def check_generalized_BAC_CAB_formulas():
Print_Function()
(a,b,c,d) = MV.setup('a b c d')
print('g_{ij} =',MV.metric)
print('\\bm{a|(b*c)} =',a|(b*c))
print('\\bm{a|(b^c)} =',a|(b^c))
print('\\bm{a|(b^c^d)} =',a|(b^c^d))
print('\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',(a|(b^c))+(c|(a^b))+(b|(c^a)))
print('\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',a*(b^c)-b*(a^c)+c*(a^b))
print('\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c))
print('\\bm{(a^b)|(c^d)} =',(a^b)|(c^d))
print('\\bm{((a^b)|c)|d} =',((a^b)|c)|d)
print('\\bm{(a^b)\\times (c^d)} =',Ga.com(a^b,c^d))
return
def check_generalized_BAC_CAB_formulas():
(a,b,c,d,e) = MV.setup('a b c d e')
print('g_{ij} =\n',MV.metric)
print('a|(b*c) =',a|(b*c))
print('a|(b^c) =',a|(b^c))
print('a|(b^c^d) =',a|(b^c^d))
print('a|(b^c)+c|(a^b)+b|(c^a) =',(a|(b^c))+(c|(a^b))+(b|(c^a)))
print('a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b))
print('a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c) =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c))
print('(a^b)|(c^d) =',(a^b)|(c^d))
print('((a^b)|c)|d =',((a^b)|c)|d)
print('(a^b)x(c^d) =',Ga.com(a^b,c^d))
print('(a|(b^c))|(d^e) =',(a|(b^c))|(d^e))
return
def test_check_generalized_BAC_CAB_formulas(self):
(a,b,c,d,e) = Ga('a b c d e').mv()
assert str(a|(b*c)) == '-(a.c)*b + (a.b)*c'
assert str(a|(b^c)) == '-(a.c)*b + (a.b)*c'
assert str(a|(b^c^d)) == '(a.d)*b^c - (a.c)*b^d + (a.b)*c^d'
expr = (a|(b^c))+(c|(a^b))+(b|(c^a)) # = (a.b)*c - (b.c)*a - ((a.b)*c - (b.c)*a)
assert str(expr.simplify()) == '0'
assert str(a*(b^c)-b*(a^c)+c*(a^b)) == '3*a^b^c'
assert str(a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)) == '4*a^b^c^d'
assert str((a^b)|(c^d)) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(((a^b)|c)|d) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(Ga.com(a^b,c^d)) == '-(b.d)*a^c + (b.c)*a^d + (a.d)*b^c - (a.c)*b^d'
assert str((a|(b^c))|(d^e)) == '(-(a.b)*(c.e) + (a.c)*(b.e))*d + ((a.b)*(c.d) - (a.c)*(b.d))*e'
return
def test_check_generalized_BAC_CAB_formulas(self):
(a,b,c,d,e) = Ga('a b c d e').mv()
assert str(a|(b*c)) == '-(a.c)*b + (a.b)*c'
assert str(a|(b^c)) == '-(a.c)*b + (a.b)*c'
assert str(a|(b^c^d)) == '(a.d)*b^c - (a.c)*b^d + (a.b)*c^d'
expr = (a|(b^c))+(c|(a^b))+(b|(c^a)) # = (a.b)*c - (b.c)*a - ((a.b)*c - (b.c)*a)
assert str(expr.simplify()) == '0'
assert str(a*(b^c)-b*(a^c)+c*(a^b)) == '3*a^b^c'
assert str(a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)) == '4*a^b^c^d'
assert str((a^b)|(c^d)) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(((a^b)|c)|d) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(Ga.com(a^b,c^d)) == '-(b.d)*a^c + (b.c)*a^d + (a.d)*b^c - (a.c)*b^d'
assert str((a|(b^c))|(d^e)) == '(-(a.b)*(c.e) + (a.c)*(b.e))*d + ((a.b)*(c.d) - (a.c)*(b.d))*e'
return
def check_generalized_BAC_CAB_formulas():
Print_Function()
g5d = Ga('a b c d e')
