Abstract
It is shown that the mixed states of a closed dynamics supports a reduplicated symmetry, which is reduced back to the subgroup of the original symmetry group when the dynamics is open. The elementary components of the open dynamics are defined as operators of the Liouville space in the irreducible representations of the symmetry of the open system. These are tensor operators in the case of rotational symmetry. The case of translation symmetry is discussed in more detail for harmonic systems.
Highlights
The obvious disadvantage of this strategy is that not all irreducible spaces represent possible physical states. This problem can partially be removed by noting that any symmetry group G of a closed dynamics, |ψi → U ( g)|ψi g ∈ G, extends to the symmetry G ⊗ G acting on the Liouville space, A → U ( g−, g+ ) A = U ( g− ) AU † ( g+ ) with g± ∈ G, since it can be realized on the bras and the kets independently
The definition of the elementary pure state can formally be generalized for the mixed states of closed and open dynamics: the elementary closed mixed state components are the elements of the irreducible representation of G ⊗ G (G) in the Liouville space
The symmetry properties of the open dynamics were discussed in the present work
Summary
The obvious disadvantage of this strategy is that not all irreducible spaces represent possible physical states This problem can partially be removed by noting that any symmetry group G of a closed dynamics, |ψi → U ( g)|ψi g ∈ G, extends to the symmetry G ⊗ G acting on the Liouville space, A → U ( g− , g+ ) A = U ( g− ) AU † ( g+ ) with g± ∈ G, since it can be realized on the bras and the kets independently. The decomposition of the mixed states into the sum of open elementary components offers a representation in terms of elementary blocks, evolving independently of each other This structure is introduced below, starting with a brief introduction of the symmetry group G ⊗ G of the mixed state of a closed dynamics, followed by Section 3, with the reduction of the symmetry to a diagonal subgroup by the open interaction channels.
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