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)
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.)
-
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 :
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)
-
class
sympy.tensor.tensor.
TensorSymmetry
[source]¶ Monoterm symmetry of a tensor
- Parameters
bsgs : tuple
(base, sgs)
BSGS of the symmetry 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.
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')
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 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 tensor
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')
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 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
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 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)
, 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.
-
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_matrix
()[source]¶ DEPRECATED: do not use.
Returns ndarray components data as a matrix, if components data are available and ndarray dimension does not exceed 2.
-
replace_with_arrays
(replacement_dict, indices)[source]¶ 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 isL
, 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
andj
:>>> expr.replace_with_arrays(repl, [j, i]) [[0, -b/2 + c/2], [b/2 - c/2, 0]]
-
-
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 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]
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 Butler-Portugal 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 contracted
Notes
see the
TensorIndexType
docstring for the contraction conventions
-
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_0, l) + 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 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
-
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)
-
get_free_indices
()[source]¶ 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]
-
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] >>> t2 = p(m1)*g(-m1, m2) >>> t2.get_indices() [L_0, -L_0, m2]
-
perm2tensor
(g, is_canon_bp=False)[source]¶ Returns the tensor corresponding to the permutation
g
For further details, see the method in
TIDS
with the same name.
-
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