Apply the linear transformation of the scalar fields to either time- or space-derivative product. Its coefficients become

α γ

multiplying the square of the first derivative,

β δ

multiplying the square of the second, and

α δ + β γ

multiplying their cross product. Invariance for arbitrary fields therefore requires

α γ = 0, β δ = 0, α δ + β γ = 1.

The complete real solution is

(α γ β δ) = (r 0 0 r^{- 1}) or (0 r^{- 1} r 0), r \in R ∖ {0} .

There are four connected components, distinguished by the diagonal/antidiagonal choice and the sign of

r

. Only the positive diagonal family is in the identity component; write

r = e^{s}

to obtain

ϕ_{1} \mapsto e^{s} ϕ_{1}

ϕ_{2} \mapsto e^{- s} ϕ_{2}

. This is the reciprocal rescaling symmetry of two cross-coupled scalar fields. The antidiagonal transformations exchange the fields as well as rescale them.

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