X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.cpp;h=b2e56224e11ee9518ccfe87fdffea1968686591c;hp=74465926988156902934c0616ca84cfb1a1a7d43;hb=d7d0bcda91b647db9588f3aa1a465f1570d088c4;hpb=591b85b0697370f2f5f25a29a1e94ff831a02c12 diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp index 74465926..b2e56224 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) 1999-2001 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by @@ -21,6 +21,7 @@ */ #include "clifford.h" + #include "ex.h" #include "idx.h" #include "ncmul.h" @@ -29,40 +30,65 @@ #include "symmetry.h" #include "lst.h" #include "relational.h" -#include "print.h" +#include "operators.h" +#include "add.h" +#include "mul.h" +#include "power.h" +#include "matrix.h" #include "archive.h" -#include "debugmsg.h" #include "utils.h" -#include - namespace GiNaC { -GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed) -GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor) -GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor) -GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor) +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed, + print_func(&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 constructor, destructor, copy constructor assignment operator and helpers +// default constructors ////////// -clifford::clifford() : representation_label(0) +static ex default_metric() { - debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT); - tinfo_key = TINFO_clifford; + static ex m = (new minkmetric)->setflag(status_flags::dynallocated); + return m; } -void clifford::copy(const clifford & other) +clifford::clifford() : representation_label(0), metric(default_metric()) { - inherited::copy(other); - representation_label = other.representation_label; + tinfo_key = TINFO_clifford; } -DEFAULT_DESTROY(clifford) -DEFAULT_CTORS(diracone) -DEFAULT_CTORS(diracgamma) -DEFAULT_CTORS(diracgamma5) +DEFAULT_CTOR(diracone) +DEFAULT_CTOR(cliffordunit) +DEFAULT_CTOR(diracgamma) +DEFAULT_CTOR(diracgamma5) +DEFAULT_CTOR(diracgammaL) +DEFAULT_CTOR(diracgammaR) ////////// // other constructors @@ -71,31 +97,28 @@ DEFAULT_CTORS(diracgamma5) /** Construct object without any indices. This constructor is for internal * use only. Use the dirac_ONE() function instead. * @see dirac_ONE */ -clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl) +clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0) { - debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT); tinfo_key = TINFO_clifford; } /** Construct object with one Lorentz index. This constructor is for internal - * use only. Use the dirac_gamma() function instead. + * use only. Use the clifford_unit() or dirac_gamma() functions instead. + * @see clifford_unit * @see dirac_gamma */ -clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl) +clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr) { - debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT); - GINAC_ASSERT(is_ex_of_type(mu, varidx)); + GINAC_ASSERT(is_a(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, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr) { - debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT); tinfo_key = TINFO_clifford; } -clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl) +clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr) { - debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT); tinfo_key = TINFO_clifford; } @@ -103,32 +126,51 @@ clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), // archiving ////////// -clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst) +clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst) { - debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT); unsigned rl; n.find_unsigned("label", rl); representation_label = rl; + n.find_ex("metric", metric, sym_lst); } void clifford::archive(archive_node &n) const { inherited::archive(n); n.add_unsigned("label", representation_label); + n.add_ex("metric", metric); } DEFAULT_UNARCHIVE(clifford) DEFAULT_ARCHIVING(diracone) +DEFAULT_ARCHIVING(cliffordunit) DEFAULT_ARCHIVING(diracgamma) DEFAULT_ARCHIVING(diracgamma5) +DEFAULT_ARCHIVING(diracgammaL) +DEFAULT_ARCHIVING(diracgammaR) ////////// -// functions overriding virtual functions from bases classes +// functions overriding virtual functions from base classes ////////// +ex clifford::get_metric(const ex & i, const ex & j) const +{ + return indexed(metric, symmetric2(), i, j); +} + +bool clifford::same_metric(const ex & other) const +{ + if (is_a(other)) { + return get_metric().is_equal(ex_to(other).get_metric()); + } else if (is_a(other)) { + return get_metric(other.op(1), other.op(2)).is_equal(other); + } else + return false; +} + int clifford::compare_same_type(const basic & other) const { - GINAC_ASSERT(other.tinfo() == TINFO_clifford); + GINAC_ASSERT(is_a(other)); const clifford &o = static_cast(other); if (representation_label != o.representation_label) { @@ -139,81 +181,297 @@ int clifford::compare_same_type(const basic & other) const return inherited::compare_same_type(other); } +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) && same_metric(o); +} + +static bool is_dirac_slash(const ex & seq0) +{ + return !is_a(seq0) && !is_a(seq0) && + !is_a(seq0) && !is_a(seq0) && + !is_a(seq0); +} + +void clifford::do_print_dflt(const print_dflt & c, unsigned level) const +{ + // dirac_slash() object is printed differently + if (is_dirac_slash(seq[0])) { + seq[0].print(c, level); + c.s << "\\"; + } else + this->print_dispatch(c, level); +} + +void clifford::do_print_latex(const print_latex & c, unsigned level) const +{ + // dirac_slash() object is printed differently + if (is_dirac_slash(seq[0])) { + c.s << "{"; + seq[0].print(c, level); + c.s << "\\hspace{-1.0ex}/}"; + } else + this->print_dispatch(c, level); +} + DEFAULT_COMPARE(diracone) +DEFAULT_COMPARE(cliffordunit) DEFAULT_COMPARE(diracgamma) DEFAULT_COMPARE(diracgamma5) +DEFAULT_COMPARE(diracgammaL) +DEFAULT_COMPARE(diracgammaR) DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}") +DEFAULT_PRINT_LATEX(cliffordunit, "e", "e") DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma") DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}") +DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}") +DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}") + +/** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */ +static void base_and_index(const ex & c, ex & b, ex & i) +{ + GINAC_ASSERT(is_a(c)); + GINAC_ASSERT(c.nops() == 2); + + 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 + i = _ex0; + b = _ex1; + } else { // slash object, generate new dummy index + varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to(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 { + 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 { - GINAC_ASSERT(is_ex_of_type(*self, clifford)); - GINAC_ASSERT(is_ex_of_type(*other, indexed)); - GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma)); + GINAC_ASSERT(is_a(*self)); + GINAC_ASSERT(is_a(*other)); + GINAC_ASSERT(is_a(self->op(0))); unsigned char rl = ex_to(*self).get_representation_label(); - if (is_ex_of_type(*other, clifford)) { + ex dim = ex_to(self->op(1)).get_dim(); + if (other->nops() > 1) + dim = minimal_dim(dim, ex_to(other->op(1)).get_dim()); + + if (is_a(*other)) { + + // Contraction only makes sense if the represenation labels are equal + if (ex_to(*other).get_representation_label() != rl) + return false; - ex dim = ex_to(self->op(1)).get_dim(); + size_t num = other - self; // gamma~mu gamma.mu = dim ONE - if (other - self == 1) { + if (num == 1) { *self = dim; *other = dirac_ONE(rl); return true; // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha - } else if (other - self == 2 - && is_ex_of_type(self[1], clifford)) { + } else if (num == 2 + && is_a(self[1])) { *self = 2 - dim; - *other = _ex1(); + *other = _ex1; return true; // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta - } else if (other - self == 3 - && is_ex_of_type(self[1], clifford) - && is_ex_of_type(self[2], clifford)) { - *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2]; - self[1] = _ex1(); - self[2] = _ex1(); - *other = _ex1(); + } else if (num == 3 + && is_a(self[1]) + && is_a(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(self[1]) + && is_a(self[2]) + && is_a(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(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 { + 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(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[i]) && is_a(v[i]) + && ex_to(c).same_metric(v[i]) + && (ex_to(c.op(1)) == ex_to(v[i]).get_indices()[0] + || ex_to(c.op(1)).toggle_variance() == ex_to(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(*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; + + // 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(ex_to(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(self[1])) { + + const ex & ia = self[1].op(1); + const ex & ib = self[1].op(1); + if (is_a(unit.get_metric())) + *self = 2 - unit.get_metric(self->op(1), other->op(1)); + else if (prev_square != 0) { + *self = 2-squared_metric; + v[prev_square-1] = _ex1; + } else + *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1)); + *other = _ex1; + return true; + + // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha + // (commutate contracted indices towards each other, simplify_indexed() + // will re-expand and re-run the simplification) } else { exvector::iterator it = self + 1, next_to_last = other - 1; while (it != other) { - if (!is_ex_of_type(*it, clifford)) + if (!is_a(*it)) return false; - it++; + ++it; } it = self + 1; - ex S = _ex1(); + ex S = _ex1; while (it != next_to_last) { S *= *it; - *it++ = _ex1(); + *it++ = _ex1; } - *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last); - *next_to_last = _ex1(); - *other = _ex1(); + const ex & ia = next_to_last->op(1); + const ex & ib = next_to_last->op(1); + if (is_a(unit.get_metric())) + *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last); + else if (prev_square != 0) { + *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1)); + v[prev_square-1] = _ex1; + } else + *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last); + *next_to_last = _ex1; + *other = _ex1; return true; } - } + + } return false; } /** Perform automatic simplification on noncommutative product of clifford - * objects. This removes superfluous ONEs, permutes gamma5's to the front + * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front * and removes squares of gamma objects. */ -ex clifford::simplify_ncmul(const exvector & v) const +ex clifford::eval_ncmul(const exvector & v) const { exvector s; s.reserve(v.size()); @@ -221,7 +479,7 @@ ex clifford::simplify_ncmul(const exvector & v) const // Remove superfluous ONEs exvector::const_iterator cit = v.begin(), citend = v.end(); while (cit != citend) { - if (!is_ex_of_type(cit->op(0), diracone)) + if (!is_a(*cit) || !is_a(cit->op(0))) s.push_back(*cit); cit++; } @@ -229,69 +487,177 @@ ex clifford::simplify_ncmul(const exvector & v) const bool something_changed = false; int sign = 1; - // Anticommute gamma5's to the front + // Anticommutate gamma5/L/R's to the front if (s.size() >= 2) { exvector::iterator first = s.begin(), next_to_last = s.end() - 2; while (true) { exvector::iterator it = next_to_last; while (true) { exvector::iterator it2 = it + 1; - if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) { - it->swap(*it2); - sign = -sign; - something_changed = true; + if (is_a(*it) && is_a(*it2)) { + ex e1 = it->op(0), e2 = it2->op(0); + + if (is_a(e2)) { + + if (is_a(e1) || is_a(e1)) { + + // gammaL/R gamma5 -> gamma5 gammaL/R + it->swap(*it2); + something_changed = true; + + } else if (!is_a(e1)) { + + // gamma5 gamma5 -> gamma5 gamma5 (do nothing) + // x gamma5 -> -gamma5 x + it->swap(*it2); + sign = -sign; + something_changed = true; + } + + } else if (is_a(e2)) { + + if (is_a(e1)) { + + // gammaR gammaL -> 0 + return _ex0; + + } else if (!is_a(e1) && !is_a(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(*it).get_representation_label()); + something_changed = true; + } + + } else if (is_a(e2)) { + + if (is_a(e1)) { + + // gammaL gammaR -> 0 + return _ex0; + + } else if (!is_a(e1) && !is_a(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(*it).get_representation_label()); + something_changed = true; + } + } } if (it == first) break; - it--; + --it; } if (next_to_last == first) break; - next_to_last--; + --next_to_last; } } - // Remove squares of gamma5 - while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) { - s.erase(s.begin(), s.begin() + 2); - something_changed = true; - } - // Remove equal adjacent gammas if (s.size() >= 2) { - exvector::iterator it = s.begin(), itend = s.end() - 1; - while (it != itend) { + exvector::iterator it, itend = s.end() - 1; + for (it = s.begin(); it != itend; ++it) { ex & a = it[0]; ex & b = it[1]; - if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) { + if (!is_a(a) || !is_a(b)) + continue; + + const ex & ag = a.op(0); + const ex & bg = b.op(0); + 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)) { + const ex & ia = a.op(1); const ex & ib = b.op(1); - if (ia.is_equal(ib)) { - a = lorentz_g(ia, ib); + if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha + a = ex_to(a).get_metric(ia, ib); b = dirac_ONE(representation_label); something_changed = true; } + + } else if ((is_a(ag) && is_a(bg))) { + + // Remove squares of gamma5 + a = dirac_ONE(representation_label); + b = dirac_ONE(representation_label); + something_changed = true; + + } else if ((is_a(ag) && is_a(bg)) + || (is_a(ag) && is_a(bg))) { + + // Remove squares of gammaL/R + b = dirac_ONE(representation_label); + something_changed = true; + + } else if (is_a(ag) && is_a(bg)) { + + // gammaL and gammaR are orthogonal + return _ex0; + + } else if (is_a(ag) && is_a(bg)) { + + // gamma5 gammaL -> -gammaL + a = dirac_ONE(representation_label); + sign = -sign; + something_changed = true; + + } else if (is_a(ag) && is_a(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(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; } - it++; } } - if (s.size() == 0) + if (s.empty()) return clifford(diracone(), representation_label) * sign; if (something_changed) - return nonsimplified_ncmul(s) * sign; + return reeval_ncmul(s) * sign; else - return simplified_ncmul(s) * sign; + return hold_ncmul(s) * sign; +} + +ex clifford::thiscontainer(const exvector & v) const +{ + return clifford(representation_label, get_metric(), v); +} + +ex clifford::thiscontainer(std::auto_ptr vp) const +{ + return clifford(representation_label, get_metric(), vp); +} + +ex diracgamma5::conjugate() const +{ + return _ex_1 * (*this); } -ex clifford::thisexprseq(const exvector & v) const +ex diracgammaL::conjugate() const { - return clifford(representation_label, v); + return (new diracgammaR)->setflag(status_flags::dynallocated); } -ex clifford::thisexprseq(exvector * vp) const +ex diracgammaR::conjugate() const { - return clifford(representation_label, vp); + return (new diracgammaL)->setflag(status_flags::dynallocated); } ////////// @@ -300,36 +666,54 @@ ex clifford::thisexprseq(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); +} + +ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl) +{ + static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated); + + if (!is_a(mu)) + throw(std::invalid_argument("index of Clifford unit must be of type varidx")); + + return clifford(unit, mu, metr, rl); } ex dirac_gamma(const ex & mu, unsigned char rl) { - if (!is_ex_of_type(mu, varidx)) + static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated); + + if (!is_a(mu)) throw(std::invalid_argument("index of Dirac gamma must be of type varidx")); - return clifford(diracgamma(), mu, rl); + return clifford(gamma, mu, default_metric(), rl); } ex dirac_gamma5(unsigned char rl) { - return clifford(diracgamma5(), rl); + static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated); + return clifford(gamma5, rl); } -ex dirac_gamma6(unsigned char rl) +ex dirac_gammaL(unsigned char rl) { - return clifford(diracone(), rl) + clifford(diracgamma5(), rl); + static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated); + return clifford(gammaL, rl); } -ex dirac_gamma7(unsigned char rl) +ex dirac_gammaR(unsigned char rl) { - return clifford(diracone(), rl) - clifford(diracgamma5(), rl); + static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated); + return