- dirac_trace() handles arbitrary expressions (even unexpanded), including
[ginac.git] / ginac / clifford.cpp
index 94f434a7710da07db1625332d7fae96b8968c095..8a8bfe09294e942e88023d0770784f09d5d9f289 100644 (file)
@@ -24,6 +24,8 @@
 #include "ex.h"
 #include "idx.h"
 #include "ncmul.h"
+#include "symbol.h"
+#include "print.h"
 #include "archive.h"
 #include "debugmsg.h"
 #include "utils.h"
@@ -35,54 +37,61 @@ namespace GiNaC {
 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
 
 //////////
 // default constructor, destructor, copy constructor assignment operator and helpers
 //////////
 
-clifford::clifford()
+clifford::clifford() : representation_label(0)
 {
        debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_clifford;
 }
 
-DEFAULT_COPY(clifford)
+void clifford::copy(const clifford & other)
+{
+       inherited::copy(other);
+       representation_label = other.representation_label;
+}
+
 DEFAULT_DESTROY(clifford)
 DEFAULT_CTORS(diracone)
 DEFAULT_CTORS(diracgamma)
+DEFAULT_CTORS(diracgamma5)
 
 //////////
 // other constructors
 //////////
 
+/** Construct object without any indices. This constructor is for internal
+ *  use only. Use the dirac_ONE() function instead.
+ *  @see dirac_ONE */
+clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
+{
+       debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
+       tinfo_key = TINFO_clifford;
+}
+
 /** Construct object with one Lorentz index. This constructor is for internal
  *  use only. Use the dirac_gamma() function instead.
  *  @see dirac_gamma */
-clifford::clifford(const ex & b, const ex & mu) : inherited(b, mu)
+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));
        tinfo_key = TINFO_clifford;
 }
 
-/** Construct object without any indices. This constructor is for internal
- *  use only. Use the dirac_one() function instead.
- *  @see dirac_one */
-clifford::clifford(const ex & b) : inherited(b)
-{
-       debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
-       tinfo_key = TINFO_clifford;
-}
-
-clifford::clifford(const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable)
+clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
 {
-       debugmsg("clifford constructor from exvector", LOGLEVEL_CONSTRUCT);
+       debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_clifford;
 }
 
-clifford::clifford(exvector * vp) : inherited(indexed::unknown, vp)
+clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
 {
-       debugmsg("clifford constructor from exvector *", LOGLEVEL_CONSTRUCT);
+       debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_clifford;
 }
 
@@ -90,9 +99,24 @@ clifford::clifford(exvector * vp) : inherited(indexed::unknown, vp)
 // archiving
 //////////
 
-DEFAULT_ARCHIVING(clifford)
+clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+{
+       debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
+       unsigned rl;
+       n.find_unsigned("label", rl);
+       representation_label = rl;
+}
+
+void clifford::archive(archive_node &n) const
+{
+       inherited::archive(n);
+       n.add_unsigned("label", representation_label);
+}
+
+DEFAULT_UNARCHIVE(clifford)
 DEFAULT_ARCHIVING(diracone)
 DEFAULT_ARCHIVING(diracgamma)
+DEFAULT_ARCHIVING(diracgamma5)
 
 //////////
 // functions overriding virtual functions from bases classes
@@ -100,32 +124,69 @@ DEFAULT_ARCHIVING(diracgamma)
 
 int clifford::compare_same_type(const basic & other) const
 {
+       GINAC_ASSERT(other.tinfo() == TINFO_clifford);
+       const clifford &o = static_cast<const clifford &>(other);
+
+       if (representation_label != o.representation_label) {
+               // different representation label
+               return representation_label < o.representation_label ? -1 : 1;
+       }
+
        return inherited::compare_same_type(other);
 }
 
 DEFAULT_COMPARE(diracone)
 DEFAULT_COMPARE(diracgamma)
+DEFAULT_COMPARE(diracgamma5)
+
 DEFAULT_PRINT(diracone, "ONE")
 DEFAULT_PRINT(diracgamma, "gamma")
+DEFAULT_PRINT(diracgamma5, "gamma5")
 
 /** 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, indexed));
+       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));
+       unsigned char rl = ex_to_clifford(*self).get_representation_label();
 
        if (is_ex_of_type(other->op(0), diracgamma)) {
 
+               ex dim = ex_to_idx(self->op(1)).get_dim();
+
                // gamma~mu*gamma.mu = dim*ONE
                if (other - self == 1) {
-                       *self = ex_to_idx(self->op(1)).get_dim();
-                       *other = dirac_one();
+                       *self = dim;
+                       *other = dirac_ONE(rl);
                        return true;
 
                // gamma~mu*gamma~alpha*gamma.mu = (2-dim)*gamma~alpha
-               } else if (other - self == 2) {
-                       *self = 2 - ex_to_idx(self->op(1)).get_dim();
+               } else if (other - self == 2
+                       && is_ex_of_type(self[1], clifford)) {
+                       *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 * metric_tensor(self[1].op(1), self[2].op(1)) * 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+(4-dim)*gamma~alpha*gamma~beta*gamma~delta
+               } else if (other - self == 4
+                       && is_ex_of_type(self[1], clifford)
+                       && is_ex_of_type(self[2], clifford)
+                       && is_ex_of_type(self[3], clifford)) {
+                       *self = -2 * self[3] * self[2] * self[1] + (4 - dim) * self[1] * self[2] * self[3];
+                       self[1] = _ex1();
+                       self[2] = _ex1();
+                       self[3] = _ex1();
                        *other = _ex1();
                        return true;
                }
@@ -135,52 +196,232 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other
 }
 
 /** Perform automatic simplification on noncommutative product of clifford
- *  objects. This removes superfluous ONEs. */
+ *  objects. This removes superfluous ONEs, permutes gamma5's to the front
+ *  and removes squares of gamma objects. */
 ex clifford::simplify_ncmul(const exvector & v) const
 {
        exvector s;
        s.reserve(v.size());
+       unsigned rl = ex_to_clifford(v[0]).get_representation_label();
+
+       // Remove superfluous ONEs
+       exvector::const_iterator cit = v.begin(), citend = v.end();
+       while (cit != citend) {
+               if (!is_ex_of_type(cit->op(0), diracone))
+                       s.push_back(*cit);
+               cit++;
+       }
+
+       bool something_changed = false;
+       int sign = 1;
+
+       // Anticommute gamma5's to the front
+       if (s.size() >= 2) {
+               exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
+               while (true) {
+                       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)) {
+                                       it->swap(*it2);
+                                       sign = -sign;
+                                       something_changed = true;
+                               }
+                               if (it == first)
+                                       break;
+                               it--;
+                       }
+                       if (next_to_last == first)
+                               break;
+                       next_to_last--;
+               }
+       }
+
+       // 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)) {
+               s.erase(s.begin(), s.begin() + 2);
+               something_changed = true;
+       }
 
-       exvector::const_iterator it = v.begin(), itend = v.end();
-       while (it != itend) {
-               if (!is_ex_of_type(it->op(0), diracone))
-                       s.push_back(*it);
-               it++;
+       // Remove equal adjacent gammas
+       if (s.size() >= 2) {
+               exvector::iterator it = s.begin(), itend = s.end() - 1;
+               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)) {
+                               const ex & ia = a.op(1);
+                               const ex & ib = b.op(1);
+                               if (ia.is_equal(ib)) {
+                                       a = lorentz_g(ia, ib);
+                                       b = dirac_ONE(rl);
+                                       something_changed = true;
+                               }
+                       }
+                       it++;
+               }
        }
 
        if (s.size() == 0)
-               return clifford(diracone());
-       else if (s.size() == v.size())
-               return simplified_ncmul(v);
+               return clifford(diracone(), rl) * sign;
+       if (something_changed)
+               return nonsimplified_ncmul(s) * sign;
        else
-               return simplified_ncmul(s);
+               return simplified_ncmul(s) * sign;
 }
 
 ex clifford::thisexprseq(const exvector & v) const
 {
-       return clifford(v);
+       return clifford(representation_label, v);
 }
 
 ex clifford::thisexprseq(exvector * vp) const
 {
-       return clifford(vp);
+       return clifford(representation_label, vp);
 }
 
 //////////
 // global functions
 //////////
 
-ex dirac_one(void)
+ex dirac_ONE(unsigned char rl)
 {
-       return clifford(diracone());
+       return clifford(diracone(), rl);
 }
 
-ex dirac_gamma(const ex & mu)
+ex dirac_gamma(const ex & mu, unsigned char rl)
 {
        if (!is_ex_of_type(mu, varidx))
                throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 
-       return clifford(diracgamma(), mu);
+       return clifford(diracgamma(), mu, rl);
+}
+
+ex dirac_gamma5(unsigned char rl)
+{
+       return clifford(diracgamma5(), 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);
+}
+
+/** Check whether a given tinfo key (as returned by return_type_tinfo()
+ *  is that of a clifford object with the specified representation label. */
+static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
+{
+       return ti == (TINFO_clifford + rl);
+}
+
+ex dirac_trace(const ex & e, unsigned char rl)
+{
+       if (is_ex_of_type(e, clifford)) {
+
+               if (ex_to_clifford(e).get_representation_label() == rl
+                && is_ex_of_type(e.op(0), diracone))
+                       return _ex4();
+               else
+                       return _ex0();
+
+       } else if (is_ex_exactly_of_type(e, add)) {
+
+               // Trace of sum = sum of traces
+               ex sum = _ex0();
+               for (unsigned i=0; i<e.nops(); i++)
+                       sum += dirac_trace(e.op(i), rl);
+               return sum;
+
+       } else if (is_ex_exactly_of_type(e, mul)) {
+
+               // Trace of product: pull out non-clifford factors
+               ex prod = _ex1();
+               for (unsigned i=0; i<e.nops(); i++) {
+                       const ex &o = e.op(i);
+                       unsigned ti = o.return_type_tinfo();
+                       if (is_clifford_tinfo(o.return_type_tinfo(), rl))
+                               prod *= dirac_trace(o, rl);
+                       else
+                               prod *= o;
+               }
+               return prod;
+
+       } else if (is_ex_exactly_of_type(e, ncmul)) {
+
+               if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
+                       return _ex0();
+
+               // Expand product, if necessary
+               ex e_expanded = e.expand();
+               if (!is_ex_of_type(e_expanded, ncmul))
+                       return dirac_trace(e_expanded, rl);
+
+               // gamma5 gets moved to the front so this check is enough
+               bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
+               unsigned num = e.nops();
+
+               if (has_gamma5) {
+
+                       // Trace of gamma5 * odd number of gammas and trace of
+                       // gamma5 * gamma_mu * gamma_nu are zero
+                       if ((num & 1) == 0 || num == 3)
+                               return _ex0();
+
+                       // Tr gamma5 S_2k =
+                       //   epsilon0123_mu1_mu2_mu3_mu4 * Tr gamma_mu1 gamma_mu2 gamma_mu3 gamma_mu4 S_2k
+                       ex dim = ex_to_idx(e.op(1).op(1)).get_dim();
+                       varidx mu1((new symbol)->setflag(status_flags::dynallocated), dim),
+                              mu2((new symbol)->setflag(status_flags::dynallocated), dim),
+                              mu3((new symbol)->setflag(status_flags::dynallocated), dim),
+                              mu4((new symbol)->setflag(status_flags::dynallocated), dim);
+                       exvector v;
+                       v.reserve(num + 3);
+                       v.push_back(dirac_gamma(mu1, rl));
+                       v.push_back(dirac_gamma(mu2, rl));
+                       v.push_back(dirac_gamma(mu3, rl));
+                       v.push_back(dirac_gamma(mu4, rl));
+                       for (int i=1; i<num; i++)
+                               v.push_back(e.op(i));
+
+                       return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
+                               dirac_trace(ncmul(v), rl)).simplify_indexed() / 24;
+
+               } else { // no gamma5
+
+                       // Trace of odd number of gammas is zero
+                       if ((num & 1) == 1)
+                               return _ex0();
+
+                       // Tr gamma_mu gamma_nu = 4 g_mu_nu
+                       if (num == 2)
+                               return 4 * lorentz_g(e.op(0).op(1), e.op(1).op(1));
+
+                       // Traces of 4 or more gammas are computed recursively:
+                       // Tr gamma_mu1 gamma_mu2 ... gamma_mun =
+                       //   + eta_mu1_mu2 * Tr gamma_mu3 ... gamma_mun
+                       //   - eta_mu1_mu3 * Tr gamma_mu2 gamma_mu4 ... gamma_mun
+                       //   + eta_mu1_mu4 * Tr gamma_mu3 gamma_mu3 gamma_mu5 ... gamma_mun
+                       //   - ...
+                       //   + eta_mu1_mun * Tr gamma_mu2 ... gamma_mu(n-1)
+                       exvector v(num - 2);
+                       int sign = 1;
+                       const ex &ix1 = e.op(0).op(1);
+                       ex result;
+                       for (int i=1; i<num; i++) {
+                               for (int n=1, j=0; n<num; n++) {
+                                       if (n == i)
+                                               continue;
+                                       v[j++] = e.op(n);
+                               }
+                               result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl);
+                               sign = -sign;
+                       }
+                       return result;
+               }
+       }
+
+       return _ex0();
 }
 
 } // namespace GiNaC