Elementary open quantum states

It is shown that the mixed states of a closed dynamics support 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.


I. INTRODUCTION
One of the surprises of quantum mechanics is the emergence of elementary excitations, the discrete building blocks of the state of a dynamical system. This results from representing the physical states by vectors in a Hilbert space [1]. In fact, the symmetries of the dynamics are realized in the Hilbert space in such a manner that they can be broken up into the direct sum of irreducible representations. The importance of the irreducibility is that the subspaces of these representations are the smallest linear subspaces, which remain closed The present work is an attempt to see to what extent these expectations are fulfilled.
One notices at the very beginning that the elementary states of an open system are a more involved mathematical concept than that of a closed dynamics. This has actually nothing to do with the openness of the dynamics. The mixed states of an open system are represented by density matrices, which can be constructed along two physically different routes. On the one hand, an uncertainty about the pure state, expressed by a particular probability distribution over several pure states, leads to the density matrix representation. On the other hand, mixed states are generated when the system interacts with its environment. The first view of the mixed states shows clearly that they are not elementary. However, a particular dynamics, appearing in the second view, may generate its own irreducible representation spaces. This problem can partially be removed by noting that any symmetry group G of a closed dynamics, |ψ → U(g)|ψ 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. Such a doubled symmetry is reduced back to at most the diagonal subgroup g + = g − ∈ G when the dynamics becomes open. Such a symmetry breaking by the open interaction channels can be used to establish the relation between the closed and open irreducible subspaces. The reduplication of the symmetry has already been spotted in a number of open systems, in particular at finite temperature [2], in hydrodynamics [3,4], in the extension of the BRST symmetry [5], and in identifying Goldstone bosons [6][7][8].
The present work aims at a simpler problem, the use of the reduplicated symmetry for the construction of elementary states in nonrelativistic (first quantized) quantum mechanics. in Section V. The summary of the results is given in Section VI.

II. CLOSED SYSTEMS
We start with a closed quantum system whose states are represented by the vectors of a Hilbert space H and dynamics is given by the Schrödinger equation, i ∂ t |ψ = H|ψ . The ket |ψ ∈ H and bra ψ| ∈ H † stand for the same physical state. The sum over the basis vectors in the expectation value in a pure state, represents the sum over quantum fluctuations, filtered by a given basis set. Since the coefficient of m|A|n in the sum, the pure state density matrix, is factorized, n|ρ|m = n|ψ ψ|m , the quantum fluctuations of the bra and the ket components are independent of each other in a pure state. The bra and the ket are related by Hermitian conjugation; hence, it is sufficient to follow the time dependence only for one of them.
We further suppose that a symmetry group G of the closed dynamics is represented by the basis transformations of H; there are unitary operators acting on the pure states, |ψ → U(g)|ψ , ψ| → ψ|U † (g), and operators A → U(g)AU † (g), g ∈ G in such a manner that they preserve the scalar product, the matrix elements, and leave the form of the Schrödinger equation invariant, H = U(g)HU † (g). Thus, the symmetry group of the closed dynamics on pure states is G.
The mixed states are represented by density matrices, represented by certain elements The density matrices belong to A ρ , a convex subset of A, containing the positive, Hermitian operators with unit trace. The elements of A are called state components. The expectation value of the observable A = A † in the state ρ ∈ A ρ is given by the Liouville space scalar In the case of a symmetry, we introduce the extended basis transformations, which act with different symmetry group elements on the bra and the ket, It is easy to see that the scalar product is preserved by these transformations and that the Neumann reflecting the independence of the bra and ket dynamics of closed dynamics. Hence, the symmetry group of the mixed state descriptions of a closed dynamics is G ⊗ G.

III. OPEN SYSTEMS
We extend our description from the observed system to its environment by assuming that the full system obeys a closed dynamics, defined by the total Hamiltonian, H tot = H s + H e + H i , where the terms in the sum correspond to the system, the environment, and their interactions. The pure states make up the direct product Hilbert space, where the first and the second factor denote the space of system and environment states, respectively. The state of the observed system is given by the reduced density matrix, . A further assumption is that there are no correlations between the system and the environment in the initial state, i.e., the initial density matrix is factorized, Let us consider a symmetry group G of the full closed dynamics, represented by basis transformations, H tot = U(g)H tot U † (g). While the Neumann equation of the full dynamics is symmetrical with respect to the extended transformations, the symmetry group of the effective system dynamics is usually more restricted. To find out the latter, we assume that U(g) acts without mixing the system and the environment states, if U s,± and U e are symmetries of each term in the Hamiltonian one-by-one, The necessity of the last condition shows that the symmetry with respect to the extended transformation, ρ s → U s,− ρ s U † s,+ with U s,+ = U s,− , is usually no longer a symmetry in the presence of system-environment interactions.
The condition of preserving the full symmetry group for the diagonal basis transformations, g + = g − , is the symmetry of the environment initial condition, ρ e,i = U e (g)ρ e,i U † e (g).
. It is natural to use a continuous quantum number k to define a translation-independent basis in A q , Note that the quantum number q is not related to the momentum. In fact, the momentum operator is the sum of two terms: with ∇ ± = ∂/∂x ± . The contribution of the first term to the expectation value of a momentum-dependent observable is vanishing for normalizable states, and a simplified qindependent momentum operator, −i ∇ d , can be used in coordinate-independent observables. In the case of translation noninvariant observables, the term with ∇ is needed, and it takes care of the coordinate dependence of the observables.
with ∇ = ∂/∂x and ∇ d = ∂/∂x d . The first two terms correspond to the Neumann equation of a closed system under the influence of a dragging force f . The parameter ν generates Newton's friction force; d 0 > 0 and d 2 > 0 control the decoherence; ξ is the coefficient of a total time derivative term in the effective Lagrangian and represents no dynamical process [11]. The positivity of the density matrix is preserved for weak friction as long as The solution of this master equation or one of its simplified forms has been studied extensively [12,13], and here, we confine ourselves only to some simple remarks about the time dependence of the elementary open components.
The elementary state components of the translation-invariant dynamics are labeled by the wave vector q, which might be complex. It is easy to see that the master Equation (9) preserves such an x-dependence; hence, the time evolution keeps the subspaces A q of the form ρ(x, x d ) = e iqx χ(x d ) closed. This expression might be used in the full coordinate space or only in a restricted region, depending on the actual problem. The master equation for the factor χ(x d ), can easily be solved, where the initial condition χ i (x d ) = χ(0, x d ) is used and: We make here only a few general comments about these elementary components, and their application to simple problems was discussed in [14]. The solution gives the elementary components as the product of two factors: one is the initial condition, χ i (x 0 ), considered at the initial coordinate of the characteristic passing x d at t. This factor displays the presence of the diffusive forces: the information of the initial state is reduced to a single number, χ i (i q/mν), as t → ∞. The other multiplicative factor, written in the form of an exponential function, represents the physical processes building up on the characteristic curve during the time evolution and contains three terms. The O(x 0 d ) piece provides a time-dependent weight factor, It is instructive to look into the structure of the time-dependent weight factor, which results from two processes: one is driven by the external force and the other by the environment, represented by the first and the second term in the parentheses of the exponent of (13), respectively. Let us first take the case of a real wave vector when the contribution to the probability density of the coordinate, ρ(x, 0), is a standing wave. The decoherence, implied by the presence of d 0 in the exponent of (13), tends to suppress this state component, and the concomitant smearing of the probability distribution is slowed down by increasing ν, the strength of the dissipative force. The case of the imaginary wave number, q = iq i with real q i , is more interesting because ρ(x, 0) can be interpreted as a probability distribution displaying an exponential coordinate dependence, e −q i x and used in the coordinate space segment x ±,j sign(q i,j ) ≥ 0, j = 1, 2, 3. To support such an inhomogeneous particle density against diffusion, one needs an outflux of particles at x = 0. Such an outflux decreases the norm of this state component because the contribution of the compensating influx to the norm at x ±,j sign(q i,j ) = ∞ is suppressed. To find the necessary flux to keep the state in equilibrium, we chose the external force to compensate this flux in (13), The time dependence of a generic state, starting with a regular, real q dependence around evolves into: One finds after a long time evolution, tν ≫ 1, where the decoherence suppresses the elementary components with q = 0, yielding: The exponential environment-induced suppression of a fixed q is weakened into a prefactor td 0 /mν 2 by the small q contributions, and the drift due to the external force is generated, together the Gaussian decoherence. The impact of the external force is enhanced by the small q contributions, and the emerging singularity as the decoherence is turned off is expected due to the instabilities such a force generates in closed dynamics.
It is instructive to look into the norm of the state for νt ≫ 1, A spread of ∆q of χ i,q (x d ) around q = 0 makes the norm suppressed for t ≫ m 2 ν 2 /d 0 ∆q 2 .
The only possibility to preserve the norm is to have a singular peak at q = 0, χ i,q (i q/mν) ∼ δ(q). In other words, elementary components with any inhomogeneity are depopulated by the combined effect of decoherence and dissipation.

VI. SUMMARY
The symmetry properties of the open dynamics were discussed in the present work. It was pointed out that the mixed states of a closed system enjoy a reduplicated symmetry, G → G ⊗ G. Such an enlarged symmetry is reduced back to the original symmetry group or to one of its subgroups when the interaction with the environment is taken into account.
The The irreducible representations of the symmetries were discussed in the context of nonrelativistic quantum mechanics, and it is natural to raise the question about many-body systems, handled by quantum field theory. There seems to be no conceptual difficulty to extend the discussion to the Fock space of free quantum field theories after the representations have been found in first quantized quantum mechanics. However, interactions may generate unexpected representations and open up an interesting, difficult problem.