3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2002 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"
42 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
43 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
44 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
45 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
48 // default ctor, dtor, copy ctor, assignment operator and helpers
51 clifford::clifford() : representation_label(0)
53 tinfo_key = TINFO_clifford;
56 void clifford::copy(const clifford & other)
58 inherited::copy(other);
59 representation_label = other.representation_label;
62 DEFAULT_DESTROY(clifford)
63 DEFAULT_CTORS(diracone)
64 DEFAULT_CTORS(diracgamma)
65 DEFAULT_CTORS(diracgamma5)
71 /** Construct object without any indices. This constructor is for internal
72 * use only. Use the dirac_ONE() function instead.
74 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
76 tinfo_key = TINFO_clifford;
79 /** Construct object with one Lorentz index. This constructor is for internal
80 * use only. Use the dirac_gamma() function instead.
82 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
84 GINAC_ASSERT(is_a<varidx>(mu));
85 tinfo_key = TINFO_clifford;
88 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
90 tinfo_key = TINFO_clifford;
93 clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
95 tinfo_key = TINFO_clifford;
102 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
105 n.find_unsigned("label", rl);
106 representation_label = rl;
109 void clifford::archive(archive_node &n) const
111 inherited::archive(n);
112 n.add_unsigned("label", representation_label);
115 DEFAULT_UNARCHIVE(clifford)
116 DEFAULT_ARCHIVING(diracone)
117 DEFAULT_ARCHIVING(diracgamma)
118 DEFAULT_ARCHIVING(diracgamma5)
121 // functions overriding virtual functions from base classes
124 int clifford::compare_same_type(const basic & other) const
126 GINAC_ASSERT(is_a<clifford>(other));
127 const clifford &o = static_cast<const clifford &>(other);
129 if (representation_label != o.representation_label) {
130 // different representation label
131 return representation_label < o.representation_label ? -1 : 1;
134 return inherited::compare_same_type(other);
137 bool clifford::match_same_type(const basic & other) const
139 GINAC_ASSERT(is_a<clifford>(other));
140 const clifford &o = static_cast<const clifford &>(other);
142 return representation_label == o.representation_label;
145 void clifford::print(const print_context & c, unsigned level) const
147 if (!is_a<diracgamma5>(seq[0]) && !is_a<diracgamma>(seq[0]) && !is_a<diracone>(seq[0])) {
149 // dirac_slash() object is printed differently
150 if (is_a<print_tree>(c))
151 inherited::print(c, level);
152 else if (is_a<print_latex>(c)) {
154 seq[0].print(c, level);
155 c.s << "\\hspace{-1.0ex}/}";
157 seq[0].print(c, level);
162 inherited::print(c, level);
165 DEFAULT_COMPARE(diracone)
166 DEFAULT_COMPARE(diracgamma)
167 DEFAULT_COMPARE(diracgamma5)
169 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
170 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
171 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
173 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
174 static void base_and_index(const ex & c, ex & b, ex & i)
176 GINAC_ASSERT(is_a<clifford>(c));
177 GINAC_ASSERT(c.nops() == 2);
179 if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
182 } else if (is_a<diracgamma5>(c.op(0))) { // gamma5
185 } else { // slash object, generate new dummy index
186 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
187 b = indexed(c.op(0), ix.toggle_variance());
192 /** Contraction of a gamma matrix with something else. */
193 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
195 GINAC_ASSERT(is_a<clifford>(*self));
196 GINAC_ASSERT(is_a<indexed>(*other));
197 GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
198 unsigned char rl = ex_to<clifford>(*self).get_representation_label();
200 if (is_a<clifford>(*other)) {
202 // Contraction only makes sense if the represenation labels are equal
203 if (ex_to<clifford>(*other).get_representation_label() != rl)
206 ex dim = ex_to<idx>(self->op(1)).get_dim();
208 // gamma~mu gamma.mu = dim ONE
209 if (other - self == 1) {
211 *other = dirac_ONE(rl);
214 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
215 } else if (other - self == 2
216 && is_a<clifford>(self[1])) {
221 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
222 } else if (other - self == 3
223 && is_a<clifford>(self[1])
224 && is_a<clifford>(self[2])) {
226 base_and_index(self[1], b1, i1);
227 base_and_index(self[2], b2, i2);
228 *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
234 // 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
235 } else if (other - self == 4
236 && is_a<clifford>(self[1])
237 && is_a<clifford>(self[2])
238 && is_a<clifford>(self[3])) {
239 *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
246 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
247 // (commutate contracted indices towards each other, simplify_indexed()
248 // will re-expand and re-run the simplification)
250 exvector::iterator it = self + 1, next_to_last = other - 1;
251 while (it != other) {
252 if (!is_a<clifford>(*it))
259 while (it != next_to_last) {
264 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
265 *next_to_last = _ex1;
274 /** Perform automatic simplification on noncommutative product of clifford
275 * objects. This removes superfluous ONEs, permutes gamma5's to the front
276 * and removes squares of gamma objects. */
277 ex clifford::simplify_ncmul(const exvector & v) const
282 // Remove superfluous ONEs
283 exvector::const_iterator cit = v.begin(), citend = v.end();
284 while (cit != citend) {
285 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
290 bool something_changed = false;
293 // Anticommute gamma5's to the front
295 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
297 exvector::iterator it = next_to_last;
299 exvector::iterator it2 = it + 1;
300 if (is_a<clifford>(*it) && is_a<clifford>(*it2) && !is_a<diracgamma5>(it->op(0)) && is_a<diracgamma5>(it2->op(0))) {
303 something_changed = true;
309 if (next_to_last == first)
315 // Remove equal adjacent gammas
317 exvector::iterator it, itend = s.end() - 1;
318 for (it = s.begin(); it != itend; ++it) {
321 if (!is_a<clifford>(a) || !is_a<clifford>(b))
323 bool a_is_diracgamma = is_a<diracgamma>(a.op(0));
324 bool b_is_diracgamma = is_a<diracgamma>(b.op(0));
325 if (a_is_diracgamma && b_is_diracgamma) {
326 const ex & ia = a.op(1);
327 const ex & ib = b.op(1);
328 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
329 a = lorentz_g(ia, ib);
330 b = dirac_ONE(representation_label);
331 something_changed = true;
333 } else if (is_a<diracgamma5>(a.op(0)) && is_a<diracgamma5>(b.op(0))) {
334 // Remove squares of gamma5
335 a = dirac_ONE(representation_label);
336 b = dirac_ONE(representation_label);
337 something_changed = true;
338 } else if (!a_is_diracgamma && !b_is_diracgamma) {
339 const ex & ba = a.op(0);
340 const ex & bb = b.op(0);
341 if (ba.is_equal(bb)) { // a\ a\ -> a^2
342 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
343 a = indexed(ba, ix) * indexed(bb, ix.toggle_variance());
344 b = dirac_ONE(representation_label);
345 something_changed = true;
352 return clifford(diracone(), representation_label) * sign;
353 if (something_changed)
354 return nonsimplified_ncmul(s) * sign;
356 return simplified_ncmul(s) * sign;
359 ex clifford::thisexprseq(const exvector & v) const
361 return clifford(representation_label, v);
364 ex clifford::thisexprseq(exvector * vp) const
366 return clifford(representation_label, vp);
373 ex dirac_ONE(unsigned char rl)
375 return clifford(diracone(), rl);
378 ex dirac_gamma(const ex & mu, unsigned char rl)
380 if (!is_a<varidx>(mu))
381 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
383 return clifford(diracgamma(), mu, rl);
386 ex dirac_gamma5(unsigned char rl)
388 return clifford(diracgamma5(), rl);
391 ex dirac_gamma6(unsigned char rl)
393 return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
396 ex dirac_gamma7(unsigned char rl)
398 return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
401 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
403 // Slashed vectors are actually stored as a clifford object with the
404 // vector as its base expression and a (dummy) index that just serves
405 // for storing the space dimensionality
406 return clifford(e, varidx(0, dim), rl);
409 /** Check whether a given tinfo key (as returned by return_type_tinfo()
410 * is that of a clifford object with the specified representation label. */
411 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
413 return ti == (TINFO_clifford + rl);
416 /** Check whether a given tinfo key (as returned by return_type_tinfo()
417 * is that of a clifford object (with an arbitrary representation label). */
418 static bool is_clifford_tinfo(unsigned ti)
420 return (ti & ~0xff) == TINFO_clifford;
423 /** Take trace of a string of an even number of Dirac gammas given a vector
425 static ex trace_string(exvector::const_iterator ix, unsigned num)
427 // Tr gamma.mu gamma.nu = 4 g.mu.nu
429 return lorentz_g(ix[0], ix[1]);
431 // 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
433 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
434 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
435 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
437 // Traces of 6 or more gammas are computed recursively:
438 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
439 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
440 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
441 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
443 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
447 for (unsigned i=1; i<num; i++) {
448 for (unsigned n=1, j=0; n<num; n++) {
453 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
459 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
461 if (is_a<clifford>(e)) {
463 if (ex_to<clifford>(e).get_representation_label() == rl
464 && is_a<diracone>(e.op(0)))
469 } else if (is_ex_exactly_of_type(e, mul)) {
471 // Trace of product: pull out non-clifford factors
473 for (unsigned i=0; i<e.nops(); i++) {
474 const ex &o = e.op(i);
475 if (is_clifford_tinfo(o.return_type_tinfo(), rl))
476 prod *= dirac_trace(o, rl, trONE);
482 } else if (is_ex_exactly_of_type(e, ncmul)) {
484 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
487 // Expand product, if necessary
488 ex e_expanded = e.expand();
489 if (!is_a<ncmul>(e_expanded))
490 return dirac_trace(e_expanded, rl, trONE);
492 // gamma5 gets moved to the front so this check is enough
493 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
494 unsigned num = e.nops();
498 // Trace of gamma5 * odd number of gammas and trace of
499 // gamma5 * gamma.mu * gamma.nu are zero
500 if ((num & 1) == 0 || num == 3)
503 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
504 // (the epsilon is always 4-dimensional)
506 ex b1, i1, b2, i2, b3, i3, b4, i4;
507 base_and_index(e.op(1), b1, i1);
508 base_and_index(e.op(2), b2, i2);
509 base_and_index(e.op(3), b3, i3);
510 base_and_index(e.op(4), b4, i4);
511 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();
515 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
516 // (the epsilon is always 4-dimensional)
517 exvector ix(num-1), bv(num-1);
518 for (unsigned i=1; i<num; i++)
519 base_and_index(e.op(i), bv[i-1], ix[i-1]);
521 int *iv = new int[num];
523 for (unsigned i=0; i<num-3; i++) {
525 for (unsigned j=i+1; j<num-2; j++) {
527 for (unsigned k=j+1; k<num-1; k++) {
529 for (unsigned l=k+1; l<num; l++) {
531 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
534 for (unsigned n=0, t=4; n<num; n++) {
535 if (n == i || n == j || n == k || n == l)
540 int sign = permutation_sign(iv, iv + num);
541 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))
542 * trace_string(v.begin(), num - 4);
548 return trONE * I * result * mul(bv);
550 } else { // no gamma5
552 // Trace of odd number of gammas is zero
556 // Tr gamma.mu gamma.nu = 4 g.mu.nu
559 base_and_index(e.op(0), b1, i1);
560 base_and_index(e.op(1), b2, i2);
561 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
564 exvector iv(num), bv(num);
565 for (unsigned i=0; i<num; i++)
566 base_and_index(e.op(i), bv[i], iv[i]);
568 return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
571 } else if (e.nops() > 0) {
573 // Trace maps to all other container classes (this includes sums)
574 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
581 ex canonicalize_clifford(const ex & e)
583 // Scan for any ncmul objects
585 ex aux = e.to_rational(srl);
586 for (unsigned i=0; i<srl.nops(); i++) {
588 ex lhs = srl.op(i).lhs();
589 ex rhs = srl.op(i).rhs();
591 if (is_ex_exactly_of_type(rhs, ncmul)
592 && rhs.return_type() == return_types::noncommutative
593 && is_clifford_tinfo(rhs.return_type_tinfo())) {
595 // Expand product, if necessary
596 ex rhs_expanded = rhs.expand();
597 if (!is_a<ncmul>(rhs_expanded)) {
598 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
601 } else if (!is_a<clifford>(rhs.op(0)))
605 v.reserve(rhs.nops());
606 for (unsigned j=0; j<rhs.nops(); j++)
607 v.push_back(rhs.op(j));
609 // Stupid recursive bubble sort because we only want to swap adjacent gammas
610 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
611 if (is_a<diracgamma5>(it->op(0)))
613 while (it != next_to_last) {
614 if (it[0].compare(it[1]) > 0) {
615 ex save0 = it[0], save1 = it[1];
617 base_and_index(it[0], b1, i1);
618 base_and_index(it[1], b2, i2);
619 it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
624 sum -= ncmul(v, true);
625 srl.let_op(i) = (lhs == canonicalize_clifford(sum));
633 return aux.subs(srl).simplify_indexed();