Fermionic scalar field

Understand

We reexamine the connection between spin and statistics through the quantization of a complex scalar field, using the formulation with the property that the hermitian conjugate of canonical momentum for a variable is just the canonical momentum for the hermitian conjugate of the variable.

• Starting from an ordinary Lagrangian density and imposing the anti-commutation relations on the field, we find that the difficulty stems from not the ill-definiteness (or unboundedness) of the energy and the breakdown of the causality but the appearance of states with negative norms.
• It is overcome by introducing an ordinary scalar field to form a doublet of fermionic symmetries, although the system becomes empty leaving the vacuum state alone.
• These features also hold for the system with a spinor field imposing the commutation relations on.