Tensor¶

class
sympy.tensor.tensor.
_TensorManager
[source]¶ 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 tensor1
tensors anticommuting among themselves2
tensors not commuting, apart with those withcomm=0
Other groups can be defined using
set_comm
; tensors in those groups commute with those withcomm=0
; by default they do not commute with any other group.
comm_symbols2i
(i)[source]¶ get the commutation group number corresponding to
i
i
can be a symbol or a number or a stringIf
i
is not already defined its commutation group number is set.

get_comm
(i, j)[source]¶ Return the commutation parameter for commutation group numbers
i, j
see
_TensorManager.set_comm

set_comm
(i, j, c)[source]¶ set the commutation parameter
c
for commutation groupsi, j
Parameters: i, j : symbols representing commutation groups
c : group commutation number
Notes
i, j
can be symbols, strings or numbers, apart from0, 1
and2
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 defaultc=None
.The group commutation number
c
is assigned in correspondence to the group commutation symbols; it can be0 commuting
1 anticommuting
None no commutation property
Examples
G
andGH
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)


class
sympy.tensor.tensor.
TensorIndexType
[source]¶ 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
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 metricmetric
can be an object havingname
andantisym
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 theKronecker delta
(seeTensorIndex
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’; forN > 2
they are linearly independent.eps_dim
is by default equal todim
, if the latter is an integer; else it can be assigned (for use in naive dimensional regularization); ifeps_dim
is not an integerepsilon
isNone
.Examples
>>> from sympy.tensor.tensor import TensorIndexType >>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L') >>> Lorentz.metric metric(Lorentz,Lorentz)
Examples with metric components data added, this means it is working on a fixed basis:
>>> Lorentz.data = [1, 1, 1, 1] >>> Lorentz TensorIndexType(Lorentz, 0) >>> Lorentz.data [[1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]]
Attributes
name
metric_name
(it is ‘metric’ or metric.name) metric_antisym
metric
(the metric tensor) delta
( Kronecker delta
)epsilon
(the LeviCivita epsilon
tensor)dim
dim_eps
dummy_fmt
data
(a property to add ndarray
values, to work in a specified basis.)

class
sympy.tensor.tensor.
TensorIndex
[source]¶ Represents an abstract tensor index.
Parameters: name : name of the index, or
True
if you want it to be automatically assignedtensortype :
TensorIndexType
of the indexis_up : flag for contravariant index
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 thename
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)
Attributes
name
tensortype
is_up

sympy.tensor.tensor.
tensor_indices
(s, typ)[source]¶ Returns list of tensor indices given their names and their types
Parameters: s : string of comma separated names of indices
typ : list of
TensorIndexType
of the indicesExamples
>>> 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)

class
sympy.tensor.tensor.
TensorSymmetry
[source]¶ Monoterm symmetry of a tensor
Parameters: bsgs : tuple (base, sgs)
BSGS of the symmetry of the tensorNotes
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.
Examples
Define a symmetric tensor
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, 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')
Attributes
base
(base of the BSGS) generators
(generators of the BSGS) rank
(rank of the tensor)

sympy.tensor.tensor.
tensorsymmetry
(*args)[source]¶ Return a
TensorSymmetry
object.One can represent a tensor with any monoterm slot symmetry group using a BSGS.
args
can be a BSGSargs[0]
baseargs[1]
sgsUsually tensors are in (direct products of) representations of the symmetric group;
args
can be a list of lists representing the shapes of Young tableauxNotes
For instance:
[[1]]
vector[[1]*n]
symmetric tensor of rankn
[[n]]
antisymmetric tensor of rankn
[[2, 2]]
monoterm slot symmetry of the Riemann tensor[[1],[1]]
vector*vector[[2],[1],[1]
(antisymmetric tensor)*vector*vectorNotice 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 TensorSymmetry, get_symmetric_group_sgs >>> sym2 = tensorsymmetry(*get_symmetric_group_sgs(2)) >>> S2 = TensorType([Lorentz]*2, sym2) >>> V = S2('V')

class
sympy.tensor.tensor.
TensorType
[source]¶ Class of tensor types.
Parameters: index_types : list of
TensorIndexType
of the tensor indicessymmetry :
TensorSymmetry
of the tensorExamples
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')
Attributes
index_types
symmetry
types
(list of TensorIndexType
without repetitions)

