examples/qft.py

   1 #!/usr/bin/env python
   2 import iam_sympy_example
   3
   4 from sympy import Basic,exp,Symbol,sin,Rational,I,Mul,NCSymbol, Matrix, \
   5     gamma, sigma, one, Pauli
   6
   7 #gamma^mu
   8 gamma0=gamma(0)
   9 gamma1=gamma(1)
  10 gamma2=gamma(2)
  11 gamma3=gamma(3)
  12 gamma5=gamma(5)
  13
  14 #sigma_i
  15 sigma1=sigma(1)
  16 sigma2=sigma(2)
  17 sigma3=sigma(3)
  18
  19 a=Symbol("a")
  20 b=Symbol("b")
  21 c=Symbol("c")
  22
  23 E = Symbol("E")
  24 m = Symbol("m")
  25
  26 def u(p,r):
  27     """ p = (p1, p2, p3); r = 0,1 """
  28     assert r in [1,2]
  29     p1,p2,p3 = p
  30     if r == 1:
  31         ksi = Matrix([ [1],[0] ])
  32     else:
  33         ksi = Matrix([ [0],[1] ])
  34     a = (sigma1*p1 + sigma2*p2 + sigma3*p3) / (E+m) * ksi
  35     if a ==0:
  36         a = zeronm(2,1)
  37     return (E+m).sqrt() * Matrix([ [ksi[0,0]], [ksi[1,0]], [a[0,0]], [a[1,0]] ])
  38
  39 def v(p,r):
  40     """ p = (p1, p2, p3); r = 0,1 """
  41     assert r in [1,2]
  42     p1,p2,p3 = p
  43     if r == 1:
  44         ksi = Matrix([ [1],[0] ])
  45     else:
  46         ksi = -Matrix([ [0],[1] ])
  47     a = (sigma1*p1 + sigma2*p2 + sigma3*p3) / (E+m) * ksi
  48     if a ==0:
  49         a = zeronm(2,1)
  50     return sqrt(E+m) * Matrix([ [a[0,0]], [a[1,0]], [ksi[0,0]], [ksi[1,0]] ])
  51
  52 def pslash(p):
  53     p1,p2,p3 = p
  54     p0 = sqrt(m**2+p1**2+p2**2+p3**2)
  55     return gamma0*p0-gamma1*p1-gamma2*p2-gamma3*p3
  56
  57 def Tr(M):
  58     assert M.lines == M.cols
  59     t = 0
  60     for i in range(M.lines):
  61         t+=M[i,i]
  62     return t
  63
  64 p = (a,b,c)
  65
  66 assert u(p, 1).D * u(p, 2) == 0
  67 assert u(p, 2).D * u(p, 1) == 0
  68
  69 p1,p2,p3 =[Symbol(x) for x in ["p1","p2","p3"]]
  70 pp1,pp2,pp3 =[Symbol(x) for x in ["pp1","pp2","pp3"]]
  71 k1,k2,k3 =[Symbol(x) for x in ["k1","k2","k3"]]
  72 kp1,kp2,kp3 =[Symbol(x) for x in ["kp1","kp2","kp3"]]
  73
  74 p = (p1,p2,p3)
  75 pp = (pp1,pp2,pp3)
  76
  77 k = (k1,k2,k3)
  78 kp = (kp1,kp2,kp3)
  79
  80 mu = Symbol("mu")
  81
  82 #e = (pslash(p)+m*one(4))*(pslash(k)-m*one(4))
  83 #f = pslash(p)+m*one(4)
  84 #g = pslash(p)-m*one(4)
  85 #print Tr(f*g)
  86 #print Tr(pslash(p) * pslash(k)).expand()
  87
  88 #M0 = [ ( v(pp, 1).D * gamma(mu) * u(p, 1) ) * ( u(k, 1).D * gamma(mu,True) * \
  89 #        v(kp, 1) ) for mu in range(4)]
  90 #M = M0[0]+M0[1]+M0[2]+M0[3]
  91 #assert isinstance(M, Basic)
  92
  93 #d=Symbol("d",True) #d=E+m
  94
  95 #print M
  96 #print "-"*40
  97 #M = ((M.subs(E,d-m)).expand() * d**2 ).expand()
  98 #print "1/(E+m)**2 * ",M
  99 #print "-"*40
 100 #x,y= get_re_im(M)
 101 #print x,y
 102 #e = x**2+y**2
 103 #print e
 104
 105 print Pauli(1)*Pauli(1)
 106 #print Pauli(1)**2
 107 #print Pauli(1)*2*Pauli(1)