the space of field configurations (a field configuration is the spacetime history of a particular solution, e.g. the electromagnetic field in all of space over all time is one field configuration) resembles the action on the Hilbert spaces of quantum mechanics, except that the commutator brackets are replaced by field theoretical Poisson brackets. Relativistic quantum mechanics For the present purposes the following definition is made: A relativistic wave function is a set of functions on spacetime which transforms under an arbitrary proper Lorentz transformation as where is an -dimensional matrix representative of belonging to some direct sum of the representations to be introduced below. The most useful relativistic quantum mechanics one-particle theories (there are no fully consistent such theories) are the Klein–Gordon equation and the Dirac equation in their original setting. They are relativistically invariant and their solutions transform under the Lorentz group as Lorentz scalars () and bispinors respectively (). The electromagnetic field is a relativistic wave function according to this definition, transforming under . The infinite-dimensional representations may be used in the analysis of scattering. Quantum field theory In quantum field theory, the demand for relativistic invariance enters, among other ways in that the S-matrix necessarily must be Poincaré invariant. This has the implication that there is one or more infinite-dimensional representation of the Lorentz group acting on Fock space. One way to guarantee the existence of such representations is the existence of a Lagrangian description (with modest requirements imposed, see the reference) of the system using the canonical formalism, from which a realization of the generators of the Lorentz group may be deduced. The transformations of field operators illustrate the complementary role played by the finite-dimensional representations of the Lorentz group and the infinite-dimensional unitary representations of the Poincare group, witnessing the deep unity between mathematics and physics. For illustration, consider the definition an -component field operator: A relativistic field operator is a set of operator valued functions on spacetime which transforms under proper Poincaré transformations according to Here is the unitary operator representing on the Hilbert space on which is defined and is an -dimensional representation of the Lorentz group. The transformation rule is the second Wightman axiom of quantum field theory. By considerations of differential constraints that the field operator must be subjected to in order to describe a single particle with definite mass and spin (or helicity), it is deduced that where are interpreted