* Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
/*
- * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2006 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
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 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 "power.h"
+#include "matrix.h"
#include "archive.h"
#include "utils.h"
print_func<print_dflt>(&clifford::do_print_dflt).
print_func<print_latex>(&clifford::do_print_latex))
+const tinfo_static_t clifford::return_type_tinfo_static[256] = {{}};
+
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(diracgamma, tensor,
+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))
// default constructors
//////////
-clifford::clifford() : representation_label(0)
+clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
{
- tinfo_key = TINFO_clifford;
+ tinfo_key = &clifford::tinfo_static;
}
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), commutator_sign(-1)
{
- tinfo_key = TINFO_clifford;
+ tinfo_key = &clifford::tinfo_static;
}
/** 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, int comm_sign) : inherited(b, mu), representation_label(rl), metric(metr), commutator_sign(comm_sign)
{
GINAC_ASSERT(is_a<varidx>(mu));
- tinfo_key = TINFO_clifford;
+ tinfo_key = &clifford::tinfo_static;
}
-clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
+clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), commutator_sign(comm_sign)
{
- tinfo_key = TINFO_clifford;
+ tinfo_key = &clifford::tinfo_static;
}
-clifford::clifford(unsigned char rl, std::auto_ptr<exvector> vp) : inherited(sy_none(), vp), representation_label(rl)
+clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), commutator_sign(comm_sign)
{
- tinfo_key = TINFO_clifford;
+ tinfo_key = &clifford::tinfo_static;
}
//////////
// archiving
//////////
-clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
+clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
{
unsigned rl;
n.find_unsigned("label", rl);
representation_label = rl;
+ n.find_ex("metric", metric, sym_lst);
+ n.find_unsigned("commutator_sign+1", rl);
+ commutator_sign = rl - 1;
}
-void clifford::archive(archive_node &n) const
+void clifford::archive(archive_node & n) const
{
inherited::archive(n);
n.add_unsigned("label", representation_label);
+ n.add_ex("metric", metric);
+ n.add_unsigned("commutator_sign+1", commutator_sign+1);
}
DEFAULT_UNARCHIVE(clifford)
DEFAULT_ARCHIVING(diracone)
+DEFAULT_ARCHIVING(cliffordunit)
DEFAULT_ARCHIVING(diracgamma)
DEFAULT_ARCHIVING(diracgamma5)
DEFAULT_ARCHIVING(diracgammaL)
DEFAULT_ARCHIVING(diracgammaR)
+
+ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
+{
+ if (is_a<indexed>(metric)) {
+ if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
+ if (is_a<matrix>(metric.op(0))) {
+ return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
+ symmetric2(), i, j);
+ } else {
+ return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
+ }
+ } else {
+ return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern);
+ }
+ } else {
+ exvector indices = metric.get_free_indices();
+ if (symmetrised)
+ return _ex1_2*simplify_indexed(metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern)
+ + metric.subs(lst(indices[0] == j, indices[1] == i), subs_options::no_pattern));
+ else
+ return metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern);
+ }
+}
+
+bool clifford::same_metric(const ex & other) const
+{
+ ex metr;
+ if (is_a<clifford>(other))
+ metr = ex_to<clifford>(other).get_metric();
+ else
+ metr = other;
+
+ if (is_a<indexed>(metr))
+ return metr.op(0).is_equal(get_metric().op(0));
+ else {
+ exvector indices = metr.get_free_indices();
+ return (indices.size() == 2)
+ && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
+ }
+}
+
//////////
// functions overriding virtual functions from base classes
//////////
+ex clifford::op(size_t i) const
+{
+ GINAC_ASSERT(i<nops());
+ if (nops()-i == 1)
+ return representation_label;
+ else
+ return inherited::op(i);
+}
+
+ex & clifford::let_op(size_t i)
+{
+ GINAC_ASSERT(i<nops());
+
+ static ex rl = numeric(representation_label);
+ ensure_if_modifiable();
+ if (nops()-i == 1)
+ return rl;
+ else
+ return inherited::let_op(i);
+}
+
+ex clifford::subs(const exmap & m, unsigned options) const
+{
+ ex subsed = inherited::subs(m, options);
+ if(is_a<clifford>(subsed)) {
+ ex prevmetric = ex_to<clifford>(subsed).metric;
+ ex newmetric = prevmetric.subs(m, options);
+ if(!are_ex_trivially_equal(prevmetric, newmetric)) {
+ clifford c = ex_to<clifford>(subsed);
+ c.metric = newmetric;
+ subsed = c;
+ }
+ }
+ return subsed;
+}
+
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) && (commutator_sign == o.get_commutator_sign()) && 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<diracgamma>(seq0) &&
+ !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
!is_a<diracone>(seq0);
}
{
// dirac_slash() object is printed differently
if (is_dirac_slash(seq[0])) {
- seq[0].print(c, level);
+ seq[0].print(c, precedence());
c.s << "\\";
} else
this->print_dispatch<inherited>(c, level);
// dirac_slash() object is printed differently
if (is_dirac_slash(seq[0])) {
c.s << "{";
- seq[0].print(c, level);
+ seq[0].print(c, precedence());
c.s << "\\hspace{-1.0ex}/}";
- } else
+ } else {
+ c.s << "\\clifford[" << int(representation_label) << "]";
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(diracone, "ONE", "\\mathbf{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}")
static void base_and_index(const ex & c, ex & b, ex & i)
{
GINAC_ASSERT(is_a<clifford>(c));
- GINAC_ASSERT(c.nops() == 2);
+ GINAC_ASSERT(c.nops() == 2+1);
- 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
}
}
+/** 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
{
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
+ } else if (num == 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
+ } else if (num == 3
&& is_a<clifford>(self[1])
&& is_a<clifford>(self[2])) {
ex b1, i1, b2, i2;
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
+ } else if (num == 4
&& is_a<clifford>(self[1])
&& is_a<clifford>(self[2])
&& is_a<clifford>(self[3])) {
*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 {
- exvector::iterator it = self + 1, next_to_last = other - 1;
- while (it != other) {
- if (!is_a<clifford>(*it))
- return false;
- ++it;
- }
+ if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
+ return false;
- it = self + 1;
- ex S = _ex1;
- while (it != next_to_last) {
- S *= *it;
- *it++ = _ex1;
- }
+ 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);
- *next_to_last = _ex1;
- *other = _ex1;
+ std::fill(self + 1, other + 1, _ex1);
return true;
}
return false;
}
+/** 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;
+
+ exvector::iterator before_other = other - 1;
+ ex mu = self->op(1);
+ ex mu_toggle = other->op(1);
+ ex alpha = before_other->op(1);
+
+ // e~mu e.mu = Tr ONE
+ if (other - self == 1) {
+ *self = unit.get_metric(mu, mu_toggle, true);
+ *other = dirac_ONE(rl);
+ return true;
+
+ } else if (other - self == 2) {
+ if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
+ // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
+ *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
+ *before_other = _ex1;
+ *other = _ex1;
+ return true;
+
+ } else {
+ // e~mu S e.mu = Tr S ONE
+ *self = unit.get_metric(mu, mu_toggle, true);
+ *other = dirac_ONE(rl);
+ return true;
+ }
+ } else {
+ // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
+ // (commutate contracted indices towards each other, simplify_indexed()
+ // will re-expand and re-run the simplification)
+ if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
+ return false;
+ }
+
+ ex S = ncmul(exvector(self + 1, before_other), true);
+
+ if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
+ *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
+ } else {
+ // simply commutes
+ *self = (*self) * S * (*other) * (*before_other);
+ }
+
+ std::fill(self + 1, other + 1, _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. */
bool something_changed = false;
int sign = 1;
- // Anticommute gamma5/L/R'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) {
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) {
+ 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)
+ && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
+ // This is done only for Clifford algebras
+
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, true);
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;
}
if (s.empty())
- return clifford(diracone(), representation_label) * sign;
+ return dirac_ONE(representation_label) * sign;
if (something_changed)
return reeval_ncmul(s) * sign;
else
ex clifford::thiscontainer(const exvector & v) const
{
- return clifford(representation_label, v);
+ return clifford(representation_label, metric, commutator_sign, v);
}
ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
{
- return clifford(representation_label, vp);
+ return clifford(representation_label, metric, commutator_sign, vp);
}
ex diracgamma5::conjugate() const
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<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) && is_a<varidx>(indices[0]) && is_a<varidx>(indices[1])) {
+ 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 varidx xi((new symbol)->setflag(status_flags::dynallocated), n),
+ chi((new symbol)->setflag(status_flags::dynallocated), n);
+ if ((n == M.cols()) && (n == ex_to<idx>(mu).get_dim())) {
+ for (unsigned i = 0; i < n; i++) {
+ for (unsigned j = i+1; j < n; j++) {
+ if (M(i, j) != 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<varidx>(mu).get_dim()),
+ chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(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("index of Dirac gamma must be of type varidx"));
+ throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
- return clifford(diracgamma(), mu, rl);
+ 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)
{
- return clifford(diracgamma5(), rl);
+ static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
+ return clifford(gamma5, rl);
}
ex dirac_gammaL(unsigned char rl)
{
- return clifford(diracgammaL(), rl);
+ static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
+ return clifford(gammaL, rl);
}
ex dirac_gammaR(unsigned char rl)
{
- return clifford(diracgammaR(), 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
- return clifford(e, varidx(0, dim), rl);
+
+ 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);
}
/** 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)
+ * is that of a clifford object (with an arbitrary representation label). */
+bool is_clifford_tinfo(tinfo_t ti)
{
- return ti == (TINFO_clifford + rl);
+ p_int start_loc=(p_int)&clifford::return_type_tinfo_static;
+ return (p_int)ti>=start_loc && (p_int)ti<start_loc+256;
}
-/** 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)
+/** Extract representation label from tinfo key (as returned by
+ * return_type_tinfo()). */
+static unsigned char get_representation_label(tinfo_t ti)
{
- return (ti & ~0xff) == TINFO_clifford;
+ return (unsigned char)((p_int)ti-(p_int)&clifford::return_type_tinfo_static);
}
/** Take trace of a string of an even number of Dirac gammas given a vector
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_a<clifford>(e)) {
- if (!ex_to<clifford>(e).get_representation_label() == rl)
- return _ex0;
+ 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;
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(), rl))
- prod *= dirac_trace(o, rl, trONE);
+ if (is_clifford_tinfo(o.return_type_tinfo()))
+ prod *= dirac_trace(o, rls, trONE);
else
prod *= o;
}
} else if (is_exactly_a<ncmul>(e)) {
- if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
- return _ex0;
+ 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_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, rl, trONE);
+ 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));
} 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);
+ 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 canonicalize_clifford(const ex & e)
+ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
{
- // Scan for any ncmul objects
- exmap srl;
- ex aux = e.to_rational(srl);
- for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
+ // 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());
+ }
- ex lhs = i->first;
- ex rhs = i->second;
+ return dirac_trace(e, rls, trONE);
+}
- if (is_exactly_a<ncmul>(rhs)
- && rhs.return_type() == return_types::noncommutative
- && is_clifford_tinfo(rhs.return_type_tinfo())) {
+ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
+{
+ // Convert label to set
+ std::set<unsigned char> rls;
+ rls.insert(rl);
- // Expand product, if necessary
- ex rhs_expanded = rhs.expand();
- if (!is_a<ncmul>(rhs_expanded)) {
- i->second = canonicalize_clifford(rhs_expanded);
- continue;
+ return dirac_trace(e, rls, trONE);
+}
- } else if (!is_a<clifford>(rhs.op(0)))
- continue;
- exvector v;
- v.reserve(rhs.nops());
- for (size_t 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] = 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);
- i->second = canonicalize_clifford(sum);
- goto next_sym;
+ex canonicalize_clifford(const ex & e_)
+{
+ pointer_to_map_function fcn(canonicalize_clifford);
+
+ if (is_a<matrix>(e_) // || is_a<pseries>(e) || is_a<integral>(e)
+ || e_.info(info_flags::list)) {
+ return e_.map(fcn);
+ } else {
+ ex e=simplify_indexed(e_);
+ // Scan for any ncmul objects
+ exmap srl;
+ ex aux = e.to_rational(srl);
+ for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
+
+ ex lhs = i->first;
+ ex rhs = i->second;
+
+ 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_a<ncmul>(rhs_expanded)) {
+ i->second = canonicalize_clifford(rhs_expanded);
+ continue;
+
+ } else if (!is_a<clifford>(rhs.op(0)))
+ continue;
+
+ exvector v;
+ v.reserve(rhs.nops());
+ for (size_t 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);
+ // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
+ it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
+ it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
+ ex sum = ncmul(v);
+ it[0] = save1;
+ it[1] = save0;
+ sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
+ i->second = canonicalize_clifford(sum);
+ goto next_sym;
+ }
+ ++it;
}
- ++it;
- }
next_sym: ;
+ }
}
+ return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
}
- 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) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
+ || is_a<matrix>(e) || e.info(info_flags::list)) {
+ return e.map(fcn);
+ } else if (is_a<power>(e)) {
+ return pow(clifford_prime(e.op(0)), e.op(1));
+ } else
+ return e;
+}
+
+ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
+{
+ pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
+ bool need_reevaluation = false;
+ ex e1 = e;
+ if (! (options & 1) ) { // is not a child
+ if (options & 2)
+ e1 = expand_dummy_sum(e, true);
+ e1 = canonicalize_clifford(e1);
+ }
+
+ if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
+ if (is_a<diracone>(e1.op(0)))
+ return 1;
+ else
+ throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
+ } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
+ || is_a<matrix>(e1) || e1.info(info_flags::list)) {
+ if (options & 3) // is a child or was already expanded
+ return e1.map(fcn);
+ else
+ try {
+ return e1.map(fcn);
+ } catch (std::exception &p) {
+ need_reevaluation = true;
+ }
+ } else if (is_a<power>(e1)) {
+ if (options & 3) // is a child or was already expanded
+ return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
+ else
+ try {
+ return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
+ } catch (std::exception &p) {
+ need_reevaluation = true;
+ }
+ }
+ if (need_reevaluation)
+ return remove_dirac_ONE(e, rl, options | 2);
+ return e1;
+}
+
+char clifford_max_label(const ex & e, bool ignore_ONE)
+{
+ if (is_a<clifford>(e))
+ if (ignore_ONE && is_a<diracone>(e.op(0)))
+ return -1;
+ else
+ return ex_to<clifford>(e).get_representation_label();
+ else {
+ char rl = -1;
+ for (size_t i=0; i < e.nops(); i++)
+ rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
+ return rl;
+ }
+}
+
+ex clifford_norm(const ex & e)
+{
+ return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
+}
+
+ex clifford_inverse(const ex & e)
+{
+ ex norm = clifford_norm(e);
+ if (!norm.is_zero())
+ return clifford_bar(e) / pow(norm, 2);
+ else
+ throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
+}
+
+ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
+{
+ if (!ex_to<idx>(mu).is_dim_numeric())
+ throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
+ ex e = clifford_unit(mu, metr, rl);
+ return lst_to_clifford(v, e);
+}
+
+ex lst_to_clifford(const ex & v, const ex & e) {
+ unsigned min, max;
+
+ if (is_a<clifford>(e)) {
+ ex mu = e.op(1);
+ ex mu_toggle
+ = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
+ unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
+
+ 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)
+ return indexed(v, mu_toggle) * e;
+ else
+ throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
+ } else
+ throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
+ } else if (v.info(info_flags::list)) {
+ if (dim == ex_to<lst>(v).nops())
+ return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
+ else
+ throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
+ } else
+ throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
+ } else
+ throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
+}
+
+/** Auxiliary structure to define a function for striping one Clifford unit
+ * from vectors. Used in clifford_to_lst(). */
+static ex get_clifford_comp(const ex & e, const ex & c)
+{
+ pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
+ int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
+
+ if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
+ || is_a<matrix>(e))
+ return e.map(fcn);
+ else if (is_a<ncmul>(e) || is_a<mul>(e)) {
+ // find a Clifford unit with the same metric, delete it and substitute its index
+ size_t ind = e.nops() + 1;
+ for (size_t j = 0; j < e.nops(); j++)
+ if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
+ if (ind > e.nops())
+ ind = j;
+ else
+ throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
+ if (ind < e.nops()) {
+ ex S = 1;
+ bool same_value_index, found_dummy;
+ same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
+ && (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
+ found_dummy = same_value_index;
+ for(size_t j=0; j < e.nops(); j++)
+ if (j != ind)
+ if (same_value_index)
+ S = S * e.op(j);
+ else {
+ exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
+ if (ind_vec.size() > 0) {
+ found_dummy = true;
+ exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
+ while (it != itend) {
+ ex curridx = *it;
+ ex curridx_toggle = is_a<varidx>(curridx)
+ ? ex_to<varidx>(curridx).toggle_variance()
+ : curridx;
+ S = S * e.op(j).subs(lst(curridx == ival,
+ curridx_toggle == ival), subs_options::no_pattern);
+ ++it;
+ }
+ } else
+ S = S * e.op(j);
+ }
+ return (found_dummy ? S : 0);
+ } else
+ throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
+ } else if (e.is_zero())
+ return e;
+ else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
+ if ( ex_to<idx>(e.op(1)).is_numeric() &&
+ (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
+ return 0;
+ else
+ return 1;
+ else
+ throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
+}
+
+
+lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
+{
+ GINAC_ASSERT(is_a<clifford>(c));
+ ex mu = c.op(1);
+ if (! ex_to<idx>(mu).is_dim_numeric())
+ throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
+ unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
+
+ if (algebraic) // check if algebraic method is applicable
+ for (unsigned int i = 0; i < D; i++)
+ if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
+ or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
+ algebraic = false;
+ lst V;
+ if (algebraic) {
+ for (unsigned int i = 0; i < D; i++)
+ V.append(remove_dirac_ONE(
+ simplify_indexed(canonicalize_clifford(e * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e))
+ / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
+ } else {
+ ex e1 = canonicalize_clifford(e);
+ try {
+ for (unsigned int i = 0; i < D; i++)
+ V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
+ } catch (std::exception &p) {
+ /* Try to expand dummy summations to simplify the expression*/
+ e1 = canonicalize_clifford(expand_dummy_sum(e1, true));
+ for (unsigned int i = 0; i < D; i++)
+ V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
+ }
+ }
+ return V;
+}
+
+
+ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl)
+{
+ ex x, D, cu;
+
+ if (! is_a<matrix>(v) && ! v.info(info_flags::list))
+ throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
+
+ if (is_a<clifford>(G)) {
+ cu = G;
+ } else {
+ if (is_a<indexed>(G))
+ D = ex_to<varidx>(G.op(1)).get_dim();
+ else if (is_a<matrix>(G))
+ D = ex_to<matrix>(G).rows();
+ else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
+
+ varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
+ cu = clifford_unit(mu, G, rl);
+ }
+
+ x = lst_to_clifford(v, cu);
+ ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
+ return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
+}
+
+ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
+{
+ if (is_a<matrix>(M))
+ return clifford_moebius_map(ex_to<matrix>(M)(0,0), ex_to<matrix>(M)(0,1),
+ ex_to<matrix>(M)(1,0), ex_to<matrix>(M)(1,1), v, G, rl);
+ else
+ throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a matrix"));
}
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff