/** @file clifford.cpp
*
* Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
/*
* GiNaC Copyright (C) 1999-2007 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
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#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 "operators.h"
#include "add.h"
#include "mul.h"
#include "power.h"
#include "matrix.h"
#include "archive.h"
#include "utils.h"
namespace GiNaC {
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
//////////
static ex default_metric()
{
static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
return m;
}
clifford::clifford() : representation_label(0), metric(default_metric())
{
tinfo_key = TINFO_clifford;
}
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), metric(0)
{
tinfo_key = TINFO_clifford;
}
/** Construct object with one Lorentz index. This constructor is for internal
* 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, 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 ex & metr, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr)
{
tinfo_key = TINFO_clifford;
}
clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr)
{
tinfo_key = TINFO_clifford;
}
//////////
// archiving
//////////
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);
}
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 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(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) && 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, precedence());
c.s << "\\";
} else { // We do not print representation label if it is 0
if (representation_label == 0) {
this->print_dispatch<inherited>(c, level);
} else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp
if (precedence() <= level) {
c.s << '(';
}
seq[0].print(c, precedence());
c.s << '[' << int(representation_label) << ']';
printindices(c, level);
if (precedence() <= level) {
c.s << ')';
}
}
}
}
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, precedence());
c.s << "\\hspace{-1.0ex}/}";
} 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", "\\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}")
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_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>(other->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 (num == 1) {
*self = dim;
*other = dirac_ONE(rl);
return true;
// gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
} 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 (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>(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_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()))
*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. */
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_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
s.push_back(*cit);
cit++;
}
bool something_changed = false;
int sign = 1;
// 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_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_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)) { // 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;
}
}
}
if (s.empty())
return clifford(diracone(), representation_label) * sign;
if (something_changed)
return reeval_ncmul(s) * sign;
else
return hold_ncmul(s) * sign;
}
ex clifford::thiscontainer(const exvector & v) const
{
return clifford(representation_label, get_metric(), v);
}
ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
{
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);
}
//////////
// global functions
//////////
ex dirac_ONE(unsigned char 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"));
if (is_a<indexed>(metr))
return clifford(unit, mu, metr.op(0), rl);
else if(is_a<tensmetric>(metr) || is_a<matrix>(metr))
return clifford(unit, mu, metr, rl);
else
throw(std::invalid_argument("metric for Clifford unit must be of type indexed, tensormetric or matrix"));
}
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"));
return clifford(gamma, mu, default_metric(), rl);
}
ex dirac_gamma5(unsigned char rl)
{
static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
return clifford(gamma5, rl);
}
ex dirac_gammaL(unsigned char rl)
{
static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
return clifford(gammaL, rl);
}
ex dirac_gammaR(unsigned char 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), default_metric(), rl);
}
/** Check whether a given tinfo key (as returned by return_type_tinfo()
* is that of a clifford object with the specified representation label. */
static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
{
return ti == (TINFO_clifford + rl);
}
/** Check whether a given tinfo key (as returned by return_type_tinfo()
* is that of a clifford object (with an arbitrary representation label). */
static bool is_clifford_tinfo(unsigned ti)
{
return (ti & ~0xff) == TINFO_clifford;
}
/** 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, 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 )
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 (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];
}
result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
sign = -sign;
}
return result;
}
ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
{
if (is_a<clifford>(e)) {
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;
else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
return trONE/2;
else
return _ex0;
} else if (is_exactly_a<mul>(e)) {
// Trace of product: pull out non-clifford factors
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()))
prod *= dirac_trace(o, rls, trONE);
else
prod *= o;
}
return prod;
} else if (is_exactly_a<ncmul>(e)) {
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_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_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;
// 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 (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 (size_t i=0; i<num-3; i++) {
ex idx1 = ix[i];
for (size_t j=i+1; j<num-2; j++) {
ex idx2 = ix[j];
for (size_t k=j+1; k<num-1; k++) {
ex idx3 = ix[k];
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 (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 * 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 (size_t 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<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 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_)
{
pointer_to_map_function fcn(canonicalize_clifford);
if (is_a<matrix>(e_) // || is_a<pseries>(e) || is_a<integral>(e)
|| is_a<lst>(e_)) {
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);
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);
i->second = canonicalize_clifford(sum);
goto next_sym;
}
++it;
}
next_sym: ;
}
}
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) || is_a<lst>(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 remove_dirac_ONE(const ex & e)
{
return remove_dirac_ONE(e, 0);
}
ex remove_dirac_ONE(const ex & e, unsigned char rl)
{
pointer_to_map_function_1arg<unsigned char> fcn(remove_dirac_ONE, rl);
if (is_a<clifford>(e) && ex_to<clifford>(e).get_representation_label() >= rl) {
if (is_a<diracone>(e.op(0)))
return 1;
else
throw(std::invalid_argument("Expression is a non-scalar Clifford number!"));
} 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) || is_a<lst>(e)) {
return e.map(fcn);
} else if (is_a<power>(e)) {
return pow(remove_dirac_ONE(e.op(0)), e.op(1));
} else
return e;
}
ex clifford_norm(const ex & e)
{
return sqrt(remove_dirac_ONE(canonicalize_clifford(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);
else
throw(std::invalid_argument("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("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)) {
varidx mu = ex_to<varidx>(e.op(1));
unsigned dim = (ex_to<numeric>(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, ex_to<varidx>(mu).toggle_variance()) * e;
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()) * e;
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"));
} else
throw(std::invalid_argument("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<varidx>(c.op(1)).get_value()).to_int();
if (is_a<add>(e) || is_a<lst>(e) // || 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("Expression is a Clifford multi-vector"));
if (ind < e.nops()) {
ex S = 1;
bool same_value_index, found_dummy;
same_value_index = ( ex_to<varidx>(e.op(ind).op(1)).is_numeric()
&& (ival == ex_to<numeric>(ex_to<varidx>(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) {
S = S * e.op(j).subs(lst(ex_to<varidx>(*it) == ival, ex_to<varidx>(*it).toggle_variance() == ival), subs_options::no_pattern);
++it;
}
} else
S = S * e.op(j);
}
return (found_dummy ? S : 0);
} else
throw(std::invalid_argument("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<varidx>(e.op(1)).is_numeric() &&
(ival != ex_to<numeric>(ex_to<varidx>(e.op(1)).get_value()).to_int()) )
return 0;
else
return 1;
else
throw(std::invalid_argument("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));
varidx mu = ex_to<varidx>(c.op(1));
if (! mu.is_dim_numeric())
throw(std::invalid_argument("Index should have a numeric dimension"));
unsigned int D = ex_to<numeric>(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), 2).is_zero()
or (not is_a<numeric>(pow(c.subs(mu == i), 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) + c.subs(mu == i) * e))
/ (2*pow(c.subs(mu == i), 2))));
else {
ex e1 = canonicalize_clifford(e);
for (unsigned int i = 0; i < D; i++)
V.append(get_clifford_comp(e1, c.subs(c.op(1) == i)));
}
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) && ! is_a<lst>(v))
throw(std::invalid_argument("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("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 = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d)));
return clifford_to_lst(e, cu, false);
}
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("parameter M should be a matrix"));
}
ex clifford_moebius_map(const ex & M, const ex & v, const ex & G)
{
return clifford_moebius_map(M, v, G, 0);
}
ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G)
{
return clifford_moebius_map(a, b, c, d, v, G, 0);
}
} // namespace GiNaC
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