3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2003 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
31 #include "numeric.h" // for I
34 #include "relational.h"
35 #include "operators.h"
42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
43 print_func<print_dflt>(&clifford::do_print_dflt).
44 print_func<print_latex>(&clifford::do_print_latex))
46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
47 print_func<print_dflt>(&diracone::do_print).
48 print_func<print_latex>(&diracone::do_print_latex))
50 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, tensor,
51 print_func<print_dflt>(&diracgamma::do_print).
52 print_func<print_latex>(&diracgamma::do_print_latex))
54 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
55 print_func<print_dflt>(&diracgamma5::do_print).
56 print_func<print_latex>(&diracgamma5::do_print_latex))
58 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
59 print_func<print_context>(&diracgammaL::do_print).
60 print_func<print_latex>(&diracgammaL::do_print_latex))
62 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
63 print_func<print_context>(&diracgammaR::do_print).
64 print_func<print_latex>(&diracgammaR::do_print_latex))
67 // default constructors
70 clifford::clifford() : representation_label(0)
72 tinfo_key = TINFO_clifford;
75 DEFAULT_CTOR(diracone)
76 DEFAULT_CTOR(diracgamma)
77 DEFAULT_CTOR(diracgamma5)
78 DEFAULT_CTOR(diracgammaL)
79 DEFAULT_CTOR(diracgammaR)
85 /** Construct object without any indices. This constructor is for internal
86 * use only. Use the dirac_ONE() function instead.
88 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
90 tinfo_key = TINFO_clifford;
93 /** Construct object with one Lorentz index. This constructor is for internal
94 * use only. Use the dirac_gamma() function instead.
96 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
98 GINAC_ASSERT(is_a<varidx>(mu));
99 tinfo_key = TINFO_clifford;
102 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
104 tinfo_key = TINFO_clifford;
107 clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
109 tinfo_key = TINFO_clifford;
116 clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
119 n.find_unsigned("label", rl);
120 representation_label = rl;
123 void clifford::archive(archive_node &n) const
125 inherited::archive(n);
126 n.add_unsigned("label", representation_label);
129 DEFAULT_UNARCHIVE(clifford)
130 DEFAULT_ARCHIVING(diracone)
131 DEFAULT_ARCHIVING(diracgamma)
132 DEFAULT_ARCHIVING(diracgamma5)
133 DEFAULT_ARCHIVING(diracgammaL)
134 DEFAULT_ARCHIVING(diracgammaR)
137 // functions overriding virtual functions from base classes
140 int clifford::compare_same_type(const basic & other) const
142 GINAC_ASSERT(is_a<clifford>(other));
143 const clifford &o = static_cast<const clifford &>(other);
145 if (representation_label != o.representation_label) {
146 // different representation label
147 return representation_label < o.representation_label ? -1 : 1;
150 return inherited::compare_same_type(other);
153 bool clifford::match_same_type(const basic & other) const
155 GINAC_ASSERT(is_a<clifford>(other));
156 const clifford &o = static_cast<const clifford &>(other);
158 return representation_label == o.representation_label;
161 static bool is_dirac_slash(const ex & seq0)
163 return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
164 !is_a<diracgammaR>(seq0) && !is_a<diracgamma>(seq0) &&
165 !is_a<diracone>(seq0);
168 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
170 // dirac_slash() object is printed differently
171 if (is_dirac_slash(seq[0])) {
172 seq[0].print(c, level);
175 this->print_dispatch<inherited>(c, level);
178 void clifford::do_print_latex(const print_latex & c, unsigned level) const
180 // dirac_slash() object is printed differently
181 if (is_dirac_slash(seq[0])) {
183 seq[0].print(c, level);
184 c.s << "\\hspace{-1.0ex}/}";
186 this->print_dispatch<inherited>(c, level);
189 DEFAULT_COMPARE(diracone)
190 DEFAULT_COMPARE(diracgamma)
191 DEFAULT_COMPARE(diracgamma5)
192 DEFAULT_COMPARE(diracgammaL)
193 DEFAULT_COMPARE(diracgammaR)
195 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
196 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
197 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
