return (ti & ~0xff) == TINFO_clifford;
}
+/** Take trace of a string of an even number of Dirac gammas given a vector
+ * of indices. */
+static ex trace_string(exvector::const_iterator ix, unsigned num)
+{
+ // Tr gamma.mu gamma.nu = 4 g.mu.nu
+ if (num == 2)
+ return lorentz_g(ix[0], ix[1]);
+
+ // Tr gamma.mu gamma.nu gamma.rho gamma.sig = 4 (g.mu.nu g.rho.sig + g.nu.rho g.mu.sig - g.mu.rho g.nu.sig
+ else if (num == 4)
+ return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
+ + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
+ - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
+
+ // Traces of 6 or more gammas are computed recursively:
+ // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
+ // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
+ // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
+ // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
+ // - ...
+ // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
+ exvector v(num - 2);
+ int sign = 1;
+ ex result;
+ for (int i=1; i<num; i++) {
+ for (int n=1, j=0; n<num; n++) {
+ if (n == i)
+ continue;
+ v[j++] = ix[n];
+ }
+ result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
+ sign = -sign;
+ }
+ return result;
+}
+
ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
{
if (is_ex_of_type(e, clifford)) {
if (num == 5)
return trONE * I * eps0123(e.op(1).op(1), e.op(2).op(1), e.op(3).op(1), e.op(4).op(1));
- // Tr gamma5 gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 gamma.mu5 gamma.mu6 = ...
- if (num == 7) {
- ex i1 = e.op(1).op(1), i2 = e.op(2).op(1),
- i3 = e.op(3).op(1), i4 = e.op(4).op(1),
- i5 = e.op(5).op(1), i6 = e.op(6).op(1);
- return trONE * I * (lorentz_g(i1, i2) * eps0123(i3, i4, i5, i6)
- - lorentz_g(i1, i3) * eps0123(i2, i4, i5, i6)
- + lorentz_g(i1, i4) * eps0123(i2, i3, i5, i6)
- - lorentz_g(i1, i5) * eps0123(i2, i3, i4, i6)
- + lorentz_g(i1, i6) * eps0123(i2, i3, i4, i5)
- + lorentz_g(i2, i3) * eps0123(i1, i4, i5, i6)
- - lorentz_g(i2, i4) * eps0123(i1, i3, i5, i6)
- + lorentz_g(i2, i5) * eps0123(i1, i3, i4, i6)
- - lorentz_g(i2, i6) * eps0123(i1, i3, i4, i5)
- + lorentz_g(i3, i4) * eps0123(i1, i2, i5, i6)
- - lorentz_g(i3, i5) * eps0123(i1, i2, i4, i6)
- + lorentz_g(i3, i6) * eps0123(i1, i2, i4, i5)
- + lorentz_g(i4, i5) * eps0123(i1, i2, i3, i6)
- - lorentz_g(i4, i6) * eps0123(i1, i2, i3, i5)
- + lorentz_g(i5, i6) * eps0123(i1, i2, i3, i4));
- }
-
- // Tr gamma5 S_2k =
+ // Tr gamma5 S_2k =
// I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
+ exvector ix;
+ ix.reserve(num - 1);
+ for (unsigned i=1; i<num; i++)
+ ix.push_back(e.op(i).op(1));
+ num--;
+ int *iv = new int[num];
ex result;
- for (int i=1; i<num-3; i++) {
- ex idx1 = e.op(i).op(1);
+ for (int i=0; i<num-3; i++) {
+ ex idx1 = ix[i];
for (int j=i+1; j<num-2; j++) {
- ex idx2 = e.op(j).op(1);
+ ex idx2 = ix[j];
for (int k=j+1; k<num-1; k++) {
- ex idx3 = e.op(k).op(1);
+ ex idx3 = ix[k];
for (int l=k+1; l<num; l++) {
- ex idx4 = e.op(l).op(1);
- vector<int> iv;
- iv.reserve(num-1);
+ ex idx4 = ix[l];
+ iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
exvector v;
- v.reserve(num-1);
- iv.push_back(i); iv.push_back(j); iv.push_back(k); iv.push_back(l);
- for (int n=1; n<num; n++) {
+ v.reserve(num - 4);
+ for (int n=0, t=4; n<num; n++) {
if (n == i || n == j || n == k || n == l)
continue;
- iv.push_back(n);
- v.push_back(e.op(n));
+ iv[t++] = n;
+ v.push_back(ix[n]);
}
- int sign = permutation_sign(iv.begin(), iv.end());
+ int sign = permutation_sign(iv, iv + num);
result += sign * eps0123(idx1, idx2, idx3, idx4)
- * dirac_trace(ncmul(v, true), rl, trONE);
+ * trace_string(v.begin(), num - 4);
}
}
}
}
- return result * I;
+ delete[] iv;
+ return trONE * I * result;
} else { // no gamma5
if (num == 2)
return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
- // Tr gamma.mu gamma.nu gamma.rho gamma.sig = 4 (g.mu.nu g.rho.sig + g.nu.rho g.mu.sig - g.mu.rho g.nu.sig
- if (num == 4)
- return trONE * (lorentz_g(e.op(0).op(1), e.op(1).op(1)) * lorentz_g(e.op(2).op(1), e.op(3).op(1))
- + lorentz_g(e.op(1).op(1), e.op(2).op(1)) * lorentz_g(e.op(0).op(1), e.op(3).op(1))
- - lorentz_g(e.op(0).op(1), e.op(2).op(1)) * lorentz_g(e.op(1).op(1), e.op(3).op(1)));
-
- // Traces of 6 or more gammas are computed recursively:
- // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
- // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
- // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
- // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
- // - ...
- // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
- exvector v(num - 2);
- int sign = 1;
- const ex &ix1 = e.op(0).op(1);
- ex result;
- for (int i=1; i<num; i++) {
- for (int n=1, j=0; n<num; n++) {
- if (n == i)
- continue;
- v[j++] = e.op(n);
- }
- result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl, trONE);
- sign = -sign;
- }
- return result;
+ exvector iv;
+ iv.reserve(num);
+ for (unsigned i=0; i<num; i++)
+ iv.push_back(e.op(i).op(1));
+
+ return trONE * trace_string(iv.begin(), num);
}
}
for (unsigned j=0; j<rhs.nops(); j++)
v.push_back(rhs.op(j));
- // Stupid bubble sort because we only want to swap adjacent gammas
+ // Stupid recursive bubble sort because we only want to swap adjacent gammas
exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
if (is_ex_of_type(it->op(0), diracgamma5))
it++;
git://www.ginac.de / ginac.git / blobdiff