#include "ex.h"
#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"
DEFAULT_COMPARE(diracgamma)
DEFAULT_COMPARE(diracgamma5)
-DEFAULT_PRINT(diracone, "ONE")
-DEFAULT_PRINT(diracgamma, "gamma")
-DEFAULT_PRINT(diracgamma5, "gamma5")
+DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
+DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
/** 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->op(0), diracgamma));
unsigned char rl = ex_to_clifford(*self).get_representation_label();
- if (is_ex_of_type(other->op(0), diracgamma)) {
+ if (is_ex_of_type(*other, clifford)) {
ex dim = ex_to_idx(self->op(1)).get_dim();
- // gamma~mu*gamma.mu = dim*ONE
+ // gamma~mu gamma.mu = dim ONE
if (other - self == 1) {
*self = dim;
*other = dirac_ONE(rl);
return true;
- // gamma~mu*gamma~alpha*gamma.mu = (2-dim)*gamma~alpha
+ // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
} else if (other - self == 2
&& is_ex_of_type(self[1], clifford)) {
*self = 2 - dim;
*other = _ex1();
return true;
- // gamma~mu*gamma~alpha*gamma~beta*gamma.mu = 4*g~alpha~beta+(dim-4)*gamam~alpha*gamma~beta
+ // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
} 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(rl) + (dim - 4) * self[1] * self[2];
+ *self = 4 * lorentz_g(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();
return true;
- // 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();
+ // 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)
+ } else {
+ exvector::iterator it = self + 1, next_to_last = other - 1;
+ while (it != other) {
+ if (!is_ex_of_type(*it, clifford))
+ return false;
+ it++;
+ }
+
+ it = self + 1;
+ ex S = _ex1();
+ while (it != next_to_last) {
+ S *= *it;
+ *it++ = _ex1();
+ }
+
+ *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
+ *next_to_last = _ex1();
*other = _ex1();
return true;
}
{
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_trace(const ex & e, unsigned char rl = 0)
+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 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);
+}
+
+/** 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;
+}
+
+/** 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 (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)) {
ex prod = _ex1();
for (unsigned i=0; i<e.nops(); i++) {
const ex &o = e.op(i);
- if (is_ex_of_type(o, clifford)
- && ex_to_clifford(o).get_representation_label() == rl)
- prod *= dirac_trace(o, rl);
- else if (is_ex_of_type(o, ncmul)
- && is_ex_of_type(o.op(0), clifford)
- && ex_to_clifford(o.op(0)).get_representation_label() == rl)
- prod *= dirac_trace(o, rl);
+ unsigned ti = o.return_type_tinfo();
+ if (is_clifford_tinfo(o.return_type_tinfo(), rl))
+ prod *= dirac_trace(o, rl, trONE);
else
prod *= o;
}
} else if (is_ex_exactly_of_type(e, ncmul)) {
- if (!is_ex_of_type(e.op(0), clifford)
- || ex_to_clifford(e.op(0)).get_representation_label() != rl)
+ 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, 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);
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 == 2)
+ // 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)
+ 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 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=0; i<num-3; i++) {
+ ex idx1 = ix[i];
+ for (int j=i+1; j<num-2; j++) {
+ ex idx2 = ix[j];
+ for (int k=j+1; k<num-1; k++) {
+ ex idx3 = ix[k];
+ for (int 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 (int 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 * eps0123(idx1, idx2, idx3, idx4)
+ * trace_string(v.begin(), num - 4);
+ }
+ }
+ }
+ }
+ delete[] iv;
+ return trONE * I * result;
+
} 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
+ // 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 trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
+
+ exvector iv;
+ iv.reserve(num);
+ for (unsigned i=0; i<num; i++)
+ iv.push_back(e.op(i).op(1));
- throw (std::logic_error("dirac_trace: don't know how to compute trace"));
+ return trONE * trace_string(iv.begin(), num);
+ }
}
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 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++;
+ 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