# Dynamics V: Construction of local covariant ffields

# Dynamics V: Construction of local covariant ffields

In many of the standard texts on quantum field theory, the introduction of fields representing particles of low spin (zero, ½, or one — there is no direct phenomenological evidence for elementary particles of any higher spin) is a fairly *ad hoc* matter. Relativistic wave equations are introduced here and shown to have ‘nice’ covariance properties. A Lagrangian formalism is then constructed for which these equations are just the Euler–Lagrange equations of the theory, corresponding to the extremal condition on the classical action. Finally, a canonical quantization procedure is carried out: conjugate momentum fields are introduced, and the resultant Hamiltonian is shown to be the appropriate energy operator for particles of the desired mass and spin. This chapter eschews this *ad hoc* methodology in favour of a more direct, constructive approach. The relativistic wave equations satisfied by the covariant fields representing particles of low spin are shown to be automatic consequences of the representation theory of the Poincaré group, which can be used to write a completely general expression for the fields transforming according to an arbitrary finite-dimensional representation of the Lorentz group and representing particles of arbitrary mass and spin.

*Keywords:*
quantum field theory, low spin particles, Hamiltoninans, relativistic wave equations, covariatnt fields, Poincaré group, Lorentz group

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