X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?a=blobdiff_plain;ds=sidebyside;f=ginac%2Fclifford.cpp;h=eaea353605375c2cad14c571f593eabba01ab6f4;hb=e6dca6e2347de9341b95e0dd259eedf54c0baff4;hp=e6f9c2914be0a638f0e76cabf4879ca2f18fb727;hpb=6872b42cd09e72e7ca9906f4fc2fe775ace2f043;p=ginac.git diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp index e6f9c291..eaea3536 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) 1999-2004 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2005 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 @@ -20,10 +20,8 @@ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ -#include -#include - #include "clifford.h" + #include "ex.h" #include "idx.h" #include "ncmul.h" @@ -33,7 +31,10 @@ #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" @@ -47,7 +48,11 @@ GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor, print_func(&diracone::do_print). print_func(&diracone::do_print_latex)) -GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, tensor, +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)) @@ -67,12 +72,19 @@ GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor, // default constructors ////////// -clifford::clifford() : representation_label(0) +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) @@ -85,26 +97,27 @@ 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) { tinfo_key = TINFO_clifford; } /** Construct object with one Lorentz index. This constructor is for internal - * use only. Use the dirac_gamma() function instead. + * use only. Use the clifford_unit() or dirac_gamma() functions instead. + * @see clifford_unit * @see dirac_gamma */ -clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl) +clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr) { GINAC_ASSERT(is_a(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) { tinfo_key = TINFO_clifford; } -clifford::clifford(unsigned char rl, std::auto_ptr 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) { tinfo_key = TINFO_clifford; } @@ -113,21 +126,24 @@ clifford::clifford(unsigned char rl, std::auto_ptr vp) : inherited(sy_ // archiving ////////// -clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst) +clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst) { unsigned rl; n.find_unsigned("label", rl); representation_label = rl; + n.find_ex("metric", metric, sym_lst); } -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); } DEFAULT_UNARCHIVE(clifford) DEFAULT_ARCHIVING(diracone) +DEFAULT_ARCHIVING(cliffordunit) DEFAULT_ARCHIVING(diracgamma) DEFAULT_ARCHIVING(diracgamma5) DEFAULT_ARCHIVING(diracgammaL) @@ -137,6 +153,21 @@ 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(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(is_a(other)); @@ -155,13 +186,13 @@ 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) && 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) && !is_a(seq0) && !is_a(seq0); } @@ -169,7 +200,7 @@ 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); + seq[0].print(c, precedence()); c.s << "\\"; } else this->print_dispatch(c, level); @@ -180,19 +211,23 @@ 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); + seq[0].print(c, precedence()); c.s << "\\hspace{-1.0ex}/}"; - } else + } 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}") @@ -204,7 +239,7 @@ 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 + 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 @@ -217,6 +252,14 @@ 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 { @@ -235,21 +278,23 @@ 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; @@ -262,7 +307,7 @@ 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])) { @@ -273,9 +318,124 @@ 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 { + 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(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) { @@ -291,19 +451,21 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other *it++ = _ex1; } - *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last); + 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; } - } 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; } @@ -327,7 +489,7 @@ 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) { @@ -410,15 +572,15 @@ 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); + bool a_is_cliffordunit = is_a(ag); + bool b_is_cliffordunit = is_a(bg); - if (a_is_diracgamma && b_is_diracgamma) { + 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)) { // gamma~alpha gamma~alpha -> g~alpha~alpha - a = lorentz_g(ia, ib); + a = ex_to(a).get_metric(ia, ib); b = dirac_ONE(representation_label); something_changed = true; } @@ -455,10 +617,11 @@ 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; @@ -476,12 +639,12 @@ ex clifford::eval_ncmul(const exvector & v) const ex clifford::thiscontainer(const exvector & v) const { - return clifford(representation_label, v); + return clifford(representation_label, get_metric(), v); } ex clifford::thiscontainer(std::auto_ptr vp) const { - return clifford(representation_label, vp); + return clifford(representation_label, get_metric(), vp); } ex diracgamma5::conjugate() const @@ -505,30 +668,51 @@ ex diracgammaR::conjugate() const ex dirac_ONE(unsigned char rl) { - return clifford(diracone(), rl); + static ex ONE = (new diracone)->setflag(status_flags::dynallocated); + return clifford(ONE, rl); +} + +ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl) +{ + static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated); + + if (!is_a(mu)) + throw(std::invalid_argument("index of Clifford unit must be of type varidx")); + + if (is_a(metr)) + return clifford(unit, mu, metr.op(0), rl); + else if(is_a(metr) || is_a(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(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_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) @@ -536,7 +720,7 @@ 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); + return clifford(e, varidx(0, dim), default_metric(), rl); } /** Check whether a given tinfo key (as returned by return_type_tinfo() @@ -553,6 +737,13 @@ 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) @@ -589,12 +780,17 @@ 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; @@ -609,8 +805,8 @@ 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( @@ -627,7 +826,7 @@ 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)); @@ -711,66 +910,301 @@ 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 - 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; + // 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()); + } - if (is_exactly_a(rhs) - && rhs.return_type() == return_types::noncommutative - && is_clifford_tinfo(rhs.return_type_tinfo())) { + return dirac_trace(e, rls, trONE); +} - // Expand product, if necessary - ex rhs_expanded = rhs.expand(); - if (!is_a(rhs_expanded)) { - i->second = canonicalize_clifford(rhs_expanded); - continue; +ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) +{ + // Convert label to set + std::set rls; + rls.insert(rl); - } else if (!is_a(rhs.op(0))) - continue; + return dirac_trace(e, rls, trONE); +} - 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] = 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); - i->second = 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) + || is_a(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(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); + 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); + 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) || 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 remove_dirac_ONE(const ex & e, unsigned char rl) +{ + pointer_to_map_function_1arg fcn(remove_dirac_ONE, rl); + if (is_a(e) && ex_to(e).get_representation_label() >= rl) { + if (is_a(e.op(0))) + return 1; + else + throw(std::invalid_argument("Expression is a non-scalar Clifford number!")); + } else if (is_a(e) || is_a(e) || is_a(e) // || is_a(e) || is_a(e) + || is_a(e) || is_a(e)) { + return e.map(fcn); + } else if (is_a(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(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(e)) { + varidx mu = ex_to(e.op(1)); + unsigned dim = (ex_to(mu.get_dim())).to_int(); + + 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, ex_to(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(v)) { + if (dim == ex_to(v).nops()) + return indexed(matrix(dim, 1, ex_to(v)), ex_to(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 fcn(get_clifford_comp, c); + int ival = ex_to(ex_to(c.op(1)).get_value()).to_int(); + + if (is_a(e) || is_a(e) // || 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("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) { + S = S * e.op(j).subs(lst(ex_to(*it) == ival, ex_to(*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(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("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)); + varidx mu = ex_to(c.op(1)); + if (! mu.is_dim_numeric()) + throw(std::invalid_argument("Index should have a numeric dimension")); + unsigned int D = 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), 2).is_zero() + or (not is_a(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 aux.subs(srl, subs_options::no_pattern).simplify_indexed(); + 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) && ! is_a(v)) + throw(std::invalid_argument("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(); + else if (is_a(G)) + D = ex_to(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(M)) + return clifford_moebius_map(ex_to(M)(0,0), ex_to(M)(0,1), + ex_to(M)(1,0), ex_to(M)(1,1), v, G, rl); + else + throw(std::invalid_argument("parameter M should be a matrix")); } } // namespace GiNaC