Scalar product Hermitian

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... 

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... 

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