Variational Wave Equations for Fermions Interacting via Scalar and Vector Fields

  • J. W. Darewych
  • A. Duviryak
Conference paper
Part of the Few-Body Systems book series (FEWBODY, volume 14)


We consider a method for deriving relativistic two-body wave equations for fermions with scalar, pseudoscalar, vector, pseudovector and tensor coupling. The Lagrangian of the theory is reformulated by partially eliminating the mediating field by means of covariant Green’s functions. The reformulated Lagrangian \(\mathcal{L} = \sum\nolimits_{{a = 1}}^{2} {\mathcal{L}_{a}^{{free}} + {{\mathcal{L}}^{{\operatorname{int} }}}}\) contains local free terms
$$\mathcal{L}_{a}^{{free}} = {{\bar{\psi }}_{a}}(x)\{ i{{\gamma }^{\mu }}{{\partial }_{\mu }} - {{m}_{a}}\} {{\psi }_{a}}(x)$$
and nonlocal interaction terms of the type
$${{\mathcal{L}}^{{\operatorname{int} }}} = - \tfrac{1}{2}\int {{{d}^{4}}x\prime J(x)D(x - x\prime )J(x\prime )}$$
in which the mediating-field propagators D(x - x′) are the symmetric Green’s functions, sandwiched between the fermionic particle currents \(J(x) = {{\sum\nolimits_{a} {{{g}_{a}}\bar{\psi }} }_{a}}(x)\Gamma {{\psi }_{a}}(x)\); here m a are masses of fermions, g a are coupling constants, and the matrices Γ depend on the tensor structure of the interaction (1 for scalar, γ5 for pseudoscalar, γμ for vector, γμγ5 for pseudovector and σμν for tensor interactions).


Wave Equation Tensor Structure Canonical Quantization Tensor Interaction Tensor Coupling 
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Copyright information

© Springer-Verlag/Wien 2003

Authors and Affiliations

  • J. W. Darewych
    • 1
  • A. Duviryak
    • 2
  1. 1.York UniversityTorontoCanada
  2. 2.Inst. of Condensed Matter Phys.LvivUkraine

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