Physical stress, mass, and energy for non-relativistic matter

Journal of High Energy Physics, Jun 2017

For theories of relativistic matter fields there exist two possible definitions of the stress-energy tensor, one defined by a variation of the action with the coframes at fixed connection, and the other at fixed torsion. These two stress-energy tensors do not necessarily coincide and it is the latter that corresponds to the Cauchy stress measured in the lab. In this note we discuss the corresponding issue for non-relativistic matter theories. We point out that while the physical non-relativistic stress, momentum, and mass currents are defined by a variation of the action at fixed torsion, the energy current does not admit such a description and is naturally defined at fixed connection. Any attempt to define an energy current at fixed torsion results in an ambiguity which cannot be resolved from the background spacetime data or conservation laws. We also provide computations of these quantities for some simple non-relativistic actions.

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Physical stress, mass, and energy for non-relativistic matter

Received: November Physical stress, mass, and energy for non-relativistic matter Michael Geracie 0 1 4 Kartik Prabhu 0 1 2 Matthew M. Roberts 0 1 3 0 Open Access , c The Authors 1 University of California , Davis, CA 95616 , U.S.A 2 Cornell Laboratory for Accelerator-based Sciences and Education (CLASSE), Cornell University 3 Kadanoff Center for Theoretical Physics, University of Chicago 4 Center for Quantum Mathematics and Physics (QMAP), Department of Physics For theories of relativistic matter fields there exist two possible definitions of the stress-energy tensor, one defined by a variation of the action with the coframes at fixed connection, and the other at fixed torsion. These two stress-energy tensors do not necessarily coincide and it is the latter that corresponds to the Cauchy stress measured in the lab. In this note we discuss the corresponding issue for non-relativistic matter theories. We point out that while the physical non-relativistic stress, momentum, and mass currents are defined by a variation of the action at fixed torsion, the energy current does not admit such a description and is naturally defined at fixed connection. Any attempt to define an energy current at fixed torsion results in an ambiguity which cannot be resolved from the background spacetime data or conservation laws. We also provide computations of these quantities for some simple non-relativistic actions. ArXiv ePrint: 1609.06729 matter; Space-Time Symmetries; Differential and Algebraic Geometry; Effective Field - Spinful Schr¨odinger field Non-relativistic Dirac field A Symmetries of the Riemann tensor Ward identities The stress-energy tensor The Cauchy stress-mass tensor Energy currents and the Hamiltonian Galilean Ward identity Diffeomorphism and U(1)M Ward identity 1 Introduction Bargmann spacetimes The Galilean group and its representations The physics of Bargmann geometries Stress-energy for non-relativistic matter fields In relativistic theories with spinful matter there are two possible definitions of the stressenergy tensor [1–3]. One can vary the matter action considering the coframes eA ≡ eµAdxµ S = Z dd+1x|e|(−T˜µ AδeµA + sµAB δωµAB ) to define the stress-energy tensor T˜µ A and the spin current sµAB . Alternatively, since there exists a unique torsion-free metric compatible connection — the Levi-Civita connection and the contorsion CAB ≡ ωAB − ω(LC)AB as independent S = Z dd+1x|e|(−T µ AδeAµ + sµAB δCµAB ) and we have used the Lorentzian coframes and frames to convert the internal frame indices to spacetime indices. In the relativistic case, (1.2) is equivalent to considering the coframes and the torsion T A ≡ 2 1 T Aµν dxµ ∧ dxν as the independent variables S = dd+1x|e|(−T µ AδeµA + SAµν δT Aµν ) . = ηABeλBSλµν is algebraically related (and thus, equivalent) to sµAB . But even on torsionless background spacetimes, the “new” gets additional contributions from the derivatives of the spin when the matter fields carry spin. physical problems. The Noether identity corresponding to local Lorentz transformations tensor which is the relevant physical quantity when considering shearing or straining the system. More directly, stresses in lattice systems are induced by spatial deformations of the system without introducing dislocations i.e. varying the spatial geometry at fixed torsion.1 to gravity through the Einstein equation (see [1, 2]). The main goal of this paper is to investigate a similar issue that arises for nonrelativistic Galilean invariant