Physical stress, mass, and energy for nonrelativistic matter
Received: November
Physical stress, mass, and energy for nonrelativistic 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 Acceleratorbased 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 stressenergy tensor, one defined by a variation of the action with the coframes at fixed connection, and the other at fixed torsion. These two stressenergy 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 nonrelativistic matter theories. We point out that while the physical nonrelativistic 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 nonrelativistic actions. ArXiv ePrint: 1609.06729
matter; SpaceTime Symmetries; Differential and Algebraic Geometry; Effective Field

Spinful Schr¨odinger field
Nonrelativistic Dirac field
A Symmetries of the Riemann tensor
Ward identities
The stressenergy tensor
The Cauchy stressmass 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
Stressenergy for nonrelativistic 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+1xe(−T˜µ AδeµA + sµAB δωµAB )
to define the stressenergy tensor T˜µ A and the spin current sµAB . Alternatively, since there
exists a unique torsionfree metric compatible connection — the LeviCivita connection
and the contorsion CAB ≡ ωAB − ω(LC)AB as independent
S =
Z dd+1xe(−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+1xe(−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 nonrelativistic 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 nonrelativistic spacetimes following the formulation introduced
in [6, 7], called Bargmann spacetimes.3 For matter fields on a Bargmann spacetime, the
covariant nonrelativistic stressenergy is a tensor of the form
the coframes and Galilean connection as independent variables (i.e. through the analogue
S =
dd+1xe(−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 NewtonCartan 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 LeviCivita connection
in the relativistic case) and so the Cauchy stressenergy must be computed by varying the
coframes and torsion as independent variables (similar to (1.4))
S =
However, due the nonrelativistic 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 nonrelativistic theories:
the Cauchy stress, momentum, and mass current can be collected into a covariant Cauchy
stressmass 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 workenergy equation (see (4.22)) as well as a conservation law for the
physical stressmass tensor
(∇ν − T λλν )T νµ = F µ ν jν + ΞˆAµ νλSAνλ
4A more precise, but unwieldy name, would be the stressmassmomentum tensor, but as we will show a
Noether identity equates the momentum with the mass current for the physical Cauchy stressmass 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
nonrelativistic 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 stressmass tensor in section 4. In section 5 we provide examples of Cauchy stress
and mass tensors for nonrelativistic field theories. Appendix A collects the symmetry
properties of the nonrelativistic 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
NewtonCartan 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], nonrelativistic
fluids [6, 15–17], the quantum Hall effect [18–22], as well as nonrelativistic 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 NewtonCartan 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 extendedvalued 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 Galileancovariant 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
oneforms 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 clockform 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 NewtonCartan conditions
= 0
The extended coframe contains the metric data of a NewtonCartan 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 nonrelativistic 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 1form 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 NewtonCartan 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) .
Stressenergy for nonrelativistic matter fields
In this section we define the stressenergy tensor for nonrelativistic 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 nonrelativistic stressenergy
tions. This is due to the fact that in nonrelativistic 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 stressenergy 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 stressmass 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 stressenergy 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 spinboost current sµAB
collects together the boost current bµa and the spin
current sµab .
stressmass 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
stressenergy 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
stressenergy 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 LeviCivita 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 nonrelativistic 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 stressenergy tensor without additional data. Let us see how this works. We try to
write the variation of the action in the form
S =
dd+1xe(−τ µ I δeµI T + SAµν δTˆAµν ) .
This is what we would like to consider the physical (or Cauchy) stressenergy tensor.
As before, we can isolate the stressmass components
As we shall show (see (4.8)), this is symmetric on torsionless spacetimes, so that the stress
call the stressmass tensor T AB obtained from the variation (3.16) the Cauchy stressmass
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+1xe 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 stressmass 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 noncovariant 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 stressmass 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 stressmass on torsionless
spacetimes. Under infinitesimal Galilean transformations we have
Local Galilean invariance of the action then implies
S =
dd+1xe(−τ˜µ I ΠAI Θˆ ABeµB − sµAB Dµ Θˆ AB)
dd+1xeΘˆ 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 nonrelativistic physics [39].
virtue of (4.7) and (3.29) satisfies the Ward identity
authors [40] to impose Galilean invariance to the case of multiconstituent 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 stressenergy 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 stressenergy, 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 stressenergy. It
Unlike the Cauchy equation (4.17), the workenergy 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 stressmass 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 workenergy 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 workenergy 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 stressmass for various nonrelativistic 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 nonrelativistic
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 WenZee 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 +
δe = eeµAδeµA = eΠµ 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 onshell 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 stressmass tensor in a manifestly
symmetric form as implied by the onshell 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 onshell
that can be interpreted as the mass flow due to the nonuniform spin of matter. Note
in particular that even in the singleconstituent 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 nonuniform spin current. Of course,
we do not display the momentum current since it is equal to the mass current onshell.
∂µ sµi = 0 .
in a 3+1dimensional spacetime. We note that, since Galilean boosts are noncompact they
do not have finitedimensional unitary representations and one cannot use the Schr¨odinger
We consider the spin 12 representation8 originally discovered by L´evyLeblond [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 + 1dimensional relativistic Dirac
dimensional relativistic Dirac action (see also [45]).
8See [42, 43] for higher spin representations.
From the action (5.19) we find the stressenergy 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 wellknown gfactor coupling of the spin to the magnetic field, and also
induces a nonminimal 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 gfactor attaches to particles. Note that the
mass magnetization enters as though it had gfactor 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 nonminimal couplings to magnetic field and
curvature found in (5.25).
WenZee term
We conclude with the WenZee term in 2 + 1dimensions, an important example from
effective field theory for gapped systems in the presence of external curvature and
electromagnetic field. The action for the WenZee 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 nonrelativistic 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 WenZee 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 timeindependent curved
geome
The induced charge current is straightforward to calculate
jµ =
j0 =
The stressenergy vanishes since it is defined at fixed connection
identically conserved Dµ sµAB
= 0.
The Cauchy stressmass is then
so that in particular we see that the WenZee 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 3velocity
A =
where we have discarded a term due to the conservation of the spin current yielding a
manifestly symmetric stressmass 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 stressmass tensor needs to be identically symmetric, not simply symmetric onshell,
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 nonrelativistic scale anomalies for spinful
nonrelativistic fields following [50–54]. Another potentially interesting application would
be to examine the linear response in Son’s Dirac theory of the halffilled Landaulevel [55].
Acknowledgments
This work is supported in part by the NSF grants DMRMRSEC 1420709, PHY 1202718
and PHY 1505124. M.G. is supported in part by the University of California. K.P. is
supported in part by the NSF grant PHY1404105. M.M.R. is supported in part by the
DOE grant DEFG0213ER41958.
Symmetries of the Riemann tensor
In the main text, we required the symmetries of the NewtonCartan 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 antisymmetric 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 pseudoRiemannian 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 twoform, 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.
To prove (A.1c) we begin with
which written in tensor notation reads
may check by an explicit computation in components that
This is simply the extended index version of (A.1b), which one can obtain from here by
noting that both sides are nI orthogonal in all their indices. (A.1c) then follows exactly as
in the pseudoRiemannian case by repeated applications of this equation along with
R(IJ)KL = RIJ(KL) = 0 .
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
DT I = RI J ∧ eJ ,
RI[JKL] =
(DTI )JKL .
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