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Divided SU3 routines in different files
This commit is contained in:
Alberto Ramos
parent
9de5e4ad59
commit
d11f7eefa3
6 changed files with 371 additions and 328 deletions
184
src/Groups/AlgebraSU3.jl
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184
src/Groups/AlgebraSU3.jl
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@ -0,0 +1,184 @@
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###
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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: AlgebraSU3.jl
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### created: Sun Oct 3 09:13:07 2021
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###
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function projalg(a::SU3{T}) where T <: AbstractFloat
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sr3ov2::T = 0.866025403784438646763723170752
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ditr = ( imag(a.u11) + imag(a.u22) + 2.0*imag(a.u11*a.u22 - a.u12*a.u21) )/3.0
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m12 = (a.u12 - conj(a.u21))/2.0
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m13 = (a.u13 - (a.u12*a.u23 - a.u13*a.u22) )/2.0
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m23 = (a.u23 - (a.u13*a.u21 - a.u11*a.u23) )/2.0
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return SU3alg{T}(imag( m12 ), imag( m13 ), imag( m23 ),
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real( m12 ), real( m13 ), real( m23 ),
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(imag(a.u11)-imag(a.u22))/2.0,
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sr3ov2*(ditr))
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end
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function projalg(a::M3x3{T}) where T <: AbstractFloat
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sr3ov2::T = 0.866025403784438646763723170752
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ditr = ( imag(a.u11) + imag(a.u22) + 2.0*imag(a.u33) )/3.0
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m12 = (a.u12 - conj(a.u21))/2.0
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m13 = (a.u13 - conj(a.u31))/2.0
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m23 = (a.u23 - conj(a.u32))/2.0
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return SU3alg{T}(imag( m12 ), imag( m13 ), imag( m23 ),
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real( m12 ), real( m13 ), real( m23 ),
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(imag(a.u11)-imag(a.u22))/2.0,
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sr3ov2*(ditr))
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end
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dot(a::SU3alg{T},b::SU3alg{T}) where T <: AbstractFloat = a.t1*b.t1 + a.t2*b.t2 + a.t3*b.t3 + a.t4*b.t4 + a.t5*b.t5 + a.t6*b.t6 + a.t7*b.t7 + a.t8*b.t8
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norm2(a::SU3alg{T}) where T <: AbstractFloat = a.t1^2 + a.t2^2 + a.t3^2 + a.t4^2 + a.t5^2 + a.t6^2 + a.t7^2 + a.t8^2
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norm(a::SU3alg{T}) where T <: AbstractFloat = sqrt(a.t1^2 + a.t2^2 + a.t3^2 + a.t4^2 + a.t5^2 + a.t6^2 + a.t7^2 + a.t8^2)
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Base.zero(::Type{SU3alg{T}}) where T <: AbstractFloat = SU3alg{T}(zero(T),zero(T),zero(T),zero(T),zero(T),zero(T),zero(T),zero(T))
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Base.:+(a::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(a.t1,a.t2,a.t3,a.t4,a.t5,a.t6,a.t7,a.t8)
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Base.:-(a::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(-a.t1,-a.t2,-a.t3,-a.t4,-a.t5,-a.t6,-a.t7,-a.t8)
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Base.:+(a::SU3alg{T},b::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(a.t1+b.t1,a.t2+b.t2,a.t3+b.t3,a.t4+b.t4,a.t5+b.t5,a.t6+b.t6,a.t7+b.t7,a.t8+b.t8)
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Base.:-(a::SU3alg{T},b::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(a.t1-b.t1,a.t2-b.t2,a.t3-b.t3,a.t4-b.t4,a.t5-b.t5,a.t6-b.t6,a.t7-b.t7,a.t8-b.t8)
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Base.:*(a::SU3alg{T},b::Number) where T <: AbstractFloat = SU3alg{T}(b*a.t1,b*a.t2,b*a.t3,b*a.t4,b*a.t5,b*a.t6,b*a.t7,b*a.t8)
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Base.:*(b::Number,a::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(b*a.t1,b*a.t2,b*a.t3,b*a.t4,b*a.t5,b*a.t6,b*a.t7,b*a.t8)
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Base.:/(a::SU3alg{T},b::Number) where T <: AbstractFloat = SU3alg{T}(a.t1/b,a.t2/b,a.t3/b,a.t4/b,a.t5/b,a.t6/b,a.t7/b,a.t8/b)
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function alg2mat(a::SU3alg{T}) where T <: AbstractFloat
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two::T = 2.0
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rct::T = 3.46410161513775458
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x8p::T = a.t8/rct
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x7p::T = a.t7/two
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u11::Complex{T} = complex(0.0, x7p + x8p)
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u22::Complex{T} = complex(0.0,-x7p + x8p)
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u33::Complex{T} = complex(0.0,-2.0*x8p)
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u12::Complex{T} = complex(a.t4,a.t1)/two
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u13::Complex{T} = complex(a.t5,a.t2)/two
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u23::Complex{T} = complex(a.t6,a.t3)/two
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u21::Complex{T} = -conj(u12)
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u31::Complex{T} = -conj(u13)
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u32::Complex{T} = -conj(u23)
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return M3x3{T}(u11,u12,u13, u21,u22,u23, u31,u32,u33)
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end
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Base.:*(a::SU3alg,b::SU3) = alg2mat(a)*b
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Base.:*(a::SU3,b::SU3alg) = a*alg2mat(b)
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Base.:/(a::SU3alg,b::SU3) = alg2mat(a)/b
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Base.:\(a::SU3,b::SU3alg) = a\alg2mat(b)
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@inline function exp_iter(dch::Complex{T}, tch::T) where T <: AbstractFloat
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c::NTuple{22, T} = ( 1.957294106339126128e-20, 4.110317623312164853e-19,
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8.220635246624329711e-18, 1.561920696858622643e-16,
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2.811457254345520766e-15, 4.779477332387385293e-14,
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7.647163731819816473e-13, 1.147074559772972473e-11,
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1.605904383682161451e-10, 2.087675698786809894e-09,
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2.505210838544171879e-08, 2.755731922398589067e-07,
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2.755731922398589065e-06, 2.480158730158730158e-05,
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1.984126984126984127e-04, 1.388888888888888888e-03,
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8.333333333333333333e-03, 4.166666666666666666e-02,
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1.666666666666666666e-01, 0.5, 1.0, 1.0 )
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q0 = complex(c[1])
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q1 = complex(0.0)
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q2 = complex(0.0)
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@inbounds for i in 2:length(c)
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qt0 = q0
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qt1 = q1
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q0 = complex(c[i]) + dch*q2
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q1 = qt0 - tch*q2
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q2 = qt1
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end
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return q0, q1, q2
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end
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function expm(g::SU3{T}, a::SU3alg{T}, t::Number) where T <: AbstractFloat
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tpw = t^2
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M = alg2mat(a)
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Msq = M*M
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dch::Complex{T} = tpw*t*(M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 +
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M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 -
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M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31)
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tch::T = -tpw*(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0
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q0, q1, q2 = exp_iter(dch, tch)
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q1 = t*q1
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q2 = tpw*q2
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g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13,
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q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23)
