#include "ex.h"
#include "idx.h"
#include "ncmul.h"
+#include "symbol.h"
+#include "numeric.h" // for I
+#include "lst.h"
+#include "relational.h"
+#include "print.h"
#include "archive.h"
#include "debugmsg.h"
#include "utils.h"
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;
}
// 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
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_PRINT(diracone, "ONE")
-DEFAULT_PRINT(diracgamma, "gamma")
+DEFAULT_COMPARE(diracgamma5)
+
+DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
+DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
/** 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)) {
+ if (is_ex_of_type(*other, clifford)) {
ex dim = ex_to_idx(self->op(1)).get_dim();
- // gamma~mu*gamma.mu = dim*ONE
+ // gamma~mu gamma.mu = dim ONE
if (other - self == 1) {
*self = dim;
- *other = dirac_one();
+ *other = dirac_ONE(rl);
return true;
- // gamma~mu*gamma~alpha*gamma.mu = (2-dim)*gamma~alpha
+ // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
} 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
+ // 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() + (dim - 4) * self[1] * self[2];
+ *self = 4 * lorentz_g(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();
+ // 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))
+ return false;
+ it++;
+ }
+
+ it = self + 1;
+ ex S = _ex1();
+ while (it != next_to_last) {
+ S *= *it;
+ *it++ = _ex1();
+ }
+
+ *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
+ *next_to_last = _ex1();
*other = _ex1();
return true;
}
}
/** 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());
- 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 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;
+ }
+
+ // 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(representation_label);
+ something_changed = true;
+ }
+ }
+ it++;
+ }
}
if (s.size() == 0)
- return clifford(diracone());
- else if (s.size() == v.size())
- return simplified_ncmul(v);
+ return clifford(diracone(), representation_label) * 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_gamma6(unsigned char rl)
+{
+ return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
+}
+
+ex dirac_gamma7(unsigned char rl)
+{
+ return clifford(diracone(), rl) - 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);
+}
+
+/** Check whether a given tinfo key (as returned by return_type_tinfo()
+ * is that of a clifford object (with an arbitrary representation label). */
+static bool is_clifford_tinfo(unsigned ti)
+{
+ return (ti & ~0xff) == TINFO_clifford;
+}
+
+ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
+{
+ 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 trONE;
+ 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, trONE);
+ 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, trONE);
+ 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, 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);
+ 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 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));
+
+ // Tr gamma5 gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 gamma.mu5 gamma.mu6 = ...
+ if (num == 7) {
+ ex i1 = e.op(1).op(1), i2 = e.op(2).op(1),
+ i3 = e.op(3).op(1), i4 = e.op(4).op(1),
+ i5 = e.op(5).op(1), i6 = e.op(6).op(1);
+ return trONE * I * (lorentz_g(i1, i2) * eps0123(i3, i4, i5, i6)
+ - lorentz_g(i1, i3) * eps0123(i2, i4, i5, i6)
+ + lorentz_g(i1, i4) * eps0123(i2, i3, i5, i6)
+ - lorentz_g(i1, i5) * eps0123(i2, i3, i4, i6)
+ + lorentz_g(i1, i6) * eps0123(i2, i3, i4, i5)
+ + lorentz_g(i2, i3) * eps0123(i1, i4, i5, i6)
+ - lorentz_g(i2, i4) * eps0123(i1, i3, i5, i6)
+ + lorentz_g(i2, i5) * eps0123(i1, i3, i4, i6)
+ - lorentz_g(i2, i6) * eps0123(i1, i3, i4, i5)
+ + lorentz_g(i3, i4) * eps0123(i1, i2, i5, i6)
+ - lorentz_g(i3, i5) * eps0123(i1, i2, i4, i6)
+ + lorentz_g(i3, i6) * eps0123(i1, i2, i4, i5)
+ + lorentz_g(i4, i5) * eps0123(i1, i2, i3, i6)
+ - lorentz_g(i4, i6) * eps0123(i1, i2, i3, i5)
+ + lorentz_g(i5, i6) * eps0123(i1, i2, i3, i4));
+ }
+
+ // Tr gamma5 S_2k =
+ // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
+ ex result;
+ for (int i=1; i<num-3; i++) {
+ ex idx1 = e.op(i).op(1);
+ for (int j=i+1; j<num-2; j++) {
+ ex idx2 = e.op(j).op(1);
+ for (int k=j+1; k<num-1; k++) {
+ ex idx3 = e.op(k).op(1);
+ for (int l=k+1; l<num; l++) {
+ ex idx4 = e.op(l).op(1);
+ vector<int> iv;
+ iv.reserve(num-1);
+ exvector v;
+ v.reserve(num-1);
+ iv.push_back(i); iv.push_back(j); iv.push_back(k); iv.push_back(l);
+ for (int n=1; n<num; n++) {
+ if (n == i || n == j || n == k || n == l)
+ continue;
+ iv.push_back(n);
+ v.push_back(e.op(n));
+ }
+ int sign = permutation_sign(iv.begin(), iv.end());
+ result += sign * eps0123(idx1, idx2, idx3, idx4)
+ * dirac_trace(ncmul(v, true), rl, trONE);
+ }
+ }
+ }
+ }
+ return result * I;
+
+ } 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 trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
+
+ // 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
+ if (num == 4)
+ return trONE * (lorentz_g(e.op(0).op(1), e.op(1).op(1)) * lorentz_g(e.op(2).op(1), e.op(3).op(1))
+ + lorentz_g(e.op(1).op(1), e.op(2).op(1)) * lorentz_g(e.op(0).op(1), e.op(3).op(1))
+ - lorentz_g(e.op(0).op(1), e.op(2).op(1)) * lorentz_g(e.op(1).op(1), e.op(3).op(1)));
+
+ // Traces of 6 or more gammas are computed recursively:
+ // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
+ // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
+ // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
+ // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
+ // - ...
+ // + g.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, trONE);
+ sign = -sign;
+ }
+ return result;
+ }
+ }
+
+ return _ex0();
+}
+
+ex canonicalize_clifford(const ex & e)
+{
+ // Scan for any ncmul objects
+ lst srl;
+ ex aux = e.to_rational(srl);
+ for (unsigned i=0; i<srl.nops(); i++) {
+
+ ex lhs = srl.op(i).lhs();
+ ex rhs = srl.op(i).rhs();
+
+ if (is_ex_exactly_of_type(rhs, ncmul)
+ && rhs.return_type() == return_types::noncommutative
+ && is_clifford_tinfo(rhs.return_type_tinfo())) {
+
+ // Expand product, if necessary
+ ex rhs_expanded = rhs.expand();
+ if (!is_ex_of_type(rhs_expanded, ncmul)) {
+ srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
+ continue;
+
+ } else if (!is_ex_of_type(rhs.op(0), clifford))
+ continue;
+
+ exvector v;
+ v.reserve(rhs.nops());
+ for (unsigned j=0; j<rhs.nops(); j++)
+ v.push_back(rhs.op(j));
+
+ // Stupid 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))
+ it++;
+ while (it != next_to_last) {
+ if (it[0].op(1).compare(it[1].op(1)) > 0) {
+ ex save0 = it[0], save1 = it[1];
+ it[0] = lorentz_g(it[0].op(1), it[1].op(1));
+ it[1] = _ex2();
+ ex sum = ncmul(v);
+ it[0] = save1;
+ it[1] = save0;
+ sum -= ncmul(v, true);
+ srl.let_op(i) = (lhs == canonicalize_clifford(sum));
+ goto next_sym;
+ }
+ it++;
+ }
+next_sym: ;
+ }
+ }
+ return aux.subs(srl);
}
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff