3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2004 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 /** Contraction of a gamma matrix with something else. */
254 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
256 GINAC_ASSERT(is_a<clifford>(*self));
257 GINAC_ASSERT(is_a<indexed>(*other));
258 GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
259 unsigned char rl = ex_to<clifford>(*self).get_representation_label();
261 ex dim = ex_to<idx>(self->op(1)).get_dim();
262 if (other->nops() > 1)
263 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
265 if (is_a<clifford>(*other)) {
267 // Contraction only makes sense if the represenation labels are equal
268 if (ex_to<clifford>(*other).get_representation_label() != rl)
271 // gamma~mu gamma.mu = dim ONE
272 if (other - self == 1) {
274 *other = dirac_ONE(rl);
277 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
278 } else if (other - self == 2
279 && is_a<clifford>(self[1])) {
284 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
285 } else if (other - self == 3
286 && is_a<clifford>(self[1])
287 && is_a<clifford>(self[2])) {
289 base_and_index(self[1], b1, i1);
290 base_and_index(self[2], b2, i2);
291 *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
297 // 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
298 } else if (other - self == 4
299 && is_a<clifford>(self[1])
300 && is_a<clifford>(self[2])
301 && is_a<clifford>(self[3])) {
302 *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
309 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
310 // (commutate contracted indices towards each other, simplify_indexed()
311 // will re-expand and re-run the simplification)
313 exvector::iterator it = self + 1, next_to_last = other - 1;
314 while (it != other) {
315 if (!is_a<clifford>(*it))
322 while (it != next_to_last) {
327 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
328 *next_to_last = _ex1;
333 } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
335 // x.mu gamma~mu -> x-slash
336 *self = dirac_slash(other->op(0), dim, rl);
344 /** An utility function looking for a given metric within an exvector,
345 * used in cliffordunit::contract_with(). */
346 static int find_same_metric(exvector & v, ex & c)
348 for (int i=0; i<v.size();i++) {
349 if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i])
350 && ex_to<clifford>(c).same_metric(v[i])
351 && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
352 || ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])) {
353 return ++i; // next to found
356 return 0; //nothing found
359 /** Contraction of a Clifford unit with something else. */
360 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
362 GINAC_ASSERT(is_a<clifford>(*self));
363 GINAC_ASSERT(is_a<indexed>(*other));
364 GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
365 clifford unit = ex_to<clifford>(*self);
366 unsigned char rl = unit.get_representation_label();
368 if (is_a<clifford>(*other)) {
369 // Contraction only makes sense if the represenation labels are equal
370 // and the metrics are the same
371 if ((ex_to<clifford>(*other).get_representation_label() != rl)
372 && unit.same_metric(*other))
375 // Find if a previous contraction produces the square of self
376 int prev_square = find_same_metric(v, self[0]);
377 varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(ex_to<idx>(self->op(1)).get_dim()));
378 ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
380 // e~mu e.mu = Tr ONE
381 if (other - self == 1) {
382 if (prev_square != 0) {
383 *self = squared_metric;
384 v[prev_square-1] = _ex1;
386 *self = unit.get_metric(self->op(1), other->op(1));
387 *other = dirac_ONE(rl);
390 // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
391 } else if (other - self == 2
392 && is_a<clifford>(self[1])) {
394 const ex & ia = self[1].op(1);
395 const ex & ib = self[1].op(1);
396 if (is_a<tensmetric>(unit.get_metric()))
397 *self = 2 - unit.get_metric(self->op(1), other->op(1));
398 else if (prev_square != 0) {
399 *self = 2-squared_metric;
400 v[prev_square-1] = _ex1;
402 *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
406 // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha
407 // (commutate contracted indices towards each other, simplify_indexed()
408 // will re-expand and re-run the simplification)
410 exvector::iterator it = self + 1, next_to_last = other - 1;
411 while (it != other) {
412 if (!is_a<clifford>(*it))
419 while (it != next_to_last) {
424 const ex & ia = next_to_last->op(1);
425 const ex & ib = next_to_last->op(1);
426 if (is_a<tensmetric>(unit.get_metric()))
427 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
428 else if (prev_square != 0) {
429 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
430 v[prev_square-1] = _ex1;
432 *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
433 *next_to_last = _ex1;
443 /** Perform automatic simplification on noncommutative product of clifford
444 * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
445 * and removes squares of gamma objects. */
446 ex clifford::eval_ncmul(const exvector & v) const
451 // Remove superfluous ONEs
452 exvector::const_iterator cit = v.begin(), citend = v.end();
453 while (cit != citend) {
454 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
459 bool something_changed = false;
462 // Anticommute gamma5/L/R's to the front
464 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
466 exvector::iterator it = next_to_last;
468 exvector::iterator it2 = it + 1;
469 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
470 ex e1 = it->op(0), e2 = it2->op(0);
472 if (is_a<diracgamma5>(e2)) {
474 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
476 // gammaL/R gamma5 -> gamma5 gammaL/R
478 something_changed = true;
480 } else if (!is_a<diracgamma5>(e1)) {
482 // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
483 // x gamma5 -> -gamma5 x
486 something_changed = true;
489 } else if (is_a<diracgammaL>(e2)) {
491 if (is_a<diracgammaR>(e1)) {
493 // gammaR gammaL -> 0
496 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
498 // gammaL gammaL -> gammaL gammaL (do nothing)
499 // gamma5 gammaL -> gamma5 gammaL (do nothing)
500 // x gammaL -> gammaR x
502 *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
503 something_changed = true;
506 } else if (is_a<diracgammaR>(e2)) {
508 if (is_a<diracgammaL>(e1)) {
510 // gammaL gammaR -> 0
513 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
515 // gammaR gammaR -> gammaR gammaR (do nothing)
516 // gamma5 gammaR -> gamma5 gammaR (do nothing)
517 // x gammaR -> gammaL x
519 *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
520 something_changed = true;
528 if (next_to_last == first)
534 // Remove equal adjacent gammas
536 exvector::iterator it, itend = s.end() - 1;
537 for (it = s.begin(); it != itend; ++it) {
540 if (!is_a<clifford>(a) || !is_a<clifford>(b))
543 const ex & ag = a.op(0);
544 const ex & bg = b.op(0);
545 bool a_is_cliffordunit = is_a<cliffordunit>(ag);
546 bool b_is_cliffordunit = is_a<cliffordunit>(bg);
548 if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
550 const ex & ia = a.op(1);
551 const ex & ib = b.op(1);
552 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
553 a = ex_to<clifford>(a).get_metric(ia, ib);
554 b = dirac_ONE(representation_label);
555 something_changed = true;
558 } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
560 // Remove squares of gamma5
561 a = dirac_ONE(representation_label);
562 b = dirac_ONE(representation_label);
563 something_changed = true;
565 } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
566 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
568 // Remove squares of gammaL/R
569 b = dirac_ONE(representation_label);
570 something_changed = true;
572 } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
574 // gammaL and gammaR are orthogonal
577 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
579 // gamma5 gammaL -> -gammaL
580 a = dirac_ONE(representation_label);
582 something_changed = true;
584 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
586 // gamma5 gammaR -> gammaR
587 a = dirac_ONE(representation_label);
588 something_changed = true;
590 } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
593 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
595 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
596 b = dirac_ONE(representation_label);
597 something_changed = true;
603 return clifford(diracone(), representation_label) * sign;
604 if (something_changed)
605 return reeval_ncmul(s) * sign;
607 return hold_ncmul(s) * sign;
610 ex clifford::thiscontainer(const exvector & v) const
612 return clifford(representation_label, get_metric(), v);
615 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
617 return clifford(representation_label, get_metric(), vp);
620 ex diracgamma5::conjugate() const
622 return _ex_1 * (*this);
625 ex diracgammaL::conjugate() const
627 return (new diracgammaR)->setflag(status_flags::dynallocated);
630 ex diracgammaR::conjugate() const
632 return (new diracgammaL)->setflag(status_flags::dynallocated);
639 ex dirac_ONE(unsigned char rl)
641 static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
642 return clifford(ONE, rl);
645 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
647 static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
649 if (!is_a<varidx>(mu))
650 throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
652 return clifford(unit, mu, metr, rl);
655 ex dirac_gamma(const ex & mu, unsigned char rl)
657 static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
659 if (!is_a<varidx>(mu))
660 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
662 return clifford(gamma, mu, default_metric(), rl);
665 ex dirac_gamma5(unsigned char rl)
667 static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
668 return clifford(gamma5, rl);
671 ex dirac_gammaL(unsigned char rl)
673 static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
674 return clifford(gammaL, rl);
677 ex dirac_gammaR(unsigned char rl)
679 static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
680 return clifford(gammaR, rl);
683 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
685 // Slashed vectors are actually stored as a clifford object with the
686 // vector as its base expression and a (dummy) index that just serves
687 // for storing the space dimensionality
688 return clifford(e, varidx(0, dim), rl);
691 /** Check whether a given tinfo key (as returned by return_type_tinfo()
692 * is that of a clifford object with the specified representation label. */
693 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
695 return ti == (TINFO_clifford + rl);
698 /** Check whether a given tinfo key (as returned by return_type_tinfo()
699 * is that of a clifford object (with an arbitrary representation label). */
700 static bool is_clifford_tinfo(unsigned ti)
702 return (ti & ~0xff) == TINFO_clifford;
705 /** Take trace of a string of an even number of Dirac gammas given a vector
707 static ex trace_string(exvector::const_iterator ix, size_t num)
709 // Tr gamma.mu gamma.nu = 4 g.mu.nu
711 return lorentz_g(ix[0], ix[1]);
713 // 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 )
715 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
716 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
717 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
719 // Traces of 6 or more gammas are computed recursively:
720 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
721 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
722 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
723 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
725 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
729 for (size_t i=1; i<num; i++) {
730 for (size_t n=1, j=0; n<num; n++) {
735 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
741 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
743 if (is_a<clifford>(e)) {
745 if (!ex_to<clifford>(e).get_representation_label() == rl)
747 const ex & g = e.op(0);
748 if (is_a<diracone>(g))
750 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
755 } else if (is_exactly_a<mul>(e)) {
757 // Trace of product: pull out non-clifford factors
759 for (size_t i=0; i<e.nops(); i++) {
760 const ex &o = e.op(i);
761 if (is_clifford_tinfo(o.return_type_tinfo(), rl))
762 prod *= dirac_trace(o, rl, trONE);
768 } else if (is_exactly_a<ncmul>(e)) {
770 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
773 // Substitute gammaL/R and expand product, if necessary
774 ex e_expanded = e.subs(lst(
775 dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
776 dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
777 ), subs_options::no_pattern).expand();
778 if (!is_a<ncmul>(e_expanded))
779 return dirac_trace(e_expanded, rl, trONE);
781 // gamma5 gets moved to the front so this check is enough
782 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
783 size_t num = e.nops();
787 // Trace of gamma5 * odd number of gammas and trace of
788 // gamma5 * gamma.mu * gamma.nu are zero
789 if ((num & 1) == 0 || num == 3)
792 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
793 // (the epsilon is always 4-dimensional)
795 ex b1, i1, b2, i2, b3, i3, b4, i4;
796 base_and_index(e.op(1), b1, i1);
797 base_and_index(e.op(2), b2, i2);
798 base_and_index(e.op(3), b3, i3);
799 base_and_index(e.op(4), b4, i4);
800 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();
804 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
805 // (the epsilon is always 4-dimensional)
806 exvector ix(num-1), bv(num-1);
807 for (size_t i=1; i<num; i++)
808 base_and_index(e.op(i), bv[i-1], ix[i-1]);
810 int *iv = new int[num];
812 for (size_t i=0; i<num-3; i++) {
814 for (size_t j=i+1; j<num-2; j++) {
816 for (size_t k=j+1; k<num-1; k++) {
818 for (size_t l=k+1; l<num; l++) {
820 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
823 for (size_t n=0, t=4; n<num; n++) {
824 if (n == i || n == j || n == k || n == l)
829 int sign = permutation_sign(iv, iv + num);
830 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))
831 * trace_string(v.begin(), num - 4);
837 return trONE * I * result * mul(bv);
839 } else { // no gamma5
841 // Trace of odd number of gammas is zero
845 // Tr gamma.mu gamma.nu = 4 g.mu.nu
848 base_and_index(e.op(0), b1, i1);
849 base_and_index(e.op(1), b2, i2);
850 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
853 exvector iv(num), bv(num);
854 for (size_t i=0; i<num; i++)
855 base_and_index(e.op(i), bv[i], iv[i]);
857 return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
860 } else if (e.nops() > 0) {
862 // Trace maps to all other container classes (this includes sums)
863 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
870 ex canonicalize_clifford(const ex & e)
872 // Scan for any ncmul objects
874 ex aux = e.to_rational(srl);
875 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
880 if (is_exactly_a<ncmul>(rhs)
881 && rhs.return_type() == return_types::noncommutative
882 && is_clifford_tinfo(rhs.return_type_tinfo())) {
884 // Expand product, if necessary
885 ex rhs_expanded = rhs.expand();
886 if (!is_a<ncmul>(rhs_expanded)) {
887 i->second = canonicalize_clifford(rhs_expanded);
890 } else if (!is_a<clifford>(rhs.op(0)))
894 v.reserve(rhs.nops());
895 for (size_t j=0; j<rhs.nops(); j++)
896 v.push_back(rhs.op(j));
898 // Stupid recursive bubble sort because we only want to swap adjacent gammas
899 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
900 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
902 while (it != next_to_last) {
903 if (it[0].compare(it[1]) > 0) {
904 ex save0 = it[0], save1 = it[1];
906 base_and_index(it[0], b1, i1);
907 base_and_index(it[1], b2, i2);
908 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
909 it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
913 sum -= ncmul(v, true);
914 i->second = canonicalize_clifford(sum);
922 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
925 ex clifford_prime(const ex &e)
927 pointer_to_map_function fcn(clifford_prime);
928 if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
930 } else if (is_a<add>(e)) {
932 } else if (is_a<ncmul>(e)) {
934 } else if (is_a<power>(e)) {
935 return pow(clifford_prime(e.op(0)), e.op(1));
940 ex delete_ONE(const ex &e)
942 pointer_to_map_function fcn(delete_ONE);
943 if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
945 } else if (is_a<add>(e)) {
947 } else if (is_a<ncmul>(e)) {
949 } else if (is_a<mul>(e)) {
951 } else if (is_a<power>(e)) {
952 return pow(delete_ONE(e.op(0)), e.op(1));
957 ex clifford_norm(const ex &e)
959 return sqrt(delete_ONE((e * clifford_bar(e)).simplify_indexed()));
962 ex clifford_inverse(const ex &e)
964 ex norm = clifford_norm(e);
966 return clifford_bar(e) / pow(norm, 2);
969 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
972 if (!ex_to<idx>(mu).is_dim_numeric())
973 throw(std::invalid_argument("Index should have a numeric dimension"));
974 unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
975 ex c = clifford_unit(mu, metr, rl);
977 if (is_a<matrix>(v)) {
978 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
979 min = ex_to<matrix>(v).rows();
980 max = ex_to<matrix>(v).cols();
982 min = ex_to<matrix>(v).cols();
983 max = ex_to<matrix>(v).rows();
987 if (is_a<varidx>(mu)) // need to swap variance
988 return indexed(v, ex_to<varidx>(mu).toggle_variance()) * c;
990 return indexed(v, mu) * c;
992 throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
994 throw(std::invalid_argument("First argument should be a vector vector"));
995 } else if (is_a<lst>(v)) {
996 if (dim == ex_to<lst>(v).nops())
997 return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * c;
999 throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
1001 throw(std::invalid_argument("Cannot construct from anything but list or vector"));
1004 } // namespace GiNaC