. 1a. In the case of a medium, one can compare the template functions for different propagation speeds ν and constant energy gap, as depicted in Fig. 2. One observes that the template functions converge for small p/m (i.e., for small momenta or suitably massive detectors) and also finds new transient behaviour before converging to the vacuum case. Additionally, while the relativistic firstquantized template function diverges for large p/m, the corresponding semi-and non-relativistic template functions asymptote to the field propagation speed which was also observed for the vacuum case in Fig. (1a). Due to the presence of the field's propagation speed ν, one obtains a local maximum in the second-quantized case, most easily seen in Fig. 2b, which corresponds to the asymptotic behaviour observed for the semi-and nonrelativistic COM limiting cases. Moreover, this local maximum is still present even for field propagation speeds very close to the speed of light, with large ν translating the peaks to larger p/m.

IV. COMPARISON OF SPONTANEOUS EMISSION RATES
As defined in the previous section, the transition rate (37) is a functional of the detector's initial wavefunction, and is given bẏ where the dynamics are entirely described by a respective template function. In the subsequent analysis of the spontaneous emission rate, it is simplest to consider a detector initially in a Gaussian state with width L, which in momentum space is described by where the detector is centered about p = p D .
Analytic results for the relativistic first-and secondquantized cases can be obtained in the case of a vacuum with mean momentum p D = 0. Substituting the Gaussian wavefunction into the transition rate (37) alongside the relativistic template functions (38), the respective integrals can be evaluated by Euler substitution. For the relativistic first-quantized case, this giveṡ where K ν (z) is the modified Bessel function of the second kind. In the second-quantized case, one obtainṡ where U (a, b, z) is the confluent hypergeometric function of the second kind, and we have expressed the above result in terms of the first-quantized coupling λ ≡ λ (1st) by employing Eq. (40). While analytic results can be easily obtained in the vacuum case, for the semi-and non-relativistic regimes and generally in a medium, the spontaneous emission rate is evaluated numerically. In Fig. 3, we plot the spontaneous emission rates for the relativistic first-and secondquantized models for both a vacuum and medium. One observes that for the system in a medium, the transition rate is re-scaled when compared to the vacuum case.
Comparing the transitions rates for the relativistic, classical and semi-and non-relativistic COM in a vacuum, we analyze the effects of the Gaussian width L on the results, given in Fig. 4. The relativistic localizations converge for large energy gap E/m, but fall short of the value predicted in the case of a classical COM. More precisely, in the limit of small rest mass m, one finds that while for a classical COM, whose template function (45) is independent of momentum, the transition rate trivially evaluates toṖ from which it is clear that the classical and relativistic COM disagree in the regime of small mass detectors, which is unsurprising given the mass defect between the ground and excited energy levels. All models coincide for sufficiently massive detectors, i.e., when L/λ c 1 and E/m 1, where the free dynamics of the COM become trivial; this regime can be most easily seen in Fig. 4b, where the models coincide for small energy gap E/m. In the semi-and non-relativistic regimes, both of the transition rates converge to zero in the limit of small rest mass, that is As seen in Fig. 4a, the semi-and non-relativistic results are particularly discrepant in comparison to the relativistic and classical COM cases, with significant disagreement for small Gaussian widths, which is most likely due to the use of the Galilean group for the detector [53].
Notably, in Fig. 4b for the regime where the Gaussian width is large L/λ c 1, one sees that both localizations for the relativistic COM are in close agreement, and perfectly coincide in the infinite width limit L M e → ∞. Taking this limit in the case of a vacuum, we find To better understand this convergence between the firstand second-quantized localizations, one finds that the two localizations scale inversely with respect to the spread of the Gaussian wavefunction, which can be seen in Fig. 5. For small Gaussian widths, the transition rate for the first-quantized COM is amplified, while for the second-quantized COM it is suppressed, which ultimately leads to the observed difference in sign between the second-order terms.
Quantitatively, for a hydrogen atom with L on the or- der of the Bohr radius, the fractional difference between the two rates is of the order ∼ 10 −10 , which would be very difficult to distinguish empirically; we briefly comment on potential experimental tests in the following section.

V. DISCUSSION
Given the results of the previous sections, a pertinent and obvious question to ask at this stage would be which of the two localizations, the first-or second-quantized, is the "true" or "correct" one. Due to their different predictions (namely the different emission rates), this question can and should ultimately be resolved by experiment. Note that analytical results for the emission rates presented here are for the detector with vanishing mean momentum, which is expected to minimize the difference between these two models.
Indeed, the marked difference in the dependence of the first-and second-quantized template functions on momentum, e.g., Eqs. (39a) and (41b), suggests that the predictions of the two models would diverge further for initial states with non-vanishing momentum. Choosing an optimal initial state and exploring other processes (such as absorption) to empirically distinguish between these two models is a worthwhile topic for further research.
Throughout this analysis we have considered a relativistic "first-quantized" regime, which some will object to on the grounds that RQM can be consistently formalized only in terms of a relativistic QFT, i.e., the second-quantized model outlined in Sec. II C. Such an argument is often made on the basis of relativistic covariance, namely that both space and time must be treated equivalently under the Lorentz group. While this is an important point, it must be stressed that the issue of localization exists independently of covariance. Both a clear physical explanation of the discrepancy between the two localizations, and a theoretical argument for which of the predictions would correspond to empirical observation, are missing. Filling this gap is necessary for a complete understanding of RQM.

A. Position operator and localization
The problem of localization in RQM has a long history, with its initial study originating in the seminal paper by Pryce [54], covering the different possible definitions of a relativistic COM. As opposed to the Newtonian case under the Galilean group, one finds many different competing definitions of a COM in relativity, whether it be classical or quantum. Shortly after Pryce's analysis, Newton and Wigner [51] investigated possible definitions for a relativistic position operator by imposing a number of invariance conditions, from which they derived their Newton-Wigner position operator, which is equivalent to the standard position operator of NRQM.
Subsequent analyses (see e.g., [24] and references within) have further investigated the problem of localization, namely whether one can introduce a covariant position operator and whether localized states are consistent with RQM and requirements of causality. The UDW model provides an especially simple and convenient theoretical framework to study this question, and particularly the consequences of what we refer to as the first-and second-quantized localizations. Moreover, in the massive spin-0 case, the different possible definitions for a relativistic COM are comparatively simpler and in this instance equivalent [55,56], which is particularly convenient for studying and interpreting the first-and secondquantized localizations.
In the first-quantized model, we required that the respective position states be identical to that of NRQM, which is equivalent to imposing the Newton-Wigner lo-calization and defining a corresponding position operator that obeysx NW |x D = x |x D [51]. As discussed previously in Sec. II C, by imposing this localization scheme one obtains orthogonal position states, unlike that obtained for the second-quantized model. The difficulty of these second-quantized "position states" (22) is that they are non-orthogonal, which can be most easily seen by taking the inner product with indices denoting the internal degrees of freedom. Working in spherical coordinates, and suppressing indices, the above integral can be evaluated to which can alternatively be rewritten in terms of the Compton wavelength λ c = 1/M with respect to the detector mass M Following [48], the dependence on the Compton scale can be more easily seen by Taylor expanding the integration measure E(p) = p 2 + M 2 about |p| = 0 where one finds that the position states are orthogonal up to second order in the Compton wavelength. The first-and second-quantized localizations are equivalent provided that the Compton scale is sufficiently small with respect to spread of the wavefunction in position space, which was observed in the UDW model for large Gaussian widths in Fig. 4b.

B. Foldy-Wouthuysen (FW) transformation
The results of Newton and Wigner [51] were subsequently extended by Foldy and Wouthuysen [57], who found the representation of the Newton-Wigner position operator for the spin-1/2 case when the positive and negative energy states are decoupled. Moreover, Foldy and Wouthuysen found their eponymous unitary transformation relating the two representations, where the decoupled representation gives a Newton-Wigner position operator identical to the non-relativistic position operator [58].
In the relativistic second-quantized UDW model, the non-orthogonal position states are transformed into the first-quantized position states by the action of an FW transformation, which maps between the respective firstand second-quantized localizations, and decouples the internal states of the detector. Moreover, by transforming to the first-quantized representation, one loses manifest Lorentz covariance but obtains a position operator with corresponding position eigenstates.
While in the literature it is common to take the secondquantized localization as more fundamental, and thus as more correct in any relativistic context, it may be premature here. Note that we are concerned with an effective description of a composite system, such as an atom, whose many degrees of freedom are reduced to just the COM and two internal states; taking such an effective model as fundamental by treating it within QFT appears inconsistent. Indeed, in scenarios like ours, the predictions from the first-quantized model may have a greater credence than those obtained by modelling the complexities of an atom as a two-component scalar field.
Further theoretical work combined with experiments are necessary to distinguish between the first-and second-quantized localizations and their corresponding representations. In particular, while the FW transformation is commonly interpreted as a purely mathematical transformation which decouples the internal states, it may be possible to give it a physical interpretation, and understand its role both in mapping between the two different localizations and in the loss of covariance that results from the first-quantized representation.

C. Time and covariance
In the traditional UDW model, one conventionally parameterizes the classical trajectory of the detector by its proper time; however, in the case of a relativistic detector with quantized COM, the treatment of time is conceptually more difficult due to various problems of time in RQM. One particular strategy to formalizing a firstquantized RQM is by introducing an additional invariant evolution parameter, such as a proper-time, which parameterizes the spacetime coordinates. This approach to relativistic dynamics was started in the twentieth century by a number of eminent physicists, and over the last several decades has since been developed considerably, with one of its most recent realizations being the Stueckelberg-Horwitz-Piron (SHP) theory [59,60]. While the relativistic first-quantized formulation we have considered in this paper has been sufficient, it is worth exploring a proper-time framework or equivalent, which would provide a manifestly covariant description of the detector's dynamics. This would be particularly beneficial for more general treatments of the UDW model, namely in the study of non-inertial detectors and curved spacetimes.
Closely related to the aforementioned problems of localization and time is the difficulty in defining a covariant position operator and state with respect to the Lorentz transformations. In conventional NRQM, one can identify a wavefunction in the position basis parameterized by time, and obtain a covariant wavefunction under the Galilean group. However, the relativistic extension is highly non-trivial due to the mixing of space and time under Lorentz transformations, and problematizes a relativistic first-quantized approach. This is avoided in a relativistic QFT, where instead one defines a field operator as a function of spacetime, which is manifestly covariant and bypasses this particular problem of time in a first-quantized formalism.

D. Mass-energy operator
In traditional models, mass-energy (mass) is treated as a c-number and is formally described as the central charge of the Poincaré (Galilean) group. In many relativistic extensions to quantum mechanics, one considers a central extension of the underlying group structure and treats mass as a dynamical variable [60,61], although we do not consider such a treatment here. In our relativistic first-quantized model, we promote mass-energy to an operator, such that the dispersion relation for the secondquantized detector with two mass components (18) has in flat spacetime the same dispersion relation given by Eq. (29) for a mass operator defined by Eq. (31).
The motivations for introducing a mass-energy operator are numerous, and it has been introduced multiple times independently in the literature. A mass-energy operator naturally follows from a proper-time dynamical framework of RQM [20,[62][63][64]; or within the context of a dynamical mass approach, it can be obtained through canonical quantization by introducing a conjugate variable with respect to the mass [65]. More recently, studies of the Einstein equivalence principle in the quantum regime have also motivated the introduction of quantized mass-energy [66]; and the discovery of neutrino oscillations has found an explanation in terms of superpositions of mass eigenstates [67], which provides a compelling empirical motivation for introducing a mass operator.

VI. CONCLUSION
In this paper, we proposed and studied an extension of the UDW detector model which incorporates a firstquantized relativistic COM. By studying the process of spontaneous emission, we have found that the transition rates are different between our first-quantized description and the second-quantized model, and likewise for the cases of a classical, semi-and non-relativistic COM. Furthermore, we have studied the detector-field system in both a vacuum and medium, obtaining analytic results in the former case.
The different predictions between the relativistic models are ultimately due to the different localization schemes implied by the first-and second-quantized approaches. Significantly, for these two cases we find that the dynamics encapsulated by the "template functions" are notably distinct, which suggests that the two localizations can be empirically studied. In the interest of future experiments, these results can and should be considered for other physical processes such as absorption and stimulated emission; likewise, one should also search for optimal detector states in order to find more easily testable parameter regimes.
On the theoretical side, we indicate that the two localizations are related by an FW transformation, the physical role of which must still be clarified given the importance of its mapping between the two schemes. Extending the model to include the spin of the detector and field would also be beneficial for future investigations, since the inclusion of angular momentum in the model may lead to additional empirically distinguishable effects resulting from the two localizations.
For the models we have developed in this paper, we have only considered inertial detectors in Minkowski spacetime. It is of obvious interest to extend this study to non-inertial trajectories and curved spacetimes, of which the former can be modelled by introducing an electromagnetic field to accelerate (e.g., a charged) detector. One could then study the impact of the detector's localization on the Unruh effect and further explore the FW transformation in the context of curved spacetimes.
Lastly, while one can provide a relativistically covariant description for the detector following a secondquantized approach, it is worthwhile exploring whether a first-quantized model can also be formalized covariantly. Such a project could be developed via a proper-time dynamical framework, which can be developed in several ways, such as in terms of a path-integral formulation or alternatively in terms of a relativistic dynamical approach, e.g., using the SHP theory. Such an investigation can help to clarify the relation between the Lorentz and FW transformations, and may have interesting implications for the study of quantum reference frames.

ACKNOWLEDGMENTS
We thank C. Wood, F. Costa, J. Foo, L. Henderson, V. Baccetti and N. Menicucci for helpful discussions and advice.
This work was supported by the ARC Future Fellowship grant FT210100675, and through the ARC Centre of Excellence for Engineered Quantum Systems (EQUS) CE170100009.
The authors acknowledge the Turrbal and Jagera peoples, who are the traditional owners of the land on which the University of Queensland is situated. We pay our respects to their Elders, past and present.

Appendix: Scalar field, conventions and derivation
In the interest of providing a self-contained analysis, we derive the field operator for a scalar field following similar treatments given by Birrell and Davies [4] and Peskin and Schroeder [45], employing the same conventions commonly used in the UDW literature. As opposed to the main body of the paper, here we consider a (d + 1)dimensional Minkowski spacetime.
A real-valued scalar field φ(x) has Lagrangian density from which one can derive the Klein-Gordon equation where ≡ ∂ µ ∂ µ denotes the d'Alembert operator and m is the mass. The field equation is solved by a set of solutions for positive and negative frequencies u (±) k (x) with momentum k. Following convention, we choose a planewave basis for the field modes such that where k · x = ω(k)t − k · x, ω(k) = √ k 2 + m 2 and f (k) is an arbitrary normalization factor. To ensure orthonormal u (±) k (x) modes, we employ the symmetric Fourier convention (2π) −d/2 and non-covariant integration measure (2ω(k)) −1/2 , such that where φ (±) (x) can in turn be expressed as a superposition of the corresponding u (±) k (x) modes, such that φ(x) = d d k a (+) (k)u (+) k (x) + a (−) (k)u (−) k (x) . (A.6) The field φ(x) can be quantized by introducing a canonically conjugate field π(x) = ∂L(x) ∂(∂ t φ(x)) = ∂ t φ(x) , (A.7) such thatφ(x) andπ(x) are promoted to operators which obey the equal-time canonical commutation relations The field operatorφ(x) may now be expanded in terms of the field modeŝ where u k (x) and u * k (x) are orthonormal modes corresponding respectively to the positive and negative frequency solutions of the classical field, andâ † (k) andâ(k) are creation and annihilation operators which obey the non-covariant commutation relations