content_image/358110.jpg content_image/358111.jpg content_image/358109.jpg content_image/358108.jpg content_image/358107.jpg
## 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 . 

<img src='content_image/358110.jpg'>

<img src='content_image/358111.jpg'>

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. 

<img src='content_image/358109.jpg'>

In terms of the shifted coordinate r ′ the Euclidean action becomes that of a simple harmonic oscillator: 

<img src='content_image/358108.jpg'>

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 

<img src='content_image/358107.jpg'>

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 