198 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
199 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
201 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
202 static void base_and_index(const ex & c, ex & b, ex & i)
204 GINAC_ASSERT(is_a<clifford>(c));
205 GINAC_ASSERT(c.nops() == 2);
207 if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
210 } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
213 } else { // slash object, generate new dummy index
214 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
215 b = indexed(c.op(0), ix.toggle_variance());
220 /** Contraction of a gamma matrix with something else. */
221 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
223 GINAC_ASSERT(is_a<clifford>(*self));
224 GINAC_ASSERT(is_a<indexed>(*other));
225 GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
226 unsigned char rl = ex_to<clifford>(*self).get_representation_label();
228 ex dim = ex_to<idx>(self->op(1)).get_dim();
229 if (other->nops() > 1)
230 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
232 if (is_a<clifford>(*other)) {
234 // Contraction only makes sense if the represenation labels are equal
235 if (ex_to<clifford>(*other).get_representation_label() != rl)
238 // gamma~mu gamma.mu = dim ONE
239 if (other - self == 1) {
241 *other = dirac_ONE(rl);
244 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
245 } else if (other - self == 2
246 && is_a<clifford>(self[1])) {
251 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
252 } else if (other - self == 3
253 && is_a<clifford>(self[1])
254 && is_a<clifford>(self[2])) {
256 base_and_index(self[1], b1, i1);
257 base_and_index(self[2], b2, i2);
258 *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
264 // 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
265 } else if (other - self == 4
266 && is_a<clifford>(self[1])
267 && is_a<clifford>(self[2])
268 && is_a<clifford>(self[3])) {
269 *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
276 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
277 // (commutate contracted indices towards each other, simplify_indexed()
278 // will re-expand and re-run the simplification)
280 exvector::iterator it = self + 1, next_to_last = other - 1;
281 while (it != other) {
282 if (!is_a<clifford>(*it))
289 while (it != next_to_last) {
294 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
295 *next_to_last = _ex1;
300 } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
302 // x.mu gamma~mu -> x-slash
303 *self = dirac_slash(other->op(0), dim, rl);
311 /** Perform automatic simplification on noncommutative product of clifford
312 * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
313 * and removes squares of gamma objects. */
314 ex clifford::eval_ncmul(const exvector & v) const
319 // Remove superfluous ONEs
320 exvector::const_iterator cit = v.begin(), citend = v.end();
321 while (cit != citend) {
322 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
327 bool something_changed = false;
330 // Anticommute gamma5/L/R's to the front
332 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
334 exvector::iterator it = next_to_last;
336 exvector::iterator it2 = it + 1;
337 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
338 ex e1 = it->op(0), e2 = it2->op(0);
340 if (is_a<diracgamma5>(e2)) {
342 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
344 // gammaL/R gamma5 -> gamma5 gammaL/R
346 something_changed = true;
348 } else if (!is_a<diracgamma5>(e1)) {
350 // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
351 // x gamma5 -> -gamma5 x
354 something_changed = true;
357 } else if (is_a<diracgammaL>(e2)) {
359 if (is_a<diracgammaR>(e1)) {
361 // gammaR gammaL -> 0
364 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
366 // gammaL gammaL -> gammaL gammaL (do nothing)
367 // gamma5 gammaL -> gamma5 gammaL (do nothing)
368 // x gammaL -> gammaR x
370 *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
371 something_changed = true;
374 } else if (is_a<diracgammaR>(e2)) {
376 if (is_a<diracgammaL>(e1)) {
378 // gammaL gammaR -> 0
381 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
383 // gammaR gammaR -> gammaR gammaR (do nothing)
384 // gamma5 gammaR -> gamma5 gammaR (do nothing)
