Wave-functional formulation of dissipative CSL models

Abstract

We formulate minimal and dissipative Continuous Spontaneous Localization (CSL) dynamics in the functional Schrödinger representation for a non-relativistic bosonic field. In this framework, the Fock-space state is encoded in a wave functional, and fixed particle-number wave functions are obtained by sector projection. For the minimal CSL coupling to the smeared mass density, this projection gives the standard nonlinear stochastic dynamics in each \(N\)-particle sector, with the collapse operator acting on the total smeared density of the configuration. This makes the amplification mechanism transparent and allows us to discuss sector superpositions, local probability balance, and the status of Bohmian equivariance at the wave-function level. We then consider a dissipative extension in which the collapse operator includes a smeared current contribution. The one-particle sector reproduces the expected dissipative CSL energy balance, while fixed many-body sectors contain additional collective momentum shifts and pair-mixing terms that are not reducible, in general, to independent one-particle contributions. Within a leading compact closure, the collective pair friction produces a non-extensive stationary mean kinetic energy: in three dimensions and for weak dissipation, $T_N^{\rm comp}\simeq 2T_β/N$, whereas the corresponding dilute energy remains extensive.