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 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