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latticegpu.jl/src/Groups/AlgebraSU3.jl

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###
### "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: AlgebraSU3.jl
### created: Sun Oct 3 09:13:07 2021
###
function projalg(a::SU3{T}) where T <: AbstractFloat
sr3ov2::T = 0.866025403784438646763723170752
ditr = ( imag(a.u11) + imag(a.u22) + 2.0*imag(a.u11*a.u22 - a.u12*a.u21) )/3.0
m12 = (a.u12 - conj(a.u21))/2.0
m13 = (a.u13 - (a.u12*a.u23 - a.u13*a.u22) )/2.0
m23 = (a.u23 - (a.u13*a.u21 - a.u11*a.u23) )/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 projalg(z::Complex{T}, a::SU3{T}) where T <: AbstractFloat
sr3ov2::T = 0.866025403784438646763723170752
zu11 = z*a.u11
zu12 = z*a.u12
zu13 = z*a.u13
zu21 = z*a.u21
zu22 = z*a.u22
zu23 = z*a.u23
ditr = ( imag(zu11) + imag(zu22) - 2.0*imag(z*conj(a.u11*a.u22 - a.u12*a.u21)) )/3.0
m12 = (zu12 - conj(zu21))/2.0
m13 = (zu13 - conj(z)*(a.u12*a.u23 - a.u13*a.u22) )/2.0
m23 = (zu23 - conj(z)*(a.u13*a.u21 - a.u11*a.u23) )/2.0
return SU3alg{T}(imag( m12 ), imag( m13 ), imag( m23 ),
real( m12 ), real( m13 ), real( m23 ),
(imag(zu11)-imag(zu22))/2.0,
sr3ov2*(ditr))
end
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
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
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)
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))
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)
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)
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)
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)
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)
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)
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)
Base.:*(a::SU3alg{T},b::SU3alg{T}) where T = convert(M3x3{T}, a)*convert(M3x3{T}, a)
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
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