matter fields with spin and highlight some subtleties not present in the relativistic case.2 As many non-relativistic systems are constructed out of particles with spin, this is a crucial step in describing their physical properties in a covariant manner. We summarize the main arguments and results in the following. We work with the covariant construction of non-relativistic spacetimes following the formulation introduced in [6, 7], called Bargmann spacetimes.3 For matter fields on a Bargmann spacetime, the covariant non-relativistic stress-energy is a tensor of the form the coframes and Galilean connection as independent variables (i.e. through the analogue S = dd+1x|e|(−τ˜µ I δeI µ + sµAB δωµAB ) . 1See [4] and references therein for discussions on computing stress response from a lattice theory. 2Similar results were obtained, using different methods, by [5]. 3A more thorough list of references for Newton-Cartan geometry and its applications is provided in Here eI contains not just the spacetime coframes eA but also the Newtonian potential a. We show, for spinful matter fields on torsionless background spacetimes, using the decomsymmetric i.e. it does not correspond to the Cauchy stress tensor, and (2) the momentum need not coincide with the mass current. In contrast to the relativistic case, in general torsionful Bargmann spacetimes one does not have a natural unique reference Galilean connection (unlike the Levi-Civita connection in the relativistic case) and so the Cauchy stress-energy must be computed by varying the coframes and torsion as independent variables (similar to (1.4)) S = However, due the non-relativistic nature of the spacetime, the variations of the coframes and torsion are not independent but have to satisfy a covariant constraint (see (2.23)). This constrained variation leads to the following novel feature in non-relativistic theories: the Cauchy stress, momentum, and mass current can be collected into a covariant Cauchy stress-mass tensor,4 T AB = which guarantees that the Cauchy stress tensor T ab is symmetric when the torsion vanishes Further, the Noether identities for diffeomorphisms give conservation law − eµI (Dν − T λλν )τ˜ν I = Fµν jν + RABµν sνAB which contains the work-energy equation (see (4.22)) as well as a conservation law for the physical stress-mass tensor (∇ν − T λλν )T νµ = F µ ν jν + ΞˆAµ νλSAνλ 4A more precise, but unwieldy name, would be the stress-mass-momentum tensor, but as we will show a Noether identity equates the momentum with the mass current for the physical Cauchy stress-mass tensor. 5The kinetic energy current can not be defined in a frame independent way, and in a given local Galilean which is unambigously defined in complete analogy with the relativistic case (see (1.3)) antisymmetric tensor (see (3.21)). We argue that this ambiguity is unphysical as the relativistic system. We then show that the Noether identity for local Galilean transformations is torsionful terms are essential for studying energy response [8, 9] and for applications in non-relativistic fluid dynamics [6], they do of course vanish in the real world.6 In this case these identities take the simpler form The remainder of the paper details the above results and is organized as follows. We begin in section 2 with a summary of Bargmann spacetimes and the relevant geometric data. Section 3 gives explicit formulae for the Cauchy stress, momentum, and mass current attempting to define a “Cauchy energy current”. We give the Noether identities for the Cauchy stress-mass tensor in section 4. In section 5 we provide examples of Cauchy stress and mass tensors for non-relativistic field theories. Appendix A collects the symmetry properties of the non-relativistic Reimann tensor in the presence of torsion, which we use to simplify some of the formulae in the main body of the paper. Bargmann spacetimes Newton-Cartan geometry was originally developed by Cartan to describe Newtonian gravity within a geometric framework similar to that of General Relativity [10, 11] (see also [12, 13]). Recently, it has been used in the condensed matter literature as the natural setting for Galilean invariant physics, with applications that include cold atoms [14], non-relativistic fluids [6, 15–17], the quantum Hall effect [18–22], as well as non-relativistic holographic systems [23–27]. It is well recognized in the literature that it is necessary to couple these systems to torsionful geometries to define the full suite of currents available in a nonrelativistic system and to study their linear response [9, 23, 24, 26, 28]. Hence in this section and the next, all formulae will be written for the most