Nonunitary transformations

Let us consider transformations that preserve the Hermitian nonsingular metric tnatrix S  

To proceed, we first introduce the general nonintegral power A of a nonsingular matrix 

By substitution, it is now easily verified that W may be written in terms of a unitaiy matrix U as 

Since X is anti-Hermitian, exp(—X)Aexp(X) represents a unitary transformation. Carrying out a standard BCH expansion, we obtain 

This expansion is identical to a conventional BCH expansion except that the metric S has been inserted between each pair of adjacent matrices. Introducing the S commutators 

---

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 situation changes drastically for nonintegrable systems. As we will see, the transformation U = U is replaced by a nonunitary transformation At A. ... 

From the above expressions presented in this section, we can already conclude several interesting properties of the transformation A. First, Eqs. (27) and (28) show that it is nonunitary transformation, A A . Acmally, one can show that this transformation satisfies a more general symmetric property called star unitarity (see Refs. 5, 10, 13, and 14) ... 

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

The dots on the solid line are the numerical result of the inverse transformation. This is obtained by applying the exphcit theoretical form of A on the numerical values of Ji(t) at several points of time. The figure shows that by applying A to Ji t), we indeed go back to Ji(t). This numerically demonstrates that there is no loss of information by the nonunitary transformation. 

As is well known, CASSCF wavefunctions are invariant to general (i.e. nonunitary) linear transformations of the active orbitals. As such, we may seek alternative, but equivalent, representations in which a small number of configurations are dominant. This is achieved in our case by means of efficient computational schemes for carrying out exactly the transformations of full-CI spaces induced by nonunitary transformations of orbital spaces [9]. 

The first difficulty referred to above is the source of numerous technical problems in many perturbation theory applications. For the systems treated in this review, these difficulties can be avoided by exploiting the Dalgarno and Lewis (1955) procedure (see also Schiffi, 1968, p. 263). However, it is a remarkable aspect of the Lie algebraic approach that the continuum problem can be simply avoided by introducing a nonunitary transformation, which can... 

In order to apply the algebraic methods based on so(4, 2) it is necessary to carry out a noncanonical and nonunitary transformation of Eq. (249). Thus, multiplying on the left by r and applying the scaling transformation (cf. Section V and Appendix B) to operators and functions... 

Since f is not anti-Hermitian, it gives rise to a nonunitary transformation and the similarity-transformed Hamiltonian operator is therefore non-Hermitian. Naively, we would expect the BCH expansion (3.1.7) of the similarity-transformed Hamiltonian to yield an infinite sequence of nested commutators. Nevertheless, we shall see that the expansion terminates after five terms. 

Formally, this expansion may be obtained from the BCH expansion of exp(—X)A exp(X) by padding X with S everywhere except next to A. The theory of nonunitary transformations is used in the development of density-based Hartree-Fock theory in Section 10.7. 

---