Hilbert space mapping operators

On the other hand, the duality mapping I is defined by the scalar product in the Hilbert space V. Assume that the operator A is self-conjugate. Then we can define the scalar product in V as follows ... 

Consider a deteriiiinistic local reversible CA i.o. start with an infinite array of sites, T, arranged in some regular fashion, and a.ssume each site can be any of N states labeled by 0 < cr x) < N. If the number of sites is Af, the Hilbert space spanned by the states <7-(x is N- dimensional. The state at time t + 1, cTf+i(a ) depends only on the values cri x ) that are in the immediate neighborhood of X. Because the cellular automata is reversible, the mapping ai x) crt+i x ) is assumed to have a unique inveuse and the evolution operator U t,t + 1) in this Hilbert space is unitary,... 

If A induces a one-to-one mapping of the Hilbert space on itself then the inverse operator A 1 exists. It is an antilinear operator with the property that... 

The mapping preserves the commutation relations (70) of the spin operators. As can be seen from (75d), the image of the 2s + 1)-dimensional spin Hilbert space is the subspace of the two-oscillator Hilbert space with 2s quanmm of excitation— the so-called physical subspace [218, 220], This subspace is invariant under the action of any operator which results by the mapping (75a)-(75d) from an arbitrary spin operator A(5 i, S2, S3). Thus, starting in this subspace the system will always remain in it. As a consequence, the mapping yields the following identity for the matrix elements of an operator A ... 

In the Schwinger representation the identity operator in the spin Hilbert space is mapped onto the constant of motion a a + a a2. The existence of this constant of motion is utilized by the Holstein-Primakoff transformation to eliminate one boson DoF, thus representing the spin DoF by a single oscillator [97] ... 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

The image of the A-level Hilbert space is the subspace of the A-oscUlator Hilbert space with a single quantum excitation. Again, this (physical) subspace is invariant under the action of any operator which results by the mapping (80a) from an arbitrary A-level system operator. As a consequence we obtain the propagator identity... 

Here, for notational convenience, we have assumed that Vnm = We would like to emphasize that the mapping to the continuous Hamiltonian (88) does not involve any approximation, but merely represents the discrete Hamiltonian (1) in an extended Hilbert space. The quantum dynamics generated by both Hamilton operators is thus equivalent. The Hamiltonian (88) describes a general vibronically coupled molecular system, whereby both electronic and nuclear DoF are represented by continuous variables. Contrary to Eq. (1), the quantum-mechanical system described by Eq. (88) therefore has a well-defined classical analog. 

The norm A of an operator is (A, A)112. The operators whose norm is finite constitute a super Hilbert space. A linear mapping of operators A->B,... 

Our discussion is complicated by the fact that, in the definition Eq. (2.27) of the transformed operator T, the operator T occurs in the middle. If 4 is an element of the L2 Hilbert space, then the result 4"" = T P is obtained by three successive mappings,... 

We denote linear operators acting on the Hilbert space by capital letters with a hat, such as UAt, whereas linear maps on operators (operators in the Liouville space) are denoted by calligraphic letters such as Uat, where the argument is often placed within braces in order to avoid ambiguity. 

Such an operator space T is another realization of the abstract Hilbert space, and the superoperators til are then conveniently defined as mappings of this space. In a previous paper,4 it has been shown that—by using this general approach—one may generalize most of the L2 methods earlier developed to solve the Schrodinger equation to solve also the Liouville equation. The purpose of the present article is to give some additional aspects of this problem, and particularly to show the connection with some of the so-called propagator methods. 

The specification super-operator is common in quantum chemical emd physical literature for linear mappings of Fock-space operators. It is very helpful to transfer this concept to the extended states A, B) and define the application of super-operators by the action on the operators A and B. We will see later how this definition helps for a compeict notation of iterated equations of motion and perturbation expansions. In certain cases, however, the action of a super-operator is fully equivalent to the action of an operator in the Hilbert space Y. The alternative concept of Y-space operators allows to introduce approximations by finite basis set representations of operators in a well-defined and lucid way. 

We also note that, in contrast to the Pegg-Bamett formalism [45], we consider an extended space of states, including the Hilbert-Fock state of photons as well as the space of atomic states [36,46,53,54]. The quantum phase of radiation is defined, in this case, by mapping of corresponding operators from the atomic space of states to the whole Hilbert-Fock space of photons. This procedure does not lead to any violation of the algebraic properties of multipole photons and therefore gives an adequate picture of quantum phase fluctuations [46],... 

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