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S1g. Quantum-classical mechanics
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Quantum mechanics and classical mechanics are very close relatives.
There are analogous objects for everything of relevance in
classical and quantum statistical mechanics.

Observable f: 
   classical - real phase space function f(x,p) 
   quantum   - Hermitian linear operator or sesquilinear form f
Lie product f \lp g:
            read \lp as 'Lie', and visualize it as inverted, stylized L;
            Macro for LaTeX:
            \def\lp{\mbox{\Large$\,_\urcorner\,$}}  
   classical: f \lp g = {g,f} in terms of the Poisson bracket 
   quantum:   f \lp g = i/hbar [f,g] in terms of the commutator
The Lie product is bilinear in the arguments and satisfies
   f \lp g = - g \lp f
   f \lp gh = (f \lp g)h + g(f \lp h)                   (Leibniz)
   f \lp (g \lp h) = (f \lp g) \lp h + g \lp (f \lp h)  (Jacobi)
Invariant measure:
   classical - integral f := integral dxdp f(x,p)
   quantum   - integral f := trace f
Integrability: integral |f| finite
   quantum integrable <==> f trace class 
Partial integration formula:
   integral f \lp g = 0.
Dynamics: df/dt = X_H f := H \lp f with Hermitian H
canonical transformations = mappings exp(tX_H) with Hermitian H
   Liouville's theorem says that 
      integral f = integral exp(tX_H)f
   The infinitesimal form of this is the partial integration formula.
State rho:
   classical - real integrable phase space function rho(x,p)>=0
   quantum   - Hermitian positive semidefinite trace class operator rho
   both normalized to integral rho = 1.
expectation of f in state rho:
   <f> = integral rho f