3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
29 #include "numeric.h" // for I
32 #include "relational.h"
33 #include "operators.h"
43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
44 print_func<print_dflt>(&clifford::do_print_dflt).
45 print_func<print_latex>(&clifford::do_print_latex))
47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
48 print_func<print_dflt>(&diracone::do_print).
49 print_func<print_latex>(&diracone::do_print_latex))
51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
52 print_func<print_dflt>(&cliffordunit::do_print).
53 print_func<print_latex>(&cliffordunit::do_print_latex))
55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
56 print_func<print_dflt>(&diracgamma::do_print).
57 print_func<print_latex>(&diracgamma::do_print_latex))
59 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
60 print_func<print_dflt>(&diracgamma5::do_print).
61 print_func<print_latex>(&diracgamma5::do_print_latex))
63 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
64 print_func<print_context>(&diracgammaL::do_print).
65 print_func<print_latex>(&diracgammaL::do_print_latex))
67 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
68 print_func<print_context>(&diracgammaR::do_print).
69 print_func<print_latex>(&diracgammaR::do_print_latex))
72 // default constructors
75 static ex default_metric()
77 static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
81 clifford::clifford() : representation_label(0), metric(default_metric())
83 tinfo_key = TINFO_clifford;
86 DEFAULT_CTOR(diracone)
87 DEFAULT_CTOR(cliffordunit)
88 DEFAULT_CTOR(diracgamma)
89 DEFAULT_CTOR(diracgamma5)
90 DEFAULT_CTOR(diracgammaL)
91 DEFAULT_CTOR(diracgammaR)
97 /** Construct object without any indices. This constructor is for internal
98 * use only. Use the dirac_ONE() function instead.
100 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0)
102 tinfo_key = TINFO_clifford;
105 /** Construct object with one Lorentz index. This constructor is for internal
106 * use only. Use the clifford_unit() or dirac_gamma() functions instead.
108 * @see dirac_gamma */
109 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr)
111 GINAC_ASSERT(is_a<varidx>(mu));
112 tinfo_key = TINFO_clifford;
115 clifford::clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr)
117 tinfo_key = TINFO_clifford;
120 clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr)
122 tinfo_key = TINFO_clifford;
129 clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
132 n.find_unsigned("label", rl);
133 representation_label = rl;
134 n.find_ex("metric", metric, sym_lst);
137 void clifford::archive(archive_node &n) const
139 inherited::archive(n);
140 n.add_unsigned("label", representation_label);
141 n.add_ex("metric", metric);
144 DEFAULT_UNARCHIVE(clifford)
145 DEFAULT_ARCHIVING(diracone)
146 DEFAULT_ARCHIVING(cliffordunit)
147 DEFAULT_ARCHIVING(diracgamma)
148 DEFAULT_ARCHIVING(diracgamma5)
149 DEFAULT_ARCHIVING(diracgammaL)
150 DEFAULT_ARCHIVING(diracgammaR)
153 // functions overriding virtual functions from base classes
156 ex clifford::get_metric(const ex & i, const ex & j) const
158 return indexed(metric, symmetric2(), i, j);
161 bool clifford::same_metric(const ex & other) const
163 if (is_a<clifford>(other)) {
164 return get_metric().is_equal(ex_to<clifford>(other).get_metric());
165 } else if (is_a<indexed>(other)) {
166 return get_metric(other.op(1), other.op(2)).is_equal(other);
171 int clifford::compare_same_type(const basic & other) const
173 GINAC_ASSERT(is_a<clifford>(other));
174 const clifford &o = static_cast<const clifford &>(other);
176 if (representation_label != o.representation_label) {
177 // different representation label
178 return representation_label < o.representation_label ? -1 : 1;
181 return inherited::compare_same_type(other);
184 bool clifford::match_same_type(const basic & other) const
186 GINAC_ASSERT(is_a<clifford>(other));
187 const clifford &o = static_cast<const clifford &>(other);
189 return (representation_label == o.representation_label) && same_metric(o);
192 static bool is_dirac_slash(const ex & seq0)
194 return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
195 !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
196 !is_a<diracone>(seq0);
199 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
201 // dirac_slash() object is printed differently
202 if (is_dirac_slash(seq[0])) {
203 seq[0].print(c, level);
206 this->print_dispatch<inherited>(c, level);
209 void clifford::do_print_latex(const print_latex & c, unsigned level) const
211 // dirac_slash() object is printed differently
212 if (is_dirac_slash(seq[0])) {
214 seq[0].print(c, level);
215 c.s << "\\hspace{-1.0ex}/}";
217 this->print_dispatch<inherited>(c, level);
220 DEFAULT_COMPARE(diracone)
