def check_generalized_BAC_CAB_formulas():
Print_Function()
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)} =',com(a^b,c^d)
return
def check_generalized_BAC_CAB_formulas():
Print_Function()
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)} =',com(a^b,c^d)
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) =',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()
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) =',com(a^b,c^d))
print('(a|(b^c))|(d^e) =',(a|(b^c))|(d^e))
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)} =',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()
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)} =', 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()
return