Can the synthetic gauge field be dynamical? 
Can the synthetic gauge field be dynamical gauge field?

Early idea of Wilczek and Zee stated that (non-Abelian) gauge fields
arise in the adiabatic development of simple quantum mechanical systems. Characteristics of the gauge fields are related to energy splittings, which may be observable in real systems. However, Wilczek-Zee's work is on the non-dynamical gauge field associated to Berry connection (similar to a probed background field). 
I believe that it also occurs in the nearly degenerate ground states such as the systems with intrinsic topological orders. But in this case, the gauge field can be emergent and dynamical.
I think the idea of gauge fields in Wilczek and Zee, is based on the three key elements: quantum system, and adiabatic slow evolution, and energy levels are close. But they claim that similar phenomena are found for suitable classical systems


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*[Classical synthetic gauge field]: It is said that in Wiki, in Atomic-Molecule-Optics (AMO) Cold Atoms, people introduce synthetic gauge field, either by rotation of a gas (resulting from the correspondence between the Lorentz force and the Coriolis force) or by imprinting a spatially varying geometric phase through an atom-laser interaction scheme. So is the AMO synthetic gauge field a classical gauge field?

*[Quantum synthetic gauge field]: Does the synthetic gauge field in AMO Cold Atoms also require the conditions? So can the AMO synthetic gauge field be a quantum gauge field?

*[Dynamical synthetic gauge field]: The Lorentz force and the Coriolis force by AMO does not seem to be a dynamical way to generate dynamical gauge field. By dynamical gauge field $A$, we mean that the gauge field needs to be sum over in the path integral measure $[DA]$ for all possible configurations (thus dynamical):
$$
Z[A, \psi, \dots] =\int [DA][D \psi] e^{i S[A, \psi, \dots]}.
$$
In otherwords, the AMO effective classical force method is only the probed field $A_{probe}$:
$$
Z[ \psi, \dots] =\int [D \psi] e^{i S[A_{probe}, \psi, \dots]}.
$$ So $A_{probe}$ is only a probe field but non-dynamical.
Can the synthetic gauge field, such as those in AMO be dynamical gauge fields?
 A: Synthetic gauge fields should become dynamical at least in certain circumstances. The synthetic gauge fields are given in general by nonlinear functions of parameter manifolds, active subspaces, fast coordinates etc. For example in the case of the $CP^N$ model, the emergent Abelian gauge field has the form:
$$A_{\mu} = \frac{\phi^{\dagger} \partial_{\mu}\phi - \partial_{\mu}\phi^{\dagger}  \phi}{\phi^{\dagger}  \phi}$$
Where: $\phi$ is an $N$-component scalar field
When the scalar field $\phi$ fluctuates, i.e., becomes a quantum field, we should expect that the composite gauge, the field $A_{\mu}$ to fluctuate also. In order to know if the gauge field develops a Maxwell kinetic term, one should check if the propagator:
$$ \langle \hat{A}_{\mu}(p) \hat{A}_{\nu}(0)\rangle$$
develops a pole at $p=0$. ($\hat{A}_{\mu}(p)$ is the Fourier transformed field in the momentum space).
The propagator can be computed only approximately and in certain approximations (some of them nonperturbative) such as the large $N$ or lattice gauge theory, such a pole has been obtained. 
As far as I know, the above situation summarizes the state of the current knowledge, namely, there is no realistic model where the dynamical term was exactly computed or proven to exist beyond any approximation. 
Please see for example the following two  recent  works and references therein treating this matter. 
There are  recent  suggestions for an experimental proof using ultracold atoms in optical lattices.
There is another issue that one needs to consider in relativistic field theories: The Weinberg-Witten theorem  which prevents the existence of a massless gauge boson when there is a Lorentz covariant conserved Noether current.  As explained in the Wikipedia text, theories of Yang-Mills types evade this theorem because the gauge current can be made either closed but not gauge invariant or vice versa.
