Many-particle relativistic equations for fermions

F. Sanchez-Martin, F. Lopez-Gordo

Jörg Resag, Dieter Schütte

We give a derivation of the particle-hole RPA equations for an interacting multi-fermion system by applying the instantaneous approximation to the amputated two-fermion propagator of the system. In relativistic field theory the same approximation leads from the fermion-antifermion Bethe-Salpeter equation to the Salpeter equation. We show that RPA equations and Salpeter equation are indeed equivalent.

T. S. Chang

The relativistic second quantization of free bosons is extended to fermions. To know how relativistic creation and annihilation operators operate on bra and ket vectors, it is necessary to have a relativistic scalar product which in the case of free particles can be constructed. For particles interacting with each other or capable of emitting and absorbing particles of the same kind, it is pointed out that adequate wave equations for an indefinite number of particles each taking a separate time coordinate have not yet been found and so no scalar product can be found. Thus creation and annihilation operators can be defined only by their operation on a bra vector or a ket vector but not by both. With the help of these, the relativistic wave equations referred to above are proposed and their consistency conditions studied. For the particular case of scalar particles, an illustration is given that such a theory may admit certain expressions as probabilities.

Benjamin Simmen

Jau Tang

We present a physically intuitive and mathematically rigorous framework for deriving the relativistic quantum wave equations, dynamics, and topological structures of bosons and fermions in 1+1-dimensional spacetime. Starting from a simple flip-flop dual-component system that models internal oscillations, we show how first-order linear rate equations naturally give rise to the Klein-Gordon and Dirac equations. For bosons, the system leads to the familiar Klein-Gordon equation, while for spin-½ fermions—augmented by an internal clock degree of freedom—the Dirac equation emerges in 1+1D. The topological distinction between bosons and fermions is revealed through their rotational symmetry: bosons follow a 360° closed loop structure, while fermions are represented by a Möbius band, requiring a 720° rotation to return to their original state. We also introduce two distinct Lorentz transformation structures: hyperbolic (sinh–cosh) for bosons and trigonometric (sin–cos) for fermions. This approach provides a clear, unified, and pedagogical interpretation of relativistic quantum dynamics and internal particle structure.

Egon Marx