/** @file clifford.cpp
*
- * Implementation of GiNaC's clifford objects.
- * No real implementation yet, to be done. */
+ * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
-#include <string>
-
#include "clifford.h"
#include "ex.h"
+#include "idx.h"
#include "ncmul.h"
-#include "utils.h"
+#include "symbol.h"
+#include "numeric.h" // for I
+#include "print.h"
+#include "archive.h"
#include "debugmsg.h"
+#include "utils.h"
+
+#include <stdexcept>
-#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
-#endif // ndef NO_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
//////////
-// public
-
-clifford::clifford()
+clifford::clifford() : representation_label(0)
{
- debugmsg("clifford default constructor",LOGLEVEL_CONSTRUCT);
- serial=next_serial++;
- name=autoname_prefix()+ToString(serial);
- tinfo_key=TINFO_clifford;
+ debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
+ tinfo_key = TINFO_clifford;
}
-clifford::~clifford()
+void clifford::copy(const clifford & other)
{
- debugmsg("clifford destructor",LOGLEVEL_DESTRUCT);
- destroy(false);
+ inherited::copy(other);
+ representation_label = other.representation_label;
}
-clifford::clifford(const clifford & other)
+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 copy constructor",LOGLEVEL_CONSTRUCT);
- copy (other);
+ debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
+ tinfo_key = TINFO_clifford;
}
-const clifford & clifford::operator=(const clifford & other)
+/** 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, unsigned char rl) : inherited(b, mu), representation_label(rl)
{
- debugmsg("clifford operator=",LOGLEVEL_ASSIGNMENT);
- if (this != &other) {
- destroy(true);
- copy(other);
- }
- return *this;
+ debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
+ GINAC_ASSERT(is_ex_of_type(mu, varidx));
+ tinfo_key = TINFO_clifford;
}
-// protected
-
-void clifford::copy(const clifford & other)
+clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
{
- indexed::copy(other);
- name=other.name;
- serial=other.serial;
+ debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
+ tinfo_key = TINFO_clifford;
}
-void clifford::destroy(bool call_parent)
+clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
{
- if (call_parent) {
- indexed::destroy(call_parent);
- }
+ debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
+ tinfo_key = TINFO_clifford;
}
//////////
-// other constructors
+// archiving
//////////
-// public
+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;
+}
-clifford::clifford(const std::string & initname)
+void clifford::archive(archive_node &n) const
{
- debugmsg("clifford constructor from string",LOGLEVEL_CONSTRUCT);
- name=initname;
- serial=next_serial++;
- tinfo_key=TINFO_clifford;
+ 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
//////////
-// public
-
-basic * clifford::duplicate() const
+int clifford::compare_same_type(const basic & other) const
{
- debugmsg("clifford duplicate",LOGLEVEL_DUPLICATE);
- return new clifford(*this);
-}
+ GINAC_ASSERT(other.tinfo() == TINFO_clifford);
+ const clifford &o = static_cast<const clifford &>(other);
-void clifford::printraw(std::ostream & os) const
-{
- debugmsg("clifford printraw",LOGLEVEL_PRINT);
- os << "clifford(" << "name=" << name << ",serial=" << serial
- << ",indices=";
- printrawindices(os);
- os << ",hash=" << hashvalue << ",flags=" << flags << ")";
-}
+ if (representation_label != o.representation_label) {
+ // different representation label
+ return representation_label < o.representation_label ? -1 : 1;
+ }
-void clifford::printtree(std::ostream & os, unsigned indent) const
-{
- debugmsg("clifford printtree",LOGLEVEL_PRINT);
- os << std::string(indent,' ') << name << " (clifford): "
- << "serial=" << serial << ","
- << seq.size() << "indices=";
- printtreeindices(os, indent);
- os << ", hash=" << hashvalue
- << " (0x" << std::hex << hashvalue << std::dec << ")"
- << ", flags=" << flags << std::endl;
+ return inherited::compare_same_type(other);
}
-void clifford::print(std::ostream & os, unsigned upper_precedence) const
-{
- debugmsg("clifford print",LOGLEVEL_PRINT);
- os << name;
- printindices(os);
-}
+DEFAULT_COMPARE(diracone)
+DEFAULT_COMPARE(diracgamma)
+DEFAULT_COMPARE(diracgamma5)
+
+DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
+DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
-void clifford::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
+/** Contraction of a gamma matrix with something else. */
+bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
- debugmsg("clifford print csrc",LOGLEVEL_PRINT);
- print(os,upper_precedence);
+ 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, clifford)) {
+
+ ex dim = ex_to_idx(self->op(1)).get_dim();
+
+ // gamma~mu gamma.mu = dim ONE
+ if (other - self == 1) {
+ *self = dim;
+ *other = dirac_ONE(rl);
+ return true;
+
+ // 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
+ } 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];
+ self[1] = _ex1();
+ self[2] = _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))
+ 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;
+ }
+ }
+
+ return false;
}
-bool clifford::info(unsigned inf) const
+/** Perform automatic simplification on noncommutative product of clifford
+ * 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
{
- return indexed::info(inf);
-}
+ 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++;
+ }
-// protected
+ 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--;
+ }
+ }
-int clifford::compare_same_type(const basic & other) const
-{
- GINAC_ASSERT(other.tinfo() == TINFO_clifford);
- const clifford *o = static_cast<const clifford *>(&other);
- if (serial==o->serial) {
- return indexed::compare_same_type(other);
+ // 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(rl);
+ something_changed = true;
+ }
+ }
+ it++;
+ }
}
- return serial < o->serial ? -1 : 1;
+
+ if (s.size() == 0)
+ return clifford(diracone(), rl) * sign;
+ if (something_changed)
+ return nonsimplified_ncmul(s) * sign;
+ else
+ return simplified_ncmul(s) * sign;
}
-ex clifford::simplify_ncmul(const exvector & v) const
+ex clifford::thisexprseq(const exvector & v) const
{
- return simplified_ncmul(v);
+ return clifford(representation_label, v);
}
-unsigned clifford::calchash(void) const
+ex clifford::thisexprseq(exvector * vp) const
{
- hashvalue=golden_ratio_hash(golden_ratio_hash(0x55555556U ^
- golden_ratio_hash(tinfo_key) ^
- serial));
- setflag(status_flags::hash_calculated);
- return hashvalue;
+ return clifford(representation_label, vp);
}
//////////
-// virtual functions which can be overridden by derived classes
+// global functions
//////////
-// none
-
-//////////
-// non-virtual functions in this class
-//////////
+ex dirac_ONE(unsigned char rl)
+{
+ return clifford(diracone(), rl);
+}
-void clifford::setname(const std::string & n)
+ex dirac_gamma(const ex & mu, unsigned char rl)
{
- name = n;
+ if (!is_ex_of_type(mu, varidx))
+ throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
+
+ return clifford(diracgamma(), mu, rl);
}
-// private
+ex dirac_gamma5(unsigned char rl)
+{
+ return clifford(diracgamma5(), rl);
+}
-std::string & clifford::autoname_prefix(void)
+ex dirac_gamma6(unsigned char rl)
{
- static std::string * s = new std::string("clifford");
- return *s;
+ return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
}
-//////////
-// static member variables
-//////////
+ex dirac_gamma7(unsigned char rl)
+{
+ return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
+}
-// private
+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);
+}
-unsigned clifford::next_serial=0;
+/** 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);
+}
-//////////
-// global constants
-//////////
+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 S_2k =
+ // I/4! * 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, trONE)).simplify_indexed() * I / 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 trONE * 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 =
+ // + 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;
+ }
+ }
-const clifford some_clifford;
-const type_info & typeid_clifford=typeid(some_clifford);
+ return _ex0();
+}
-#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff