Galilean and gauge invariance

In the previous Sec. 3.1, the functional has been required to be consistent with time reversal invariance, invariance under space reflections and rotational invariance. These constraints arise from symmetries of the NN interaction, see e.g. Refs. [32,31]. Derivations above were much easier to perform in a general form, without imposing any other additional symmetry conditions. In this section, we treat such additional constraints coming from imposing the Galilean and gauge invariance.

Assumption of the Galilean instead of Lorentz invariance goes hand in hand with using the Schrödinger equation as a starting point and relies on the assumption that relativistic effects are negligible. This symmetry ensures that the collective translational mass, calculated within the time-dependent HF or random-phase approximations, is correctly equal to the total mass, $M=Am$. Therefore, in principle, the Galilean invariance should always be imposed. However, in many phenomenological approaches, like the non-interacting or interacting shell model, the Galilean symmetry is not considered, because the translational motion is not within the scope of such models. The question of whether the Galilean symmetry must be imposed in phenomenological models is not yet resolved, and in the present study we keep this question open.

For a local interaction, $v(\vec {r}_{1}',\vec {r}_{2}',\vec {r}_{1},\vec {r}_{2})=\delta(\vec {r}_{1}' -\vec {r}_{1})\delta(\vec {r}_{2}'-\vec {r}_{2})v(\vec {r}_{1},\vec {r}_{2})$, the HF interaction energy is invariant with respect to the local gauge [36]. Therefore, for the total energy (1) obtained within the EDF method, one may also consider constraints resulting from assuming local gauge invariance. As mentioned, this symmetry is only fulfilled when the forces involved are local. An example of a local approximation is the well known local one pion-exchange (OPE) potential, which is only an approximate representation of the correct nonlocal OPE Feynman amplitude [33]. This approximation is good as long as the relative momenta of interacting particles are about the same in the initial and final states (see Fig. 10 in Ref. [33]). Some of the fitted NN potentials like the Argonne $V_{18}$, Nijm-II, and Reid93 use this local approximation while others (CD-Bonn) use the full nonlocal OPE amplitude [33]. The most important nonlocal term is the two-body spin orbit interaction [32], which violates the assumption of gauge invariance. However, in this case a gauge invariant spin-orbit term (used in the Skyrme and Gogny forces) can be obtained in the short-range limit [34,32,36].

For the EDF derived in this work it is, however, the symmetries of effective forces rather than bare forces that should be considered. One of the methods to obtain an effective NN force from the bare NN force is the unitary correlation operator method (UCOM) [35]. The use of the UCOM scheme, however, leads to a nonlocal effective interaction even if the bare interaction would have been a local one.

But rather than discussing to which extent gauge symmetry is conserved or broken in nuclei we aim to provide a theoretical framework where different choices can be accommodated. Because several successful phenomenological forces (e.g. Skyrme and Gogny [36]) are invariant under the gauge transformation, this symmetry constitute a natural starting point in the search of improved EDFs. To which extent gauge symmetry is violated for effective renormalized interactions is a question that can be investigated by comparing models using preserved and broken symmetries (see e.g. Ref. [37]).