+ex dirac_ONE(unsigned char rl)
+{
+ static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
+ return clifford(ONE, rl);
+}
+
+static unsigned get_dim_uint(const ex& e)
+{
+ if (!is_a<idx>(e))
+ throw std::invalid_argument("get_dim_uint: argument is not an index");
+ ex dim = ex_to<idx>(e).get_dim();
+ if (!dim.info(info_flags::posint))
+ throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
+ unsigned d = ex_to<numeric>(dim).to_int();
+ return d;
+}
+
+ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
+{
+ //static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
+ ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
+
+ if (!is_a<idx>(mu))
+ throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
+
+ exvector indices = metr.get_free_indices();
+
+ if (indices.size() == 2) {
+ return clifford(unit, mu, metr, rl);
+ } else if (is_a<matrix>(metr)) {
+ matrix M = ex_to<matrix>(metr);
+ unsigned n = M.rows();
+ bool symmetric = true;
+
+ //static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
+ // chi((new symbol)->setflag(status_flags::dynallocated), n);
+ idx xi((new symbol)->setflag(status_flags::dynallocated), n),
+ chi((new symbol)->setflag(status_flags::dynallocated), n);
+ if ((n == M.cols()) && (n == get_dim_uint(mu))) {
+ for (unsigned i = 0; i < n; i++) {
+ for (unsigned j = i+1; j < n; j++) {
+ if (!M(i, j).is_equal(M(j, i))) {
+ symmetric = false;
+ }
+ }
+ }
+ return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
+ } else {
+ throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
+ }
+ } else if (indices.size() == 0) { // a tensor or other expression without indices
+ //static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
+ // chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
+ varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
+ chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
+ return clifford(unit, mu, indexed(metr, xi, chi), rl);
+ } else
+ throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
+}
+
+ex dirac_gamma(const ex & mu, unsigned char rl)
+{
+ static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
+
+ if (!is_a<varidx>(mu))
+ throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
+
+ static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
+ chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
+ return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
+}
+
+ex dirac_gamma5(unsigned char rl)
+{
+ static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
+ return clifford(gamma5, rl);
+}
+
+ex dirac_gammaL(unsigned char rl)
+{
+ static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
+ return clifford(gammaL, rl);
+}
+
+ex dirac_gammaR(unsigned char rl)
+{
+ static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
+ return clifford(gammaR, rl);
+}
+
+ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
+{
+ // Slashed vectors are actually stored as a clifford object with the
+ // vector as its base expression and a (dummy) index that just serves
+ // for storing the space dimensionality
+
+ static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
+ chi((new symbol)->setflag(status_flags::dynallocated), dim);
+ return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
+}
+
+/** Extract representation label from tinfo key (as returned by
+ * return_type_tinfo()). */
+static unsigned char get_representation_label(const return_type_t& ti)
+{
+ return (unsigned char)ti.rl;
+}
+
+/** 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, size_t 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 (size_t i=1; i<num; i++) {
+ for (size_t 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, const std::set<unsigned char> & rls, const ex & trONE)
+{
+ if (is_a<clifford>(e)) {
+
+ unsigned char rl = ex_to<clifford>(e).get_representation_label();
+
+ // Are we taking the trace over this object's representation label?
+ if (rls.find(rl) == rls.end())
+ return e;
+
+ // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
+ const ex & g = e.op(0);
+ if (is_a<diracone>(g))
+ return trONE;
+ else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
+ return trONE/2;
+ else
+ return _ex0;
+
+ } else if (is_exactly_a<mul>(e)) {
+
+ // Trace of product: pull out non-clifford factors
+ ex prod = _ex1;
+ for (size_t i=0; i<e.nops(); i++) {
+ const ex &o = e.op(i);
+ if (is_clifford_tinfo(o.return_type_tinfo()))
+ prod *= dirac_trace(o, rls, trONE);
+ else
+ prod *= o;
+ }
+ return prod;
+
+ } else if (is_exactly_a<ncmul>(e)) {
+
+ unsigned char rl = get_representation_label(e.return_type_tinfo());
+
+ // Are we taking the trace over this string's representation label?
+ if (rls.find(rl) == rls.end())
+ return e;
+
+ // Substitute gammaL/R and expand product, if necessary
+ ex e_expanded = e.subs(lst(
+ dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
+ dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
+ ), subs_options::no_pattern).expand();
+ if (!is_a<ncmul>(e_expanded))
+ return dirac_trace(e_expanded, rls, trONE);
+
+ // gamma5 gets moved to the front so this check is enough
+ bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
+ size_t 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 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
+ // (the epsilon is always 4-dimensional)
+ if (num == 5) {
+ ex b1, i1, b2, i2, b3, i3, b4, i4;
+ base_and_index(e.op(1), b1, i1);
+ base_and_index(e.op(2), b2, i2);
+ base_and_index(e.op(3), b3, i3);
+ base_and_index(e.op(4), b4, i4);
+ return trONE * I * (lorentz_eps(ex_to<idx>(i1).replace_dim(_ex4), ex_to<idx>(i2).replace_dim(_ex4), ex_to<idx>(i3).replace_dim(_ex4), ex_to<idx>(i4).replace_dim(_ex4)) * b1 * b2 * b3 * b4).simplify_indexed();
+ }
+
+ // Tr gamma5 S_2k =
+ // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
+ // (the epsilon is always 4-dimensional)
+ exvector ix(num-1), bv(num-1);
+ for (size_t i=1; i<num; i++)
+ base_and_index(e.op(i), bv[i-1], ix[i-1]);
+ num--;
+ int *iv = new int[num];
+ ex result;
+ for (size_t i=0; i<num-3; i++) {
+ ex idx1 = ix[i];
+ for (size_t j=i+1; j<num-2; j++) {
+ ex idx2 = ix[j];
+ for (size_t k=j+1; k<num-1; k++) {
+ ex idx3 = ix[k];
+ for (size_t l=k+1; l<num; l++) {
+ ex idx4 = ix[l];
+ iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
+ exvector v;
+ v.reserve(num - 4);
+ for (size_t n=0, t=4; n<num; n++) {
+ if (n == i || n == j || n == k || n == l)
+ continue;
+ iv[t++] = n;
+ v.push_back(ix[n]);
+ }
+ int sign = permutation_sign(iv, iv + num);
+ result += sign * lorentz_eps(ex_to<idx>(idx1).replace_dim(_ex4), ex_to<idx>(idx2).replace_dim(_ex4), ex_to<idx>(idx3).replace_dim(_ex4), ex_to<idx>(idx4).replace_dim(_ex4))
+ * trace_string(v.begin(), num - 4);
+ }
+ }
+ }
+ }
+ delete[] iv;
+ return trONE * I * result * mul(bv);
+
+ } 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) {
+ ex b1, i1, b2, i2;
+ base_and_index(e.op(0), b1, i1);
+ base_and_index(e.op(1), b2, i2);
+ return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
+ }
+
+ exvector iv(num), bv(num);
+ for (size_t i=0; i<num; i++)
+ base_and_index(e.op(i), bv[i], iv[i]);
+
+ return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
+ }
+
+ } else if (e.nops() > 0) {
+
+ // Trace maps to all other container classes (this includes sums)
+ pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
+ return e.map(fcn);
+
+ } else
+ return _ex0;
+}
+
+ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)