Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Symplectic Field Theory of the Galilean Covariant Scalar and Spinor
  Representations   Exploring the concept of the extended Galilei group $\mathcal{G}$, a
representation for the symplectic quantum mechanics in the manifold of
$\mathcal{G}$, written in the light-cone of a five-dimensional De Sitter
space-time, is derived consistently with the method of the Wigner function. A
Hilbert space is constructed endowed with a simplectic structure, studying
unitary operators describing rotations and translations, whose generators
satisfy the Lie algebra of $\mathcal {G}$. This representation gives rise to
the Schr\"{o}dinger (Klein-Gordon-like) equation for the wave functions in
phase-space, such that the dependent variables have the position and linear
momentum contents. Wave functions are associated with the Wigner function
through the Moyal product, such that the wave functions represent a
quasi-amplitude of probability. We construct the Pauli-Schr\"{o}dinger
(Dirac-like) equation in phase-space in its explicitly covariant form. Finally,
we show the equivalence between the five dimensional formalism of phase-space
with the usual formalism, proposing a solution that recover the non-covariant
form of the Pauli-Schr\"{o}dinger equation in phase-space.

3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol