Unitary space

The diabatic electronic functions are related to the adiabatic functions by unitary transformations at each point in coordinate space... 

Thus Ap is a unitary matrix at any point in configuration space. 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

The unitary transform does the same thing as a similarity transform, except that it operates in a complex space rather than a real space. Thinking in terms of an added imaginary dimension for each real dimension, the space of the unitary matrix is a 2m-dimensionaI space. The unitary transform is introduced here because atomic or molecular wave functions may be complex. 

The importance of the characters of the symmetry operations lies in the fact that they do not depend on the specific basis used to form them. That is, they are invariant to a unitary or orthorgonal transformation of the objects used to define the matrices. As a result, they contain information about the symmetry operation itself and about the space spanned by the set of objects. The significance of this observation for our symmetry adaptation process will become clear later. 

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,... 

We shall only mention the fact, that a unitary representation of the inhomogeneous proper Lorentz group is exhibited in this Hilbert space through the following identification of the generators of the... 

Let us next adopt the Schrodinger-type description. The statement that quantum electrodynamics is invariant under space inversion can now be translated into the statement that there exists a unitary operator U(it) such that... 

The existence of a unitary transformation U(a,A) which relates the field operators in the two frames imposes certain conditions on the operators themselves they must satisfy certain commutation rules with P and Muv. Consider first the case of a space-time translation... 

As indicated at the beginning of the last section, to say that quantum electrodynamics is invariant under space inversion (x = ijX) means that we can find new field operators tfi (x ),A v x ) expressible in terms of fj(x) and A nix) which satisfy the same equations of motion and commutation rules with respect to the primed coordinate system (a = igx) as did tf/(x) and Av(x) in terms of x. Since the commutation rules are to be the same for both sets of operators and the set of realizable states must be invariant, there must exist a unitary (or anti-unitary) transformation connecting these two sets of operators if the theory is invariant. For the case of space inversions, such a unitary operator is... 

Lorentz group, inhomogeneous proper, unitary representation in Hilbert space, 497... 

However, to determine the number of real pieces of information required to fix the projection from an M-dimensional space onto an /V-dimensional subspace spanned, not by the particular (occupied) basis in which P is diagonal, but by any basis of the subspace, it is necessary to subtract the numberof real parameters required to fix a particular basis in the /V-dimensional subspace from the total Kcy, such a number corresponds to the N2 real conditions that are necessary to fix a unitary transformation [11] in the subspace. But, as the phases ofthe eigenstates, < , are arbitrary as far as the physical state is concerned [4, 12], this latter number is reduced by N, the number of eigenstates belonging to the projection space. Hence, the number of independent real parameters in the unitary transformation which fixes the basis spanning the... 

Of course, there is an infinity of unitary transformations in the space we are dealing with, that satisfies this equation. 

The special class of transformation, known as symmetry (or unitary) transformation, preserves the shape of geometrical objects, and in particular the norm (length) of individual vectors. For this class of transformation the symmetry operation becomes equivalent to a transformation of the coordinate system. Rotation, translation, reflection and inversion are obvious examples of such transformations. If the discussion is restricted to real vector space the transformations are called orthogonal. 

A vector space L defined over a field F is further called an inner-product space or unitary space if its elements satisfy one more condition ... 

A complete unitary space is called a Hilbert space. The unitary spaces of finite dimension are necessarily complete. For reasons of completeness the vector space of all n-tuplets of rational numbers is not a Hilbert space, since it is not complete. For instance, it is possible to define a sequence of rational numbers that approaches the irrational number y/2 as a limit. The set of all rational numbers therefore does not define a Hilbert space. Similar arguments apply to the set of all n-tuplets of rational numbers. 

Integration of the phase density over classical phase space corresponds to finding the trace of the density matrix in quantum mechanics. Transition to a new basis is achieved by unitary transformation... 

The previous argument is valid for all observables, each represented by a characteristic operator X with experimental uncertainty AX. The problem is to identify an elementary cell within the energy shell, to be consistent with the macroscopic operators. This cell would constitute a linear sub-space over the Hilbert space in which all operators commute with the Hamiltonian. In principle each operator may be diagonalized by unitary transformation and only those elements within a narrow range along the diagonal that represents the minimum uncertainties would differ perceptibly from zero. 

The matrix T is clearly stochastic, as YJj=i Tij = 1 due to the unitarity of SG the set of transition matrices related to a unitary matrix as defined in (6) is a subset of the set of all stochastic transition matrices, referred to as the set of unitary-stochastic matrices. The topology of the set in the space of all stochastic matrices is in fact quite complicated, see Pakonski et.al. (2001). In what follows, we will only use that T... 

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

This w -algebra structure can be used to develop a representation theory of symmetry groups, taking H as a representation space for Lie algebras. As before let g be a Lie algebra specified by giOgj = C gu-A unitary representation of g in H is then given by... 

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