* Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2004 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
*/
#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 "print.h"
+#include "operators.h"
+#include "add.h"
+#include "mul.h"
+#include "power.h"
+#include "matrix.h"
#include "archive.h"
-#include "debugmsg.h"
#include "utils.h"
-#include <stdexcept>
-
namespace GiNaC {
-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_OPT(clifford, indexed,
+ print_func<print_dflt>(&clifford::do_print_dflt).
+ print_func<print_latex>(&clifford::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
+ print_func<print_dflt>(&diracone::do_print).
+ print_func<print_latex>(&diracone::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
+ print_func<print_dflt>(&cliffordunit::do_print).
+ print_func<print_latex>(&cliffordunit::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
+ print_func<print_dflt>(&diracgamma::do_print).
+ print_func<print_latex>(&diracgamma::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
+ print_func<print_dflt>(&diracgamma5::do_print).
+ print_func<print_latex>(&diracgamma5::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
+ print_func<print_context>(&diracgammaL::do_print).
+ print_func<print_latex>(&diracgammaL::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
+ print_func<print_context>(&diracgammaR::do_print).
+ print_func<print_latex>(&diracgammaR::do_print_latex))
//////////
-// default constructor, destructor, copy constructor assignment operator and helpers
+// default constructors
//////////
-clifford::clifford() : representation_label(0)
+static ex default_metric()
{
- debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
- tinfo_key = TINFO_clifford;
+ static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
+ return m;
}
-void clifford::copy(const clifford & other)
+clifford::clifford() : representation_label(0), metric(default_metric())
{
- inherited::copy(other);
- representation_label = other.representation_label;
+ tinfo_key = TINFO_clifford;
}
-DEFAULT_DESTROY(clifford)
-DEFAULT_CTORS(diracone)
-DEFAULT_CTORS(diracgamma)
-DEFAULT_CTORS(diracgamma5)
+DEFAULT_CTOR(diracone)
+DEFAULT_CTOR(cliffordunit)
+DEFAULT_CTOR(diracgamma)
+DEFAULT_CTOR(diracgamma5)
+DEFAULT_CTOR(diracgammaL)
+DEFAULT_CTOR(diracgammaR)
//////////
// other constructors
/** 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)
+clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0)
{
- debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_clifford;
}
/** Construct object with one Lorentz index. This constructor is for internal
- * use only. Use the dirac_gamma() function instead.
+ * use only. Use the clifford_unit() or dirac_gamma() functions instead.
+ * @see clifford_unit
* @see dirac_gamma */
-clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
+clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr)
{
- debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
- GINAC_ASSERT(is_ex_of_type(mu, varidx));
+ GINAC_ASSERT(is_a<varidx>(mu));
tinfo_key = TINFO_clifford;
}
-clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
+clifford::clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr)
{
- debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_clifford;
}
-clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
+clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr)
{
- debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_clifford;
}
// archiving
//////////
-clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+clifford::clifford(const archive_node &n, 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;
+ n.find_ex("metric", metric, sym_lst);
}
void clifford::archive(archive_node &n) const
{
inherited::archive(n);
n.add_unsigned("label", representation_label);
+ n.add_ex("metric", metric);
}
DEFAULT_UNARCHIVE(clifford)
DEFAULT_ARCHIVING(diracone)
+DEFAULT_ARCHIVING(cliffordunit)
DEFAULT_ARCHIVING(diracgamma)
DEFAULT_ARCHIVING(diracgamma5)
+DEFAULT_ARCHIVING(diracgammaL)
+DEFAULT_ARCHIVING(diracgammaR)
//////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
//////////
+ex clifford::get_metric(const ex & i, const ex & j) const
+{
+ return indexed(metric, symmetric2(), i, j);
+}
+
+bool clifford::same_metric(const ex & other) const
+{
+ if (is_a<clifford>(other)) {
+ return get_metric().is_equal(ex_to<clifford>(other).get_metric());
+ } else if (is_a<indexed>(other)) {
+ return get_metric(other.op(1), other.op(2)).is_equal(other);
+ } else
+ return false;
+}
+
int clifford::compare_same_type(const basic & other) const
{
- GINAC_ASSERT(other.tinfo() == TINFO_clifford);
+ GINAC_ASSERT(is_a<clifford>(other));
const clifford &o = static_cast<const clifford &>(other);
