Unitary basis

In EucUdean space, orthonormal bases help both to simplify calculations and to prove theorems. Unitary bases, also called complex orthonormal bases, play the same role in complex scalar product spaces. To define a unitary basis for arbitrary (including infinite-dimensional) complex scalar product spaces, we first define spanning. 

Eor example, if we consider C" with the standard complex scalar product, then the set ck k = 1,..., , where Ck denotes the vector whose kth entry is 1 and all of whose other entries are 0, is a unitary basis of V. A more sophisticated example (left to readers in Exercise 3.14) is that the set of functions... 

Proposition 3.2 Suppose V is a finite-dimensional complex scalar product space. Suppose T V —> V is a linear operator. Then T is unitary if and only if the columns of its matrix in any unitary basis form a unitary basis. 

Proof. First suppose T is unitary and suppose that B = bi,..., b is a unitary basis of V. Then the /cth column of the matrix of T in the basis B consists of the coefficients of the vector Tbk in the basis B. In other words,... 

Exercise 3.15 Consider the set B = iei, ie2, ie in where e, 02 and ej, are the standard basis vectors. Show that B is a unitary basis. Show that... 

So when there is one fixed complex scalar product on a vector space V, it is consistent to use the notation v for both dual and adjoint. In a unitary basis, the asterisk means coordinate transpose. 

In group theory the most important cases are those of an orthogonal or unitary basis when M is the 3 x 3 unit matrix, and consequently... 

This situation is of course not satisfactory as observable quantities should be invariant with respect to unitary basis transformations. " Here, we outline the adiabatic route to a basis-invariant formulation of the theory. 

Proteins are major components in dendritic nerve membranes and may exhibit electroactivity—i.e., the characteristic of being switched between two states of differing ionic conductivities. Such electro-activity is interesting because the electricity of the nerve impulse, the unitary basis of information encoding in neural systems, is generated in the dendritic membrane, which is composed of electro-chemically active proteins in a lipid bilayer. Thus, by interacting with neuroscientists in the investigation of neural information code(s), electrochemists may make fundamental contributions to the molecular elucidation of the human brain and the nervous systems of other major animal species (6). 

The transformation of Eq. (16.4) is not really a quasiparticle transformation, since it just reflects a transformation of the underlying orbital space If creates an electron on orbital Xi than creates one on = X/t ikXk- The canonical condition of Eq. (16.5) means that the transformation matrix A is unitary. In Sect. 13 we have also seen non-unitary basis set transformations of the form of Eq. (16.4), for which Eq. (16.5) does not hold, and which do not leave the anticommutation rules invariant. 

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