B = GA.MV('B','vector',fct=True)
E = GA.MV('E','vector',fct=True)
B.set_coef(1,0,0)
E.set_coef(1,0,0)
B *= gamma_t
E *= gamma_t
B.convert_to_blades()
E.convert_to_blades()
J = GA.MV('J','vector',fct=True)
print '$B$ Magnetic Field Bi-Vector'
print 'B = Bvec gamma_0 =',B
print '$E$ Electric Field Bi-Vector'
print 'E = Evec gamma_0 =',E
F = E+I*B
print '$E+IB$ Electo-Magnetic Field Bi-Vector'
print 'F = E+IB =',F
print '$J$ Four Current'
print 'J =',J
gradF = F.grad()
gradF.convert_to_blades()
print 'Geometric Derivative of Electo-Magnetic Field Bi-Vector'
tex.MV_format(3)
print '\\nabla F =',gradF
print 'All Maxwell Equations are'
print '\\nabla F = J'
print 'Div $E$ and Curl $H$ Equations'
print '<\\nabla F>_1 -J =',gradF.project(1)-J,' = 0'
print 'Curl $E$ and Div $B$ equations'
print '<\\nabla F>_3 =',gradF.project(3),' = 0'
tex.xdvi(filename='Maxwell.tex')
import sys, sympy
import sympy.galgebra.latex_ex as tex
if __name__ == '__main__':
tex.Format()
xbm,alpha_1,delta__nugamma_r = sympy.symbols('xbm alpha_1 delta__nugamma_r')
x = alpha_1*xbm/delta__nugamma_r
print 'x =',x
tex.xdvi()
B = GA.MV('B', 'vector', fct=True)
E = GA.MV('E', 'vector', fct=True)
B.set_coef(1, 0, 0)
E.set_coef(1, 0, 0)
B *= gamma_t
E *= gamma_t
B.convert_to_blades()
E.convert_to_blades()
J = GA.MV('J', 'vector', fct=True)
print '$B$ Magnetic Field Bi-Vector'
print 'B = Bvec gamma_0 =', B
print '$E$ Electric Field Bi-Vector'
print 'E = Evec gamma_0 =', E
F = E + I * B
print '$E+IB$ Electo-Magnetic Field Bi-Vector'
print 'F = E+IB =', F
print '$J$ Four Current'
print 'J =', J
gradF = F.grad()
gradF.convert_to_blades()
print 'Geometric Derivative of Electo-Magnetic Field Bi-Vector'
tex.MV_format(3)
print '\\nabla F =', gradF
print 'All Maxwell Equations are'
print '\\nabla F = J'
print 'Div $E$ and Curl $H$ Equations'
print '<\\nabla F>_1 -J =', gradF.project(1) - J, ' = 0'
print 'Curl $E$ and Div $B$ equations'
print '<\\nabla F>_3 =', gradF.project(3), ' = 0'
tex.xdvi(filename='Maxwell.tex')
'0 -1 0 0,'+\
'0 0 -1 0,'+\
'0 0 0 -1'
vars = GA.make_symbols('t x y z')
GA.MV.setup('gamma_t gamma_x gamma_y gamma_z',metric,True,vars)
parms = GA.make_symbols('m e')
tex.Format()
I = GA.MV(GA.ONE,'pseudo')
nvars = len(vars)
psi = GA.MV('psi','spinor',fct=True)
psi.convert_to_blades()
A = GA.MV('A','vector',fct=True)
sig_x = gamma_x*gamma_t
sig_y = gamma_y*gamma_t
sig_z = gamma_z*gamma_t
print '$A$ is 4-vector potential'
print A
print r'$\bm{\psi}$ is 8-component real spinor (even multi-vector)'
print psi
dirac_eq = psi.grad()*I*sig_z-e*A*psi-m*psi*gamma_t
dirac_eq.simplify()
dirac_eq.convert_to_blades()
print 'Dirac equation in terms of real geometric algebra/calculus '+\
r'$\lp\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi} = m\bm{\psi}\gamma_{t}\rp$'
print 'Spin measured with respect to $z$ axis'
tex.MV_format(3)
print r'\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi}-m\bm{\psi}\gamma_{t} = ',dirac_eq,' = 0'
tex.xdvi(filename='Dirac.tex')
'gamma_x gamma_y gamma_z', metric, True)
tex.Format()
coords = r, theta, phi = sympy.symbols('r theta phi')
x = r*(sympy.cos(theta)*gamma_z + sympy.sin(theta)*
(sympy.cos(phi)*gamma_x + sympy.sin(phi)*gamma_y))
x.set_name('x')
GA.MV.rebase(x, coords, 'e', True)
psi = GA.MV('psi', 'scalar', fct=True)
dpsi = psi.grad()
print 'Gradient of Scalar Function $\\psi$'
print '\\nabla\\psi =', dpsi
A = GA.MV('A', 'vector', fct=True)
print 'Div and Curl of Vector Function $A$'
print A
gradA = A.grad()
I = GA.MV(GA.ONE, 'pseudo')
divA = A.grad_int()
curlA = -I*A.grad_ext()
print '\\nabla \\cdot A =', divA
tex.MV_format(3)
print '-I\\lp\\nabla \\W A\\rp =', curlA
tex.xdvi(filename='coords.tex')
True)
tex.Format()
coords = r, theta, phi = sympy.symbols('r theta phi')
x = r * (sympy.cos(theta) * gamma_z + sympy.sin(theta) *
(sympy.cos(phi) * gamma_x + sympy.sin(phi) * gamma_y))
x.set_name('x')
GA.MV.rebase(x, coords, 'e', True)
psi = GA.MV('psi', 'scalar', fct=True)
dpsi = psi.grad()
print 'Gradient of Scalar Function $\\psi$'
print '\\nabla\\psi =', dpsi
A = GA.MV('A', 'vector', fct=True)
print 'Div and Curl of Vector Function $A$'
print A
gradA = A.grad()
I = GA.MV(GA.ONE, 'pseudo')
divA = A.grad_int()
curlA = -I * A.grad_ext()
print '\\nabla \\cdot A =', divA
tex.MV_format(3)
print '-I\\lp\\nabla \\W A\\rp =', curlA
tex.xdvi(filename='coords.tex')
B = GA.MV("B", "vector", fct=True)
E = GA.MV("E", "vector", fct=True)
B.set_coef(1, 0, 0)
E.set_coef(1, 0, 0)
B *= gamma_t
E *= gamma_t
B.convert_to_blades()
E.convert_to_blades()
J = GA.MV("J", "vector", fct=True)
print "$B$ Magnetic Field Bi-Vector"
print "B = Bvec gamma_0 =", B
print "$E$ Electric Field Bi-Vector"
print "E = Evec gamma_0 =", E
F = E + I * B
print "$E+IB$ Electo-Magnetic Field Bi-Vector"
print "F = E+IB =", F
print "$J$ Four Current"
print "J =", J
gradF = F.grad()
gradF.convert_to_blades()
print "Geometric Derivative of Electo-Magnetic Field Bi-Vector"
tex.MV_format(3)
print "\\nabla F =", gradF
print "All Maxwell Equations are"
print "\\nabla F = J"
print "Div $E$ and Curl $H$ Equations"
print "<\\nabla F>_1 -J =", gradF.project(1) - J, " = 0"
print "Curl $E$ and Div $B$ equations"
print "<\\nabla F>_3 =", gradF.project(3), " = 0"
tex.xdvi(filename="Maxwell.tex")
'0 -1 0 0,'+\
'0 0 -1 0,'+\
'0 0 0 -1'
vars = GA.make_symbols('t x y z')
GA.MV.setup('gamma_t gamma_x gamma_y gamma_z', metric, True, vars)
parms = GA.make_symbols('m e')
tex.Format()
I = GA.MV(GA.ONE, 'pseudo')
nvars = len(vars)
psi = GA.MV('psi', 'spinor', fct=True)
psi.convert_to_blades()
A = GA.MV('A', 'vector', fct=True)
sig_x = gamma_x * gamma_t
sig_y = gamma_y * gamma_t
sig_z = gamma_z * gamma_t
print '$A$ is 4-vector potential'
print A
print r'$\bm{\psi}$ is 8-component real spinor (even multi-vector)'
print psi
dirac_eq = psi.grad() * I * sig_z - e * A * psi - m * psi * gamma_t
dirac_eq.simplify()
dirac_eq.convert_to_blades()
print 'Dirac equation in terms of real geometric algebra/calculus '+\
r'$\lp\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi} = m\bm{\psi}\gamma_{t}\rp$'
print 'Spin measured with respect to $z$ axis'
tex.MV_format(3)
print r'\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi}-m\bm{\psi}\gamma_{t} = ', dirac_eq, ' = 0'
tex.xdvi(filename='Dirac.tex')
gamma_x, gamma_y, gamma_z = GA.MV.setup("gamma_x gamma_y gamma_z", metric, True)
tex.Format()
coords = r, theta, phi = sympy.symbols("r theta phi")
x = r * (sympy.cos(theta) * gamma_z + sympy.sin(theta) * (sympy.cos(phi) * gamma_x + sympy.sin(phi) * gamma_y))
x.set_name("x")
GA.MV.rebase(x, coords, "e", True)
psi = GA.MV("psi", "scalar", fct=True)
dpsi = psi.grad()
print "Gradient of Scalar Function $\\psi$"
print "\\nabla\\psi =", dpsi
A = GA.MV("A", "vector", fct=True)
print "Div and Curl of Vector Function $A$"
print A
gradA = A.grad()
I = GA.MV(GA.ONE, "pseudo")
divA = A.grad_int()
curlA = -I * A.grad_ext()
print "\\nabla \\cdot A =", divA
tex.MV_format(3)
print "-I\\lp\\nabla \\W A\\rp =", curlA
tex.xdvi(filename="coords.tex")
import sys, sympy
import sympy.galgebra.latex_ex as tex
if __name__ == '__main__':
tex.Format()
xbm, alpha_1, delta__nugamma_r = sympy.symbols(
'xbm alpha_1 delta__nugamma_r')
x = alpha_1 * xbm / delta__nugamma_r
print 'x =', x
tex.xdvi()
if __name__ == "__main__":
metric = "1 0 0 0," + "0 -1 0 0," + "0 0 -1 0," + "0 0 0 -1"
vars = GA.make_symbols("t x y z")
GA.MV.setup("gamma_t gamma_x gamma_y gamma_z", metric, True, vars)
parms = GA.make_symbols("m e")
tex.Format()
I = GA.MV(GA.ONE, "pseudo")
nvars = len(vars)
psi = GA.MV("psi", "spinor", fct=True)
psi.convert_to_blades()
A = GA.MV("A", "vector", fct=True)
sig_x = gamma_x * gamma_t
sig_y = gamma_y * gamma_t
sig_z = gamma_z * gamma_t
print "$A$ is 4-vector potential"
print A
print r"$\bm{\psi}$ is 8-component real spinor (even multi-vector)"
print psi
dirac_eq = psi.grad() * I * sig_z - e * A * psi - m * psi * gamma_t
dirac_eq.simplify()
dirac_eq.convert_to_blades()
print "Dirac equation in terms of real geometric algebra/calculus " + r"$\lp\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi} = m\bm{\psi}\gamma_{t}\rp$"
print "Spin measured with respect to $z$ axis"
tex.MV_format(3)
print r"\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi}-m\bm{\psi}\gamma_{t} = ", dirac_eq, " = 0"
tex.xdvi(filename="Dirac.tex")
