if ((num & 1) == 0 || num == 3)
return _ex0();
+ // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
+ 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 =
// I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
- ex dim = ex_to_idx(e.op(1).op(1)).get_dim();
- varidx mu1((new symbol)->setflag(status_flags::dynallocated), dim),
- mu2((new symbol)->setflag(status_flags::dynallocated), dim),
- mu3((new symbol)->setflag(status_flags::dynallocated), dim),
- mu4((new symbol)->setflag(status_flags::dynallocated), dim);
- exvector v;
- v.reserve(num + 3);
- v.push_back(dirac_gamma(mu1, rl));
- v.push_back(dirac_gamma(mu2, rl));
- v.push_back(dirac_gamma(mu3, rl));
- v.push_back(dirac_gamma(mu4, rl));
- for (int i=1; i<num; i++)
- v.push_back(e.op(i));
-
- return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
- dirac_trace(ncmul(v), rl, trONE)).simplify_indexed() * I / 24;
+ ex result;
+ for (int i=1; i<num-3; i++) {
+ ex idx1 = e.op(i).op(1);
+ for (int j=i+1; j<num-2; j++) {
+ ex idx2 = e.op(j).op(1);
+ for (int k=j+1; k<num-1; k++) {
+ ex idx3 = e.op(k).op(1);
+ for (int l=k+1; l<num; l++) {
+ ex idx4 = e.op(l).op(1);
+ vector<int> iv;
+ iv.reserve(num-1);
+ 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++) {
+ if (n == i || n == j || n == k || n == l)
+ continue;
+ iv.push_back(n);
+ v.push_back(e.op(n));
+ }
+ int sign = permutation_sign(iv);
+ result += sign * eps0123(idx1, idx2, idx3, idx4)
+ * dirac_trace(ncmul(v), rl, trONE);
+ }
+ }
+ }
+ }
+ return result * I;
} else { // no gamma5
if (num == 2)
return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
- // Traces of 4 or more gammas are computed recursively:
+ // 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
exvector free_indices, dummy_indices;
find_free_and_dummy(all_indices, free_indices, dummy_indices);
- // d.abc*d.abc=40/3
+ // d.abc d.abc = 40/3
if (dummy_indices.size() == 3) {
*self = numeric(40, 3);
*other = _ex1();
return true;
- // d.akl*d.bkl=5/3*delta.ab
+ // d.akl d.bkl = 5/3 delta.ab
} else if (dummy_indices.size() == 2) {
exvector a;
back_insert_iterator<exvector> ita(a);
*other = _ex1();
return true;
}
+
+ } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
+
+ // d.abc T.b T.c = 5/6 T.a
+ if (other+1 != v.end()
+ && is_ex_exactly_of_type(other[1].op(0), su3t)
+ && ex_to_indexed(*self).has_dummy_index_for(other[1].op(1))) {
+
+ exvector self_indices = ex_to_indexed(*self).get_indices();
+ exvector dummy_indices;
+ dummy_indices.push_back(other[0].op(1));
+ dummy_indices.push_back(other[1].op(1));
+ int sig;
+ ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
+ *self = numeric(5, 6);
+ other[0] = color_T(a, ex_to_color(other[0]).get_representation_label());
+ other[1] = _ex1();
+ return true;
+ }
}
return false;
exvector dummy_indices;
dummy_indices = ex_to_indexed(*self).get_dummy_indices(ex_to_indexed(*other));
- // f.abc*f.abc=24
+ // f.abc f.abc = 24
if (dummy_indices.size() == 3) {
*self = 24;
*other = _ex1();
return true;
- // f.akl*f.bkl=3*delta.ab
+ // f.akl f.bkl = 3 delta.ab
} else if (dummy_indices.size() == 2) {
int sign1, sign2;
ex a = permute_free_index_to_front(ex_to_indexed(*self).get_indices(), dummy_indices, sign1);
*other = _ex1();
return true;
}
+
+ } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
+
+ // f.abc T.b T.c = 3/2 I T.a
+ if (other+1 != v.end()
+ && is_ex_exactly_of_type(other[1].op(0), su3t)
+ && ex_to_indexed(*self).has_dummy_index_for(other[1].op(1))) {
+
+ exvector self_indices = ex_to_indexed(*self).get_indices();
+ exvector dummy_indices;
+ dummy_indices.push_back(other[0].op(1));
+ dummy_indices.push_back(other[1].op(1));
+ int sig;
+ ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
+ *self = numeric(3, 2) * sig * I;
+ other[0] = color_T(a, ex_to_color(other[0]).get_representation_label());
+ other[1] = _ex1();
+ return true;
+ }
}
return false;
return dummy_indices;
}
+bool indexed::has_dummy_index_for(const ex & i) const
+{
+ exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
+ while (it != itend) {
+ if (is_dummy_pair(*it, i))
+ return true;
+ it++;
+ }
+ return false;
+}
+
exvector indexed::get_free_indices(void) const
{
exvector free_indices, dummy_indices;
if (!is_ex_of_type(*it1, indexed))
continue;
+ bool first_noncommutative = (it1->return_type() != return_types::commutative);
+
// Indexed factor found, get free indices and look for contraction
// candidates
exvector free1, dummy1;
if (!is_ex_of_type(*it2, indexed))
continue;
+ bool second_noncommutative = (it2->return_type() != return_types::commutative);
+
// Find free indices of second factor and merge them with free
// indices of first factor
exvector un;
}
if (contracted) {
contraction_done:
- if (non_commutative
+ if (first_noncommutative || second_noncommutative
|| is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
|| is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
|| is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {