def main():
a=Symbol("a", real=True)
b=Symbol("b", real=True)
c=Symbol("c", real=True)
p = (a,b,c)
assert u(p, 1).D * u(p, 2) == Matrix(1, 1, [0])
assert u(p, 2).D * u(p, 1) == Matrix(1, 1, [0])
p1,p2,p3 =[Symbol(x, real=True) for x in ["p1","p2","p3"]]
pp1,pp2,pp3 =[Symbol(x, real=True) for x in ["pp1","pp2","pp3"]]
k1,k2,k3 =[Symbol(x, real=True) for x in ["k1","k2","k3"]]
kp1,kp2,kp3 =[Symbol(x, real=True) for x in ["kp1","kp2","kp3"]]
p = (p1,p2,p3)
pp = (pp1,pp2,pp3)
k = (k1,k2,k3)
kp = (kp1,kp2,kp3)
mu = Symbol("mu")
e = (pslash(p)+m*ones(4))*(pslash(k)-m*ones(4))
f = pslash(p)+m*ones(4)
g = pslash(p)-m*ones(4)
#pprint(e)
xprint( 'Tr(f*g)', Tr(f*g) )
#print Tr(pslash(p) * pslash(k)).expand()
M0 = [ ( v(pp, 1).D * mgamma(mu) * u(p, 1) ) * ( u(k, 1).D * mgamma(mu,True) * \
v(kp, 1) ) for mu in range(4)]
M = M0[0]+M0[1]+M0[2]+M0[3]
M = M[0]
assert isinstance(M, Basic)
#print M
#print simplify(M)
d=Symbol("d", real=True) #d=E+m
xprint('M', M)
print "-"*40
M = ((M.subs(E,d-m)).expand() * d**2 ).expand()
xprint('M2', 1/(E+m)**2 * M)
print "-"*40
x,y= M.as_real_imag()
xprint('Re(M)', x)
xprint('Im(M)', y)
e = x**2+y**2
xprint('abs(M)**2', e)
print "-"*40
xprint('Expand(abs(M)**2)', e.expand())
def main():
a = Symbol("a", real=True)
b = Symbol("b", real=True)
c = Symbol("c", real=True)
p = (a, b, c)
assert u(p, 1).D * u(p, 2) == Matrix(1, 1, [0])
assert u(p, 2).D * u(p, 1) == Matrix(1, 1, [0])
p1, p2, p3 = [Symbol(x, real=True) for x in ["p1", "p2", "p3"]]
pp1, pp2, pp3 = [Symbol(x, real=True) for x in ["pp1", "pp2", "pp3"]]
k1, k2, k3 = [Symbol(x, real=True) for x in ["k1", "k2", "k3"]]
kp1, kp2, kp3 = [Symbol(x, real=True) for x in ["kp1", "kp2", "kp3"]]
p = (p1, p2, p3)
pp = (pp1, pp2, pp3)
k = (k1, k2, k3)
kp = (kp1, kp2, kp3)
mu = Symbol("mu")
e = (pslash(p) + m * ones(4)) * (pslash(k) - m * ones(4))
f = pslash(p) + m * ones(4)
g = pslash(p) - m * ones(4)
xprint("Tr(f*g)", Tr(f * g))
M0 = [(v(pp, 1).D * mgamma(mu) * u(p, 1)) *
(u(k, 1).D * mgamma(mu, True) * v(kp, 1)) for mu in range(4)]
M = M0[0] + M0[1] + M0[2] + M0[3]
M = M[0]
if not isinstance(M, Basic):
raise TypeError("Invalid type of variable")
d = Symbol("d", real=True) # d=E+m
xprint("M", M)
print("-" * 40)
M = ((M.subs(E, d - m)).expand() * d**2).expand()
xprint("M2", 1 / (E + m)**2 * M)
print("-" * 40)
x, y = M.as_real_imag()
xprint("Re(M)", x)
xprint("Im(M)", y)
e = x**2 + y**2
xprint("abs(M)**2", e)
print("-" * 40)
xprint("Expand(abs(M)**2)", e.expand())
scattering amplitude of the process:
electron + positron -> photon -> electron + positron
in QED (http://en.wikipedia.org/wiki/Quantum_electrodynamics). The aim is to be
able to do any kind of calculations in QED or standard model in SymPy, but
that's a long journey.
"""
from sympy import Basic,exp,Symbol,sin,Rational,I,Mul, Matrix, \
ones, sqrt, pprint, simplify, Eq, sympify
from sympy.physics import msigma, mgamma
#gamma^mu
gamma0=mgamma(0)
gamma1=mgamma(1)
gamma2=mgamma(2)
gamma3=mgamma(3)
gamma5=mgamma(5)
#sigma_i
sigma1=msigma(1)
sigma2=msigma(2)
sigma3=msigma(3)
E = Symbol("E", real=True)
m = Symbol("m", real=True)
def u(p,r):
""" p = (p1, p2, p3); r = 0,1 """
