The geometry of coherent states

We examine the geometry of the state space of a relativistic quantum field. The mathematical tools used involve complex algebraic geometry and Hilbert space theory. We consider the Kähler geometry of the state space of any quantum field theory based on a linear classical field equation. The state space is viewed as an infinite-dimensional complex projective space. In the case of boson fields, a special role is played by the coherent states, the totality of which constitutes a nonlinear submanifold 𝒞 of the projective Fock space PF. We derive the metric on 𝒞 induced from the ambient Fubini–Study metric on PF. Arguments from differential geometric, algebraic, and Kählerian points of view are presented, leading to the result that the induced metric is flat, and that the intrinsic geometry of 𝒞 is Euclidean. The coordinates for the single-particle Hilbert space of solutions are shown to be complex Euclidean coordinates for 𝒞. A transversal intersection property of complex projective lines in PF with 𝒞 is derived, and it is shown that the intrinsic geodesic distance between any two coherent states is strictly greater than the corresponding geodesic distance in the ambient Fubini–Study geometry. The functional metric norm of a difference field is shown to give the intrinsic geodesic distance between two coherent states, and the metric overlap expression is shown to measure the angle subtended by two coherent states at the vacuum, which acts as a preferred origin in the Euclidean geometry of 𝒞. Using the flatness of 𝒞 we demonstrate the relationship between the manifold complex structure on PF and the quantum complex structure viewed as an active transformation on the single-particle Hilbert space. These properties of 𝒞 hold independently of the specific details of the single-particle Hilbert space. We show how 𝒞 arises as the affine part of its compactification obtained by setting the vacuum part of the state vector to zero. We discuss the relationship between unitary orbits and geodesics on 𝒞 and on PF. We show that for a Fock space in which the expectation of the total number operator is bounded above, the coherent state submanifold is Kähler and has finite conformal curvature.