385 // x gammaR -> gammaL x
387 *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
388 something_changed = true;
396 if (next_to_last == first)
402 // Remove equal adjacent gammas
404 exvector::iterator it, itend = s.end() - 1;
405 for (it = s.begin(); it != itend; ++it) {
408 if (!is_a<clifford>(a) || !is_a<clifford>(b))
411 const ex & ag = a.op(0);
412 const ex & bg = b.op(0);
413 bool a_is_diracgamma = is_a<diracgamma>(ag);
414 bool b_is_diracgamma = is_a<diracgamma>(bg);
416 if (a_is_diracgamma && b_is_diracgamma) {
418 const ex & ia = a.op(1);
419 const ex & ib = b.op(1);
420 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
421 a = lorentz_g(ia, ib);
422 b = dirac_ONE(representation_label);
423 something_changed = true;
426 } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
428 // Remove squares of gamma5
429 a = dirac_ONE(representation_label);
430 b = dirac_ONE(representation_label);
431 something_changed = true;
433 } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
434 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
436 // Remove squares of gammaL/R
437 b = dirac_ONE(representation_label);
438 something_changed = true;
440 } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
442 // gammaL and gammaR are orthogonal
445 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
447 // gamma5 gammaL -> -gammaL
448 a = dirac_ONE(representation_label);
450 something_changed = true;
452 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
454 // gamma5 gammaR -> gammaR
455 a = dirac_ONE(representation_label);
456 something_changed = true;
458 } else if (!a_is_diracgamma && !b_is_diracgamma && ag.is_equal(bg)) {
461 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
462 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
463 b = dirac_ONE(representation_label);
464 something_changed = true;
470 return clifford(diracone(), representation_label) * sign;
471 if (something_changed)
472 return reeval_ncmul(s) * sign;
474 return hold_ncmul(s) * sign;
477 ex clifford::thiscontainer(const exvector & v) const
479 return clifford(representation_label, v);
482 ex clifford::thiscontainer(exvector * vp) const
484 return clifford(representation_label, vp);
491 ex dirac_ONE(unsigned char rl)
493 return clifford(diracone(), rl);
496 ex dirac_gamma(const ex & mu, unsigned char rl)
498 if (!is_a<varidx>(mu))
499 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
501 return clifford(diracgamma(), mu, rl);
504 ex dirac_gamma5(unsigned char rl)
506 return clifford(diracgamma5(), rl);
509 ex dirac_gammaL(unsigned char rl)
511 return clifford(diracgammaL(), rl);
514 ex dirac_gammaR(unsigned char rl)
516 return clifford(diracgammaR(), rl);
519 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
521 // Slashed vectors are actually stored as a clifford object with the
522 // vector as its base expression and a (dummy) index that just serves
523 // for storing the space dimensionality
524 return clifford(e, varidx(0, dim), rl);
527 /** Check whether a given tinfo key (as returned by return_type_tinfo()
528 * is that of a clifford object with the specified representation label. */
529 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
531 return ti == (TINFO_clifford + rl);
534 /** Check whether a given tinfo key (as returned by return_type_tinfo()
535 * is that of a clifford object (with an arbitrary representation label). */
536 static bool is_clifford_tinfo(unsigned ti)
538 return (ti & ~0xff) == TINFO_clifford;
541 /** Take trace of a string of an even number of Dirac gammas given a vector
543 static ex trace_string(exvector::const_iterator ix, size_t num)
545 // Tr gamma.mu gamma.nu = 4 g.mu.nu
547 return lorentz_g(ix[0], ix[1]);
549 // 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 )
551 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
552 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
553 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
555 // Traces of 6 or more gammas are computed recursively:
556 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
557 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
558 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
559 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
561 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
565 for (size_t i=1; i<num; i++) {
566 for (size_t n=1, j=0; n<num; n++) {
571 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
577 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
579 if (is_a<clifford>(e)) {
581 if (!ex_to<clifford>(e).get_representation_label() == rl)
583 const ex & g = e.op(0);
584 if (is_a<diracone>(g))
586 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
591 } else if (is_exactly_a<mul>(e)) {
593 // Trace of product: pull out non-clifford factors
595 for (size_t i=0; i<e.nops(); i++) {
596 const ex &o = e.op(i);
597 if (is_clifford_tinfo(o.return_type_tinfo(), rl))
598 prod *= dirac_trace(o, rl, trONE);
604 } else if (is_exactly_a<ncmul>(e)) {
606 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
609 // Substitute gammaL/R and expand product, if necessary
610 ex e_expanded = e.subs(lst(
611 dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
612 dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
613 ), subs_options::no_pattern).expand();
614 if (!is_a<ncmul>(e_expanded))
615 return dirac_trace(e_expanded, rl, trONE);
617 // gamma5 gets moved to the front so this check is enough
618 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
619 size_t num = e.nops();
623 // Trace of gamma5 * odd number of gammas and trace of
624 // gamma5 * gamma.mu * gamma.nu are zero
625 if ((num & 1) == 0 || num == 3)
628 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
629 // (the epsilon is always 4-dimensional)
631 ex b1, i1, b2, i2, b3, i3, b4, i4;
632 base_and_index(e.op(1), b1, i1);
633 base_and_index(e.op(2), b2, i2);
634 base_and_index(e.op(3), b3, i3);
635 base_and_index(e.op(4), b4, i4);
636 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();
640 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
641 // (the epsilon is always 4-dimensional)
642 exvector ix(num-1), bv(num-1);
643 for (size_t i=1; i<num; i++)
644 base_and_index(e.op(i), bv[i-1], ix[i-1]);
646 int *iv = new int[num];
648 for (size_t i=0; i<num-3; i++) {
650 for (size_t j=i+1; j<num-2; j++) {
652 for (size_t k=j+1; k<num-1; k++) {
654 for (size_t l=k+1; l<num; l++) {
656 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
659 for (size_t n=0, t=4; n<num; n++) {
660 if (n == i || n == j || n == k || n == l)
665 int sign = permutation_sign(iv, iv + num);
666 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))
667 * trace_string(v.begin(), num - 4);
673 return trONE * I * result * mul(bv);
675 } else { // no gamma5
677 // Trace of odd number of gammas is zero
681 // Tr gamma.mu gamma.nu = 4 g.mu.nu
684 base_and_index(e.op(0), b1, i1);
685 base_and_index(e.op(1), b2, i2);
686 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
689 exvector iv(num), bv(num);
690 for (size_t i=0; i<num; i++)
691 base_and_index(e.op(i), bv[i], iv[i]);
693 return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
696 } else if (e.nops() > 0) {
698 // Trace maps to all other container classes (this includes sums)
699 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
706 ex canonicalize_clifford(const ex & e)
708 // Scan for any ncmul objects
710 ex aux = e.to_rational(srl);
711 for (size_t i=0; i<srl.nops(); i++) {
717 if (is_exactly_a<ncmul>(rhs)
718 && rhs.return_type() == return_types::noncommutative
719 && is_clifford_tinfo(rhs.return_type_tinfo())) {
721 // Expand product, if necessary
722 ex rhs_expanded = rhs.expand();
723 if (!is_a<ncmul>(rhs_expanded)) {
724 srl[i] = (lhs == canonicalize_clifford(rhs_expanded));
727 } else if (!is_a<clifford>(rhs.op(0)))
731 v.reserve(rhs.nops());
732 for (size_t j=0; j<rhs.nops(); j++)
733 v.push_back(rhs.op(j));
735 // Stupid recursive bubble sort because we only want to swap adjacent gammas
736 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
737 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
739 while (it != next_to_last) {
740 if (it[0].compare(it[1]) > 0) {
741 ex save0 = it[0], save1 = it[1];
743 base_and_index(it[0], b1, i1);
744 base_and_index(it[1], b2, i2);
745 it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
750 sum -= ncmul(v, true);
751 srl[i] = (lhs == canonicalize_clifford(sum));
759 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();