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LittleSchwinger
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- Would anybody add anything to this account of Decoherence
When I'm teaching Advanced QM, I like to include how to describe some processes in the Heisenberg picture (e.g. double slit) so that a student's thinking isn't overly attached to the "dynamics of the quantum state", but they can also understand effects involving operator evolution. This is a sketch of how I go about decoherence, I was wondering if anybody has any other ideas. I assume familiarity with the mathematics of decoherence.
So we have a system ##S## and it's environment ##E## with Hamiltonian:
##H = H_{S}\otimes\mathbb{I}_{E} + \mathbb{I}_{S}\otimes H_{E} + H_{I}##
In the Heisenberg picture we then obtain the evolution operator for the observable algebra ##\mathcal{O}_{S}## alone via:
##\mathcal{T}_{t}\left(A_{S}\right) = P_{E}\left(e^{itH}A_{S}\otimes\mathbb{I}_{E} e^{-itH}\right)##
This operator gives us the "environment traced out" evolution of the operators. You can make quick arguments that for certain models ##T_{t} = e^{tG}## with ##G## given by a Markov Master equation you tend to get in decoherence studies.
We can then prove that the observable algebra splits as follows:
##\mathcal{O}_{S} = \mathcal{M}_{1} \oplus \mathcal{M}_{2}##
With ##T_{t}## reversible on ##\mathcal{M}_{1}##, but for ##\mathcal{M}_{2}## we have:
##\lim_{t\rightarrow\infty} Tr\left(\rho N\right) = 0, \quad \forall \rho, N \in \mathcal{M}_{2}##
Long story short that in the Heisenberg picture Decoherence causes operators which don't commute with the macroscopic collective coordinates to have all their statistical moments suppressed and effectively converge (in Trace Norm) to the zero operator.
I was wondering if anybody has any other ideas. Thanks.
So we have a system ##S## and it's environment ##E## with Hamiltonian:
##H = H_{S}\otimes\mathbb{I}_{E} + \mathbb{I}_{S}\otimes H_{E} + H_{I}##
In the Heisenberg picture we then obtain the evolution operator for the observable algebra ##\mathcal{O}_{S}## alone via:
##\mathcal{T}_{t}\left(A_{S}\right) = P_{E}\left(e^{itH}A_{S}\otimes\mathbb{I}_{E} e^{-itH}\right)##
This operator gives us the "environment traced out" evolution of the operators. You can make quick arguments that for certain models ##T_{t} = e^{tG}## with ##G## given by a Markov Master equation you tend to get in decoherence studies.
We can then prove that the observable algebra splits as follows:
##\mathcal{O}_{S} = \mathcal{M}_{1} \oplus \mathcal{M}_{2}##
With ##T_{t}## reversible on ##\mathcal{M}_{1}##, but for ##\mathcal{M}_{2}## we have:
##\lim_{t\rightarrow\infty} Tr\left(\rho N\right) = 0, \quad \forall \rho, N \in \mathcal{M}_{2}##
Long story short that in the Heisenberg picture Decoherence causes operators which don't commute with the macroscopic collective coordinates to have all their statistical moments suppressed and effectively converge (in Trace Norm) to the zero operator.
I was wondering if anybody has any other ideas. Thanks.
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