spherical.modes.derivatives

Lminus

Lminus(self)

Source: spherical/modes/derivatives.py

Lowering operator for Lz We define Lminus to be the lowering operator for the left Lie derivative with respect to rotation about z, Lz. By definition, this means that [Lz, Lminus] = -Lminus, which allows us to derive Lminus = Lx - 1j * Ly. In terms of the SWSHs, we can write the action of Lminus as Lminus {s}Y{l,m} = sqrt((l+m)(l-m+1)) * {s}Y{l,m-1} Consequently, the modes of a function are affected as {Lminus f}{s, l, m} = sqrt((l-m)(l+m+1)) * f{s,l,m+1}

Lplus

Lplus(self)

Source: spherical/modes/derivatives.py

Raising operator for Lz We define Lplus to be the raising operator for the left Lie derivative with respect to rotation about z, Lz. By definition, this means that [Lz, Lplus] = Lplus, which allows us to derive Lplus = Lx + 1j * Ly. In terms of the SWSHs, we can write the action of Lplus as Lplus {s}Y{l,m} = sqrt((l-m)(l+m+1)) {s}Y{l,m+1} Consequently, the modes of a function are affected as {Lplus f}{s, l, m} = sqrt((l+m)(l-m+1)) * f{s,l,m-1}

Lsquared

Lsquared(self)

Source: spherical/modes/derivatives.py

Total angular-momentum operator This is the standard L^2 operator, familiar from basic physics, extended to work with SWSHs. It is also known as the Casimir operator, and is equal to L^2 = LxLx + LyLy + LzLz = 0.5(L+L- + L-L+) + LzLz Note that these are the left Lie derivatives, but L^2 = R^2, where R is the right Lie derivative. The left Lie derivative of a function f(Q) over the unit quaternions with respect to a generator of rotation g is defined as Lg(f){Q} = -0.5j df{exp(t*g) * Q} / dt |t=0 This agrees with the usual angular-momentum operators familiar from spherical-harmonic theory, and reduces to it when the function has spin weight 0, but also applies to functions of general spin weight. In terms of the SWSHs, we can write the action of Lsquared as Lsquared {s}Y{l,m} = l * (l+1) * {s}Y{l,m}

Lz

Lz(self)

Source: spherical/modes/derivatives.py

Left Lie derivative with respect to rotation about z The left Lie derivative of a function f(Q) over the unit quaternions with respect to a generator of rotation g is defined as Lg(f){Q} = -0.5j df{exp(t*g) * Q} / dt |t=0 This agrees with the usual angular-momentum operators familiar from spherical-harmonic theory, and reduces to it when the function has spin weight 0, but also applies to functions of general spin weight. In terms of the SWSHs, we can write the action of Lz as Lz {s}Y{l,m} = m * {s}Y{l,m}

Rminus

Rminus(self)

Source: spherical/modes/derivatives.py

Lowering operator for Rz We define Rminus to be the lowering operator for the right Lie derivative with respect to rotation about z, Rz. By definition, this means that [Rz, Rminus] = -Rminus, which allows us to derive Rminus = Rx + 1j * Ry. In terms of the SWSHs, we can write the action of Rminus as Rminus {s}Y{l,m} = sqrt((l-s)(l+s+1)) {s+1}Y{l,m} Consequently, the modes of a function are affected as {Rminus f} {s,l,m} = sqrt((l+s)(l-s+1)) f{s-1,l,m} Again, because of the unfortunate choice of the sign of s made in the original paper by Newman and Penrose, this looks like a raising operator for s. But it really is a lowering operator for Rz, and lowers the eigenvalue of the corresponding Wigner matrix - though that raises the value of s.

Rplus

Rplus(self)

Source: spherical/modes/derivatives.py

Raising operator for Rz We define Rplus to be the raising operator for the right Lie derivative with respect to rotation about z, Rz. By definition, this means that [Rz, Rplus] = Rplus, which allows us to derive Rplus = Rx - 1j * Ry. In terms of the SWSHs, we can write the action of Rplus as Rplus {s}Y{l,m} = sqrt((l+s)(l-s+1)) {s-1}Y{l,m} Consequently, the modes of a function are affected as {Rplus f} {s,l,m} = sqrt((l-s)(l+s+1)) f{s+1,l,m} Again, because of the unfortunate choice of the sign of s made in the original paper by Newman and Penrose, this looks like a lowering operator for s. But it really is a raising operator for Rz, and raises the eigenvalue of the corresponding Wigner matrix - though that lowers the value of s.

Rsquared

Rsquared(self)

Source: spherical/modes/derivatives.py

Total angular-momentum operator This is the R^2 operator, much like the L^2 operator familiar from basic physics, but using the right Lie derivative, and extended to work with SWSHs. It is also known as the Casimir operator, and is equal to R^2 = RxRx + RyRy + RzRz = 0.5(R+R- + R-R+) + RzRz Note that these are the right Lie derivatives, but L^2 = R^2, where L is the left Lie derivative. The right Lie derivative of a function f(Q) over the unit quaternions with respect to a generator of rotation g is defined as Rg(f){Q} = -0.5j df{Q * exp(t*g)} / dt |t=0 This is unlike the usual angular-momentum operators Lz, etc., familiar from spherical-harmonic theory because the exponential is on the right-hand side of the argument. This operator is less common in physics because it represents the dependence of the function on the choice of frame. In terms of the SWSHs, we can write the action of Rsquared as Rsquared {s}Y{l,m} = l * (l+1) * {s}Y{l,m}

Rz

Rz(self)

Source: spherical/modes/derivatives.py

Right Lie derivative with respect to rotation about z The right Lie derivative of a function f(Q) over the unit quaternions with respect to a generator of rotation g is defined as Rg(f){Q} = -0.5j df{Q * exp(t*g)} / dt |t=0 This is unlike the usual angular-momentum operators Lz, etc., familiar from spherical-harmonic theory because the exponential is on the right-hand side of the argument. This operator is less common in physics because it represents the dependence of the function on the choice of frame. In terms of the SWSHs, we can write the action of Rz as Rz {s}Y{l,m} = -s * {s}Y{l,m} Equivalently, the modes of a function are affected as {Rz f} {s,l,m} = -s * f{s,l,m} Note the unfortunate sign of s, which seems to be opposite to what we expect, and arises from the choice of definition of s in the original paper by Newman and Penrose.