Abstract
A Schrödinger equation proposed for the Girvin-MacDonald-Platzman gapped spin-2 mode of fractional quantum Hall states is found from a novel nonrelativistic 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 nonlinear Einstein equations, a confining harmonic oscillator potential for the individual particles.
Highlights
A Schrödinger equation proposed for the Girvin-MacDonald-Platzman gapped spin-2 mode of fractional quantum Hall states is found from a novel nonrelativistic limit, applicable only in 2 þ 1 dimensions, of the massive spin-2 Fierz-Pauli field equations
Einstein’s theory of general relativity ceases to be a theory of gravity when considered in a 3D spacetime (i.e., 2 þ 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
Summary
A Schrödinger equation proposed for the Girvin-MacDonald-Platzman gapped spin-2 mode of fractional quantum Hall states is found from a novel nonrelativistic limit, applicable only in 2 þ 1 dimensions, of the massive spin-2 Fierz-Pauli field equations. 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].
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