i

Thus we get, for each term of this sum, products of single particle operators which are solely functions of the energy (namely operators exp (- z E1)) and of single and two-body operators which are functions of both the energy and posi- 2

tion coordinates (namely the operators exp(- z E)E2 x2 and exp(- z E ̧)E ̧

i

For the first kind of factors which depend only upon the energy we can use the number operator N(p) of section IV.2,

We shall need in fact the more general expression, where n =

0, 1, 2

n

E"

=

2

Sp2dp
p2dp N(p)e-ZEE1

The corresponding invariant matrix elements for spin s = 0, 1 Bosons and spin
S = 1/2 Fermions are given by the expressions (4.5), (4.7) and (4.9) where the

factor p2 in the integrand of Eq. (4.5) is replaced by e

- zEn

'E'

i.e.,

In order to evaluate the mixed energy-coordinate terms we shall first consider the simpler case of the spin O Boson fields. The extension of the calculational method to the other fields will be readily made thereafter.

Let & be a mixed energy coordinate operators, where & is a function of the energy. This mixed operator must be symmetrized since & and x do not necessarily commute. Thus we have to evaluate in terms of the field operators for Bosons

Each of the terms of the right hand expression is then separated into two parts by inserting the unit operator. For example

It differs from the non-relativistic expression by the square root of the energies associated with the orthogonality relation (3.30) or equivalently the com

mutators (3.28). Of course instead of the factor E/E2 one can employ √2/E1,

2

and we use this fact in evaluating (4.30). Thus for example

In the field momentum representation the mixed energy-coordinate invariant matrix element is

After insertion of the unit operator (4.29) we obtain the separated form

f

2 (P2)j2 (P2x)

(4.31)

In this expression the matrix elements of the energy functions & have been given in Eq. (4.26). We are left with the calculation of the matrix elements of the coordinate operator .

Here again as in figu re 4.2 the isospin part separates simply and we are using a schematic graphical representation for the isospin coupling.

For the spin 1 Boson fields with isospin t the position operator for the different multipolarities κ is

Thus the invariant matrix elements are, for the magnetic multipoles