A Field Theory of Extended Particles Based on Covariant Harmonic Oscillator Wavefunctions

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## Abstract

We attempt to combine the covariant harmonic oscillator (CHO) quark model with second quantized field theory. We review the CHO formalism for a system of two quarks (meson). We introduce a mesonic field Φ(x1 ,x2) that depends on the position of both quarks, and then derive the field equations from a covariant lagrangian L(x1, x2). The CHO equation allows a complete separation of the average meson coordinate X from the relative quark coordinate ξ. The CHO wavefunction in the field expresses the extended size and internal structure of the meson. Φ, describes mesons in the ground state and any excited state , with angular momentum ∞ mass^2. From Φ we construct conserved tensors like P^μ the meson momentum. We second quantize Φ in the X variable only and discuss the extended particle commutation relations. We investigate a Φ^3-type meson interaction where the vertex function is an overlap integral of the wavefunctions entering the interaction region. We derive a nonlinear integrodifferential equation for the U matrix , linearize and solve it by perturbation theory. The result is simple diagramatic rules for the S matrix. The S matrix is covariant and unitary. We do not find any contradiction between the principles of QFT and the CHO quark model. The Φ field theory includes scalar meson(point particle)theory as a special case, while its greater generality illuminates the difference between point and extended particles.