3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2001 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
28 #include "numeric.h" // for I
30 #include "relational.h"
40 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
41 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
42 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
43 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
46 // default constructor, destructor, copy constructor assignment operator and helpers
49 clifford::clifford() : representation_label(0)
51 debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
52 tinfo_key = TINFO_clifford;
55 void clifford::copy(const clifford & other)
57 inherited::copy(other);
58 representation_label = other.representation_label;
61 DEFAULT_DESTROY(clifford)
62 DEFAULT_CTORS(diracone)
63 DEFAULT_CTORS(diracgamma)
64 DEFAULT_CTORS(diracgamma5)
70 /** Construct object without any indices. This constructor is for internal
71 * use only. Use the dirac_ONE() function instead.
73 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
75 debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
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 debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
85 GINAC_ASSERT(is_ex_of_type(mu, varidx));
86 tinfo_key = TINFO_clifford;
89 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
91 debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
92 tinfo_key = TINFO_clifford;
95 clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
97 debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
98 tinfo_key = TINFO_clifford;
105 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
107 debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
109 n.find_unsigned("label", rl);
110 representation_label = rl;
113 void clifford::archive(archive_node &n) const
115 inherited::archive(n);
116 n.add_unsigned("label", representation_label);
119 DEFAULT_UNARCHIVE(clifford)
120 DEFAULT_ARCHIVING(diracone)
121 DEFAULT_ARCHIVING(diracgamma)
122 DEFAULT_ARCHIVING(diracgamma5)
125 // functions overriding virtual functions from bases classes
128 int clifford::compare_same_type(const basic & other) const
130 GINAC_ASSERT(other.tinfo() == TINFO_clifford);
131 const clifford &o = static_cast<const clifford &>(other);
133 if (representation_label != o.representation_label) {
134 // different representation label
135 return representation_label < o.representation_label ? -1 : 1;
138 return inherited::compare_same_type(other);
141 DEFAULT_COMPARE(diracone)
142 DEFAULT_COMPARE(diracgamma)
143 DEFAULT_COMPARE(diracgamma5)
145 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
146 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
147 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
149 /** Contraction of a gamma matrix with something else. */
150 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
152 GINAC_ASSERT(is_ex_of_type(*self, clifford));
153 GINAC_ASSERT(is_ex_of_type(*other, indexed));
154 GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
155 unsigned char rl = ex_to_clifford(*self).get_representation_label();
157 if (is_ex_of_type(*other, clifford)) {
159 ex dim = ex_to_idx(self->op(1)).get_dim();
161 // gamma~mu gamma.mu = dim ONE
162 if (other - self == 1) {
164 *other = dirac_ONE(rl);
167 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
168 } else if (other - self == 2
169 && is_ex_of_type(self[1], clifford)) {
174 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
175 } else if (other - self == 3
176 && is_ex_of_type(self[1], clifford)
177 && is_ex_of_type(self[2], clifford)) {
178 *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
184 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
185 // (commutate contracted indices towards each other, simplify_indexed()
186 // will re-expand and re-run the simplification)
188 exvector::iterator it = self + 1, next_to_last = other - 1;
189 while (it != other) {
190 if (!is_ex_of_type(*it, clifford))
197 while (it != next_to_last) {
202 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
203 *next_to_last = _ex1();
212 /** Perform automatic simplification on noncommutative product of clifford
213 * objects. This removes superfluous ONEs, permutes gamma5's to the front
214 * and removes squares of gamma objects. */
215 ex clifford::simplify_ncmul(const exvector & v) const
220 // Remove superfluous ONEs
221 exvector::const_iterator cit = v.begin(), citend = v.end();
222 while (cit != citend) {
223 if (!is_ex_of_type(cit->op(0), diracone))
228 bool something_changed = false;
231 // Anticommute gamma5's to the front
233 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
235 exvector::iterator it = next_to_last;
237 exvector::iterator it2 = it + 1;
238 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
241 something_changed = true;
247 if (next_to_last == first)
253 // Remove squares of gamma5
254 while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
255 s.erase(s.begin(), s.begin() + 2);
256 something_changed = true;
259 // Remove equal adjacent gammas
261 exvector::iterator it = s.begin(), itend = s.end() - 1;
262 while (it != itend) {
265 if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
266 const ex & ia = a.op(1);
267 const ex & ib = b.op(1);
268 if (ia.is_equal(ib)) {
269 a = lorentz_g(ia, ib);
270 b = dirac_ONE(representation_label);
271 something_changed = true;
279 return clifford(diracone(), representation_label) * sign;
280 if (something_changed)
281 return nonsimplified_ncmul(s) * sign;
283 return simplified_ncmul(s) * sign;
286 ex clifford::thisexprseq(const exvector & v) const
288 return clifford(representation_label, v);
