chapter  16
12 Pages

Massive vector field

To describe a particle with the spin 1 (vector particle), we need a field function having three independent components only. For a free-moving particle of nonzero mass the rest reference frame is always available. Obviously, just in this system the intrinsic symmetry properties of a particle itself are revealed. In this case we should consider symmetry regarding all possible rotations around the center, that is, symmetry regarding the whole group of spherical symmetry. The symmetry properties of a particle relative to this group are characterized by its spin J , determining the number 2J +1 of the wavefunction components transformed through each other. When the four-dimensional vector is used as a wave function, in a rest reference frame the vector particle is associated with three space components of this vector. In order to make certain that this is the case, we introduce the projection operators:

Λ(1)µν (∂) = δ µ ν +

∂µ∂ν m2

, Λ(0)µν (∂) = − ∂µ∂ν m2

, (16.1)

having the properties:

Λ (1)µ ν (∂) + Λ

(0)µ ν (∂) = δµν , Λ

(1)ν σ (∂) = Λ

Λ (0)µ ν (∂)Λ

(0)ν σ (∂) = Λ

(0)µ σ (∂), Λ

(1)ν σ (∂) = 0.