(a, b, c, d, e) = g5d.mv()
print('g_{ij} =\n', g5d.g)
print('a|(b*c) =', a | (b * c))
print('a|(b^c) =', a | (b ^ c))
print('a|(b^c^d) =', a | (b ^ c ^ d))
print('a|(b^c)+c|(a^b)+b|(c^a) =', (a | ( b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)))
print('a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b))
print('a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c) =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c))
print('(a^b)|(c^d) =',(a^b)|(c^d))
print('((a^b)|c)|d =',((a^b)|c)|d)
print('(a^b)x(c^d) =',Ga.com(a^b,c^d))
print('(a|(b^c))|(d^e) =',(a|(b^c))|(d^e))
return
def check_generalized_BAC_CAB_formulas():
Print_Function()
g4d = Ga('a b c d e')
(a, b, c, d, e) = g4d.mv()
print('g_{ij} =', g4d.g)
print('\\bm{a|(b*c)} =', a | (b * c))
print('\\bm{a|(b^c)} =', a | (b ^ c))
print('\\bm{a|(b^c^d)} =', a | (b ^ c ^ d))
print('\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',
(a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)))
print('\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',
a * (b ^ c) - b * (a ^ c) + c * (a ^ b))
print(
'\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',
a * (b ^ c ^ d) - b * (a ^ c ^ d) + c * (a ^ b ^ d) - d * (a ^ b ^ c))
print('\\bm{(a^b)|(c^d)} =', (a ^ b) | (c ^ d))
print('\\bm{((a^b)|c)|d} =', ((a ^ b) | c) | d)
print('\\bm{(a^b)\\times (c^d)} =', Ga.com(a ^ b, c ^ d))
print('\\bm{(a^b^c)(d^e)} =', ((a ^ b ^ c) * (d ^ e)).Fmt(2))
return
def main():
Format()
(g3d, ex, ey, ez) = Ga.build('e*x|y|z')
A = g3d.mv('A', 'mv')
print(r'\bm{A} =', A)
A.Fmt(2, r'\bm{A}')
A.Fmt(3, r'\bm{A}')
X = (x, y, z) = symbols('x y z', real=True)
o3d = Ga('e_x e_y e_z', g=[1, 1, 1], coords=X)
(ex, ey, ez) = o3d.mv()
f = o3d.mv('f', 'scalar', f=True)
A = o3d.mv('A', 'vector', f=True)
B = o3d.mv('B', 'bivector', f=True)
print(r'\bm{A} =', A)
print(r'\bm{B} =', B)
print('grad*f =', o3d.grad * f)
print(r'grad|\bm{A} =', o3d.grad | A)
print(r'grad*\bm{A} =', o3d.grad * A)
print(r'-I*(grad^\bm{A}) =', -o3d.i * (o3d.grad ^ A))
print(r'grad*\bm{B} =', o3d.grad * B)
print(r'grad^\bm{B} =', o3d.grad ^ B)
print(r'grad|\bm{B} =', o3d.grad | B)
g4d = Ga('a b c d')
(a, b, c, d) = g4d.mv()
print('g_{ij} =', g4d.g)
print('\\bm{a|(b*c)} =', a | (b * c))
print('\\bm{a|(b^c)} =', a | (b ^ c))
print('\\bm{a|(b^c^d)} =', a | (b ^ c ^ d))
print('\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',
(a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)))
print('\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',
a * (b ^ c) - b * (a ^ c) + c * (a ^ b))
print(
'\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',
a * (b ^ c ^ d) - b * (a ^ c ^ d) + c * (a ^ b ^ d) - d * (a ^ b ^ c))
print('\\bm{(a^b)|(c^d)} =', (a ^ b) | (c ^ d))
print('\\bm{((a^b)|c)|d} =', ((a ^ b) | c) | d)
print('\\bm{(a^b)\\times (c^d)} =', Ga.com(a ^ b, c ^ d))
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
