Chapter 44

Symmetries and Relativistic

Scalar Quantum Fields

Just as for non-relativistic quantum fields, the theory of free relativistic scalar
quantum fields starts by taking as phase space an infinite dimensional space of
solutions of an equation of motion. Quantization of this phase space proceeds by
constructing field operators which provide a representation of the corresponding
Heisenberg Lie algebra, using an infinite dimensional version of the Bargmann-
Fock construction. In both cases the equation of motion has a representation-
theoretical significance: it is an eigenvalue equation for the Casimir operator
of a group of space-time symmetries, picking out an irreducible representation
of that group. In the non-relativistic case, the Laplacian ∆ was the Casimir
operator, the symmetry group was the Euclidean groupE(3) and one got an
irreducible representation for fixed energy. In the relativistic case the Casimir
operator is the Minkowski space version of the Laplacian

−

∂^2

∂t^2

+ ∆

the space-time symmetry group is the Poincar ́e group, and the eigenvalue of the
Casimir ism^2.
The Poincar ́e group acts on the phase space of solutions to the Klein-Gordon
equation, preserving the Poisson bracket. The same general methods as in
the finite dimensional and non-relativistic quantum field theory cases can be
used to get a representation of the Poincar ́e group by intertwining operators
for the Heisenberg Lie algebra representation (the representation given by the
field operators). These methods give a representation of the Lie algebra of the
Poincar ́e group in terms of quadratic combinations of the field operators.
We’ll begin though with the case of an even simpler group action on the
phase space, that coming from an “internal symmetry” of multi-component
scalar fields, with an orthogonal group or unitary group acting on the real or
complex vector space in which the classical fields take their values. For the