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"""Known matrices related to physics"""
from sympy.core.numbers import I
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.utilities.decorator import deprecated
def msigma(i):
r"""Returns a Pauli matrix `\sigma_i` with `i=1,2,3`.
References
==========
.. [1] https://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)
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)
def mgamma(mu, lower=False):
r"""Returns a Dirac gamma matrix `\gamma^\mu` in the standard
(Dirac) representation.
Explanation
===========
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] https://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 mu not 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)
))
@deprecated(
"""
The sympy.physics.matrices.mdft method is deprecated. Use
sympy.DFT(n).as_explicit() instead.
""",
deprecated_since_version="1.9",
active_deprecations_target="deprecated-physics-mdft",
)
def mdft(n):
r"""
.. deprecated:: 1.9
Use DFT from sympy.matrices.expressions.fourier instead.
To get identical behavior to ``mdft(n)``, use ``DFT(n).as_explicit()``.
"""
from sympy.matrices.expressions.fourier import DFT
return DFT(n).as_mutable()
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