tensor.cpp

Go to the documentation of this file.

 
/*
 *  GiNaC Copyright (C) 1999-2026 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
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program.  If not, see <https://www.gnu.org/licenses/>.
 */
 
#include "tensor.h"
#include "idx.h"
#include "indexed.h"
#include "symmetry.h"
#include "relational.h"
#include "operators.h"
#include "lst.h"
#include "numeric.h"
#include "matrix.h"
#include "archive.h"
#include "utils.h"
 
#include <stdexcept>
#include <vector>
 
namespace GiNaC {
 
GINAC_IMPLEMENT_REGISTERED_CLASS(tensor, basic)
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(tensdelta, tensor,
  print_func<print_dflt>(&tensdelta::do_print).
  print_func<print_latex>(&tensdelta::do_print_latex))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(tensmetric, tensor,
  print_func<print_dflt>(&tensmetric::do_print).
  print_func<print_latex>(&tensmetric::do_print))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(minkmetric, tensmetric,
  print_func<print_dflt>(&minkmetric::do_print).
  print_func<print_latex>(&minkmetric::do_print_latex))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(spinmetric, tensmetric,
  print_func<print_dflt>(&spinmetric::do_print).
  print_func<print_latex>(&spinmetric::do_print_latex))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(tensepsilon, tensor,
  print_func<print_dflt>(&tensepsilon::do_print).
  print_func<print_latex>(&tensepsilon::do_print_latex))
 
// constructors
 
tensor::tensor()
{
    setflag(status_flags::evaluated | status_flags::expanded);
}
 
DEFAULT_CTOR(tensdelta)
DEFAULT_CTOR(tensmetric)
 
minkmetric::minkmetric() : pos_sig(false)
{
}
 
spinmetric::spinmetric()
{
}
 
minkmetric::minkmetric(bool ps) : pos_sig(ps)
{
}
 
tensepsilon::tensepsilon() : minkowski(false), pos_sig(false)
{
}
 
tensepsilon::tensepsilon(bool mink, bool ps) : minkowski(mink), pos_sig(ps)
{
}
 
// archiving
 
void minkmetric::read_archive(const archive_node& n, lst& sym_lst)
{
    inherited::read_archive(n, sym_lst);
    n.find_bool("pos_sig", pos_sig);
}
GINAC_BIND_UNARCHIVER(minkmetric);
 
void minkmetric::archive(archive_node &n) const
{
    inherited::archive(n);
    n.add_bool("pos_sig", pos_sig);
}
 
void tensepsilon::read_archive(const archive_node& n, lst& sym_lst)
{
    inherited::read_archive(n, sym_lst);
    n.find_bool("minkowski", minkowski);
    n.find_bool("pos_sig", pos_sig);
}
GINAC_BIND_UNARCHIVER(tensepsilon);
 
void tensepsilon::archive(archive_node &n) const
{
    inherited::archive(n);
    n.add_bool("minkowski", minkowski);
    n.add_bool("pos_sig", pos_sig);
}
 
GINAC_BIND_UNARCHIVER(tensdelta);
GINAC_BIND_UNARCHIVER(tensmetric);
GINAC_BIND_UNARCHIVER(spinmetric);
 
// functions overriding virtual functions from base classes
 
DEFAULT_COMPARE(tensor)
DEFAULT_COMPARE(tensdelta)
DEFAULT_COMPARE(tensmetric)
DEFAULT_COMPARE(spinmetric)
 
bool tensdelta::info(unsigned inf) const
{
    if(inf == info_flags::real)
        return true;
 
    return false;
}
 
bool tensmetric::info(unsigned inf) const
{
    if(inf == info_flags::real)
        return true;
 
    return false;
}
 
int minkmetric::compare_same_type(const basic & other) const
{
    GINAC_ASSERT(is_a<minkmetric>(other));
    const minkmetric &o = static_cast<const minkmetric &>(other);
 
    if (pos_sig != o.pos_sig)
        return pos_sig ? -1 : 1;
    else
        return inherited::compare_same_type(other);
}
 
bool minkmetric::info(unsigned inf) const
{
    if(inf == info_flags::real)
        return true;
 
    return false;
}
 
int tensepsilon::compare_same_type(const basic & other) const
{
    GINAC_ASSERT(is_a<tensepsilon>(other));
    const tensepsilon &o = static_cast<const tensepsilon &>(other);
 
    if (minkowski != o.minkowski)
        return minkowski ? -1 : 1;
    else if (pos_sig != o.pos_sig)
        return pos_sig ? -1 : 1;
    else
        return inherited::compare_same_type(other);
}
 
bool tensepsilon::info(unsigned inf) const
{
    if(inf == info_flags::real)
        return true;
 
    return false;
}
 
bool spinmetric::info(unsigned inf) const
{
    if(inf == info_flags::real)
        return true;
 
    return false;
}
 
DEFAULT_PRINT_LATEX(tensdelta, "delta", "\\delta")
DEFAULT_PRINT(tensmetric, "g")
DEFAULT_PRINT_LATEX(minkmetric, "eta", "\\eta")
DEFAULT_PRINT_LATEX(spinmetric, "eps", "\\varepsilon")
DEFAULT_PRINT_LATEX(tensepsilon, "eps", "\\varepsilon")
 
ex tensdelta::eval_indexed(const basic & i) const
{
    GINAC_ASSERT(is_a<indexed>(i));
    GINAC_ASSERT(i.nops() == 3);
    GINAC_ASSERT(is_a<tensdelta>(i.op(0)));
 
    const idx & i1 = ex_to<idx>(i.op(1));
    const idx & i2 = ex_to<idx>(i.op(2));
 
    // The dimension of the indices must be equal, otherwise we use the minimal
    // dimension
    if (!i1.get_dim().is_equal(i2.get_dim())) {
        ex min_dim = i1.minimal_dim(i2);
        exmap m;
        m[i1] = i1.replace_dim(min_dim);
        m[i2] = i2.replace_dim(min_dim);
        return i.subs(m, subs_options::no_pattern);
    }
 
    // Trace of delta tensor is the (effective) dimension of the space
    if (is_dummy_pair(i1, i2)) {
        try {
            return i1.minimal_dim(i2);
        } catch (std::exception &e) {
            return i.hold();
        }
    }
 
    // Numeric evaluation
    if (static_cast<const indexed &>(i).all_index_values_are(info_flags::integer)) {
        int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
        if (n1 == n2)
            return _ex1;
        else
            return _ex0;
    }
 
    // No further simplifications
    return i.hold();
}
 
ex tensmetric::eval_indexed(const basic & i) const
{
    GINAC_ASSERT(is_a<indexed>(i));
    GINAC_ASSERT(i.nops() == 3);
    GINAC_ASSERT(is_a<tensmetric>(i.op(0)));
    GINAC_ASSERT(is_a<varidx>(i.op(1)));
    GINAC_ASSERT(is_a<varidx>(i.op(2)));
 
    const varidx & i1 = ex_to<varidx>(i.op(1));
    const varidx & i2 = ex_to<varidx>(i.op(2));
 
    // The dimension of the indices must be equal, otherwise we use the minimal
    // dimension
    if (!i1.get_dim().is_equal(i2.get_dim())) {
        ex min_dim = i1.minimal_dim(i2);
        exmap m;
        m[i1] = i1.replace_dim(min_dim);
        m[i2] = i2.replace_dim(min_dim);
        return i.subs(m, subs_options::no_pattern);
    }
 
    // A metric tensor with one covariant and one contravariant index gets
    // replaced by a delta tensor
    if (i1.is_covariant() != i2.is_covariant())
        return delta_tensor(i1, i2);
 
    // No further simplifications
    return i.hold();
}
 
ex minkmetric::eval_indexed(const basic & i) const
{
    GINAC_ASSERT(is_a<indexed>(i));
    GINAC_ASSERT(i.nops() == 3);
    GINAC_ASSERT(is_a<minkmetric>(i.op(0)));
    GINAC_ASSERT(is_a<varidx>(i.op(1)));
    GINAC_ASSERT(is_a<varidx>(i.op(2)));
 
    const varidx & i1 = ex_to<varidx>(i.op(1));
    const varidx & i2 = ex_to<varidx>(i.op(2));
 
    // 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 == 0)
            return pos_sig ? _ex_1 : _ex1;
        else
            return pos_sig ? _ex1 : _ex_1;
    }
 
    // Perform the usual evaluations of a metric tensor
    return inherited::eval_indexed(i);
}
 
ex spinmetric::eval_indexed(const basic & i) const
{
    GINAC_ASSERT(is_a<indexed>(i));
    GINAC_ASSERT(i.nops() == 3);
    GINAC_ASSERT(is_a<spinmetric>(i.op(0)));
    GINAC_ASSERT(is_a<spinidx>(i.op(1)));
    GINAC_ASSERT(is_a<spinidx>(i.op(2)));
 
    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().empty()))
        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();
}
 
ex tensepsilon::eval_indexed(const basic & i) const
{
    GINAC_ASSERT(is_a<indexed>(i));
    GINAC_ASSERT(i.nops() > 1);
    GINAC_ASSERT(is_a<tensepsilon>(i.op(0)));
 
    // Convolutions are zero
    if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
        return _ex0;
 
    // Numeric evaluation
    if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 
        // Get sign of index permutation (the indices should already be in
        // a canonic order but we can't assume what exactly that order is)
        std::vector<int> v;
        v.reserve(i.nops() - 1);
        for (size_t j=1; j<i.nops(); j++)
            v.push_back(ex_to<numeric>(ex_to<idx>(i.op(j)).get_value()).to_int());
        int sign = permutation_sign(v.begin(), v.end());
 
        // In a Minkowski space, check for covariant indices
        if (minkowski) {
            for (size_t j=1; j<i.nops(); j++) {
                const ex & x = i.op(j);
                if (!is_a<varidx>(x)) {
                    throw(std::runtime_error("indices of epsilon tensor in Minkowski space must be of type varidx"));
                }
                if (ex_to<varidx>(x).is_covariant()) {
                    if (ex_to<idx>(x).get_value().is_zero()) {
                        sign = (pos_sig ? -sign : sign);
                    }
                    else {
                        sign = (pos_sig ? sign : -sign);
                    }
                }
            }
        }
 
        return sign;
    }
 
    // No further simplifications
    return i.hold();
}
 
bool tensor::replace_contr_index(exvector::iterator self, exvector::iterator other) const
{
    // Try to contract the 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;
 
again:
    if (self_idx->is_symbolic()) {
        for (size_t i=1; i<other->nops(); i++) {
            if (! is_a<idx>(other->op(i)))
                continue;
            const idx &other_idx = ex_to<idx>(other->op(i));
            if (is_dummy_pair(*self_idx, other_idx)) {
 
                // Contraction found, remove this tensor and substitute the
                // index in the second object
                try {
                    // minimal_dim() throws an exception when index dimensions are not comparable
                    ex min_dim = self_idx->minimal_dim(other_idx);
                    *other = other->subs(other_idx == free_idx->replace_dim(min_dim));
                    *self = _ex1; // *other is assigned first because assigning *self invalidates free_idx
                    return true;
                } catch (std::exception &e) {
                    return false;
                }
            }
        }
    }
 
    if (!first_index_tried) {
 
        // No contraction with the first index found, try the second index
        self_idx = &ex_to<idx>(self->op(2));
        free_idx = &ex_to<idx>(self->op(1));
        first_index_tried = true;
        goto again;
    }
 
    return false;
}
 
bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
    GINAC_ASSERT(is_a<indexed>(*self));
    GINAC_ASSERT(is_a<indexed>(*other));
    GINAC_ASSERT(self->nops() == 3);
    GINAC_ASSERT(is_a<tensdelta>(self->op(0)));
 
    // Replace the dummy index with this tensor's other index and remove
    // the tensor (this is valid for contractions with all other tensors)
    return replace_contr_index(self, other);
}
 
bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
    GINAC_ASSERT(is_a<indexed>(*self));
    GINAC_ASSERT(is_a<indexed>(*other));
    GINAC_ASSERT(self->nops() == 3);
    GINAC_ASSERT(is_a<tensmetric>(self->op(0)));
 
    // If contracting with the delta tensor, let the delta do it
    // (don't raise/lower delta indices)
    if (is_a<tensdelta>(other->op(0)))
        return false;
 
    // Replace the dummy index with this tensor's other index and remove
    // the tensor
    return replace_contr_index(self, other);
}
 
bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
    GINAC_ASSERT(is_a<indexed>(*self));
    GINAC_ASSERT(is_a<indexed>(*other));
    GINAC_ASSERT(self->nops() == 3);
    GINAC_ASSERT(is_a<spinmetric>(self->op(0)));
 
    // Contractions between spinor metrics
    if (is_a<spinmetric>(other->op(0))) {
        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_a<tensdelta>(other->op(0)))
        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 (size_t 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 (assign *self last because this
                // invalidates free_idx)
                *other = other->subs(other_idx == *free_idx);
                *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
                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;
}
 
bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
    GINAC_ASSERT(is_a<indexed>(*self));
    GINAC_ASSERT(is_a<indexed>(*other));
    GINAC_ASSERT(is_a<tensepsilon>(self->op(0)));
    size_t num = self->nops() - 1;
 
    if (is_exactly_a<tensepsilon>(other->op(0)) && num+1 == other->nops()) {
 
        // Contraction of two epsilon tensors is a determinant
        bool variance = is_a<varidx>(self->op(1));
        matrix M(num, num);
        for (size_t i=0; i<num; i++) {
            for (size_t j=0; j<num; j++) {
                if (minkowski)
                    M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
                else if (variance)
                    M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
                else
                    M(i, j) = delta_tensor(self->op(i+1), other->op(j+1));
            }
        }
        int sign = minkowski ? -1 : 1;
        *self = sign * M.determinant().simplify_indexed();
        *other = _ex1;
        return true;
    }
 
    return false;
}
 
// global functions
 
ex delta_tensor(const ex & i1, const ex & i2)
{
    static ex delta = dynallocate<tensdelta>();
 
    if (!is_a<idx>(i1) || !is_a<idx>(i2))
        throw(std::invalid_argument("indices of delta tensor must be of type idx"));
 
    return indexed(delta, symmetric2(), i1, i2);
}
 
ex metric_tensor(const ex & i1, const ex & i2)
{
    static ex metric = dynallocate<tensmetric>();
 
    if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
        throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 
    return indexed(metric, symmetric2(), i1, i2);
}
 
ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
{
    static ex metric_neg = dynallocate<minkmetric>(false);
    static ex metric_pos = dynallocate<minkmetric>(true);
 
    if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
        throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 
    return indexed(pos_sig ? metric_pos : metric_neg, symmetric2(), i1, i2);
}
 
ex spinor_metric(const ex & i1, const ex & i2)
{
    static ex metric = dynallocate<spinmetric>();
 
    if (!is_a<spinidx>(i1) || !is_a<spinidx>(i2))
        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(metric, antisymmetric2(), i1, i2);
}
 
ex epsilon_tensor(const ex & i1, const ex & i2)
{
    static ex epsilon = dynallocate<tensepsilon>();
 
    if (!is_a<idx>(i1) || !is_a<idx>(i2))
        throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 
    ex dim = ex_to<idx>(i1).get_dim();
    if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
        throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
    if (!ex_to<idx>(i1).get_dim().is_equal(_ex2))
        throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 
    if(is_a<wildcard>(i1.op(0))||is_a<wildcard>(i2.op(0)))
        return indexed(epsilon, antisymmetric2(), i1, i2).hold();
 
    return indexed(epsilon, antisymmetric2(), i1, i2);
}
 
ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
{
    static ex epsilon = dynallocate<tensepsilon>();
 
    if (!is_a<idx>(i1) || !is_a<idx>(i2) || !is_a<idx>(i3))
        throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 
    ex dim = ex_to<idx>(i1).get_dim();
    if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()))
        throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
    if (!ex_to<idx>(i1).get_dim().is_equal(_ex3))
        throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 
    if(is_a<wildcard>(i1.op(0))||is_a<wildcard>(i2.op(0))||is_a<wildcard>(i3.op(0)))
        return indexed(epsilon, antisymmetric3(), i1, i2, i3).hold();
 
    return indexed(epsilon, antisymmetric3(), i1, i2, i3);
}
 
ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
{
    static ex epsilon_neg = dynallocate<tensepsilon>(true, false);
    static ex epsilon_pos = dynallocate<tensepsilon>(true, true);
 
    if (!is_a<varidx>(i1) || !is_a<varidx>(i2) || !is_a<varidx>(i3) || !is_a<varidx>(i4))
        throw(std::invalid_argument("indices of Lorentz epsilon tensor must be of type varidx"));
 
    ex dim = ex_to<idx>(i1).get_dim();
    if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()) || !dim.is_equal(ex_to<idx>(i4).get_dim()))
        throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
    if (!ex_to<idx>(i1).get_dim().is_equal(_ex4))
        throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 
    if(is_a<wildcard>(i1.op(0))||is_a<wildcard>(i2.op(0))||is_a<wildcard>(i3.op(0))||is_a<wildcard>(i4.op(0)))
        return indexed(pos_sig ? epsilon_pos : epsilon_neg, antisymmetric4(), i1, i2, i3, i4).hold();
 
    return indexed(pos_sig ? epsilon_pos : epsilon_neg, antisymmetric4(), i1, i2, i3, i4);
}
 
} // namespace GiNaC