A. The reduced density matrix Without loss of generality, consider a system consisting of a heavy particle of mass M which we will call the system, minimally coupled to a harmonic oscillator of mass m . We analytically continue this Lagrangian to τ = it . We can simplify this Lagrangian by making a change of variables. We subtract the particular solution to the classical equations of motion as determined by Eq. 2. In terms of the shifted coordinate r ′ the Euclidean action becomes that of a simple harmonic oscillator: As always, the propagator of a system for imaginary time β = − i/T gives matrix elements of the thermal density operator. In our example, the density matrix has 4 indices, 2 for each degree of freedom of the system. In the example given above the density matrix is given as If we were never interested in measurements of the degree of freedom r , we could take the trace over the indices corresponding to this degree of freedom, and work with a density 4 ρ ( x$_{i}$, r$_{i}$ ; x$_{f}$, r$_{f}$ ; β ) = ∫ x ( β )= x$_{f}$ x (0)= x$_{i}$ D x$^{∫}$ r ( β )= r$_{f}$ r (0)= r$_{i}$ D r exp $^{(}$− S E S [ x ] − S E I [ x, r ] $^{)}$. (5) S E I [ x, r ] = ∫ β 0 $^{[}$1 2 m ˙ r ′ 2 + 1 2 mω $^{2}$r ′ 2 + 1 2 Cx ( τ ) A [ x, τ ]$^{]}$ dτ + m ˙ A [ x, β ]$^{(}$ r $^{′}$( β ) + 1 2 A [ x, β ]$^{)}$ ≡ S ′ E I [ x, r $^{′}$] . (4) r ( τ ) ≡ r $^{′}$( τ ) + C mω ∫ τ 0 dτ $^{′}$x ( τ $^{′}$) sinh [ ω ( τ − τ $^{′}$)] ≡ r $^{′}$( τ ) + A [ x, τ ] (3) S E [ x ] = ∫ β 0 $^{[}$1 2 M ˙ x 2 + V ( x )$^{]}$ , S E I [ x, r ] = ∫ β 0 $^{[}$1 2 m ˙ r 2 + 1 2 mω $^{2}$r2 + Cxr$^{]}$ dτ (2) L = L$_{S}$ + L$_{I}$ ; L$_{S}$ = 1 2 M ˙ x 2 − V ( x ) , L$_{I}$ = 1 2 m ˙ r 2 − 1 2 mω $^{2}$r2 − Cxr. (1)