general case of unconstrained A manifestly Galilean covariant definition of torsionful Newton-Cartan geometries was given in [7] (related constructions can be found in [28–34]). These geometries are called Bargmann geometries and this section is dedicated to a brief review of their features. In section 2.1 we introduce the necessary background, formally define a Bargmann geometry, and collect the identities that will be used repeatedly throughout this note. Section 2.2 then recaps the physics of Bargmann geometries. The Galilean group and its representations The Galilean group Gal(d), is the set of matrices of the form ering Euclidean statistical path integrals with inhomogeneous temperature. −ka Rab where Rab are spatial rotation matrices in SO(d) and ka parametrize Galilean boosts. Our conventions are that capital Latin indices A, B, . . . transform in the vector representations of Gal(d), while lower case Latin indices a, b, . . . transform under the SO(d) subgroup. The Galilean group preserves the invariant tensors nA = 1 0 , hAB = Here nA is called the internal clock form, hAB the internal spatial metric, and ǫA0...Ad is the There is another (d + 2)-dimensional representation of Gal(d) given by hABnB = 0 . − 12 k2 kcRcb 1 eI = ea , ωAB → ΛAC (ωC D + δC Dd)(Λ−1)DB . This representation will prove useful in what follows and we call it the extended representation. It preserves an extended version of the clock form nI as well as a (d + 2)-dimensional internal metric of Lorentzian signature which we shall use to raise and lower extended nI = 1 0 0 , The defining and extended representations also together preserve a mixed invariant that may be used to project from the extended to the vector representation, or pull back from the covector to the extended representation. For instance A Bargmann geometry then consists of an extended-valued coframe eI and a Galilean We could alternatively present the spin connection in the extended representation as −̟b 0 By virtue of being in the Lie algebra of the Galilean group, the connection satisfies the One can then use the Galilean connection to define a Galilean-covariant exterior derivative D under which the Galilean invariant tensors considered above are covariantly constant DnA = 0 , DhAB = 0 , DnI = 0 , DgIJ = 0 , Given this data we may naturally define an extended torsion tensor T I = DeI , which in components reads and transforms covariantly T I T a = dea + ωab ∧ eb + ̟a da − ̟a ∧ e ∧ n , RAB = dωAB + ωAC ∧ ωC B . To conclude this section we collect a few further identities that we will use extensively in what follows. First, note that the defining and extended representations of the Galilean connection (2.8) both contain precisely the same data as a totally antisymmetric matrix of one-forms with lowered indices −̟b! , and indeed, they can both be written as ωˆAB → (Λ−1)C A(Λ−1)DBωˆCD − dΘˆ AB , nµ = nAeµA , These are the clock-form and spatial metric found in standard treatments of NewtonCartan geometry [10, 11] and are used to measure elapsed times and spatial distances These satisfy the Newton-Cartan conditions = 0 The extended coframe contains the metric data of a Newton-Cartan geometry in its vecwhose components form a basis for the cotangent space of the Galilean spacetime. We can then form the Galilean invariant tensor fields by virtue of the identities (2.12). The vector component T A of the extended torsion gives One of the key features of non-relativistic geometries is that the derivative operator equation of pure constraint nAT A = nADeA =⇒ T 0 = dn . the Galilean connection in terms of a and f . Henceforth we will assume that the derivative The 1-form a is the Newtonian gravitational vector potential and it is through the derivative’s dependence on a that the geometry encodes Newtonian gravity. To see this, ξν ∇ν ξµ = 0 =⇒ ξ˙i + ξj ∂j ξi + ∂iφ = 0 , and this is the manner in which a Newton-Cartan geometry encodes Newtonian gravity (see chapter 12 of [13] for a textbook discussion). The extended component of the torsion f is zero on physical, torsionless spacetimes, but is necessary to discuss torsionful spacetimes in a Galilean covariant way. It acts on matter as an external field strength exerting a Lorentz Finally, a Bargmann spacetime also admits a natural volume element ∧ · · · ∧ eAd , where ǫ01···d = 1 . which may be used to define integration over spacetime. There is similarly a “volume element” with raised indices εµ 0···µ d = ǫA0···Ad eµA00 · · · eµAdd , where ǫ01···d = 1 . coordinate components where |e| = det(eµA) . Stress-energy for non-relativistic matter fields In this section we define the stress-energy tensor for non-relativistic