X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.cpp;h=3b4684c2165f07dda393d241401caeff9ba9db23;hp=0b3801e154efc15b6cdea5a0cecd3c648d1fe53a;hb=1602530f716ba1d425a0667b897182b99c374823;hpb=b301f03a61bc9f72b27940ca7fe1f8d0b343a4e2 diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp index 0b3801e1..3b4684c2 100644 --- a/ginac/clifford.cpp +++ b/ginac/clifford.cpp @@ -3,7 +3,7 @@ * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */ /* - * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2009 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 @@ -17,13 +17,11 @@ * * 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 -#include - #include "clifford.h" + #include "ex.h" #include "idx.h" #include "ncmul.h" @@ -33,30 +31,55 @@ #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" +#include + 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(&clifford::do_print_dflt). + print_func(&clifford::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor, + print_func(&diracone::do_print). + print_func(&diracone::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor, + print_func(&cliffordunit::do_print). + print_func(&cliffordunit::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit, + print_func(&diracgamma::do_print). + print_func(&diracgamma::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor, + print_func(&diracgamma5::do_print). + print_func(&diracgamma5::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor, + print_func(&diracgammaL::do_print). + print_func(&diracgammaL::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor, + print_func(&diracgammaR::do_print). + print_func(&diracgammaR::do_print_latex)) ////////// // default constructors ////////// -clifford::clifford() : representation_label(0) +clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1) { - tinfo_key = TINFO_clifford; } DEFAULT_CTOR(diracone) +DEFAULT_CTOR(cliffordunit) DEFAULT_CTOR(diracgamma) DEFAULT_CTOR(diracgamma5) DEFAULT_CTOR(diracgammaL) @@ -69,58 +92,143 @@ DEFAULT_CTOR(diracgammaR) /** 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; } /** 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(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 ex & metr, int comm_sign, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), commutator_sign(comm_sign) +{ +} + +clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), commutator_sign(comm_sign) { - tinfo_key = TINFO_clifford; } -clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl) +return_type_t clifford::return_type_tinfo() const { - tinfo_key = TINFO_clifford; + return make_return_type_t(representation_label); } ////////// // archiving ////////// -clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst) +void clifford::read_archive(const archive_node& n, lst& sym_lst) { + inherited::read_archive(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); +} + +GINAC_BIND_UNARCHIVER(clifford); +GINAC_BIND_UNARCHIVER(diracone); +GINAC_BIND_UNARCHIVER(diracgamma); +GINAC_BIND_UNARCHIVER(diracgamma5); +GINAC_BIND_UNARCHIVER(diracgammaL); +GINAC_BIND_UNARCHIVER(diracgammaR); + + +ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const +{ + if (is_a(metric)) { + if (symmetrised && !(ex_to(ex_to(metric).get_symmetry()).has_symmetry())) { + if (is_a(metric.op(0))) { + return indexed((ex_to(metric.op(0)).add(ex_to(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); + } } -DEFAULT_UNARCHIVE(clifford) -DEFAULT_ARCHIVING(diracone) -DEFAULT_ARCHIVING(diracgamma) -DEFAULT_ARCHIVING(diracgamma5) -DEFAULT_ARCHIVING(diracgammaL) -DEFAULT_ARCHIVING(diracgammaR) +bool clifford::same_metric(const ex & other) const +{ + ex metr; + if (is_a(other)) + metr = ex_to(other).get_metric(); + else + metr = other; + + if (is_a(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(subsed)) { + ex prevmetric = ex_to(subsed).metric; + ex newmetric = prevmetric.subs(m, options); + if(!are_ex_trivially_equal(prevmetric, newmetric)) { + clifford c = ex_to(subsed); + c.metric = newmetric; + subsed = c; + } + } + return subsed; +} + int clifford::compare_same_type(const basic & other) const { GINAC_ASSERT(is_a(other)); @@ -139,38 +247,61 @@ bool clifford::match_same_type(const basic & other) const GINAC_ASSERT(is_a(other)); const clifford &o = static_cast(other); - return representation_label == o.representation_label; + return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o)); } -void clifford::print(const print_context & c, unsigned level) const +static bool is_dirac_slash(const ex & seq0) { - if (!is_a(seq[0]) && !is_a(seq[0]) && - !is_a(seq[0]) && !is_a(seq[0]) && - !is_a(seq[0])) { + return !is_a(seq0) && !is_a(seq0) && + !is_a(seq0) && !is_a(seq0) && + !is_a(seq0); +} - // dirac_slash() object is printed differently - if (is_a(c)) - inherited::print(c, level); - else if (is_a(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, precedence()); + c.s << "\\"; + } else { // We do not print representation label if it is 0 + if (representation_label == 0) { + this->print_dispatch(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 << ')'; + } } + } +} - } else - inherited::print(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, precedence()); + c.s << "\\hspace{-1.0ex}/}"; + } else { + c.s << "\\clifford[" << int(representation_label) << "]"; + this->print_dispatch(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}") @@ -180,9 +311,9 @@ DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}") static void base_and_index(const ex & c, ex & b, ex & i) { GINAC_ASSERT(is_a(c)); - GINAC_ASSERT(c.nops() == 2); + GINAC_ASSERT(c.nops() == 2+1); - if (is_a(c.op(0))) { // proper dirac gamma object + if (is_a(c.op(0))) { // proper dirac gamma object or clifford unit i = c.op(1); b = _ex1; } else if (is_a(c.op(0)) || is_a(c.op(0)) || is_a(c.op(0))) { // gamma5/L/R @@ -195,6 +326,14 @@ static void base_and_index(const ex & c, ex & b, ex & i) } } +/** Predicate for finding non-clifford objects. */ +struct is_not_a_clifford : public std::unary_function { + bool operator()(const ex & e) + { + return !is_a(e); + } +}; + /** Contraction of a gamma matrix with something else. */ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const { @@ -213,21 +352,23 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other if (ex_to(*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(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(self[1]) && is_a(self[2])) { ex b1, i1, b2, i2; @@ -240,7 +381,7 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other 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(self[1]) && is_a(self[2]) && is_a(self[3])) { @@ -251,27 +392,45 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other *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(other), std::reverse_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(next_to_last), std::reverse_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(*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; } @@ -286,6 +445,71 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other 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(*self)); + GINAC_ASSERT(is_a(*other)); + GINAC_ASSERT(is_a(self->op(0))); + clifford unit = ex_to(*self); + unsigned char rl = unit.get_representation_label(); + + if (is_a(*other)) { + // Contraction only makes sense if the represenation labels are equal + // and the metrics are the same + if ((ex_to(*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(*before_other) && ex_to(*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(*before_other) && ex_to(*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. */ @@ -305,7 +529,7 @@ ex clifford::eval_ncmul(const exvector & v) const 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) { @@ -388,15 +612,17 @@ ex clifford::eval_ncmul(const exvector & v) const const ex & ag = a.op(0); const ex & bg = b.op(0); - bool a_is_diracgamma = is_a(ag); - bool b_is_diracgamma = is_a(bg); - - if (a_is_diracgamma && b_is_diracgamma) { + bool a_is_cliffordunit = is_a(ag); + bool b_is_cliffordunit = is_a(bg); + if (a_is_cliffordunit && b_is_cliffordunit && ex_to(a).same_metric(b) + && (ex_to(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(a).get_metric(ia, ib, true); b = dirac_ONE(representation_label); something_changed = true; } @@ -433,10 +659,11 @@ ex clifford::eval_ncmul(const exvector & v) const 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(a.op(1)).minimal_dim(ex_to(b.op(1)))); + a = indexed(ag, ix) * indexed(ag, ix.toggle_variance()); b = dirac_ONE(representation_label); something_changed = true; @@ -445,7 +672,7 @@ ex clifford::eval_ncmul(const exvector & v) const } if (s.empty()) - return clifford(diracone(), representation_label) * sign; + return dirac_ONE(representation_label) * sign; if (something_changed) return reeval_ncmul(s) * sign; else @@ -454,12 +681,27 @@ ex clifford::eval_ncmul(const exvector & v) const 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 vp) const +{ + return clifford(representation_label, metric, commutator_sign, vp); +} + +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); } ////////// @@ -468,30 +710,92 @@ ex clifford::thiscontainer(exvector * vp) const ex dirac_ONE(unsigned char rl) { - return clifford(diracone(), rl); + static ex ONE = (new diracone)->setflag(status_flags::dynallocated); + return clifford(ONE, rl); +} + +static unsigned get_dim_uint(const ex& e) +{ + if (!is_a(e)) + throw std::invalid_argument("get_dim_uint: argument is not an index"); + ex dim = ex_to(e).get_dim(); + if (!dim.info(info_flags::posint)) + throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer"); + unsigned d = ex_to(dim).to_int(); + return d; +} + +ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl) +{ + //static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated); + ex unit = (new cliffordunit)->setflag(status_flags::dynallocated); + + if (!is_a(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) { + return clifford(unit, mu, metr, rl); + } else if (is_a(metr)) { + matrix M = ex_to(metr); + unsigned n = M.rows(); + bool symmetric = true; + + //static idx xi((new symbol)->setflag(status_flags::dynallocated), n), + // chi((new symbol)->setflag(status_flags::dynallocated), n); + idx xi((new symbol)->setflag(status_flags::dynallocated), n), + chi((new symbol)->setflag(status_flags::dynallocated), n); + if ((n == M.cols()) && (n == get_dim_uint(mu))) { + for (unsigned i = 0; i < n; i++) { + for (unsigned j = i+1; j < n; j++) { + if (!M(i, j).is_equal(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(mu).get_dim()), + // chi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()); + varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()), + chi((new symbol)->setflag(status_flags::dynallocated), ex_to(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(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(mu).get_dim()), + chi((new symbol)->setflag(status_flags::dynallocated), ex_to(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) @@ -499,21 +803,17 @@ 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); -} -/** 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); + 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 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(const return_type_t& ti) { - return (ti & ~0xff) == TINFO_clifford; + return (unsigned char)ti.rl; } /** Take trace of a string of an even number of Dirac gammas given a vector @@ -552,12 +852,17 @@ static ex trace_string(exvector::const_iterator ix, size_t num) return result; } -ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) +ex dirac_trace(const ex & e, const std::set & rls, const ex & trONE) { if (is_a(e)) { - if (!ex_to(e).get_representation_label() == rl) - return _ex0; + unsigned char rl = ex_to(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(g)) return trONE; @@ -572,8 +877,8 @@ ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) ex prod = _ex1; for (size_t i=0; i(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( @@ -590,7 +898,7 @@ ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2 ), subs_options::no_pattern).expand(); if (!is_a(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(e.op(0).op(0)); @@ -674,67 +982,381 @@ ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) } else if (e.nops() > 0) { // Trace maps to all other container classes (this includes sums) - pointer_to_map_function_2args fcn(dirac_trace, rl, trONE); + pointer_to_map_function_2args &, 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 - lst srl; - ex aux = e.to_rational(srl); - for (size_t i=0; i rls; + for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) { + if (i->info(info_flags::nonnegint)) + rls.insert(ex_to(*i).to_int()); + } - ex o = srl.op(i); - ex lhs = o.lhs(); - ex rhs = o.rhs(); + return dirac_trace(e, rls, trONE); +} - if (is_exactly_a(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 rls; + rls.insert(rl); - // Expand product, if necessary - ex rhs_expanded = rhs.expand(); - if (!is_a(rhs_expanded)) { - srl[i] = (lhs == canonicalize_clifford(rhs_expanded)); - continue; + return dirac_trace(e, rls, trONE); +} - } else if (!is_a(rhs.op(0))) - continue; - exvector v; - v.reserve(rhs.nops()); - for (size_t j=0; j(it->op(0)) || is_a(it->op(0)) || is_a(it->op(0))) - ++it; - while (it != next_to_last) { - if (it[0].compare(it[1]) > 0) { - ex save0 = it[0], save1 = it[1]; - ex b1, i1, b2, i2; - base_and_index(it[0], b1, i1); - base_and_index(it[1], b2, i2); - it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed(); - it[1] = _ex2; - ex sum = ncmul(v); - it[0] = save1; - it[1] = save0; - sum -= ncmul(v, true); - srl[i] = (lhs == canonicalize_clifford(sum)); - goto next_sym; +ex canonicalize_clifford(const ex & e_) +{ + pointer_to_map_function fcn(canonicalize_clifford); + + if (is_a(e_) // || is_a(e) || is_a(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(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(rhs_expanded)) { + i->second = canonicalize_clifford(rhs_expanded); + continue; + + } else if (!is_a(rhs.op(0))) + continue; + + exvector v; + v.reserve(rhs.nops()); + for (size_t j=0; j(it->op(0)) || is_a(it->op(0)) || is_a(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(save0).get_metric(i1, i2, ex_to(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed(); + it[1] = v.size() ? _ex2 * dirac_ONE(ex_to(save0).get_representation_label()) : _ex2; + ex sum = ncmul(v); + it[0] = save1; + it[1] = save0; + sum += ex_to(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(); + } +} + +ex clifford_prime(const ex & e) +{ + pointer_to_map_function fcn(clifford_prime); + if (is_a(e) && is_a(e.op(0))) { + return -e; + } else if (is_a(e) || is_a(e) || is_a(e) //|| is_a(e) || is_a(e) + || is_a(e) || e.info(info_flags::list)) { + return e.map(fcn); + } else if (is_a(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 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); } - return aux.subs(srl, subs_options::no_pattern).simplify_indexed(); + + if (is_a(e1) && ex_to(e1).get_representation_label() >= rl) { + if (is_a(e1.op(0))) + return 1; + else + throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!")); + } else if (is_a(e1) || is_a(e1) || is_a(e1) + || is_a(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(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(e)) + if (ignore_ONE && is_a(e.op(0))) + return -1; + else + return ex_to(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(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(e)) { + ex mu = e.op(1); + ex mu_toggle + = is_a(mu) ? ex_to(mu).toggle_variance() : mu; + unsigned dim = get_dim_uint(mu); + + if (is_a(v)) { + if (ex_to(v).cols() > ex_to(v).rows()) { + min = ex_to(v).rows(); + max = ex_to(v).cols(); + } else { + min = ex_to(v).cols(); + max = ex_to(v).rows(); + } + if (min == 1) { + if (dim == max) + return indexed(v, mu_toggle) * e; + else if (max - dim == 1) { + if (ex_to(v).cols() > ex_to(v).rows()) + return v.op(0) * dirac_ONE(ex_to(e).get_representation_label()) + indexed(sub_matrix(ex_to(v), 0, 1, 1, dim), mu_toggle) * e; + else + return v.op(0) * dirac_ONE(ex_to(e).get_representation_label()) + indexed(sub_matrix(ex_to(v), 1, dim, 0, 1), 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(v).nops()) + return indexed(matrix(dim, 1, ex_to(v)), mu_toggle) * e; + else if (ex_to(v).nops() - dim == 1) + return v.op(0) * dirac_ONE(ex_to(e).get_representation_label()) + indexed(sub_matrix(matrix(dim+1, 1, ex_to(v)), 1, dim, 0, 1), 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 fcn(get_clifford_comp, c); + int ival = ex_to(ex_to(c.op(1)).get_value()).to_int(); + + if (is_a(e) || e.info(info_flags::list) // || is_a(e) || is_a(e) + || is_a(e)) + return e.map(fcn); + else if (is_a(e) || is_a(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(e.op(j)) && ex_to(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(e.op(ind).op(1)).is_numeric() + && (ival == ex_to(ex_to(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(e.op(j)).get_dummy_indices(ex_to(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(curridx) + ? ex_to(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(e) && ex_to(e).same_metric(c)) + if ( ex_to(e.op(1)).is_numeric() && + (ival != ex_to(ex_to(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(c)); + ex mu = c.op(1); + if (! ex_to(mu).is_dim_numeric()) + throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension")); + unsigned int D = ex_to(ex_to(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(pow(c.subs(mu == i, subs_options::no_pattern), 2)))) + algebraic = false; + lst V; + ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2; + if (not v0.is_zero()) + V.append(v0); + ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to(c).get_representation_label())); + if (algebraic) { + for (unsigned int i = 0; i < D; i++) + V.append(remove_dirac_ONE( + simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e1)) + / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2)))); + } else { + 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(e, true)); + V.remove_all(); + v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2; + if (not v0.is_zero()) { + V.append(v0); + e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to(c).get_representation_label())); + } + 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(v) && ! v.info(info_flags::list)) + throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list")); + + if (is_a(G)) { + cu = G; + } else { + if (is_a(G)) { + D = ex_to(G.op(1)).get_dim(); + varidx mu((new symbol)->setflag(status_flags::dynallocated), D); + cu = clifford_unit(mu, G, rl); + } else if (is_a(G)) { + D = ex_to(G).rows(); + idx mu((new symbol)->setflag(status_flags::dynallocated), D); + cu = clifford_unit(mu, G, rl); + } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit")); + + } + + 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(v) ? matrix(ex_to(v).rows(), ex_to(v).cols(), ex_to(e)) : e); +} + +ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl) +{ + if (is_a(M) && (ex_to(M).rows() == 2) && (ex_to(M).cols() == 2)) + return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl); + else + throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix")); } } // namespace GiNaC