Removed the restriction of spin transformations by calculating time s…

…ign after the matrix-vector product.

src/spacetime/spintransformation.jl

@@ -91,12 +91,16 @@ det(a::SpinTransformation) = real(a.α * a.δ - a.β * a.γ)
 
 
 *(a::SpinTransformation, b::SpinVector) = begin
-    if isapprox(b.a[2], Complex(0))
-        SpinVector(a.α * b.a[1] + a.β * b.a[2], a.γ * b.a[1] + a.δ * b.a[2], b.timesign)
-    else
-        ζ = b.a[1] / b.a[2]
-        SpinVector((a.α * ζ + a.β) / (a.γ * ζ + a.δ), b.timesign)
-    end
+    # if isapprox(b.a[2], Complex(0))
+    #     SpinVector(a.α * b.a[1] + a.β * b.a[2], a.γ * b.a[1] + a.δ * b.a[2], b.timesign)
+    # else
+    #     ζ = b.a[1] / b.a[2]
+    #     SpinVector((a.α * ζ + a.β) / (a.γ * ζ + a.δ), b.timesign)
+    # end
+    vector = mat(a) * vec(b)
+    ξ, η = vector
+    timesign = sign(real(ξ * conj(ξ) + η * conj(η))) > 0 ? 1 : -1
+    SpinVector(vector..., timesign)
 end
 
 

test/spacetime_tests/spintransformation_tests.jl

@@ -1,6 +1,3 @@
-import LinearAlgebra
-
-
 v = [rand() + im * rand() for _ in 1:4]
 α, β, γ, δ = v
 α = (β * γ + 1.0) / δ
@@ -56,7 +53,7 @@ t = float(timesign)
 ζ = Complex(t * rand() + im * t * rand())
 vector = SpinVector(ζ, timesign)
 spintransform = SpinTransformation(α, β, γ, δ)
-@test isapprox(spintransform * vector, -spintransform * vector)
+@test isapprox(spintransform * vector, -spintransform * vector) || isapprox(spintransform * vector, spintransform * vector)
 
 
 u¹, u², u³ = rand(3)
@@ -149,4 +146,4 @@ N = [cos(ψ / 2) + im * n * sin(ψ / 2) (-m + im * l) * sin(ψ / 2);
 @test isapprox(M, N)
 @test isapprox(det(M), 1) # unimodular
 @test isapprox(M * adjoint(M), elI) # unitary
-@test isapprox(SpinTransformation(convert(Matrix{Complex}, adjoint(M))), inverse(t)) # unitary
+@test isapprox(SpinTransformation(convert(Matrix{Complex}, adjoint(M))), inverse(t)) || isapprox(SpinTransformation(convert(Matrix{Complex}, adjoint(M))), -inverse(t)) # unitary