221 DEFAULT_COMPARE(cliffordunit)
222 DEFAULT_COMPARE(diracgamma)
223 DEFAULT_COMPARE(diracgamma5)
224 DEFAULT_COMPARE(diracgammaL)
225 DEFAULT_COMPARE(diracgammaR)
227 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
228 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
229 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
230 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
231 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
232 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
234 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
235 static void base_and_index(const ex & c, ex & b, ex & i)
237 GINAC_ASSERT(is_a<clifford>(c));
238 GINAC_ASSERT(c.nops() == 2);
240 if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
243 } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
246 } else { // slash object, generate new dummy index
247 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
248 b = indexed(c.op(0), ix.toggle_variance());
253 /** Predicate for finding non-clifford objects. */
254 struct is_not_a_clifford : public std::unary_function<ex, bool> {
255 bool operator()(const ex & e)
257 return !is_a<clifford>(e);
261 /** Contraction of a gamma matrix with something else. */
262 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
264 GINAC_ASSERT(is_a<clifford>(*self));
265 GINAC_ASSERT(is_a<indexed>(*other));
266 GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
267 unsigned char rl = ex_to<clifford>(*self).get_representation_label();
269 ex dim = ex_to<idx>(self->op(1)).get_dim();
270 if (other->nops() > 1)
271 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
273 if (is_a<clifford>(*other)) {
275 // Contraction only makes sense if the represenation labels are equal
276 if (ex_to<clifford>(*other).get_representation_label() != rl)
279 size_t num = other - self;
281 // gamma~mu gamma.mu = dim ONE
284 *other = dirac_ONE(rl);
287 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
289 && is_a<clifford>(self[1])) {
294 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
296 && is_a<clifford>(self[1])
297 && is_a<clifford>(self[2])) {
299 base_and_index(self[1], b1, i1);
300 base_and_index(self[2], b2, i2);
301 *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
307 // 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
309 && is_a<clifford>(self[1])
310 && is_a<clifford>(self[2])
311 && is_a<clifford>(self[3])) {
312 *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
319 // gamma~mu Sodd gamma.mu = -2 Sodd_R
320 // (Chisholm identity in 4 dimensions)
321 } else if (!((other - self) & 1) && dim.is_equal(4)) {
322 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
325 *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
326 std::fill(self + 1, other, _ex1);
330 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
331 // (commutate contracted indices towards each other, then use
332 // Chisholm identity in 4 dimensions)
333 } else if (((other - self) & 1) && dim.is_equal(4)) {
334 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
337 exvector::iterator next_to_last = other - 1;
338 ex S = ncmul(exvector(self + 1, next_to_last), true);
339 ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
341 *self = (*next_to_last) * S + SR * (*next_to_last);
342 std::fill(self + 1, other, _ex1);
346 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
347 // (commutate contracted indices towards each other, simplify_indexed()
348 // will re-expand and re-run the simplification)
350 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
353 exvector::iterator next_to_last = other - 1;
354 ex S = ncmul(exvector(self + 1, next_to_last), true);
356 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
357 std::fill(self + 1, other + 1, _ex1);
361 } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
363 // x.mu gamma~mu -> x-slash
364 *self = dirac_slash(other->op(0), dim, rl);
372 /** An utility function looking for a given metric within an exvector,
373 * used in cliffordunit::contract_with(). */
374 static int find_same_metric(exvector & v, ex & c)
376 for (int i=0; i<v.size();i++) {
377 if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i])
378 && ex_to<clifford>(c).same_metric(v[i])
379 && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
380 || ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])) {
381 return ++i; // next to found
384 return 0; //nothing found
387 /** Contraction of a Clifford unit with something else. */
388 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
390 GINAC_ASSERT(is_a<clifford>(*self));
391 GINAC_ASSERT(is_a<indexed>(*other));
392 GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
393 clifford unit = ex_to<clifford>(*self);
394 unsigned char rl = unit.get_representation_label();
396 if (is_a<clifford>(*other)) {
397 // Contraction only makes sense if the represenation labels are equal
398 // and the metrics are the same
399 if ((ex_to<clifford>(*other).get_representation_label() != rl)
400 && unit.same_metric(*other))
403 // Find if a previous contraction produces the square of self
404 int prev_square = find_same_metric(v, self[0]);
405 varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim());
406 ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
408 // e~mu e.mu = Tr ONE
409 if (other - self == 1) {
410 if (prev_square != 0) {
411 *self = squared_metric;
412 v[prev_square-1] = _ex1;
414 *self = unit.get_metric(self->op(1), other->op(1));
415 *other = dirac_ONE(rl);
418 // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
419 } else if (other - self == 2
420 && is_a<clifford>(self[1])) {
422 const ex & ia = self[1].op(1);
423 const ex & ib = self[1].op(1);
424 if (is_a<tensmetric>(unit.get_metric()))
425 *self = 2 - unit.get_metric(self->op(1), other->op(1));
426 else if (prev_square != 0) {
427 *self = 2-squared_metric;
428 v[prev_square-1] = _ex1;
430 *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
434 // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha
435 // (commutate contracted indices towards each other, simplify_indexed()
436 // will re-expand and re-run the simplification)
438 exvector::iterator it = self + 1, next_to_last = other - 1;
439 while (it != other) {
440 if (!is_a<clifford>(*it))
447 while (it != next_to_last) {
452 const ex & ia = next_to_last->op(1);
453 const ex & ib = next_to_last->op(1);
454 if (is_a<tensmetric>(unit.get_metric()))
455 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
456 else if (prev_square != 0) {
457 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
458 v[prev_square-1] = _ex1;
460 *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
461 *next_to_last = _ex1;
471 /** Perform automatic simplification on noncommutative product of clifford
472 * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
473 * and removes squares of gamma objects. */
474 ex clifford::eval_ncmul(const exvector & v) const
479 // Remove superfluous ONEs
480 exvector::const_iterator cit = v.begin(), citend = v.end();
481 while (cit != citend) {
482 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
487 bool something_changed = false;
490 // Anticommutate gamma5/L/R's to the front
492 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
494 exvector::iterator it = next_to_last;
496 exvector::iterator it2 = it + 1;
497 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
498 ex e1 = it->op(0), e2 = it2->op(0);
500 if (is_a<diracgamma5>(e2)) {
502 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
504 // gammaL/R gamma5 -> gamma5 gammaL/R
506 something_changed = true;
508 } else if (!is_a<diracgamma5>(e1)) {
510 // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
511 // x gamma5 -> -gamma5 x
514 something_changed = true;
517 } else if (is_a<diracgammaL>(e2)) {
519 if (is_a<diracgammaR>(e1)) {
521 // gammaR gammaL -> 0
524 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
526 // gammaL gammaL -> gammaL gammaL (do nothing)
527 // gamma5 gammaL -> gamma5 gammaL (do nothing)
528 // x gammaL -> gammaR x
530 *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
531 something_changed = true;
534 } else if (is_a<diracgammaR>(e2)) {
536 if (is_a<diracgammaL>(e1)) {
538 // gammaL gammaR -> 0
541 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
543 // gammaR gammaR -> gammaR gammaR (do nothing)
544 // gamma5 gammaR -> gamma5 gammaR (do nothing)
545 // x gammaR -> gammaL x
547 *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
548 something_changed = true;
556 if (next_to_last == first)
562 // Remove equal adjacent gammas
564 exvector::iterator it, itend = s.end() - 1;
565 for (it = s.begin(); it != itend; ++it) {
568 if (!is_a<clifford>(a) || !is_a<clifford>(b))
571 const ex & ag = a.op(0);
572 const ex & bg = b.op(0);
573 bool a_is_cliffordunit = is_a<cliffordunit>(ag);
574 bool b_is_cliffordunit = is_a<cliffordunit>(bg);
576 if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
578 const ex & ia = a.op(1);
579 const ex & ib = b.op(1);
580 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
581 a = ex_to<clifford>(a).get_metric(ia, ib);
582 b = dirac_ONE(representation_label);
583 something_changed = true;
586 } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
588 // Remove squares of gamma5
589 a = dirac_ONE(representation_label);
590 b = dirac_ONE(representation_label);
591 something_changed = true;
593 } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
594 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
596 // Remove squares of gammaL/R
597 b = dirac_ONE(representation_label);
598 something_changed = true;
600 } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
602 // gammaL and gammaR are orthogonal
605 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
607 // gamma5 gammaL -> -gammaL
608 a = dirac_ONE(representation_label);
610 something_changed = true;
612 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
614 // gamma5 gammaR -> gammaR
615 a = dirac_ONE(representation_label);
616 something_changed = true;
618 } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
621 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
623 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
624 b = dirac_ONE(representation_label);
625 something_changed = true;
631 return clifford(diracone(), representation_label) * sign;
632 if (something_changed)
633 return reeval_ncmul(s) * sign;
635 return hold_ncmul(s) * sign;
638 ex clifford::thiscontainer(const exvector & v) const
640 return clifford(representation_label, get_metric(), v);
643 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
645 return clifford(representation_label, get_metric(), vp);
648 ex diracgamma5::conjugate() const
650 return _ex_1 * (*this);
653 ex diracgammaL::conjugate() const
655 return (new diracgammaR)->setflag(status_flags::dynallocated);
658 ex diracgammaR::conjugate() const
660 return (new diracgammaL)->setflag(status_flags::dynallocated);
667 ex dirac_ONE(unsigned char rl)
669 static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
670 return clifford(ONE, rl);
673 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
675 static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
677 if (!is_a<varidx>(mu))
678 throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
680 if (is_a<indexed>(metr))
681 return clifford(unit, mu, metr.op(0), rl);
682 else if(is_a<tensmetric>(metr) || is_a<matrix>(metr))
683 return clifford(unit, mu, metr, rl);
685 throw(std::invalid_argument("metric for Clifford unit must be of type indexed, tensormetric or matrix"));
688 ex dirac_gamma(const ex & mu, unsigned char rl)
690 static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
692 if (!is_a<varidx>(mu))
693 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
695 return clifford(gamma, mu, default_metric(), rl);
698 ex dirac_gamma5(unsigned char rl)
700 static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
701 return clifford(gamma5, rl);
704 ex dirac_gammaL(unsigned char rl)
706 static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
707 return clifford(gammaL, rl);
710 ex dirac_gammaR(unsigned char rl)
712 static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
713 return clifford(gammaR, rl);
716 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
718 // Slashed vectors are actually stored as a clifford object with the
719 // vector as its base expression and a (dummy) index that just serves
720 // for storing the space dimensionality
721 return clifford(e, varidx(0, dim), default_metric(), rl);
724 /** Check whether a given tinfo key (as returned by return_type_tinfo()
725 * is that of a clifford object with the specified representation label. */
726 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
728 return ti == (TINFO_clifford + rl);
731 /** Check whether a given tinfo key (as returned by return_type_tinfo()
732 * is that of a clifford object (with an arbitrary representation label). */
733 static bool is_clifford_tinfo(unsigned ti)
735 return (ti & ~0xff) == TINFO_clifford;
738 /** Extract representation label from tinfo key (as returned by
739 * return_type_tinfo()). */
740 static unsigned char get_representation_label(unsigned ti)
745 /** Take trace of a string of an even number of Dirac gammas given a vector
747 static ex trace_string(exvector::const_iterator ix, size_t num)
749 // Tr gamma.mu gamma.nu = 4 g.mu.nu
751 return lorentz_g(ix[0], ix[1]);
753 // 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 )
755 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
756 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
757 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
759 // Traces of 6 or more gammas are computed recursively:
760 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
761 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
762 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
763 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
765 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
769 for (size_t i=1; i<num; i++) {
770 for (size_t n=1, j=0; n<num; n++) {
775 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
781 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
783 if (is_a<clifford>(e)) {
785 unsigned char rl = ex_to<clifford>(e).get_representation_label();
787 // Are we taking the trace over this object's representation label?
788 if (rls.find(rl) == rls.end())
791 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
792 const ex & g = e.op(0);
793 if (is_a<diracone>(g))
795 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
800 } else if (is_exactly_a<mul>(e)) {
802 // Trace of product: pull out non-clifford factors
804 for (size_t i=0; i<e.nops(); i++) {
805 const ex &o = e.op(i);
806 if (is_clifford_tinfo(o.return_type_tinfo()))
807 prod *= dirac_trace(o, rls, trONE);
813 } else if (is_exactly_a<ncmul>(e)) {
815 unsigned char rl = get_representation_label(e.return_type_tinfo());
817 // Are we taking the trace over this string's representation label?
818 if (rls.find(rl) == rls.end())
821 // Substitute gammaL/R and expand product, if necessary
822 ex e_expanded = e.subs(lst(
823 dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
824 dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
825 ), subs_options::no_pattern).expand();
826 if (!is_a<ncmul>(e_expanded))
827 return dirac_trace(e_expanded, rls, trONE);
829 // gamma5 gets moved to the front so this check is enough
830 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
831 size_t num = e.nops();
835 // Trace of gamma5 * odd number of gammas and trace of
836 // gamma5 * gamma.mu * gamma.nu are zero
837 if ((num & 1) == 0 || num == 3)
840 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
841 // (the epsilon is always 4-dimensional)
843 ex b1, i1, b2, i2, b3, i3, b4, i4;
844 base_and_index(e.op(1), b1, i1);
845 base_and_index(e.op(2), b2, i2);
846 base_and_index(e.op(3), b3, i3);
847 base_and_index(e.op(4), b4, i4);
848 return trONE * I * (lorentz_eps(ex_to<idx>(i1).replace_dim(_ex4), ex_to<idx>(i2).replace_dim(_ex4), ex_to<idx>(i3).replace_dim(_ex4), ex_to<idx>(i4).replace_dim(_ex4)) * b1 * b2 * b3 * b4).simplify_indexed();
852 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
853 // (the epsilon is always 4-dimensional)
854 exvector ix(num-1), bv(num-1);
855 for (size_t i=1; i<num; i++)
856 base_and_index(e.op(i), bv[i-1], ix[i-1]);
858 int *iv = new int[num];
860 for (size_t i=0; i<num-3; i++) {
862 for (size_t j=i+1; j<num-2; j++) {
864 for (size_t k=j+1; k<num-1; k++) {
866 for (size_t l=k+1; l<num; l++) {
868 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
871 for (size_t n=0, t=4; n<num; n++) {
872 if (n == i || n == j || n == k || n == l)
877 int sign = permutation_sign(iv, iv + num);
878 result += sign * lorentz_eps(ex_to<idx>(idx1).replace_dim(_ex4), ex_to<idx>(idx2).replace_dim(_ex4), ex_to<idx>(idx3).replace_dim(_ex4), ex_to<idx>(idx4).replace_dim(_ex4))
879 * trace_string(v.begin(), num - 4);
885 return trONE * I * result * mul(bv);
887 } else { // no gamma5
889 // Trace of odd number of gammas is zero
893 // Tr gamma.mu gamma.nu = 4 g.mu.nu
896 base_and_index(e.op(0), b1, i1);
897 base_and_index(e.op(1), b2, i2);
898 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
901 exvector iv(num), bv(num);
902 for (size_t i=0; i<num; i++)
903 base_and_index(e.op(i), bv[i], iv[i]);
905 return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
908 } else if (e.nops() > 0) {
910 // Trace maps to all other container classes (this includes sums)
911 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
918 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
920 // Convert list to set
921 std::set<unsigned char> rls;
922 for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
923 if (i->info(info_flags::nonnegint))
924 rls.insert(ex_to<numeric>(*i).to_int());
927 return dirac_trace(e, rls, trONE);
930 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
932 // Convert label to set
933 std::set<unsigned char> rls;
936 return dirac_trace(e, rls, trONE);
940 ex canonicalize_clifford(const ex & e)
942 pointer_to_map_function fcn(canonicalize_clifford);
944 if (is_a<matrix>(e) // || is_a<pseries>(e) || is_a<integral>(e)
948 // Scan for any ncmul objects
950 ex aux = e.to_rational(srl);
951 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
956 if (is_exactly_a<ncmul>(rhs)
957 && rhs.return_type() == return_types::noncommutative
958 && is_clifford_tinfo(rhs.return_type_tinfo())) {
960 // Expand product, if necessary
961 ex rhs_expanded = rhs.expand();
962 if (!is_a<ncmul>(rhs_expanded)) {
963 i->second = canonicalize_clifford(rhs_expanded);
966 } else if (!is_a<clifford>(rhs.op(0)))
970 v.reserve(rhs.nops());
971 for (size_t j=0; j<rhs.nops(); j++)
972 v.push_back(rhs.op(j));
974 // Stupid recursive bubble sort because we only want to swap adjacent gammas
975 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
976 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
978 while (it != next_to_last) {
979 if (it[0].compare(it[1]) > 0) {
980 ex save0 = it[0], save1 = it[1];
982 base_and_index(it[0], b1, i1);
983 base_and_index(it[1], b2, i2);
984 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
985 it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
989 sum -= ncmul(v, true);
990 i->second = canonicalize_clifford(sum);
998 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1002 ex clifford_prime(const ex & e)
1004 pointer_to_map_function fcn(clifford_prime);
1005 if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1007 } else if (is_a<add>(e) || is_a<ncmul>(e) // || is_a<pseries>(e) || is_a<integral>(e)
1008 || is_a<matrix>(e) || is_a<lst>(e)) {
1010 } else if (is_a<power>(e)) {
1011 return pow(clifford_prime(e.op(0)), e.op(1));
1016 ex remove_dirac_ONE(const ex & e)
1018 pointer_to_map_function fcn(remove_dirac_ONE);
1019 if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
1021 } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) // || is_a<pseries>(e) || is_a<integral>(e)
1022 || is_a<matrix>(e) || is_a<lst>(e)) {
1024 } else if (is_a<power>(e)) {
1025 return pow(remove_dirac_ONE(e.op(0)), e.op(1));
1030 ex clifford_norm(const ex & e)
1032 return sqrt(remove_dirac_ONE(canonicalize_clifford(e * clifford_bar(e)).simplify_indexed()));
1035 ex clifford_inverse(const ex & e)
1037 ex norm = clifford_norm(e);
1038 if (!norm.is_zero())
1039 return clifford_bar(e) / pow(norm, 2);
1041 throw(std::invalid_argument("Cannot find inverse of Clifford number with zero norm!"));
1044 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1047 if (!ex_to<idx>(mu).is_dim_numeric())
1048 throw(std::invalid_argument("Index should have a numeric dimension"));
1049 unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
1050 ex c = clifford_unit(mu, metr, rl);
1052 if (is_a<matrix>(v)) {
1053 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1054 min = ex_to<matrix>(v).rows();
1055 max = ex_to<matrix>(v).cols();
1057 min = ex_to<matrix>(v).cols();
1058 max = ex_to<matrix>(v).rows();
1062 if (is_a<varidx>(mu)) // need to swap variance
1063 return indexed(v, ex_to<varidx>(mu).toggle_variance()) * c;
1065 return indexed(v, mu) * c;
1067 throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
1069 throw(std::invalid_argument("First argument should be a vector vector"));
1070 } else if (is_a<lst>(v)) {
1071 if (dim == ex_to<lst>(v).nops())
1072 return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * c;
1074 throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
1076 throw(std::invalid_argument("Cannot construct from anything but list or vector"));
1079 /** Auxiliary structure to define a function for striping one Clifford unit
1080 * from vectors. Used in clifford_to_lst(). */
1081 static ex get_clifford_comp(const ex & e, const ex & c)
1083 pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1084 int ival = ex_to<numeric>(ex_to<varidx>(c.op(1)).get_value()).to_int();
1086 if (is_a<add>(e) || is_a<lst>(e) // || is_a<pseries>(e) || is_a<integral>(e)
1089 else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1090 // find a Clifford unit with the same metric, delete it and substitute its index
1091 size_t ind = e.nops() + 1;
1092 for (size_t j = 0; j < e.nops(); j++)
1093 if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1097 throw(std::invalid_argument("Expression is a Clifford multi-vector"));
1098 if (ind < e.nops()) {
1100 bool same_value_index, found_dummy;
1101 same_value_index = ( ex_to<varidx>(e.op(ind).op(1)).is_numeric()
1102 && (ival == ex_to<numeric>(ex_to<varidx>(e.op(ind).op(1)).get_value()).to_int()) );
1103 found_dummy = same_value_index;
1104 for(size_t j=0; j < e.nops(); j++)
1106 if (same_value_index)
1109 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1110 if (ind_vec.size() > 0) {
1112 exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1113 while (it != itend) {
1114 S = S * e.op(j).subs(lst(ex_to<varidx>(*it) == ival, ex_to<varidx>(*it).toggle_variance() == ival), subs_options::no_pattern);
1120 return (found_dummy ? S : 0);
1122 throw(std::invalid_argument("Expression is not a Clifford vector to the given units"));
1123 } else if (e.is_zero())
1125 else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1126 if ( ex_to<varidx>(e.op(1)).is_numeric() &&
1127 (ival != ex_to<numeric>(ex_to<varidx>(e.op(1)).get_value()).to_int()) )
1132 throw(std::invalid_argument("Expression is not usable as a Clifford vector"));
1136 lst clifford_to_lst (const ex & e, const ex & c, bool algebraic)
1138 GINAC_ASSERT(is_a<clifford>(c));
1139 varidx mu = ex_to<varidx>(c.op(1));
1140 if (! mu.is_dim_numeric())
1141 throw(std::invalid_argument("Index should have a numeric dimension"));
1142 unsigned int D = ex_to<numeric>(mu.get_dim()).to_int();
1144 if (algebraic) // check if algebraic method is applicable
1145 for (unsigned int i = 0; i < D; i++)
1146 if (pow(c.subs(mu == i), 2) == 0)
1150 for (unsigned int i = 0; i < D; i++)
1151 V.append(remove_dirac_ONE(
1152 simplify_indexed(canonicalize_clifford(e * c.subs(mu == i) + c.subs(mu == i) * e))
1153 / (2*pow(c.subs(mu == i), 2))));
1155 ex e1 = canonicalize_clifford(e);
1156 for (unsigned int i = 0; i < D; i++)
1157 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i)));
1163 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)
1166 if (is_a<indexed>(G))
1167 D = ex_to<varidx>(G.op(1)).get_dim();
1168 else if (is_a<matrix>(G))
1169 D = ex_to<matrix>(G).rows();
1171 throw(std::invalid_argument("metric should be an indexed object or matrix"));
1173 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1175 if (! is_a<matrix>(v) && ! is_a<lst>(v))
1176 throw(std::invalid_argument("parameter v should be either vector or list"));
1178 x = lst_to_clifford(v, mu, G, rl);
1179 ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d)));
1180 ex cu = clifford_unit(mu, G);
1181 return clifford_to_lst(e, cu, false);
1184 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1186 if (is_a<matrix>(M))
1187 return clifford_moebius_map(ex_to<matrix>(M)(0,0), ex_to<matrix>(M)(0,1),
1188 ex_to<matrix>(M)(1,0), ex_to<matrix>(M)(1,1), v, G, rl);
1190 throw(std::invalid_argument("parameter M should be a matrix"));
1193 } // namespace GiNaC