*other = _ex1();
return true;
-#if 0
- // 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;
-#endif
-
// 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)
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 newly constructed object */
ex dirac_gamma5(unsigned char rl = 0);
+/** This returns (dirac_ONE(rl) + dirac_gamma5(rl)). */
+ex dirac_gamma6(unsigned char rl = 0);
+
+/** This returns (dirac_ONE(rl) - dirac_gamma5(rl)). */
+ex dirac_gamma7(unsigned char rl = 0);
+
/** Create a term of the form e_mu * gamma~mu with a unique index mu.
*
* @param dim Dimension of index
* objects. This removes superfluous ONEs. */
ex color::simplify_ncmul(const exvector & v) const
{
- //!! TODO: sort by representation label
exvector s;
s.reserve(v.size());
+ unsigned rl = ex_to_color(v[0]).get_representation_label();
+ // Remove superfluous ONEs
exvector::const_iterator it = v.begin(), itend = v.end();
while (it != itend) {
if (!is_ex_of_type(it->op(0), su3one))
}
if (s.size() == 0)
- return color(su3one());
- else if (s.size() == v.size())
- return simplified_ncmul(v);
+ return color(su3one(), rl);
else
return simplified_ncmul(s);
}
}
+/** Contraction of generator with something else. */
+bool su3t::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(*other, indexed));
+ GINAC_ASSERT(self->nops() == 2);
+ GINAC_ASSERT(is_ex_of_type(self->op(0), su3t));
+ unsigned char rl = ex_to_color(*self).get_representation_label();
+
+ if (is_ex_exactly_of_type(other->op(0), su3t)) {
+
+ // T.a T.a = 4/3 ONE
+ if (other - self == 1) {
+ *self = numeric(4, 3);
+ *other = color_ONE(rl);
+ return true;
+
+ // T.a T.b T.a = -1/6 T.b
+ } else if (other - self == 2
+ && is_ex_of_type(self[1], color)) {
+ *self = numeric(-1, 6);
+ *other = _ex1();
+ return true;
+
+ // T.a S T.a = 1/2 Tr(S) - 1/6 S
+ } else {
+ exvector::iterator it = self + 1;
+ while (it != other) {
+ if (!is_ex_of_type(*it, color)) {
+ return false;
+ }
+ it++;
+ }
+
+ it = self + 1;
+ ex S = _ex1();
+ while (it != other) {
+ S *= *it;
+ *it++ = _ex1();
+ }
+
+ *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
+ *other = _ex1();
+ return true;
+ }
+ }
+
+ return false;
+}
+
/** Contraction of an indexed symmetric structure constant with something else. */
bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
// functions overriding virtual functions from bases classes
public:
void print(const print_context & c, unsigned level = 0) const;
+ bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
};
/** This class represents the tensor of antisymmetric su(3) structure
GINAC_IMPLEMENT_REGISTERED_CLASS(idx, basic)
GINAC_IMPLEMENT_REGISTERED_CLASS(varidx, idx)
+GINAC_IMPLEMENT_REGISTERED_CLASS(spinidx, varidx)
//////////
// default constructor, destructor, copy constructor assignment operator and helpers
tinfo_key = TINFO_varidx;
}
+spinidx::spinidx() : dotted(false)
+{
+ debugmsg("spinidx default constructor", LOGLEVEL_CONSTRUCT);
+ tinfo_key = TINFO_spinidx;
+}
+
void idx::copy(const idx & other)
{
inherited::copy(other);
covariant = other.covariant;
}
+void spinidx::copy(const spinidx & other)
+{
+ inherited::copy(other);
+ dotted = other.dotted;
+}
+
DEFAULT_DESTROY(idx)
DEFAULT_DESTROY(varidx)
+DEFAULT_DESTROY(spinidx)
//////////
// other constructors
tinfo_key = TINFO_varidx;
}
+spinidx::spinidx(const ex & v, const ex & d, bool cov, bool dot) : inherited(v, d, cov), dotted(dot)
+{
+ debugmsg("spinidx constructor from ex,ex,bool,bool", LOGLEVEL_CONSTRUCT);
+ tinfo_key = TINFO_spinidx;
+}
+
//////////
// archiving
//////////
n.find_bool("covariant", covariant);
}
+spinidx::spinidx(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+{
+ debugmsg("spinidx constructor from archive_node", LOGLEVEL_CONSTRUCT);
+ n.find_bool("dotted", dotted);
+}
+
void idx::archive(archive_node &n) const
{
inherited::archive(n);
n.add_bool("covariant", covariant);
}
+void spinidx::archive(archive_node &n) const
+{
+ inherited::archive(n);
+ n.add_bool("dotted", dotted);
+}
+
DEFAULT_UNARCHIVE(idx)
DEFAULT_UNARCHIVE(varidx)
+DEFAULT_UNARCHIVE(spinidx)
//////////
// functions overriding virtual functions from bases classes
}
}
+void spinidx::print(const print_context & c, unsigned level) const
+{
+ debugmsg("spinidx print", LOGLEVEL_PRINT);
+
+ if (is_of_type(c, print_tree)) {
+
+ c.s << std::string(level, ' ') << class_name()
+ << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
+ << (covariant ? ", covariant" : ", contravariant")
+ << (dotted ? ", dotted" : ", undotted")
+ << std::endl;
+ unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
+ value.print(c, level + delta_indent);
+ dim.print(c, level + delta_indent);
+
+ } else {
+
+ bool is_tex = is_of_type(c, print_latex);
+ if (!is_tex) {
+ if (covariant)
+ c.s << ".";
+ else
+ c.s << "~";
+ }
+ if (dotted) {
+ if (is_tex)
+ c.s << "\\dot{";
+ else
+ c.s << "*";
+ }
+ bool need_parens = !(is_ex_exactly_of_type(value, numeric) || is_ex_of_type(value, symbol));
+ if (need_parens)
+ c.s << "(";
+ value.print(c);
+ if (need_parens)
+ c.s << ")";
+ if (is_tex && dotted)
+ c.s << "}";
+ }
+}
+
bool idx::info(unsigned inf) const
{
if (inf == info_flags::idx)
return 0;
}
+int spinidx::compare_same_type(const basic & other) const
+{
+ GINAC_ASSERT(is_of_type(other, spinidx));
+ const spinidx &o = static_cast<const spinidx &>(other);
+
+ int cmpval = inherited::compare_same_type(other);
+ if (cmpval)
+ return cmpval;
+
+ // Check variance and dottedness last so dummy indices will end up next to each other
+ if (covariant != o.covariant)
+ return covariant ? -1 : 1;
+ if (dotted != o.dotted)
+ return dotted ? -1 : 1;
+
+ return 0;
+}
+
ex idx::subs(const lst & ls, const lst & lr) const
{
GINAC_ASSERT(ls.nops() == lr.nops());
return inherited::is_dummy_pair_same_type(other);
}
+bool spinidx::is_dummy_pair_same_type(const basic & other) const
+{
+ const spinidx &o = static_cast<const spinidx &>(other);
+
+ // Dottedness must be the same
+ if (dotted != o.dotted)
+ return false;
+
+ return inherited::is_dummy_pair_same_type(other);
+}
+
+
//////////
// non-virtual functions
//////////
return i_copy->setflag(status_flags::dynallocated);
}
+ex spinidx::toggle_dot(void) const
+{
+ spinidx *i_copy = static_cast<spinidx *>(duplicate());
+ i_copy->dotted = !i_copy->dotted;
+ i_copy->clearflag(status_flags::hash_calculated);
+ return i_copy->setflag(status_flags::dynallocated);
+}
+
+ex spinidx::toggle_variance_dot(void) const
+{
+ spinidx *i_copy = static_cast<spinidx *>(duplicate());
+ i_copy->covariant = !i_copy->covariant;
+ i_copy->dotted = !i_copy->dotted;
+ i_copy->clearflag(status_flags::hash_calculated);
+ return i_copy->setflag(status_flags::dynallocated);
+}
+
//////////
// global functions
//////////
};
+/** This class holds a spinor index that can be dotted or undotted and that
+ * also has a variance. This is used in the Weyl-van-der-Waerden formalism
+ * where the dot indicates complex conjugation. There is an associated
+ * (asymmetric) metric tensor that can be used to raise/lower spinor
+ * indices. */
+class spinidx : public varidx
+{
+ GINAC_DECLARE_REGISTERED_CLASS(spinidx, varidx)
+
+ // other constructors
+public:
+ /** Construct index with given value, dimension, variance and dot.
+ *
+ * @param v Value of index (numeric or symbolic)
+ * @param dim Dimension of index space (numeric or symbolic)
+ * @param covariant Make covariant index (default is contravariant)
+ * @param dotted Make covariant dotted (default is undotted)
+ * @return newly constructed index */
+ spinidx(const ex & v, const ex & dim = 2, bool covariant = false, bool dotted = false);
+
+ // functions overriding virtual functions from bases classes
+public:
+ void print(const print_context & c, unsigned level = 0) const;
+ bool is_dummy_pair_same_type(const basic & other) const;
+
+ // non-virtual functions in this class
+public:
+ /** Check whether the index is dotted. */
+ bool is_dotted(void) const {return dotted;}
+
+ /** Check whether the index is not dotted. */
+ bool is_undotted(void) const {return !dotted;}
+
+ /** Make a new index with the same value and variance but the opposite
+ * dottedness. */
+ ex toggle_dot(void) const;
+
+ /** Make a new index with the same value but opposite variance and
+ * dottedness. */
+ ex toggle_variance_dot(void) const;
+
+ // member variables
+protected:
+ bool dotted;
+};
+
+
// utility functions
inline const idx &ex_to_idx(const ex & e)
{
return static_cast<const varidx &>(*e.bp);
}
+inline const spinidx &ex_to_spinidx(const ex & e)
+{
+ return static_cast<const spinidx &>(*e.bp);
+}
+
/** Check whether two indices form a dummy pair. */
bool is_dummy_pair(const idx & i1, const idx & i2);
}
if (contracted) {
contraction_done:
- if (is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
+ if (non_commutative
+ || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
|| is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
|| is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
// One of the factors became a sum or product:
// re-expand expression and run again
+ // Non-commutative products are always re-expanded to give
+ // simplify_ncmul() the chance to re-order and canonicalize
+ // the product
ex r = (non_commutative ? ex(ncmul(v)) : ex(mul(v)));
return simplify_indexed(r, free_indices, sp);
}
spm[make_key(v1, v2)] = sp;
}
+void scalar_products::add_vectors(const lst & l)
+{
+ // Add all possible pairs of products
+ unsigned num = l.nops();
+ for (unsigned i=0; i<num; i++) {
+ ex a = l.op(i);
+ for (unsigned j=0; j<num; j++) {
+ ex b = l.op(j);
+ add(a, b, a*b);
+ }
+ }
+}
+
void scalar_products::clear(void)
{
spm.clear();
/** Register scalar product pair and its value. */
void add(const ex & v1, const ex & v2, const ex & sp);
+ /** Register list of vectors. This adds all possible pairs of products
+ * a.i * b.i with the value a*b (note that this is not a scalar vector
+ * product but an ordinary product of scalars). */
+ void add_vectors(const lst & l);
+
/** Clear all registered scalar products. */
void clear(void);
GINAC_IMPLEMENT_REGISTERED_CLASS(tensdelta, tensor)
GINAC_IMPLEMENT_REGISTERED_CLASS(tensmetric, tensor)
GINAC_IMPLEMENT_REGISTERED_CLASS(minkmetric, tensmetric)
+GINAC_IMPLEMENT_REGISTERED_CLASS(spinmetric, tensmetric)
GINAC_IMPLEMENT_REGISTERED_CLASS(tensepsilon, tensor)
//////////
DEFAULT_CTORS(tensor)
DEFAULT_CTORS(tensdelta)
DEFAULT_CTORS(tensmetric)
+DEFAULT_COPY(spinmetric)
+DEFAULT_DESTROY(spinmetric)
DEFAULT_DESTROY(minkmetric)
DEFAULT_DESTROY(tensepsilon)
tinfo_key = TINFO_minkmetric;
}
+spinmetric::spinmetric()
+{
+ debugmsg("spinmetric default constructor", LOGLEVEL_CONSTRUCT);
+ tinfo_key = TINFO_spinmetric;
+}
+
minkmetric::minkmetric(bool ps) : pos_sig(ps)
{
debugmsg("minkmetric constructor from bool", LOGLEVEL_CONSTRUCT);
DEFAULT_ARCHIVING(tensor)
DEFAULT_ARCHIVING(tensdelta)
DEFAULT_ARCHIVING(tensmetric)
+DEFAULT_ARCHIVING(spinmetric)
DEFAULT_UNARCHIVE(minkmetric)
DEFAULT_UNARCHIVE(tensepsilon)
DEFAULT_COMPARE(tensor)
DEFAULT_COMPARE(tensdelta)
DEFAULT_COMPARE(tensmetric)
+DEFAULT_COMPARE(spinmetric)
int minkmetric::compare_same_type(const basic & other) const
{
DEFAULT_PRINT_LATEX(tensdelta, "delta", "\\delta")
DEFAULT_PRINT(tensmetric, "g")
DEFAULT_PRINT_LATEX(minkmetric, "eta", "\\eta")
-DEFAULT_PRINT_LATEX(tensepsilon, "eps", "\\epsilon")
+DEFAULT_PRINT_LATEX(spinmetric, "eps", "\\varepsilon")
+DEFAULT_PRINT_LATEX(tensepsilon, "eps", "\\varepsilon")
/** Automatic symbolic evaluation of an indexed delta tensor. */
ex tensdelta::eval_indexed(const basic & i) const
return inherited::eval_indexed(i);
}
+/** Automatic symbolic evaluation of an indexed metric tensor. */
+ex spinmetric::eval_indexed(const basic & i) const
+{
+ GINAC_ASSERT(is_of_type(i, indexed));
+ GINAC_ASSERT(i.nops() == 3);
+ GINAC_ASSERT(is_ex_of_type(i.op(0), spinmetric));
+ GINAC_ASSERT(is_ex_of_type(i.op(1), spinidx));
+ GINAC_ASSERT(is_ex_of_type(i.op(2), spinidx));
+
+ const spinidx & i1 = ex_to_spinidx(i.op(1));
+ const spinidx & i2 = ex_to_spinidx(i.op(2));
+
+ // Convolutions are zero
+ if (static_cast<const indexed &>(i).get_dummy_indices().size() != 0)
+ return _ex0();
+
+ // Numeric evaluation
+ if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
+ int n1 = ex_to_numeric(i1.get_value()).to_int(), n2 = ex_to_numeric(i2.get_value()).to_int();
+ if (n1 == n2)
+ return _ex0();
+ else if (n1 < n2)
+ return _ex1();
+ else
+ return _ex_1();
+ }
+
+ // No further simplifications
+ return i.hold();
+}
+
/** Automatic symbolic evaluation of an indexed epsilon tensor. */
ex tensepsilon::eval_indexed(const basic & i) const
{
return false;
}
+/** Contraction of an indexed spinor metric with something else. */
+bool spinmetric::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(*other, indexed));
+ GINAC_ASSERT(self->nops() == 3);
+ GINAC_ASSERT(is_ex_of_type(self->op(0), spinmetric));
+
+ // Contractions between spinor metrics
+ if (is_ex_of_type(other->op(0), spinmetric)) {
+ const idx &self_i1 = ex_to_idx(self->op(1));
+ const idx &self_i2 = ex_to_idx(self->op(2));
+ const idx &other_i1 = ex_to_idx(other->op(1));
+ const idx &other_i2 = ex_to_idx(other->op(2));
+
+ if (is_dummy_pair(self_i1, other_i1)) {
+ if (is_dummy_pair(self_i2, other_i2))
+ *self = _ex2();
+ else
+ *self = delta_tensor(self_i2, other_i2);
+ *other = _ex1();
+ return true;
+ } else if (is_dummy_pair(self_i1, other_i2)) {
+ if (is_dummy_pair(self_i2, other_i1))
+ *self = _ex_2();
+ else
+ *self = -delta_tensor(self_i2, other_i1);
+ *other = _ex1();
+ return true;
+ } else if (is_dummy_pair(self_i2, other_i1)) {
+ *self = -delta_tensor(self_i1, other_i2);
+ *other = _ex1();
+ return true;
+ } else if (is_dummy_pair(self_i2, other_i2)) {
+ *self = delta_tensor(self_i1, other_i1);
+ *other = _ex1();
+ return true;
+ }
+ }
+
+ // If contracting with the delta tensor, let the delta do it
+ // (don't raise/lower delta indices)
+ if (is_ex_of_type(other->op(0), tensdelta))
+ return false;
+
+ // Try to contract first index
+ const idx *self_idx = &ex_to_idx(self->op(1));
+ const idx *free_idx = &ex_to_idx(self->op(2));
+ bool first_index_tried = false;
+ int sign = 1;
+
+again:
+ if (self_idx->is_symbolic()) {
+ for (int i=1; i<other->nops(); i++) {
+ const idx &other_idx = ex_to_idx(other->op(i));
+ if (is_dummy_pair(*self_idx, other_idx)) {
+
+ // Contraction found, remove metric tensor and substitute
+ // index in second object
+ *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
+ *other = other->subs(other_idx == *free_idx);
+ return true;
+ }
+ }
+ }
+
+ if (!first_index_tried) {
+
+ // No contraction with first index found, try second index
+ self_idx = &ex_to_idx(self->op(2));
+ free_idx = &ex_to_idx(self->op(1));
+ first_index_tried = true;
+ sign = -sign;
+ goto again;
+ }
+
+ return false;
+}
+
//////////
// global functions
//////////
return indexed(minkmetric(pos_sig), indexed::symmetric, i1, i2);
}
+ex spinor_metric(const ex & i1, const ex & i2)
+{
+ if (!is_ex_of_type(i1, spinidx) || !is_ex_of_type(i2, spinidx))
+ throw(std::invalid_argument("indices of spinor metric must be of type spinidx"));
+ if (!ex_to_idx(i1).get_dim().is_equal(2) || !ex_to_idx(i2).get_dim().is_equal(2))
+ throw(std::runtime_error("index dimension for spinor metric must be 2"));
+
+ return indexed(spinmetric(), indexed::antisymmetric, i1, i2);
+}
+
ex epsilon_tensor(const ex & i1, const ex & i2)
{
if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx))
};
+/** This class represents an antisymmetric spinor metric tensor which
+ * can be used to raise/lower indices of 2-component Weyl spinors. If
+ * indexed, it must have exactly two indices of the same type which
+ * must be of class spinidx or a subclass and have dimension 2. */
+class spinmetric : public tensmetric
+{
+ GINAC_DECLARE_REGISTERED_CLASS(spinmetric, tensmetric)
+
+ // functions overriding virtual functions from bases classes
+public:
+ void print(const print_context & c, unsigned level = 0) const;
+ ex eval_indexed(const basic & i) const;
+ bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
+};
+
+
/** This class represents the totally antisymmetric epsilon tensor. If
* indexed, all indices must be of the same type and their number must
* be equal to the dimension of the index space. */
* @return newly constructed Lorentz metric tensor */
ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig = false);
+/** Create a spinor metric tensor with specified indices. The indices must be
+ * of class spinidx or a subclass and have a dimension of 2. The spinor
+ * metric is an antisymmetric tensor with a matrix representation of
+ * [[ [[ 0, 1 ]], [[ -1, 0 ]] ]].
+ *
+ * @param i1 First index
+ * @param i2 Second index
+ * @return newly constructed spinor metric tensor */
+ex spinor_metric(const ex & i1, const ex & i2);
+
/** Create an epsilon tensor in a Euclidean space with two indices. The
* indices must be of class idx or a subclass, and have a dimension of 2.
*
const unsigned TINFO_idx = 0x000d0001U;
const unsigned TINFO_varidx = 0x000d1001U;
+const unsigned TINFO_spinidx = 0x000d2001U;
const unsigned TINFO_tensor = 0x000e0001U;
const unsigned TINFO_tensdelta = 0x000e1001U;
const unsigned TINFO_tensmetric = 0x000e1002U;
const unsigned TINFO_minkmetric = 0x000e2001U;
+const unsigned TINFO_spinmetric = 0x000e2002U;
const unsigned TINFO_tensepsilon = 0x000e1003U;
const unsigned TINFO_su3one = 0x000e1008U;
const unsigned TINFO_su3t = 0x000e1009U;
git://www.ginac.de / ginac.git / commitdiff