ng3d = Ga('e1 e2 e3', g=g)
(e1, e2, e3) = ng3d.mv()
E = e1 ^ e2 ^ e3
Esq = (E * E).scalar()
print('E =', E)
print('%E^{2} =', Esq)
Esq_inv = 1 / Esq
E1 = (e2 ^ e3) * E
E2 = (-1) * (e1 ^ e3) * E
E3 = (e1 ^ e2) * E
print('E1 = (e2^e3)*E =', E1)
print('E2 =-(e1^e3)*E =', E2)
print('E3 = (e1^e2)*E =', E3)
print('E1|e2 =', (E1 | e2).expand())
print('E1|e3 =', (E1 | e3).expand())
print('E2|e1 =', (E2 | e1).expand())
print('E2|e3 =', (E2 | e3).expand())
print('E3|e1 =', (E3 | e1).expand())
print('E3|e2 =', (E3 | e2).expand())
w = ((E1 | e1).expand()).scalar()
Esq = expand(Esq)
print('%(E1\\cdot e1)/E^{2} =', simplify(w / Esq))
w = ((E2 | e2).expand()).scalar()
print('%(E2\\cdot e2)/E^{2} =', simplify(w / Esq))
w = ((E3 | e3).expand()).scalar()
print('%(E3\\cdot e3)/E^{2} =', simplify(w / Esq))
X = (r, th, phi) = symbols('r theta phi')
s3d = Ga('e_r e_theta e_phi',
g=[1, r**2, r**2 * sin(th)**2],
coords=X,
norm=True)
(er, eth, ephi) = s3d.mv()
f = s3d.mv('f', 'scalar', f=True)
A = s3d.mv('A', 'vector', f=True)
B = s3d.mv('B', 'bivector', f=True)
print('A =', A)
print('B =', B)
print('grad*f =', s3d.grad * f)
print('grad|A =', s3d.grad | A)
print('-I*(grad^A) =', -s3d.i * (s3d.grad ^ A))
print('grad^B =', s3d.grad ^ B)
coords = symbols('t x y z')
m4d = Ga('gamma*t|x|y|z', g=[1, -1, -1, -1], coords=coords)
(g0, g1, g2, g3) = m4d.mv()
I = m4d.i
B = m4d.mv('B', 'vector', f=True)
E = m4d.mv('E', 'vector', f=True)
B.set_coef(1, 0, 0)
E.set_coef(1, 0, 0)
B *= g0
E *= g0
J = m4d.mv('J', 'vector', f=True)
F = E + I * B
print('B = \\bm{B\\gamma_{t}} =', B)
print('E = \\bm{E\\gamma_{t}} =', E)
print('F = E+IB =', F)
print('J =', J)
gradF = m4d.grad * F
gradF.Fmt(3, 'grad*F')
print('grad*F = J')
(gradF.get_grade(1) - J).Fmt(3, '%\\grade{\\nabla F}_{1} -J = 0')
(gradF.get_grade(3)).Fmt(3, '%\\grade{\\nabla F}_{3} = 0')
(alpha, beta, gamma) = symbols('alpha beta gamma')
(x, t, xp, tp) = symbols("x t x' t'")
m2d = Ga('gamma*t|x', g=[1, -1])
(g0, g1) = m2d.mv()
R = cosh(alpha / 2) + sinh(alpha / 2) * (g0 ^ g1)
X = t * g0 + x * g1
Xp = tp * g0 + xp * g1
print('R =', R)
print(
r"#%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} = t'\bm{\gamma'_{t}}+x'\bm{\gamma'_{x}} = R\lp t'\bm{\gamma_{t}}+x'\bm{\gamma_{x}}\rp R^{\dagger}"
)
Xpp = R * Xp * R.rev()
Xpp = Xpp.collect()
Xpp = Xpp.trigsimp()
print(r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =", Xpp)
Xpp = Xpp.subs({sinh(alpha): gamma * beta, cosh(alpha): gamma})
print(r'%\f{\sinh}{\alpha} = \gamma\beta')
print(r'%\f{\cosh}{\alpha} = \gamma')
print(r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =", Xpp.collect())
coords = symbols('t x y z')
m4d = Ga('gamma*t|x|y|z', g=[1, -1, -1, -1], coords=coords)
(g0, g1, g2, g3) = m4d.mv()
I = m4d.i
(m, e) = symbols('m e')
psi = m4d.mv('psi', 'spinor', f=True)
A = m4d.mv('A', 'vector', f=True)
sig_z = g3 * g0
print('\\bm{A} =', A)
print('\\bm{\\psi} =', psi)
dirac_eq = (m4d.grad * psi) * I * sig_z - e * A * psi - m * psi * g0
dirac_eq.simplify()
dirac_eq.Fmt(
3,
r'\nabla \bm{\psi} I \sigma_{z}-e\bm{A}\bm{\psi}-m\bm{\psi}\gamma_{t} = 0'
)
# xpdf()
xpdf(pdfprog=None)
return
from __future__ import absolute_import, division
from __future__ import print_function
from galgebra.printer import Format, xpdf
from galgebra.ga import Ga
Format()
g4d = Ga('a b c d')
(a, b, c, d) = g4d.mv()
print('g_{ij} =', g4d.g)
print('\\bm{a|(b*c)} =', a | (b * c))
print('\\bm{a|(b^c)} =', a | (b ^ c))
print('\\bm{a|(b^c^d)} =', a | (b ^ c ^ d))
# FIXME:FIXED this should print 0, got blank
print('\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',
(a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)))
print('\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',
a * (b ^ c) - b * (a ^ c) + c * (a ^ b))
print('\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',
a * (b ^ c ^ d) - b * (a ^ c ^ d) + c * (a ^ b ^ d) - d * (a ^ b ^ c))
print('\\bm{(a^b)|(c^d)} =', (a ^ b) | (c ^ d))
print('\\bm{((a^b)|c)|d} =', ((a ^ b) | c) | d)
print('\\bm{(a^b)\\times (c^d)} =', Ga.com(a ^ b, c ^ d))
xpdf(paper='letter', prog=True)
def main():
Format()
(ex, ey, ez) = MV.setup('e*x|y|z')
A = MV('A', 'mv')
print(r'\bm{A} =', A)
A.Fmt(2, r'\bm{A}')
A.Fmt(3, r'\bm{A}')
X = (x, y, z) = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e_x e_y e_z', metric='[1,1,1]', coords=X)
f = MV('f', 'scalar', fct=True)
A = MV('A', 'vector', fct=True)
B = MV('B', 'grade2', fct=True)
print(r'\bm{A} =', A)
print(r'\bm{B} =', B)
print('grad*f =', grad * f)
print(r'grad|\bm{A} =', grad | A)
print(r'grad*\bm{A} =', grad * A)
print(r'-I*(grad^\bm{A}) =', -MV.I * (grad ^ A))
print(r'grad*\bm{B} =', grad * B)
print(r'grad^\bm{B} =', grad ^ B)
print(r'grad|\bm{B} =', grad | B)
(a, b, c, d) = MV.setup('a b c d')
print('g_{ij} =', MV.metric)
print('\\bm{a|(b*c)} =', a | (b * c))
print('\\bm{a|(b^c)} =', a | (b ^ c))
print('\\bm{a|(b^c^d)} =', a | (b ^ c ^ d))
print('\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',
(a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)))
print('\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',
a * (b ^ c) - b * (a ^ c) + c * (a ^ b))
print(
'\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',
a * (b ^ c ^ d) - b * (a ^ c ^ d) + c * (a ^ b ^ d) - d * (a ^ b ^ c))
print('\\bm{(a^b)|(c^d)} =', (a ^ b) | (c ^ d))
print('\\bm{((a^b)|c)|d} =', ((a ^ b) | c) | d)
print('\\bm{(a^b)\\times (c^d)} =', Ga.com(a ^ b, c ^ d))
metric = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
(e1, e2, e3) = MV.setup('e1 e2 e3', metric)
E = e1 ^ e2 ^ e3
Esq = (E * E).scalar()
print('E =', E)
print('%E^{2} =', Esq)
Esq_inv = 1 / Esq
E1 = (e2 ^ e3) * E
E2 = (-1) * (e1 ^ e3) * E
E3 = (e1 ^ e2) * E
print('E1 = (e2^e3)*E =', E1)
print('E2 =-(e1^e3)*E =', E2)
print('E3 = (e1^e2)*E =', E3)
print('E1|e2 =', (E1 | e2).expand())
print('E1|e3 =', (E1 | e3).expand())
print('E2|e1 =', (E2 | e1).expand())
print('E2|e3 =', (E2 | e3).expand())
print('E3|e1 =', (E3 | e1).expand())
print('E3|e2 =', (E3 | e2).expand())
w = ((E1 | e1).expand()).scalar()
Esq = expand(Esq)
print('%(E1\\cdot e1)/E^{2} =', simplify(w / Esq))
w = ((E2 | e2).expand()).scalar()
print('%(E2\\cdot e2)/E^{2} =', simplify(w / Esq))
w = ((E3 | e3).expand()).scalar()
print('%(E3\\cdot e3)/E^{2} =', simplify(w / Esq))
X = (r, th, phi) = symbols('r theta phi')
curv = [[r * cos(phi) * sin(th), r * sin(phi) * sin(th), r * cos(th)],
[1, r, r * sin(th)]]
(er, eth, ephi, grad) = MV.setup('e_r e_theta e_phi',
metric='[1,1,1]',
coords=X,
curv=curv)
f = MV('f', 'scalar', fct=True)
A = MV('A', 'vector', fct=True)
B = MV('B', 'grade2', fct=True)
print('A =', A)
print('B =', B)
print('grad*f =', grad * f)
print('grad|A =', grad | A)
print('-I*(grad^A) =', -MV.I * (grad ^ A))
print('grad^B =', grad ^ B)
vars = symbols('t x y z')
(g0, g1, g2, g3, grad) = MV.setup('gamma*t|x|y|z',
metric='[1,-1,-1,-1]',
coords=vars)
I = MV.I
B = MV('B', 'vector', fct=True)
E = MV('E', 'vector', fct=True)
B.set_coef(1, 0, 0)
E.set_coef(1, 0, 0)
B *= g0
E *= g0
J = MV('J', 'vector', fct=True)
F = E + I * B
print('B = \\bm{B\\gamma_{t}} =', B)
print('E = \\bm{E\\gamma_{t}} =', E)
print('F = E+IB =', F)
print('J =', J)
gradF = grad * F
gradF.Fmt(3, 'grad*F')
print('grad*F = J')
(gradF.grade(1) - J).Fmt(3, '%\\grade{\\nabla F}_{1} -J = 0')
(gradF.grade(3)).Fmt(3, '%\\grade{\\nabla F}_{3} = 0')
(alpha, beta, gamma) = symbols('alpha beta gamma')
(x, t, xp, tp) = symbols("x t x' t'")
(g0, g1) = MV.setup('gamma*t|x', metric='[1,-1]')
R = cosh(alpha / 2) + sinh(alpha / 2) * (g0 ^ g1)
X = t * g0 + x * g1
Xp = tp * g0 + xp * g1
print('R =', R)
print(
r"#%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} = t'\bm{\gamma'_{t}}+x'\bm{\gamma'_{x}} = R\lp t'\bm{\gamma_{t}}+x'\bm{\gamma_{x}}\rp R^{\dagger}"
)
Xpp = R * Xp * R.rev()
Xpp = Xpp.collect()
Xpp = Xpp.subs({
2 * sinh(alpha / 2) * cosh(alpha / 2): sinh(alpha),
sinh(alpha / 2)**2 + cosh(alpha / 2)**2: cosh(alpha)
})
print(r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =", Xpp)
Xpp = Xpp.subs({sinh(alpha): gamma * beta, cosh(alpha): gamma})
print(r'%\f{\sinh}{\alpha} = \gamma\beta')
print(r'%\f{\cosh}{\alpha} = \gamma')
print(r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =", Xpp.collect())
vars = symbols('t x y z')
(g0, g1, g2, g3, grad) = MV.setup('gamma*t|x|y|z',
metric='[1,-1,-1,-1]',
coords=vars)
I = MV.I
(m, e) = symbols('m e')
psi = MV('psi', 'spinor', fct=True)
A = MV('A', 'vector', fct=True)
sig_z = g3 * g0
print('\\bm{A} =', A)
print('\\bm{\\psi} =', psi)
dirac_eq = (grad * psi) * I * sig_z - e * A * psi - m * psi * g0
dirac_eq.simplify()
dirac_eq.Fmt(
3,
r'\nabla \bm{\psi} I \sigma_{z}-e\bm{A}\bm{\psi}-m\bm{\psi}\gamma_{t} = 0'
)
# xpdf()
xpdf(pdfprog=None)
return
