Since we are aiming at a formulation of the secular problem which will lead to the diagonalization of finite matrices, we must now re-express Eq. (1.30) in terms of discretized basis states. Before doing it let us note that discretization of relativistic fields is somewhat different from the non-relativistic case. Namely in non-relativistic theories one can use a fictitious potential to generate a complete set of eigenstates with discrete energy eigenvalues. The solution of the true Hamiltonian then can be expanded in terms of these discrete states. In a relativistic theory one cannot use the non-relativistic concept of potential. Consequently one has to start from the free field solutions themselves for basis states, as it has been done above. The way to achieve a discrete secular problem is then to construct wave packets over the energy. This will be done in detail in chapter III. by applying a unitary transformation on the field operators, where the energy functions f (c) with discrete index κ form an orthonormal set A convenient choice is for example harmonic oscillator functions. Inserting the discretized fields (1.35) into the definitions of basis state vectors, section 1.3.2., by means of the inverse transformations we shall obtain discretized basis configurational states r). The corresponding x(r) for the solution amplitudes n discretized system (S) Y (t) are now given by the completely n [ {<r │H。 € \r') + <r\V\r') } x (x') n = 0 (1.37) In words, the time independent description of quantum field theory leads to a secular equation which is formally indistinguishable from that of the classical theory. 1.3.5 - Nature of the solutions The solution n (t)) contains an undefined number of hadrons, i.e. it is expanded on configurational states with different numbers of Bosons. Unless the employed truncation energy is above the baryon - antibaryon threshold, the number of baryons of course is identical in all components, but their intrinsic quantum numbers may be different. The meaning of a stationary solution is then one in which the relative admixtures of parton components do not change in time. A pictorial representation of the effect of the creation and annihilation operators in time is given in figure 1.1. There a system is described as a mixture of a single Fermion f with different numbers of Bosons b. Thus at time t the state (S) vector (t)) is a mixture of the components f, fb, fbb ... with amplitudes (r) X (r= 1,2,3.. respectively). After the infinitesimal time At, several processes have taken place due to the time evolution Hamiltonian, where a single Bos on has been created or absorbed. These processes are associated with the field interaction matrix elements (r' |V|r). At time t + At the configurational mixture must, however, remain unchanged in order for the state vector to be stationary. We have n n which is the relation between the amplitudes given by Eq. (1.37). Assuming for simplicity H to be diagonal on the representation r r = 1 may via the free field Hamiltonian H with the Graphically (see fig.1.1) when going from t to t + At n the component Ο via an interaction V where a Boson is created, amplitude 61 x(1) or (amplitude (2 V 1) x(1). Likewise the component r = 2 may either go to r = 1 via the absorption of a meson (amplitude (1 |V|2) x(2) into n n r = 2 via H or = 3 via the creation of a Boson (amplitude (3 |V|2) x(2) etc. n O into r for the composite state to be stationary all the amplitude flows (indicated by the arrows) must be such that they leave all the ratios (r)x(r unchanged. n n This is precisely the meaning of the linear relations (1.37) or (1..40). There are fundamental differences between the present approximation scheme, based on the hierarchy of the basis configurational states in the expansion of Eq. (1.29) with a cut-off energy or truncation of the Hilbert space, and the usual time-dependent perturbation treatment generally carried out in the interaction picture: i) The energy matrix of the secular problem requires only the calculation of the interaction operator V at a single time point, between of course many-body basis state vectors. The proper statistics between the particles is readily achieved with well known techniques. This is in contrast to the ill-defined nature of the chronological products of interaction operators at contact points, see for example ref. [7] page 220, and to the rapidly increasing difficulty in calculating irreducible diagrams of increasing order. ii) Upon diagonalization of the energy matrix all processes which can exist in the chosen configuration space are automatically generated and treated to all orders, with only the restriction of the cut-off energy. In a timedependent treatment only selected irreducible graphs are calculated. iii) The solutions of the Schrödinger equation are of course unitary in the truncated space. iv) The divergences of the strong interaction theories which in the timedependent treatments show up when integrating over the four-momenta of the intermediate virtual states, appear here as a divergent dependence of the solutions upon the cut-off energy of the truncated space. As usual when working in a truncated Hilbert space, the solutions of the secular problem of Eq. (1.37) are each a mixture of states with different center of mass motion. In order to obtain solutions with a well-defined C.M. motion, viz. Os, a standard procedure of non-relativistic quantum mechanics consists in adding an artificial C.M. energy operator to the Hamiltonian *C.M. - // 5 (B2 + n2 R2) (1.41) where and are the center of mass momentum and position operators. This C.M. Hamiltonian is used without ascribing a physical meaning to it but as a device to split apart the solutions of H+CM into groups with each a given C.M. motion. This splitting increases with increasing values of the parameter §. Non-relativistically this procedure is exact besides the difficulties associated with truncation Note that a large value of does not imply a high velocity of the C.M. motion. The parameter only changes the scale of the C.M. level spacing. It is the frequency or size parameter n which determines the velocity of the C.M. motion. We shall use this method in the present work. Relativistically the and the evaluation of the matrix elements of 2 on the relativistic basis state vectors presents no particular difficulty. However in relativistic kinematics [8] where the ε s are the energies of the partons making up the configuration i (1.44) (1.45) (1.46) in the C.M. Hamiltonian (1.41) is in fact not only a sum of many-body operators 2 because of the denominators ( )2 Σ εκ k but is non-separable in the individual energy variables. This typically relativistic difficulty can be circumvented by the use of the transformation which factorizes, after substitution, each term of (1.47) into products of oneor two-body operators. Thus (z = + = 2) Ζε Ex) ( 2 z ↑ e e i#j k#i,j - Z The invariant matrix elements of the pseudo-C.M. Hamiltonian (1.41) can thus be evaluated simply as shown in detail in Chapter IV. |