Source code for sympy.physics.matrices

"""Known matrices related to physics"""

from __future__ import print_function, division

from sympy import Matrix, I


[docs]def msigma(i): r"""Returns a Pauli matrix `\sigma_i` with `i=1,2,3` References ========== .. [1] http://en.wikipedia.org/wiki/Pauli_matrices Examples ======== >>> from sympy.physics.matrices import msigma >>> msigma(1) Matrix([ [0, 1], [1, 0]]) """ if i == 1: mat = ( ( (0, 1), (1, 0) ) ) elif i == 2: mat = ( ( (0, -I), (I, 0) ) ) elif i == 3: mat = ( ( (1, 0), (0, -1) ) ) else: raise IndexError("Invalid Pauli index") return Matrix(mat)
[docs]def pat_matrix(m, dx, dy, dz): """Returns the Parallel Axis Theorem matrix to translate the inertia matrix a distance of `(dx, dy, dz)` for a body of mass m. Examples ======== To translate a body having a mass of 2 units a distance of 1 unit along the `x`-axis we get: >>> from sympy.physics.matrices import pat_matrix >>> pat_matrix(2, 1, 0, 0) Matrix([ [0, 0, 0], [0, 2, 0], [0, 0, 2]]) """ dxdy = -dx*dy dydz = -dy*dz dzdx = -dz*dx dxdx = dx**2 dydy = dy**2 dzdz = dz**2 mat = ((dydy + dzdz, dxdy, dzdx), (dxdy, dxdx + dzdz, dydz), (dzdx, dydz, dydy + dxdx)) return m*Matrix(mat)
[docs]def mgamma(mu, lower=False): r"""Returns a Dirac gamma matrix `\gamma^\mu` in the standard (Dirac) representation. If you want `\gamma_\mu`, use ``gamma(mu, True)``. We use a convention: `\gamma^5 = i \cdot \gamma^0 \cdot \gamma^1 \cdot \gamma^2 \cdot \gamma^3` `\gamma_5 = i \cdot \gamma_0 \cdot \gamma_1 \cdot \gamma_2 \cdot \gamma_3 = - \gamma^5` References ========== .. [1] http://en.wikipedia.org/wiki/Gamma_matrices Examples ======== >>> from sympy.physics.matrices import mgamma >>> mgamma(1) Matrix([ [ 0, 0, 0, 1], [ 0, 0, 1, 0], [ 0, -1, 0, 0], [-1, 0, 0, 0]]) """ if not mu in [0, 1, 2, 3, 5]: raise IndexError("Invalid Dirac index") if mu == 0: mat = ( (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, -1, 0), (0, 0, 0, -1) ) elif mu == 1: mat = ( (0, 0, 0, 1), (0, 0, 1, 0), (0, -1, 0, 0), (-1, 0, 0, 0) ) elif mu == 2: mat = ( (0, 0, 0, -I), (0, 0, I, 0), (0, I, 0, 0), (-I, 0, 0, 0) ) elif mu == 3: mat = ( (0, 0, 1, 0), (0, 0, 0, -1), (-1, 0, 0, 0), (0, 1, 0, 0) ) elif mu == 5: mat = ( (0, 0, 1, 0), (0, 0, 0, 1), (1, 0, 0, 0), (0, 1, 0, 0) ) m = Matrix(mat) if lower: if mu in [1, 2, 3, 5]: m = -m return m #Minkowski tensor using the convention (+,-,-,-) used in the Quantum Field #Theory
minkowski_tensor = Matrix( ( (1, 0, 0, 0), (0, -1, 0, 0), (0, 0, -1, 0), (0, 0, 0, -1) ))