- epsilon*epsilon contractions work
authorChristian Bauer <Christian.Bauer@uni-mainz.de>
Thu, 26 Jul 2001 23:28:08 +0000 (23:28 +0000)
committerChristian Bauer <Christian.Bauer@uni-mainz.de>
Thu, 26 Jul 2001 23:28:08 +0000 (23:28 +0000)
- 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
ginac/clifford.h
ginac/indexed.cpp
ginac/tensor.cpp

index 8d8c2c6..8687f96 100644 (file)
@@ -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<varidx>(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<diracgamma5>(seq[0]) && !is_a<diracgamma>(seq[0]) && !is_a<diracone>(seq[0])) {
+
+               // dirac_slash() object is printed differently
+               if (is_a<print_tree>(c))
+                       inherited::print(c, level);
+               else if (is_a<print_latex>(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<clifford>(c));
+       GINAC_ASSERT(c.nops() == 2);
+
+       if (is_a<diracgamma>(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<idx>(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<clifford>(*self));
+       GINAC_ASSERT(is_a<indexed>(*other));
+       GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
        unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 
-       if (is_ex_of_type(*other, clifford)) {
+       if (is_a<clifford>(*other)) {
 
                ex dim = ex_to<idx>(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<clifford>(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<clifford>(self[1])
+                       && is_a<clifford>(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<clifford>(self[1])
+                       && is_a<clifford>(self[2])
+                       && is_a<clifford>(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<clifford>(*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<diracone>(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<diracgamma5>(it->op(0)) && is_a<diracgamma5>(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<diracgamma5>(s[0].op(0)) && is_a<diracgamma5>(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<diracgamma>(a.op(0)) && is_a<diracgamma>(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<diracgamma>(a.op(0)) && !is_a<diracgamma>(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<idx>(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<varidx>(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<clifford>(e)) {
 
                if (ex_to<clifford>(e).get_representation_label() == rl
-                && is_ex_of_type(e.op(0), diracone))
+                && is_a<diracone>(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<ncmul>(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<diracgamma5>(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<num; i++)
-                               ix.push_back(e.op(i).op(1));
+                               base_and_index(e.op(i), bv[i-1], ix[i-1]);
                        num--;
                        int *iv = new int[num];
                        ex result;
@@ -472,7 +540,7 @@ ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
                                }
                        }
                        delete[] iv;
-                       return trONE * I * result;
+                       return trONE * I * result * mul(bv);
 
                } else { // no gamma5
 
@@ -481,15 +549,18 @@ ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
                                return _ex0();
 
                        // Tr gamma.mu gamma.nu = 4 g.mu.nu
-                       if (num == 2)
-                               return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
+                       if (num == 2) {
+                               ex b1, i1, b2, i2;
+                               base_and_index(e.op(0), b1, i1);
+                               base_and_index(e.op(1), b2, i2);
+                               return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
+                       }
 
-                       exvector iv;
-                       iv.reserve(num);
+                       exvector iv(num), bv(num);
                        for (unsigned i=0; i<num; i++)
-                               iv.push_back(e.op(i).op(1));
+                               base_and_index(e.op(i), bv[i], iv[i]);
 
-                       return trONE * trace_string(iv.begin(), num);
+                       return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
                }
 
        } else if (e.nops() > 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<ncmul>(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<clifford>(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<diracgamma5>(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
index 0c8573a..50a0c70 100644 (file)
@@ -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;
index 56fd0b6..bc6aa83 100644 (file)
@@ -668,29 +668,6 @@ try_again:
                                }
                        }
 
-                       // Contraction of symmetric with antisymmetric object is zero
-                       if (num_dummies > 1
-                        && ex_to<symmetry>(ex_to<indexed>(*it1).symtree).has_symmetry()
-                        && ex_to<symmetry>(ex_to<indexed>(*it2).symtree).has_symmetry()) {
-
-                               // Check all pairs of dummy indices
-                               for (unsigned idx1=0; idx1<num_dummies-1; idx1++) {
-                                       for (unsigned idx2=idx1+1; idx2<num_dummies; idx2++) {
-
-                                               // Try and swap the index pair and check whether the
-                                               // relative sign changed
-                                               lst subs_lst(dummy[idx1].op(0), dummy[idx2].op(0)), repl_lst(dummy[idx2].op(0), dummy[idx1].op(0));
-                                               ex swapped1 = it1->subs(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; i<local_dummy_indices.size(); i++)
+                       dummy_syms.append(local_dummy_indices[i].op(0));
+               if (r.symmetrize(dummy_syms).is_zero()) {
+                       free_indices.clear();
+                       return _ex0();
+               }
+       }
+
        // Dummy index renaming
        r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 
index 28a7d7b..d294c72 100644 (file)
@@ -514,6 +514,44 @@ bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator othe
                *self = sign * M.determinant().simplify_indexed();
                *other = _ex1();
                return true;
+
+       } else if (other->return_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; i<num; i++) {
+                       ex idx = self->op(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<indexed>(*ait) && ait->return_type() == return_types::commutative && ex_to<indexed>(*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;