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return g2*g
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end
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function expm(g::SU3{T}, a::SU3alg{T}) where T <: AbstractFloat
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M = alg2mat(a)
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Msq = M*M
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dch::Complex{T} = M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 +
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M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 -
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M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31
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tch::T = -(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0
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q0, q1, q2 = exp_iter(dch, tch)
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g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13,
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q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23)
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return g2*g
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end
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function Base.exp(a::SU3alg{T}) where T <: AbstractFloat
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M = alg2mat(a)
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Msq = M*M
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dch::Complex{T} = M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 +
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M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 -
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M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31
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tch::T = -(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0
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q0, q1, q2 = exp_iter(dch, tch)
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g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13,
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q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23)
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return g2
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end
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function Base.exp(a::SU3alg{T}, t::Number) where T <: AbstractFloat
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tpw = t^2
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M = alg2mat(a)
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Msq = M*M
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dch::Complex{T} = tpw*t*(M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 +
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M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 -
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M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31)
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tch::T = -tpw*(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0
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q0, q1, q2 = exp_iter(dch, tch)
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q1 = t*q1
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q2 = tpw*q2
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g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13,
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q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23)
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return g2
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end
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@ -9,32 +9,9 @@
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### created: Sun Jul 11 17:23:02 2021
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###
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#
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# Use memory efficient representation: Only store
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# first two rows. Third row constructed on the fly.
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#
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# a.u31 = conj(a.u12*a.u23 - a.u13*a.u22)
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# a.u32 = conj(a.u13*a.u21 - a.u11*a.u23)
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# a.u33 = conj(a.u11*a.u22 - a.u12*a.u21)
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#
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using Random
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import Base.:*, Base.:+, Base.:-,Base.:/,Base.:\,Base.one,Base.zero
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import Random.rand
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struct SU3{T} <: Group
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u11::Complex{T}
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u12::Complex{T}
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u13::Complex{T}
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u21::Complex{T}
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u22::Complex{T}
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u23::Complex{T}
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end
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inverse(a::SU3{T}) where T <: AbstractFloat = SU3{T}(conj(a.u11),conj(a.u21),(a.u12*a.u23 - a.u13*a.u22), conj(a.u12),conj(a.u22),(a.u13*a.u21 - a.u11*a.u23))
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dag(a::SU3{T}) where T <: AbstractFloat = inverse(a)
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tr(a::SU3{T}) where T <: AbstractFloat = a.u11+a.u22+conj(a.u11*a.u22 - a.u12*a.u21)
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Base.one(::Type{SU3{T}}) where T <: AbstractFloat = SU3{T}(one(T),zero(T),zero(T),zero(T),one(T),zero(T))
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Random.rand(rng::AbstractRNG, ::Random.SamplerType{SU3{T}}) where T <: AbstractFloat = exp(SU3alg{T}(randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T)))
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function Base.:*(a::SU3{T},b::SU3{T}) where T <: AbstractFloat
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@ -96,306 +73,3 @@ function isgroup(a::SU3{T}) where T <: AbstractFloat
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end
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end
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struct SU3alg{T} <: Algebra
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t1::T
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t2::T
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t3::T
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t4::T
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t5::T
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t6::T
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t7::T
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t8::T
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end
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function projalg(a::SU3{T}) where T <: AbstractFloat
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sr3ov2::T = 0.866025403784438646763723170752
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ditr = ( imag(a.u11) + imag(a.u22) + 2.0*imag(a.u11*a.u22 - a.u12*a.u21) )/3.0
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m12 = (a.u12 - conj(a.u21))/2.0
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m13 = (a.u13 - (a.u12*a.u23 - a.u13*a.u22) )/2.0
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m23 = (a.u23 - (a.u13*a.u21 - a.u11*a.u23) )/2.0
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return SU3alg{T}(imag( m12 ), imag( m13 ), imag( m23 ),
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real( m12 ), real( m13 ), real( m23 ),
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(imag(a.u11)-imag(a.u22))/2.0,
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sr3ov2*(ditr))
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end
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dot(a::SU3alg{T},b::SU3alg{T}) where T <: AbstractFloat = a.t1*b.t1 + a.t2*b.t2 + a.t3*b.t3 + a.t4*b.t4 + a.t5*b.t5 + a.t6*b.t6 + a.t7*b.t7 + a.t8*b.t8
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norm2(a::SU3alg{T}) where T <: AbstractFloat = a.t1^2 + a.t2^2 + a.t3^2 + a.t4^2 + a.t5^2 + a.t6^2 + a.t7^2 + a.t8^2
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norm(a::SU3alg{T}) where T <: AbstractFloat = sqrt(a.t1^2 + a.t2^2 + a.t3^2 + a.t4^2 + a.t5^2 + a.t6^2 + a.t7^2 + a.t8^2)
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Base.zero(::Type{SU3alg{T}}) where T <: AbstractFloat = SU3alg{T}(zero(T),zero(T),zero(T),zero(T),zero(T),zero(T),zero(T),zero(T))
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Random.rand(rng::AbstractRNG, ::Random.SamplerType{SU3alg{T}}) where T <: AbstractFloat = SU3alg{T}(randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T))
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Base.:+(a::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(a.t1,a.t2,a.t3,a.t4,a.t5,a.t6,a.t7,a.t8)
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Base.:-(a::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(-a.t1,-a.t2,-a.t3,-a.t4,-a.t5,-a.t6,-a.t7,-a.t8)
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Base.:+(a::SU3alg{T},b::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(a.t1+b.t1,a.t2+b.t2,a.t3+b.t3,a.t4+b.t4,a.t5+b.t5,a.t6+b.t6,a.t7+b.t7,a.t8+b.t8)
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Base.:-(a::SU3alg{T},b::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(a.t1-b.t1,a.t2-b.t2,a.t3-b.t3,a.t4-b.t4,a.t5-b.t5,a.t6-b.t6,a.t7-b.t7,a.t8-b.t8)
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Base.:*(a::SU3alg{T},b::Number) where T <: AbstractFloat = SU3alg{T}(b*a.t1,b*a.t2,b*a.t3,b*a.t4,b*a.t5,b*a.t6,b*a.t7,b*a.t8)
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Base.:*(b::Number,a::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(b*a.t1,b*a.t2,b*a.t3,b*a.t4,b*a.t5,b*a.t6,b*a.t7,b*a.t8)
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Base.:/(a::SU3alg{T},b::Number) where T <: AbstractFloat = SU3alg{T}(a.t1/b,a.t2/b,a.t3/b,a.t4/b,a.t5/b,a.t6/b,a.t7/b,a.t8/b)
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export SU3, SU3alg, inverse, dag, tr, projalg, expm, exp, norm, norm2, isgroup
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struct M3x3{T}
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u11::Complex{T}
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u12::Complex{T}
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u13::Complex{T}
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u21::Complex{T}
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u22::Complex{T}
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u23::Complex{T}
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u31::Complex{T}
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u32::Complex{T}
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u33::Complex{T}
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end
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Base.:*(a::M3x3{T},b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*b.u31,
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a.u11*b.u12 + a.u12*b.u22 + a.u13*b.u32,
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a.u11*b.u13 + a.u12*b.u23 + a.u13*b.u33,
|
||||
a.u21*b.u11 + a.u22*b.u21 + a.u23*b.u31,
|
||||
a.u21*b.u12 + a.u22*b.u22 + a.u23*b.u32,
|
||||
a.u21*b.u13 + a.u22*b.u23 + a.u23*b.u33,
|
||||
a.u31*b.u11 + a.u32*b.u21 + a.u33*b.u31,
|
||||
a.u31*b.u12 + a.u32*b.u22 + a.u33*b.u32,
|
||||
a.u31*b.u13 + a.u32*b.u23 + a.u33*b.u33)
|
||||
|
||||
function Base.:*(a::SU3{T},b::M3x3{T}) where T <: AbstractFloat
|
||||
|
||||
au31 = conj(a.u12*a.u23 - a.u13*a.u22)
|
||||
au32 = conj(a.u13*a.u21 - a.u11*a.u23)
|
||||
au33 = conj(a.u11*a.u22 - a.u12*a.u21)
|
||||
|
||||
return M3x3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*b.u31,
|
||||
a.u11*b.u12 + a.u12*b.u22 + a.u13*b.u32,
|
||||
a.u11*b.u13 + a.u12*b.u23 + a.u13*b.u33,
|
||||
a.u21*b.u11 + a.u22*b.u21 + a.u23*b.u31,
|
||||
a.u21*b.u12 + a.u22*b.u22 + a.u23*b.u32,
|
||||
a.u21*b.u13 + a.u22*b.u23 + a.u23*b.u33,
|
||||
au31*b.u11 + au32*b.u21 + au33*b.u31,
|
||||
au31*b.u12 + au32*b.u22 + au33*b.u32,
|
||||
au31*b.u13 + au32*b.u23 + au33*b.u33)
|
||||
end
|
||||
|
||||
|
||||
function Base.:*(a::M3x3{T},b::SU3{T}) where T <: AbstractFloat
|
||||
|
||||
bu31 = conj(b.u12*b.u23 - b.u13*b.u22)
|
||||
bu32 = conj(b.u13*b.u21 - b.u11*b.u23)
|
||||
bu33 = conj(b.u11*b.u22 - b.u12*b.u21)
|
||||
|
||||
return M3x3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*bu31,
|
||||
a.u11*b.u12 + a.u12*b.u22 + a.u13*bu32,
|
||||
a.u11*b.u13 + a.u12*b.u23 + a.u13*bu33,
|
||||
a.u21*b.u11 + a.u22*b.u21 + a.u23*bu31,
|
||||
a.u21*b.u12 + a.u22*b.u22 + a.u23*bu32,
|
||||
a.u21*b.u13 + a.u22*b.u23 + a.u23*bu33,
|
||||
a.u31*b.u11 + a.u32*b.u21 + a.u33*bu31,
|
||||
a.u31*b.u12 + a.u32*b.u22 + a.u33*bu32,
|
||||
a.u31*b.u13 + a.u32*b.u23 + a.u33*bu33)
|
||||
end
|
||||
|
||||
function Base.:/(a::M3x3{T},b::SU3{T}) where T <: AbstractFloat
|
||||
|
||||
bu31 = (b.u12*b.u23 - b.u13*b.u22)
|
||||
bu32 = (b.u13*b.u21 - b.u11*b.u23)
|
||||
bu33 = (b.u11*b.u22 - b.u12*b.u21)
|
||||
|
||||
return M3x3{T}(a.u11*conj(b.u11) + a.u12*conj(b.u12) + a.u13*conj(b.u13),
|
||||
a.u11*conj(b.u21) + a.u12*conj(b.u22) + a.u13*conj(b.u23),
|
||||
a.u11*(bu31) + a.u12*(bu32) + a.u13*(bu33),
|
||||
a.u21*conj(b.u11) + a.u22*conj(b.u12) + a.u23*conj(b.u13),
|
||||
a.u21*conj(b.u21) + a.u22*conj(b.u22) + a.u23*conj(b.u23),
|
||||
a.u21*(bu31) + a.u22*(bu32) + a.u23*(bu33),
|
||||
a.u31*conj(b.u11) + a.u32*conj(b.u12) + a.u33*conj(b.u13),
|
||||
a.u31*conj(b.u21) + a.u32*conj(b.u22) + a.u33*conj(b.u23),
|
||||
a.u31*(bu31) + a.u32*(bu32) + a.u33*(bu33))
|
||||
end
|
||||
|
||||
function Base.:\(a::SU3{T},b::M3x3{T}) where T <: AbstractFloat
|
||||
|
||||
au31 = (a.u12*a.u23 - a.u13*a.u22)
|
||||
au32 = (a.u13*a.u21 - a.u11*a.u23)
|
||||
au33 = (a.u11*a.u22 - a.u12*a.u21)
|
||||
|
||||
return M3x3{T}(conj(a.u11)*b.u11 + conj(a.u21)*b.u21 + (au31)*b.u31,
|
||||
conj(a.u11)*b.u12 + conj(a.u21)*b.u22 + (au31)*b.u32,
|
||||
