* Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
/*
- * GiNaC Copyright (C) 1999-2003 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
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
-#include <iostream>
-#include <stdexcept>
-
#include "clifford.h"
+
#include "ex.h"
#include "idx.h"
#include "ncmul.h"
#include "lst.h"
#include "relational.h"
#include "operators.h"
+#include "add.h"
#include "mul.h"
-#include "print.h"
+#include "power.h"
+#include "matrix.h"
#include "archive.h"
#include "utils.h"
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(diracgammaL, tensor)
-GINAC_IMPLEMENT_REGISTERED_CLASS(diracgammaR, 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 constructors
//////////
-clifford::clifford() : representation_label(0)
+clifford::clifford() : representation_label(0), metric(lorentz_g(varidx((new symbol)->setflag(status_flags::dynallocated),4),varidx((new symbol)->setflag(status_flags::dynallocated),4)))
{
tinfo_key = TINFO_clifford;
}
DEFAULT_CTOR(diracone)
+DEFAULT_CTOR(cliffordunit)
DEFAULT_CTOR(diracgamma)
DEFAULT_CTOR(diracgamma5)
DEFAULT_CTOR(diracgammaL)
/** 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)
{
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)
{
GINAC_ASSERT(is_a<varidx>(mu));
tinfo_key = TINFO_clifford;
}
-clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
+clifford::clifford(unsigned char rl, const exvector & v, bool discardable, const ex & metr) : inherited(sy_none(), v, discardable), representation_label(rl), metric(metr)
{
tinfo_key = TINFO_clifford;
}
-clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
+clifford::clifford(unsigned char rl, std::auto_ptr<exvector> vp, const ex & metr) : inherited(sy_none(), vp), representation_label(rl), metric(metr)
{
tinfo_key = TINFO_clifford;
}
unsigned rl;
n.find_unsigned("label", rl);
representation_label = rl;
+ ex m;
+ n.find_ex("metric", m, sym_lst);
+ metric = m;
}
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)
// functions overriding virtual functions from base classes
//////////
+ex clifford::get_metric(const ex & i, const ex & j) const
+{
+ return metric.subs(metric.op(1) == i).subs(metric.op(2) == j);
+}
+
+bool clifford::same_metric(const ex & other) const
+{
+ if (is_a<clifford>(other)) {
+ ex m = get_metric();
+ return m.is_equal(ex_to<clifford>(other).get_metric(m.op(1), m.op(2)));
+ } else if (is_a<indexed>(other)) {
+ ex m = get_metric(other.op(1), other.op(2));
+ return m.is_equal(other);
+ } else
+ return false;
+}
+
int clifford::compare_same_type(const basic & other) const
{
GINAC_ASSERT(is_a<clifford>(other));
GINAC_ASSERT(is_a<clifford>(other));
const clifford &o = static_cast<const clifford &>(other);
- return representation_label == o.representation_label;
+ return (representation_label == o.representation_label) && same_metric(o);
}
-void clifford::print(const print_context & c, unsigned level) const
+static bool is_dirac_slash(const ex & seq0)
{
- 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])) {
+ return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
+ !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
+ !is_a<diracone>(seq0);
+}
- // 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 << "\\";
- }
+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
- inherited::print(c, level);
+ 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}")
GINAC_ASSERT(is_a<clifford>(c));
GINAC_ASSERT(c.nops() == 2);
- if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
+ 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
return false;
}
+/** An utility function looking for given metric within exvector,
+ * used in cliffordunit::contract_with() */
+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 units 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().op(0)))
+ *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_a<clifford>(*it))
+ return false;
+ ++it;
+ }
+
+ it = self + 1;
+ ex S = _ex1;
+ while (it != next_to_last) {
+ S *= *it;
+ *it++ = _ex1;
+ }
+
+ const ex & ia = next_to_last->op(1);
+ const ex & ib = next_to_last->op(1);
+ if (is_a<tensmetric>(unit.get_metric().op(0)))
+ *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/L/R's to the front
* and removes squares of gamma objects. */
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);
+ bool a_is_cliffordunit = is_a<cliffordunit>(ag);
+ bool b_is_cliffordunit = is_a<cliffordunit>(bg);
- if (a_is_diracgamma && b_is_diracgamma) {
+ 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)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
- a = lorentz_g(ia, ib);
+ a = ex_to<clifford>(a).get_metric(ia,ib);
b = dirac_ONE(representation_label);
something_changed = true;
}
a = dirac_ONE(representation_label);
something_changed = true;
- } else if (!a_is_diracgamma && !b_is_diracgamma && ag.is_equal(bg)) {
+ } 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;
ex clifford::thiscontainer(const exvector & v) const
{
- return clifford(representation_label, v);
+ return clifford(representation_label, v, false, get_metric());
+}
+
+ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
+{
+ return clifford(representation_label, vp, get_metric());
+}
+
+ex diracgamma5::conjugate() const
+{
+ return _ex_1 * (*this);
+}
+
+ex diracgammaL::conjugate() const
+{
+ return (new diracgammaR)->setflag(status_flags::dynallocated);
}
-ex clifford::thiscontainer(exvector * vp) const
+ex diracgammaR::conjugate() const
{
- return clifford(representation_label, vp);
+ return (new diracgammaL)->setflag(status_flags::dynallocated);
}
//////////
return clifford(diracone(), rl);
}
+ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
+{
+ if (!is_a<varidx>(mu))
+ throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
+ if (!is_a<indexed>(metr))
+ throw(std::invalid_argument("metric for Clifford unit must be of type indexed"));
+ exvector d = ex_to<indexed>(metr).get_indices();
+ if (d.size() > 2 || ex_to<idx>(d[0]).get_dim() != ex_to<idx>(d[1]).get_dim())
+ //|| ex_to<idx>(d[0]).get_dim() != ex_to<idx>(mu).get_dim())
+ throw(std::invalid_argument("metric is not square"));
+ else
+ ex_to<idx>(mu).replace_dim(ex_to<idx>(d[0]).get_dim());
+ return clifford(cliffordunit(), mu, metr, rl);
+}
+
ex dirac_gamma(const ex & mu, unsigned char rl)
{
if (!is_a<varidx>(mu))
throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
- return clifford(diracgamma(), mu, rl);
+ ex dim = ex_to<idx>(mu).get_dim();
+ return clifford(diracgamma(), mu, lorentz_g(varidx((new symbol)->setflag(status_flags::dynallocated), dim),varidx((new symbol)->setflag(status_flags::dynallocated), dim)), rl);
}
ex dirac_gamma5(unsigned char rl)
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();
+ ), subs_options::no_pattern).expand();
if (!is_a<ncmul>(e_expanded))
return dirac_trace(e_expanded, rl, trONE);
ex canonicalize_clifford(const ex & e)
{
// Scan for any ncmul objects
- lst srl;
+ exmap srl;
ex aux = e.to_rational(srl);
- for (size_t i=0; i<srl.nops(); i++) {
+ for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
- ex o = srl.op(i);
- ex lhs = o.lhs();
- ex rhs = o.rhs();
+ ex lhs = i->first;
+ ex rhs = i->second;
if (is_exactly_a<ncmul>(rhs)
&& rhs.return_type() == return_types::noncommutative
// Expand product, if necessary
ex rhs_expanded = rhs.expand();
if (!is_a<ncmul>(rhs_expanded)) {
- srl[i] = (lhs == canonicalize_clifford(rhs_expanded));
+ i->second = canonicalize_clifford(rhs_expanded);
continue;
} else if (!is_a<clifford>(rhs.op(0)))
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;
+ 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[i] = (lhs == canonicalize_clifford(sum));
+ i->second = canonicalize_clifford(sum);
goto next_sym;
}
++it;
next_sym: ;
}
}
- return aux.subs(srl).simplify_indexed();
+ return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
}
-} // namespace GiNaC
+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 (-1)*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
+
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