Symmetry
From Physics wiki
Spacetime symmetries
A field that assigns a value (or tuple of values) to every point in spacetime should transform under a particular representation of the symmetry group of the spacetime. This is a consistency condition. Consider a field on Minkowski space and let us restrict ourselves to the Lorentz group for this spacetime. When we change from one reference frame to another our coordinate system undergoes a Lorentz boost: 
. (We are using matrix notation for conciseness.) If the field contains only one component, i.e. is a scalar field, then we expect the field to transform as a scalar: 
. On the other hand, if the field contains multiple components then can expect mixing between these components (If there is no mixing then we can simply treat the field as a set of independent scalar fields). The field then transforms as:
.
We have used M to denote the matrix that transforms the components of φ, i.e. we are assuming the field transforms linearly. Our consistency condition requires that two consecutive changes of reference frame be indistinguishable from a single combined change:
.
The argument of φ trivially satisfies this requirement, because it is a vector that already transforms under a representation of the Lorentz group. In other words, 
. The non-trivial part is to require that M form a representation of the Lorentz group. In other words:
Since the field transforms under a particular representation of the group, we can classify it according to the representation. The most important non-trivial representations of the Lorentz group are the scalar, vector and spinor representations. Example of fields that transform under these representations are the Klein Gordon field (scalar), the Electromagnetic field (vector) and the Dirac field (spinor).