conj(a.u11)*b.u13 + conj(a.u21)*b.u23 + (au31)*b.u33,
|
||||
conj(a.u12)*b.u11 + conj(a.u22)*b.u21 + (au32)*b.u31,
|
||||
conj(a.u12)*b.u12 + conj(a.u22)*b.u22 + (au32)*b.u32,
|
||||
conj(a.u12)*b.u13 + conj(a.u22)*b.u23 + (au32)*b.u33,
|
||||
conj(a.u13)*b.u11 + conj(a.u23)*b.u21 + (au33)*b.u31,
|
||||
conj(a.u13)*b.u12 + conj(a.u23)*b.u22 + (au33)*b.u32,
|
||||
conj(a.u13)*b.u13 + conj(a.u23)*b.u23 + (au33)*b.u33)
|
||||
|
||||
end
|
||||
|
||||
Base.:*(a::Number,b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a*b.u11, a*b.u12, a*bu13,
|
||||
a*b.u21, a*b.u22, a*bu23,
|
||||
a*b.u31, a*b.u32, a*bu33)
|
||||
|
||||
Base.:*(b::M3x3{T},a::Number) where T <: AbstractFloat = M3x3{T}(a*b.u11, a*b.u12, a*bu13,
|
||||
a*b.u21, a*b.u22, a*bu23,
|
||||
a*b.u31, a*b.u32, a*bu33)
|
||||
|
||||
Base.:+(a::M3x3{T},b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a.u11+b.u11, a.u12+b.u12, a.u13+bu13,
|
||||
a.u21+b.u21, a.u22+b.u22, a.u23+bu23,
|
||||
a.u31+b.u31, a.u32+b.u32, a.u33+bu33)
|
||||
|
||||
Base.:-(a::M3x3{T},b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a.u11-b.u11, a.u12-b.u12, a.u13-bu13,
|
||||
a.u21-b.u21, a.u22-b.u22, a.u23-bu23,
|
||||
a.u31-b.u31, a.u32-b.u32, a.u33-bu33)
|
||||
|
||||
Base.:-(b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(-b.u11, -b.u12, -bu13,
|
||||
-b.u21, -b.u22, -bu23,
|
||||
-b.u31, -b.u32, -bu33)
|
||||
|
||||
Base.:+(b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(b.u11, b.u12, bu13,
|
||||
b.u21, b.u22, bu23,
|
||||
b.u31, b.u32, bu33)
|
||||
function projalg(a::M3x3{T}) where T <: AbstractFloat
|
||||
|
||||
sr3ov2::T = 0.866025403784438646763723170752
|
||||
|
||||
ditr = ( imag(a.u11) + imag(a.u22) + 2.0*imag(a.u33) )/3.0
|
||||
m12 = (a.u12 - conj(a.u21))/2.0
|
||||
m13 = (a.u13 - conj(a.u31))/2.0
|
||||
m23 = (a.u23 - conj(a.u32))/2.0
|
||||
|
||||
return SU3alg{T}(imag( m12 ), imag( m13 ), imag( m23 ),
|
||||
real( m12 ), real( m13 ), real( m23 ),
|
||||
(imag(a.u11)-imag(a.u22))/2.0,
|
||||
sr3ov2*(ditr))
|
||||
end
|
||||
|
||||
function alg2mat(a::SU3alg{T}) where T <: AbstractFloat
|
||||
|
||||
two::T = 2.0
|
||||
rct::T = 3.46410161513775458
|
||||
|
||||
x8p::T = a.t8/rct
|
||||
x7p::T = a.t7/two
|
||||
u11::Complex{T} = complex(0.0, x7p + x8p)
|
||||
u22::Complex{T} = complex(0.0,-x7p + x8p)
|
||||
u33::Complex{T} = complex(0.0,-2.0*x8p)
|
||||
u12::Complex{T} = complex(a.t4,a.t1)/two
|
||||
u13::Complex{T} = complex(a.t5,a.t2)/two
|
||||
u23::Complex{T} = complex(a.t6,a.t3)/two
|
||||
u21::Complex{T} = -conj(u12)
|
||||
u31::Complex{T} = -conj(u13)
|
||||
u32::Complex{T} = -conj(u23)
|
||||
|
||||
return M3x3{T}(u11,u12,u13, u21,u22,u23, u31,u32,u33)
|
||||
end
|
||||
|
||||
Base.:*(a::SU3alg,b::SU3) = alg2mat(a)*b
|
||||
Base.:*(a::SU3,b::SU3alg) = a*alg2mat(b)
|
||||
Base.:/(a::SU3alg,b::SU3) = alg2mat(a)/b
|
||||
Base.:\(a::SU3,b::SU3alg) = a\alg2mat(b)
|
||||
|
||||
@inline function exp_iter(dch::Complex{T}, tch::T) where T <: AbstractFloat
|
||||
|
||||
c::NTuple{22, T} = ( 1.957294106339126128e-20, 4.110317623312164853e-19,
|
||||
8.220635246624329711e-18, 1.561920696858622643e-16,
|
||||
2.811457254345520766e-15, 4.779477332387385293e-14,
|
||||
7.647163731819816473e-13, 1.147074559772972473e-11,
|
||||
1.605904383682161451e-10, 2.087675698786809894e-09,
|
||||
2.505210838544171879e-08, 2.755731922398589067e-07,
|
||||
2.755731922398589065e-06, 2.480158730158730158e-05,
|
||||
1.984126984126984127e-04, 1.388888888888888888e-03,
|
||||
8.333333333333333333e-03, 4.166666666666666666e-02,
|
||||
1.666666666666666666e-01, 0.5, 1.0, 1.0 )
|
||||
|
||||
q0 = complex(c[1])
|
||||
q1 = complex(0.0)
|
||||
q2 = complex(0.0)
|
||||
@inbounds for i in 2:length(c)
|
||||
qt0 = q0
|
||||
qt1 = q1
|
||||
q0 = complex(c[i]) + dch*q2
|
||||
q1 = qt0 - tch*q2
|
||||
q2 = qt1
|
||||
end
|
||||
|
||||
return q0, q1, q2
|
||||
end
|
||||
|
||||
|
||||
function expm(g::SU3{T}, a::SU3alg{T}, t::Number) where T <: AbstractFloat
|
||||
|
||||
tpw = t^2
|
||||
M = alg2mat(a)
|
||||
Msq = M*M
|
||||
dch::Complex{T} = tpw*t*(M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 +
|
||||
M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 -
|
||||
M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31)
|
||||
tch::T = -tpw*(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0
|
||||
|
||||
q0, q1, q2 = exp_iter(dch, tch)
|
||||
|
||||
q1 = t*q1
|
||||
q2 = tpw*q2
|
||||
|
||||
g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13,
|
||||
q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23)
|
||||
|
||||
return g2*g
|
||||
end
|
||||
|
||||
function expm(g::SU3{T}, a::SU3alg{T}) where T <: AbstractFloat
|
||||
|
||||
M = alg2mat(a)
|
||||
Msq = M*M
|
||||
dch::Complex{T} = M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 +
|
||||
M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 -
|
||||
