Editing Wigner's classification (section)

== Massive scalar fields ==

As an example, let us visualize the irreducible unitary representation with <math>~ m > 0 ~,</math> and <math>~ s = 0~.</math> It corresponds to the space of [[scalar field|massive scalar field]]s.

Let {{mvar|M}} be the hyperboloid sheet defined by:

:<math>~ P_0^2 - P_1^2 - P_2^2 - P_3^2 = m^2 ~, \quad</math> <math>~P_0 > 0~.</math>

The Minkowski metric restricts to a [[Riemannian metric]] on {{mvar|M}}, giving {{mvar|M}} the metric structure of a [[hyperbolic space]], in particular it is the [[hyperboloid model]] of hyperbolic space,  see [[Minkowski space#Geometry|geometry of Minkowski space]] for proof. The Poincare group {{mvar|''P''}} acts on {{mvar|M}} because (forgetting the action of the translation subgroup {{math|ℝ<sup>4</sup>}} with addition inside {{mvar|P}}) it preserves the [[Minkowski inner product]], and an element {{mvar|x}} of the translation subgroup {{math|ℝ<sup>4</sup>}} of the Poincare group acts on
<math>~ L^2(M) ~</math>
by multiplication by suitable phase multipliers
<math>~ \exp \left( -i \vec{p} \cdot \vec{x} \right) ~,</math>
where <math>~ p \in M ~.</math> These two actions can be combined in a clever way using [[induced representations]] to obtain an action
of {{mvar|P}} acting on <math>~ L^2(M) ~,</math> that combines motions of {{mvar|M}} and phase multiplication.

This yields an action of the Poincare group on the space of square-integrable functions defined on the hypersurface {{mvar|M}} in Minkowski space. These may be viewed as measures defined on Minkowski space that are concentrated on the set {{mvar|M}} defined by

:<math>E^2 - P_1^2 - P_2^2 - P_3^2 = m^2~, \quad </math> <math>~E ~\equiv~ P_0 > 0~.</math>

The Fourier transform (in all four variables) of such measures yields positive-energy,{{clarify|reason=What does this mean?|date=October 2016}} finite-energy solutions of the [[Klein–Gordon equation]] defined on Minkowski space, namely

:<math> \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + m^2 \psi = 0,</math>

without physical units. In this way, the <math>~ m > 0, \quad s = 0 ~</math> irreducible representation of the Poincare group is realized by its action on a suitable space of solutions of a linear wave equation.