  The most common quantization scheme in which to study a conformal field
theory is radial quantization, wherein a Hilbert space of states is defined on
a sphere, whose Hamiltonian when mapped to the plane is the dilatation
generator and which boasts a state/operator correspondence. In this paper, we
consider an alternative quantization scheme for 2d CFTs in which the plane is
foliated by constant-angle slices, as opposed to concentric circles, whose
Hamiltonian is the rotation generator. In this angular quantization, there is
no state/operator correspondence but instead an "asymptotics/operator
correspondence". A central feature is that the quantization slice ends on two
operators, and a regulator must be chosen by excising holes around each
operator and imposing suitable boundary conditions such that the holes shrink
to the desired local operators. This angular quantization may be viewed as
constructing CFTs in Minkowski space, or separately as studying non-conformal
(but local) boundary conditions in CFTs. We provide explicit Fock space
constructions for various free 2d CFTs. In addition to the motivation of
applying angular quantization in string theory situations where traditional
radial quantization is insufficient, we comment on its relation to modular
Hamiltonian approaches to entanglement entropy.
