"asymptotic" region, the system must break apart into a pion and a nucleon.

Here

We now discuss the form of the solution at this asymptotic separation. It

can be constructed as follows.

Consider the states Y and Y of the nucleon
N

π

and m-meson, boosted to P and -P respectively. According to (B.8) the boosted intrinsic parts of the configurations are given by

Here and below p and r are the relative momentum and coordinate, respectively.

->

They are obtained by putting in (A.24) through (A.31) 1 = IN = nucleon CM

r = pion CM position, etc., and taking = = 0 since here

we are considering the scattering state in the intrinsic, i.e., the CM system.

Finally, the

are measured from r and the

i

= FN

from. The relative motion is provided by the expansion of a plane wave into multipoles in which in the radial

part j1(pr) is replaced by

δ

are the phase shifts for a given 1,j scattering state. A particular λ, j 入手

system then can be extracted by angular projection.

Thus, a general intrinsic scattering state (the CM wave function is still absent) is of the form

and thus the intrinsic configurational scattering state of a given multipolarity is of the form

means of (A.3), (A.32), (B. 3) by the parton laboratory coordinates, equation (C.7) is already almost in the form (B.2). To achieve that form one must rewrite the multipoles making up the nucleon system and the pion system: they are still written about the points r, for the nucleon and for the pion. They can be

->

N

-> π

[17]

rewritten about the center of mass point by means of the translation operator. The state which one obtains in the diagonalization of the Hamiltonian, say Y n' thus is (implicitly) identical to the state, say Y which one obtains from the rewritten equation (C.7) when multiplying each configuration (C.7) with the CM functions

S'

(^)^

(R), multiplying with the amplitudes X

νμ'

N

and X
x(H) (cf. Eq. 1.37)

ᅲ

of the solutions for the nucleon and pion, respectively, and summing over the

n

n

(C.9)

(c.10)

and computing the overlaps of (()) and (YY)). The ratio of these β two overlaps equals -cotổ, 1/2,

λ =

1.)

; cf.(C.3). (Of course, in our example j =

We recall here that this is true only for non-relativistic CM motion.

The relative motion between the scattering particles, however, is allowed to be relativistic. Note, that for non-relativistic relative mot ion no Lorentz contraction takes place and the sum over k, & in Eq. (C.7) collapses into a single

е

term (k = 1/2, = 0 for the considered nucleon-pion system).

n

We shall not discuss here the analysis of Y in the case of several open channels since no new problems arise in that case. Finally, of course,

the

scattering states are correct as far as the CM motion is concerned. No recoil

or relativistic corrections have to be made.

The lowest eigenstate, say |V), obtained in diagonalizing the pseudoHamiltonian for a system having the quantum numbers of the vacuum is the (not manifestly covariant) physical vacuum state of the model field theory. energies have to be measured from the expectation value of the Hamiltonian for

All

The vacuum state arises as a consequence of the matrix elements of the kind

of Fig. 5.7(a) and its time reversed form, or of the similar matrix elements of Fig. 5.13, i.e., of those matrix elements which connect the "ground configuration" 10) with a configuration containing partons. (These matrix elements thus are proportional to the P = 0 component of the part on configuration.)

This situation

is familiar in the non-relativistic shell model in which the ground state of a many-body system is not equal to the ground configuration.

Owing to the finite size of the model Hilbert space it is always possible to construct a unitary operator which allows to represent the eigenstates on the basis of the physical vacuum state, V), instead of on the basis of the ground configuration |0). We now list the principal characteristics of IV) :

This defines implicitly the physical state vector o*. Note that an explicit

definition is not easy since only (0|[v,v*]_|0) = 1 while [v,v] is not a

+

c-number. At any rate, the physical state vector o shows which parton configurations in excess of the physical vacuum make up the physical particle. (Note, that in non-relativistic physics the definition of an operator anologous to (D.7) is possible, and, in fact, particularly in nuclear physics might be an interesting quantity to investigate.)

П

In the model of Chapter VI, |V) will have the configurations: |0), |π2), |π2), |πo), |πw), 2p). At higher truncation energy more configurations will participate. If a neutral scalar meson field (o-model) is included, |V) would contain likewise terms of the form o), lo2), lo3), etc. In writing out the configuration mixture of |V) one must supply a particular creation operator,

to generate the amplitude of the ground configuration |0) in |V), in order

Equation (D. 8) above has been written symbolically, omitting the integrations over momenta and summations over spin and isospin, etc.

In time-dependent perturbation theory the structure of the physical vacuum can be eliminated from explicit treatment by omitting the disconnected graphs and by ignoring the Pauli principle within the diagrams, and by dividing the S-matrix elements by the vacuum expectation value of the S-matrix.

Ref. 5, Chapter 7.2.)

(See, e.g.,