class
sympy.tensor.tensor.
TensorHead
[source]¶ Tensor head of the tensor
Parameters: name : name of the tensor
typ : list of TensorIndexType
comm : commutation group number
Notes
A
TensorHead
belongs to a commutation group, defined by a symbol on numbercomm
(see_TensorManager.set_comm
); tensors in a commutation group have the same commutation properties; by defaultcomm
is0
, 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 fullycontravariant representation. In case it is necessary to assign components data which represents the values of a nonfully covariant tensor, see the other examples.>>> from sympy.tensor.tensor import tensor_indices, tensorhead >>> Lorentz.data = [1, 1, 1, 1] >>> i0, i1 = tensor_indices('i0:2', Lorentz) >>> A.data = [[j+2*i for j in range(4)] for i in range(4)]
in order to retrieve data, it is also necessary to specify abstract indices enclosed by round brackets, then numerical indices inside square brackets.
>>> A(i0, i1)[0, 0] 0 >>> A(i0, i1)[2, 3] == 3+2*2 True
Notice that square brackets create a valued tensor expression instance:
>>> A(i0, i1) A(i0, i1)
To view the data, just type:
>>> A.data [[0 1 2 3] [2 3 4 5] [4 5 6 7] [6 7 8 9]]
Turning to a tensor expression, covariant indices get the corresponding components data corrected by the metric:
>>> A(i0, i1).data [[0 1 2 3] [2 3 4 5] [4 5 6 7] [6 7 8 9]]
>>> A(i0, i1).data [[0 1 2 3] [2 3 4 5] [4 5 6 7] [6 7 8 9]]
while if all indices are contravariant, the
ndarray
remains the same>>> A(i0, i1).data [[0 1 2 3] [2 3 4 5] [4 5 6 7] [6 7 8 9]]
When all indices are contracted and components data are added to the tensor, accessing the data will return a scalar, no numpy object. In fact, numpy ndarrays are dropped to scalars if they contain only one element.
>>> A(i0, i0) A(L_0, L_0) >>> A(i0, i0).data 18
It is also possible to assign components data to an indexed tensor, i.e. a tensor with specified covariant and contravariant components. In this example, the covariant components data of the Electromagnetic tensor are injected into \(A\):
>>> 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('A', [Lorentz, Lorentz], [[2]]) >>> F(i0, i1).data = [ ... [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).data [[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 contravariantcovariant form:
>>> F(i0, i1).data [[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]]
To convert the numpy’s ndarray to a sympy matrix, just cast:
>>> from sympy import Matrix >>> Matrix(F.data) Matrix([ [ 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]])
Still notice, in this last example, that accessing components data from a tensor without specifying the indices is equivalent to assume that all indices are contravariant.
It is also possible to store symbolic components data inside a tensor, for example, define a fourmomentumlike tensor:
>>> from sympy import symbols >>> P = tensorhead('P', [Lorentz], [[1]]) >>> E, px, py, pz = symbols('E p_x p_y p_z', positive=True) >>> P.data = [E, px, py, pz]
The contravariant and covariant components are, respectively:
>>> P(i0).data [E p_x p_y p_z] >>> P(i0).data [E p_x p_y p_z]
The contraction of a 1index tensor by itself is usually indicated by a power by two:
>>> P(i0)**2 E**2  p_x**2  p_y**2  p_z**2
As the power by two is clearly identical to \(P_\mu P^\mu\), it is possible to simply contract the
TensorHead
object, without specifying the indices>>> P**2 E**2  p_x**2  p_y**2  p_z**2
Attributes
name
index_types
rank
types
( equal to typ.types
)symmetry
(equal to typ.symmetry
)comm
(commutation group)

class
sympy.tensor.tensor.
TensExpr
[source]¶ 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 aTensAdd
or aTensMul
.TensAdd
objects are put in canonical form using the ButlerPortugal 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)
, whereicomp1
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.

