In case of spin- ⁠ 1 / 2 ⁠ particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product that is preserved by this representation. We associate a 4-component Dirac spinor ψ with each particle. These spinors transform under Lorentz transformations according to the law ψ ψ ′ = exp ⁡ ⁡ ( 1 8 ω ω μ μ ν ν [ γ γ μ μ , γ γ ν ν ] ) ψ ψ , where γ ν are gamma matrices, and ω μν is an antisymmetric 4 × 4 matrix parametrizing the transformation. It can be shown that the scalar product ⟨ ⟨ ψ ψ | ϕ ϕ ⟩ ⟩ = ψ ψ ¯ ¯ ϕ ϕ = ψ ψ † † γ γ 0 ϕ ϕ is preserved. It is not, however, positive-definite, so the representation is not unitary.