theories and discuss the difference between the Cauchy stress and the stress defined at fixed connection. As originally presented in [6], and as we shall recap in section 3.1, the non-relativistic stress-energy tions. This is due to the fact that in non-relativistic theories, energy and mass are not identified and are independent quantities. In addition to the stress and energy currents, this object also contains information on the flow of momentum and mass. The other key difference with the relativistic case is the constraint T 0 = dn . This leads to several complications, as it does not allow us to define a “Cauchy stressenergy” directly: any variation of the metric data that includes the clock form n by necessity cannot be done at fixed torsion. There is however an invariant way to isolate the stress, mass, and momentum parts of the stress-energy tensor, each of which admits improvement to the physical tensors. However, as we shall see in section 3.1, the full stressenergy cannot be improved and any attempt to do so results in an unresolvable ambiguity. We will then demonstrate a way to define the Cauchy stress-mass tensor in section 3.2. In section 3.3 we discuss why “improving” the energy current is unnecessary, as the energy density defined at fixed connection already corresponds to the Hamiltonian density (less terms coupling the system to external potentials). The stress-energy tensor background Bargmann geometry given by the extended coframes eI and the connection S = dd+1x(−τ˜µ I δeI µ + sµAB δωˆµAB ) . Expanding this definition in components (2.8) S = dd+1x(−ε˜µ δnµ + T˜iaδeia + p˜aδeta + ρ˜µ δaµ + sµab δωµab + bµa δ̟µa ) , − 2 tensor, though as we have seen it contains far more information than the name suggests. Similarly the spin-boost current sµAB collects together the boost current bµa and the spin current sµab . stress-mass tensor may be defined as current can be isolated as currents under a local Galilean boost transformation This is to be expected on physical grounds since the energy current also includes the kinetic energy of the system, which depends explicitly on a notion of rest frame. one may discuss the energy current as measured by these observers as follows. Let ˚vI denote the unique null extension of vµ to the extended representation. That is ˚vI ˚vI = 0 , =⇒ ˚vI =  v − 21 v2 Then, the energy current measured by the observer moving with velocity vµ is given by While compact, this definition may seem somewhat obtuse. To lend some motivation, we compute the relationship between the energy measured by an observer vµ as defined stress-energy tensor. For simplicity we will consider the flat, spinless case, and so drop the In writing this we have also used the upcoming Ward identity (4.8) for local Galilean and the above simplifies to do not measure. As we will show in (4.7), for spinful matter fields on torsionless spacetimes, the Ward identity for local Galilean transformations (on torsionless spacetimes) is For spinful matter, we see that the stress tensor T˜ij need not be symmetric and the mass stress-energy we can proceed in analogy to the relativistic case discussed in section 1. However, on Bargmann spacetimes with torsion there is no natural reference connection analogous to the Levi-Civita connection in the relativistic case; hence, there is no analog of the variation section 1.2. Thus, to get the symmetric (i.e. Cauchy) stress tensor for non-relativistic fields we should vary the action considering the extended coframe and extended torsion as the independent geometric variables (the analogue of (1.4)). However, due to the identity (2.13) (in particular (2.23)), this amounts to doing a constrained variation since the variations To carry out this constrained variation of the action we first note that, from (3.14), we can get the explicit expression Now we would like to vary the coframes at fixed T I . However, fixing the torsion implies fix the stress-energy tensor without additional data. Let us see how this works. We try to write the variation of the action in the form S = dd+1x|e|(−τ µ I δeµI |T + SAµν δTˆAµν ) . This is what we would like to consider the physical (or Cauchy) stress-energy tensor. As before, we can isolate the stress-mass components As we shall show (see (4.8)), this is symmetric on torsionless spacetimes, so that the stress call the stress-mass tensor T AB obtained from the variation (3.16) the Cauchy stress-mass following term, which does not affect the variation of the action since we must vary the torsion keeping the coframes fixed = 2 1 T λνµ δnλ . Integrating by parts and ignoring the boundary term, this becomes dd+1x|e| 2(∇ν − T λλν )Hµν + T µ νλHνλ δnµ . τ µ I → τ µ I + 2(∇ν − T λλν )Hµν + T µ νλHνλ nI . Since the ambiguity is proportional to nI this only affects the energy current components → εµ + 2(∇ν − T λλν )Hµν + T µ νλHνλ, conservation laws might resolve this ambiguity, but from (4.22) and the fact that the ambiguity is the divergence of an antisymmetric tensor we find it does not. Thus, while the physical stress, mass, and momentum are contained within the Cauchy transforms under internal Galilean transformations according to (3.8). Fortunately we will derive a relationship between the components of T˜AB and T AB and therefore can rewrite (3.8) purely in terms of physical quantities. Let us now consider the stress-mass tensor, whose story is straightforward. The physical currents are to be defined at fixed torsion, so we will require that the variation of eI does not involve the clock form, which is fine as we are neglecting energy currents. This in turn Similarly we also have by the variation nI δeI = 0 =⇒ δeI = ΠAI δeˆA . nI δT I = 0 =⇒ δT I = ΠAI δTˆA S = To perform the translation between (3.3) and (3.24) we shall need the variation of the structure equation T I = DeI , which gives which after some algebraic rearrangement gives − (δTˆA)BC − (δTˆB)CA + (δTˆC )AB . along with the relations S = ρµ = ρ˜µ − 2(∇λ − T ρρλ)Sνµλ nν − T µ λρSνλρnν . response to a gravitational perturbation at fixed torsion. energy density of a simple spinful theory, the spinful Schro¨dinger equation, whose action S = Z dd+1x√hnt ∂µ − iqAµ − imaµ − 2 and J ab are the spin representation matrices. One then finds whereas the Hamiltonian density for this system is H = Diψ†Dj ψ − qAtψ†ψ − matψ†ψ − 2 ωtabψ†J abψ . so corresponds to the internal kinetic and interaction energy of a system. While we have motivated this in the specific case of the Schr¨odinger theory, a similar a generalization of (3.33) for arbitrary theories. Ward identities by virtue of the action begin invariant under diffeomorphisms, local U(1)M transformations, and local Galilean transformations. These were computed in a manifestly covariant form in section 5 of [6], following derivations in flat space in [35, 36] and in non-covariant form on curved space in [21]. In the spinful case, these Ward identities were for the unimproved currents defined at fixed connection. In this section we present the corresponding identities In [6], we found that invariance of the action under local U(1)M transformations and diffeomorphisms gives us the conservation laws −eµI (Dν − T λλν )τ˜ν I = Fµν jν + RABµν sνAB µI + (aµ − eµ0 )nI we also find for the stress-mass tensor (∇ν − T λλν )T˜νµ = F µ ν jν + RABµ ν sνAB The equation (4.1a) is simply the conservation of mass on torsionful spacetimes, while (4.1b) is a covariant version of energy conservation and the continuum version of Newton’s second law (also called the Cauchy momentum equation; see (4.4c)). To make this more transparent, restrict to spinless matter on flat, torsionless spacetimes in Cartesian components of (4.1a), (4.1b), we find p˙i + ∂j T ij = Eijt + ǫijkjj Bk − ∂iφ ρt. where Ei and Bi are the external electric and magnetic fields respectively. Galilean Ward identity In this section, we consider the Ward identity that follows from the invariance of the action under local Galilean transformations. This has previously been discussed in a noncoviariant form in [6, 21, 37], and we take to opportunity here to finally state the covariant version, from which we derive the symmetry of the Cauchy stress-mass on torsionless spacetimes. Under infinitesimal Galilean transformations we have Local Galilean invariance of the action then implies S = dd+1x|e|(−τ˜µ I ΠAI Θˆ ABeµB − sµAB Dµ Θˆ AB) dd+1x|e|Θˆ AB T˜AB + (Dµ − T ν νµ )sµAB from which we find the Ward identity T˜[µν ] = −(∇λ − T ρρλ)sλµν . Thus, for spinful matter, even on torsionless backgrounds that preserve local rotational fails to be symmetric. T˜ij is then not the Cauchy stress tensor used commonly in physics and engineering applications [38]. Moreover, the momentum p˜i need violating a common constraint assumed in non-relativistic physics [39]. virtue of (4.7) and (3.29) satisfies the Ward identity authors [40] to impose Galilean invariance to the case of multi-constituent systems. Note that due to a manifestly covariant formalism this relationship is guaranteed and we do not need to impose it as a functional constraint on the effective action as in [39]. Diffeomorphism and U(1)M Ward identity We would now like to state the diffeomorphism Ward identity − eµI (Dν − T λλν )τ˜ν I = Fµν jν + RABµν sνAB in terms of the physical currents as much as possible. There is unfortunately nothing that can be done about the full equation as it stands since, as we have seen, there is no way to improve the stress-energy tensor as a whole. We can however do so for the Cauchymomentum equation (∇ν − T λλν )T˜νµ = F µ ν jν + RABµ ν sνAB Using (3.29), we find this reads (∇ν − T λλν )T νµ = F µ ν jν + (2Rˆρνλµ − Rµ ρνλ)sρνλ hAE RˆEBCD given by We now simplify the second term on the right hand side of (4.11) using the symmetry of the Riemann tensor under exchange of the first and second pairs of indices. This identity is slightly more subtle than the usual relativistic case since we do not have an invertible metric tensor. We first note that nI DT I = d2n = 0 =⇒ df − ̟b ∧ Tb, dTa − Tb ∧ ωba + T 0 ∧ ̟a . Using this, the identity for the symmetry of the Riemann tensor under exchange of the first and second pairs of indices is given by RˆABCD = RˆCDAB + where ΞˆABCD = (ΞˆA)µνρ eµ eν eρ . The interested reader can find the proof of (4.16) in B C D Using (4.16) we then find that (4.11) simplifies to (∇ν − T λλν )T νµ = F µ ν jν + ΞˆAµ νλsAνλ which is the covariant generalization of the Cauchy momentum equation to unconstrained Bargmann spacetimes. In particular we see that there are external forces exerted by extended torsion on spin current and stress-energy, in addition to the usual Lorentz force on converted everything else to the physical currents, this is something we must simply accept as we have shown there is no unambiguous way to improve it. We simply observe that the external force exerted by extended torsion couples to the unimproved stress-energy. It Unlike the Cauchy equation (4.17), the work-energy equation cannot be isolated in a Galilean frame independent manner. The problem is that observed in the discussion following (3.7): while one can invariantly isolate the stress-mass part of the the stressenergy tensor, there is no observer independent definition of energy. This is to be expected on physical grounds since the energy current also includes the kinetic energy of the system, which must be defined with respect to some notion of rest. However, given a family of as measured by these observers to be (3.10), which we reproduce here Now we saw previously in (4.4) that the temporal component of the diffeomorphism Ward identity contains the work-energy equation. Given a family of observers, we can obtain the covariant version of this by contracting the Ward identity with some frame vµ . In doing so, the following identity is useful eµI vµ = ˚vI + ˚vJ eµJ vµ nI . Using this equation and mass conservation, one finds ˚vI (Dµ − T ν νµ )τ˜µ I = Fµν jµ vν + RABµν sµAB vν + T I µν vµ τ˜ν I , =⇒ (∇µ − T ν νµ )ε˜µ = Fµν jµ vν + RABµν sµAB vν + T I µν vµ τ˜ν I − τ µ I Dµ ˚v . I find that tµ ν = ∇µ vν and so Plugging this in, one finds the work-energy equation for the comoving energy current is Finally, let us turn to a few examples. In this section we collect computations for the stressenergy, spin current, and Cauchy stress-mass for various non-relativistic field theories. The principle aim of this discussion will be to derive covariant formulae for these objects and to demonstrate how to carry out the computation maintaining manifest covariance We begin with the spinful Schr¨odinger field in section 5.1. The formulae (5.11) we derive, in their flat space component form (5.12), should for the most part be familiar, but also include spin contributions to the Cauchy stress tensor and mass current which to our knowledge are not present in the literature. In section 5.2 we consider the non-relativistic Dirac theory which is a Galilean invariant theory for matter charged under both boosts and spatial rotations and is first order in both time and spatial derivatives. We conclude with the Wen-Zee term which arises in the effective actions for describing quantum Hall Spinful Schro¨dinger field J AB =  for d = 2 for d ≥ 3 and it can be verified that these satisfy the standard commutation relation of the Galilean ∂µ − iqAµ − imaµ − 2 ωˆµAB J AB discussed in [6], this derivative is not covariant under local Galilean boosts. The Galileancovariant derivative acting on massive fields is given by curved spacetime as there do not exist global inertal coordinates. As detailed in section 1.2 of [41] and 2.2 is trivial under local Galilean boosts. The Schr¨odinger action for such fields can then be written in a manifestly invariant form as [6, 7] S = − 2m which one may check reduces to the standard Schr¨odinger action in flat spacetime after expanding in components. We are now in a position to perform a covariant calculation of the various currents defined in this note. from which we find δ(eµADµ ψ) = −δeνBeνAeµBDµ ψ − imδaν eνAψ = −δeIν eνADI ψ . Including the variation of the spin connection and electromagnetic gauge field then gives Similarly, the variation of the volume element is δDI ψ = −Πµ I δeµJ DJ ψ + δ|e| = |e|eµAδeµA = |e|Πµ I δeµI . Using these, a straightforward computation gives the currents (on torsionless back = − 4m jµ = − 2m D Aψ = ΠAI DI ψ = Before writing the Cauchy currents, we note that the spin current (5.8a) for the Schr¨odinger field is conserved on-shell i.e. Dµ sµAB = 0 which can be shown as follows. = − 4m = − 4m = − 4m = − 8m ψ†J µν D↔λψ − ψ†J νλD↔µ ψ − ψ†J λµ D↔ν ψ , 1 D (µ ψ†D ν)ψ − 2m where in (5.11b) we have used (5.10) to write the stress-mass tensor in a manifestly symmetric form as implied by the on-shell Ward identity for local Galilean transformations (4.8). We also give component expressions of the above equations for a spin- 12 particle in flat, can be written as The physical Cauchy currents are obtained from (3.29) giving T ij = − 2m sµj = jµ = spin density of standard quantum mechanics and whose current may also be interpreted along the lines of the energy current as it is half the anticommutator of the spin and velocity. By virtue of (5.10), this spin current is conserved on-shell that can be interpreted as the mass flow due to the non-uniform spin of matter. Note in particular that even in the single-constituent case in the presence of spinful matter, the charge and mass currents need not be aligned since the inhomogenous spin carries momentum. Finally, the stress tensor is the standard stress tensor for spinless Schr¨odinger fields, plus a contribution ǫkl(i∂ksl j) arising from any non-uniform spin current. Of course, we do not display the momentum current since it is equal to the mass current on-shell. ∂µ sµi = 0 . in a 3+1-dimensional spacetime. We note that, since Galilean boosts are non-compact they do not have finite-dimensional unitary representations and one cannot use the Schr¨odinger We consider the spin- 12 representation8 originally discovered by L´evy-Leblond [44]; see action of the Galilean algebra generators is given by Ka = under rotations and transform into each other under Galilean boosts i.e. transformation as 1 A−1γ0(γ0 + γ4)A , matrices found in equation (10) of [43], given by10 1 I − 2i I − 2i I 21 I M = 0 −2I S = Given the relations (5.17) the action (5.19) is the 4 + 1-dimensional relativistic Dirac dimensional relativistic Dirac action (see also [45]). 8See [42, 43] for higher spin representations. From the action (5.19) we find the stress-energy and spin current as the components of M ABC from (5.14) and (5.18), we find that M ABC is in fact totally antisymmetric in its indices. Further, using the relations in (5.17), the total antisymmetry of M CAB, and the equaThe physical currents in torsionless backgrounds are = − 2 = − 8 = − 2 To cast the above currents in a more familiar form, we use the decompositon (5.15) to exapnd the action (5.19) as S = Plugging this in, on torsionless backgrounds and after integration by parts, gives S = gives rise to the well-known g-factor coupling of the spin to the magnetic field, and also induces a non-minimal coupling to the Ricci scalar R. Using this we write the currents as sµj = T ij = jµ = − 41m ψ†DkDkψ − 41m DkDkψ†ψ − mq Biψ†Siψ! − 41m Diψ†Dtψ + 2mi ǫijkDj ψ†SkDtψ + c.c. − 2imq ψ†Diψ + mq ǫijk∂j (ψ†Skψ) In particular, we have the standard charge and mass currents, plus magnetization currents arising from the magnetic moments that the g-factor attaches to particles. Note that the mass magnetization enters as though it had g-factor 1. In comparison to the Schr¨odinger case (5.12), these currents also have additional terms (modulo the equations of motion) in the energy current and stress arising from the non-minimal couplings to magnetic field and curvature found in (5.25). Wen-Zee term We conclude with the Wen-Zee term in 2 + 1-dimensions, an important example from effective field theory for gapped systems in the presence of external curvature and electromagnetic field. The action for the Wen-Zee term is S = This term famously encodes the Hall viscosity of quantum Hall systems [46]. Gauge invariby a continuous deformation of the microscopic parameters of a system that does not close the gap and so characterizes topological phases of matter. Examination of quantum Hall effective actions with constraints from non-relativistic symmetries was initiated by Hoyos and Son in [18]. In this work the authors also impose additional symmetries owing to the single component nature of the quantum Hall fluid (see [22] for a manifestly covariant way of implementing these symmetries). In this section we will reproduce known results for the stress and energy current induced by the Wen-Zee term as a simple example in the use of the formalism given above (also see [47] for a similar computation). For those interested in a full general effective action the results can be found in [41]. It would be interesting to examine, which, if any, of these terms require the introduction of gapless edge modes, along the lines of [48]. is the torsionfree connection purely for spatial rotations defined in terms of the spatial coframes. However to preserve Galilean invariance we use the full spacetime spin connection which reduces to the one used by [46] when restricted to time-independent curved geome The induced charge current is straightforward to calculate jµ = j0 = The stress-energy vanishes since it is defined at fixed connection identically conserved Dµ sµAB = 0. The Cauchy stress-mass is then so that in particular we see that the Wen-Zee term makes no contribution to the physical energy density or current, as in [47]. The spin current is where we have introduced the covariant drift 3-velocity A = where we have discarded a term due to the conservation of the spin current yielding a manifestly symmetric stress-mass tensor.11 The first term is the standard Hall viscosity term, while the final term gives a mass magnetization current in the presence of inhomogeneities in the external magnetic field 11Note that the stress-mass tensor needs to be identically symmetric, not simply symmetric on-shell, since we have integrated out the matter fields. Working with a manifestly covariant geometric description, given by Bargmann spacetimes, we define the physical energy current, stress tensor, and mass current for any Galilean invariant physical system with spin. We find that when the stress, mass and momentum are appropriately defined, the stress tensor is symmetric, and momentum and mass currents coincide as a consequence of manifest local Galilean invariance. We also argue that the physical energy current is naturally defined via variation at fixed connection, not fixed torsion. While we have worked out some illustrative examples, it would be of interest to use this formalism to extend the analysis of [6, 37, 49] to spinful fluids, and that of [22] to spinful electrons. One could also investigate non-relativistic scale anomalies for spinful non-relativistic fields following [50–54]. Another potentially interesting application would be to examine the linear response in Son’s Dirac theory of the half-filled Landau-level [55]. Acknowledgments This work is supported in part by the NSF grants DMR-MRSEC 1420709, PHY 12-02718 and PHY 15-05124. M.G. is supported in part by the University of California. K.P. is supported in part by the NSF grant PHY-1404105. M.M.R. is supported in part by the DOE grant DE-FG02-13ER41958. Symmetries of the Riemann tensor In the main text, we required the symmetries of the Newton-Cartan Riemann tensor to derive equation (4.17). These identities involve a few subtleties not present in the pseudoRiemannian case, so we collect their derivations here. Since we are interested in the Ward identities on unrestricted Bargmann geometries, we will present these symmetries on spacetimes with general extended torsion T I (the torsionless case can be found in [12]). They are RA[BCD] = RIJKL = RKLIJ + (DTI )JKL + (DTJ )KIL + (DTK )ILJ + (DTL)IJK , Where we have defined RˆABCD as the unique object anti-symmetric in it’s first two indices such that RABCD = hAE RˆEBCD. Equivalently 2R[µν ] = 3∇[µ T λλν] + T λλρT ρµν , The derivation of (4.17) requires only the first three of these identities, but we include the which was used in obtaining equation (4.11). The first identity follows trivially from the definition of RˆAB while the derivations of (A.1b) and (A.1) from DT A = RAB ∧ eB, DRˆAB = 0 are identical to the pseudo-Riemannian case. The only identity that requires some care is (A.1c), which is most easily stated when valued in the extended representation. By RI JKL we mean the curvature two-form, valued in the extended representation of gal(d), with spacetime indices pulled back to the extended representation using the Galilean invariant projector Since all indices in this equation are nI orthogonal, it is simply the pullback of an equation valued in the covector representation of Gal(d) RˆABCD = RˆCDAB + identity is most naturally carried out in it’s extended form. 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Michael Geracie, Kartik Prabhu, Matthew M. Roberts. Physical stress, mass, and energy for non-relativistic matter, Journal of High Energy Physics, 2017, 1-29, DOI: 10.1007/JHEP06(2017)089