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S11f. Master equation and pointer variables
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On an approximate level, the preferred basis problem is approached
via quantum master equations.
A quantum master equation is a dynamical equation for the density matrix
of a dissipative quantum systems, which approximates a quantum system
weakly coupled to an environment at time scales long compared to the
typical interaction time but short enough to avoid recurrence effects.
More precisely, the dynamics is given by a completely positive
Markovian semigroup in a representation named after Lindblad,
wo discovered its general form.
For a classical damped linear system xdot(t)=Ax(t) with a matrix A
whose spectrum is in the left complex half plane, the contribution of x
in the invariant subspace corresponding to eigenvalues which are not
purely imaginary decays to zero, so that at large times t,
x(t) essentially approaches the invariant subspace corresponding to
purely imaginary eigenvalues.
For a quantum master equation, a similar analysis holds and shows that
(under suitable conditions) the density matrix at times much larger
than the so-called decoherence time approaches a block diagonal form
in a suitable basis. Thus it (almost) commutes with a special set
of observables, which define the 'pointer variables' of the system.
These pointer variables therefore behave essentially classically.
If the pointer variables form a complete set of commuting variables,
the density matrix approaches a diagonal matrix, and the basis in
which this happens is called the 'preferred basis'.
For details, see, e.g., cond-mat/0011204 or gr-qc/9406054