clifford(gammaR, rl); } ex dirac_slash(const ex & e, const ex & dim, unsigned char rl) { - varidx mu((new symbol)->setflag(status_flags::dynallocated), dim); - return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl); + // Slashed vectors are actually stored as a clifford object with the + // vector as its base expression and a (dummy) index that just serves + // for storing the space dimensionality + return clifford(e, varidx(0, dim), default_metric(), rl); } /** Check whether a given tinfo key (as returned by return_type_tinfo() @@ -346,15 +730,22 @@ 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, unsigned num) +static ex trace_string(exvector::const_iterator ix, size_t num) { // Tr gamma.mu gamma.nu = 4 g.mu.nu if (num == 2) return lorentz_g(ix[0], ix[1]); - // Tr gamma.mu gamma.nu gamma.rho gamma.sig = 4 (g.mu.nu g.rho.sig + g.nu.rho g.mu.sig - g.mu.rho g.nu.sig + // Tr gamma.mu gamma.nu gamma.rho gamma.sig = 4 (g.mu.nu g.rho.sig + g.nu.rho g.mu.sig - g.mu.rho g.nu.sig ) else if (num == 4) return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3]) + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3]) @@ -370,8 +761,8 @@ static ex trace_string(exvector::const_iterator ix, unsigned num) exvector v(num - 2); int sign = 1; ex result; - for (int i=1; i & rls, const ex & trONE) { - if (is_ex_of_type(e, clifford)) { + if (is_a(e)) { - if (ex_to(e).get_representation_label() == rl - && is_ex_of_type(e.op(0), diracone)) + 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; + else if (is_a(g) || is_a(g)) + return trONE/2; else - return _ex0(); + return _ex0; - } else if (is_ex_exactly_of_type(e, mul)) { + } else if (is_exactly_a(e)) { // Trace of product: pull out non-clifford factors - ex prod = _ex1(); - for (unsigned i=0; i(e)) { + + unsigned char rl = get_representation_label(e.return_type_tinfo()); - if (!is_clifford_tinfo(e.return_type_tinfo(), rl)) - return _ex0(); + // Are we taking the trace over this string's representation label? + if (rls.find(rl) == rls.end()) + return e; - // Expand product, if necessary - ex e_expanded = e.expand(); - if (!is_ex_of_type(e_expanded, ncmul)) - return dirac_trace(e_expanded, rl, trONE); + // Substitute gammaL/R and expand product, if necessary + ex e_expanded = e.subs(lst( + dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2, + dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2 + ), subs_options::no_pattern).expand(); + if (!is_a(e_expanded)) + return dirac_trace(e_expanded, rls, trONE); // gamma5 gets moved to the front so this check is enough - bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5); - unsigned num = e.nops(); + bool has_gamma5 = is_a(e.op(0).op(0)); + size_t num = e.nops(); if (has_gamma5) { // Trace of gamma5 * odd number of gammas and trace of // gamma5 * gamma.mu * gamma.nu are zero if ((num & 1) == 0 || num == 3) - return _ex0(); + return _ex0; // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma) - if (num == 5) - return trONE * I * eps0123(e.op(1).op(1), e.op(2).op(1), e.op(3).op(1), e.op(4).op(1)); + // (the epsilon is always 4-dimensional) + if (num == 5) { + ex b1, i1, b2, i2, b3, i3, b4, i4; + base_and_index(e.op(1), b1, i1); + base_and_index(e.op(2), b2, i2); + base_and_index(e.op(3), b3, i3); + base_and_index(e.op(4), b4, i4); + return trONE * I * (lorentz_eps(ex_to(i1).replace_dim(_ex4), ex_to(i2).replace_dim(_ex4), ex_to(i3).replace_dim(_ex4), ex_to(i4).replace_dim(_ex4)) * b1 * b2 * b3 * b4).simplify_indexed(); + } // Tr gamma5 S_2k = // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k - exvector ix; - ix.reserve(num - 1); - for (unsigned i=1; i(idx1).replace_dim(_ex4), ex_to(idx2).replace_dim(_ex4), ex_to(idx3).replace_dim(_ex4), ex_to(idx4).replace_dim(_ex4)) * trace_string(v.begin(), num - 4); } } } } delete[] iv; - return trONE * I * result; + return trONE * I * result * mul(bv); } else { // no gamma5 // Trace of odd number of gammas is zero if ((num & 1) == 1) - return _ex0(); + return _ex0; // Tr gamma.mu gamma.nu = 4 g.mu.nu - if (num == 2) - return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1)); + if (num == 2) { + ex b1, i1, b2, i2; + base_and_index(e.op(0), b1, i1); + base_and_index(e.op(1), b2, i2); + return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed(); + } - exvector iv; - iv.reserve(num); - for (unsigned i=0; i 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(); + return _ex0; } +ex dirac_trace(const ex & e, const lst & rll, const ex & trONE) +{ + // Convert list to set + std::set 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()); + } + + return dirac_trace(e, rls, trONE); +} + +ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) +{ + // Convert label to set + std::set rls; + rls.insert(rl); + + return dirac_trace(e, rls, trONE); +} + + ex canonicalize_clifford(const ex & e) { // Scan for any ncmul objects - lst srl; + exmap srl; ex aux = e.to_rational(srl); - for (unsigned i=0; ifirst; + ex rhs = i->second; - if (is_ex_exactly_of_type(rhs, ncmul) + 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_ex_of_type(rhs_expanded, ncmul)) { - srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded)); + if (!is_a(rhs_expanded)) { + i->second = canonicalize_clifford(rhs_expanded); continue; - } else if (!is_ex_of_type(rhs.op(0), clifford)) + } else if (!is_a(rhs.op(0))) continue; exvector v; v.reserve(rhs.nops()); - for (unsigned j=0; jop(0), diracgamma5)) - it++; + if (is_a(it->op(0)) || is_a(it->op(0)) || is_a(it->op(0))) + ++it; while (it != next_to_last) { - if (it[0].op(1).compare(it[1].op(1)) > 0) { + if (it[0].compare(it[1]) > 0) { ex save0 = it[0], save1 = it[1]; - it[0] = lorentz_g(it[0].op(1), it[1].op(1)); - it[1] = _ex2(); + ex b1, i1, b2, i2; + base_and_index(it[0], b1, i1); + base_and_index(it[1], b2, i2); + it[0] = (ex_to(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed(); + it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to(it[1]).get_representation_label()) : _ex2; ex sum = ncmul(v); it[0] = save1; it[1] = save0; sum -= ncmul(v, true); - srl.let_op(i) = (lhs == canonicalize_clifford(sum)); + i->second = canonicalize_clifford(sum); goto next_sym; } - it++; + ++it; } next_sym: ; } } - return aux.subs(srl); + return aux.subs(srl, subs_options::no_pattern).simplify_indexed(); } +ex clifford_prime(const ex &e) +{ + pointer_to_map_function fcn(clifford_prime); + if (is_a(e) && is_a(e.op(0))) { + return -e; + } else if (is_a(e)) { + return e.map(fcn); + } else if (is_a(e)) { + return e.map(fcn); + } else if (is_a(e)) { + return pow(clifford_prime(e.op(0)), e.op(1)); + } else + return e; +} + +ex delete_ONE(const ex &e) +{ + pointer_to_map_function fcn(delete_ONE); + if (is_a(e) && is_a(e.op(0))) { + return 1; + } else if (is_a(e)) { + return e.map(fcn); + } else if (is_a(e)) { + return e.map(fcn); + } else if (is_a(e)) { + return e.map(fcn); + } else if (is_a(e)) { + return pow(delete_ONE(e.op(0)), e.op(1)); + } else + return e; +} + +ex clifford_norm(const ex &e) +{ + return sqrt(delete_ONE((e * clifford_bar(e)).simplify_indexed())); +} + +ex clifford_inverse(const ex &e) +{ + ex norm = clifford_norm(e); + if (!norm.is_zero()) + return clifford_bar(e) / pow(norm, 2); +} + +ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl) +{ + unsigned min, max; + if (!ex_to(mu).is_dim_numeric()) + throw(std::invalid_argument("Index should have a numeric dimension")); + unsigned dim = (ex_to(ex_to(mu).get_dim())).to_int(); + ex c = clifford_unit(mu, metr, rl); + + 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) + if (is_a(mu)) // need to swap variance + return indexed(v, ex_to(mu).toggle_variance()) * c; + else + return indexed(v, mu) * c; + else + throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch")); + } else + throw(std::invalid_argument("First argument should be a vector vector")); + } else if (is_a(v)) { + if (dim == ex_to(v).nops()) + return indexed(matrix(dim, 1, ex_to(v)), ex_to(mu).toggle_variance()) * c; + else + throw(std::invalid_argument("List length and dimension of clifford unit mismatch")); + } else + throw(std::invalid_argument("Cannot construct from anything but list or vector")); +} + } // namespace GiNaC