291 ex clifford::thisexprseq(exvector * vp) const
293 return clifford(representation_label, vp);
300 ex dirac_ONE(unsigned char rl)
302 return clifford(diracone(), rl);
305 ex dirac_gamma(const ex & mu, unsigned char rl)
307 if (!is_ex_of_type(mu, varidx))
308 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
310 return clifford(diracgamma(), mu, rl);
313 ex dirac_gamma5(unsigned char rl)
315 return clifford(diracgamma5(), rl);
318 ex dirac_gamma6(unsigned char rl)
320 return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
323 ex dirac_gamma7(unsigned char rl)
325 return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
328 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
330 varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
331 return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
334 /** Check whether a given tinfo key (as returned by return_type_tinfo()
335 * is that of a clifford object with the specified representation label. */
336 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
338 return ti == (TINFO_clifford + rl);
341 /** Check whether a given tinfo key (as returned by return_type_tinfo()
342 * is that of a clifford object (with an arbitrary representation label). */
343 static bool is_clifford_tinfo(unsigned ti)
345 return (ti & ~0xff) == TINFO_clifford;
348 /** Take trace of a string of an even number of Dirac gammas given a vector
350 static ex trace_string(exvector::const_iterator ix, unsigned num)
352 // Tr gamma.mu gamma.nu = 4 g.mu.nu
354 return lorentz_g(ix[0], ix[1]);
356 // 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
358 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
359 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
360 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
362 // Traces of 6 or more gammas are computed recursively:
363 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
364 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
365 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
366 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
368 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
372 for (int i=1; i<num; i++) {
373 for (int n=1, j=0; n<num; n++) {
378 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
384 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
386 if (is_ex_of_type(e, clifford)) {
388 if (ex_to_clifford(e).get_representation_label() == rl
389 && is_ex_of_type(e.op(0), diracone))
394 } else if (is_ex_exactly_of_type(e, add)) {
396 // Trace of sum = sum of traces
398 for (unsigned i=0; i<e.nops(); i++)
399 sum += dirac_trace(e.op(i), rl, trONE);
402 } else if (is_ex_exactly_of_type(e, mul)) {
404 // Trace of product: pull out non-clifford factors
406 for (unsigned i=0; i<e.nops(); i++) {
407 const ex &o = e.op(i);
408 unsigned ti = o.return_type_tinfo();
409 if (is_clifford_tinfo(o.return_type_tinfo(), rl))
410 prod *= dirac_trace(o, rl, trONE);
416 } else if (is_ex_exactly_of_type(e, ncmul)) {
418 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
421 // Expand product, if necessary
422 ex e_expanded = e.expand();
423 if (!is_ex_of_type(e_expanded, ncmul))
424 return dirac_trace(e_expanded, rl, trONE);
426 // gamma5 gets moved to the front so this check is enough
427 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
428 unsigned num = e.nops();
432 // Trace of gamma5 * odd number of gammas and trace of
433 // gamma5 * gamma.mu * gamma.nu are zero
434 if ((num & 1) == 0 || num == 3)
437 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
439 return trONE * I * eps0123(e.op(1).op(1), e.op(2).op(1), e.op(3).op(1), e.op(4).op(1));
442 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
445 for (unsigned i=1; i<num; i++)
446 ix.push_back(e.op(i).op(1));
448 int *iv = new int[num];
450 for (int i=0; i<num-3; i++) {
452 for (int j=i+1; j<num-2; j++) {
454 for (int k=j+1; k<num-1; k++) {
456 for (int l=k+1; l<num; l++) {
458 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
461 for (int n=0, t=4; n<num; n++) {
462 if (n == i || n == j || n == k || n == l)
467 int sign = permutation_sign(iv, iv + num);
468 result += sign * eps0123(idx1, idx2, idx3, idx4)
469 * trace_string(v.begin(), num - 4);
475 return trONE * I * result;
477 } else { // no gamma5
479 // Trace of odd number of gammas is zero
483 // Tr gamma.mu gamma.nu = 4 g.mu.nu
485 return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
489 for (unsigned i=0; i<num; i++)
490 iv.push_back(e.op(i).op(1));
492 return trONE * trace_string(iv.begin(), num);
499 ex canonicalize_clifford(const ex & e)
501 // Scan for any ncmul objects
503 ex aux = e.to_rational(srl);
504 for (unsigned i=0; i<srl.nops(); i++) {
506 ex lhs = srl.op(i).lhs();
507 ex rhs = srl.op(i).rhs();
509 if (is_ex_exactly_of_type(rhs, ncmul)
510 && rhs.return_type() == return_types::noncommutative
511 && is_clifford_tinfo(rhs.return_type_tinfo())) {
513 // Expand product, if necessary
514 ex rhs_expanded = rhs.expand();
515 if (!is_ex_of_type(rhs_expanded, ncmul)) {
516 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
519 } else if (!is_ex_of_type(rhs.op(0), clifford))
523 v.reserve(rhs.nops());
524 for (unsigned j=0; j<rhs.nops(); j++)
525 v.push_back(rhs.op(j));
527 // Stupid recursive bubble sort because we only want to swap adjacent gammas
528 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
529 if (is_ex_of_type(it->op(0), diracgamma5))
531 while (it != next_to_last) {
532 if (it[0].op(1).compare(it[1].op(1)) > 0) {
533 ex save0 = it[0], save1 = it[1];
534 it[0] = lorentz_g(it[0].op(1), it[1].op(1));
539 sum -= ncmul(v, true);
540 srl.let_op(i) = (lhs == canonicalize_clifford(sum));
548 return aux.subs(srl);