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We therefore discuss below the basic facts connected with the defect of invariance. , xn ) into itself. , n). We write the basic infinitesimal operators of H in the form ∂ Xα = ξαi (x) i (α = 1, . . , r). H. IBRAGIMOV SELECTED WORKS, VOL. I Consider a manifold N ⊂ E(x) of dimension dimN = n − s, regularly defined [140] by a system of equations ψ σ (x) = 0 (σ = 1, . . , s), (3) and introduce the notation Ta (N ) = ∪ Ta x, x∈N H(N ) = ∪ Ta (N ). Ta ∈H (4) If M ⊂ E(x) is some manifold which contains N and has the property H(M) = M (in particular, H(N ) is such a manifold), then H(N ) ⊂ M.

2 Since µ is arbitrary, equations (28) lead to (24). It follows from equations (24) that ai = b,i (i = 1, . . , n) (28) (29) with a certain function b(x). Now the equivalence transformation (17) with f (x) = 21 b(x) maps our equation to an equivalent equation with ai = 0 (i = 1, . . , n). (30) Therefore, we can assume that equation (8) has the following form: g ij (x)ϕ,ij + c(x)ϕ = 0. (31) This equation is written in the covariant form, and hence it holds in an arbitrary coordinate system. Therefore, it follows from equation (21) and the definition of the function H that c(x) = n−2 R.

G. for the operator S1 . The one-parameter transformation group for this operator has the form (13). Substituting (13) into (1), (2) and using (9) one obtains ∂ + m)ψ (γµ ∂xµ ∂ ∂ + m)ψ + sinh α(γµ µ + m)γ4 γ2 ψ¯ µ ∂x ∂x ∂ ∂ ∂ ∂ = sinh α · γ4 γ2 (γ1 1 − γ2 2 + γ3 3 − γ4 4 + m)ψ¯ ∂x ∂x ∂x ∂x ∂ = sinh α · γ4 γ2 (−γµT µ + m)ψ¯ = 0, ∂x = cosh α(γµ and (γµT ∂ − m)ψ¯ ∂xµ ∂ ∂ − m)ψ¯ − sinh α(γµT µ − m)γ4 γ2 ψ µ ∂x ∂x ∂ = sinh α · γ4 γ2 (γµ µ + m)ψ = 0. ∂x These equations prove the invariance of Dirac’s equations.

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