From 834bacb25c9427fb03b8f9b34605738fb59202e1 Mon Sep 17 00:00:00 2001 From: Christian Bauer Date: Thu, 26 Jul 2001 23:28:08 +0000 Subject: [PATCH] - epsilon*epsilon contractions work - dirac_slash(a, D) now constructs a special kind of clifford object (printed as "a\") instead of creating a "a.symbol42*gamma~symbol42" product. This helps avoiding problems when doing something like ex prop = (dirac_slash(p, D) - m * dirac_ONE()); ex I = prop * dirac_slash(l, D) * prop; where the same dummy index would be used for both p-slash in I. Now it's perfectly safe to do this. Dummy indices are only created when taking traces or contracting indices. A nice side-effect of this is that "a\*a\" immediately gets simplified to a^2. - The "contraction of symmetric and antisymmetric objects is zero" rule in simplify_indexed() has been generalized. Now expressions like "epsilon.i.j.k*A.j*A.k" also get simplified to zero (what GiNaC does is to assert dummy index exchange symmetry). --- ginac/clifford.cpp | 154 +++++++++++++++++++++++++++++++++------------ ginac/clifford.h | 3 + ginac/indexed.cpp | 36 ++++------- ginac/tensor.cpp | 38 +++++++++++ 4 files changed, 168 insertions(+), 63 deletions(-) diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp index 8d8c2c63..8687f969 100644 --- a/ginac/clifford.cpp +++ b/ginac/clifford.cpp @@ -29,6 +29,7 @@ #include "symmetry.h" #include "lst.h" #include "relational.h" +#include "mul.h" #include "print.h" #include "archive.h" #include "debugmsg.h" @@ -83,7 +84,7 @@ clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representatio clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl) { debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT); - GINAC_ASSERT(is_ex_of_type(mu, varidx)); + GINAC_ASSERT(is_a(mu)); tinfo_key = TINFO_clifford; } @@ -147,6 +148,26 @@ bool clifford::match_same_type(const basic & other) const return representation_label == o.representation_label; } +void clifford::print(const print_context & c, unsigned level = 0) const +{ + if (!is_a(seq[0]) && !is_a(seq[0]) && !is_a(seq[0])) { + + // dirac_slash() object is printed differently + if (is_a(c)) + inherited::print(c, level); + else if (is_a(c)) { + c.s << "{"; + seq[0].print(c, level); + c.s << "\\hspace{-1.0ex}/}"; + } else { + seq[0].print(c, level); + c.s << "\\"; + } + + } else + inherited::print(c, level); +} + DEFAULT_COMPARE(diracone) DEFAULT_COMPARE(diracgamma) DEFAULT_COMPARE(diracgamma5) @@ -155,15 +176,31 @@ DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}") DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma") DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}") +/** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */ +static void base_and_index(const ex & c, ex & b, ex & i) +{ + GINAC_ASSERT(is_a(c)); + GINAC_ASSERT(c.nops() == 2); + + if (is_a(c.op(0))) { // proper dirac gamma object + i = c.op(1); + b = _ex1(); + } else { // slash object, generate new dummy index + varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to(c.op(1)).get_dim()); + b = indexed(c.op(0), ix.toggle_variance()); + i = ix; + } +} + /** Contraction of a gamma matrix with something else. */ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const { - GINAC_ASSERT(is_ex_of_type(*self, clifford)); - GINAC_ASSERT(is_ex_of_type(*other, indexed)); - GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma)); + GINAC_ASSERT(is_a(*self)); + GINAC_ASSERT(is_a(*other)); + GINAC_ASSERT(is_a(self->op(0))); unsigned char rl = ex_to(*self).get_representation_label(); - if (is_ex_of_type(*other, clifford)) { + if (is_a(*other)) { ex dim = ex_to(self->op(1)).get_dim(); @@ -175,28 +212,43 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha } else if (other - self == 2 - && is_ex_of_type(self[1], clifford)) { + && is_a(self[1])) { *self = 2 - dim; *other = _ex1(); return true; // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta } else if (other - self == 3 - && is_ex_of_type(self[1], clifford) - && is_ex_of_type(self[2], clifford)) { - *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2]; + && is_a(self[1]) + && is_a(self[2])) { + ex b1, i1, b2, i2; + base_and_index(self[1], b1, i1); + base_and_index(self[2], b2, i2); + *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2]; self[1] = _ex1(); self[2] = _ex1(); *other = _ex1(); return true; + // 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 + } else if (other - self == 4 + && is_a(self[1]) + && is_a(self[2]) + && is_a(self[3])) { + *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3]; + self[1] = _ex1(); + self[2] = _ex1(); + self[3] = _ex1(); + *other = _ex1(); + return true; + // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha // (commutate contracted indices towards each other, simplify_indexed() // will re-expand and re-run the simplification) } else { exvector::iterator it = self + 1, next_to_last = other - 1; while (it != other) { - if (!is_ex_of_type(*it, clifford)) + if (!is_a(*it)) return false; ++it; } @@ -229,7 +281,7 @@ ex clifford::simplify_ncmul(const exvector & v) const // Remove superfluous ONEs exvector::const_iterator cit = v.begin(), citend = v.end(); while (cit != citend) { - if (!is_ex_of_type(cit->op(0), diracone)) + if (!is_a(cit->op(0))) s.push_back(*cit); cit++; } @@ -244,7 +296,7 @@ ex clifford::simplify_ncmul(const exvector & v) const exvector::iterator it = next_to_last; while (true) { exvector::iterator it2 = it + 1; - if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) { + if (!is_a(it->op(0)) && is_a(it2->op(0))) { it->swap(*it2); sign = -sign; something_changed = true; @@ -260,7 +312,7 @@ ex clifford::simplify_ncmul(const exvector & v) const } // Remove squares of gamma5 - while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) { + while (s.size() >= 2 && is_a(s[0].op(0)) && is_a(s[1].op(0))) { s.erase(s.begin(), s.begin() + 2); something_changed = true; } @@ -271,14 +323,23 @@ ex clifford::simplify_ncmul(const exvector & v) const while (it != itend) { ex & a = it[0]; ex & b = it[1]; - if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) { + if (is_a(a.op(0)) && is_a(b.op(0))) { const ex & ia = a.op(1); const ex & ib = b.op(1); - if (ia.is_equal(ib)) { + if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha a = lorentz_g(ia, ib); b = dirac_ONE(representation_label); something_changed = true; } + } else if (!is_a(a.op(0)) && !is_a(b.op(0))) { + const ex & ba = a.op(0); + const ex & bb = b.op(0); + if (ba.is_equal(bb)) { // a\ a\ -> a^2 + varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to(a.op(1)).get_dim()); + a = indexed(ba, ix) * indexed(bb, ix.toggle_variance()); + b = dirac_ONE(representation_label); + something_changed = true; + } } ++it; } @@ -313,7 +374,7 @@ ex dirac_ONE(unsigned char rl) ex dirac_gamma(const ex & mu, unsigned char rl) { - if (!is_ex_of_type(mu, varidx)) + if (!is_a(mu)) throw(std::invalid_argument("index of Dirac gamma must be of type varidx")); return clifford(diracgamma(), mu, rl); @@ -336,8 +397,10 @@ ex dirac_gamma7(unsigned char rl) ex dirac_slash(const ex & e, const ex & dim, unsigned char rl) { - varidx mu((new symbol)->setflag(status_flags::dynallocated), dim); - return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl); + // Slashed vectors are actually stored as a clifford object with the + // vector as its base expression and a (dummy) index that just serves + // for storing the space dimensionality + return clifford(e, varidx(0, dim), rl); } /** Check whether a given tinfo key (as returned by return_type_tinfo() @@ -392,10 +455,10 @@ static ex trace_string(exvector::const_iterator ix, unsigned num) ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) { - if (is_ex_of_type(e, clifford)) { + if (is_a(e)) { if (ex_to(e).get_representation_label() == rl - && is_ex_of_type(e.op(0), diracone)) + && is_a(e.op(0))) return trONE; else return _ex0(); @@ -420,11 +483,11 @@ ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) // Expand product, if necessary ex e_expanded = e.expand(); - if (!is_ex_of_type(e_expanded, ncmul)) + if (!is_a(e_expanded)) return dirac_trace(e_expanded, rl, trONE); // gamma5 gets moved to the front so this check is enough - bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5); + bool has_gamma5 = is_a(e.op(0).op(0)); unsigned num = e.nops(); if (has_gamma5) { @@ -435,15 +498,20 @@ ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) return _ex0(); // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma) - if (num == 5) - return trONE * I * eps0123(e.op(1).op(1), e.op(2).op(1), e.op(3).op(1), e.op(4).op(1)); + if (num == 5) { + ex b1, i1, b2, i2, b3, i3, b4, i4; + base_and_index(e.op(1), b1, i1); + base_and_index(e.op(2), b2, i2); + base_and_index(e.op(3), b3, i3); + base_and_index(e.op(4), b4, i4); + return trONE * I * (eps0123(i1, i2, i3, i4) * b1 * b2 * b3 * b4).simplify_indexed(); + } // Tr gamma5 S_2k = // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k - exvector ix; - ix.reserve(num - 1); + exvector ix(num-1), bv(num-1); for (unsigned i=1; i 0) { @@ -518,11 +589,11 @@ ex canonicalize_clifford(const ex & e) // Expand product, if necessary ex rhs_expanded = rhs.expand(); - if (!is_ex_of_type(rhs_expanded, ncmul)) { + if (!is_a(rhs_expanded)) { srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded)); continue; - } else if (!is_ex_of_type(rhs.op(0), clifford)) + } else if (!is_a(rhs.op(0))) continue; exvector v; @@ -532,12 +603,15 @@ ex canonicalize_clifford(const ex & e) // Stupid recursive bubble sort because we only want to swap adjacent gammas exvector::iterator it = v.begin(), next_to_last = v.end() - 1; - if (is_ex_of_type(it->op(0), diracgamma5)) + if (is_a(it->op(0))) ++it; while (it != next_to_last) { - if (it[0].op(1).compare(it[1].op(1)) > 0) { + if (it[0].compare(it[1]) > 0) { ex save0 = it[0], save1 = it[1]; - it[0] = lorentz_g(it[0].op(1), it[1].op(1)); + ex b1, i1, b2, i2; + base_and_index(it[0], b1, i1); + base_and_index(it[1], b2, i2); + it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed(); it[1] = _ex2(); ex sum = ncmul(v); it[0] = save1; @@ -551,7 +625,7 @@ ex canonicalize_clifford(const ex & e) next_sym: ; } } - return aux.subs(srl); + return aux.subs(srl).simplify_indexed(); } } // namespace GiNaC diff --git a/ginac/clifford.h b/ginac/clifford.h index 0c8573ae..50a0c70c 100644 --- a/ginac/clifford.h +++ b/ginac/clifford.h @@ -48,6 +48,9 @@ public: clifford(unsigned char rl, exvector * vp); // vp will be deleted // functions overriding virtual functions from base classes +public: + void print(const print_context & c, unsigned level = 0) const; + protected: bool match_same_type(const basic & other) const; ex simplify_ncmul(const exvector & v) const; diff --git a/ginac/indexed.cpp b/ginac/indexed.cpp index 56fd0b6b..bc6aa838 100644 --- a/ginac/indexed.cpp +++ b/ginac/indexed.cpp @@ -668,29 +668,6 @@ try_again: } } - // Contraction of symmetric with antisymmetric object is zero - if (num_dummies > 1 - && ex_to(ex_to(*it1).symtree).has_symmetry() - && ex_to(ex_to(*it2).symtree).has_symmetry()) { - - // Check all pairs of dummy indices - for (unsigned idx1=0; idx1subs(subs_lst, repl_lst); - ex swapped2 = it2->subs(subs_lst, repl_lst); - if (it1->is_equal(swapped1) && it2->is_equal(-swapped2) - || it1->is_equal(-swapped1) && it2->is_equal(swapped2)) { - free_indices.clear(); - return _ex0(); - } - } - } - } - // Try to contract the first one with the second one contracted = it1->op(0).bp->contract_with(it1, it2, v); if (!contracted) { @@ -749,6 +726,19 @@ contraction_done: else r = e; + // The result should be symmetric with respect to exchange of dummy + // indices, so if the symmetrization vanishes, the whole expression is + // zero. This detects things like eps.i.j.k * p.j * p.k = 0. + if (local_dummy_indices.size() >= 2) { + lst dummy_syms; + for (int i=0; ireturn_type() == return_types::commutative) { + +#if 0 + // This handles eps.i.j.k * p.j * p.k = 0 + // Maybe something like this should go to simplify_indexed() because + // such relations are true for any antisymmetric tensors... + exvector c; + + // Handle all indices of the epsilon tensor + for (int i=0; iop(i+1); + + // Look whether there's a contraction with this index + exvector::const_iterator ait, aitend = v.end(); + for (ait = v.begin(); ait != aitend; ait++) { + if (ait == self) + continue; + if (is_a(*ait) && ait->return_type() == return_types::commutative && ex_to(*ait).has_dummy_index_for(idx) && ait->nops() == 2) { + + // Yes, did we already have another contraction with the same base expression? + ex base = ait->op(0); + if (std::find_if(c.begin(), c.end(), bind2nd(ex_is_equal(), base)) == c.end()) { + + // No, add the base expression to the list + c.push_back(base); + + } else { + + // Yes, the contraction is zero + *self = _ex0(); + *other = _ex0(); + return true; + } + } + } + } +#endif } return false; -- 2.44.0