PHYSICAL REVIEW D 98, 016011 (2018)

Relativistic corrections to nonrelativistic effective field theories

† ‡ Mohammad Hossein Namjoo,* Alan H. Guth, and David I. Kaiser Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 13 June 2018; published 20 July 2018)

In this paper we develop a formalism for studying the nonrelativistic limit of relativistic field theories in a systematic way. By introducing a simple, nonlocal field redefinition, we transform a given relativistic theory, describing a real, self-interacting scalar field, into an equivalent theory, describing a complex scalar field that encodes at each time both the original field and its conjugate momentum. Our low-energy effective theory incorporates relativistic corrections to the kinetic energy as well as the backreaction of fast-oscillating terms on the behavior of the dominant, slowly varying component of the field. Possible applications of our new approach include axion , though the methods developed here should be applicable to the low-energy limits of other field theories as well.

DOI: 10.1103/PhysRevD.98.016011

I. INTRODUCTION condensed-matter systems, and to specific phenomena such as “oscillons” [9–11] and “superradiance” [12,13]. Understanding the nature of dark matter remains a major Axions are an attractive candidate for dark matter. challenge at the intersections of astrophysics, cosmology, The hypothetical particles were originally introduced to and . Multiple lines of observational evidence address the strong CP problem in QCD [14–16], but their indicate that dark matter should be plentiful throughout the expected mass and self-coupling make them well-suited universe, contributing roughly five times more to the energy for cold dark matter as well. In particular, axions, with density of the universe than ordinary (baryonic) matter, and ∼ 10−4 − 10−5 that the dark matter should be cold and collisionless [1]. m eV, are expected to be produced early in From the standpoint of particle theory, the puzzle of dark cosmic history, around the time of the QCD phase matter includes at least two components: identifying a transition. At that time the typical wavenumber was ∼ ∼ 10−11 ∼ 2 plausible dark-matter candidate within realistic models of k HQCD eV [17], where HQCD TQCD=Mpl is ∼ particle physics, and developing an accurate, theoretical the Hubble scale at the time of the QCD transition, TQCD 0.1 GeV is the temperature at which the transition occurs, description that is suitable for low-energy phenomena pffiffiffiffiffiffiffiffiffi ≡ 1 8π ∼ 1018 associated with cold matter. For the latter goal, it is important and Mpl = G GeV is the reduced Planck to develop a means of characterizing the nonrelativistic limit mass. Axions are expected to become accelerated gravita- of relativistic quantum field theories in a systematic way. tionally during large-scale structure formation, up to speeds In this paper we develop an v=c ∼ Oð10−3Þ within galactic halos [17]. Hence even at approach for the nonrelativistic limit of relativistic field late times they should remain squarely within the non- theories. We focus on the nonrelativistic behavior of scalar relativistic limit, albeit in a regime in which relativistic fields in Minkowski spacetime, incorporating systematic corrections may be competitive with other nontrivial relativistic corrections as well as corrections from the corrections, such as from higher-order interaction terms. ’ fields (weakly coupled) self-interactions. We have in mind In the nonrelativistic limit, fluctuations that oscillate – applications to axion dark matter [2 5], though the methods rapidly on timescales m−1 (where m is the mass of the developed here should be applicable to the low-energy scalar field) may be expected to average to zero over – limits of other field theories, such as QED, QCD [6 8], and timescales Δt ≫ m−1. However, nonlinear self-couplings can induce a backreaction of the fast-oscillating terms on * the dominant, slowly varying component of the nonrela- [email protected] † [email protected] tivistic field, affecting its behavior. Such coupling of fast- ‡ [email protected] and slow-oscillating terms produces measurable effects in many physical systems, such as neuronal processes related Published by the American Physical Society under the terms of to memory formation [18,19]. We develop an iterative, the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to perturbative procedure to incorporate this backreaction. the author(s) and the published article’s title, journal citation, Our approach complements other recent work, such as and DOI. Funded by SCOAP3. Refs. [11,17,20]. In particular, the authors of Ref. [20]

2470-0010=2018=98(1)=016011(17) 016011-1 Published by the American Physical Society NAMJOO, GUTH, and KAISER PHYS. REV. D 98, 016011 (2018) develop a nonrelativistic effective field theory for describ- where ϕ is a real-valued scalar field, and we take the ing axions by calculating scattering amplitudes for n → n Minkowski metric to be ημν ¼ diag½−1; 1; 1; 1. The body scattering in the full, relativistic theory, and then canonical momentum field is π ¼ ϕ_ , where an overdot taking the low-energy limit of those amplitudes to match denotes a derivative with respect to time, and the coefficients in a series expansion of the effective potential. Hamiltonian is given by In Ref. [11], the authors developed an effective field theory Z by using diagrammatic techniques to integrate out the high- 3 H x momentum modes. In this paper, we develop an effective H ¼ d x ð Þ; description for the nonrelativistic limit with no need to 1 1 1 1 calculate scattering amplitudes in the corresponding rela- H π2 ∇ϕ 2 2ϕ2 λϕ4 ¼ 2 þ 2 ð Þ þ 2 m þ 4! : ð2Þ tivistic theory. Our method yields the same coefficients for the leading higher-order interaction terms in the low-energy The equations of motion take the form limit as those found in Refs. [11,20], while also incorpo- rating systematic relativistic corrections which do not δ ϕ_ H π appear in the previous analysis. ¼ δπ ¼ ; In Sec. II we introduce a convenient field redefinition to δH 1 relate the real-valued scalar field described by the relativ- π_ − ∇2 − 2 ϕ − λϕ3 ¼ δϕ ¼ð m Þ 3! : ð3Þ istic theory to a complex scalar field more appropriate to the nonrelativistic limit. In particular, we introduce a In contrast, the standard Lagrangian for a free, non- nonlocal field redefinition rather than the local redefinition that one typically finds in the literature. Although we relativistic field may be written expect the resulting descriptions of the low-energy limit to i 1 be equivalent in either redefinition, we find that the non- L ψψ_ − ψψ_ − ∇ψ∇ψ ¼ 2 ð Þ 2 ; ð4Þ local redefinition considerably simplifies the derivation. m In Sec. III we develop an effective field theory for the where ψ is a complex scalar field, and overdots again nonrelativistic field, which incorporates contributions from denote derivatives with respect to time. The kinetic term is fast-oscillating terms on the evolution of the principal, normalized so that the field ψ obeys the standard Poisson slowly varying portion of the field. Concluding remarks bracket relations with ψ , so that when quantized the two follow in Sec. IV. In Appendix A we demonstrate that our fields obey the standard commutation relations, as given transformation from the real-valued relativistic field ϕðt; xÞ – ψ x below in Eq. (10) and discussed in Eqs. (14) (20). Whereas to the complex-valued nonrelativistic field ðt; Þ can be the Lagrangian in Eq. (1) yields a second-order equation of constructed as a canonical transformation. In Appendix B motion for ϕ, the Lagrangian in Eq. (4) yields first order we demonstrate that the low-energy effective field theory equations of motion for the real and imaginary parts of ψ. based on our nonlocal field redefinition explicitly matches The Lagrangian in Eq. (4) has a global Uð1Þ symmetry; what one would calculate with the local field redefinition the associated conserved charge is simply the number of in two regimes of interest. In Appendix C we compare our particles, results with the recent calculation in Ref. [11] and dem- Z onstrate that the two results are equivalent, related to each 3 2 other by a field redefinition. N ¼ d xjψj ; ð5Þ

II. FIELD REDEFINITION FOR A confirming the usual expectation that particle number NONRELATIVISTIC FORMULATION should be conserved in a nonrelativistic theory, appropriate In this section we begin with the Lagrangian for a for energy scales E ≪ m. relativistic field theory that describes a scalar field in Previous authors (see, e.g., Refs. [17,20]) have typically Minkowski spacetime, and introduce a suitable field related the nonrelativistic field ψ to the relativistic field ϕ redefinition with which we may consider the nonrelativistic by using the relations limit systematically. Our goal is to obtain an expression for 1 the Lagrangian that yields the Schrödinger equation as the ϕðt; xÞ¼pffiffiffiffiffiffiffi ½e−imtψðt; xÞþeimtψ ðt; xÞ; ð6aÞ effective equation of motion for the redefined field in the 2m extreme nonrelativistic limit. rffiffiffiffi We consider a relativistic scalar field of mass m with a m −imt imt 4 πðt; xÞ¼−i ½e ψðt; xÞ − e ψ ðt; xÞ: ð6bÞ λϕ interaction, described by the Lagrangian density 2 1 1 1 μν 2 2 4 Note that the quantities in brackets in Eqs. (6a) and (6b) L ¼ − η ∂μϕ∂νϕ − m ϕ − λϕ ; ð1Þ 2 2 4! could have been written as Re½e−imtψ and Im½e−imtψ,

016011-2 RELATIVISTIC CORRECTIONS TO NONRELATIVISTIC … PHYS. REV. D 98, 016011 (2018) respectively, so the equations are independent. Equation (6b) is a consequence of the fact that the nonlocal operators on is not ordinarily written explicitly, but this equation or the two lines of Eq. (7) are the inverses of each other. something similar has to be assumed if ψðt; xÞ is to be Given Eqs. (3) and (9), it is straightforward to work out uniquely defined. At any fixed time t,Eqs.(6a) and (6b), the equation of motion for ψðt; xÞ: taken together, give a one-to-one mapping between the complex-valued ψðt; xÞ and the two real-valued functions iψ_ ¼ mðP − 1Þψ ϕ x π x ðt; Þ and ðt; Þ. If we use Eqs. (3) and (6) to derive λeimt equations of motion for ψ, and then set λ ¼ 0 and neglect all þ P−1=2½e−imtP−1=2ψ þ eimtP−1=2ψ 3: ð11Þ 4!m2 rapidly oscillating terms (proportional to eimt), we find equations of motion consistent with the Lagrangian of A key step in this calculation, which motivates the nonlocal Eq. (4). However, for the purpose of systematically obtaining operator P, is the calculation of the time derivative of the relativistic corrections to the nonrelativistic theory, ϕ þ i P−1π: we find it more convenient to start from a nonlocal field m   redefinition. In place of Eqs. (6),wewrite i i 1 ϕ_ þ P−1π_ ¼ π þ P−1 ð∇2 − m2Þϕ − λϕ3 m m 3! 1 −1 2 − ϕðt; xÞ¼pffiffiffiffiffiffiffi P = ½e imtψðt; xÞþeimtψ ðt; xÞ; ð7aÞ iλ 2m π − Pϕ − P−1ϕ3 ¼ im 3!  m  rffiffiffiffi λ m − P ϕ i P−1π − i P−1ϕ3 π x − P1=2 −imtψ x − imtψ x ¼ im þ : ð12Þ ðt; Þ¼ i 2 ½e ðt; Þ e ðt; Þ; ð7bÞ m 3!m The definition of P was chosen so that the first term on the wherewehavedefined right-hand side of the last line above is proportional to rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ þ i P−1π ∝ ψ. If we used the local definition instead, the ∇2 m P ≡ 1 − right-hand side would have also contained a term propor- 2: ð8Þ m tional to e2imtψ , which would lead to rapidly oscillating terms even in the free field theory (λ ¼ 0). With the Note that mP corresponds to the total energy of a free, nonlocal field definition of Eqs. (7), the free theory leads relativistic particle. Equations (7) can be inverted to obtain ψ ϕ π to no rapidly oscillating terms. an equation for in terms of and : It is for some purposes useful to write a Lagrangian rffiffiffiffi   density for the nonrelativistic formulation, so we note that m i ψðt; xÞ¼ eimtP1=2 ϕðt; xÞþ P−1πðt; xÞ : ð9Þ the equation of motion in Eq. (11) can be derived from the 2 m Lagrangian density

Although our field redefinition in Eqs. (7) involves i L ¼ ðψψ_ − ψψ_ Þ − mψ ðP − 1Þψ nonlocal operators, the new fields ψ and ψ are well 2 P behaved in the nonrelativistic limit, in which the operator λ − −1 2 −1 2 4 2 2 − ½e imtP = ψ þ eimtP = ψ : ð13Þ can be expanded in powers of ∇ =m . The leading term 4 · 4!m2 matches the local definitions of Eq. (6). Furthermore, even though the ψ field has a nonlocal relation to ϕ, the ψ field is Note that the free field theory terms in Eq. (13), corre- local with respect to itself. When quantized, the commu- sponding to λ ¼ 0, show a manifest global Uð1Þ symmetry, tator of ψðt; xÞ and ψ ðt; yÞ becomes ψ → eiθψ. This symmetry is associated with the conserva- tion of particle number, which is exact in the free theory ½ψðt; xÞ; ψ ðt; yÞ even when energies are relativistic.  To construct the corresponding Hamiltonian density, we m 1=2 1=2 i −1 ¼ Px Py ϕðt; xÞ − Px πðt; xÞ; ϕðt; yÞ explicitly decompose the field into its real and imaginary 2 m ψ ≡ ψ ψ  parts, R þ i I. The kinetic terms of the Lagrangian i density then become þ P−1πðt; yÞ m y i δ3 x − y L ψψ_ − ψψ_ ψ_ ψ − ψ ψ_ ¼ ð Þ; ð10Þ kinetic ¼ 2 ð Þ¼ R I R I: ð14Þ P where the subscripts on indicate the coordinates on If we take ψ R to be the canonical field, then ψ I will become which they act, and we assumed of course that the canonical momentum ∂L=∂ψ_ R. If we proceeded ½ϕðt; xÞ; πðt; yÞ ¼ iδ3ðx − yÞ. The simplicity of this result directly to construct the Hamiltonian density in the standard

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H πψ_ − L _ way, ¼ R , the second term on the right of Eq. (1) by the following procedure. Replace ϕ in the −ψ ψ_ −ψ π_ _ Eq. (14) would be rewritten as R I ¼ R . This, Lagrangian by an auxiliary field using χ ¼ ϕ, and impose however, would take us outside the standard Hamiltonian the relation via a Lagrange multiplier: L½ϕ; ϕ_ → procedure, in which the Hamiltonian is assumed to be a L ϕ χ ζ χ − ϕ_ ζ function of the fields and their canonical momenta, but not ½ ; þ ð Þ, where is a Lagrange multiplier. We next note that χ is a nondynamical variable in the resulting the time derivatives of canonical momenta. To avoid this L problem, we can add a total time derivative to the Lagrangian. We remove it from the theory by varying with respect to χ, which yields an expression for χ, which Lagrangian density, which does not change the equations L of motion. So we replace L by we then substitute back into . We use similar relations kinetic as in Eqs. (7) to relate the set fϕ; ζg (instead of fϕ; πg)to ψ ψ d the set f ; g, and substitute the relations back into the L0 L ψ ψ 2ψ_ ψ ¼ kinetic þ ð R IÞ¼ R I: ð15Þ Lagrangian. kinetic dt We close this section with a final observation about the To absorb the factor of 2 we find it convenient to define the field redefinitions in Eqs. (7), compared to the traditional canonical field ψ c to be relation in Eqs. (6). The usual relation of Eqs. (6) has the advantage of being local, but the resulting Lagrangian— pffiffiffi even for free fields—would no longer preserve the manifest ψ cðt; xÞ ≡ 2ψ Rðt; xÞ; ð16Þ Uð1Þ symmetry, thereby obscuring the fact that such from which it follows that models should conserve particle number. Moreover, if we had used Eqs. (6), the fast oscillatory factors ðeimtÞ, ∂L pffiffiffi π x ≡ 2ψ x which are absent in the free field theory using our nonlocal cðt; Þ ¼ Iðt; Þ: ð17Þ formulation, would have appeared in the free-field ∂ψ_ cðt; xÞ Lagrangian. We build upon these advantages of the Thus, we have relations in Eqs. (7) in the following section. 1 ffiffiffi III. EFFECTIVE FIELD THEORY IN THE ψðt; xÞ¼p ðψ cðt; xÞþiπcðt; xÞÞ: ð18Þ 2 NONRELATIVISTIC LIMIT ψ The Hamiltonian density is then given by Given the Lagrangian and equation of motion for in Eqs. (13) and (11), we may now consider the effective description for such a model in the nonrelativistic limit. H ¼ πcψ_ c −L λ We aim to take the nonrelativistic limit in a way that ψ P −1 ψ −imtP−1=2ψ imtP−1=2ψ 4 ¼ m ð Þ þ 2 ½e þe ; enables us to incorporate relativistic corrections system- 4·4!m atically. One step will obviously be to expand the nonlocal ð19Þ operator P in powers of ∇2=m2. However, we must take additional steps in order to recover an appropriate descrip- where ψ is given by Eq. (18). The canonical quantization of tion in the nonrelativistic limit. In particular, we must find this Hamiltonian would give some way to incorporate the effects of fast-oscillating terms on the behavior of the slowly varying field. 3 ½ψ cðt; xÞ; πcðt; yÞ ¼ iδ ðx − yÞ; ð20Þ Consider, as a first step, the equation of motion of Eq. (11) in the limit in which we may ignore all higher which is equivalent to Eq. (10). The discussion here has spatial-derivative terms arising from the expansion of P. shown that the Hamiltonian of Eq. (19) gives the correct Then we find equation of motion for the field ψðt; xÞ. In Appendix A,we show that the transformation from ϕðt; xÞ to ψðt; xÞ can be 1 λ iψ_ ≃− ∇2ψ þ jψj2ψ constructed as a canonical transformation, which guaran- 2m 8m2 tees that the canonical commutators will be preserved, as λ þ ½e−2imtψ 3 þ e4imtψ 3 þ 3e2imtjψj2ψ : ð21Þ we have found. 4!m2 Our findings are in contrast with the recent claim in Ref. [21], in which the authors found it problematic to The first two terms on the right-hand side consist of the derive a Lagrangian for a complex, nonrelativistic field usual terms in the Schrödinger equation for a model from a real-valued relativistic one. Based on the canonical incorporating self-interactions [17,20]. The remaining transformation we have explicitly formulated, we do not terms are usually neglected in the nonrelativistic limit, believe there should be any difficulty in constructing because they are proportional to fast-oscillating factors, and Lðψ; ψ_ Þ. It is interesting to note that we may obtain the in the limit of large m, such terms might be expected to Lagrangian in Eq. (13) directly from the Lagrangian in average to zero if the system were observed over timescales

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−1 Δt ≫ m . However, such oscillatory terms must be treated where each ψ νðt; xÞ will be slowly varying on a timescale with care. Suppose, for example, that the full solution of of order m−1. We label the mode with ν ¼ 0 as the slowly Eq. (21) consists of a slowly varying contribution ðψ sÞ and varying contribution to the field, 2imt a small term that does not vary slowly: ψ ¼ ψ s þ δψe . ψ x ≡ ψ x Inserting this ansatz back into Eq. (21), keeping only terms ν¼0ðt; Þ sðt; Þ: ð25Þ up to linear order in δψ, and retaining terms that vary in time more slowly than eimt, we obtain We assume that in the nonrelativistic limit, jψ − ψ sj ≪ jψ sj. 1 λ We may rewrite the equation of motion for ψ in Eq. (11) iψ_ ≃− ∇2ψ þ jψ j2ψ s 2m s 8m2 s s in the form λ ψ ψ δψ 2ψ δψ … þ 2 sð s þ s Þþ ð22Þ λ 8m iψ_ ðt; xÞ¼mðP − 1Þψðt; xÞþ G˜ ðt; xÞ; ð26Þ 4!m2 We find that the fast-oscillating portion of the solution where we have defined ðδψÞ backreacts on the slowly varying portion ðψ sÞ, contributing to the dynamics of ψ in a nontrivial way. s ˜ x ≡ P−1=2 imt −imtP−1=2ψ imtP−1=2ψ 3 (Cf. Refs. [18,19].) This simple example illustrates that we Gðt; Þ e ½e þ e : ð27Þ must take into account the fast-oscillating contributions of ˜ the full solution to obtain an effective description of the We expand G in a series akin to the expansion for ψ: slowly varying portion. The removal of the fast-oscillating X∞ contributions is analogous to integrating out the high- ˜ ˜ iνmt Gðt; xÞ¼ Gνðt; xÞe : ð28Þ frequency components of a field in a path integral. ν¼−∞ We construct a perturbative approach with which to account for the contribution of the fast-oscillating terms For a given mode ν, Eq. (26) then takes the form to the time evolution of the slowly varying portion of the ψ field . In general there are three small quantities to λ ˜ iψ_ ν − νmψ ν ¼ mðP − 1Þψ ν þ Gν ð29Þ consider for our perturbative treatment. For any quantity 4!m2 Fðt; xÞ, we may consider spatial variations compared to the length-scale m−1; time variation compared to the timescale with m−1; and self-interactions mediated by the coupling con- X λ ˜ x P−1=2 Ψ Ψ Ψ ΨΨ Ψ stant . We parametrize the magnitude of spatial and Gνðt; Þ¼ f μ μ0 2þν−μ−μ0 þ μ μ0 4−ν−μ−μ0 temporal variations as μ;μ0 þ 3ΨμΨμ0 Ψ 0 þ 3ΨμΨ 0 Ψν−2 μ μ0 g; ð30Þ ∇2 _ μþμ −ν μ þ þ F ∼ ϵ F ∼ ϵ 2 xF; tF: ð23Þ m m where we have made use of the convenient notation

For weakly interacting systems in the nonrelativistic limit, −1=2 Ψνðt; xÞ ≡ P ψ νðt; xÞ: ð31Þ we assume that ϵx, ϵt, λ ≪ 1. Given our nonlocal field redefinition in terms of the operator P, we may manipulate For ν ¼ 0, Eq. (29) gives us the equation obeyed by quantities to arbitrary order in ϵx during intermediate steps ψ sðt; xÞ, which defines the low-energy effective field and expand the P operators to the desired order at the end. ˜ We may then Taylor expand various quantities in powers of theory that we seek. However, to evaluate Gν, we need to (perturbatively) calculate ψ νðt; xÞ for all other values ϵt and λ and iteratively track their effects on the slowly P of ν. Before proceeding, we note that for each ν, the mode varying portion of the field. Upon expanding the ψ operators, we may use the equation of motion (at appro- ν carries a definite charge. In particular, if we assign 1 − ν ψ ¯ − ψ priate, iterative order) to relate higher-order terms in ϵ to charges Qν ¼ to ν and Qν ¼ Qν to ν, then total t charge remains conserved because all terms in Eq. (29) corresponding higher-order terms in ϵx and λ. In the end this yields a controlled expansion in two small parameters, carry the same charge. This implies that the mode of ψ ϵ and λ. interest, s, carries a conserved charge, and hence we x ψ We construct the solution for ψðt; xÞ in the form of an expect the effective field theory governing s to obey a 1 expansion in an infinite series of harmonics, global Uð Þ symmetry, to all orders in the perturbative expansion, in spite of the fact that the exact theory violates X∞ this symmetry. Below we will explicitly confirm that iνmt ψðt; xÞ¼ ψ νðt; xÞe ; ð24Þ particle number conservation is exact for the leading ν¼−∞ perturbative corrections.

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ðnÞ Equation (29) is a first-order differential equation in where Gν ðt; xÞ consists of all terms in Eq. (34) which are ψ x time, which in principle defines νðt; Þ up to an arbitrary proportional to the nth power of the small quantities ϵ x t function of . However, as long as we can construct a and λ. From Eq. (32), we have at first order function ψ νðt; xÞ which satisfies Eq. (29), then the full series of Eq. (24) will satisfy the equation of motion, ð1Þ ð0Þ Eq. (11), so it will be sufficient for us to construct a Ψν ¼ λGν for ν ≠ 0; ð38Þ particular solution to Eq. (29). To do so, we multiply both sides of Eq. (29) by P−1=2 and rearrange terms to write and for higher orders we have

i _ Ψν ¼ − ΓνΨν þ λGν; ð32Þ ðnÞ i _ ðn−1Þ ðn−1Þ m Ψν ¼ − ΓνΨν þλGν for ν ≠ 0 and n>1: ð39Þ m where we have defined The first term on the right-hand side is suppressed relative Γ ≡ 1 − ν − P −1 ðn−1Þ ν ð Þ ð33Þ to Ψν by ϵt, and the second term is suppressed relative to ðn−1Þ Gν by λ. Thus, from Eq. (39), we see by induction that and ðnÞ Ψν will have a total power in the small quantities ϵt and λ of exactly n. Γ P−1=2 x ν ˜ x Our aim is to calculate the effective equation of motion Gνðt; Þ¼ 3 Gνðt; Þ ψ 4!m for s, which is given by Eqs. (29) and (34) as Γ P−1 X ν ¼ fΨμΨμ0 Ψ2 ν−μ−μ0 þ ΨμΨ 0 Ψ 0 4!m3 þ μ 4−ν−μ−μ ψ_ P − 1 ψ λΓ−1P1=2 μ;μ0 i s ¼ mð Þ s þ m 0 G0; ð40Þ 3Ψ Ψ Ψ 3ΨΨ Ψ þ μ μ0 μþμ0−ν þ μ μ0 ν−2þμþμg: ð34Þ with G0 expanded to some perturbative order, as in Eq. (37). Working to order n ¼ 1 we find Equation (32) holds to any order in ϵx, ϵt, and λ. We note, however, that the first term on the right-hand side is −1 suppressed relative to Ψν by a factor of ϵt and the second 1 Γ P ð0Þ ð1Þ ν 3 3 2 λ Gν ¼ Ψν ¼ fΨ δν −2 þ Ψ δν 4 þ 3jΨ j Ψ δν 2 term is suppressed relative to Gν by a factor of ,sowemay λ 4!m3 s ; s ; s s ; treat the right-hand side as a perturbative source for Ψν. 2 3 Ψ Ψ δν 0 ; Therefore we can solve Eq. (32) iteratively, starting with the þ j sj s ; g −1 zeroth-order approximation ð1Þ 3ΓνP 2 ð1Þ 2 ð1Þ 2 ð1Þ Gν Ψ Ψ Ψ Ψ Ψ Ψ−ν ¼ 4! 3 f s 2þν þ s 4−ν þ s  m Ψ x ν 0 2 ð1Þ 2 ð1Þ 2 ð1Þ ð0Þ sðt; Þ if ¼ ; þ 2jΨ j Ψν þ Ψ Ψ þ 2jΨ j Ψ g; ð41Þ Ψν ðt; xÞ¼ ð35Þ s s ν−2 s 2−ν 0 if ν ≠ 0: where δ is the Kronecker delta function and, as discussed ðnÞ i;j We denote by Ψν the correction to Ψν obtained at the nth ðnÞ after Eq. (36), we set Ψ ¼ 0. Using the expressions in iteration, which in our construction will always be propor- 0 Eq. (41), we may expand Eq. (40) to order n ¼ 1, which tional to a total of n powers of the small quantities ϵ and λ. t yields We expand the modes with ν ≠ 0 in the series

X∞ λP−1=2 Ψ x ΨðnÞ x ν ≠ 0 iψ_ ¼ mðP − 1Þψ þ jΨ j2Ψ νðt; Þ¼ ν ðt; Þ for ; ð36Þ s s 8m2 s s n 0 ¼ 3λ2P−1=2 3Ψ2Γ P−1 Ψ 2Ψ Ψ2Γ P−1 Ψ3 þ 2 5 f s 2 ðj sj sÞþ s 4 ð s Þ ðnÞ ð4!Þ m where the terms Ψν will be determined iteratively, as 2 −1 3 2 −1 2 ðnÞ þ Ψs Γ−2P ðΨs Þþ6jΨsj Γ2P ðjΨsj ΨsÞg described below, and Ψ0 ≡ 0 for n>0, since we are not Ψ ≡ Ψ 3 3 2 2 expanding 0 s in a power series. We also expand þ O½λ ; ϵt ; λ ϵt; λϵt : ð42Þ X∞ x ðnÞ x 2 4 4 Gνðt; Þ¼ Gν ðt; Þ; ð37Þ We may now expand the P operators to order ϵx ∼ ∇ =m , n¼0 which yields

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1 λ We note that a single self-interaction term in the iψ_ ≃− ∇2ψ þ jψ j2ψ s 2m s 8m2 s s relativistic Lagrangian (in this case, λϕ4=4!) yields more h 1 λ than one interaction term in the nonrelativistic effective − ∇4ψ ψ 2∇2ψ 2 ψ 2∇2ψ 3 s þ 4 s s þ j sj s ψ 8m 32m theory. In particular, the effective Lagrangian for s in i 17λ2 Eq. (44) includes interaction terms that yield both 2 → 2 þ ∇2ðjψ j2ψ Þ − jψ j4ψ scattering and 3 → 3 scattering; including more terms in s s 768m5 s s the iterative expansion for ψ ν (with ν ≠ 0) would yield O λ3; ϵ3; ϵ3; λ2ϵ ; λ2ϵ ; λϵ2; λϵ2; λϵ ϵ ; ϵ ϵ2; ϵ2ϵ : L → 1 þ ½ t x t x t x t x t x t x operators in eff for each n n scattering, for n> .On L 1 ð43Þ the other hand, eff obeys a global Uð Þ symmetry, a feature which holds to all orders in the perturbative expansion, as discussed after Eq. (31). Thus, particle ΨðnÞ To obtain Eq. (43), we only needed to calculate the ν number is conserved to all orders in the nonrelativistic 1 up to n ¼ , so it was sufficient to use Eq. (38), and the effective field theory. A process that may occur in the time derivative in Eq. (39) never appeared. But of course if relativistic theory, such as 4 → 2 scattering in which four ðn−1Þ we continued to the next order, the time derivative of Ψν low-energy particles annihilate to produce two relativistic would appear. The iteration of this procedure will lead to ones, involves energies E>mand hence lies beyond the L time derivatives higher than the first, which will give at range of validity of eff in Eq. (44). least the appearance of introducing new degrees of freedom We have obtained the effective theory in Eq. (44) starting (d.o.f.). There exists, however, a well-defined procedure for from the nonlocal field redefinition of Eqs. (7).Our treating these higher time derivatives perturbatively, so that procedure was relatively straightforward because the equa- no new d.o.f. are introduced. We illustrate these techniques tion of motion in the free-field limit does not contain fast- in Appendices B and C. The higher time derivatives can be oscillating factors, and all of the spatial-derivative operators removed later, but in the context of the iteration procedure may be manipulated in terms of the nonlocal operator P. described here, they can be eliminated at each iteration. To Whereas these steps simplify the calculation when working ðn−1Þ do this, start by writing Ψν in terms of ψ s.Forn ¼ 2, with the nonlocal field redefinition of Eqs. (7), one may this would be the top line of Eq. (41). At each stage, we will follow similar steps to derive an equivalent effective field ðn−1Þ theory starting from the local field redefinition of Eqs. (6), be able to express Ψν in a perturbative expansion, with as we demonstrate in Appendix B. all terms up to a total power of n − 1 in the small quantities We may apply Eqs. (43) and (44) to the case of QCD ϵ and λ. Then we can differentiate this expression with t axions in the nonrelativistic limit, and compare our effec- respect to t, evaluating ψ_ according to Eq. (40).Even s tive description with other recent treatments [17,20].We Ψðn−1Þ − 1 though ν has a total power of n in the small begin with the usual relativistic potential, quantities ϵt and λ, its time derivative can be calculated up to one power higher, since the time derivative brings an 4 VðϕÞ¼Λ ½1 − cosðϕ=faÞ; ð45Þ extra factor of ϵt. Thus the first term on the right-hand side of Eq. (39) can be evaluated as a term with no time where Λ ∼ 0.1 GeV is associated with the QCD scale, and derivatives, and with a total power of n in the small f sets the scale for Peccei–Quinn symmetry breaking; we ϵ λ a quantities t and , which is exactly what is wanted. expect f ∼ 1011 − 1012 GeV for the typical Peccei–Quinn The first line of Eq. (43) is the usual Schrödinger a model [14–17]. For field values well below fa,wemay equation for a self-interacting scalar field in the non- expand VðϕÞ as relativistic limit, and the other lines include the lowest- order relativistic corrections. The terms that include the 1 λ ϕ 2ϕ2 ϕ4 O ϕ 6 Laplacian operator come from expanding the operator P, Vð Þ¼2 m þ 4! þ ½ð =faÞ ; ð46Þ and the final term comes from the contribution of the fast- ð1Þ 2 4 4 oscillating terms ψ ν (for ν ≠ 0). Given the equation of where m ¼ Λ =fa and λ ¼ −Λ =fa < 0. Given expected −4 −5 motion (43), we can find an effective Lagrangian which values for Λ and fa, these correspond to m∼10 –10 eV produces this equation of motion: and jλj ∼ 10−48–10−52. Remarkably, the final term in Eq. (43), proportional to i 1 λ λ2, exactly matches the corresponding term found in L ¼ ðψ_ ψ − ψ ψ_ Þ − ∇ψ ∇ψ − jψ j4 eff 2 s s s s 2m s s 16m2 s Ref. [20], following quite a different procedure. (In 1 λ Ref. [20], the authors fixed the coefficients in a series þ ∇2ψ ∇2ψ − jψ j2ðψ ∇2ψ þ ψ ∇2ψ Þ 8m3 s s 32m4 s s s s s expansion for the low-energy effective potential by calcu- 17λ2 lating scattering amplitudes for various n-body scattering ψ 6: 44 processes in the full, relativistic theory, and then took the þ 9 28 5 j sj ð Þ × m low-energy limit of those amplitudes.) On the other hand,

016011-7 NAMJOO, GUTH, and KAISER PHYS. REV. D 98, 016011 (2018) the analysis in Ref. [20] does not capture the relativistic IV. CONCLUSIONS ∇4 λ∇2 corrections present in Eq. (43), proportional to and . In this paper we have developed a self-consistent frame- λ∇2 λ2 We may compare magnitudes for the and terms: work for obtaining an effective field theory to describe the    nonrelativistic limit of a relativistic field theory. The lowest- 17λ2 ψ 4ψ 32 2 j sj s m order corrections to the ultranonrelativistic limit arise both 768m5 λjψ j2∇2ψ from expanding the kinetic energy as well as from incor-  s s  2 porating the backreaction from fast-oscillating terms on the λjψ j ψ 2m λN O 1 s s ∼ ; ¼ ð Þ 8 2 ∇2ψ 3 2 ð47Þ dominant, slowly varying portion of the field. Our results m s m v are largely consistent with the complementary analyses of Refs. [11,20]. Our approach is perhaps simpler, working N ψ 2 where ¼j sj is the axion number density, and we have directly with the equations of motion in the nonrelativistic 2 2 2 used ∇ =m ∼ v . Thus we see that whenever the contri- limit. In addition, our approach incorporates nontrivial butions arising from kinetic and potential energy to the relativistic corrections in a systematic way, which had been zeroth-order equation of motion are of comparable magni- neglected in previous analyses. These additional terms may tude to each other—that is, the two terms on the right-hand be comparable in magnitude to the other, known terms in side of the top line of Eq. (43)—then the leading-order various physical situations of interest. correction terms, on the second and third lines of Eq. (43), Rather than begin with the usual relation of Eqs. (6) will also be of comparable magnitude to each other. In such between the real, relativistic scalar field and the complex, situations, it is inconsistent to retain (for example) the λ2 nonrelativistic field, we introduce a fairly simple, nonlocal corrections while neglecting the λ∇2 corrections. Put field redefinition, as in Eqs. (7). This new field redefinition another way, one must retain each of the corrections on considerably simplifies the treatment for free fields, and the second and third lines of Eq. (43) for situations in which enables us to construct an iterative, perturbative procedure the axions have virialized, with comparable (time-aver- in the presence of interactions. aged) values of kinetic and potential energy [22].We Other than imposing the standard commutation relations, 2 further note that in Ref. [20], the term mjψ sj appears in our treatment relies on manipulating classical fields. Even their low-energy Hamiltonian [Eqs. (26) and (27) of their so, we believe our formalism captures the relevant dynam- paper], whereas we find no such term in ours. As we show ics of the low-energy effective — 2 in Appendix A [see Eqs. (A3) and (A11)], the mjψ sj term including higher-order corrections in the coupling constant is exactly canceled by a compensating term arising from the λ. The tree-level diagrams of the quantum theory directly canonical transformation. reflect the equations of motion of the classical theory, and We may also compare our results in Eq. (43) with the we would expect that any low-energy effective Lagrangian recent calculation in Ref. [11]. Superficially our equations that produces the correct tree-level diagrams will also of motion appear to disagree at order λ2.However,as produce the correct loop diagrams. In our formalism we demonstrated in Appendix C, our results are completely find that particle number is conserved to all orders in the consistent (at least to this perturbative order), as can be nonrelativistic, perturbative description; an interesting shown by performing a nonlinear field redefinition. question is how best to incorporate particle-number-violat- Finally, we note that if one considers corrections from ing processes in a low-energy effective description, in a ðnÞ self-consistent way. (Cf. Refs. [11,28].) modes ψ ν with ν ≠ 0 and n ≥ 2, then one will generically We have focused on the dynamics of a single real, self- find higher-order time-derivative terms in the effective interacting scalar field. Applications of interest, which equation of motion for ψ , as discussed after Eq. (43). s we intend to explore in future work, include the behavior This is because the mode functions for fast-oscillating ðjÞ ðiÞ ðiÞ of axion dark matter in galactic halos (building on terms ψ ν depend, in general, on both ψ ν and ψ_ ν for Refs. [17,29] and references therein), as well as in i

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This work is supported in part by the U.S. Department of We can then algebraically manipulate these equations to Energy under grant Contract No. DE-SC0012567. find expressions for πðxÞ and ψ cðxÞ in terms of ϕ and πc,to compare with Eqs. (A3): Note added.—As we were finalizing revisions to this rffiffiffiffi m paper, Ref. [37] appeared, which reaches similar conclu- ψ x C P1=2 ϕ x cðt; Þ¼ 1 2 secðmtÞ ðt; Þ sions regarding relations among various approaches to the  low-energy effective descriptions of a given relativistic 1 − tanðmtÞπcðt; xÞ ; quantum field theory. C2 1 pffiffiffiffiffiffiffi 1=2 πðt; xÞ¼ 2mP secðmtÞπcðt; xÞ APPENDIX A: CANONICAL DERIVATION OF C2 THE FIELD REDEFINITION − mP tanðmtÞϕðt; xÞ: ðA5Þ Our calculations have been based on the field redefini- 1 2 tion of Eqs. (7), and their inverse given by Eq. (9).In To be as clear as possible, we express P and P = as integral Eq. (13) we constructed a Lagrangian that gives the correct operators, equation of motion for ψ, but in this Appendix we show Z how to construct the field redefinition as a canonical ðPfÞðxÞ ≡ d3yPðx; yÞfðyÞ; transformation. Canonical transformations are guaranteed Z to preserve the Poisson bracket relations, which become the P1=2 x ≡ 3 P1=2 x y commutators upon quantization. ð fÞð Þ d y ð ; ÞfðyÞ; ðA6Þ We start by adopting the real part of ψ as the new 1=2 1=2 canonical field ψ c, but we allow a normalization factor C1 where Pðx; yÞ and P ðx; yÞ are both symmetric, and P that can later be adjusted for convenience: is the operator square root of P, defined by Z ψ ¼ C1ψ ; ðA1Þ 3 1 2 1 2 c R d zP = ðx; zÞP = ðz; yÞ¼Pðx; yÞ: ðA7Þ where ψ ≡ ψ R þ iψ I. The canonical momentum conjugate to ψ c will be called πc, which we expect to be proportional It can then be seen that it is possible to find F2 by to ψ I: functionally integrating the first two of Eqs. (A3), where the consistency of the term involving both ϕ and π requires pffiffiffi c πc ¼ C2ψ I: ðA2Þ that C1C2 ¼ 2. We choose C1 ¼ C2 ¼ 2, so that

ϕ and π will be used to denote the original relativistic field 1 ψðt; xÞ¼pffiffiffi ðψ ðt; xÞþiπ ðt; xÞÞ; ðA8Þ and its canonical conjugate. To describe the canonical 2 c c transformation, we adapt the discrete-variable formalism of Goldstein, Poole, and Safko [38] to the case of continuous in agreement with Eq. (18). F2 is then given by fields. The canonical transformation is then defined in Z pffiffiffiffi terms of a generating functional F2½ϕ; πc;t, and the 3 3 1=2 F2½ϕ;πc;t¼ msecðmtÞ d xd yϕðt;xÞP ðx;yÞπcðt;yÞ canonical transformation is described by Z 1 δ δ ∂ − mtanðmtÞ d3xd3yϕðt;xÞPðx;yÞϕðt;yÞ π F2 ψ F2 F2 2 ðxÞ¼δϕ ; cðxÞ¼δπ ;Hnew ¼ H þ ∂ ; Z ðxÞ cðxÞ t 1 − 3 π2 x ðA3Þ 2tanðmtÞ d x cðt; Þ: ðA9Þ where the partial derivatives in Ref. [38] have been replaced To calculate the change in the Hamiltonian from the third of by functional derivatives. To try to match these equations, Eqs. (A3), one differentiates Eq. (A9) with respect to t and we first rewrite the transformation in terms of the real and then replaces ϕðt; xÞ by using imaginary parts of ψ: 1 rffiffiffiffi P1=2ϕ t; x ffiffiffiffi cos mt ψ t; x sin mt π t; x ; 2 ð Þð Þ¼p ½ ð Þ cð Þþ ð Þ cð Þ −1=2 m ϕðt; xÞ¼ P ½cosðmtÞψ Rðt; xÞþsinðmtÞψ Iðt; xÞ; m A10 pffiffiffiffiffiffiffi ð Þ 1=2 πðt; xÞ¼ 2mP ½cosðmtÞψ Iðt; xÞ − sinðmtÞψ Rðt; xÞ: which follows from Eqs. (A4), along with Eqs. (A1) ðA4Þ and (A2). The result is

016011-9 NAMJOO, GUTH, and KAISER PHYS. REV. D 98, 016011 (2018) Z ∂F2 1 1 ΔH ¼ ¼ − m d3xfψ 2ðt; xÞþπ2ðt; xÞg: ðA11Þ ϕðt; xÞ¼pffiffiffiffiffiffiffi ½e−imtχðt; xÞþeimtχðt; xÞ; ∂t 2 c c 2 rmffiffiffiffi m To find the new Hamiltonian, we must also express the π x − −imtχ x − imtχ x ðt; Þ¼ i 2 ½e ðt; Þ e ðt; Þ: ðB1Þ original Hamiltonian, given by Eq. (2), in terms of the new variables. Note that Z Z By comparing these equations with Eqs. (7), one can see 1 1 that χ is related to the field ψ that we used in the main body d3x m2ϕ2 ∇ϕ 2 d3x m2ϕ2 − ϕ∇2ϕ 2 f þð Þ g¼2 f g of this article by the field redefinition Z m2 ¼ d3xϕP2ϕ: ðA12Þ 1 1 2 χ P−1=2 P1=2 ψ 2imt P−1=2 −P1=2 ψ ¼ 2ð þ Þ þ2e ð Þ ; ðB2aÞ It is then straightforward to show that the free part of the original Hamiltonian can be written as 1 1 ψ P1=2 P−1=2 χ 2imt P1=2 −P−1=2 χ Z ¼ 2ð þ Þ þ2e ð Þ : ðB2bÞ 1 3 π Pπ ψ Pψ Hfree ¼ m d xf c c þ c cg: ðA13Þ 2 The equation of motion for χ takes the form The full new Hamiltonian, as prescribed by the third of 1 e2imt λ Eqs. (A3), is then given by χ_ x − ∇2χ − ∇2χ ˜ loc x i ðt; Þ¼ þ 2 G ðt; Þ; ðB3Þ Z 2m 2m 4!m 3 H Hnew ¼ d x new; where we have defined λ H ψ P −1 ψ −imtψ imtψ 4 new ¼ m ð Þ þ 2 ½e þe ; ðA14Þ ˜ loc x ≡ imt −imtχ imtχ 3 96m G ðt; Þ e ½e þ e ; ðB4Þ ψ where is given by Eq. (A8). Note that this result is in which matches to the similar object, G˜ , defined in Eq. (27), complete agreement with Eq. (19). in the limit P → 1. Note from Eq. (B3) that the global Uð1Þ symmetry is already broken at the free-field level, obscur- APPENDIX B: LOCAL VS NONLOCAL ing the conservation of particle number in the nonrelativ- FIELD REDEFINITION istic limit. As before, we decompose the field into different modes by In order to obtain the effective field theory for the nonrelativistic field we have used the nonlocal field X∞ iνmt redefinition of Eqs. (7), in contrast to the local redefinition χðt; xÞ¼ χνðt; xÞe ; ðB5Þ of Eqs. (6), which is more typically found in the literature. ν¼−∞ As discussed in Secs. II and III, the nonlocal field redefinition makes the computations easier, although it is and we define the mode with ν ¼ 0 as the slowly varying not fundamentally necessary. In this appendix, we begin portion of the field (in whose evolution we are interested), with the local field redefinition of Eqs. (6) and obtain the same low-energy effective theory at order n ¼ 1.Asa χ x ≡ χ x ν¼0ðt; Þ sðt; Þ: ðB6Þ separate, nontrivial consistency check, we compute the effective field theory for free fields up to order ϵ5 ∼ðk=mÞ10 x We then expand G˜ loc in a series similar to χ (in Fourier space) and again find results compatible with those obtained with the nonlocal field redefinition, even X∞ though, a priori, the results appear rather different. To ˜ loc ˜ loc iνmt G ðt; xÞ¼ Gν ðt; xÞe : ðB7Þ demonstrate the equivalence, we remove higher-order time- ν¼−∞ derivative operators that appear in the EFT in favor of spatial operators, using the equations of motion. For a given mode ν, Eq. (B3) then takes the form

1. Effective field theory through n =1 1 1 λ 2 2 ˜ loc iχ_ν −νmχν ¼ − ∇ χν − ∇ χ þ Gν ; ðB8Þ We begin with a local field redefinition, in which the 2m 2m 2−ν 4!m2 relativistic field ϕ and the nonrelativistic field χ are related by with

016011-10 RELATIVISTIC CORRECTIONS TO NONRELATIVISTIC … PHYS. REV. D 98, 016011 (2018) X ˜ loc x χ χ χ χχ χ ˜ locðnÞ Gν ðt; Þ¼ f μ μ0 2þν−μ−μ0 þ μ μ0 4−ν−μ−μ0 where Gν ðt; xÞ consists of all terms in Eq. (B11) which μ;μ0 are proportional to a total of n powers of the small 3χ χ χ 3χχ χ quantities ϵt, ϵx, and λ. Note that we decompose χν only þ μ μ0 μ μ0−ν þ μ μ0 ν−2þμþμ0 g: ðB9Þ þ ð0Þ ðnÞ for ν ≠ 0, and, as in Sec. III, we set χ0 ≡ χs, χ0 ¼ 0 for χ ð0Þ For the slowly varying mode, s, Eqs. (B8) and (B9) n>0, and χν ¼ 0 for ν ≠ 0. Expanding (B12) yields yield 1 λ ν χð1Þ ∇2χδ − ˜ locð0Þ ν ≠ 0 1 1 λ m ν ¼ 2 s 2;ν 4! 2 Gν for ; χ_ − ∇2χ − ∇2χ ˜ loc m m i s ¼ s 2 þ 2 G0 ; ðB10Þ 2m 2m 4!m ðnÞ ðn−1Þ 1 2 ðn−1Þ 1 2 ðn−1Þ νmχν ¼ iχ_ν þ ∇ χν þ ∇ χ 2m 2m 2−ν where λ ˜ locðn−1Þ − Gν ν ≠ 0 n ≥ 2: X 4! 2 for and ˜ loc χ χ χ χχ χ 3χ χ χ m G0 ¼ f μ μ0 2−μ−μ0 þ μ μ0 4−μ−μ0 þ μ μ0 μþμ0 μ;μ0 ðB15Þ 3χχ χ þ μ μ0 μþμ0−2g: ðB11Þ From this iterative expansion one can obtain the effective equation for χs: Equations (B8) and (B10) reveal that even at linear order in X∞ X∞ 1 1 λ ðnÞ the modes χν, different modes couple to each other. This is iχ_ ¼ − ∇2χ − ∇2 χðnÞ þ G˜ loc : ðB16Þ s 2m s 2m 2 4! 2 0 one source of complication in the local field redefinition n¼1 m n¼0 approach. The other difficulty is that powers of the spatial As in Sec. III, we only compute the leading-order correc- Laplacian appear at each order of the iteration, rather than tions to the Schrödinger equation for χ . From Eq. (B9), having them all contained in the operator P. As a result, we s we have must expand in powers of all three small quantities (ϵt, ϵx, and λ) from the beginning of our iterative computations. 0 ˜ locð Þ χ3δ χ3δ 3 χ 2χδ 3 χ 2χ δ This is in contrast with the calculation in Sec. III, in which Gν ¼ s ν;−2 þ s ν;4 þ j sj s ν;2 þ j sj s ν;0; we used the nonlocal field redefinition, which enabled us ðB17Þ to conduct most of the iterative calculation, until the final step, in terms of just two small parameters (ϵt and λ). Apart and then to order n ¼ 1, we find from these technical difficulties, however, nothing prevents  1 λ 1 χ ð1Þ 2 3 3 us from obtaining an effective field theory for s using χν ¼ ∇ χ δν 2 þ χ δν −2 − χ δν 4 – 4m2 s ; 2 × 4!m3 s ; 2 s ; Eqs. (B8) (B11).  It is instructive to rewrite (B8) as 2 − 3jχ j χ δν 2 for ν ≠ 0: ðB18Þ 1 1 λ s s ; ν χ χ_ ∇2χ ∇2χ − ˜ loc m ν ¼ i ν þ ν þ 2−ν 2 Gν ; ðB12Þ 2m 2m 4!m Again from Eq. (B9), we find where, in contrast with the nonlocal case in Eq. (32),we ˜ locð1Þ 3 χ2χð1Þ χ2χð1Þ χ2χð1Þ 2 χ 2χð1Þ have kept the prefactors on the left-hand side to avoid Gν ¼ ½ s 2þν þ s 4−ν þ s −ν þ j sj ν ν 0 the apparent divergence for ¼ . The right-hand side of 2 ð1Þ 2 ð1Þ þ χs χν−2 þ 2jχsj χ2−ν ; ðB19Þ Eq. (B12) can be thought of as a perturbative source for χν: the first term is suppressed relative to χν by ϵ , the second t so and third terms are suppressed relative to χν and χ2−ν by ϵx, ˜ loc and the last term is suppressed relative to Gν by λ. 1 3λ 1 1 iχ_ ¼ − ∇2χ þ jχ j2χ − ∇2χð Þ Therefore, as noted above, our expansion in this case is s 2m s 4!m2 s s 2m 2 in powers of all three small quantities. 3λ h i 2 ð1Þ 2 ð1Þ 2 ð1Þ 2 ð1Þ ν ≠ 0 þ χ χ2 þ χ χ4 þ χ χ−2 þ 2jχ j χ2 We next expand the modes with : 4!m2 s s s s X∞ O ϵ2 ðnÞ þ ð Þ: ðB20Þ χνðt; xÞ¼ χν ðt; xÞ for ν ≠ 0; ðB13Þ n¼1 For convenience, we have introduced the shorthand ðnÞ notation where the terms χν are to be determined by iterative ˜ loc approximation, as described below. We also expand Gν : 2 2 2 2 Oðϵ Þ ≡ O½ϵx; ϵt ; λ ; ϵxϵt; ϵxλ; ϵtλ; X∞ Oðϵ3Þ ≡ O½ϵ3; ϵ3; λ3; ϵ2ϵ ; ϵ2λ; ϵ ϵ2; ϵ λ2; ϵ ϵ λ; ϵ2λ; ϵ λ2: ˜ loc ˜ locðnÞ x t x t x x t x x t t t Gν ðt; xÞ¼ Gν ðt; xÞ; ðB14Þ n¼0 ðB21Þ

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ð1Þ Next we substitute χν from Eq. (B18) into Eq. (B20), not be straightforward. We also show that the final result, which yields after removing the higher-order time derivatives, is of exactly the form one would obtain for ψ, as defined via 1 λ 1 the nonlocal field redefinition. We pursue the comparison χ_ ≃− ∇2χ χ 2χ − ∇4χ i s s þ 2 j sj s 3 s ϵ5 ∼ 10 2m 8m 8m up to order x ðk=mÞ . λ þ ½χ2∇2χ þ 2jχ j2∇2χ þ ∇2ðjχ j2χ Þ 32m4 s s s s s s 2. Higher-order time-derivative terms 17λ2 in the free-field limit − χ 4χ O ϵ3 5 j sj s þ ð Þ: ðB22Þ χ 768m In the free-field limit the equation of motion for , Eq. (B3), simplifies to Equation (B22) for χs exactly matches Eq. (43) for ψ s, which is related to the relativistic field ϕ via the nonlocal 1 e2imt iχ_ ¼ − ∇2χ − ∇2χ: ðB25Þ field redefinition of Eqs. (7). 2m 2m Since χ and ψ are related to each other by the nontrivial field redefinition of Eqs. (B2), there was no guarantee that Equation (B25) (together with its complex conjugate) can χs and ψ s should obey the same equation. The two field be solved exactly, with a solution of the form of Eq. (B2a). theories must be equivalent, since they are both equivalent As discussed above, the solution is the sum of two terms, to the low-energy effective field theory derived from the one that varies slowly, and the other that varies rapidly. 4 Thus it is straightforward to write an exact expression relativistic ϕ theory, but χs and ψ s could be related by a field redefinition that causes them to have different equa- for the slow mode, Eq. (B23). However, here we will put tions of motion. We have found, however, that through aside the exact solution, and use this case to further 2 illustrate the iterative approximation technique, as was Oðϵ Þ, the equations for χs and ψ s are identical. We do not know if this relation will continue to hold at higher orders. used in Appendix B1. χ However, for the special case of the free field theory, λ ¼ 0, After decomposing the field into modes as in Eq. (B5) and again assigning χν 0 ≡ χ , the equation of motion for the relation between χs and ψ s is simple. For the free field ¼ s χ takes the form theory, ψ itself is slowly varying, so ψ s ¼ ψ. Then in the s χ ψ Eq. (B2a), which expresses in terms of , the first term is 1 1 2 2 purely slowly varying, while the second term is the product iχ_ ¼ − ∇ χ − ∇ χ2: ðB26Þ 2 s 2 s 2 of e imt and a slowly varying function. Thus, the first term m m is the slowly varying part of χ: Therefore, among all modes χν with ν ≠ 0, only χ2 contributes to the evolution of χs in the limit λ → 0.So 1 −1 2 1 2 χ P = P = ψ we only need to find an appropriate substitution for χ2 in s ¼ 2 ð þ Þ s: ðB23Þ order to obtain the EFT for χs in this limit. From Eq. (B15) Since ðP−1=2 þ P1=2Þ commutes with the differential oper- we may write χ ators in the equation of motion, s obeys the same equation 1 1 2 2 of motion as ϕ , iχ_2 − 2mχ2 ¼ − ∇ χ2 − ∇ χ : ðB27Þ s 2m 2m s χ_ P − 1 χ i s ¼ mð Þ s: ðB24Þ Decomposing χ2 as in Eq. (B13) it is easy to obtain

The iterative procedure used above can be continued 1 1 χð Þ ¼ ∇2χ; ðB28Þ to obtain higher- and higher-order corrections to the 2 4m2 s Schrödinger equation. Generically, higher-order terms con- tain higher-order time-derivative operators, as expected in while for higher-order iterations we have an EFT framework, and as was discussed after Eq. (43)   and at the end of Sec. III. The higher-order time-derivative 1 1 −1 χðnÞ ¼ ∇2 þ i∂ χðn Þ; for n ≥ 2 ðB29Þ terms do not introduce new d.o.f., because such terms must 2 2m 2m t 2 be considered perturbations around the zeroth-order equa- tions. In fact, one can use the equations of motion to all of which can be encapsulated in the relation remove the higher-order time-derivative terms and replace   them with lower-order terms [25]. (See also [26,27].) We 1 1 i n−1 χðnÞ ¼ ∇2 ∇2 þ ∂ χ: ðB30Þ demonstrate this explicitly in the following subsection for χ 2 4m2 4m2 2m t s defined via the local field redefinition of Eq. (B1), in the free-field limit (taking λ → 0). This limit is sufficiently Having obtained χ2 at arbitrary iteration n, one may try to nontrivial that a generalization to the interacting case would formally resum the infinite series to obtain

016011-12 RELATIVISTIC CORRECTIONS TO NONRELATIVISTIC … PHYS. REV. D 98, 016011 (2018)   X∞ 1 1 −1 2 4 6 4 8 ðnÞ 2 2 i k k k k k χ2 ¼ χ ¼ ∇ 1− ∇ − ∂ χ : ðB31Þ iχ_ ≃ χ − χ þ χ þ χ̈ − χ 2 4 2 4 2 2m t s s 2m s 8m3 s 16m5 s 32m5 s 32m7 s n¼1 m m 4 6 10 ik … 3k k − χ − χ̈ þ χ þ O½ðk=mÞ12: Substituting this relation back to Eq. (B26) yields 64m6 s 27m7 s 64m9 s   ðB35Þ 1 1 1 i −1 iχ_ ¼ − ∇2χ − ∇4 1 − ∇2 þ ∂ χ : s 2m s 8m3 4m2 2m t s The right-hand side of Eq. (B35) contains no first time- derivative terms, though it still contains higher-order ones. ðB32Þ To remove the χ̈s terms, we take a time derivative of Eq. (B35) once and remove terms containing χ_s using Although the infinite series has been resummed, one can Eq. (B35) itself, again only keeping terms up to the desired check that the exact solution of Eq. (B26) would satisfy order in (k=m). This yields an algebraic equation for χ̈s that Eq. (B32). Here we will only consider the first few terms does not contain χ_s. Using the resulting equation to remove among the infinite tower of terms in order to demonstrate χ̈s from Eq. (B35) yields how higher-order time derivatives enter the calculation and may be systematically removed. To do so, it is easiest to k2 k4 k6 5k8 work in Fourier space, within which we expand Eq. (B32) iχ_ ≃ χ − χ þ χ − χ 10 s 2 s 3 s 5 s 27 7 s up to the order ðk=mÞ : m 8m 16m m 4 10 ik … 13k − χ þ χ : ðB36Þ k2 64m6 s 29m9 s iχ_ ≃ χ s 2m s 4 By this series of steps, we have removed χ̈ from the k s … − χs equation. We repeat the same procedure to remove χ by 8m3 s 4 6 taking the time derivative of Eq. …(B36) twice and sub- ik χ_ k χ stituting the resulting expression for χ into Eq. (B36). This þ 4 s þ 5 s s 16m 32m finally yields 4 6 8 k χ̈ − ik χ_ − k χ   þ 32 5 s 32 6 s 27 7 s k2 k4 k6 5k8 7k10 m m m iχ_ ¼ − þ − þ χ 4 6 8 10 s 2 8 3 16 4 128 7 256 9 s ik 3k 3ik k m m m m m − χ − χ̈ þ χ_ þ χ 64m6 s 27m7 s 28m8 s 29m9 s þ O½ðk=mÞ12; ðB37Þ þ O½ðk=mÞ12: ðB33Þ which no longer includes any higher-order time-derivative In Eq. (B33) we have arranged terms of similar magnitude terms. The form of Eq. (B37) is in complete agreement to be in the same line. Notice that from the equation (up to the working order) with the expansion of the of motion, Eq. (B26), one can conclude that the time- equation of motion in the free-field limit that one would derivative operator and the spatial Laplacian are at the same obtain by using the nonlocal field redefinition, namely ∂ ∼ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! order (i.e., in Fourier space, t k =m) so that, effectively, 2 our expansion is in powers of (k=m). ψ_ 1 k − 1 ψ i s ¼ m þ 2 s; ðB38Þ Equation (B33) is a higher-order equation in time m derivatives. But, as mentioned above, since all terms from the second to the fifth line should be considered as which is the Fourier transform of Eq. (40) in the perturbations, we can replace higher-order time-derivative limit λ → 0. terms with lower-order terms, upon using the equation of motion and its time derivatives. The procedure is simple. First, consider Eq. (B33) as an algebraic equation for χ_s; APPENDIX C: COMPARISON WITH THE move all terms involving χ_s to the left-hand side; divide RECENT CALCULATION BY MUKAIDA, both sides by the resulting coefficient: TAKIMOTO, AND YAMADA   In this Appendix we compare our EFT with the recent 4 6 3 8 1 − k k − k calculation by Mukaida, Takimoto, and Yamada in Ref. [11]. 4 þ 6 8 8 ; ðB34Þ 16m 32m 2 m We show that the two resulting low-energy descriptions are equivalent, related by a field redefinition. and expand up to the desired order (here Oðk=mÞ10). This In Ref. [11] the authors begin by separating the real-valued yields scalar field of the relativistic theory, ϕ, into a nonrelativistic

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ϕ ϕ δϕ P χ component and fluctuations, ðxÞ¼ NRðxÞþ ðxÞ.The operator . However, according to Eq. (B18), the 2 term ϕ nonrelativistic field NR is defined as acquires an extra contribution that exactly compensates the Z missing term due to the absence of P from the relation σ χ ϕ 4 −iK·xϕ between and . NRðxÞ¼ d Ke ðKÞ; ðC1Þ K∈NR To demonstrate that the low-energy effective descrip- tions for σ and ψ s are indeed related through the field where K ∈ NR indicates the region of four-momentum redefinition of Eq. (C5), we first consider the equations of μ 0 0 2 2 K ¼ðk ;kÞ with jk j∼mc þOðmv Þ and 0 ≤ jkj ≤ mv, motion. In Eq. (2.15) of Ref. [11] the authors present their with v ≪ c. (The fluctuation δϕ is defined as the Fourier effective Lagrangian, which is given by integral over the complementary range of Kμ.) Next the authors parameterize the nonrelativistic component as 1 L 2 σσ_ − σσ̈ σ∇2σ − ¼ 4 ½ im þ Veff; ðC6Þ 1 ϕ x σ x −imt σ x imt NRðt; Þ¼2 ½ ðt; Þe þ ðt; Þe : ðC2Þ where, from Eq. (2.17) of Ref. [11], (The field we have labeled σ is denoted by Ψ in Ref. [11]; 2 we use σ to avoid confusion with the modes Ψν defined in 3g4 g4 V ðσ; σÞ¼− jσj4 þ jσj6: ðC7Þ Eq. (31).) In contrast with our approach, the authors of eff 8 128m2 Ref. [11] first separate the relativistic field ϕ into components with small and large spatial momenta, σ and δϕ (respectively), Here g4 is the coupling constant of the quartic interaction in and then construct an EFT for σ, whereas we relate the real- the original, relativistic theory, and we have set the cubic valued relativistic field ϕ to the complex field ψ via Eqs. (7) interaction to zero, to match the form of VðϕÞ that we have and construct an EFT for the slowly varying portion, ψ s, considered throughout our analysis. The quartic coupling identified as the ν ¼ 0 mode of Eq. (24). This suggests that constant g4 of Ref. [11] is related to the coupling λ we the two resulting low-energy effective descriptions might introduced in Eq. (1) by g4 ¼ −λ=6. be related by a field redefinition. In particular, substituting The equation of motion for σ, up to the order that the mode decomposition of Eq. (24) into Eq. (7),wemay matches our analysis, can be obtained by varying the action relate σ to ψ: with respect to σ, which yields 1 −imt imt 1 1 1 ½σðt; xÞe þ σðt; xÞe σ_ σ̈− ∇2σ 2 i ¼ 2 2 þ 2 Veff;σ : ðC8Þ 1 m m m ffiffiffiffiffiffiffi −1=2 −imt imt ¼ p P ½ðψ s þ ψ 2Þe þðψ s þ ψ 2Þe : ðC3Þ 2m Equation (C8) is second order in time derivatives. Upon substituting the field redefinition of Eq. (C5), we find an Since σ, ψ s,andψ 2 are each slowly varying functions of t, equation of motion for ψ s which is also second order in Eq. (C3) implies that time derivatives: rffiffiffiffi m −1=2 1 λ σ ¼ P ðψ þ ψ Þ: ðC4Þ 2 2 2 s 2 iψ_ ¼ − ∇ ψ þ jψ j ψ s 2m s 8m2 s s 1 i iλ At leading order in ϵ , ϵ ,andλ,thisreducesto ψ̈ − ∇2ψ_ ψ 2ψ_ ψ ψ ψ_ x t þ 2 s 4 2 s þ 16 3 sð s s þ s s Þ rffiffiffiffi   m m m 1 λ 1 λ m 2 2 2 − ∇4ψ ψ 2∇2ψ 2 ψ 2∇2ψ σ ¼ 1þ ∇ ψ − jψ j ψ þOðϵ Þ; ðC5Þ 3 s þ 4 ½ s s þ j sj s 2 4m2 s 16m3 s s 8m 32m 17λ2 ∇2 ψ 2ψ − ψ 4ψ O ϵ3 P þ ðj sj sÞ 5 j sj s þ ð Þ: ðC9Þ where we have expanded the nonlocal operator ,usedthe 768m relations in Eqs. (38) and (41) to replace ψ 2,andadoptedthe 2 notation Oðϵ Þ of Eq. (B21) to indicate terms that are at least Equation (C9) does not appear to match the equation of ϵ ϵ λ second order in the small quantities x, t,and . motion for ψ s which we found above, in Eq. (43).In Before showing the equivalence of the low-energy particular, the two equations differ by the presence of the σ ψ effective descriptions for and s, we note that one can terms in the second line of Eq. (C9). However, to the derive the same relation as in Eq. (C5), starting from the perturbative order to which we are working, it is straight- locally defined field χ of Eq. (B1)p.ffiffiffiffiffiffiffiffiffi In that case, the relation forward to show that the new terms that appear in Eq. (C9) between the two fields becomes m=2σ ¼ χs þ χ2, which vanish. In particular, we may take one time derivative of differs from Eq. (C4) by the absence of the nonlocal Eq. (C9) to find

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1 λ obtained in Eq. (44), although it is easy to show that iψ̈ ¼ − ∇2ψ_ þ ψ ð2ψ_ ψ þ ψ ψ_ ÞþOðϵ3Þ: s 2m s 8m2 s s s s s Eq. (C12) gives rise to the same equations of motion. To ðC10Þ demonstrate the equivalence between the Lagrangians in Eqs. (44) and (C12), we perform another field redefinition: Substituting Eq. (C10) into Eq. (C9) yields   1 1 λ ψ ψ˜ − ψ˜_ ∇2ψ˜ − ψ˜ 2ψ˜ 1 λ 1 s ¼ i þ 2 j j : ðC13Þ iψ_ ¼ − ∇2ψ þ jψ j2ψ − ∇4ψ 4m 2m 8m s 2m s 8m2 s s 8m3 s λ þ ½ψ 2∇2ψ þ 2jψ j2∇2ψ þ ∇2ðjψ j2ψ Þ Substituting Eq. (C13) into Eq. (C12) yields 32m4 s s s s s s 17λ2 4 3 1 λ − ψ ψ O ϵ : C11 i _ _ 4 768 5 j sj s þ ð Þ ð Þ L½ψ˜ ¼ ðψ˜ ψ˜ − ψ˜ ψ˜ Þ − ∇ψ˜ ∇ψ˜ − jψ˜ j m 2 2m 16m2 1 λ To the order to which we have been working, Eq. (C11) ∇2ψ˜ ∇2ψ˜ − ψ˜ 2 ψ˜ ∇2ψ˜ ψ˜ ∇2ψ˜ þ 3 4 j j ð þ Þ exactly matches Eq. (43). We therefore find that the equations 8m 32m of motion for the two low-energy effective descriptions are 17λ2 þ jψ˜ j6: ðC14Þ indeed equivalent, at least up to the working order. 9 × 28m5 So far our discussion has established the equivalence between the two low-energy descriptions at the classical This is precisely the form of the Lagrangian of Eq. (44), level. To analyze the equivalence even for matters concern- after trivial relabeling of the dynamical field ψ s → ψ˜ . ing quantization, we next consider the relevant Lagrangians. Since the Lagrangians of Eqs. (C6), (C12),and(C14) are Substituting the field redefinition of Eq. (C5) into the related to each other via field redefinitions, we expect their Lagrangian of Eq. (C6) and performing some straightfor- corresponding S-matrices to remain equivalent as well, at ward algebra, we find least up to the perturbative order to which we have been – i 1 λ working [25 27]. L ψ ψ_ ψ − ψ ψ_ − ∇ψ ∇ψ − ψ 4 ½ s¼2ð s s s sÞ 2 s s 16 2 j sj We close with some comments on the field redefinitions m m in Eqs. (C5) and (C13). Both of the field redefinitions may 1 i 1 − ψ ψ̈ − ψ_ ∇2ψ ∇2ψ ∇2ψ be written in the form ψ → ψ ϵT ψ , where T ψ is a s s 2 s s þ 3 s s þ ð Þ ð Þ 2m 2m 4m local function of ψ and its derivatives. As demonstrated by iλ λ − jψ j2ψ ψ_ − jψ j2ðψ ∇2ψ þ ψ ∇2ψ Þ Arzt in Ref. [27] (see also Refs. [25,26]), within an EFT 8m3 s s s 16m4 s s s s s context, such redefinitions do not change the S-matrix. We 2 35λ 6 therefore conclude that the low-energy effective description þ jψ j ; ðC12Þ ψ 9 × 28m5 s we have derived for the field s is equivalent, to our working order in perturbation theory, to the low-energy where we have neglected some boundary terms. The description derived in Ref. [11], at both the classical and Lagrangian of Eq. (C12) is different from the one we quantum levels [39].

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Wilson, Coordinated interactions 3rd ed. (Addison Wesley, Reading, MA, 2002), p. 373. between hippocampal ripples and cortical spindles during [39] Recently a paper appeared [40], which criticizes our slow-wave sleep, Neuron 21, 1123 (1998). approach to constructing nonrelativistic effective field [19] Z. Clemens, M. Möll, L. Eröss, R. Jakus, G. Rásonyi, P. theories. The authors raise a number of objections to Halász, and J. Born, Fine-tuned coupling between human the nonlocal field redefinition of our Eqs. (7), but they all parahippocampal ripples and sleep spindles, Eur. J. Neuro- appear to be based on misunderstandings. First, the sci. 33, 511 (2011). authors say that it is enough to define a new field ψ by [20] E. Braaten, A. Mohapatra, and H. Zhang, Nonrelativistic our Eq. (6a), but that is not true. Equation (6a) cannot be effective field theory for axions, Phys. Rev. D 94, 076004 inverted to define the new field ψ in terms of the original (2016). relativistic field ϕ. One way to see this is to notice that [21] R. Banerjee and P. Mukherjee, On the subtelties of non- Eq. (6a) depends only on the real part of e−imtψ,and relativistic reduction and applications, arXiv:1801.08373. therefore says nothing about the imaginary part. As a [22] Axion dark matter is expected to be virialized in galaxies, consequence, replacing ψ by ψ → ψ 0 ¼ ψ þ ieimtρðt; xÞ, though in that case the dominant contribution to the for any real-valued function ρðt; xÞ, results in no change in potential energy arises from gravitational self-interactions ϕðt; xÞ.Tofixψðt; xÞ uniquely, at any given time, both of rather than from the λϕ4 interaction that we study here. See, Eqs. (6a) and (6b) are needed. In this paper, however, we e.g., Refs. [23,24]. showed that there are advantages to using the nonlocal [23] F. Zwicky, Die Rotverschiebung von extragalaktischen definition of Eqs. (7a) and (7b),althoughweshowedin Nebeln, Helv. Phys. Acta 6, 110 (1933); F. Zwicky, On Appendix B that the local definition could alternatively be the masses of nebulae and of clusters of nebulae, Astrophys. used. The arguments in Ref. [40] seem inconsistent with J. 86, 217 (1937). the fact that Eqs. (7a) and (7b) together allow a unique [24] J. Binney and S. Tremaine, Galactic Dynamics,2nded. determination of ψðt; xÞ at any given time, in terms of (Princeton University Press, Princeton, NJ, 2008), ϕðt; xÞ and πðt; xÞ,whereψðt; xÞ is given explicitly by our Chap. 9. Eq. (9), which is an exact inverse to Eqs. (7a) and (7b). [25] C. Grosse-Knetter, Effective Lagrangians and higher order Using this one-to-one mapping of field configurations, derivatives, Phys. Rev. D 49, 6709 (1994). replacing the two real functions ϕðt; xÞ and πðt; xÞ with [26] H. Georgi, On-shell effective field theory, Nucl. Phys. B361, one complex function ψðt; xÞ, we construct a theory for 339 (1991). ψðt; xÞ that is exactly equivalent to the original relativistic [27] C. Arzt, Reduced effective Lagrangians, Phys. Lett. B 342, theory. After that, we make approximations that are 189 (1995). appropriate in the nonrelativistic regime. The authors [28] E. Braaten, A. Mohapatra, and H. Zhang, Emission of go on to argue that since our equation of motion is first photons and relativistic axions from axion stars, Phys. Rev. order in the time derivative but infinite-order in spatial D 96, 031901(R) (2017). derivatives, it requires an infinite number of initial con- [29] H. Deng, M. P. Hertzberg, M. H. Namjoo, and A. Masoumi, ditions. But our initial value problem is exactly equivalent Can light dark matter solve the core-cusp problem?, to the standard case: given an initial value for the spatial arXiv:1804.05921. function ψðti; xÞ (which is equivalent to knowing ϕðti; xÞ [30] C. J. Hogan and M. J. Rees, Axion miniclusters, Phys. Lett. and πðti; xÞ), the differential equation determines the time B 205, 228 (1988). derivative, so one can find the function at a slightly later [31] E. W. Kolb and I. I. Tkachev, Axion Miniclusters and Bose time. Next the authors point out that in the free-field Stars, Phys. Rev. Lett. 71, 3051 (1993). theory case (λ ¼ 0), our equation of motion gives only a

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positive frequency solution for ψðt; xÞ. They claim that ψðt; xÞ, and vice versa. We also point out that our therefore our treatment does not contain all the informa- equation of motion for ψ sðt; xÞ, through second order tion of the complete theory. However, the negative in small quantities, agrees exactly with Refs. [11] and frequency solutions are all contained in ψ ðt; xÞ, and both [20]. ψðt; xÞ and ψ ðt; xÞ are used to reconstruct ϕðt; xÞ.Again, [40] J. T. Firouzjaee, A. Rostami, and M. Bahmanabadi, Revis- there is an exact one-to-one mapping, so any solution for iting nonrelativistic limits of field theory: Antiparticles, ϕðt; xÞ and πðt; xÞ can be reconstructed by knowing arXiv:1804.08600.

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