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80 lines
2.3 KiB
Julia
80 lines
2.3 KiB
Julia
###
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### "THE BEER-WARE LICENSE":
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### Alberto Ramos wrote this file. As long as you retain this
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### notice you can do whatever you want with this stuff. If we meet some
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### day, and you think this stuff is worth it, you can buy me a beer in
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### return. <alberto.ramos@cern.ch>
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###
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### file: GroupSU2.jl
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### created: Sun Jul 11 17:23:12 2021
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###
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#
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# SU(2) group elements represented trough Cayley-Dickson
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# construction
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# https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction
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using CUDA, Random
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SU2(a::T, b::T) where T <: AbstractFloat = SU2{T}(complex(a), complex(b))
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"""
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inverse(g::T) where T <: Group
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Returns the group inverse of `g`.
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"""
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inverse(b::SU2{T}) where T <: AbstractFloat = SU2{T}(conj(b.t1), -b.t2)
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"""
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dag(g::T) where T <: Group
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Returns the group inverse of `g`.
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"""
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dag(a::SU2{T}) where T <: AbstractFloat = inverse(a)
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norm(a::SU2{T}) where T <: AbstractFloat = sqrt(abs2(a.t1) + abs2(a.t2))
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norm2(a::SU2{T}) where T <: AbstractFloat = abs2(a.t1) + abs2(a.t2)
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"""
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tr(g::T) where T <: Group
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Returns the trace of the group element `g`.
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"""
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tr(g::SU2{T}) where T <: AbstractFloat = complex(2*real(g.t1), 0.0)
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"""
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dev_one(T) where T <: Group
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Returns the distance to the unit group element
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"""
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dev_one(g::SU2{T}) where T <: AbstractFloat = sqrt(( abs2(g.t1 - one(T)) + abs2(g.t2))/2)
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"""
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unitarize(a::T) where {T <: Group}
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Return a unitarized element of the group.
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"""
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function unitarize(a::SU2{T}) where T <: AbstractFloat
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dr = sqrt(abs2(a.t1) + abs2(a.t2))
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if (dr == 0.0)
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return SU2{T}(0.0,0.0)
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end
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return SU2{T}(a.t1/dr,a.t2/dr)
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end
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Base.:*(a::SU2{T},b::SU2{T}) where T <: AbstractFloat = SU2{T}(a.t1*b.t1-a.t2*conj(b.t2),a.t1*b.t2+a.t2*conj(b.t1))
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Base.:/(a::SU2{T},b::SU2{T}) where T <: AbstractFloat = SU2{T}(a.t1*conj(b.t1)+a.t2*conj(b.t2),-a.t1*b.t2+a.t2*b.t1)
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Base.:\(a::SU2{T},b::SU2{T}) where T <: AbstractFloat = SU2{T}(conj(a.t1)*b.t1+a.t2*conj(b.t2),conj(a.t1)*b.t2-a.t2*conj(b.t1))
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"""
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isgroup(g::T) where T <: Group
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Returns `true` if `g` is a group element, `false` otherwise.
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"""
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function isgroup(a::SU2{T}) where T <: AbstractFloat
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tol = 1.0E-10
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if (abs2(a.t1) + abs2(a.t2) - 1.0 < 1.0E-10)
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return true
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else
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return false
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end
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end
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