  In this paper we study gapless fermionic and bosonic systems in
$d$-dimensional continuum space with $U(1)$ particle-number conservation and
$\mathbb{R}^d$ translation symmetry. We write down low energy effective field
theories for several gapless phases where $U(1)\times \mathbb{R}^d$ is viewed
as internal symmetry. The $U(1)\times \mathbb{R}^d$ symmetry, when viewed as an
internal symmetry, has a mixed anomaly, and the different effective field
theories for different phases must have the same mixed anomaly. Such a mixed
anomaly is proportional to the particle number density, and can be measured
from the distribution of the total momentum $\boldsymbol{k}_\text{tot}$ for low
energy many-body states (\ie how such a distribution is shifted by $U(1)$
symmetry twist $\boldsymbol{a}$), as well as some other low energy universal
properties of the systems. In particular, we write down low energy effective
field theory for Fermi liquid with infinite number of fields, in the presence
of both real space magnetic field and $\boldsymbol{k}$-space "magnetic" field.
The effective field theory also captures the mixed anomaly, which constraints
the low energy dynamics, such as determine the volume of Fermi surface (which
is another formulation of Luttinger-Ward-Oshikawa theorem).