get_matrix
()[source]¶ Returns ndarray components data as a matrix, if components data are available and ndarray dimension does not exceed 2.
Examples
>>> from sympy.tensor.tensor import TensorIndexType, tensorsymmetry, TensorType >>> from sympy import ones >>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L') >>> sym2 = tensorsymmetry([1]*2) >>> S2 = TensorType([Lorentz]*2, sym2) >>> A = S2('A')
The tensor
A
is symmetric in its indices, as can be deduced by the[1, 1]
Young tableau when constructing \(sym2\). One has to be careful to assign symmetric component data toA
, as the symmetry properties of data are currently not checked to be compatible with the defined tensor symmetry.>>> from sympy.tensor.tensor import tensor_indices, tensorhead >>> Lorentz.data = [1, 1, 1, 1] >>> i0, i1 = tensor_indices('i0:2', Lorentz) >>> A.data = [[j+i for j in range(4)] for i in range(4)] >>> A(i0, i1).get_matrix() Matrix([ [0, 1, 2, 3], [1, 2, 3, 4], [2, 3, 4, 5], [3, 4, 5, 6]])
It is possible to perform usual operation on matrices, such as the matrix multiplication:
>>> A(i0, i1).get_matrix()*ones(4, 1) Matrix([ [ 6], [10], [14], [18]])


class
sympy.tensor.tensor.
TensAdd
[source]¶ Sum of tensors
Parameters: free_args : list of the free indices 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 eye >>> Lorentz.data = [1, 1, 1, 1] >>> a, b = tensor_indices('a, b', Lorentz) >>> p.data = [2, 3, 2, 7] >>> q.data = [2, 3, 2, 7] >>> t = p(a) + q(a); t p(a) + q(a) >>> t(b) p(b) + q(b)
The following are: 2**2  3**2  2**2  7**2 ==> 58
>>> (p(a)*p(a)).data 58 >>> p(a)**2 58
Attributes
args
(tuple of addends) rank
(rank of the tensor) free_args
(list of the free indices in sorted order) 
canon_bp
()[source]¶ canonicalize using the ButlerPortugal algorithm for canonicalization under monoterm symmetries.

contract_metric
(g)[source]¶ Raise or lower indices with the metric
g
Parameters: g : metric
contract_all : if True, eliminate all
g
which are contractedNotes
see the
TensorIndexType
docstring for the contraction conventions

static
from_TIDS_list
(coeff, tids_list)[source]¶ Given a list of coefficients and a list of
TIDS
objects, construct aTensAdd
instance, equivalent to the one that would result from creating single instances ofTensMul
and then adding them.Examples
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, TensAdd >>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L') >>> i, j = tensor_indices('i,j', Lorentz) >>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2]) >>> eA = 3*A(i, j) >>> eB = 2*B(j, i) >>> t1 = eA._tids >>> t2 = eB._tids >>> c1 = eA.coeff >>> c2 = eB.coeff >>> TensAdd.from_TIDS_list([c1, c2], [t1, t2]) 2*B(i, j) + 3*A(i, j)
If the coefficient parameter is a scalar, then it will be applied as a coefficient on all
TIDS
objects.>>> TensAdd.from_TIDS_list(4, [t1, t2]) 4*A(i, j) + 4*B(i, j)

fun_eval
(*index_tuples)[source]¶ 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, L_0) + A(k, l)

substitute_indices
(*index_tuples)[source]¶ 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)


class
sympy.tensor.tensor.
TensMul
[source]¶ Product of tensors
Parameters: coeff : SymPy coefficient of the tensor
args
Notes
args[0]
list ofTensorHead
of the component tensors.args[1]
list of(ind, ipos, icomp)
whereind
is a free index,ipos
is the slot position ofind
in theicomp
th component tensor.args[2]
list of tuples representing dummy indices.(ipos1, ipos2, icomp1, icomp2)
indicates that the contravariant dummy index is theipos1
th slot position in theicomp1
th component tensor; the corresponding covariant index is in theipos2
slot position in theicomp2
th component tensor.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)
canon_bp
()[source]¶ Canonicalize using the ButlerPortugal 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

contract_metric
(g)[source]¶ Raise or lower indices with the metric
g
Parameters: g : metric Notes
see the
TensorIndexType
docstring for the contraction conventionsExamples
>>> 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)

fun_eval
(*index_tuples)[source]¶ 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)

get_indices
()[source]¶ 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]

perm2tensor
(g, canon_bp=False)[source]¶ Returns the tensor corresponding to the permutation
g
For further details, see the method in
TIDS
with the same name.

sorted_components
()[source]¶ Returns a tensor with sorted components calling the corresponding method in a
TIDS
object.

split
()[source]¶ 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)]


sympy.tensor.tensor.
riemann_cyclic_replace
(t_r)[source]¶ 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)

sympy.tensor.tensor.
riemann_cyclic
(t2)[source]¶ 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