A Time-Dependent Canonical Transformation between Bateman and Doubled Caldirola--Kanai Systems for a Homogeneous Massive Scalar Field on a Prescribed FLRW Background

Abstract

Dissipative equations admit distinct variational descriptions in the Bateman and Caldirola--Kanai (CK) formalisms. The classical correspondence between them is extended to a homogeneous canonical scalar field on a prescribed spatially flat Friedmann--Lemaître--Robertson--Walker (FLRW) background, where the expansion produces the time-dependent damping coefficient $3H(t)$. A multiplier action yields the Klein--Gordon equation and a complementary anti-damped equation containing the term $-3\dot{H}(t)\ddot{I}‡$. A first-order Bateman Lagrangian derived from the same multiplier action reproduces this physical--auxiliary pair for a general potential. Specializing to a free massive field gives the Bateman and doubled CK Lagrangians and Hamiltonians used in the canonical comparison. The factors $a^3(t)$ and $a^{-3}(t)$ generate the damped and anti-damped CK sectors, respectively. An explicit time-dependent canonical transformation, generated by a function linear in the Bateman momenta, maps the complete doubled CK system to the Bateman system. For this point transformation, the terms proportional to $\dot{H}(t)$ are required for Hamiltonian equivalence. In rotated variables, the Bateman scalar-field Hamiltonian takes the difference form $H_{B,\mathrm{SF}}=E_u-E_v$. It is conserved for constant $H$ and generally varies with time otherwise. For the power-law background $a(t)\propto t^p$, however, a correlated family at $p=2/3$ has conserved $H_{B,\mathrm{SF}}$ despite the time dependence of $H(t)$. These results concern classical homogeneous fields on a prescribed FLRW background and exclude the gravitational phase space.

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