The Coherent State Projection Eigenvalue Theorem

Abstract

Recent papers have developed a cyclic symmetry projected extension for coherent state representations of bosonic Hilbert space. This extension can greatly reduce sampling errors in phase-space simulations of bosonic quantum systems that have symmetries. Here we prove an eigenvalue theorem for the continuum limit of the projected coherent states. Our intuitively appealing result is that in the continuum limit the projected coherent states are eigenstates of the symmetry generating operator. This result gives a differential identity in phase-space that includes the conserved eigenvalues. Detailed applications of these differential identities to obtaining improved Fokker-Planck and stochastic equations in many-body physics will be given elsewhere.

The paper asks whether cyclic symmetry projections of bosonic coherent states converge, as the cyclic order increases, to eigenstates of the corresponding continuous-symmetry generators. It considers finitely many bosonic modes and mutually commuting generators $\hat g_\nu=\hat{\boldsymbol a}^\dagger g_\nu\hat{\boldsymbol a}$, where the one-particle matrices are Hermitian with integer eigenvalues. For a fixed integer sector label $p$, it proves that $\Pi_p^{(\mathcal M)}\lVert\boldsymbol\alpha\rangle\to E_p\lVert\boldsymbol\alpha\rangle$ in Hilbert-space norm, where $E_p$ is the generator’s spectral projector. It also establishes $\|(\hat g-p)\Pi_p^{(\mathcal M)}\lVert\boldsymbol\alpha\rangle\|\to0$. If the limiting projection is nonzero, normalized projected states converge to its normalization. The proof uses the exact identity $\Pi_p^{(\mathcal M)}=\sum_{\ell\in\mathbb Z}E_{p+\ell\mathcal M}$: unwanted sectors lie at $|m|\ge\mathcal M-|p|$, so their total weight vanishes as a spectral tail. Finite coherent-state generator moments establish the corresponding convergence for the unbounded generator. The argument extends to multiple commuting projections and limits involving only a subset of them. The limiting eigenvalue relations yield first- and second-order differential identities intended to modify drift and diffusion in phase-space simulations. The convergence is strong, not generally in operator norm; generators, amplitudes, and sector labels remain fixed. The construction assumes commuting integer-spectrum one-particle generators, and a finite lattice’s discrete translation symmetry does not itself admit the continuum limit used here. Explicit improved stochastic equations, numerical demonstrations, and sampling-error reductions are deferred to other work.