Abstract
In this short note we perform canonical analysis of Schrodinger field and non-relativistic electrodynamics coupled to Newton-Cartan gravity. We identify physical degrees of freedom and analyze constraints structure of these theories.
Highlights
AND SUMMARYThere has been a renewed interest in Newton-Cartan (NC) geometry in recent years
The first significant paper was Ref. [1] which introduced NC geometry to field theory to analyze strongly correlated electrons. It was further shown in Refs. [2,3] that NC geometry with torsion naturally emerges as the background boundary geometry in holography for z 1⁄4 2 Lifshitz geometries; for relevant works, see Refs. [4,5,6,7] and for a review and extensive list of references, see Ref. [8]
NC geometry is a nonrelativistic background geometry to which nonrelativistic field theories can be covariantly coupled; see e.g., Refs. [5,9,10,11,12,13]. It was shown in a significant paper [11] how nonrelativistic electrodynamics can couple to the most general NC geometry with torsion
Summary
There has been a renewed interest in Newton-Cartan (NC) geometry in recent years. The first significant paper was Ref. [1] which introduced NC geometry to field theory to analyze strongly correlated electrons. Nonrelativistic scalar fields coupled to NC geometry and a background electromagnetic field were analyzed there Since these results are very interesting nonrelativistic theories in the NC background certainly deserve to be studied in more detail. Since canonical analysis is based on an existence of Lagrangian we start with the nonrelativistic electrodynamics action in the NC background that is derived using null-dimensional reduction [11]. We obtain the Hamiltonian form of nonrelativistic theories on the NC background and we determine the physical degrees of freedom (d.o.f.) This is a very important result since we show that in the case of nonrelativistic electrodynamics the only physical d.o.f. are the scalar field and conjugate momenta. In the Appendix we study constrained systems with explicit time dependence and discuss their properties
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