- dirac_trace() handles arbitrary expressions (even unexpanded), including
[ginac.git] / ginac / clifford.cpp
index 06d9e5415a361d95b3305133e1032f84c10bae52..8a8bfe09294e942e88023d0770784f09d5d9f289 100644 (file)
@@ -24,6 +24,7 @@
 #include "ex.h"
 #include "idx.h"
 #include "ncmul.h"
+#include "symbol.h"
 #include "print.h"
 #include "archive.h"
 #include "debugmsg.h"
@@ -145,9 +146,10 @@ DEFAULT_PRINT(diracgamma5, "gamma5")
 /** Contraction of a gamma matrix with something else. */
 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 {
-       GINAC_ASSERT(is_ex_of_type(*self, indexed));
+       GINAC_ASSERT(is_ex_of_type(*self, clifford));
        GINAC_ASSERT(is_ex_of_type(*other, indexed));
        GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
+       unsigned char rl = ex_to_clifford(*self).get_representation_label();
 
        if (is_ex_of_type(other->op(0), diracgamma)) {
 
@@ -156,7 +158,7 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other
                // gamma~mu*gamma.mu = dim*ONE
                if (other - self == 1) {
                        *self = dim;
-                       *other = dirac_ONE();
+                       *other = dirac_ONE(rl);
                        return true;
 
                // gamma~mu*gamma~alpha*gamma.mu = (2-dim)*gamma~alpha
@@ -170,7 +172,7 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other
                } else if (other - self == 3
                        && is_ex_of_type(self[1], clifford)
                        && is_ex_of_type(self[2], clifford)) {
-                       *self = 4 * metric_tensor(self[1].op(1), self[2].op(1)) * dirac_ONE() + (dim - 4) * self[1] * self[2];
+                       *self = 4 * metric_tensor(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
                        self[1] = _ex1();
                        self[2] = _ex1();
                        *other = _ex1();
@@ -200,6 +202,7 @@ ex clifford::simplify_ncmul(const exvector & v) const
 {
        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();
@@ -251,7 +254,7 @@ ex clifford::simplify_ncmul(const exvector & v) const
                                const ex & ib = b.op(1);
                                if (ia.is_equal(ib)) {
                                        a = lorentz_g(ia, ib);
-                                       b = dirac_ONE();
+                                       b = dirac_ONE(rl);
                                        something_changed = true;
                                }
                        }
@@ -260,7 +263,7 @@ ex clifford::simplify_ncmul(const exvector & v) const
        }
 
        if (s.size() == 0)
-               return clifford(diracone()) * sign;
+               return clifford(diracone(), rl) * sign;
        if (something_changed)
                return nonsimplified_ncmul(s) * sign;
        else
@@ -299,4 +302,126 @@ ex dirac_gamma5(unsigned char rl)
        return 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 indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
+}
+
+/** Check whether a given tinfo key (as returned by return_type_tinfo()
+ *  is that of a clifford object with the specified representation label. */
+static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
+{
+       return ti == (TINFO_clifford + rl);
+}
+
+ex dirac_trace(const ex & e, unsigned char rl)
+{
+       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();
+               else
+                       return _ex0();
+
+       } else if (is_ex_exactly_of_type(e, add)) {
+
+               // Trace of sum = sum of traces
+               ex sum = _ex0();
+               for (unsigned i=0; i<e.nops(); i++)
+                       sum += dirac_trace(e.op(i), rl);
+               return sum;
+
+       } else if (is_ex_exactly_of_type(e, mul)) {
+
+               // Trace of product: pull out non-clifford factors
+               ex prod = _ex1();
+               for (unsigned i=0; i<e.nops(); i++) {
+                       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);
+                       else
+                               prod *= o;
+               }
+               return prod;
+
+       } else if (is_ex_exactly_of_type(e, ncmul)) {
+
+               if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
+                       return _ex0();
+
+               // Expand product, if necessary
+               ex e_expanded = e.expand();
+               if (!is_ex_of_type(e_expanded, ncmul))
+                       return dirac_trace(e_expanded, rl);
+
+               // gamma5 gets moved to the front so this check is enough
+               bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
+               unsigned num = e.nops();
+
+               if (has_gamma5) {
+
+                       // Trace of gamma5 * odd number of gammas and trace of
+                       // gamma5 * gamma_mu * gamma_nu are zero
+                       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));
+
+                       return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
+                               dirac_trace(ncmul(v), rl)).simplify_indexed() / 24;
+
+               } else { // no gamma5
+
+                       // Trace of odd number of gammas is zero
+                       if ((num & 1) == 1)
+                               return _ex0();
+
+                       // 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));
+
+                       // Traces of 4 or more gammas are computed recursively:
+                       // Tr gamma_mu1 gamma_mu2 ... gamma_mun =
+                       //   + eta_mu1_mu2 * Tr gamma_mu3 ... gamma_mun
+                       //   - eta_mu1_mu3 * Tr gamma_mu2 gamma_mu4 ... gamma_mun
+                       //   + eta_mu1_mu4 * Tr gamma_mu3 gamma_mu3 gamma_mu5 ... gamma_mun
+                       //   - ...
+                       //   + eta_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);
+                               sign = -sign;
+                       }
+                       return result;
+               }
+       }
+
+       return _ex0();
+}
+
 } // namespace GiNaC