#include "idx.h"
#include "ncmul.h"
#include "symbol.h"
+#include "numeric.h" // for I
+#include "lst.h"
+#include "relational.h"
#include "print.h"
#include "archive.h"
#include "debugmsg.h"
*other = _ex1();
return true;
-#if 0
- // gamma~mu gamma~alpha gamma~beta gamma~delta gamma.mu = -2 gamma~delta gamma~beta gamma~alpha + (4-dim) gamma~alpha gamma~beta gamma~delta
- } else if (other - self == 4
- && is_ex_of_type(self[1], clifford)
- && is_ex_of_type(self[2], clifford)
- && is_ex_of_type(self[3], clifford)) {
- *self = -2 * self[3] * self[2] * self[1] + (4 - dim) * self[1] * self[2] * self[3];
- self[1] = _ex1();
- self[2] = _ex1();
- self[3] = _ex1();
- *other = _ex1();
- return true;
-#endif
-
// gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
// (commutate contracted indices towards each other, simplify_indexed()
// will re-expand and re-run the simplification)
{
exvector s;
s.reserve(v.size());
- unsigned rl = ex_to_clifford(v[0]).get_representation_label();
// Remove superfluous ONEs
exvector::const_iterator cit = v.begin(), citend = v.end();
const ex & ib = b.op(1);
if (ia.is_equal(ib)) {
a = lorentz_g(ia, ib);
- b = dirac_ONE(rl);
+ b = dirac_ONE(representation_label);
something_changed = true;
}
}
}
if (s.size() == 0)
- return clifford(diracone(), rl) * sign;
+ return clifford(diracone(), representation_label) * sign;
if (something_changed)
return nonsimplified_ncmul(s) * sign;
else
return clifford(diracgamma5(), rl);
}
+ex dirac_gamma6(unsigned char rl)
+{
+ return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
+}
+
+ex dirac_gamma7(unsigned char rl)
+{
+ return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
+}
+
ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
{
varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
return ti == (TINFO_clifford + rl);
}
-ex dirac_trace(const ex & e, unsigned char rl)
+/** Check whether a given tinfo key (as returned by return_type_tinfo()
+ * is that of a clifford object (with an arbitrary representation label). */
+static bool is_clifford_tinfo(unsigned ti)
+{
+ return (ti & ~0xff) == TINFO_clifford;
+}
+
+ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
{
if (is_ex_of_type(e, clifford)) {
if (ex_to_clifford(e).get_representation_label() == rl
&& is_ex_of_type(e.op(0), diracone))
- return _ex4();
+ return trONE;
else
return _ex0();
// Trace of sum = sum of traces
ex sum = _ex0();
for (unsigned i=0; i<e.nops(); i++)
- sum += dirac_trace(e.op(i), rl);
+ sum += dirac_trace(e.op(i), rl, trONE);
return sum;
} else if (is_ex_exactly_of_type(e, mul)) {
const ex &o = e.op(i);
unsigned ti = o.return_type_tinfo();
if (is_clifford_tinfo(o.return_type_tinfo(), rl))
- prod *= dirac_trace(o, rl);
+ prod *= dirac_trace(o, rl, trONE);
else
prod *= o;
}
// Expand product, if necessary
ex e_expanded = e.expand();
if (!is_ex_of_type(e_expanded, ncmul))
- return dirac_trace(e_expanded, rl);
+ return dirac_trace(e_expanded, rl, trONE);
// gamma5 gets moved to the front so this check is enough
bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
if ((num & 1) == 0 || num == 3)
return _ex0();
- // Tr gamma5 S_2k =
- // 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));
+ // 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));
+ }
- return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
- dirac_trace(ncmul(v), rl)).simplify_indexed() / 24;
+ // Tr gamma5 S_2k =
+ // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
+ 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.begin(), iv.end());
+ result += sign * eps0123(idx1, idx2, idx3, idx4)
+ * dirac_trace(ncmul(v, true), rl, trONE);
+ }
+ }
+ }
+ }
+ return result * I;
} else { // no gamma5
// Tr gamma.mu gamma.nu = 4 g.mu.nu
if (num == 2)
- return 4 * lorentz_g(e.op(0).op(1), e.op(1).op(1));
+ 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 4 or more gammas are computed recursively:
+ // 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
continue;
v[j++] = e.op(n);
}
- result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl);
+ result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl, trONE);
sign = -sign;
}
return result;
return _ex0();
}
+ex canonicalize_clifford(const ex & e)
+{
+ // Scan for any ncmul objects
+ lst srl;
+ ex aux = e.to_rational(srl);
+ for (unsigned i=0; i<srl.nops(); i++) {
+
+ ex lhs = srl.op(i).lhs();
+ ex rhs = srl.op(i).rhs();
+
+ if (is_ex_exactly_of_type(rhs, ncmul)
+ && rhs.return_type() == return_types::noncommutative
+ && is_clifford_tinfo(rhs.return_type_tinfo())) {
+
+ // Expand product, if necessary
+ ex rhs_expanded = rhs.expand();
+ if (!is_ex_of_type(rhs_expanded, ncmul)) {
+ srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
+ continue;
+
+ } else if (!is_ex_of_type(rhs.op(0), clifford))
+ continue;
+
+ exvector v;
+ v.reserve(rhs.nops());
+ 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
+ exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
+ if (is_ex_of_type(it->op(0), diracgamma5))
+ it++;
+ while (it != next_to_last) {
+ if (it[0].op(1).compare(it[1].op(1)) > 0) {
+ ex save0 = it[0], save1 = it[1];
+ it[0] = lorentz_g(it[0].op(1), it[1].op(1));
+ it[1] = _ex2();
+ ex sum = ncmul(v);
+ it[0] = save1;
+ it[1] = save0;
+ sum -= ncmul(v, true);
+ srl.let_op(i) = (lhs == canonicalize_clifford(sum));
+ goto next_sym;
+ }
+ it++;
+ }
+next_sym: ;
+ }
+ }
+ return aux.subs(srl);
+}
+
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff