"""
This module defines tensors with abstract index notation.
The abstract index notation has been first formalized by Penrose.
Tensor indices are formal objects, with a tensor type; there is no
notion of index range, it is only possible to assign the dimension,
used to trace the Kronecker delta; the dimension can be a Symbol.
The Einstein summation convention is used.
The covariant indices are indicated with a minus sign in front of the index.
For instance the tensor ``t = p(a)*A(b,c)*q(-c)`` has the index ``c``
contracted.
A tensor expression ``t`` can be called; called with its
indices in sorted order it is equal to itself:
in the above example ``t(a, b) == t``;
one can call ``t`` with different indices; ``t(c, d) == p(c)*A(d,a)*q(-a)``.
The contracted indices are dummy indices, internally they have no name,
the indices being represented by a graph-like structure.
Tensors are put in canonical form using ``canon_bp``, which uses
the Butler-Portugal algorithm for canonicalization using the monoterm
symmetries of the tensors.
If there is a (anti)symmetric metric, the indices can be raised and
lowered when the tensor is put in canonical form.
"""
from __future__ import print_function, division
from collections import defaultdict
import operator
import itertools
from sympy import Rational, prod, Integer
from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, \
bsgs_direct_product, canonicalize, riemann_bsgs
from sympy.core import Basic, Expr, sympify, Add, Mul, S
from sympy.core.compatibility import string_types, reduce, range, SYMPY_INTS
from sympy.core.containers import Tuple, Dict
from sympy.core.decorators import deprecated
from sympy.core.symbol import Symbol, symbols
from sympy.core.sympify import CantSympify, _sympify
from sympy.core.operations import AssocOp
from sympy.matrices import eye
from sympy.utilities.exceptions import SymPyDeprecationWarning
import warnings
@deprecated(useinstead=".replace_with_arrays", issue=15276, deprecated_since_version="1.4")
def deprecate_data():
pass
class _IndexStructure(CantSympify):
"""
This class handles the indices (free and dummy ones). It contains the
algorithms to manage the dummy indices replacements and contractions of
free indices under multiplications of tensor expressions, as well as stuff
related to canonicalization sorting, getting the permutation of the
expression and so on. It also includes tools to get the ``TensorIndex``
objects corresponding to the given index structure.
"""
def __init__(self, free, dum, index_types, indices, canon_bp=False):
self.free = free
self.dum = dum
self.index_types = index_types
self.indices = indices
self._ext_rank = len(self.free) + 2*len(self.dum)
self.dum.sort(key=lambda x: x[0])
@staticmethod
def from_indices(*indices):
"""
Create a new ``_IndexStructure`` object from a list of ``indices``
``indices`` ``TensorIndex`` objects, the indices. Contractions are
detected upon construction.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, _IndexStructure
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> _IndexStructure.from_indices(m0, m1, -m1, m3)
_IndexStructure([(m0, 0), (m3, 3)], [(1, 2)], [Lorentz, Lorentz, Lorentz, Lorentz])
In case of many components the same indices have slightly different
indexes:
>>> _IndexStructure.from_indices(m0, m1, -m1, m3)
_IndexStructure([(m0, 0), (m3, 3)], [(1, 2)], [Lorentz, Lorentz, Lorentz, Lorentz])
"""
free, dum = _IndexStructure._free_dum_from_indices(*indices)
index_types = [i.tensor_index_type for i in indices]
indices = _IndexStructure._replace_dummy_names(indices, free, dum)
return _IndexStructure(free, dum, index_types, indices)
@staticmethod
def from_components_free_dum(components, free, dum):
index_types = []
for component in components:
index_types.extend(component.index_types)
indices = _IndexStructure.generate_indices_from_free_dum_index_types(free, dum, index_types)
return _IndexStructure(free, dum, index_types, indices)
@staticmethod
def _free_dum_from_indices(*indices):
"""
Convert ``indices`` into ``free``, ``dum`` for single component tensor
``free`` list of tuples ``(index, pos, 0)``,
where ``pos`` is the position of index in
the list of indices formed by the component tensors
``dum`` list of tuples ``(pos_contr, pos_cov, 0, 0)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, \
_IndexStructure
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> _IndexStructure._free_dum_from_indices(m0, m1, -m1, m3)
([(m0, 0), (m3, 3)], [(1, 2)])
"""
n = len(indices)
if n == 1:
return [(indices[0], 0)], []
# find the positions of the free indices and of the dummy indices
free = [True]*len(indices)
index_dict = {}
dum = []
for i, index in enumerate(indices):
name = index._name
typ = index.tensor_index_type
contr = index._is_up
if (name, typ) in index_dict:
# found a pair of dummy indices
is_contr, pos = index_dict[(name, typ)]
# check consistency and update free
if is_contr:
if contr:
raise ValueError('two equal contravariant indices in slots %d and %d' %(pos, i))
else:
free[pos] = False
free[i] = False
else:
if contr:
free[pos] = False
free[i] = False
else:
raise ValueError('two equal covariant indices in slots %d and %d' %(pos, i))
if contr:
dum.append((i, pos))
else:
dum.append((pos, i))
else:
index_dict[(name, typ)] = index._is_up, i
free = [(index, i) for i, index in enumerate(indices) if free[i]]
free.sort()
return free, dum
def get_indices(self):
"""
Get a list of indices, creating new tensor indices to complete dummy indices.
"""
return self.indices[:]
@staticmethod
def generate_indices_from_free_dum_index_types(free, dum, index_types):
indices = [None]*(len(free)+2*len(dum))
for idx, pos in free:
indices[pos] = idx
generate_dummy_name = _IndexStructure._get_generator_for_dummy_indices(free)
for pos1, pos2 in dum:
typ1 = index_types[pos1]
indname = generate_dummy_name(typ1)
indices[pos1] = TensorIndex(indname, typ1, True)
indices[pos2] = TensorIndex(indname, typ1, False)
return _IndexStructure._replace_dummy_names(indices, free, dum)
@staticmethod
def _get_generator_for_dummy_indices(free):
cdt = defaultdict(int)
# if the free indices have names with dummy_fmt, start with an
# index higher than those for the dummy indices
# to avoid name collisions
for indx, ipos in free:
if indx._name.split('_')[0] == indx.tensor_index_type.dummy_fmt[:-3]:
cdt[indx.tensor_index_type] = max(cdt[indx.tensor_index_type], int(indx._name.split('_')[1]) + 1)
def dummy_fmt_gen(tensor_index_type):
fmt = tensor_index_type.dummy_fmt
nd = cdt[tensor_index_type]
cdt[tensor_index_type] += 1
return fmt % nd
return dummy_fmt_gen
@staticmethod
def _replace_dummy_names(indices, free, dum):
dum.sort(key=lambda x: x[0])
new_indices = [ind for ind in indices]
assert len(indices) == len(free) + 2*len(dum)
generate_dummy_name = _IndexStructure._get_generator_for_dummy_indices(free)
for ipos1, ipos2 in dum:
typ1 = new_indices[ipos1].tensor_index_type
indname = generate_dummy_name(typ1)
new_indices[ipos1] = TensorIndex(indname, typ1, True)
new_indices[ipos2] = TensorIndex(indname, typ1, False)
return new_indices
def get_free_indices(self):
"""
Get a list of free indices.
"""
# get sorted indices according to their position:
free = sorted(self.free, key=lambda x: x[1])
return [i[0] for i in free]
def __str__(self):
return "_IndexStructure({0}, {1}, {2})".format(self.free, self.dum, self.index_types)
def __repr__(self):
return self.__str__()
def _get_sorted_free_indices_for_canon(self):
sorted_free = self.free[:]
sorted_free.sort(key=lambda x: x[0])
return sorted_free
def _get_sorted_dum_indices_for_canon(self):
return sorted(self.dum, key=lambda x: x[0])
def _get_lexicographically_sorted_index_types(self):
permutation = self.indices_canon_args()[0]
index_types = [None]*self._ext_rank
for i, it in enumerate(self.index_types):
index_types[permutation(i)] = it
return index_types
def _get_lexicographically_sorted_indices(self):
permutation = self.indices_canon_args()[0]
indices = [None]*self._ext_rank
for i, it in enumerate(self.indices):
indices[permutation(i)] = it
return indices
def perm2tensor(self, g, is_canon_bp=False):
"""
Returns a ``_IndexStructure`` instance corresponding to the permutation ``g``
``g`` permutation corresponding to the tensor in the representation
used in canonicalization
``is_canon_bp`` if True, then ``g`` is the permutation
corresponding to the canonical form of the tensor
"""
sorted_free = [i[0] for i in self._get_sorted_free_indices_for_canon()]
lex_index_types = self._get_lexicographically_sorted_index_types()
lex_indices = self._get_lexicographically_sorted_indices()
nfree = len(sorted_free)
rank = self._ext_rank
dum = [[None]*2 for i in range((rank - nfree)//2)]
free = []
index_types = [None]*rank
indices = [None]*rank
for i in range(rank):
gi = g[i]
index_types[i] = lex_index_types[gi]
indices[i] = lex_indices[gi]
if gi < nfree:
ind = sorted_free[gi]
assert index_types[i] == sorted_free[gi].tensor_index_type
free.append((ind, i))
else:
j = gi - nfree
idum, cov = divmod(j, 2)
if cov:
dum[idum][1] = i
else:
dum[idum][0] = i
dum = [tuple(x) for x in dum]
return _IndexStructure(free, dum, index_types, indices)
def indices_canon_args(self):
"""
Returns ``(g, dummies, msym, v)``, the entries of ``canonicalize``
see ``canonicalize`` in ``tensor_can.py``
"""
# to be called after sorted_components
from sympy.combinatorics.permutations import _af_new
n = self._ext_rank
g = [None]*n + [n, n+1]
# ordered indices: first the free indices, ordered by types
# then the dummy indices, ordered by types and contravariant before
# covariant
# g[position in tensor] = position in ordered indices
for i, (indx, ipos) in enumerate(self._get_sorted_free_indices_for_canon()):
g[ipos] = i
pos = len(self.free)
j = len(self.free)
dummies = []
prev = None
a = []
msym = []
for ipos1, ipos2 in self._get_sorted_dum_indices_for_canon():
g[ipos1] = j
g[ipos2] = j + 1
j += 2
typ = self.index_types[ipos1]
if typ != prev:
if a:
dummies.append(a)
a = [pos, pos + 1]
prev = typ
msym.append(typ.metric_antisym)
else:
a.extend([pos, pos + 1])
pos += 2
if a:
dummies.append(a)
return _af_new(g), dummies, msym
def components_canon_args(components):
numtyp = []
prev = None
for t in components:
if t == prev:
numtyp[-1][1] += 1
else:
prev = t
numtyp.append([prev, 1])
v = []
for h, n in numtyp:
if h._comm == 0 or h._comm == 1:
comm = h._comm
else:
comm = TensorManager.get_comm(h._comm, h._comm)
v.append((h._symmetry.base, h._symmetry.generators, n, comm))
return v
class _TensorDataLazyEvaluator(CantSympify):
"""
EXPERIMENTAL: do not rely on this class, it may change without deprecation
warnings in future versions of SymPy.
This object contains the logic to associate components data to a tensor
expression. Components data are set via the ``.data`` property of tensor
expressions, is stored inside this class as a mapping between the tensor
expression and the ``ndarray``.
Computations are executed lazily: whereas the tensor expressions can have
contractions, tensor products, and additions, components data are not
computed until they are accessed by reading the ``.data`` property
associated to the tensor expression.
"""
_substitutions_dict = dict()
_substitutions_dict_tensmul = dict()
def __getitem__(self, key):
dat = self._get(key)
if dat is None:
return None
from .array import NDimArray
if not isinstance(dat, NDimArray):
return dat
if dat.rank() == 0:
return dat[()]
elif dat.rank() == 1 and len(dat) == 1:
return dat[0]
return dat
def _get(self, key):
"""
Retrieve ``data`` associated with ``key``.
This algorithm looks into ``self._substitutions_dict`` for all
``TensorHead`` in the ``TensExpr`` (or just ``TensorHead`` if key is a
TensorHead instance). It reconstructs the components data that the
tensor expression should have by performing on components data the
operations that correspond to the abstract tensor operations applied.
Metric tensor is handled in a different manner: it is pre-computed in
``self._substitutions_dict_tensmul``.
"""
if key in self._substitutions_dict:
return self._substitutions_dict[key]
if isinstance(key, TensorHead):
return None
if isinstance(key, Tensor):
# special case to handle metrics. Metric tensors cannot be
# constructed through contraction by the metric, their
# components show if they are a matrix or its inverse.
signature = tuple([i.is_up for i in key.get_indices()])
srch = (key.component,) + signature
if srch in self._substitutions_dict_tensmul:
return self._substitutions_dict_tensmul[srch]
array_list = [self.data_from_tensor(key)]
return self.data_contract_dum(array_list, key.dum, key.ext_rank)
if isinstance(key, TensMul):
tensmul_args = key.args
if len(tensmul_args) == 1 and len(tensmul_args[0].components) == 1:
# special case to handle metrics. Metric tensors cannot be
# constructed through contraction by the metric, their
# components show if they are a matrix or its inverse.
signature = tuple([i.is_up for i in tensmul_args[0].get_indices()])
srch = (tensmul_args[0].components[0],) + signature
if srch in self._substitutions_dict_tensmul:
return self._substitutions_dict_tensmul[srch]
#data_list = [self.data_from_tensor(i) for i in tensmul_args if isinstance(i, TensExpr)]
data_list = [self.data_from_tensor(i) if isinstance(i, Tensor) else i.data for i in tensmul_args if isinstance(i, TensExpr)]
coeff = prod([i for i in tensmul_args if not isinstance(i, TensExpr)])
if all([i is None for i in data_list]):
return None
if any([i is None for i in data_list]):
raise ValueError("Mixing tensors with associated components "\
"data with tensors without components data")
data_result = self.data_contract_dum(data_list, key.dum, key.ext_rank)
return coeff*data_result
if isinstance(key, TensAdd):
data_list = []
free_args_list = []
for arg in key.args:
if isinstance(arg, TensExpr):
data_list.append(arg.data)
free_args_list.append([x[0] for x in arg.free])
else:
data_list.append(arg)
free_args_list.append([])
if all([i is None for i in data_list]):
return None
if any([i is None for i in data_list]):
raise ValueError("Mixing tensors with associated components "\
"data with tensors without components data")
sum_list = []
from .array import permutedims
for data, free_args in zip(data_list, free_args_list):
if len(free_args) < 2:
sum_list.append(data)
else:
free_args_pos = {y: x for x, y in enumerate(free_args)}
axes = [free_args_pos[arg] for arg in key.free_args]
sum_list.append(permutedims(data, axes))
return reduce(lambda x, y: x+y, sum_list)
return None
@staticmethod
def data_contract_dum(ndarray_list, dum, ext_rank):
from .array import tensorproduct, tensorcontraction, MutableDenseNDimArray
arrays = list(map(MutableDenseNDimArray, ndarray_list))
prodarr = tensorproduct(*arrays)
return tensorcontraction(prodarr, *dum)
def data_tensorhead_from_tensmul(self, data, tensmul, tensorhead):
"""
This method is used when assigning components data to a ``TensMul``
object, it converts components data to a fully contravariant ndarray,
which is then stored according to the ``TensorHead`` key.
"""
if data is None:
return None
return self._correct_signature_from_indices(
data,
tensmul.get_indices(),
tensmul.free,
tensmul.dum,
True)
def data_from_tensor(self, tensor):
"""
This method corrects the components data to the right signature
(covariant/contravariant) using the metric associated with each
``TensorIndexType``.
"""
tensorhead = tensor.component
if tensorhead.data is None:
return None
return self._correct_signature_from_indices(
tensorhead.data,
tensor.get_indices(),
tensor.free,
tensor.dum)
def _assign_data_to_tensor_expr(self, key, data):
if isinstance(key, TensAdd):
raise ValueError('cannot assign data to TensAdd')
# here it is assumed that `key` is a `TensMul` instance.
if len(key.components) != 1:
raise ValueError('cannot assign data to TensMul with multiple components')
tensorhead = key.components[0]
newdata = self.data_tensorhead_from_tensmul(data, key, tensorhead)
return tensorhead, newdata
def _check_permutations_on_data(self, tens, data):
from .array import permutedims
if isinstance(tens, TensorHead):
rank = tens.rank
generators = tens.symmetry.generators
elif isinstance(tens, Tensor):
rank = tens.rank
generators = tens.components[0].symmetry.generators
elif isinstance(tens, TensorIndexType):
rank = tens.metric.rank
generators = tens.metric.symmetry.generators
# Every generator is a permutation, check that by permuting the array
# by that permutation, the array will be the same, except for a
# possible sign change if the permutation admits it.
for gener in generators:
sign_change = +1 if (gener(rank) == rank) else -1
data_swapped = data
last_data = data
permute_axes = list(map(gener, list(range(rank))))
# the order of a permutation is the number of times to get the
# identity by applying that permutation.
for i in range(gener.order()-1):
data_swapped = permutedims(data_swapped, permute_axes)
# if any value in the difference array is non-zero, raise an error:
if any(last_data - sign_change*data_swapped):
raise ValueError("Component data symmetry structure error")
last_data = data_swapped
def __setitem__(self, key, value):
"""
Set the components data of a tensor object/expression.
Components data are transformed to the all-contravariant form and stored
with the corresponding ``TensorHead`` object. If a ``TensorHead`` object
cannot be uniquely identified, it will raise an error.
"""
data = _TensorDataLazyEvaluator.parse_data(value)
self._check_permutations_on_data(key, data)
# TensorHead and TensorIndexType can be assigned data directly, while
# TensMul must first convert data to a fully contravariant form, and
# assign it to its corresponding TensorHead single component.
if not isinstance(key, (TensorHead, TensorIndexType)):
key, data = self._assign_data_to_tensor_expr(key, data)
if isinstance(key, TensorHead):
for dim, indextype in zip(data.shape, key.index_types):
if indextype.data is None:
raise ValueError("index type {} has no components data"\
" associated (needed to raise/lower index)".format(indextype))
if indextype.dim is None:
continue
if dim != indextype.dim:
raise ValueError("wrong dimension of ndarray")
self._substitutions_dict[key] = data
def __delitem__(self, key):
del self._substitutions_dict[key]
def __contains__(self, key):
return key in self._substitutions_dict
def add_metric_data(self, metric, data):
"""
Assign data to the ``metric`` tensor. The metric tensor behaves in an
anomalous way when raising and lowering indices.
A fully covariant metric is the inverse transpose of the fully
contravariant metric (it is meant matrix inverse). If the metric is
symmetric, the transpose is not necessary and mixed
covariant/contravariant metrics are Kronecker deltas.
"""
# hard assignment, data should not be added to `TensorHead` for metric:
# the problem with `TensorHead` is that the metric is anomalous, i.e.
# raising and lowering the index means considering the metric or its
# inverse, this is not the case for other tensors.
self._substitutions_dict_tensmul[metric, True, True] = data
inverse_transpose = self.inverse_transpose_matrix(data)
# in symmetric spaces, the traspose is the same as the original matrix,
# the full covariant metric tensor is the inverse transpose, so this
# code will be able to handle non-symmetric metrics.
self._substitutions_dict_tensmul[metric, False, False] = inverse_transpose
# now mixed cases, these are identical to the unit matrix if the metric
# is symmetric.
m = data.tomatrix()
invt = inverse_transpose.tomatrix()
self._substitutions_dict_tensmul[metric, True, False] = m * invt
self._substitutions_dict_tensmul[metric, False, True] = invt * m
@staticmethod
def _flip_index_by_metric(data, metric, pos):
from .array import tensorproduct, tensorcontraction
mdim = metric.rank()
ddim = data.rank()
if pos == 0:
data = tensorcontraction(
tensorproduct(
metric,
data
),
(1, mdim+pos)
)
else:
data = tensorcontraction(
tensorproduct(
data,
metric
),
(pos, ddim)
)
return data
@staticmethod
def inverse_matrix(ndarray):
m = ndarray.tomatrix().inv()
return _TensorDataLazyEvaluator.parse_data(m)
@staticmethod
def inverse_transpose_matrix(ndarray):
m = ndarray.tomatrix().inv().T
return _TensorDataLazyEvaluator.parse_data(m)
@staticmethod
def _correct_signature_from_indices(data, indices, free, dum, inverse=False):
"""
Utility function to correct the values inside the components data
ndarray according to whether indices are covariant or contravariant.
It uses the metric matrix to lower values of covariant indices.
"""
# change the ndarray values according covariantness/contravariantness of the indices
# use the metric
for i, indx in enumerate(indices):
if not indx.is_up and not inverse:
data = _TensorDataLazyEvaluator._flip_index_by_metric(data, indx.tensor_index_type.data, i)
elif not indx.is_up and inverse:
data = _TensorDataLazyEvaluator._flip_index_by_metric(
data,
_TensorDataLazyEvaluator.inverse_matrix(indx.tensor_index_type.data),
i
)
return data
@staticmethod
def _sort_data_axes(old, new):
from .array import permutedims
new_data = old.data.copy()
old_free = [i[0] for i in old.free]
new_free = [i[0] for i in new.free]
for i in range(len(new_free)):
for j in range(i, len(old_free)):
if old_free[j] == new_free[i]:
old_free[i], old_free[j] = old_free[j], old_free[i]
new_data = permutedims(new_data, (i, j))
break
return new_data
@staticmethod
def add_rearrange_tensmul_parts(new_tensmul, old_tensmul):
def sorted_compo():
return _TensorDataLazyEvaluator._sort_data_axes(old_tensmul, new_tensmul)
_TensorDataLazyEvaluator._substitutions_dict[new_tensmul] = sorted_compo()
@staticmethod
def parse_data(data):
"""
Transform ``data`` to array. The parameter ``data`` may
contain data in various formats, e.g. nested lists, sympy ``Matrix``,
and so on.
Examples
========
>>> from sympy.tensor.tensor import _TensorDataLazyEvaluator
>>> _TensorDataLazyEvaluator.parse_data([1, 3, -6, 12])
[1, 3, -6, 12]
>>> _TensorDataLazyEvaluator.parse_data([[1, 2], [4, 7]])
[[1, 2], [4, 7]]
"""
from .array import MutableDenseNDimArray
if not isinstance(data, MutableDenseNDimArray):
if len(data) == 2 and hasattr(data[0], '__call__'):
data = MutableDenseNDimArray(data[0], data[1])
else:
data = MutableDenseNDimArray(data)
return data
_tensor_data_substitution_dict = _TensorDataLazyEvaluator()
[docs]class _TensorManager(object):
"""
Class to manage tensor properties.
Notes
=====
Tensors belong to tensor commutation groups; each group has a label
``comm``; there are predefined labels:
``0`` tensors commuting with any other tensor
``1`` tensors anticommuting among themselves
``2`` tensors not commuting, apart with those with ``comm=0``
Other groups can be defined using ``set_comm``; tensors in those
groups commute with those with ``comm=0``; by default they
do not commute with any other group.
"""
def __init__(self):
self._comm_init()
def _comm_init(self):
self._comm = [{} for i in range(3)]
for i in range(3):
self._comm[0][i] = 0
self._comm[i][0] = 0
self._comm[1][1] = 1
self._comm[2][1] = None
self._comm[1][2] = None
self._comm_symbols2i = {0:0, 1:1, 2:2}
self._comm_i2symbol = {0:0, 1:1, 2:2}
@property
def comm(self):
return self._comm
[docs] def comm_symbols2i(self, i):
"""
get the commutation group number corresponding to ``i``
``i`` can be a symbol or a number or a string
If ``i`` is not already defined its commutation group number
is set.
"""
if i not in self._comm_symbols2i:
n = len(self._comm)
self._comm.append({})
self._comm[n][0] = 0
self._comm[0][n] = 0
self._comm_symbols2i[i] = n
self._comm_i2symbol[n] = i
return n
return self._comm_symbols2i[i]
[docs] def comm_i2symbol(self, i):
"""
Returns the symbol corresponding to the commutation group number.
"""
return self._comm_i2symbol[i]
[docs] def set_comm(self, i, j, c):
"""
set the commutation parameter ``c`` for commutation groups ``i, j``
Parameters
==========
i, j : symbols representing commutation groups
c : group commutation number
Notes
=====
``i, j`` can be symbols, strings or numbers,
apart from ``0, 1`` and ``2`` which are reserved respectively
for commuting, anticommuting tensors and tensors not commuting
with any other group apart with the commuting tensors.
For the remaining cases, use this method to set the commutation rules;
by default ``c=None``.
The group commutation number ``c`` is assigned in correspondence
to the group commutation symbols; it can be
0 commuting
1 anticommuting
None no commutation property
Examples
========
``G`` and ``GH`` do not commute with themselves and commute with
each other; A is commuting.
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, TensorManager
>>> Lorentz = TensorIndexType('Lorentz')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> A = tensorhead('A', [Lorentz], [[1]])
>>> G = tensorhead('G', [Lorentz], [[1]], 'Gcomm')
>>> GH = tensorhead('GH', [Lorentz], [[1]], 'GHcomm')
>>> TensorManager.set_comm('Gcomm', 'GHcomm', 0)
>>> (GH(i1)*G(i0)).canon_bp()
G(i0)*GH(i1)
>>> (G(i1)*G(i0)).canon_bp()
G(i1)*G(i0)
>>> (G(i1)*A(i0)).canon_bp()
A(i0)*G(i1)
"""
if c not in (0, 1, None):
raise ValueError('`c` can assume only the values 0, 1 or None')
if i not in self._comm_symbols2i:
n = len(self._comm)
self._comm.append({})
self._comm[n][0] = 0
self._comm[0][n] = 0
self._comm_symbols2i[i] = n
self._comm_i2symbol[n] = i
if j not in self._comm_symbols2i:
n = len(self._comm)
self._comm.append({})
self._comm[0][n] = 0
self._comm[n][0] = 0
self._comm_symbols2i[j] = n
self._comm_i2symbol[n] = j
ni = self._comm_symbols2i[i]
nj = self._comm_symbols2i[j]
self._comm[ni][nj] = c
self._comm[nj][ni] = c
[docs] def set_comms(self, *args):
"""
set the commutation group numbers ``c`` for symbols ``i, j``
Parameters
==========
args : sequence of ``(i, j, c)``
"""
for i, j, c in args:
self.set_comm(i, j, c)
[docs] def get_comm(self, i, j):
"""
Return the commutation parameter for commutation group numbers ``i, j``
see ``_TensorManager.set_comm``
"""
return self._comm[i].get(j, 0 if i == 0 or j == 0 else None)
[docs] def clear(self):
"""
Clear the TensorManager.
"""
self._comm_init()
TensorManager = _TensorManager()
[docs]class TensorIndexType(Basic):
"""
A TensorIndexType is characterized by its name and its metric.
Parameters
==========
name : name of the tensor type
metric : metric symmetry or metric object or ``None``
dim : dimension, it can be a symbol or an integer or ``None``
eps_dim : dimension of the epsilon tensor
dummy_fmt : name of the head of dummy indices
Attributes
==========
``name``
``metric_name`` : it is 'metric' or metric.name
``metric_antisym``
``metric`` : the metric tensor
``delta`` : ``Kronecker delta``
``epsilon`` : the ``Levi-Civita epsilon`` tensor
``dim``
``eps_dim``
``dummy_fmt``
``data`` : a property to add ``ndarray`` values, to work in a specified basis.
Notes
=====
The ``metric`` parameter can be:
``metric = False`` symmetric metric (in Riemannian geometry)
``metric = True`` antisymmetric metric (for spinor calculus)
``metric = None`` there is no metric
``metric`` can be an object having ``name`` and ``antisym`` attributes.
If there is a metric the metric is used to raise and lower indices.
In the case of antisymmetric metric, the following raising and
lowering conventions will be adopted:
``psi(a) = g(a, b)*psi(-b); chi(-a) = chi(b)*g(-b, -a)``
``g(-a, b) = delta(-a, b); g(b, -a) = -delta(a, -b)``
where ``delta(-a, b) = delta(b, -a)`` is the ``Kronecker delta``
(see ``TensorIndex`` for the conventions on indices).
If there is no metric it is not possible to raise or lower indices;
e.g. the index of the defining representation of ``SU(N)``
is 'covariant' and the conjugate representation is
'contravariant'; for ``N > 2`` they are linearly independent.
``eps_dim`` is by default equal to ``dim``, if the latter is an integer;
else it can be assigned (for use in naive dimensional regularization);
if ``eps_dim`` is not an integer ``epsilon`` is ``None``.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> Lorentz.metric
metric(Lorentz,Lorentz)
"""
def __new__(cls, name, metric=False, dim=None, eps_dim=None,
dummy_fmt=None):
if isinstance(name, string_types):
name = Symbol(name)
obj = Basic.__new__(cls, name, S.One if metric else S.Zero)
obj._name = str(name)
if not dummy_fmt:
obj._dummy_fmt = '%s_%%d' % obj.name
else:
obj._dummy_fmt = '%s_%%d' % dummy_fmt
if metric is None:
obj.metric_antisym = None
obj.metric = None
else:
if metric in (True, False, 0, 1):
metric_name = 'metric'
obj.metric_antisym = metric
else:
metric_name = metric.name
obj.metric_antisym = metric.antisym
sym2 = TensorSymmetry(get_symmetric_group_sgs(2, obj.metric_antisym))
S2 = TensorType([obj]*2, sym2)
obj.metric = S2(metric_name)
obj._dim = dim
obj._delta = obj.get_kronecker_delta()
obj._eps_dim = eps_dim if eps_dim else dim
obj._epsilon = obj.get_epsilon()
obj._autogenerated = []
return obj
@property
@deprecated(useinstead="TensorIndex", issue=12857, deprecated_since_version="1.1")
def auto_right(self):
if not hasattr(self, '_auto_right'):
self._auto_right = TensorIndex("auto_right", self)
return self._auto_right
@property
@deprecated(useinstead="TensorIndex", issue=12857, deprecated_since_version="1.1")
def auto_left(self):
if not hasattr(self, '_auto_left'):
self._auto_left = TensorIndex("auto_left", self)
return self._auto_left
@property
@deprecated(useinstead="TensorIndex", issue=12857, deprecated_since_version="1.1")
def auto_index(self):
if not hasattr(self, '_auto_index'):
self._auto_index = TensorIndex("auto_index", self)
return self._auto_index
@property
def data(self):
deprecate_data()
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
deprecate_data()
# This assignment is a bit controversial, should metric components be assigned
# to the metric only or also to the TensorIndexType object? The advantage here
# is the ability to assign a 1D array and transform it to a 2D diagonal array.
from .array import MutableDenseNDimArray
data = _TensorDataLazyEvaluator.parse_data(data)
if data.rank() > 2:
raise ValueError("data have to be of rank 1 (diagonal metric) or 2.")
if data.rank() == 1:
if self.dim is not None:
nda_dim = data.shape[0]
if nda_dim != self.dim:
raise ValueError("Dimension mismatch")
dim = data.shape[0]
newndarray = MutableDenseNDimArray.zeros(dim, dim)
for i, val in enumerate(data):
newndarray[i, i] = val
data = newndarray
dim1, dim2 = data.shape
if dim1 != dim2:
raise ValueError("Non-square matrix tensor.")
if self.dim is not None:
if self.dim != dim1:
raise ValueError("Dimension mismatch")
_tensor_data_substitution_dict[self] = data
_tensor_data_substitution_dict.add_metric_data(self.metric, data)
delta = self.get_kronecker_delta()
i1 = TensorIndex('i1', self)
i2 = TensorIndex('i2', self)
delta(i1, -i2).data = _TensorDataLazyEvaluator.parse_data(eye(dim1))
@data.deleter
def data(self):
deprecate_data()
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
if self.metric in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self.metric]
def _get_matrix_fmt(self, number):
return ("m" + self.dummy_fmt) % (number)
@property
def name(self):
return self._name
@property
def dim(self):
return self._dim
@property
def delta(self):
return self._delta
@property
def eps_dim(self):
return self._eps_dim
@property
def epsilon(self):
return self._epsilon
@property
def dummy_fmt(self):
return self._dummy_fmt
def get_kronecker_delta(self):
sym2 = TensorSymmetry(get_symmetric_group_sgs(2))
S2 = TensorType([self]*2, sym2)
delta = S2('KD')
return delta
def get_epsilon(self):
if not isinstance(self._eps_dim, (SYMPY_INTS, Integer)):
return None
sym = TensorSymmetry(get_symmetric_group_sgs(self._eps_dim, 1))
Sdim = TensorType([self]*self._eps_dim, sym)
epsilon = Sdim('Eps')
return epsilon
def __lt__(self, other):
return self.name < other.name
def __str__(self):
return self.name
__repr__ = __str__
def _components_data_full_destroy(self):
"""
EXPERIMENTAL: do not rely on this API method.
This destroys components data associated to the ``TensorIndexType``, if
any, specifically:
* metric tensor data
* Kronecker tensor data
"""
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def delete_tensmul_data(key):
if key in _tensor_data_substitution_dict._substitutions_dict_tensmul:
del _tensor_data_substitution_dict._substitutions_dict_tensmul[key]
# delete metric data:
delete_tensmul_data((self.metric, True, True))
delete_tensmul_data((self.metric, True, False))
delete_tensmul_data((self.metric, False, True))
delete_tensmul_data((self.metric, False, False))
# delete delta tensor data:
delta = self.get_kronecker_delta()
if delta in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[delta]
[docs]class TensorIndex(Basic):
"""
Represents an abstract tensor index.
Parameters
==========
name : name of the index, or ``True`` if you want it to be automatically assigned
tensortype : ``TensorIndexType`` of the index
is_up : flag for contravariant index
Attributes
==========
``name``
``tensortype``
``is_up``
Notes
=====
Tensor indices are contracted with the Einstein summation convention.
An index can be in contravariant or in covariant form; in the latter
case it is represented prepending a ``-`` to the index name.
Dummy indices have a name with head given by ``tensortype._dummy_fmt``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, TensorIndex, TensorSymmetry, TensorType, get_symmetric_group_sgs
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i = TensorIndex('i', Lorentz); i
i
>>> sym1 = TensorSymmetry(*get_symmetric_group_sgs(1))
>>> S1 = TensorType([Lorentz], sym1)
>>> A, B = S1('A,B')
>>> A(i)*B(-i)
A(L_0)*B(-L_0)
If you want the index name to be automatically assigned, just put ``True``
in the ``name`` field, it will be generated using the reserved character
``_`` in front of its name, in order to avoid conflicts with possible
existing indices:
>>> i0 = TensorIndex(True, Lorentz)
>>> i0
_i0
>>> i1 = TensorIndex(True, Lorentz)
>>> i1
_i1
>>> A(i0)*B(-i1)
A(_i0)*B(-_i1)
>>> A(i0)*B(-i0)
A(L_0)*B(-L_0)
"""
def __new__(cls, name, tensortype, is_up=True):
if isinstance(name, string_types):
name_symbol = Symbol(name)
elif isinstance(name, Symbol):
name_symbol = name
elif name is True:
name = "_i{0}".format(len(tensortype._autogenerated))
name_symbol = Symbol(name)
tensortype._autogenerated.append(name_symbol)
else:
raise ValueError("invalid name")
is_up = sympify(is_up)
obj = Basic.__new__(cls, name_symbol, tensortype, is_up)
obj._name = str(name)
obj._tensor_index_type = tensortype
obj._is_up = is_up
return obj
@property
def name(self):
return self._name
@property
@deprecated(useinstead="tensor_index_type", issue=12857, deprecated_since_version="1.1")
def tensortype(self):
return self.tensor_index_type
@property
def tensor_index_type(self):
return self._tensor_index_type
@property
def is_up(self):
return self._is_up
def _print(self):
s = self._name
if not self._is_up:
s = '-%s' % s
return s
def __lt__(self, other):
return (self.tensor_index_type, self._name) < (other.tensor_index_type, other._name)
def __neg__(self):
t1 = TensorIndex(self.name, self.tensor_index_type,
(not self.is_up))
return t1
[docs]def tensor_indices(s, typ):
"""
Returns list of tensor indices given their names and their types
Parameters
==========
s : string of comma separated names of indices
typ : ``TensorIndexType`` of the indices
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
"""
if isinstance(s, string_types):
a = [x.name for x in symbols(s, seq=True)]
else:
raise ValueError('expecting a string')
tilist = [TensorIndex(i, typ) for i in a]
if len(tilist) == 1:
return tilist[0]
return tilist
[docs]class TensorSymmetry(Basic):
"""
Monoterm symmetry of a tensor
Parameters
==========
bsgs : tuple ``(base, sgs)`` BSGS of the symmetry of the tensor
Attributes
==========
``base`` : base of the BSGS
``generators`` : generators of the BSGS
``rank`` : rank of the tensor
Notes
=====
A tensor can have an arbitrary monoterm symmetry provided by its BSGS.
Multiterm symmetries, like the cyclic symmetry of the Riemann tensor,
are not covered.
See Also
========
sympy.combinatorics.tensor_can.get_symmetric_group_sgs
Examples
========
Define a symmetric tensor
>>> from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = TensorSymmetry(get_symmetric_group_sgs(2))
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
"""
def __new__(cls, *args, **kw_args):
if len(args) == 1:
base, generators = args[0]
elif len(args) == 2:
base, generators = args
else:
raise TypeError("bsgs required, either two separate parameters or one tuple")
if not isinstance(base, Tuple):
base = Tuple(*base)
if not isinstance(generators, Tuple):
generators = Tuple(*generators)
obj = Basic.__new__(cls, base, generators, **kw_args)
return obj
@property
def base(self):
return self.args[0]
@property
def generators(self):
return self.args[1]
@property
def rank(self):
return self.args[1][0].size - 2
[docs]def tensorsymmetry(*args):
"""
Return a ``TensorSymmetry`` object.
One can represent a tensor with any monoterm slot symmetry group
using a BSGS.
``args`` can be a BSGS
``args[0]`` base
``args[1]`` sgs
Usually tensors are in (direct products of) representations
of the symmetric group;
``args`` can be a list of lists representing the shapes of Young tableaux
Notes
=====
For instance:
``[[1]]`` vector
``[[1]*n]`` symmetric tensor of rank ``n``
``[[n]]`` antisymmetric tensor of rank ``n``
``[[2, 2]]`` monoterm slot symmetry of the Riemann tensor
``[[1],[1]]`` vector*vector
``[[2],[1],[1]`` (antisymmetric tensor)*vector*vector
Notice that with the shape ``[2, 2]`` we associate only the monoterm
symmetries of the Riemann tensor; this is an abuse of notation,
since the shape ``[2, 2]`` corresponds usually to the irreducible
representation characterized by the monoterm symmetries and by the
cyclic symmetry.
Examples
========
Symmetric tensor using a Young tableau
>>> from sympy.tensor.tensor import TensorIndexType, TensorType, tensorsymmetry
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1, 1])
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
Symmetric tensor using a ``BSGS`` (base, strong generator set)
>>> from sympy.tensor.tensor import get_symmetric_group_sgs
>>> sym2 = tensorsymmetry(*get_symmetric_group_sgs(2))
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
"""
from sympy.combinatorics import Permutation
def tableau2bsgs(a):
if len(a) == 1:
# antisymmetric vector
n = a[0]
bsgs = get_symmetric_group_sgs(n, 1)
else:
if all(x == 1 for x in a):
# symmetric vector
n = len(a)
bsgs = get_symmetric_group_sgs(n)
elif a == [2, 2]:
bsgs = riemann_bsgs
else:
raise NotImplementedError
return bsgs
if not args:
return TensorSymmetry(Tuple(), Tuple(Permutation(1)))
if len(args) == 2 and isinstance(args[1][0], Permutation):
return TensorSymmetry(args)
base, sgs = tableau2bsgs(args[0])
for a in args[1:]:
basex, sgsx = tableau2bsgs(a)
base, sgs = bsgs_direct_product(base, sgs, basex, sgsx)
return TensorSymmetry(Tuple(base, sgs))
[docs]class TensorType(Basic):
"""
Class of tensor types.
Parameters
==========
index_types : list of ``TensorIndexType`` of the tensor indices
symmetry : ``TensorSymmetry`` of the tensor
Attributes
==========
``index_types``
``symmetry``
``types`` : list of ``TensorIndexType`` without repetitions
Examples
========
Define a symmetric tensor
>>> from sympy.tensor.tensor import TensorIndexType, tensorsymmetry, TensorType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1, 1])
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
"""
is_commutative = False
def __new__(cls, index_types, symmetry, **kw_args):
assert symmetry.rank == len(index_types)
obj = Basic.__new__(cls, Tuple(*index_types), symmetry, **kw_args)
return obj
@property
def index_types(self):
return self.args[0]
@property
def symmetry(self):
return self.args[1]
@property
def types(self):
return sorted(set(self.index_types), key=lambda x: x.name)
def __str__(self):
return 'TensorType(%s)' % ([str(x) for x in self.index_types])
def __call__(self, s, comm=0):
"""
Return a TensorHead object or a list of TensorHead objects.
``s`` name or string of names
``comm``: commutation group number
see ``_TensorManager.set_comm``
Examples
========
Define symmetric tensors ``V``, ``W`` and ``G``, respectively
commuting, anticommuting and with no commutation symmetry
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorsymmetry, TensorType, canon_bp
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> sym2 = tensorsymmetry([1]*2)
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
>>> W = S2('W', 1)
>>> G = S2('G', 2)
>>> canon_bp(V(a, b)*V(-b, -a))
V(L_0, L_1)*V(-L_0, -L_1)
>>> canon_bp(W(a, b)*W(-b, -a))
0
"""
if isinstance(s, string_types):
names = [x.name for x in symbols(s, seq=True)]
else:
raise ValueError('expecting a string')
if len(names) == 1:
return TensorHead(names[0], self, comm)
else:
return [TensorHead(name, self, comm) for name in names]
def tensorhead(name, typ, sym=None, comm=0):
"""
Function generating tensorhead(s).
Parameters
==========
name : name or sequence of names (as in ``symbol``)
typ : index types
sym : same as ``*args`` in ``tensorsymmetry``
comm : commutation group number
see ``_TensorManager.set_comm``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[1]*2])
>>> A(a, -b)
A(a, -b)
If no symmetry parameter is provided, assume there are not index
symmetries:
>>> B = tensorhead('B', [Lorentz, Lorentz])
>>> B(a, -b)
B(a, -b)
"""
if sym is None:
sym = [[1] for i in range(len(typ))]
sym = tensorsymmetry(*sym)
S = TensorType(typ, sym)
th = S(name, comm)
return th
[docs]class TensorHead(Basic):
r"""
Tensor head of the tensor
Parameters
==========
name : name of the tensor
typ : list of TensorIndexType
comm : commutation group number
Attributes
==========
``name``
``index_types``
``rank``
``types`` : equal to ``typ.types``
``symmetry`` : equal to ``typ.symmetry``
``comm`` : commutation group
Notes
=====
A ``TensorHead`` belongs to a commutation group, defined by a
symbol on number ``comm`` (see ``_TensorManager.set_comm``);
tensors in a commutation group have the same commutation properties;
by default ``comm`` is ``0``, the group of the commuting tensors.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensorhead, TensorType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> A = tensorhead('A', [Lorentz, Lorentz], [[1],[1]])
Examples with ndarray values, the components data assigned to the
``TensorHead`` object are assumed to be in a fully-contravariant
representation. In case it is necessary to assign components data which
represents the values of a non-fully covariant tensor, see the other
examples.
>>> from sympy.tensor.tensor import tensor_indices, tensorhead
>>> from sympy import diag
>>> i0, i1 = tensor_indices('i0:2', Lorentz)
Specify a replacement dictionary to keep track of the arrays to use for
replacements in the tensorial expression. The ``TensorIndexType`` is
associated to the metric used for contractions (in fully covariant form):
>>> repl = {Lorentz: diag(1, -1, -1, -1)}
Let's see some examples of working with components with the electromagnetic
tensor:
>>> from sympy import symbols
>>> Ex, Ey, Ez, Bx, By, Bz = symbols('E_x E_y E_z B_x B_y B_z')
>>> c = symbols('c', positive=True)
Let's define `F`, an antisymmetric tensor, we have to assign an
antisymmetric matrix to it, because `[[2]]` stands for the Young tableau
representation of an antisymmetric set of two elements:
>>> F = tensorhead('F', [Lorentz, Lorentz], [[2]])
Let's update the dictionary to contain the matrix to use in the
replacements:
>>> repl.update({F(-i0, -i1): [
... [0, Ex/c, Ey/c, Ez/c],
... [-Ex/c, 0, -Bz, By],
... [-Ey/c, Bz, 0, -Bx],
... [-Ez/c, -By, Bx, 0]]})
Now it is possible to retrieve the contravariant form of the Electromagnetic
tensor:
>>> F(i0, i1).replace_with_arrays(repl, [i0, i1])
[[0, -E_x/c, -E_y/c, -E_z/c], [E_x/c, 0, -B_z, B_y], [E_y/c, B_z, 0, -B_x], [E_z/c, -B_y, B_x, 0]]
and the mixed contravariant-covariant form:
>>> F(i0, -i1).replace_with_arrays(repl, [i0, -i1])
[[0, E_x/c, E_y/c, E_z/c], [E_x/c, 0, B_z, -B_y], [E_y/c, -B_z, 0, B_x], [E_z/c, B_y, -B_x, 0]]
Energy-momentum of a particle may be represented as:
>>> from sympy import symbols
>>> P = tensorhead('P', [Lorentz], [[1]])
>>> E, px, py, pz = symbols('E p_x p_y p_z', positive=True)
>>> repl.update({P(i0): [E, px, py, pz]})
The contravariant and covariant components are, respectively:
>>> P(i0).replace_with_arrays(repl, [i0])
[E, p_x, p_y, p_z]
>>> P(-i0).replace_with_arrays(repl, [-i0])
[E, -p_x, -p_y, -p_z]
The contraction of a 1-index tensor by itself:
>>> expr = P(i0)*P(-i0)
>>> expr.replace_with_arrays(repl, [])
E**2 - p_x**2 - p_y**2 - p_z**2
"""
is_commutative = False
def __new__(cls, name, typ, comm=0, **kw_args):
if isinstance(name, string_types):
name_symbol = Symbol(name)
elif isinstance(name, Symbol):
name_symbol = name
else:
raise ValueError("invalid name")
comm2i = TensorManager.comm_symbols2i(comm)
obj = Basic.__new__(cls, name_symbol, typ, **kw_args)
obj._name = obj.args[0].name
obj._rank = len(obj.index_types)
obj._symmetry = typ.symmetry
obj._comm = comm2i
return obj
@property
def name(self):
return self._name
@property
def rank(self):
return self._rank
@property
def symmetry(self):
return self._symmetry
@property
def typ(self):
return self.args[1]
@property
def comm(self):
return self._comm
@property
def types(self):
return self.args[1].types[:]
@property
def index_types(self):
return self.args[1].index_types[:]
def __lt__(self, other):
return (self.name, self.index_types) < (other.name, other.index_types)
[docs] def commutes_with(self, other):
"""
Returns ``0`` if ``self`` and ``other`` commute, ``1`` if they anticommute.
Returns ``None`` if ``self`` and ``other`` neither commute nor anticommute.
"""
r = TensorManager.get_comm(self._comm, other._comm)
return r
def _print(self):
return '%s(%s)' %(self.name, ','.join([str(x) for x in self.index_types]))
def __call__(self, *indices, **kw_args):
"""
Returns a tensor with indices.
There is a special behavior in case of indices denoted by ``True``,
they are considered auto-matrix indices, their slots are automatically
filled, and confer to the tensor the behavior of a matrix or vector
upon multiplication with another tensor containing auto-matrix indices
of the same ``TensorIndexType``. This means indices get summed over the
same way as in matrix multiplication. For matrix behavior, define two
auto-matrix indices, for vector behavior define just one.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[1]*2])
>>> t = A(a, -b)
>>> t
A(a, -b)
"""
tensor = Tensor(self, indices, **kw_args)
return tensor.doit()
def __pow__(self, other):
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
if self.data is None:
raise ValueError("No power on abstract tensors.")
deprecate_data()
from .array import tensorproduct, tensorcontraction
metrics = [_.data for _ in self.args[1].args[0]]
marray = self.data
marraydim = marray.rank()
for metric in metrics:
marray = tensorproduct(marray, metric, marray)
marray = tensorcontraction(marray, (0, marraydim), (marraydim+1, marraydim+2))
return marray ** (Rational(1, 2) * other)
@property
def data(self):
deprecate_data()
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
deprecate_data()
_tensor_data_substitution_dict[self] = data
@data.deleter
def data(self):
deprecate_data()
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def __iter__(self):
deprecate_data()
return self.data.__iter__()
def _components_data_full_destroy(self):
"""
EXPERIMENTAL: do not rely on this API method.
Destroy components data associated to the ``TensorHead`` object, this
checks for attached components data, and destroys components data too.
"""
# do not garbage collect Kronecker tensor (it should be done by
# ``TensorIndexType`` garbage collection)
if self.name == "KD":
return
# the data attached to a tensor must be deleted only by the TensorHead
# destructor. If the TensorHead is deleted, it means that there are no
# more instances of that tensor anywhere.
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def _get_argtree_pos(expr, pos):
for p in pos:
expr = expr.args[p]
return expr
[docs]class TensExpr(Expr):
"""
Abstract base class for tensor expressions
Notes
=====
A tensor expression is an expression formed by tensors;
currently the sums of tensors are distributed.
A ``TensExpr`` can be a ``TensAdd`` or a ``TensMul``.
``TensAdd`` objects are put in canonical form using the Butler-Portugal
algorithm for canonicalization under monoterm symmetries.
``TensMul`` objects are formed by products of component tensors,
and include a coefficient, which is a SymPy expression.
In the internal representation contracted indices are represented
by ``(ipos1, ipos2, icomp1, icomp2)``, where ``icomp1`` is the position
of the component tensor with contravariant index, ``ipos1`` is the
slot which the index occupies in that component tensor.
Contracted indices are therefore nameless in the internal representation.
"""
_op_priority = 12.0
is_commutative = False
def __neg__(self):
return self*S.NegativeOne
def __abs__(self):
raise NotImplementedError
def __add__(self, other):
return TensAdd(self, other).doit()
def __radd__(self, other):
return TensAdd(other, self).doit()
def __sub__(self, other):
return TensAdd(self, -other).doit()
def __rsub__(self, other):
return TensAdd(other, -self).doit()
def __mul__(self, other):
"""
Multiply two tensors using Einstein summation convention.
If the two tensors have an index in common, one contravariant
and the other covariant, in their product the indices are summed
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t1 = p(m0)
>>> t2 = q(-m0)
>>> t1*t2
p(L_0)*q(-L_0)
"""
return TensMul(self, other).doit()
def __rmul__(self, other):
return TensMul(other, self).doit()
def __div__(self, other):
other = _sympify(other)
if isinstance(other, TensExpr):
raise ValueError('cannot divide by a tensor')
return TensMul(self, S.One/other).doit()
def __rdiv__(self, other):
raise ValueError('cannot divide by a tensor')
def __pow__(self, other):
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
if self.data is None:
raise ValueError("No power without ndarray data.")
deprecate_data()
from .array import tensorproduct, tensorcontraction
free = self.free
marray = self.data
mdim = marray.rank()
for metric in free:
marray = tensorcontraction(
tensorproduct(
marray,
metric[0].tensor_index_type.data,
marray),
(0, mdim), (mdim+1, mdim+2)
)
return marray ** (Rational(1, 2) * other)
def __rpow__(self, other):
raise NotImplementedError
__truediv__ = __div__
__rtruediv__ = __rdiv__
[docs] def fun_eval(self, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
``index_types`` list of tuples ``(old_index, new_index)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.fun_eval((i, k),(-j, l))
A(k, L_0)*B(-L_0, l)
"""
expr = self.xreplace(dict(index_tuples))
expr = expr.replace(lambda x: isinstance(x, Tensor), lambda x: x.args[0](*x.args[1]))
# For some reason, `TensMul` gets replaced by `Mul`, correct it:
expr = expr.replace(lambda x: isinstance(x, (Mul, TensMul)), lambda x: TensMul(*x.args).doit())
return expr
[docs] def get_matrix(self):
"""
DEPRECATED: do not use.
Returns ndarray components data as a matrix, if components data are
available and ndarray dimension does not exceed 2.
"""
from sympy import Matrix
deprecate_data()
if 0 < self.rank <= 2:
rows = self.data.shape[0]
columns = self.data.shape[1] if self.rank == 2 else 1
if self.rank == 2:
mat_list = [] * rows
for i in range(rows):
mat_list.append([])
for j in range(columns):
mat_list[i].append(self[i, j])
else:
mat_list = [None] * rows
for i in range(rows):
mat_list[i] = self[i]
return Matrix(mat_list)
else:
raise NotImplementedError(
"missing multidimensional reduction to matrix.")
@staticmethod
def _get_indices_permutation(indices1, indices2):
return [indices1.index(i) for i in indices2]
def expand(self, **hints):
return _expand(self, **hints).doit()
def _expand(self, **kwargs):
return self
def _get_free_indices_set(self):
indset = set([])
for arg in self.args:
if isinstance(arg, TensExpr):
indset.update(arg._get_free_indices_set())
return indset
def _get_dummy_indices_set(self):
indset = set([])
for arg in self.args:
if isinstance(arg, TensExpr):
indset.update(arg._get_dummy_indices_set())
return indset
def _get_indices_set(self):
indset = set([])
for arg in self.args:
if isinstance(arg, TensExpr):
indset.update(arg._get_indices_set())
return indset
@property
def _iterate_dummy_indices(self):
dummy_set = self._get_dummy_indices_set()
def recursor(expr, pos):
if isinstance(expr, TensorIndex):
if expr in dummy_set:
yield (expr, pos)
elif isinstance(expr, (Tuple, TensExpr)):
for p, arg in enumerate(expr.args):
for i in recursor(arg, pos+(p,)):
yield i
return recursor(self, ())
@property
def _iterate_free_indices(self):
free_set = self._get_free_indices_set()
def recursor(expr, pos):
if isinstance(expr, TensorIndex):
if expr in free_set:
yield (expr, pos)
elif isinstance(expr, (Tuple, TensExpr)):
for p, arg in enumerate(expr.args):
for i in recursor(arg, pos+(p,)):
yield i
return recursor(self, ())
@property
def _iterate_indices(self):
def recursor(expr, pos):
if isinstance(expr, TensorIndex):
yield (expr, pos)
elif isinstance(expr, (Tuple, TensExpr)):
for p, arg in enumerate(expr.args):
for i in recursor(arg, pos+(p,)):
yield i
return recursor(self, ())
@staticmethod
def _match_indices_with_other_tensor(array, free_ind1, free_ind2, replacement_dict):
from .array import tensorcontraction, tensorproduct, permutedims
index_types1 = [i.tensor_index_type for i in free_ind1]
# Check if variance of indices needs to be fixed:
pos2up = []
pos2down = []
free2remaining = free_ind2[:]
for pos1, index1 in enumerate(free_ind1):
if index1 in free2remaining:
pos2 = free2remaining.index(index1)
free2remaining[pos2] = None
continue
if -index1 in free2remaining:
pos2 = free2remaining.index(-index1)
free2remaining[pos2] = None
free_ind2[pos2] = index1
if index1.is_up:
pos2up.append(pos2)
else:
pos2down.append(pos2)
else:
index2 = free2remaining[pos1]
if index2 is None:
raise ValueError("incompatible indices: %s and %s" % (free_ind1, free_ind2))
free2remaining[pos1] = None
free_ind2[pos1] = index1
if index1.is_up ^ index2.is_up:
if index1.is_up:
pos2up.append(pos1)
else:
pos2down.append(pos1)
if len(set(free_ind1) & set(free_ind2)) < len(free_ind1):
raise ValueError("incompatible indices: %s and %s" % (free_ind1, free_ind2))
# TODO: add possibility of metric after (spinors)
def contract_and_permute(metric, array, pos):
array = tensorcontraction(tensorproduct(metric, array), (1, 2+pos))
permu = list(range(len(free_ind1)))
permu[0], permu[pos] = permu[pos], permu[0]
return permutedims(array, permu)
# Raise indices:
for pos in pos2up:
metric = replacement_dict[index_types1[pos]]
metric_inverse = _TensorDataLazyEvaluator.inverse_matrix(metric)
array = contract_and_permute(metric_inverse, array, pos)
# Lower indices:
for pos in pos2down:
metric = replacement_dict[index_types1[pos]]
array = contract_and_permute(metric, array, pos)
if free_ind1:
permutation = TensExpr._get_indices_permutation(free_ind2, free_ind1)
array = permutedims(array, permutation)
if hasattr(array, "rank") and array.rank() == 0:
array = array[()]
return free_ind2, array
[docs] def replace_with_arrays(self, replacement_dict, indices):
"""
Replace the tensorial expressions with arrays. The final array will
correspond to the N-dimensional array with indices arranged according
to ``indices``.
Parameters
==========
replacement_dict
dictionary containing the replacement rules for tensors.
indices
the index order with respect to which the array is read.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices
>>> from sympy.tensor.tensor import tensorhead
>>> from sympy import symbols, diag
>>> L = TensorIndexType("L")
>>> i, j = tensor_indices("i j", L)
>>> A = tensorhead("A", [L], [[1]])
>>> A(i).replace_with_arrays({A(i): [1, 2]}, [i])
[1, 2]
>>> expr = A(i)*A(j)
>>> expr.replace_with_arrays({A(i): [1, 2]}, [i, j])
[[1, 2], [2, 4]]
For contractions, specify the metric of the ``TensorIndexType``, which
in this case is ``L``, in its covariant form:
>>> expr = A(i)*A(-i)
>>> expr.replace_with_arrays({A(i): [1, 2], L: diag(1, -1)}, [])
-3
Symmetrization of an array:
>>> H = tensorhead("H", [L, L], [[1], [1]])
>>> a, b, c, d = symbols("a b c d")
>>> expr = H(i, j)/2 + H(j, i)/2
>>> expr.replace_with_arrays({H(i, j): [[a, b], [c, d]]}, [i, j])
[[a, b/2 + c/2], [b/2 + c/2, d]]
Anti-symmetrization of an array:
>>> expr = H(i, j)/2 - H(j, i)/2
>>> repl = {H(i, j): [[a, b], [c, d]]}
>>> expr.replace_with_arrays(repl, [i, j])
[[0, b/2 - c/2], [-b/2 + c/2, 0]]
The same expression can be read as the transpose by inverting ``i`` and
``j``:
>>> expr.replace_with_arrays(repl, [j, i])
[[0, -b/2 + c/2], [b/2 - c/2, 0]]
"""
from .array import Array
replacement_dict = {tensor: Array(array) for tensor, array in replacement_dict.items()}
# Check dimensions of replaced arrays:
for tensor, array in replacement_dict.items():
if isinstance(tensor, TensorIndexType):
expected_shape = [tensor.dim for i in range(2)]
else:
expected_shape = [index_type.dim for index_type in tensor.index_types]
if len(expected_shape) != array.rank() or (not all([dim1 == dim2 if
dim1 is not None else True for dim1, dim2 in zip(expected_shape,
array.shape)])):
raise ValueError("shapes for tensor %s expected to be %s, "\
"replacement array shape is %s" % (tensor, expected_shape,
array.shape))
ret_indices, array = self._extract_data(replacement_dict)
last_indices, array = self._match_indices_with_other_tensor(array, indices, ret_indices, replacement_dict)
#permutation = self._get_indices_permutation(indices, ret_indices)
#if not hasattr(array, "rank"):
#return array
#if array.rank() == 0:
#array = array[()]
#return array
#array = permutedims(array, permutation)
return array
def _check_add_Sum(self, expr, index_symbols):
from sympy import Sum
indices = self.get_indices()
dum = self.dum
sum_indices = [ (index_symbols[i], 0,
indices[i].tensor_index_type.dim-1) for i, j in dum]
if sum_indices:
expr = Sum(expr, *sum_indices)
return expr
[docs]class TensAdd(TensExpr, AssocOp):
"""
Sum of tensors
Parameters
==========
free_args : list of the free indices
Attributes
==========
``args`` : tuple of addends
``rank`` : rank of the tensor
``free_args`` : list of the free indices in sorted order
Notes
=====
Sum of more than one tensor are put automatically in canonical form.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensorhead, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(a) + q(a); t
p(a) + q(a)
>>> t(b)
p(b) + q(b)
Examples with components data added to the tensor expression:
>>> from sympy import symbols, diag
>>> x, y, z, t = symbols("x y z t")
>>> repl = {}
>>> repl[Lorentz] = diag(1, -1, -1, -1)
>>> repl[p(a)] = [1, 2, 3, 4]
>>> repl[q(a)] = [x, y, z, t]
The following are: 2**2 - 3**2 - 2**2 - 7**2 ==> -58
>>> expr = p(a) + q(a)
>>> expr.replace_with_arrays(repl, [a])
[x + 1, y + 2, z + 3, t + 4]
"""
def __new__(cls, *args, **kw_args):
args = [_sympify(x) for x in args if x]
args = TensAdd._tensAdd_flatten(args)
obj = Basic.__new__(cls, *args, **kw_args)
return obj
def doit(self, **kwargs):
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
if not args:
return S.Zero
if len(args) == 1 and not isinstance(args[0], TensExpr):
return args[0]
# now check that all addends have the same indices:
TensAdd._tensAdd_check(args)
# if TensAdd has only 1 element in its `args`:
if len(args) == 1: # and isinstance(args[0], TensMul):
return args[0]
# Remove zeros:
args = [x for x in args if x]
# if there are no more args (i.e. have cancelled out),
# just return zero:
if not args:
return S.Zero
if len(args) == 1:
return args[0]
# Collect terms appearing more than once, differing by their coefficients:
args = TensAdd._tensAdd_collect_terms(args)
# collect canonicalized terms
def sort_key(t):
x = get_index_structure(t)
if not isinstance(t, TensExpr):
return ([], [], [])
return (t.components, x.free, x.dum)
args.sort(key=sort_key)
if not args:
return S.Zero
# it there is only a component tensor return it
if len(args) == 1:
return args[0]
obj = self.func(*args)
return obj
@staticmethod
def _tensAdd_flatten(args):
# flatten TensAdd, coerce terms which are not tensors to tensors
a = []
for x in args:
if isinstance(x, (Add, TensAdd)):
a.extend(list(x.args))
else:
a.append(x)
args = [x for x in a if x.coeff]
return args
@staticmethod
def _tensAdd_check(args):
# check that all addends have the same free indices
indices0 = set([x[0] for x in get_index_structure(args[0]).free])
list_indices = [set([y[0] for y in get_index_structure(x).free]) for x in args[1:]]
if not all(x == indices0 for x in list_indices):
raise ValueError('all tensors must have the same indices')
@staticmethod
def _tensAdd_collect_terms(args):
# collect TensMul terms differing at most by their coefficient
terms_dict = defaultdict(list)
scalars = S.Zero
if isinstance(args[0], TensExpr):
free_indices = set(args[0].get_free_indices())
else:
free_indices = set([])
for arg in args:
if not isinstance(arg, TensExpr):
if free_indices != set([]):
raise ValueError("wrong valence")
scalars += arg
continue
if free_indices != set(arg.get_free_indices()):
raise ValueError("wrong valence")
# TODO: what is the part which is not a coeff?
# needs an implementation similar to .as_coeff_Mul()
terms_dict[arg.nocoeff].append(arg.coeff)
new_args = [TensMul(Add(*coeff), t).doit() for t, coeff in terms_dict.items() if Add(*coeff) != 0]
if isinstance(scalars, Add):
new_args = list(scalars.args) + new_args
elif scalars != 0:
new_args = [scalars] + new_args
return new_args
def get_indices(self):
indices = []
for arg in self.args:
indices.extend([i for i in get_indices(arg) if i not in indices])
return indices
@property
def rank(self):
return self.args[0].rank
@property
def free_args(self):
return self.args[0].free_args
def _expand(self, **hints):
return TensAdd(*[_expand(i, **hints) for i in self.args])
def __call__(self, *indices):
"""Returns tensor with ordered free indices replaced by ``indices``
Parameters
==========
indices
Examples
========
>>> from sympy import Symbol
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> D = Symbol('D')
>>> Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> g = Lorentz.metric
>>> t = p(i0)*p(i1) + g(i0,i1)*q(i2)*q(-i2)
>>> t(i0,i2)
metric(i0, i2)*q(L_0)*q(-L_0) + p(i0)*p(i2)
>>> from sympy.tensor.tensor import canon_bp
>>> canon_bp(t(i0,i1) - t(i1,i0))
0
"""
free_args = self.free_args
indices = list(indices)
if [x.tensor_index_type for x in indices] != [x.tensor_index_type for x in free_args]:
raise ValueError('incompatible types')
if indices == free_args:
return self
index_tuples = list(zip(free_args, indices))
a = [x.func(*x.fun_eval(*index_tuples).args) for x in self.args]
res = TensAdd(*a).doit()
return res
[docs] def canon_bp(self):
"""
canonicalize using the Butler-Portugal algorithm for canonicalization
under monoterm symmetries.
"""
expr = self.expand()
args = [canon_bp(x) for x in expr.args]
res = TensAdd(*args).doit()
return res
def equals(self, other):
other = _sympify(other)
if isinstance(other, TensMul) and other._coeff == 0:
return all(x._coeff == 0 for x in self.args)
if isinstance(other, TensExpr):
if self.rank != other.rank:
return False
if isinstance(other, TensAdd):
if set(self.args) != set(other.args):
return False
else:
return True
t = self - other
if not isinstance(t, TensExpr):
return t == 0
else:
if isinstance(t, TensMul):
return t._coeff == 0
else:
return all(x._coeff == 0 for x in t.args)
def __getitem__(self, item):
deprecate_data()
return self.data[item]
def contract_delta(self, delta):
args = [x.contract_delta(delta) for x in self.args]
t = TensAdd(*args).doit()
return canon_bp(t)
[docs] def contract_metric(self, g):
"""
Raise or lower indices with the metric ``g``
Parameters
==========
g : metric
contract_all : if True, eliminate all ``g`` which are contracted
Notes
=====
see the ``TensorIndexType`` docstring for the contraction conventions
"""
args = [contract_metric(x, g) for x in self.args]
t = TensAdd(*args).doit()
return canon_bp(t)
[docs] def fun_eval(self, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
Parameters
==========
index_types : list of tuples ``(old_index, new_index)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j) + A(i, -j)
>>> t.fun_eval((i, k),(-j, l))
A(k, L_0)*B(-L_0, l) + A(k, l)
"""
args = self.args
args1 = []
for x in args:
y = x.fun_eval(*index_tuples)
args1.append(y)
return TensAdd(*args1).doit()
[docs] def substitute_indices(self, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
Parameters
==========
index_types : list of tuples ``(old_index, new_index)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.substitute_indices((i,j), (j, k))
A(j, L_0)*B(-L_0, -k)
"""
args = self.args
args1 = []
for x in args:
y = x.substitute_indices(*index_tuples)
args1.append(y)
return TensAdd(*args1).doit()
def _print(self):
a = []
args = self.args
for x in args:
a.append(str(x))
a.sort()
s = ' + '.join(a)
s = s.replace('+ -', '- ')
return s
def _extract_data(self, replacement_dict):
from sympy.tensor.array import Array, permutedims
args_indices, arrays = zip(*[
arg._extract_data(replacement_dict) if
isinstance(arg, TensExpr) else ([], arg) for arg in self.args
])
arrays = [Array(i) for i in arrays]
ref_indices = args_indices[0]
for i in range(1, len(args_indices)):
indices = args_indices[i]
array = arrays[i]
permutation = TensMul._get_indices_permutation(indices, ref_indices)
arrays[i] = permutedims(array, permutation)
return ref_indices, sum(arrays, Array.zeros(*array.shape))
@property
def data(self):
deprecate_data()
return _tensor_data_substitution_dict[self.expand()]
@data.setter
def data(self, data):
deprecate_data()
_tensor_data_substitution_dict[self] = data
@data.deleter
def data(self):
deprecate_data()
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def __iter__(self):
deprecate_data()
if not self.data:
raise ValueError("No iteration on abstract tensors")
return self.data.flatten().__iter__()
def _eval_rewrite_as_Indexed(self, *args):
return Add.fromiter(args)
class Tensor(TensExpr):
"""
Base tensor class, i.e. this represents a tensor, the single unit to be
put into an expression.
This object is usually created from a ``TensorHead``, by attaching indices
to it. Indices preceded by a minus sign are considered contravariant,
otherwise covariant.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType("Lorentz", dummy_fmt="L")
>>> mu, nu = tensor_indices('mu nu', Lorentz)
>>> A = tensorhead("A", [Lorentz, Lorentz], [[1], [1]])
>>> A(mu, -nu)
A(mu, -nu)
>>> A(mu, -mu)
A(L_0, -L_0)
"""
is_commutative = False
def __new__(cls, tensor_head, indices, **kw_args):
is_canon_bp = kw_args.pop('is_canon_bp', False)
indices = cls._parse_indices(tensor_head, indices)
obj = Basic.__new__(cls, tensor_head, Tuple(*indices), **kw_args)
obj._index_structure = _IndexStructure.from_indices(*indices)
obj._free_indices_set = set(obj._index_structure.get_free_indices())
if tensor_head.rank != len(indices):
raise ValueError("wrong number of indices")
obj._indices = indices
obj._is_canon_bp = is_canon_bp
obj._index_map = Tensor._build_index_map(indices, obj._index_structure)
return obj
@staticmethod
def _build_index_map(indices, index_structure):
index_map = {}
for idx in indices:
index_map[idx] = (indices.index(idx),)
return index_map
def doit(self, **kwargs):
args, indices, free, dum = TensMul._tensMul_contract_indices([self])
return args[0]
@staticmethod
def _parse_indices(tensor_head, indices):
if not isinstance(indices, (tuple, list, Tuple)):
raise TypeError("indices should be an array, got %s" % type(indices))
indices = list(indices)
for i, index in enumerate(indices):
if isinstance(index, Symbol):
indices[i] = TensorIndex(index, tensor_head.index_types[i], True)
elif isinstance(index, Mul):
c, e = index.as_coeff_Mul()
if c == -1 and isinstance(e, Symbol):
indices[i] = TensorIndex(e, tensor_head.index_types[i], False)
else:
raise ValueError("index not understood: %s" % index)
elif not isinstance(index, TensorIndex):
raise TypeError("wrong type for index: %s is %s" % (index, type(index)))
return indices
def _set_new_index_structure(self, im, is_canon_bp=False):
indices = im.get_indices()
return self._set_indices(*indices, is_canon_bp=is_canon_bp)
def _set_indices(self, *indices, **kw_args):
if len(indices) != self.ext_rank:
raise ValueError("indices length mismatch")
return self.func(self.args[0], indices, is_canon_bp=kw_args.pop('is_canon_bp', False)).doit()
def _get_free_indices_set(self):
return set([i[0] for i in self._index_structure.free])
def _get_dummy_indices_set(self):
dummy_pos = set(itertools.chain(*self._index_structure.dum))
return set(idx for i, idx in enumerate(self.args[1]) if i in dummy_pos)
def _get_indices_set(self):
return set(self.args[1].args)
@property
def is_canon_bp(self):
return self._is_canon_bp
@property
def indices(self):
return self._indices
@property
def free(self):
return self._index_structure.free[:]
@property
def free_in_args(self):
return [(ind, pos, 0) for ind, pos in self.free]
@property
def dum(self):
return self._index_structure.dum[:]
@property
def dum_in_args(self):
return [(p1, p2, 0, 0) for p1, p2 in self.dum]
@property
def rank(self):
return len(self.free)
@property
def ext_rank(self):
return self._index_structure._ext_rank
@property
def free_args(self):
return sorted([x[0] for x in self.free])
def commutes_with(self, other):
"""
:param other:
:return:
0 commute
1 anticommute
None neither commute nor anticommute
"""
if not isinstance(other, TensExpr):
return 0
elif isinstance(other, Tensor):
return self.component.commutes_with(other.component)
return NotImplementedError
def perm2tensor(self, g, is_canon_bp=False):
"""
Returns the tensor corresponding to the permutation ``g``
For further details, see the method in ``TIDS`` with the same name.
"""
return perm2tensor(self, g, is_canon_bp)
def canon_bp(self):
if self._is_canon_bp:
return self
expr = self.expand()
g, dummies, msym = expr._index_structure.indices_canon_args()
v = components_canon_args([expr.component])
can = canonicalize(g, dummies, msym, *v)
if can == 0:
return S.Zero
tensor = self.perm2tensor(can, True)
return tensor
@property
def index_types(self):
return list(self.component.index_types)
@property
def coeff(self):
return S.One
@property
def nocoeff(self):
return self
@property
def component(self):
return self.args[0]
@property
def components(self):
return [self.args[0]]
def split(self):
return [self]
def _expand(self, **kwargs):
return self
def sorted_components(self):
return self
def get_indices(self):
"""
Get a list of indices, corresponding to those of the tensor.
"""
return list(self.args[1])
def get_free_indices(self):
"""
Get a list of free indices, corresponding to those of the tensor.
"""
return self._index_structure.get_free_indices()
def as_base_exp(self):
return self, S.One
def substitute_indices(self, *index_tuples):
return substitute_indices(self, *index_tuples)
def __call__(self, *indices):
"""Returns tensor with ordered free indices replaced by ``indices``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> A = tensorhead('A', [Lorentz]*5, [[1]*5])
>>> t = A(i2, i1, -i2, -i3, i4)
>>> t
A(L_0, i1, -L_0, -i3, i4)
>>> t(i1, i2, i3)
A(L_0, i1, -L_0, i2, i3)
"""
free_args = self.free_args
indices = list(indices)
if [x.tensor_index_type for x in indices] != [x.tensor_index_type for x in free_args]:
raise ValueError('incompatible types')
if indices == free_args:
return self
t = self.fun_eval(*list(zip(free_args, indices)))
# object is rebuilt in order to make sure that all contracted indices
# get recognized as dummies, but only if there are contracted indices.
if len(set(i if i.is_up else -i for i in indices)) != len(indices):
return t.func(*t.args)
return t
# TODO: put this into TensExpr?
def __iter__(self):
deprecate_data()
return self.data.__iter__()
# TODO: put this into TensExpr?
def __getitem__(self, item):
deprecate_data()
return self.data[item]
def _extract_data(self, replacement_dict):
from .array import Array
for k, v in replacement_dict.items():
if isinstance(k, Tensor) and k.args[0] == self.args[0]:
other = k
array = v
break
else:
raise ValueError("%s not found in %s" % (self, replacement_dict))
# TODO: inefficient, this should be done at root level only:
replacement_dict = {k: Array(v) for k, v in replacement_dict.items()}
array = Array(array)
dum1 = self.dum
dum2 = other.dum
if len(dum2) > 0:
for pair in dum2:
# allow `dum2` if the contained values are also in `dum1`.
if pair not in dum1:
raise NotImplementedError("%s with contractions is not implemented" % other)
# Remove elements in `dum2` from `dum1`:
dum1 = [pair for pair in dum1 if pair not in dum2]
if len(dum1) > 0:
indices2 = other.get_indices()
repl = {}
for p1, p2 in dum1:
repl[indices2[p2]] = -indices2[p1]
other = other.xreplace(repl).doit()
array = _TensorDataLazyEvaluator.data_contract_dum([array], dum1, len(indices2))
free_ind1 = self.get_free_indices()
free_ind2 = other.get_free_indices()
return self._match_indices_with_other_tensor(array, free_ind1, free_ind2, replacement_dict)
@property
def data(self):
deprecate_data()
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
deprecate_data()
# TODO: check data compatibility with properties of tensor.
_tensor_data_substitution_dict[self] = data
@data.deleter
def data(self):
deprecate_data()
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
if self.metric in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self.metric]
def _print(self):
indices = [str(ind) for ind in self.indices]
component = self.component
if component.rank > 0:
return ('%s(%s)' % (component.name, ', '.join(indices)))
else:
return ('%s' % component.name)
def equals(self, other):
if other == 0:
return self.coeff == 0
other = _sympify(other)
if not isinstance(other, TensExpr):
assert not self.components
return S.One == other
def _get_compar_comp(self):
t = self.canon_bp()
r = (t.coeff, tuple(t.components), \
tuple(sorted(t.free)), tuple(sorted(t.dum)))
return r
return _get_compar_comp(self) == _get_compar_comp(other)
def contract_metric(self, g):
# if metric is not the same, ignore this step:
if self.component != g:
return self
# in case there are free components, do not perform anything:
if len(self.free) != 0:
return self
antisym = g.index_types[0].metric_antisym
sign = S.One
typ = g.index_types[0]
if not antisym:
# g(i, -i)
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
else:
# g(i, -i)
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
dp0, dp1 = self.dum[0]
if dp0 < dp1:
# g(i, -i) = -D with antisymmetric metric
sign = -sign
return sign
def contract_delta(self, metric):
return self.contract_metric(metric)
def _eval_rewrite_as_Indexed(self, tens, indices):
from sympy import Indexed
# TODO: replace .args[0] with .name:
index_symbols = [i.args[0] for i in self.get_indices()]
expr = Indexed(tens.args[0], *index_symbols)
return self._check_add_Sum(expr, index_symbols)
[docs]class TensMul(TensExpr, AssocOp):
"""
Product of tensors
Parameters
==========
coeff : SymPy coefficient of the tensor
args
Attributes
==========
``components`` : list of ``TensorHead`` of the component tensors
``types`` : list of nonrepeated ``TensorIndexType``
``free`` : list of ``(ind, ipos, icomp)``, see Notes
``dum`` : list of ``(ipos1, ipos2, icomp1, icomp2)``, see Notes
``ext_rank`` : rank of the tensor counting the dummy indices
``rank`` : rank of the tensor
``coeff`` : SymPy coefficient of the tensor
``free_args`` : list of the free indices in sorted order
``is_canon_bp`` : ``True`` if the tensor in in canonical form
Notes
=====
``args[0]`` list of ``TensorHead`` of the component tensors.
``args[1]`` list of ``(ind, ipos, icomp)``
where ``ind`` is a free index, ``ipos`` is the slot position
of ``ind`` in the ``icomp``-th component tensor.
``args[2]`` list of tuples representing dummy indices.
``(ipos1, ipos2, icomp1, icomp2)`` indicates that the contravariant
dummy index is the ``ipos1``-th slot position in the ``icomp1``-th
component tensor; the corresponding covariant index is
in the ``ipos2`` slot position in the ``icomp2``-th component tensor.
"""
identity = S.One
def __new__(cls, *args, **kw_args):
is_canon_bp = kw_args.get('is_canon_bp', False)
args = list(map(_sympify, args))
# Flatten:
args = [i for arg in args for i in (arg.args if isinstance(arg, (TensMul, Mul)) else [arg])]
args, indices, free, dum = TensMul._tensMul_contract_indices(args, replace_indices=False)
# Data for indices:
index_types = [i.tensor_index_type for i in indices]
index_structure = _IndexStructure(free, dum, index_types, indices, canon_bp=is_canon_bp)
obj = TensExpr.__new__(cls, *args)
obj._indices = indices
obj._index_types = index_types
obj._index_structure = index_structure
obj._ext_rank = len(obj._index_structure.free) + 2*len(obj._index_structure.dum)
obj._coeff = S.One
obj._is_canon_bp = is_canon_bp
return obj
@staticmethod
def _indices_to_free_dum(args_indices):
free2pos1 = {}
free2pos2 = {}
dummy_data = []
indices = []
# Notation for positions (to better understand the code):
# `pos1`: position in the `args`.
# `pos2`: position in the indices.
# Example:
# A(i, j)*B(k, m, n)*C(p)
# `pos1` of `n` is 1 because it's in `B` (second `args` of TensMul).
# `pos2` of `n` is 4 because it's the fifth overall index.
# Counter for the index position wrt the whole expression:
pos2 = 0
for pos1, arg_indices in enumerate(args_indices):
for index_pos, index in enumerate(arg_indices):
if not isinstance(index, TensorIndex):
raise TypeError("expected TensorIndex")
if -index in free2pos1:
# Dummy index detected:
other_pos1 = free2pos1.pop(-index)
other_pos2 = free2pos2.pop(-index)
if index.is_up:
dummy_data.append((index, pos1, other_pos1, pos2, other_pos2))
else:
dummy_data.append((-index, other_pos1, pos1, other_pos2, pos2))
indices.append(index)
elif index in free2pos1:
raise ValueError("Repeated index: %s" % index)
else:
free2pos1[index] = pos1
free2pos2[index] = pos2
indices.append(index)
pos2 += 1
free = [(i, p) for (i, p) in free2pos2.items()]
free_names = [i.name for i in free2pos2.keys()]
dummy_data.sort(key=lambda x: x[3])
return indices, free, free_names, dummy_data
@staticmethod
def _dummy_data_to_dum(dummy_data):
return [(p2a, p2b) for (i, p1a, p1b, p2a, p2b) in dummy_data]
@staticmethod
def _tensMul_contract_indices(args, replace_indices=True):
replacements = [{} for _ in args]
#_index_order = all([_has_index_order(arg) for arg in args])
args_indices = [get_indices(arg) for arg in args]
indices, free, free_names, dummy_data = TensMul._indices_to_free_dum(args_indices)
cdt = defaultdict(int)
def dummy_fmt_gen(tensor_index_type):
fmt = tensor_index_type.dummy_fmt
nd = cdt[tensor_index_type]
cdt[tensor_index_type] += 1
return fmt % nd
if replace_indices:
for old_index, pos1cov, pos1contra, pos2cov, pos2contra in dummy_data:
index_type = old_index.tensor_index_type
while True:
dummy_name = dummy_fmt_gen(index_type)
if dummy_name not in free_names:
break
dummy = TensorIndex(dummy_name, index_type, True)
replacements[pos1cov][old_index] = dummy
replacements[pos1contra][-old_index] = -dummy
indices[pos2cov] = dummy
indices[pos2contra] = -dummy
args = [arg.xreplace(repl) for arg, repl in zip(args, replacements)]
dum = TensMul._dummy_data_to_dum(dummy_data)
return args, indices, free, dum
@staticmethod
def _get_components_from_args(args):
"""
Get a list of ``Tensor`` objects having the same ``TIDS`` if multiplied
by one another.
"""
components = []
for arg in args:
if not isinstance(arg, TensExpr):
continue
if isinstance(arg, TensAdd):
continue
components.extend(arg.components)
return components
@staticmethod
def _rebuild_tensors_list(args, index_structure):
indices = index_structure.get_indices()
#tensors = [None for i in components] # pre-allocate list
ind_pos = 0
for i, arg in enumerate(args):
if not isinstance(arg, TensExpr):
continue
prev_pos = ind_pos
ind_pos += arg.ext_rank
args[i] = Tensor(arg.component, indices[prev_pos:ind_pos])
def doit(self, **kwargs):
is_canon_bp = self._is_canon_bp
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
args = [arg for arg in args if arg != self.identity]
# Extract non-tensor coefficients:
coeff = reduce(lambda a, b: a*b, [arg for arg in args if not isinstance(arg, TensExpr)], S.One)
args = [arg for arg in args if isinstance(arg, TensExpr)]
if len(args) == 0:
return coeff
if coeff != self.identity:
args = [coeff] + args
if coeff == 0:
return S.Zero
if len(args) == 1:
return args[0]
args, indices, free, dum = TensMul._tensMul_contract_indices(args)
# Data for indices:
index_types = [i.tensor_index_type for i in indices]
index_structure = _IndexStructure(free, dum, index_types, indices, canon_bp=is_canon_bp)
obj = self.func(*args)
obj._index_types = index_types
obj._index_structure = index_structure
obj._ext_rank = len(obj._index_structure.free) + 2*len(obj._index_structure.dum)
obj._coeff = coeff
obj._is_canon_bp = is_canon_bp
return obj
# TODO: this method should be private
# TODO: should this method be renamed _from_components_free_dum ?
@staticmethod
def from_data(coeff, components, free, dum, **kw_args):
return TensMul(coeff, *TensMul._get_tensors_from_components_free_dum(components, free, dum), **kw_args).doit()
@staticmethod
def _get_tensors_from_components_free_dum(components, free, dum):
"""
Get a list of ``Tensor`` objects by distributing ``free`` and ``dum`` indices on the ``components``.
"""
index_structure = _IndexStructure.from_components_free_dum(components, free, dum)
indices = index_structure.get_indices()
tensors = [None for i in components] # pre-allocate list
# distribute indices on components to build a list of tensors:
ind_pos = 0
for i, component in enumerate(components):
prev_pos = ind_pos
ind_pos += component.rank
tensors[i] = Tensor(component, indices[prev_pos:ind_pos])
return tensors
def _get_free_indices_set(self):
return set([i[0] for i in self.free])
def _get_dummy_indices_set(self):
dummy_pos = set(itertools.chain(*self.dum))
return set(idx for i, idx in enumerate(self._index_structure.get_indices()) if i in dummy_pos)
def _get_position_offset_for_indices(self):
arg_offset = [None for i in range(self.ext_rank)]
counter = 0
for i, arg in enumerate(self.args):
if not isinstance(arg, TensExpr):
continue
for j in range(arg.ext_rank):
arg_offset[j + counter] = counter
counter += arg.ext_rank
return arg_offset
@property
def free_args(self):
return sorted([x[0] for x in self.free])
@property
def components(self):
return self._get_components_from_args(self.args)
@property
def free(self):
return self._index_structure.free[:]
@property
def free_in_args(self):
arg_offset = self._get_position_offset_for_indices()
argpos = self._get_indices_to_args_pos()
return [(ind, pos-arg_offset[pos], argpos[pos]) for (ind, pos) in self.free]
@property
def coeff(self):
return self._coeff
@property
def nocoeff(self):
return self.func(*[t for t in self.args if isinstance(t, TensExpr)]).doit()
@property
def dum(self):
return self._index_structure.dum[:]
@property
def dum_in_args(self):
arg_offset = self._get_position_offset_for_indices()
argpos = self._get_indices_to_args_pos()
return [(p1-arg_offset[p1], p2-arg_offset[p2], argpos[p1], argpos[p2]) for p1, p2 in self.dum]
@property
def rank(self):
return len(self.free)
@property
def ext_rank(self):
return self._ext_rank
@property
def index_types(self):
return self._index_types[:]
def equals(self, other):
if other == 0:
return self.coeff == 0
other = _sympify(other)
if not isinstance(other, TensExpr):
assert not self.components
return self._coeff == other
return self.canon_bp() == other.canon_bp()
[docs] def get_indices(self):
"""
Returns the list of indices of the tensor
The indices are listed in the order in which they appear in the
component tensors.
The dummy indices are given a name which does not collide with
the names of the free indices.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m1)*g(m0,m2)
>>> t.get_indices()
[m1, m0, m2]
>>> t2 = p(m1)*g(-m1, m2)
>>> t2.get_indices()
[L_0, -L_0, m2]
"""
return self._indices
[docs] def get_free_indices(self):
"""
Returns the list of free indices of the tensor
The indices are listed in the order in which they appear in the
component tensors.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m1)*g(m0,m2)
>>> t.get_free_indices()
[m1, m0, m2]
>>> t2 = p(m1)*g(-m1, m2)
>>> t2.get_free_indices()
[m2]
"""
return self._index_structure.get_free_indices()
[docs] def split(self):
"""
Returns a list of tensors, whose product is ``self``
Dummy indices contracted among different tensor components
become free indices with the same name as the one used to
represent the dummy indices.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(a,b)*B(-b,c)
>>> t
A(a, L_0)*B(-L_0, c)
>>> t.split()
[A(a, L_0), B(-L_0, c)]
"""
if self.args == ():
return [self]
splitp = []
res = 1
for arg in self.args:
if isinstance(arg, Tensor):
splitp.append(res*arg)
res = 1
else:
res *= arg
return splitp
def _expand(self, **hints):
# TODO: temporary solution, in the future this should be linked to
# `Expr.expand`.
args = [_expand(arg, **hints) for arg in self.args]
args1 = [arg.args if isinstance(arg, (Add, TensAdd)) else (arg,) for arg in args]
return TensAdd(*[
TensMul(*i) for i in itertools.product(*args1)]
)
def __neg__(self):
return TensMul(S.NegativeOne, self, is_canon_bp=self._is_canon_bp).doit()
def __getitem__(self, item):
deprecate_data()
return self.data[item]
def _get_args_for_traditional_printer(self):
args = list(self.args)
if (self.coeff < 0) == True:
# expressions like "-A(a)"
sign = "-"
if self.coeff == S.NegativeOne:
args = args[1:]
else:
args[0] = -args[0]
else:
sign = ""
return sign, args
def _sort_args_for_sorted_components(self):
"""
Returns the ``args`` sorted according to the components commutation
properties.
The sorting is done taking into account the commutation group
of the component tensors.
"""
cv = [arg for arg in self.args if isinstance(arg, TensExpr)]
sign = 1
n = len(cv) - 1
for i in range(n):
for j in range(n, i, -1):
c = cv[j-1].commutes_with(cv[j])
# if `c` is `None`, it does neither commute nor anticommute, skip:
if c not in [0, 1]:
continue
if (cv[j-1].component.types, cv[j-1].component.name) > \
(cv[j].component.types, cv[j].component.name):
cv[j-1], cv[j] = cv[j], cv[j-1]
# if `c` is 1, the anticommute, so change sign:
if c:
sign = -sign
coeff = sign * self.coeff
if coeff != 1:
return [coeff] + cv
return cv
[docs] def sorted_components(self):
"""
Returns a tensor product with sorted components.
"""
return TensMul(*self._sort_args_for_sorted_components()).doit()
[docs] def perm2tensor(self, g, is_canon_bp=False):
"""
Returns the tensor corresponding to the permutation ``g``
For further details, see the method in ``TIDS`` with the same name.
"""
return perm2tensor(self, g, is_canon_bp=is_canon_bp)
[docs] def canon_bp(self):
"""
Canonicalize using the Butler-Portugal algorithm for canonicalization
under monoterm symmetries.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[2]])
>>> t = A(m0,-m1)*A(m1,-m0)
>>> t.canon_bp()
-A(L_0, L_1)*A(-L_0, -L_1)
>>> t = A(m0,-m1)*A(m1,-m2)*A(m2,-m0)
>>> t.canon_bp()
0
"""
if self._is_canon_bp:
return self
expr = self.expand()
if isinstance(expr, TensAdd):
return expr.canon_bp()
if not expr.components:
return expr
t = expr.sorted_components()
g, dummies, msym = t._index_structure.indices_canon_args()
v = components_canon_args(t.components)
can = canonicalize(g, dummies, msym, *v)
if can == 0:
return S.Zero
tmul = t.perm2tensor(can, True)
return tmul
def contract_delta(self, delta):
t = self.contract_metric(delta)
return t
def _get_indices_to_args_pos(self):
"""
Get a dict mapping the index position to TensMul's argument number.
"""
pos_map = dict()
pos_counter = 0
for arg_i, arg in enumerate(self.args):
if not isinstance(arg, TensExpr):
continue
assert isinstance(arg, Tensor)
for i in range(arg.ext_rank):
pos_map[pos_counter] = arg_i
pos_counter += 1
return pos_map
[docs] def contract_metric(self, g):
"""
Raise or lower indices with the metric ``g``
Parameters
==========
g : metric
Notes
=====
see the ``TensorIndexType`` docstring for the contraction conventions
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m0)*q(m1)*g(-m0, -m1)
>>> t.canon_bp()
metric(L_0, L_1)*p(-L_0)*q(-L_1)
>>> t.contract_metric(g).canon_bp()
p(L_0)*q(-L_0)
"""
expr = self.expand()
if self != expr:
expr = expr.canon_bp()
return expr.contract_metric(g)
pos_map = self._get_indices_to_args_pos()
args = list(self.args)
antisym = g.index_types[0].metric_antisym
# list of positions of the metric ``g`` inside ``args``
gpos = [i for i, x in enumerate(self.args) if isinstance(x, Tensor) and x.component == g]
if not gpos:
return self
# Sign is either 1 or -1, to correct the sign after metric contraction
# (for spinor indices).
sign = 1
dum = self.dum[:]
free = self.free[:]
elim = set()
for gposx in gpos:
if gposx in elim:
continue
free1 = [x for x in free if pos_map[x[1]] == gposx]
dum1 = [x for x in dum if pos_map[x[0]] == gposx or pos_map[x[1]] == gposx]
if not dum1:
continue
elim.add(gposx)
# subs with the multiplication neutral element, that is, remove it:
args[gposx] = 1
if len(dum1) == 2:
if not antisym:
dum10, dum11 = dum1
if pos_map[dum10[1]] == gposx:
# the index with pos p0 contravariant
p0 = dum10[0]
else:
# the index with pos p0 is covariant
p0 = dum10[1]
if pos_map[dum11[1]] == gposx:
# the index with pos p1 is contravariant
p1 = dum11[0]
else:
# the index with pos p1 is covariant
p1 = dum11[1]
dum.append((p0, p1))
else:
dum10, dum11 = dum1
# change the sign to bring the indices of the metric to contravariant
# form; change the sign if dum10 has the metric index in position 0
if pos_map[dum10[1]] == gposx:
# the index with pos p0 is contravariant
p0 = dum10[0]
if dum10[1] == 1:
sign = -sign
else:
# the index with pos p0 is covariant
p0 = dum10[1]
if dum10[0] == 0:
sign = -sign
if pos_map[dum11[1]] == gposx:
# the index with pos p1 is contravariant
p1 = dum11[0]
sign = -sign
else:
# the index with pos p1 is covariant
p1 = dum11[1]
dum.append((p0, p1))
elif len(dum1) == 1:
if not antisym:
dp0, dp1 = dum1[0]
if pos_map[dp0] == pos_map[dp1]:
# g(i, -i)
typ = g.index_types[0]
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
else:
# g(i0, i1)*p(-i1)
if pos_map[dp0] == gposx:
p1 = dp1
else:
p1 = dp0
ind, p = free1[0]
free.append((ind, p1))
else:
dp0, dp1 = dum1[0]
if pos_map[dp0] == pos_map[dp1]:
# g(i, -i)
typ = g.index_types[0]
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
if dp0 < dp1:
# g(i, -i) = -D with antisymmetric metric
sign = -sign
else:
# g(i0, i1)*p(-i1)
if pos_map[dp0] == gposx:
p1 = dp1
if dp0 == 0:
sign = -sign
else:
p1 = dp0
ind, p = free1[0]
free.append((ind, p1))
dum = [x for x in dum if x not in dum1]
free = [x for x in free if x not in free1]
# shift positions:
shift = 0
shifts = [0]*len(args)
for i in range(len(args)):
if i in elim:
shift += 2
continue
shifts[i] = shift
free = [(ind, p - shifts[pos_map[p]]) for (ind, p) in free if pos_map[p] not in elim]
dum = [(p0 - shifts[pos_map[p0]], p1 - shifts[pos_map[p1]]) for i, (p0, p1) in enumerate(dum) if pos_map[p0] not in elim and pos_map[p1] not in elim]
res = sign*TensMul(*args).doit()
if not isinstance(res, TensExpr):
return res
im = _IndexStructure.from_components_free_dum(res.components, free, dum)
return res._set_new_index_structure(im)
def _set_new_index_structure(self, im, is_canon_bp=False):
indices = im.get_indices()
return self._set_indices(*indices, is_canon_bp=is_canon_bp)
def _set_indices(self, *indices, **kw_args):
if len(indices) != self.ext_rank:
raise ValueError("indices length mismatch")
args = list(self.args)[:]
pos = 0
is_canon_bp = kw_args.pop('is_canon_bp', False)
for i, arg in enumerate(args):
if not isinstance(arg, TensExpr):
continue
assert isinstance(arg, Tensor)
ext_rank = arg.ext_rank
args[i] = arg._set_indices(*indices[pos:pos+ext_rank])
pos += ext_rank
return TensMul(*args, is_canon_bp=is_canon_bp).doit()
@staticmethod
def _index_replacement_for_contract_metric(args, free, dum):
for arg in args:
if not isinstance(arg, TensExpr):
continue
assert isinstance(arg, Tensor)
def substitute_indices(self, *index_tuples):
return substitute_indices(self, *index_tuples)
def __call__(self, *indices):
"""Returns tensor product with ordered free indices replaced by ``indices``
Examples
========
>>> from sympy import Symbol
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> D = Symbol('D')
>>> Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(i0)*q(i1)*q(-i1)
>>> t(i1)
p(i1)*q(L_0)*q(-L_0)
"""
free_args = self.free_args
indices = list(indices)
if [x.tensor_index_type for x in indices] != [x.tensor_index_type for x in free_args]:
raise ValueError('incompatible types')
if indices == free_args:
return self
t = self.fun_eval(*list(zip(free_args, indices)))
# object is rebuilt in order to make sure that all contracted indices
# get recognized as dummies, but only if there are contracted indices.
if len(set(i if i.is_up else -i for i in indices)) != len(indices):
return t.func(*t.args)
return t
def _extract_data(self, replacement_dict):
args_indices, arrays = zip(*[arg._extract_data(replacement_dict) for arg in self.args if isinstance(arg, TensExpr)])
coeff = reduce(operator.mul, [a for a in self.args if not isinstance(a, TensExpr)], S.One)
indices, free, free_names, dummy_data = TensMul._indices_to_free_dum(args_indices)
dum = TensMul._dummy_data_to_dum(dummy_data)
ext_rank = self.ext_rank
free.sort(key=lambda x: x[1])
free_indices = [i[0] for i in free]
return free_indices, coeff*_TensorDataLazyEvaluator.data_contract_dum(arrays, dum, ext_rank)
@property
def data(self):
deprecate_data()
dat = _tensor_data_substitution_dict[self.expand()]
return dat
@data.setter
def data(self, data):
deprecate_data()
raise ValueError("Not possible to set component data to a tensor expression")
@data.deleter
def data(self):
deprecate_data()
raise ValueError("Not possible to delete component data to a tensor expression")
def __iter__(self):
deprecate_data()
if self.data is None:
raise ValueError("No iteration on abstract tensors")
return self.data.__iter__()
def _eval_rewrite_as_Indexed(self, *args):
from sympy import Sum
index_symbols = [i.args[0] for i in self.get_indices()]
args = [arg.args[0] if isinstance(arg, Sum) else arg for arg in args]
expr = Mul.fromiter(args)
return self._check_add_Sum(expr, index_symbols)
class TensorElement(TensExpr):
"""
Tensor with evaluated components.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensorhead
>>> from sympy import symbols
>>> L = TensorIndexType("L")
>>> i, j, k = symbols("i j k")
>>> A = tensorhead("A", [L, L], [[1], [1]])
>>> A(i, j).get_free_indices()
[i, j]
If we want to set component ``i`` to a specific value, use the
``TensorElement`` class:
>>> from sympy.tensor.tensor import TensorElement
>>> te = TensorElement(A(i, j), {i: 2})
As index ``i`` has been accessed (``{i: 2}`` is the evaluation of its 3rd
element), the free indices will only contain ``j``:
>>> te.get_free_indices()
[j]
"""
def __new__(cls, expr, index_map):
if not isinstance(expr, Tensor):
# remap
if not isinstance(expr, TensExpr):
raise TypeError("%s is not a tensor expression" % expr)
return expr.func(*[TensorElement(arg, index_map) for arg in expr.args])
expr_free_indices = expr.get_free_indices()
name_translation = {i.args[0]: i for i in expr_free_indices}
index_map = {name_translation.get(index, index): value for index, value in index_map.items()}
index_map = {index: value for index, value in index_map.items() if index in expr_free_indices}
if len(index_map) == 0:
return expr
free_indices = [i for i in expr_free_indices if i not in index_map.keys()]
index_map = Dict(index_map)
obj = TensExpr.__new__(cls, expr, index_map)
obj._free_indices = free_indices
return obj
@property
def free(self):
return [(index, i) for i, index in enumerate(self.get_free_indices())]
@property
def dum(self):
# TODO: inherit dummies from expr
return []
@property
def expr(self):
return self._args[0]
@property
def index_map(self):
return self._args[1]
def get_free_indices(self):
return self._free_indices
def get_indices(self):
return self.get_free_indices()
def _extract_data(self, replacement_dict):
ret_indices, array = self.expr._extract_data(replacement_dict)
index_map = self.index_map
slice_tuple = tuple(index_map.get(i, slice(None)) for i in ret_indices)
ret_indices = [i for i in ret_indices if i not in index_map]
array = array.__getitem__(slice_tuple)
return ret_indices, array
[docs]def canon_bp(p):
"""
Butler-Portugal canonicalization
"""
if isinstance(p, TensExpr):
return p.canon_bp()
return p
[docs]def tensor_mul(*a):
"""
product of tensors
"""
if not a:
return TensMul.from_data(S.One, [], [], [])
t = a[0]
for tx in a[1:]:
t = t*tx
return t
[docs]def riemann_cyclic_replace(t_r):
"""
replace Riemann tensor with an equivalent expression
``R(m,n,p,q) -> 2/3*R(m,n,p,q) - 1/3*R(m,q,n,p) + 1/3*R(m,p,n,q)``
"""
free = sorted(t_r.free, key=lambda x: x[1])
m, n, p, q = [x[0] for x in free]
t0 = S(2)/3*t_r
t1 = - S(1)/3*t_r.substitute_indices((m,m),(n,q),(p,n),(q,p))
t2 = S(1)/3*t_r.substitute_indices((m,m),(n,p),(p,n),(q,q))
t3 = t0 + t1 + t2
return t3
[docs]def riemann_cyclic(t2):
"""
replace each Riemann tensor with an equivalent expression
satisfying the cyclic identity.
This trick is discussed in the reference guide to Cadabra.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, riemann_cyclic
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> R = tensorhead('R', [Lorentz]*4, [[2, 2]])
>>> t = R(i,j,k,l)*(R(-i,-j,-k,-l) - 2*R(-i,-k,-j,-l))
>>> riemann_cyclic(t)
0
"""
t2 = t2.expand()
if isinstance(t2, (TensMul, Tensor)):
args = [t2]
else:
args = t2.args
a1 = [x.split() for x in args]
a2 = [[riemann_cyclic_replace(tx) for tx in y] for y in a1]
a3 = [tensor_mul(*v) for v in a2]
t3 = TensAdd(*a3).doit()
if not t3:
return t3
else:
return canon_bp(t3)
def get_lines(ex, index_type):
"""
returns ``(lines, traces, rest)`` for an index type,
where ``lines`` is the list of list of positions of a matrix line,
``traces`` is the list of list of traced matrix lines,
``rest`` is the rest of the elements ot the tensor.
"""
def _join_lines(a):
i = 0
while i < len(a):
x = a[i]
xend = x[-1]
xstart = x[0]
hit = True
while hit:
hit = False
for j in range(i + 1, len(a)):
if j >= len(a):
break
if a[j][0] == xend:
hit = True
x.extend(a[j][1:])
xend = x[-1]
a.pop(j)
continue
if a[j][0] == xstart:
hit = True
a[i] = reversed(a[j][1:]) + x
x = a[i]
xstart = a[i][0]
a.pop(j)
continue
if a[j][-1] == xend:
hit = True
x.extend(reversed(a[j][:-1]))
xend = x[-1]
a.pop(j)
continue
if a[j][-1] == xstart:
hit = True
a[i] = a[j][:-1] + x
x = a[i]
xstart = x[0]
a.pop(j)
continue
i += 1
return a
arguments = ex.args
dt = {}
for c in ex.args:
if not isinstance(c, TensExpr):
continue
if c in dt:
continue
index_types = c.index_types
a = []
for i in range(len(index_types)):
if index_types[i] is index_type:
a.append(i)
if len(a) > 2:
raise ValueError('at most two indices of type %s allowed' % index_type)
if len(a) == 2:
dt[c] = a
#dum = ex.dum
lines = []
traces = []
traces1 = []
#indices_to_args_pos = ex._get_indices_to_args_pos()
# TODO: add a dum_to_components_map ?
for p0, p1, c0, c1 in ex.dum_in_args:
if arguments[c0] not in dt:
continue
if c0 == c1:
traces.append([c0])
continue
ta0 = dt[arguments[c0]]
ta1 = dt[arguments[c1]]
if p0 not in ta0:
continue
if ta0.index(p0) == ta1.index(p1):
# case gamma(i,s0,-s1) in c0, gamma(j,-s0,s2) in c1;
# to deal with this case one could add to the position
# a flag for transposition;
# one could write [(c0, False), (c1, True)]
raise NotImplementedError
# if p0 == ta0[1] then G in pos c0 is mult on the right by G in c1
# if p0 == ta0[0] then G in pos c1 is mult on the right by G in c0
ta0 = dt[arguments[c0]]
b0, b1 = (c0, c1) if p0 == ta0[1] else (c1, c0)
lines1 = lines[:]
for line in lines:
if line[-1] == b0:
if line[0] == b1:
n = line.index(min(line))
traces1.append(line)
traces.append(line[n:] + line[:n])
else:
line.append(b1)
break
elif line[0] == b1:
line.insert(0, b0)
break
else:
lines1.append([b0, b1])
lines = [x for x in lines1 if x not in traces1]
lines = _join_lines(lines)
rest = []
for line in lines:
for y in line:
rest.append(y)
for line in traces:
for y in line:
rest.append(y)
rest = [x for x in range(len(arguments)) if x not in rest]
return lines, traces, rest
def get_free_indices(t):
if not isinstance(t, TensExpr):
return ()
return t.get_free_indices()
def get_indices(t):
if not isinstance(t, TensExpr):
return ()
return t.get_indices()
def get_index_structure(t):
if isinstance(t, TensExpr):
return t._index_structure
return _IndexStructure([], [], [], [])
def get_coeff(t):
if isinstance(t, Tensor):
return S.One
if isinstance(t, TensMul):
return t.coeff
if isinstance(t, TensExpr):
raise ValueError("no coefficient associated to this tensor expression")
return t
def contract_metric(t, g):
if isinstance(t, TensExpr):
return t.contract_metric(g)
return t
def perm2tensor(t, g, is_canon_bp=False):
"""
Returns the tensor corresponding to the permutation ``g``
For further details, see the method in ``TIDS`` with the same name.
"""
if not isinstance(t, TensExpr):
return t
elif isinstance(t, (Tensor, TensMul)):
nim = get_index_structure(t).perm2tensor(g, is_canon_bp=is_canon_bp)
res = t._set_new_index_structure(nim, is_canon_bp=is_canon_bp)
if g[-1] != len(g) - 1:
return -res
return res
raise NotImplementedError()
def substitute_indices(t, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
``index_types`` list of tuples ``(old_index, new_index)``
Note: this method will neither raise or lower the indices, it will just replace their symbol.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.substitute_indices((i,j), (j, k))
A(j, L_0)*B(-L_0, -k)
"""
if not isinstance(t, TensExpr):
return t
free = t.free
free1 = []
for j, ipos in free:
for i, v in index_tuples:
if i._name == j._name and i.tensor_index_type == j.tensor_index_type:
if i._is_up == j._is_up:
free1.append((v, ipos))
else:
free1.append((-v, ipos))
break
else:
free1.append((j, ipos))
t = TensMul.from_data(t.coeff, t.components, free1, t.dum)
return t
def _expand(expr, **kwargs):
if isinstance(expr, TensExpr):
return expr._expand(**kwargs)
else:
return expr.expand(**kwargs)