M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31
|
||||
tch::T = -(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0
|
||||
|
||||
q0, q1, q2 = exp_iter(dch, tch)
|
||||
|
||||
g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13,
|
||||
q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23)
|
||||
|
||||
return g2*g
|
||||
end
|
||||
|
||||
function Base.exp(a::SU3alg{T}) where T <: AbstractFloat
|
||||
|
||||
M = alg2mat(a)
|
||||
Msq = M*M
|
||||
dch::Complex{T} = M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 +
|
||||
M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 -
|
||||
M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31
|
||||
tch::T = -(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0
|
||||
|
||||
q0, q1, q2 = exp_iter(dch, tch)
|
||||
|
||||
g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13,
|
||||
q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23)
|
||||
|
||||
return g2
|
||||
end
|
||||
|
||||
function Base.exp(a::SU3alg{T}, t::Number) where T <: AbstractFloat
|
||||
|
||||
tpw = t^2
|
||||
M = alg2mat(a)
|
||||
Msq = M*M
|
||||
dch::Complex{T} = tpw*t*(M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 +
|
||||
M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 -
|
||||
M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31)
|
||||
tch::T = -tpw*(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0
|
||||
|
||||
q0, q1, q2 = exp_iter(dch, tch)
|
||||
|
||||
q1 = t*q1
|
||||
q2 = tpw*q2
|
||||
|
||||
g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13,
|
||||
q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23)
|
||||
|
||||
return g2
|
||||
end
|
||||
|
||||
|
|
|
|||
|
|
@ -12,6 +12,10 @@
|
|||
|
||||
module Groups
|
||||
|
||||
using Random
|
||||
import Base.:*, Base.:+, Base.:-,Base.:/,Base.:\,Base.one,Base.zero
|
||||
import Random.rand
|
||||
|
||||
abstract type Group end
|
||||
abstract type Algebra end
|
||||
|
||||
|
|
@ -20,8 +24,16 @@ export Group, Algebra
|
|||
include("GroupSU2.jl")
|
||||
export SU2, SU2alg
|
||||
|
||||
##
|
||||
# SU(3) and 3x3 matrix operations
|
||||
##
|
||||
include("SU3Types.jl")
|
||||
export SU3, SU3alg, M3x3
|
||||
|
||||
include("GroupSU3.jl")
|
||||
export SU3, SU3alg
|
||||
include("M3x3.jl")
|
||||
include("AlgebraSU3.jl")
|
||||
## END SU(3)
|
||||
|
||||
include("GroupU1.jl")
|
||||
export U1, U1alg
|
||||
|
|
|
|||
115
src/Groups/M3x3.jl
Normal file
115
src/Groups/M3x3.jl
Normal file
|
|
@ -0,0 +1,115 @@
|
|||
###
|
||||
### "THE BEER-WARE LICENSE":
|
||||
### Alberto Ramos wrote this file. As long as you retain this
|
||||
### notice you can do whatever you want with this stuff. If we meet some
|
||||
### day, and you think this stuff is worth it, you can buy me a beer in
|
||||
### return. <alberto.ramos@cern.ch>
|
||||
###
|
||||
### file: M3x3.jl
|
||||
### created: Sun Oct 3 09:03:34 2021
|
||||
###
|
||||
|
||||
|
||||
Base.:*(a::M3x3{T},b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*b.u31,
|
||||
a.u11*b.u12 + a.u12*b.u22 + a.u13*b.u32,
|
||||
a.u11*b.u13 + a.u12*b.u23 + a.u13*b.u33,
|
||||
a.u21*b.u11 + a.u22*b.u21 + a.u23*b.u31,
|
||||
a.u21*b.u12 + a.u22*b.u22 + a.u23*b.u32,
|
||||
a.u21*b.u13 + a.u22*b.u23 + a.u23*b.u33,
|
||||
a.u31*b.u11 + a.u32*b.u21 + a.u33*b.u31,
|
||||
a.u31*b.u12 + a.u32*b.u22 + a.u33*b.u32,
|
||||
a.u31*b.u13 + a.u32*b.u23 + a.u33*b.u33)
|
||||
|
||||
function Base.:*(a::SU3{T},b::M3x3{T}) where T <: AbstractFloat
|
||||
|
||||
au31 = conj(a.u12*a.u23 - a.u13*a.u22)
|
||||
au32 = conj(a.u13*a.u21 - a.u11*a.u23)
|
||||
au33 = conj(a.u11*a.u22 - a.u12*a.u21)
|
||||
|
||||
return M3x3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*b.u31,
|
||||
a.u11*b.u12 + a.u12*b.u22 + a.u13*b.u32,
|
||||
a.u11*b.u13 + a.u12*b.u23 + a.u13*b.u33,
|
||||
a.u21*b.u11 + a.u22*b.u21 + a.u23*b.u31,
|
||||
a.u21*b.u12 + a.u22*b.u22 + a.u23*b.u32,
|
||||
a.u21*b.u13 + a.u22*b.u23 + a.u23*b.u33,
|
||||
au31*b.u11 + au32*b.u21 + au33*b.u31,
|
||||
au31*b.u12 + au32*b.u22 + au33*b.u32,
|
||||
au31*b.u13 + au32*b.u23 + au33*b.u33)
|
||||
end
|
||||
|
||||
|
||||
function Base.:*(a::M3x3{T},b::SU3{T}) where T <: AbstractFloat
|
||||
|
||||
bu31 = conj(b.u12*b.u23 - b.u13*b.u22)
|
||||
bu32 = conj(b.u13*b.u21 - b.u11*b.u23)
|
||||
bu33 = conj(b.u11*b.u22 - b.u12*b.u21)
|
||||
|
||||
return M3x3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*bu31,
|
||||
a.u11*b.u12 + a.u12*b.u22 + a.u13*bu32,
|
||||
a.u11*b.u13 + a.u12*b.u23 + a.u13*bu33,
|
||||
a.u21*b.u11 + a.u22*b.u21 + a.u23*bu31,
|
||||
a.u21*b.u12 + a.u22*b.u22 + a.u23*bu32,
|
||||
a.u21*b.u13 + a.u22*b.u23 + a.u23*bu33,
|
||||
a.u31*b.u11 + a.u32*b.u21 + a.u33*bu31,
|
||||
a.u31*b.u12 + a.u32*b.u22 + a.u33*bu32,
|
||||
a.u31*b.u13 + a.u32*b.u23 + a.u33*bu33)
|
||||
end
|
||||
|
||||
function Base.:/(a::M3x3{T},b::SU3{T}) where T <: AbstractFloat
|
||||
|
||||
bu31 = (b.u12*b.u23 - b.u13*b.u22)
|
||||
bu32 = (b.u13*b.u21 - b.u11*b.u23)
|
||||
bu33 = (b.u11*b.u22 - b.u12*b.u21)
|
||||
|
||||
return M3x3{T}(a.u11*conj(b.u11) + a.u12*conj(b.u12) + a.u13*conj(b.u13),
|
||||
a.u11*conj(b.u21) + a.u12*conj(b.u22) + a.u13*conj(b.u23),
|
||||
a.u11*(bu31) + a.u12*(bu32) + a.u13*(bu33),
|
||||
a.u21*conj(b.u11) + a.u22*conj(b.u12) + a.u23*conj(b.u13),
|
||||
a.u21*conj(b.u21) + a.u22*conj(b.u22) + a.u23*conj(b.u23),
|
||||
a.u21*(bu31) + a.u22*(bu32) + a.u23*(bu33),
|
||||
a.u31*conj(b.u11) + a.u32*conj(b.u12) + a.u33*conj(b.u13),
|
||||
a.u31*conj(b.u21) + a.u32*conj(b.u22) + a.u33*conj(b.u23),
|
||||
a.u31*(bu31) + a.u32*(bu32) + a.u33*(bu33))
|
||||
end
|
||||
|
||||
function Base.:\(a::SU3{T},b::M3x3{T}) where T <: AbstractFloat
|
||||
|
||||
au31 = (a.u12*a.u23 - a.u13*a.u22)
|
||||
au32 = (a.u13*a.u21 - a.u11*a.u23)
|
||||
au33 = (a.u11*a.u22 - a.u12*a.u21)
|
||||
|
||||
return M3x3{T}(conj(a.u11)*b.u11 + conj(a.u21)*b.u21 + (au31)*b.u31,
|
||||
conj(a.u11)*b.u12 + conj(a.u21)*b.u22 + (au31)*b.u32,
|
||||
conj(a.u11)*b.u13 + conj(a.u21)*b.u23 + (au31)*b.u33,
|
||||
conj(a.u12)*b.u11 + conj(a.u22)*b.u21 + (au32)*b.u31,
|
||||
conj(a.u12)*b.u12 + conj(a.u22)*b.u22 + (au32)*b.u32,
|
||||
conj(a.u12)*b.u13 + conj(a.u22)*b.u23 + (au32)*b.u33,
|
||||
conj(a.u13)*b.u11 + conj(a.u23)*b.u21 + (au33)*b.u31,
|
||||
conj(a.u13)*b.u12 + conj(a.u23)*b.u22 + (au33)*b.u32,
|
||||
conj(a.u13)*b.u13 + conj(a.u23)*b.u23 + (au33)*b.u33)
|
||||
|
||||
end
|
||||
|
||||
Base.:*(a::Number,b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a*b.u11, a*b.u12, a*bu13,
|
||||
a*b.u21, a*b.u22, a*bu23,
|
||||
a*b.u31, a*b.u32, a*bu33)
|
||||
|
||||
Base.:*(b::M3x3{T},a::Number) where T <: AbstractFloat = M3x3{T}(a*b.u11, a*b.u12, a*bu13,
|
||||
a*b.u21, a*b.u22, a*bu23,
|
||||
a*b.u31, a*b.u32, a*bu33)
|
||||
|
||||
Base.:+(a::M3x3{T},b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a.u11+b.u11, a.u12+b.u12, a.u13+bu13,
|
||||
a.u21+b.u21, a.u22+b.u22, a.u23+bu23,
|
||||
a.u31+b.u31, a.u32+b.u32, a.u33+bu33)
|
||||
|
||||
Base.:-(a::M3x3{T},b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a.u11-b.u11, a.u12-b.u12, a.u13-bu13,
|
||||
a.u21-b.u21, a.u22-b.u22, a.u23-bu23,
|
||||
a.u31-b.u31, a.u32-b.u32, a.u33-bu33)
|
||||
|
||||
Base.:-(b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(-b.u11, -b.u12, -bu13,
|
||||
-b.u21, -b.u22, -bu23,
|
||||
-b.u31, -b.u32, -bu33)
|
||||
|
||||
Base.:+(b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(b.u11, b.u12, bu13,
|
||||
b.u21, b.u22, bu23,
|
||||
b.u31, b.u32, bu33)
|
||||
58
src/Groups/SU3Types.jl
Normal file
58
src/Groups/SU3Types.jl
Normal file
|
|
@ -0,0 +1,58 @@
|
|||
###
|
||||
### "THE BEER-WARE LICENSE":
|
||||
### Alberto Ramos wrote this file. As long as you retain this
|
||||
### notice you can do whatever you want with this stuff. If we meet some
|
||||
### day, and you think this stuff is worth it, you can buy me a beer in
|
||||
### return. <alberto.ramos@cern.ch>
|
||||
###
|
||||
### file: SU3Types.jl
|
||||
### created: Sun Oct 3 09:05:23 2021
|
||||
###
|
||||
|
||||
#
|
||||
# Use memory efficient representation: Only store
|
||||
# first two rows. Third row constructed on the fly.
|
||||
#
|
||||
# a.u31 = conj(a.u12*a.u23 - a.u13*a.u22)
|
||||
# a.u32 = conj(a.u13*a.u21 - a.u11*a.u23)
|
||||
# a.u33 = conj(a.u11*a.u22 - a.u12*a.u21)
|
||||
#
|
||||
|
||||
struct SU3{T} <: Group
|
||||
u11::Complex{T}
|
||||
u12::Complex{T}
|
||||
u13::Complex{T}
|
||||
u21::Complex{T}
|
||||
u22::Complex{T}
|
||||
u23::Complex{T}
|
||||
end
|
||||
Base.one(::Type{SU3{T}}) where T <: AbstractFloat = SU3{T}(one(T),zero(T),zero(T),zero(T),one(T),zero(T))
|
||||
|
||||
struct M3x3{T}
|
||||
u11::Complex{T}
|
||||
u12::Complex{T}
|
||||
u13::Complex{T}
|
||||
u21::Complex{T}
|
||||
u22::Complex{T}
|
||||
u23::Complex{T}
|
||||
u31::Complex{T}
|
||||
u32::Complex{T}
|
||||
u33::Complex{T}
|
||||
end
|
||||
|
||||
struct SU3alg{T} <: Algebra
|
||||
t1::T
|
||||
t2::T
|
||||
t3::T
|
||||
t4::T
|
||||
t5::T
|
||||
t6::T
|
||||
t7::T
|
||||
t8::T
|
||||
end
|
||||
|
||||
|
||||
Random.rand(rng::AbstractRNG, ::Random.SamplerType{SU3{T}}) where T <: AbstractFloat = exp(SU3alg{T}(randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T)))
|
||||
Random.rand(rng::AbstractRNG, ::Random.SamplerType{SU3alg{T}}) where T <: AbstractFloat = SU3alg{T}(randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T))
|
||||
|
||||
|
||||
|
|
@ -16,7 +16,7 @@ include("Groups/Groups.jl")
|
|||
|
||||
using .Groups
|
||||
export Group, Algebra
|
||||
export SU2, SU2alg, SU3, SU3alg, U1, U1alg
|
||||
export SU2, SU2alg, SU3, SU3alg, M3x3, U1, U1alg
|
||||
export dot, expm, exp, dag, normalize, inverse, tr, projalg, norm, norm2, isgroup
|
||||
|
||||
include("Space/Space.jl")
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue