Defined a transform function for determining the time sign of spin-ve…

…ctors after multiplication.

src/spacetime/fig113diracsscissors.jl

@@ -31,7 +31,7 @@ sphereobservable = buildsurface(lscene, spherematrix, mask, transparency = true)
 planematrix = makestereographicprojectionplane(M, T = T, segments = segments)
 planeobservable = buildsurface(lscene, planematrix, mask, transparency = true)
 ϵ = 0.1
-transformation = SpinTransformation(ϵ + rand() * 0.1, ϵ + rand() * 0.1, ϵ + rand() * 0.1)
+transformation = SpinTransformation(rand() * ϵ, rand() * ϵ, rand() * ϵ)
 
 generate() = 2rand() - 1 + im * (2rand() - 1)
 κ = SpinVector(generate(), generate(), Int(T))
@@ -99,9 +99,9 @@ GLMakie.text!(lscene,
     markerspace = :data
 )
 
-κflagplanematrix = makeflagplane(κv, κ′v - κv, segments = segments)
+κflagplanematrix = makeflagplane(κv, κ′v - κv, T, segments = segments)
 κflagplanecolor = GLMakie.Observable(fill(GLMakie.RGBAf(0.5, 0.5, 0.5, 0.5), segments, segments))
-κflagplaneobservable = buildsurface(lscene, κflagplanematrix, κflagplanecolor, transparency = false)
+κflagplaneobservable = buildsurface(lscene, κflagplanematrix, κflagplanecolor, transparency = true)
 
 κsectional = GLMakie.Observable(GLMakie.Point3f(projectnocompression(normalize(ℍ(vec(κv))))))
 κ′sectional = GLMakie.Observable(GLMakie.Point3f(projectnocompression(normalize(ℍ(vec(κ′v))))))
@@ -142,16 +142,29 @@ animate(frame::Int) = begin
     ϕ = 0.0
     ψ = 0.0
     spintransform = SpinTransformation(θ, ϕ, ψ)
+    transform(κ, spintransform) = begin
+        vector = mat(spintransform) * vec(κ)
+        timesign = κ.timesign
+        result = SpinVector(convert(Vector{Complex}, vector)..., timesign)
+        if isapprox(result, -κ)
+            timesign = -κ.timesign
+            result = SpinVector(convert(Vector{Complex}, vector)..., timesign)
+        end
+        return result
+    end
+    κtransformed = 𝕍(transform(κ, spintransform))
+    κ′transformed = 𝕍(transform(κ′, spintransform))
+    κ″transformed = 𝕍(transform(κ″, spintransform))
+    T = Float64(transform(κ, spintransform).timesign)
+    println("T: $T")
+    northpole[] = GLMakie.Point3f(ℝ³(0.0, 0.0, T))
     spherematrix = makesphere(spintransform, T, segments = segments)
-    planematrix = makestereographicprojectionplane(spintransform, T = 1.0, segments = segments)
+    planematrix = makestereographicprojectionplane(spintransform, T = T, segments = segments)
     updatesurface!(planematrix, planeobservable)
     updatesurface!(spherematrix, sphereobservable)
-    κtransformed = 𝕍(spintransform * κ)
-    κ′transformed = 𝕍(spintransform * κ′)
-    κ″transformed = 𝕍(spintransform * κ″)
-    κflagplanematrix = makeflagplane(κtransformed, 𝕍(LinearAlgebra.normalize(vec(κ′transformed - κtransformed))), segments = segments)
+    κflagplanematrix = makeflagplane(κtransformed, 𝕍(LinearAlgebra.normalize(vec(κ′transformed - κtransformed))), T, segments = segments)
     updatesurface!(κflagplanematrix, κflagplaneobservable)
-    κflagplanecolor[] = [GLMakie.RGBAf(convert_hsvtorgb([360.0 * progress; 1.0; 1.0])..., 1.0) for i in 1:segments, j in 1:segments]
+    κflagplanecolor[] = [GLMakie.RGBAf(convert_hsvtorgb([360.0 * progress; 1.0; 1.0])..., 0.8) for i in 1:segments, j in 1:segments]
     κobservable[] = GLMakie.Point3f(projectnocompression(normalize(ℍ(vec(κtransformed)))))
     κ′observable[] = GLMakie.Point3f(projectnocompression(normalize(ℍ(vec(κ′transformed)))))
     κ″observable[] = GLMakie.Point3f(projectnocompression(normalize(ℍ(vec(κ″transformed)))))
@@ -161,10 +174,10 @@ animate(frame::Int) = begin
     κsectional[] = (κobservable[] + κprojectionobservable[]) * 0.5
     κ′sectional[] = (κ′observable[] + κ′projectionobservable[]) * 0.5
     κ″sectional[] = (κ″observable[] + κ″projectionobservable[]) * 0.5
-    for (i, scale1) in enumerate(collect(range(0.0, stop = 1.0, length = segments)))
+    for (i, scale1) in enumerate(collect(range(0.0, stop = T, length = segments)))
         _κlinepoints = GLMakie.Point3f[]
         _κlinecolors = Int[]
-        for (j, scale2) in enumerate(collect(range(0.0, stop = 1.0, length = segments)))
+        for (j, scale2) in enumerate(collect(range(0.0, stop = T, length = segments)))
             κvector = normalize(ℍ(vec(scale1 * κtransformed + scale2 * 𝕍(LinearAlgebra.normalize(vec(κ′transformed - κtransformed))))))
             κpoint = GLMakie.Point3f(projectnocompression(κvector))
             push!(_κlinepoints, κpoint)

src/spacetime/spintransformation.jl

@@ -91,16 +91,12 @@ 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
-    vector = mat(a) * vec(b)
-    ξ, η = vector
-    timesign = sign(real(ξ * conj(ξ) + η * conj(η))) > 0 ? 1 : -1
-    SpinVector(vector..., timesign)
+    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
 end
 
 

src/spacetime/spinvector.jl

@@ -110,6 +110,12 @@ end
     t = float(κ.timesign)
     v = ℝ³(κ)
     𝕍(ℝ⁴(t, vec(t * v)...))
+    # ξ, η = vec(κ)
+    # T = (1 / √2) * (ξ * conj(ξ) + η * conj(η))
+    # X = (1 / √2) * (ξ * conj(η) + η * conj(ξ))
+    # Y = (1 / (im * √2)) * (ξ * conj(η) - η * conj(ξ))
+    # Z = (1 / √2) * (ξ * conj(ξ) - η * conj(η))
+    # 𝕍(normalize(ℝ⁴(real.([T; X; Y; Z]))))
 end
 
 

src/utilities.jl

@@ -245,13 +245,13 @@ end
 
 
 """
-    makeflagplane(u, v)
+    makeflagplane(u, v, T)
 
-Make a half plane with the given 4-vectors `u` and `v`.
+Make a half plane with the given 4-vectors `u`, `v` and temporal section `T`.
 """
-function makeflagplane(u::𝕍, v::𝕍; segments::Int = 60)
-    lspace1 = range(-1.0, stop = 1.0, length = segments)
-    lspace2 = range(0.0, stop = 1.0, length = segments)
+function makeflagplane(u::𝕍, v::𝕍, T::Float64; segments::Int = 60)
+    lspace1 = range(-1.0, stop = T, length = segments)
+    lspace2 = range(0.0, stop = T, length = segments)
     matrix = [f * u + s * v for f in lspace1, s in lspace2]
     map(x -> projectnocompression(normalize(ℍ(vec(x)))), matrix)
 end