Quantization of the pionic interaction

In theoretical physics, quantization of the pionic interaction is, more precisely, quantization of the interaction constant of the cloud of virtual pions with the nucleus.

The nucleons in the nucleus interact via exchange of virtual particles, which mainly are pions. The motion of the nucleons is not relativistic; so the interaction may be regarded as apotential interaction.

The basic potential there is the Yukawa potential. The attractive forces in this potential are so strong that self energy of the pionic field is infinite, and nucleon is a point particle. If we take into account the w-field inertial forces then the field energy is finite. In this case extended Yukawa potential is

<math>P(R)\sim G^2exp\left(-{w\over R}-k^2R^2\right) <math>

where G, w, k are interaction constants.

The simplest approximation this potential is the oscillator potential

<math>P\sim G^2(1-k^2R^2) <math>

Since around 1935, such potentials have been basic to many nuclear models. But they have had little attention from specialists

The pionic field is a pseudoscalar field. When the formalism of the Clifford algebra <math>L_3 <math> is used, the coordinate vectors <math>e_k <math>are Pauli matrices. The matrix

<math>e_1e_2e_3=i1 <math>

changes sign when a right-oriented coordinate system is replaced by a left-oriented . So <math>i <math> is pseudoscalar <math>L_3 <math> algebra and the pionic potential is pure imaginary:

<math>P=iG^2(1-k^2R^2) <math>

Nuclear forces decrease at infinity. But it is known from the Coulomb nonlinear field that the potential of a field is not necessarily equal to zero on infinity. Because pionic forces are the 'tail' of the strong interaction, the vacuum potential of the basic four vector part of the nuclear forces may be not zero.

Then Schrödinger equation for the radial part of the nuclear wave function, for dimensionless variables, is

<math>\ddot F+{2\over R}\dot F=\left({l(l+1)\over R^2}+ig^2(1-k^2R^2)-b-E\right)F <math>

where <math>b=const <math> is the nucleus vacuum potential, <math>-E <math> the nucleon energy, <math>l <math> the orbital moment.

Its solutions are

<math>F\sim R^l\prod_{1}^N(R-R_n) exp\left({DR^2\over 2}\right) <math>

The equations

<math>\sum_k {{1\over x_n-x_k}+x_k+{l+1\over x_k}} =0<math>

<math>x_n=R_n\sqrt{D} <math>

determine the zeros wave function. The parity of the number <math>N=2m <math> is even.

Other parameter are

<math>D=-(+)gk exp \left(-i{\pi \over 4}\right) <math>

<math>-E=b-ig^2-D(2n+2l+3) <math>

For stable states we must have

<math>ReD<0,ImE=0 <math>

This is possible if

<math>g={k\over \sqrt{2}}(2n+2l+3) <math>

The nucleon energy levels are

<math>-E=b-{k^2\over 2}(4m+2l+3)^2 <math>

Bound levels exist if the interaction energy nucleon with the vacuum is negative.

The differences with standard oscillators models are that energy dependence from space quantum numbers <math>N,l <math> is not linear. It is because pionic interaction constant <math>g <math> became quantized.

This levels may correspond different physical situations. If <math>{k^2\over b} <math> is a small number then they are rotation levels of the whole nucleus.

If this constant depends on the numbers of protons and neutrons in the nucleus then they are levels of the shell model. Counting protons and neutrons in each shell is almost standard and gives the 'magic numbers'.

In this way, taking into account the vacuum potential of the nucleus and the pseudoscalar nature of the pionic field, the simplest nucleus models reveal new features. The main one is the quantization of the pionic interaction constant.