The Hermitian Scalar Product

Equations (25) and (26) show that the rows or the columns of a unitary matrix are orthonormal when the scalar product is defined to be the Hermitian scalar product. 

We therefore recover the result that the Hermitian scalar product is invariant under a unitary transformation. 

It has to be remembered that the ijth element of is the jtth element of U. Equation 2.88 is the Hermitian scalar product of orthonormal sets of vectors. 

The Hermitian scalar products of two different irreducible representations are also of fundamental interest. To deduce these the same procedure is employed but we now start with the matrix. 

The concept of the Hermitian scalar product of vectors as outlined in Chapter 2 is readily extended to functions of many variables. Suppose a function / contains n variables xi, Xj,. . . , x , and that each variable can take an infinite number of values within a particular interval. In short, / is a continuous function of n variables. The variables xi, x, . . . , Xn therefore define an n-fold infinite-dimensional space, and the function /(xi, X2,, x ) is a vector in that space. If g xi, x, ... 

The introduction of the Hermitian scalar product into representation theory is quite analogous to the quantum mechanical state product which is associated with numerical values in the physical theory. The quantity Pit Pi) always real and is the squared modulus of the length of a vector. It is worth noting that the Hermitian scalar ( i, i) is independent of the basis vectors in the space. Because of the relation shown in Eq. 2.67, spaces in which a Hermitian scalar product is defined are known as unitary spaces. The space defined in Equation 5.5 is the space of square-integrable functions. 

The orthonormality of functions is again analogous to the vector systems previously encountered. Usually a set of functions i, 2,. . , tpj are said to span a -fold vector space when any function in the space can be described as a linear combination of these j-functions. Two functions are orthogonal in the interval a— b when the Hermitian scalar product is zero thus 0, and 0, are orthogonal if... 

This quantity, for which we employ the bra-ket ((, )) notation due to Dirac (1958), is called the Hermitian scalar product of the two functions 0, and orthonormal functions it is then easy to show that the best fit of the function / results when... 

A Hermitian operator p is a von Neumann density if it is nonnegative and has unit trace. In more concrete terms, if is the finite-dimensional Fock space for a quantum model where electrons are distributed over a finite number of states, then p is a von Neumann density if (i) v,pv) > 0 for all operators v on and (ii) l)g = 1. By the formula v,pv) we mean the trace scalar product of the operators v and pv, that is, v,pv) = traceg(u pu) since (p, l)g = tracegp = 1 we have used this scalar product to express the trace condition. More generally. 

Definition 15 A -body operator is a Hermitian operator that can be represented as a polynomial of degree 2 A in the annihilation and creation operators, and is of even degree in these operators. In addition, a A -body operator must be orthogonal to all k — l)-body operators, all k — 2)-body operators,. .., and all scalar operators, with respect to the trace scalar product. 

Recently, a unitarily invariant decomposition of Hermitian second-order matrices of arbitrary symmetry under permutation of the indices within the row or column subsets of indices has been reported by Alcoba [77]. This decomposition, which generalizes that of Coleman, also presents three components that are mutually orthogonal with respect to the trace scalar product [77] ... 

The complex scalar product lets us dehne an analog of Euclidean orthogonal projections. First we need to dehne Hermitian operators. These are analogous to symmetric operators on R". 

If we apply the same analysis to the so(2, 1) generator T2, Eq. (75b), we find that it is not Hermitian with respect to the usual scalar product unless w = 2. However, if we define a new scalar product by... 

In a similar manner we can show that the generators T, and T3 are Hermitian with respect to the same scalar product, Eq. (79). Only the generator T3 has square integrable eigenfunctions and this is the reason why we chose to diagonalize T2 and T3 in our study of the representation theory of so(2, 1) (the notation J2, J3 was used in Section III). 

This is the usual textbook form (see, e.g., Powell and Craseman, 1961). The only difference between Eqs. (128) and (129) is a change in units, since in case w = 2 the so(2, 1) generators are Hermitian with respect to the usual scalar product [cf. Eq. (79)]. On the other hand, for the hydrogen atom the scaling parameter (i.e., X = n/Z) depends on the principal quantum number. 

Most of the properties so far described remain valid, as the formal scalar product reduces to standard integration on a distribution function in the Hermitian case. For instance, the theorem of this section remains valid, dropping the restriction that a and b are real. Of course, the peculiar properties of the zeros of orthogonal polynomials remain valid only for real polynomials. From now on we consider explicitly only the latter situation. 

Since the eigenfunction n) is not in the Hermitian domain of the Hamiltonian the definition of the inner product that we should use should be questioned. If we will keep the usual definition of the scalar product in quantum mechanics the coefficients an in Eq. 33 will get real positive values only (as well as a(e) in Eq. 32) and the possibility of interference among different resonance states which leads to the trapping of an electron due to the molecular vibrations will be eliminated. As was mentioned before the generalized definition of the inner product (.... ..) rather than the usual scalar product has to be used since the Hamiltonian is... 

When U in (10.2.4) is a general non-singular mxm matrix the infinite set U forms a matrix group, the full linear group in m dimensions, denoted by GL(m). If the matrices are chosen to be unitary (thus leaving invariant any Hermitian scalar product, as we know from Section 2.2) then we obtain the unitary group U(m) and in this case the matrices of the covariant transformation in (10.2.4b) are... 

One easily shows by differentiation with respect to t and using the hermitian property of H that this scalar product is independent of time, if and < ( )> are both solutions of (9-40). The probability... 

The operator H must be hermitian within this scalar product. One verifies that for this to be the case a and j8 must be hermitian matrices... 

In counting the number of orthonormalization conditions on C, CGM apparently did not assume the hermiticity of the scalar product in the subspace, but rather chose to impose it. Their calculation of K ran along the following lines a complex projector, which is hermitian and normalized, may be factored into [13]... 

More fundamentally, what Pecora seems to assume - although never explicitly saying so - is the following property. Since the condition CC+ = In is actually the orthonormalization constraint on the scalar product between any two wavefunctions (ft is hermitian. That is to say, it is assumed that the subspace on which the projection is made is a Hilbert subspace. 

However, if one were to exactly follow what seem to be Pecora s assumptions about the scalar product being hermitian, one would get a different result from Pecora when counting the number of real conditions on the complex P matrix, arising from the constraint + = In In fact, when the + matrix is considered to be hermitian, the normalization condition on the N complex diagonal elements of QQ+ yields N real conditions and not 2N as Pecora seemed to tacitly suppose. This is due to the fact that the diagonal elements are already known to be real since + is hermitian, and hence, Im = 0 is not a separate constraint. 

The Hermitian conjugate c (dagger) of a column vector c, is a row vector, with the components c. The scalar product of the row vector w and a column vector, v is... 

Because all of the components of J are Hermitian, and because the scalar product of any function with itself is positive semi-definite, the following identity holds ... 

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

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