Abstract
Real time evolution of classical gauge fields is relevant for a number of
applications in particle physics and cosmology, ranging from the early Universe
to dynamics of quark-gluon plasma. We present a lattice formulation of the
interaction between a $shift$-symmetric field and some $U(1)$ gauge sector,
$a(x)F_\mu\nuF^\mu\nu$, reproducing the continuum limit to order
$O(dx_\mu^2)$ and obeying the following properties: (i) the system is
gauge invariant and (ii) shift symmetry is exact on the lattice. For this end
we construct a definition of the topological number density $Q =
F_\mu\nuF^\mu\nu$ that admits a lattice total derivative
representation $Q = \Delta_\mu^+ K^\mu$, reproducing to order
$O(dx_\mu^2)$ the continuum expression $Q = \partial_K^\mu
E B$. If we consider a homogeneous field $a(x) = a(t)$,
the system can be mapped into an Abelian gauge theory with Hamiltonian
containing a Chern-Simons term for the gauge fields. This allow us to study in
an accompanying paper the real time dynamics of fermion number non-conservation
(or chirality breaking) in Abelian gauge theories at finite temperature. When
$a(x) = a(x,t)$ is inhomogeneous, the set of lattice equations of motion
do not admit however a simple explicit local solution (while preserving an
$O(dx_\mu^2)$ accuracy). We discuss an iterative scheme allowing to
overcome this difficulty.
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