if (representation_label != o.representation_label) {
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) && same_metric(o);
+}
+
+static bool is_dirac_slash(const ex & seq0)
+{
+ return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
+ !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
+ !is_a<diracone>(seq0);
+}
+
+void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
+{
+ // dirac_slash() object is printed differently
+ if (is_dirac_slash(seq[0])) {
+ seq[0].print(c, level);
+ c.s << "\\";
+ } else
+ this->print_dispatch<inherited>(c, level);
+}
+
+void clifford::do_print_latex(const print_latex & c, unsigned level) const
+{
+ // dirac_slash() object is printed differently
+ if (is_dirac_slash(seq[0])) {
+ c.s << "{";
+ seq[0].print(c, level);
+ c.s << "\\hspace{-1.0ex}/}";
+ } else
+ this->print_dispatch<inherited>(c, level);
+}
+
DEFAULT_COMPARE(diracone)
+DEFAULT_COMPARE(cliffordunit)
DEFAULT_COMPARE(diracgamma)
DEFAULT_COMPARE(diracgamma5)
+DEFAULT_COMPARE(diracgammaL)
+DEFAULT_COMPARE(diracgammaR)
DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
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<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
+ 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;
+ }
+}
+
+/** Predicate for finding non-clifford objects. */
+struct is_not_a_clifford : public std::unary_function<ex, bool> {
+ bool operator()(const ex & e)
+ {
+ return !is_a<clifford>(e);
+ }
+};
/** 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, 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();
+ 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();
- if (is_ex_of_type(*other, clifford)) {
+ ex dim = ex_to<idx>(self->op(1)).get_dim();
+ if (other->nops() > 1)
+ dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
- ex 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;
+
+ size_t num = other - self;
// gamma~mu gamma.mu = dim ONE
- if (other - self == 1) {
+ if (num == 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_ex_of_type(self[1], clifford)) {
+ } else if (num == 2
+ && is_a<clifford>(self[1])) {
*self = 2 - dim;
- *other = _ex1();
+ *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_ex_of_type(self[1], clifford)
- && is_ex_of_type(self[2], clifford)) {
- *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();
+ } else if (num == 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 (num == 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 Sodd gamma.mu = -2 Sodd_R
+ // (Chisholm identity in 4 dimensions)
+ } else if (!((other - self) & 1) && dim.is_equal(4)) {
+ if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
+ return false;
+
+ *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
+ std::fill(self + 1, other, _ex1);
+ *other = _ex_2;
+ return true;
+
+ // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
+ // (commutate contracted indices towards each other, then use
+ // Chisholm identity in 4 dimensions)
+ } else if (((other - self) & 1) && dim.is_equal(4)) {
+ if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
+ return false;
+
+ exvector::iterator next_to_last = other - 1;
+ ex S = ncmul(exvector(self + 1, next_to_last), true);
+ ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
+
+ *self = (*next_to_last) * S + SR * (*next_to_last);
+ std::fill(self + 1, other, _ex1);
+ *other = _ex2;
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 {
+ if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
+ return false;
+
+ exvector::iterator next_to_last = other - 1;
+ ex S = ncmul(exvector(self + 1, next_to_last), true);
+
+ *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
+ std::fill(self + 1, other + 1, _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;
+}
+
+/** An utility function looking for a given metric within an exvector,
+ * used in cliffordunit::contract_with(). */
+static int find_same_metric(exvector & v, ex & c)
+{
+ for (int i=0; i<v.size();i++) {
+ if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i])
+ && ex_to<clifford>(c).same_metric(v[i])
+ && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
+ || ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])) {
+ return ++i; // next to found
+ }
+ }
+ return 0; //nothing found
+}
+
+/** Contraction of a Clifford unit with something else. */
+bool cliffordunit::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<cliffordunit>(self->op(0)));
+ clifford unit = ex_to<clifford>(*self);
+ unsigned char rl = unit.get_representation_label();
+
+ if (is_a<clifford>(*other)) {
+ // Contraction only makes sense if the represenation labels are equal
+ // and the metrics are the same
+ if ((ex_to<clifford>(*other).get_representation_label() != rl)
+ && unit.same_metric(*other))
+ return false;
+
+ // Find if a previous contraction produces the square of self
+ int prev_square = find_same_metric(v, self[0]);
+ varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(ex_to<idx>(self->op(1)).get_dim()));
+ ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
+
+ // e~mu e.mu = Tr ONE
+ if (other - self == 1) {
+ if (prev_square != 0) {
+ *self = squared_metric;
+ v[prev_square-1] = _ex1;
+ } else
+ *self = unit.get_metric(self->op(1), other->op(1));
+ *other = dirac_ONE(rl);
+ return true;
+
+ // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
+ } else if (other - self == 2
+ && is_a<clifford>(self[1])) {
+
+ const ex & ia = self[1].op(1);
+ const ex & ib = self[1].op(1);
+ if (is_a<tensmetric>(unit.get_metric()))
+ *self = 2 - unit.get_metric(self->op(1), other->op(1));
+ else if (prev_square != 0) {
+ *self = 2-squared_metric;
+ v[prev_square-1] = _ex1;
+ } else
+ *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
+ *other = _ex1;
+ return true;
+
+ // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~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))
+ if (!is_a<clifford>(*it))
return false;
- it++;
+ ++it;
}
it = self + 1;
- ex S = _ex1();
+ ex S = _ex1;
while (it != next_to_last) {
S *= *it;
- *it++ = _ex1();
+ *it++ = _ex1;
}
- *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
- *next_to_last = _ex1();
- *other = _ex1();
+ const ex & ia = next_to_last->op(1);
+ const ex & ib = next_to_last->op(1);
+ if (is_a<tensmetric>(unit.get_metric()))
+ *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
+ else if (prev_square != 0) {
+ *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
+ v[prev_square-1] = _ex1;
+ } else
+ *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
+ *next_to_last = _ex1;
+ *other = _ex1;
return true;
}
- }
+
+ }
return false;
}
/** Perform automatic simplification on noncommutative product of clifford
- * objects. This removes superfluous ONEs, permutes gamma5's to the front
+ * 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
+ex clifford::eval_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_ex_of_type(cit->op(0), diracone))
+ 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's to the front
+ // Anticommutate 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_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
- it->swap(*it2);
- sign = -sign;
- something_changed = true;
+ 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--;
+ --it;
}
if (next_to_last == first)
break;
- next_to_last--;
+ --next_to_last;
}
}
- // Remove squares of gamma5
- while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
- s.erase(s.begin(), s.begin() + 2);
- something_changed = true;
- }
-
// Remove equal adjacent gammas
if (s.size() >= 2) {
- exvector::iterator it = s.begin(), itend = s.end() - 1;
- while (it != itend) {
+ exvector::iterator it, itend = s.end() - 1;
+ for (it = s.begin(); it != itend; ++it) {
ex & a = it[0];
ex & b = it[1];
- if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
+ 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_cliffordunit = is_a<cliffordunit>(ag);
+ bool b_is_cliffordunit = is_a<cliffordunit>(bg);
+
+ if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
+
const ex & ia = a.op(1);
const ex & ib = b.op(1);
- if (ia.is_equal(ib)) {
- a = lorentz_g(ia, ib);
+ if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
+ a = ex_to<clifford>(a).get_metric(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_cliffordunit && !b_is_cliffordunit && 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;
}
- it++;
}
}
- if (s.size() == 0)
+ if (s.empty())
return clifford(diracone(), representation_label) * sign;
if (something_changed)
- return nonsimplified_ncmul(s) * sign;
+ return reeval_ncmul(s) * sign;
else
- return simplified_ncmul(s) * sign;
+ return hold_ncmul(s) * sign;
}
-ex clifford::thisexprseq(const exvector & v) const
+ex clifford::thiscontainer(const exvector & v) const
{
- return clifford(representation_label, v);
+ return clifford(representation_label, get_metric(), v);
}
-ex clifford::thisexprseq(exvector * vp) const
+ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
{
- return clifford(representation_label, vp);
+ return clifford(representation_label, get_metric(), vp);
+}
+
+ex diracgamma5::conjugate() const
+{
+ return _ex_1 * (*this);
+}
+
+ex diracgammaL::conjugate() const
+{
+ return (new diracgammaR)->setflag(status_flags::dynallocated);
+}
+
+ex diracgammaR::conjugate() const
+{
+ return (new diracgammaL)->setflag(status_flags::dynallocated);
}
//////////
ex dirac_ONE(unsigned char rl)
{
- return clifford(diracone(), rl);
+ static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
+ return clifford(ONE, rl);
+}
+
+ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
+{
+ static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
+
+ if (!is_a<varidx>(mu))
+ throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
+
+ return clifford(unit, mu, metr, rl);
}
ex dirac_gamma(const ex & mu, unsigned char rl)
{
- if (!is_ex_of_type(mu, varidx))
+ static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
+
+ if (!is_a<varidx>(mu))
throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
- return clifford(diracgamma(), mu, rl);
+ return clifford(gamma, mu, default_metric(), rl);
}
ex dirac_gamma5(unsigned char rl)
{
- return clifford(diracgamma5(), rl);
+ static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
+ return clifford(gamma5, rl);
}
-ex dirac_gamma6(unsigned char rl)
+ex dirac_gammaL(unsigned char rl)
{
- return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
+ static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
+ return clifford(gammaL, rl);
}
-ex dirac_gamma7(unsigned char rl)
+ex dirac_gammaR(unsigned char rl)
{
- return clifford(diracone(), rl) - clifford(diracgamma5(), 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)
{
- varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
- return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, 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
+ return clifford(e, varidx(0, dim), default_metric(), rl);
}
/** Check whether a given tinfo key (as returned by return_type_tinfo()
return (ti & ~0xff) == TINFO_clifford;
}
+/** Extract representation label from tinfo key (as returned by
+ * return_type_tinfo()). */
+static unsigned char get_representation_label(unsigned ti)
+{
+ return ti & 0xff;
+}
+
/** 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)
+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
+ // 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])
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++) {
+ 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];
return result;
}
-ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
+ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
{
- if (is_ex_of_type(e, clifford)) {
+ if (is_a<clifford>(e)) {
- if (ex_to_clifford(e).get_representation_label() == rl
- && is_ex_of_type(e.op(0), diracone))
- return trONE;
- else
- return _ex0();
+ unsigned char rl = ex_to<clifford>(e).get_representation_label();
- } else if (is_ex_exactly_of_type(e, add)) {
+ // Are we taking the trace over this object's representation label?
+ if (rls.find(rl) == rls.end())
+ return e;
- // Trace of sum = sum of traces
- ex sum = _ex0();
- for (unsigned i=0; i<e.nops(); i++)
- sum += dirac_trace(e.op(i), rl, trONE);
- return sum;
+ // 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_ex_exactly_of_type(e, mul)) {
+ } else if (is_exactly_a<mul>(e)) {
// Trace of product: pull out non-clifford factors
- ex prod = _ex1();
- for (unsigned i=0; i<e.nops(); i++) {
+ ex prod = _ex1;
+ for (size_t 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, trONE);
+ if (is_clifford_tinfo(o.return_type_tinfo()))
+ prod *= dirac_trace(o, rls, trONE);
else
prod *= o;
}
return prod;
- } else if (is_ex_exactly_of_type(e, ncmul)) {
+ } else if (is_exactly_a<ncmul>(e)) {
+
+ unsigned char rl = get_representation_label(e.return_type_tinfo());
- if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
- return _ex0();
+ // Are we taking the trace over this string's representation label?
+ if (rls.find(rl) == rls.end())
+ return e;
- // Expand product, if necessary
- ex e_expanded = e.expand();
- if (!is_ex_of_type(e_expanded, ncmul))
- return dirac_trace(e_expanded, rl, trONE);
+ // 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_ex_of_type(e.op(0).op(0), diracgamma5);
- unsigned num = e.nops();
+ 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();
+ 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));
+ // (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
- exvector ix;
- ix.reserve(num - 1);
- for (unsigned i=1; i<num; i++)
- ix.push_back(e.op(i).op(1));
+ // (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 (int i=0; i<num-3; i++) {
+ for (size_t i=0; i<num-3; i++) {
ex idx1 = ix[i];
- for (int j=i+1; j<num-2; j++) {
+ for (size_t j=i+1; j<num-2; j++) {
ex idx2 = ix[j];
- for (int k=j+1; k<num-1; k++) {
+ for (size_t k=j+1; k<num-1; k++) {
ex idx3 = ix[k];
- for (int l=k+1; l<num; l++) {
+ 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 (int n=0, t=4; n<num; n++) {
+ 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 * eps0123(idx1, idx2, idx3, idx4)
+ 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;
+ return trONE * I * result * mul(bv);
} else { // no gamma5
// Trace of odd number of gammas is zero
if ((num & 1) == 1)
- return _ex0();
+ return _ex0;
// Tr gamma.mu gamma.nu = 4 g.mu.nu
- if (num == 2)
- return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
+ 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;
- iv.reserve(num);
- for (unsigned i=0; i<num; i++)
- iv.push_back(e.op(i).op(1));
+ 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);
+ 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)
+{
+ // Convert list to set
+ std::set<unsigned char> rls;
+ for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
+ if (i->info(info_flags::nonnegint))
+ rls.insert(ex_to<numeric>(*i).to_int());
}
- return _ex0();
+ return dirac_trace(e, rls, trONE);
}
+ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
+{
+ // Convert label to set
+ std::set<unsigned char> rls;
+ rls.insert(rl);
+
+ return dirac_trace(e, rls, trONE);
+}
+
+
ex canonicalize_clifford(const ex & e)
{
// Scan for any ncmul objects
- lst srl;
+ exmap srl;
ex aux = e.to_rational(srl);
- for (unsigned i=0; i<srl.nops(); i++) {
+ for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
- ex lhs = srl.op(i).lhs();
- ex rhs = srl.op(i).rhs();
+ ex lhs = i->first;
+ ex rhs = i->second;
- if (is_ex_exactly_of_type(rhs, ncmul)
+ if (is_exactly_a<ncmul>(rhs)
&& 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));
+ if (!is_a<ncmul>(rhs_expanded)) {
+ i->second = canonicalize_clifford(rhs_expanded);
continue;
- } else if (!is_ex_of_type(rhs.op(0), clifford))
+ } else if (!is_a<clifford>(rhs.op(0)))
continue;
exvector v;
v.reserve(rhs.nops());
- for (unsigned j=0; j<rhs.nops(); j++)
+ for (size_t j=0; j<rhs.nops(); j++)
v.push_back(rhs.op(j));
- // Stupid bubble sort because we only want to swap adjacent gammas
+ // 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++;
+ 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].op(1).compare(it[1].op(1)) > 0) {
+ if (it[0].compare(it[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 b1, i1, b2, i2;
+ base_and_index(it[0], b1, i1);
+ base_and_index(it[1], b2, i2);
+ it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
+ it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
ex sum = ncmul(v);
it[0] = save1;
it[1] = save0;
sum -= ncmul(v, true);
- srl.let_op(i) = (lhs == canonicalize_clifford(sum));
+ i->second = canonicalize_clifford(sum);
goto next_sym;
}
- it++;
+ ++it;
}
next_sym: ;
}
}
- return aux.subs(srl);
+ return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
+}
+
+ex clifford_prime(const ex &e)
+{
+ pointer_to_map_function fcn(clifford_prime);
+ if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
+ return -e;
+ } else if (is_a<add>(e)) {
+ return e.map(fcn);
+ } else if (is_a<ncmul>(e)) {
+ return e.map(fcn);
+ } else if (is_a<power>(e)) {
+ return pow(clifford_prime(e.op(0)), e.op(1));
+ } else
+ return e;
}
+ex delete_ONE(const ex &e)
+{
+ pointer_to_map_function fcn(delete_ONE);
+ if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
+ return 1;
+ } else if (is_a<add>(e)) {
+ return e.map(fcn);
+ } else if (is_a<ncmul>(e)) {
+ return e.map(fcn);
+ } else if (is_a<mul>(e)) {
+ return e.map(fcn);
+ } else if (is_a<power>(e)) {
+ return pow(delete_ONE(e.op(0)), e.op(1));
+ } else
+ return e;
+}
+
+ex clifford_norm(const ex &e)
+{
+ return sqrt(delete_ONE((e * clifford_bar(e)).simplify_indexed()));
+}
+
+ex clifford_inverse(const ex &e)
+{
+ ex norm = clifford_norm(e);
+ if (!norm.is_zero())
+ return clifford_bar(e) / pow(norm, 2);
+}
+
+ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
+{
+ unsigned min, max;
+ if (!ex_to<idx>(mu).is_dim_numeric())
+ throw(std::invalid_argument("Index should have a numeric dimension"));
+ unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
+ ex c = clifford_unit(mu, metr, rl);
+
+ if (is_a<matrix>(v)) {
+ if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
+ min = ex_to<matrix>(v).rows();
+ max = ex_to<matrix>(v).cols();
+ } else {
+ min = ex_to<matrix>(v).cols();
+ max = ex_to<matrix>(v).rows();
+ }
+ if (min == 1) {
+ if (dim == max)
+ if (is_a<varidx>(mu)) // need to swap variance
+ return indexed(v, ex_to<varidx>(mu).toggle_variance()) * c;
+ else
+ return indexed(v, mu) * c;
+ else
+ throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
+ } else
+ throw(std::invalid_argument("First argument should be a vector vector"));
+ } else if (is_a<lst>(v)) {
+ if (dim == ex_to<lst>(v).nops())
+ return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * c;
+ else
+ throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
+ } else
+ throw(std::invalid_argument("Cannot construct from anything but list or vector"));
+}
+
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff