Gravity and the spin-2 planar Schroedinger equation

A Schroedinger equation proposed for the GMP gapped spin-2 mode of fractional Quantum Hall states is found from a novel non-relativistic limit, applicable only in 2+1 dimensions, of the massive spin-2 Fierz-Pauli field equations. It is also found from a novel null reduction of the linearized Einstein field equations in 3+1 dimensions, and in this context a uniform distribution of spin-2 particles implies, via a Brinkmann-wave solution of the non-linear Einstein equations, a confining harmonic oscillator potential for the individual particles.


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Einstein's theory of General Relativity ceases to be a theory of gravity when considered in a 3D spacetime (i.e. 2+1 dimensions): there is no analog of the Newtonian force, nor gravitational waves. We call it a "3D gravity" theory mainly because it shares with 4D General Relativity the property of being a diffeomorphism invariant theory for a dynamical metric on spacetime, which makes it a useful "toy model" for considering how theories of this type might be compatible with quantum mechanics.
A simple modification of 3D General Relativity known as "New Massive Gravity" (NMG) yields a paritypreserving 3D gravity theory that does admit gravitational waves; "gravitational" in the sense that the corresponding particle excitation of the quantum theory has spin 2, although these spin-2 "gravitons" are massive rather than massless [1]. Linearization about a Minkowski vacuum yields a free field theory that is equivalent to the 3D version of the massive spin-2 field theory proposed long ago by Fierz and Pauli [2]. There are various bi-metric 3D gravity theories that have the same linearized limit [3,4], and NMG may itself be viewed as the simplest example, with an auxiliary tensor field as the second 'metric' [5].
Although NMG has no "real-world" applications as a theory of gravity, it has potential applications in the world of condensed matter systems in 2+1 dimensions. Naturally, these are typically non-relativistic, so this motivates consideration of the non-relativistic limit of NMG. Non-relativistic limits are notoriously more complicated than one would naively imagine, so it makes sense to first investigate the non-relativistic limit of the 3D Fierz-Pauli (FP) theory. One might expect to find a Schrödinger equation for a non-relativistic particle of spin 2.
As it happens, fractional Quantum Hall states have a Girvin-MacDonald-Platzman (GMP) gapped spin-2 mode [6], and a particular Schrödinger equation has been proposed as an equation governing its dynamics [7,8].
Following suggestions of a geometrical interpretation of GMP states [9], this spin-2 planar Schrödinger equation was shown to emerge upon linearization of a particular non-relativistic bi-metric theory [10,11]. We should stress that these are space metrics rather than spacetime metrics, but an obvious question is whether this bi-metric theory is the non-relativistic limit of some relativistic bi-metric theory, perhaps NMG. We do not answer this question, but we show that the Schrödinger equation proposed to describe the GMP mode is indeed a non-relativistic limit of the 3D FP theory.
The standard way in which spin is incorporated into the (time-dependent) Schrödinger equation is via a multiplet of complex wavefunctions transforming in a representation of the rotation group. This implies an SO(2) doublet for two space dimensions, and this is indeed what one finds from the standard non-relativistic limit of the 3D FP theory for a complex tensor field, but the "spin-2 Schrödinger equation" proposed to describe the GMP mode has a single complex wavefunction. What we need, although only for 2+1 dimensions, is a non-relativistic limit for a real FP tensor field.
There is a problem with the non-relativistic limit of real-field theories that propagate massive modes. This can be understood by considering the Klein-Gordon (KG) equation for a scalar field Φ of mass m. Including all factors of c and this is 1 The c → ∞ limit can be taken directly provided that the reduced Compton wavelength λ = /(mc) is held fixed, but this yields a Yukawa equation (Laplace if 1/λ = 0), which is non-dynamical. However, if Φ is complex we can set where E 0 is constant and Ψ is a new complex scalar field.
The KG equation becomes and the c → ∞ limit yields the Schrödinger equation Clearly, this procedure is not applicable for a real scalar field, and there is a group theoretical reason for this difficulty. The Bargmann symmetry group of the Schrödinger equation has one more generator than the Lorentz symmetry group of the KG equation, a central charge proportional to the mass m. This implies that the wavefunction provides only a projective representation of the Galilei group, so it must be complex, and hence the initial KG field must also be complex. The KG equation then has an additional U (1) phase invariance, so there is no longer a mismatch in the dimension of the relativistic and non-relativistic symmetry groups.
A new non-relativistic limit. The same difficulty applies to real tensor fields, such as the symmetric traceless tensor field f µν of the spin-2 FP equations; traceless in the sense that η µν f µν = 0, where η µν is the inverse of the background Minkowski metric tensor. The FP equations comprise second-order "dynamical" equations and first-order "subsidiary conditions": where ≡ η µν ∂ µ ∂ ν . Here we set = 1, in which case mc has dimensions of inverse length. Although the standard path to a non-relativistic limit of these equations requires f µν to be complex, another non-relativistic limit is possible for a Minkowski background of 2+1 dimensions. In this 3D case we have µ, ν = 0, 1, 2, the Minkowski metric matrix is diag.(−c 2 , 1, 1), and f µν has five independent components parametrizing a scalar, vector and traceless symmetric tensor of the SO(2) rotation group. The scalar is the real variable while the vector and traceless symmetric tensor are, respectively, the complex variables In terms of these variables, the subsidiary conditions arė As both f [1] and f [2] are complex, we may set Only f [2] is independent, and its dynamical equation is In terms of Ψ [2] this equation takes the form (3) and its c → ∞ limit is This Schrödinger equation is invariant under a symmetry group with one more generator than the Lorentz invariance group from which we started because the orbital angular momentum and the spin angular momentum are separately conserved in the c → ∞ limit, with a "spin rotation" becoming a phase rotation by double the angle. In fact, this Schrödinger equation is identical to the Schrödinger equation for the spin-2 GMP mode of fractional Quantum Hall states, as deduced in [11] from a non-relativistic bi-metric theory.
Parity and Time-reversal. As mentioned earlier, the FP equations for a complex tensor field allow a standard non-relativistic limit. This leads to a parity-preserving pair of Schrödinger equations: where ϕ 11 and ϕ 12 are the independent complex components of a complex symmetric traceless 2-space tensor ϕ ij . The wavefunctions Ψ[±2] are spin-2 helicity eigenstates; by "helicity" we mean the spin angular momentum while "spin" is its absolute value. The point of this discussion is that parity reverses the sign of helicity, and hence exchanges Ψ [2] with Ψ[−2]. This can be seen from the following equivalence If we choose to impose this constraint with the upper sign then we are left with the single Schrödinger equation of (12). Furthermore, the condition Ψ[−2] = 0 identifies a a spin-rotation of angle θ with a shift of the phase of Ψ [2] by angle 2θ, exactly as required.
As the real-field FP equations (5) are parity invariant, it follows that our "new non-relativistic limit" of these equations must break parity. To see why, we observe that parity for the FP equations takes f [2] →f [2], but one can see from (9) that the corresponding transformation of Ψ [2] is not defined in the c → ∞ limit. In contrast, time-reversal, which takes t → −t and f [2] →f [2] (since its action is anti-linear) is defined in the c → ∞ limit, so (12) must be, and is, time-reversal invariant.
The fact that (12) breaks parity suggests that a better starting point for this paper might have been the parity violating equations of "Topologically Massive Gravity" (TMG) [12], which has the "square-root FP" equations [13] as its linearized limit. Moreover, the self-duality condition (14) (for one choice of the sign) emerges naturally from the non-relativistic limit of the complexified "square-root FP" equations, as detailed for the spin-1 case in [14]. However, it is not clear how the required "complexification" is to be implemented for the nonlinear TMG theory. This is not a problem for NMG because complexification is not required.
Generalized null reduction. We now turn to a different derivation of the Schrödinger equation (12). It is well-known that null reduction of a Lorentz invariant theory in a 5D Minkowski spacetime yields a Galilean invariant theory in 1+3 dimensions [15][16][17]. We seek some variant procedure that will take the linearized 4D Einstein field equations to the Schrödinger equation (12). A similar issue was addressed in [18] at the level of particle mechanics: the Hamilton-Jacobi equation for a nonrelativistic particle in d space dimensions was provided with an "Eisenhart lift" to d+1 dimensions. Here we propose a quantum version in which the planar Schrödinger equation (12) is lifted to the linearized 4D Einstein equations; in reverse this becomes a generalized null reduction inspired by Scherk-Schwarz dimensional reduction [19]. In principle the idea applies in any dimension but it is only for a planar Schrödinger equation that one can lift to the real-field linearized Einstein equations.
Linearization of the 4D vacuum Einstein equations about a Minkowski vacuum with coordinates {x m ; m = 0, 1, 2, 3} and Minkowski metric η mn , yields the following equations for the metric perturbation tensor: where h m ≡ η pq ∂ p h qm and h ≡ η mn h mn . We shall choose light-cone coordinates for which x m = {x + , x − , x i }, where i = 1, 2 and x ± = (x 3 ± x 0 )/ √ 2, and units for which c = 1, but we no longer set = 1.
The standard null reduction is achieved by requiring ∂ − h mn = 0. Instead, we proceed on the assumption that ∂ − is invertible, in which case we may impose the lightcone gauge condition h m− = 0, for which The equation for h m− reduces to ∂ − (h m − ∂ m h) = 0, which implies that h m = 0 and h = 0; these equations imply that and also that the linearized Einstein equations reduce to h mn = 0. As ∂ − is assumed invertible, we may solve for the auxiliary variables h i+ and h ++ . This leaves only the traceless part of h ij , which satisfies the 4D wave equation, implying the propagation of transverse waves with two independent polarizations. So far, this is standard lightcone gauge fixing.
Next, we define The auxiliary variable equations (17) are now (19) and the wave equation for the traceless transverse metric perturbation is We now propose to effect a new null reduction by setting for positive mass m; the factor of is needed here on dimensional grounds. The equations (19) now imply that These equations are analogous to the subsidiary equations for the 3D spin-2 FP equations in the form (10). As in that case, only Ψ [2] is independent, and it satisfies whereΨ ≡ ∂ + Ψ. This is the Schrödinger equation (12), with E 0 = 0 but we address this below. One may again ask how it is that the parity invariance of our starting point is not reflected in the end result, and in this case the answer is that parity is broken by the choice of sign for the mass m appearing in (21). If we had supposed m to be negative then we would have had to take the complex conjugate of (23) to arrive at a standard Schödinger equation forΨ [2], but it follows from the definition of Ψ [2] in (18) We have now provided two distinct "gravity" interpretations of the planar Schrödinger equation that has appeared in the context of the spin-2 GMP mode of fractional Quantum Hall states. Our interpretation of it as a non-relativistic limit of the 3D FP equations is closest to "geometrical" proposals in the condensed-matter literature, but our derivation from the 4D linearized Einstein equations provides a more direct link to "gravity". In both cases the enabling feature is the fact that the relevant subgroup of the Lorentz group is U (1); this is the rotation group in 3D and the transverse rotation group in 4D, and Wigner's "little group" in both cases because in 3D the spin-2 particle is massive whereas in 4D it is massless.
Gravity and the Schrödinger potential. So far, we have discussed the planar Schrödinger equation only for a free spin-2 particle, or spin-2 GMP mode in the condensed matter context. The gravity origin of this equation becomes useful when we consider how these particles might interact. In this relativistic context, each particle will produce a gravitational field that is felt by all the others, and we can approximate the effect on any individual particle by some collective background spacetime metric; each particle then moves freely in this background. We may anticipate that this mean-field type of approximation will result in some potential for the Schrödinger Hamiltonian.
To explore this idea in the context of generalized nullreduction, we must start from some solution of the full 4D Einstein field equations: G mn = 8πG N T mn , where G mn is the Einstein tensor, G N is Newton's constant, and T mn is some specified source tensor (we again set c = 1). Given a 4-metric that solves these equations, we linearize about it to find the following equations for the metric perturbation tensor: where D is the covariant derivative with respect to the affine connection for which R p mnq is the Riemann tensor and R mn the Ricci tensor, and We choose as our background the particular Brinkmann-wave metric which also played a role in the "Eisenhart lift" of [18], and earlier in [20]. The function v is independent of x − , which ensures that ∂ − is a null Killing vector field; in particular, constant v yields Minkowski spacetime. The only non-zero components of the affine connection, up to symmetry, are and the only non-zero components of the curvature and Ricci tensors, up to symmetries, are where ∇ 2 is the Laplacian on the transverse 2-space. The background Einstein equations are satisfied if where T ++ must be the only non-zero component of T mn . As before, we impose the gauge condition h m− = 0. The Ricci tensor term in (24) is then zero. The function v does not enter into the expressions for h m and h in light-cone gauge, and neither does it enter into the "dynamical" equation for h m− , so we still have h m = h = 0, and the resulting equations (17), while the dynamical equations reduce to We need consider only the equation for h ij , which is Only the traceless part of h ij is non-zero, and we can trade this for Ψ [2] as before. Imposing the generalized null-reduction condition (21), we again recover the equations (22) determining the auxiliary fields in terms of Ψ [2], while the equation for Ψ [2] again becomes the spin-2 planar Schrödinger equation, but now with Hamiltonian where t = x + . One solution of (29) for zero source yields the linear potential V = mg · x, which is naturally interpreted as the result of a constant acceleration g.
This yields the Hamiltonian for a planar harmonic oscillator of angular frequency ω, which confines the particle to a region centered on the arbitrary point with coordinates x 0 . The natural interpretation is that of a constant uniform distribution of spin-2 particles with each particle occupying an area /(mω). An obvious question is whether this result can also be found from the non-relativistic limit of some interacting extension of the 3D spin-2 FP theory, such as NMG. In this context, the potential has an interpretation within Newton-Cartan geometry as the time component of the gauge-potential 1-form associated to the central-charge of the Bargmann algebra [21,22]; a gauge transformation preserving the form of this potential shifts v by a function of t, which corresponds to the freedom to redefine the wavefunction by a t-dependent phase factor. However, it is not clear to us at present how this modification can be implemented in the context of the new non-relativistic limit described here that avoids complexification of the FP field.
Finally, we should mention that a study by Vasiliev [23] of relativistic conformal field theories in their "unfolded" formulation led to a holographic-dual Schrödinger equation in one lower dimension, and a proposed twistor transform interpretation that was conjectured to be related to what we have called, following [18,20], the "Eisenhart lift".