  Lattice proposals for a nonperturbative formulation of the Standard Model
easily lead to a global U(1) symmetry corresponding to exactly conserved
fermion number. The absence of an anomaly in the fermion current would then
appear to inhibit anomalous processes, such as electroweak baryogenesis in the
early universe. One way to circumvent this problem is to formulate the theory
such that this U(1) symmetry is explicitly broken. However we argue that in the
framework of spectral flow, fermion creation and annihilation still in fact
occurs, despite the exact fermion number conservation. The crucial observation
is that fermions are excitations relative to the vacuum, at the surface of the
Dirac sea. The exact global U(1) symmetry prohibits a state from changing its
fermion number during time evolution, however nothing prevents the fermionic
ground state from doing so. We illustrate our reasoning with a model in two
dimensions which has axial-vector couplings, first using a sharp momentum
cutoff, then using the lattice regulator with staggered fermions. The
difference in fermion number between the time evolved state and the ground
state is indeed in agreement with the anomaly. A study of the vacuum energy
shows that the perturbative counterterm needed for restoration of gauge
invariance is insufficient in a nonperturbative setting. For reference we also
study a closely related model with vector couplings, the Schwinger model, and
we examine the emergence of the $\theta$-vacuum structure of both theories.
