/** @file clifford.cpp
*
- * Implementation of GiNaC's clifford objects.
- * No real implementation yet, to be done. */
+ * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
-#include <string>
+#include <iostream>
+#include <stdexcept>
#include "clifford.h"
#include "ex.h"
+#include "idx.h"
#include "ncmul.h"
+#include "symbol.h"
+#include "numeric.h" // for I
+#include "symmetry.h"
+#include "lst.h"
+#include "relational.h"
+#include "mul.h"
+#include "print.h"
#include "archive.h"
#include "utils.h"
-#include "debugmsg.h"
namespace GiNaC {
-GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, lortensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgammaL, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgammaR, tensor)
//////////
-// default constructor, destructor, copy constructor assignment operator and helpers
+// default ctor, dtor, copy ctor, assignment operator and helpers
//////////
-// public
-
-clifford::clifford()
+clifford::clifford() : representation_label(0)
{
- debugmsg("clifford default constructor",LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_clifford;
}
-// protected
-
void clifford::copy(const clifford & other)
{
inherited::copy(other);
+ representation_label = other.representation_label;
}
-void clifford::destroy(bool call_parent)
-{
- if (call_parent) inherited::destroy(call_parent);
-}
+DEFAULT_DESTROY(clifford)
+DEFAULT_CTORS(diracone)
+DEFAULT_CTORS(diracgamma)
+DEFAULT_CTORS(diracgamma5)
+DEFAULT_CTORS(diracgammaL)
+DEFAULT_CTORS(diracgammaR)
//////////
// other constructors
//////////
-// public
+/** Construct object without any indices. This constructor is for internal
+ * use only. Use the dirac_ONE() function instead.
+ * @see dirac_ONE */
+clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
+{
+ tinfo_key = TINFO_clifford;
+}
-clifford::clifford(const std::string & n, const ex & mu) : inherited(lortensor_symbolic, n, mu)
+/** Construct object with one Lorentz index. This constructor is for internal
+ * use only. Use the dirac_gamma() function instead.
+ * @see dirac_gamma */
+clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
{
- debugmsg("clifford constructor from string,ex",LOGLEVEL_CONSTRUCT);
- tinfo_key=TINFO_clifford;
+ GINAC_ASSERT(is_a<varidx>(mu));
+ tinfo_key = TINFO_clifford;
}
-clifford::clifford(const std::string & n, const exvector & iv) : inherited(lortensor_symbolic, n, iv)
+clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
{
- debugmsg("clifford constructor from string,exvector", LOGLEVEL_CONSTRUCT);
- GINAC_ASSERT(all_of_type_lorentzidx());
- tinfo_key=TINFO_clifford;
+ tinfo_key = TINFO_clifford;
}
-clifford::clifford(const std::string & n, exvector *ivp) : inherited(lortensor_symbolic, n, *ivp)
+clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
{
- debugmsg("clifford constructor from string,exvector", LOGLEVEL_CONSTRUCT);
- GINAC_ASSERT(all_of_type_lorentzidx());
- tinfo_key=TINFO_clifford;
+ tinfo_key = TINFO_clifford;
}
//////////
// archiving
//////////
-/** Construct object from archive_node. */
clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
{
- debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
+ unsigned rl;
+ n.find_unsigned("label", rl);
+ representation_label = rl;
}
-/** Unarchive the object. */
-ex clifford::unarchive(const archive_node &n, const lst &sym_lst)
+void clifford::archive(archive_node &n) const
{
- return (new clifford(n, sym_lst))->setflag(status_flags::dynallocated);
+ inherited::archive(n);
+ n.add_unsigned("label", representation_label);
}
-/** Archive the object. */
-void clifford::archive(archive_node &n) const
+DEFAULT_UNARCHIVE(clifford)
+DEFAULT_ARCHIVING(diracone)
+DEFAULT_ARCHIVING(diracgamma)
+DEFAULT_ARCHIVING(diracgamma5)
+DEFAULT_ARCHIVING(diracgammaL)
+DEFAULT_ARCHIVING(diracgammaR)
+
+//////////
+// functions overriding virtual functions from base classes
+//////////
+
+int clifford::compare_same_type(const basic & other) const
{
- inherited::archive(n);
+ GINAC_ASSERT(is_a<clifford>(other));
+ const clifford &o = static_cast<const clifford &>(other);
+
+ if (representation_label != o.representation_label) {
+ // different representation label
+ return representation_label < o.representation_label ? -1 : 1;
+ }
+
+ return inherited::compare_same_type(other);
+}
+
+bool clifford::match_same_type(const basic & other) const
+{
+ GINAC_ASSERT(is_a<clifford>(other));
+ const clifford &o = static_cast<const clifford &>(other);
+
+ return representation_label == o.representation_label;
+}
+
+void clifford::print(const print_context & c, unsigned level) const
+{
+ if (!is_a<diracgamma5>(seq[0]) && !is_a<diracgammaL>(seq[0]) &&
+ !is_a<diracgammaR>(seq[0]) && !is_a<diracgamma>(seq[0]) &&
+ !is_a<diracone>(seq[0])) {
+
+ // dirac_slash() object is printed differently
+ if (is_a<print_tree>(c))
+ inherited::print(c, level);
+ else if (is_a<print_latex>(c)) {
+ c.s << "{";
+ seq[0].print(c, level);
+ c.s << "\\hspace{-1.0ex}/}";
+ } else {
+ seq[0].print(c, level);
+ c.s << "\\";
+ }
+
+ } else
+ inherited::print(c, level);
+}
+
+DEFAULT_COMPARE(diracone)
+DEFAULT_COMPARE(diracgamma)
+DEFAULT_COMPARE(diracgamma5)
+DEFAULT_COMPARE(diracgammaL)
+DEFAULT_COMPARE(diracgammaR)
+
+DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
+DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
+DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
+DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
+
+/** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
+static void base_and_index(const ex & c, ex & b, ex & i)
+{
+ GINAC_ASSERT(is_a<clifford>(c));
+ GINAC_ASSERT(c.nops() == 2);
+
+ if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
+ i = c.op(1);
+ b = _ex1;
+ } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
+ i = _ex0;
+ b = _ex1;
+ } else { // slash object, generate new dummy index
+ varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
+ b = indexed(c.op(0), ix.toggle_variance());
+ i = ix;
+ }
+}
+
+/** Contraction of a gamma matrix with something else. */
+bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
+{
+ GINAC_ASSERT(is_a<clifford>(*self));
+ GINAC_ASSERT(is_a<indexed>(*other));
+ GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
+ unsigned char rl = ex_to<clifford>(*self).get_representation_label();
+
+ ex dim = ex_to<idx>(self->op(1)).get_dim();
+ if (other->nops() > 1)
+ dim = minimal_dim(dim, ex_to<idx>(self->op(1)).get_dim());
+
+ if (is_a<clifford>(*other)) {
+
+ // Contraction only makes sense if the represenation labels are equal
+ if (ex_to<clifford>(*other).get_representation_label() != rl)
+ return false;
+
+ // 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
+ } else if (other - self == 2
+ && is_a<clifford>(self[1])) {
+ *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
+ } else if (other - self == 3
+ && is_a<clifford>(self[1])
+ && is_a<clifford>(self[2])) {
+ ex b1, i1, b2, i2;
+ base_and_index(self[1], b1, i1);
+ base_and_index(self[2], b2, i2);
+ *self = 4 * lorentz_g(i1, i2) * b1 * b2 * 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 - (dim-4) gamam~alpha gamma~beta gamma~delta
+ } else if (other - self == 4
+ && is_a<clifford>(self[1])
+ && is_a<clifford>(self[2])
+ && is_a<clifford>(self[3])) {
+ *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
+ self[1] = _ex1;
+ self[2] = _ex1;
+ self[3] = _ex1;
+ *other = _ex1;
+ return true;
+
+ // 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_a<clifford>(*it))
+ 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;
+ }
+
+ } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
+
+ // x.mu gamma~mu -> x-slash
+ *self = dirac_slash(other->op(0), dim, rl);
+ *other = _ex1;
+ return true;
+ }
+
+ return false;
+}
+
+/** Perform automatic simplification on noncommutative product of clifford
+ * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
+ * and removes squares of gamma objects. */
+ex clifford::simplify_ncmul(const exvector & v) const
+{
+ exvector s;
+ s.reserve(v.size());
+
+ // Remove superfluous ONEs
+ exvector::const_iterator cit = v.begin(), citend = v.end();
+ while (cit != citend) {
+ if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
+ s.push_back(*cit);
+ cit++;
+ }
+
+ bool something_changed = false;
+ int sign = 1;
+
+ // Anticommute gamma5/L/R's to the front
+ if (s.size() >= 2) {
+ exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
+ while (true) {
+ exvector::iterator it = next_to_last;
+ while (true) {
+ exvector::iterator it2 = it + 1;
+ if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
+ ex e1 = it->op(0), e2 = it2->op(0);
+
+ if (is_a<diracgamma5>(e2)) {
+
+ if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
+
+ // gammaL/R gamma5 -> gamma5 gammaL/R
+ it->swap(*it2);
+ something_changed = true;
+
+ } else if (!is_a<diracgamma5>(e1)) {
+
+ // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
+ // x gamma5 -> -gamma5 x
+ it->swap(*it2);
+ sign = -sign;
+ something_changed = true;
+ }
+
+ } else if (is_a<diracgammaL>(e2)) {
+
+ if (is_a<diracgammaR>(e1)) {
+
+ // gammaR gammaL -> 0
+ return _ex0;
+
+ } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
+
+ // gammaL gammaL -> gammaL gammaL (do nothing)
+ // gamma5 gammaL -> gamma5 gammaL (do nothing)
+ // x gammaL -> gammaR x
+ it->swap(*it2);
+ *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
+ something_changed = true;
+ }
+
+ } else if (is_a<diracgammaR>(e2)) {
+
+ if (is_a<diracgammaL>(e1)) {
+
+ // gammaL gammaR -> 0
+ return _ex0;
+
+ } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
+
+ // gammaR gammaR -> gammaR gammaR (do nothing)
+ // gamma5 gammaR -> gamma5 gammaR (do nothing)
+ // x gammaR -> gammaL x
+ it->swap(*it2);
+ *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
+ something_changed = true;
+ }
+ }
+ }
+ if (it == first)
+ break;
+ --it;
+ }
+ if (next_to_last == first)
+ break;
+ --next_to_last;
+ }
+ }
+
+ // Remove equal adjacent gammas
+ if (s.size() >= 2) {
+ exvector::iterator it, itend = s.end() - 1;
+ for (it = s.begin(); it != itend; ++it) {
+ ex & a = it[0];
+ ex & b = it[1];
+ if (!is_a<clifford>(a) || !is_a<clifford>(b))
+ continue;
+
+ const ex & ag = a.op(0);
+ const ex & bg = b.op(0);
+ bool a_is_diracgamma = is_a<diracgamma>(ag);
+ bool b_is_diracgamma = is_a<diracgamma>(bg);
+
+ if (a_is_diracgamma && b_is_diracgamma) {
+
+ const ex & ia = a.op(1);
+ const ex & ib = b.op(1);
+ if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
+ a = lorentz_g(ia, ib);
+ b = dirac_ONE(representation_label);
+ something_changed = true;
+ }
+
+ } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
+
+ // Remove squares of gamma5
+ a = dirac_ONE(representation_label);
+ b = dirac_ONE(representation_label);
+ something_changed = true;
+
+ } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
+ || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
+
+ // Remove squares of gammaL/R
+ b = dirac_ONE(representation_label);
+ something_changed = true;
+
+ } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
+
+ // gammaL and gammaR are orthogonal
+ return _ex0;
+
+ } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
+
+ // gamma5 gammaL -> -gammaL
+ a = dirac_ONE(representation_label);
+ sign = -sign;
+ something_changed = true;
+
+ } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
+
+ // gamma5 gammaR -> gammaR
+ a = dirac_ONE(representation_label);
+ something_changed = true;
+
+ } else if (!a_is_diracgamma && !b_is_diracgamma && ag.is_equal(bg)) {
+
+ // a\ a\ -> a^2
+ varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
+ a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
+ b = dirac_ONE(representation_label);
+ something_changed = true;
+ }
+ }
+ }
+
+ if (s.empty())
+ return clifford(diracone(), representation_label) * sign;
+ if (something_changed)
+ return nonsimplified_ncmul(s) * sign;
+ else
+ return simplified_ncmul(s) * sign;
+}
+
+ex clifford::thisexprseq(const exvector & v) const
+{
+ return clifford(representation_label, v);
+}
+
+ex clifford::thisexprseq(exvector * vp) const
+{
+ return clifford(representation_label, vp);
}
//////////
-// functions overriding virtual functions from bases classes
+// global functions
//////////
-// public
+ex dirac_ONE(unsigned char rl)
+{
+ return clifford(diracone(), rl);
+}
-void clifford::printraw(std::ostream & os) const
+ex dirac_gamma(const ex & mu, unsigned char rl)
{
- debugmsg("clifford printraw",LOGLEVEL_PRINT);
- os << "clifford(" << "indices=";
- printrawindices(os);
- os << ",hash=" << hashvalue << ",flags=" << flags << ")";
+ if (!is_a<varidx>(mu))
+ throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
+
+ return clifford(diracgamma(), mu, rl);
}
-void clifford::printtree(std::ostream & os, unsigned indent) const
+ex dirac_gamma5(unsigned char rl)
{
- debugmsg("clifford printtree",LOGLEVEL_PRINT);
- os << std::string(indent,' ') << " (clifford): "
- << seq.size() << "indices=";
- printtreeindices(os, indent);
- os << ", hash=" << hashvalue
- << " (0x" << std::hex << hashvalue << std::dec << ")"
- << ", flags=" << flags << std::endl;
+ return clifford(diracgamma5(), rl);
}
-void clifford::print(std::ostream & os, unsigned upper_precedence) const
+ex dirac_gammaL(unsigned char rl)
{
- debugmsg("clifford print",LOGLEVEL_PRINT);
- os << name;
- printindices(os);
+ return clifford(diracgammaL(), rl);
}
-bool clifford::info(unsigned inf) const
+ex dirac_gammaR(unsigned char rl)
{
- return inherited::info(inf);
+ return clifford(diracgammaR(), rl);
}
-// protected
+ex dirac_gamma6(unsigned char rl)
+{
+ return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
+}
-int clifford::compare_same_type(const basic & other) const
+ex dirac_gamma7(unsigned char rl)
{
- GINAC_ASSERT(is_of_type(other,clifford));
- // only compare indices
- return exprseq::compare_same_type(other);
+ return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
}
-bool clifford::is_equal_same_type(const basic & other) const
+ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
{
- GINAC_ASSERT(is_of_type(other,clifford));
- // only compare indices
- return exprseq::is_equal_same_type(other);
+ // 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
+ return clifford(e, varidx(0, dim), rl);
}
-ex clifford::thisexprseq(const exvector & v) const
+/** 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 clifford(name, v);
+ return ti == (TINFO_clifford + rl);
}
-ex clifford::thisexprseq(exvector *vp) const
+/** 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 clifford(name, vp);
+ return (ti & ~0xff) == TINFO_clifford;
}
-ex clifford::simplify_ncmul(const exvector & v) const
+/** 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)
{
- return simplified_ncmul(v);
+ // 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 (unsigned i=1; i<num; i++) {
+ for (unsigned 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;
}
-//////////
-// friend functions
-//////////
+ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
+{
+ if (is_a<clifford>(e)) {
+
+ if (!ex_to<clifford>(e).get_representation_label() == rl)
+ return _ex0;
+ 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_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);
+ if (is_clifford_tinfo(o.return_type_tinfo(), rl))
+ prod *= dirac_trace(o, rl, trONE);
+ 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;
+
+ // 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
+ )).expand();
+ if (!is_a<ncmul>(e_expanded))
+ return dirac_trace(e_expanded, rl, trONE);
+
+ // gamma5 gets moved to the front so this check is enough
+ bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
+ 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 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 (unsigned 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 (unsigned i=0; i<num-3; i++) {
+ ex idx1 = ix[i];
+ for (unsigned j=i+1; j<num-2; j++) {
+ ex idx2 = ix[j];
+ for (unsigned k=j+1; k<num-1; k++) {
+ ex idx3 = ix[k];
+ for (unsigned 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 (unsigned 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 (unsigned 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<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
+ return e.map(fcn);
+
+ } else
+ return _ex0;
+}
-/** Construct an object representing a Dirac gamma matrix. The index must
- * be of class lorentzidx.
- *
- * @param mu Index
- * @return newly constructed object */
-clifford clifford_gamma(const ex & mu)
+ex canonicalize_clifford(const ex & e)
{
- return clifford("gamma", mu);
+ // 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_a<ncmul>(rhs_expanded)) {
+ srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
+ continue;
+
+ } else if (!is_a<clifford>(rhs.op(0)))
+ 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_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
+ ++it;
+ while (it != next_to_last) {
+ if (it[0].compare(it[1]) > 0) {
+ ex save0 = it[0], save1 = it[1];
+ ex b1, i1, b2, i2;
+ base_and_index(it[0], b1, i1);
+ base_and_index(it[1], b2, i2);
+ it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
+ 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).simplify_indexed();
}
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff