Unitary local

Focusing on strictly local (i.e. nearest neighbor) interactions. Grossing and Zeilinger [gross88a] consider the following unitary evolution operator U, approximated to first order by ... 

Having thus established at least a formal equivalency between a discretized field theory on a lattice and CA, Svozil invokes the so-called no-go theorem to show that field theory cannot be discretized in this simple fashion. The no-go theorem (see [karstSl] and [nielSl]) states essentially that under a set of only mild assumptions it is impossible to formulate a local, unitary, charge conserving lattice held theory without effectively doubling the size of the predicted fermion population (i.e. species doubling see discussion box). 

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,... 

The problem now is to find the corresponding Hamiltonian, t Hooft shows that the most obvious construction, obtained by rewriting U(t+l,t) as a product of cyclic elements, unfortunately does not work because at the end of the calculation there is no way to uniquely define the vacuum state. Given a cellular automaton with a local unitary evolution operator U = WgUg and the commutator [Ug, Ug ] 0 if [ af — af j> d for some d > 0, the real problem is therefore to find a Hamiltonian... 

With regard to the different points of view outlined in (a), (b) and (c), it should be pointed out that these differences arise mainly from the use of localized (a, LMO), or canonical (CMO, b, and c) molecular orbitals. In principle LMOs and CMOs are equivalent and are related by a unitary transformation. This can be illustrated by the C=C bonding in acetylene. 

As V is a unitary matrix, Y = VTX is just an equivalent set of Cartesian coordinates, and = UTZ is just an equivalent set of internal coordinates, simply linear combinations of the Zn. The i, , N-6, change independently, in proportion to changes in linear combinations of the Cartesian coordinates. So, locally, we have defined 3N — 6 independent internal coordinates. Every different configuration of the molecule, X, will have a different B matrix, and hence a different definition of local internal coordinates, defined automatically. 

Perez-Delgado, C.A. and Cheung, D. (2007) Local unitary quantum celular automata. Phys. Rev. A, 76, 032320. 

Step 2. The set of CMOs orthogonal molecular orbitals (LMOs) Xj using, e.g., Ruedenberg s localization criterion205. This is achieved by multiplying up with an appropriate unitary transformation matrix L ... 

Except for the initial AO —NAO transformation, which starts from non-orthogonal AOs, each step in (3.38) is a unitary transformation from one complete orthonormal set to another. Each localized set gives an exact matrix representation of any property or function that can be described by the original AO basis. 

The localized molecular orbitals (LMOs) can be defined as the unitary transformation of CMOs that (roughly speaking) makes the transformed functions as much like the localized NBOs as possible,24... 

Kellman, M. E. (1983), Dynamical Symmetries in a Unitary Algebraic Model of Coupled Local Modes of Benzene, Chem. Phys. Lett. 103,40. 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

The localized many-body perturbation theory (LMBPT) applies localized HF orbitals which are unitary transforms of the canonical ones in the diagrammatic many-body perturbation theory. The method was elaborated on models of cyclic polyenes in the Pariser-Parr-Pople (PPP) approximation. These systems are considered as not well localized so they are suitable to study the importance of non local effects. The description of LMBPT follows the main points as it was first published in 1984 (Kapuy etal, 1983). 

Transforming the occupied and the virtual single-particle functions separately by unitary transformations U and V, respectively, the localized single-particle functions ( )m will satisfy the following (non-diagonal) HF equations... 

Sumatra, by contrast, was fully exposed to the revolutionary wave of 1945-6 in favour of a unitary and democratic republic. Parties formed through wholly local initiatives to support the Republic and oppose the... 

In open-shell electronic states, the orbitals are not all doubly occupied, and the preceding procedure is not applicable. However, if the wave function can be written as a single Slater determinant, one can use a modified procedure to obtain energy-localized MOs here also. The procedure is to deal with the a spin-orbitals and the jS spin-orbitals separately, using two different unitary transformation matrices Ba and B in (2.85). 

Since the column vectors of B are orthonormal, B is a unitary matrix [Equation (2.24)]. The localized and canonical MOs are related by a unitary transformation. Since B is unitary, we can write (2.81) as can... 

We now prove that any unitary transformation of the orbitals leaves a closed-shell SCF Slater-determinant wave function unchanged, thereby showing the validity of transforming to localized MOs. Let i//loc be the SCF wave function written using localized orbitals. We wish to prove that if/loc equals can, where pc n uses the canonical MOs. In the notation of (1.260), we have... 

From any arbitrary choice of orbitals si and 38 it is possible to construct the most localized and the most delocalized orbitals through an unitary transformation. It is possible to show that the following quantities remain invariant to this transformation14 ... 

The off-diagonal elements of the matrix (3.13) contain quantities yrfa, and yrf39. Since + y2M is invariant to the unitary transformation, it is clear that for any perturbation an appropriate choice of si and 58 can make either or vanish. This will be, of course, neither the most localized nor the most delocalized orbitals. On the other hand, since has no clear physical significance, the most convenient working choice of orbitals is the one for which y m vanishes. As was already mentioned, this is the case of the most localized si = A and 58 = B and the most delocalized orbitals si = a and 58 = b. For these choices, consequently, we have to deal with two independent perturbations that are related to each other in two different basis as... 

This is accomplished by starting with an energy conserving system whose impulse response is perceptually equivalent to stationary white noise. Jot calls this a reference filter, but we will also use the term lossless prototype. Jot chooses lossless prototypes from the class of unitary feedback systems. In order to effect a frequency dependent reverberation time, absorptive filters are associated with each delay in the system. This is done in a way that eliminates coloration in the late response, by guaranteeing the local uniformity of pole modulus. 

---