Tensor

class sympy.tensor.tensor.TensorIndexType(

name,

dummy_name=None,

dim=None,

eps_dim=None,

metric_symmetry=1,

metric_name='metric',

**kwargs,

)

A TensorIndexType is characterized by its name and its metric.

Parameters:

name : name of the tensor type

dummy_name : name of the head of dummy indices

dim : dimension, it can be a symbol or an integer or None

eps_dim : dimension of the epsilon tensor

metric_symmetry : integer that denotes metric symmetry or None for no metric

metric_name : string with the name of the metric tensor

Notes

The possible values of the metric_symmetry parameter are:

1 : metric tensor is fully symmetric 0 : metric tensor possesses no index symmetry -1 : metric tensor is fully antisymmetric None: there is no metric tensor (metric equals to None)

The metric is assumed to be symmetric by default. It can also be set to a custom tensor by the .set_metric() method.

If there is a metric the metric is used to raise and lower indices.

In the case of non-symmetric metric, the following raising and lowering conventions will be adopted:

psi(a) = g(a, b)*psi(-b); chi(-a) = chi(b)*g(-b, -a)

From these it is easy to find:

g(-a, b) = delta(-a, b)

where delta(-a, b) = delta(b, -a) is the Kronecker delta (see TensorIndex for the conventions on indices). For antisymmetric metrics there is also the following equality:

g(a, -b) = -delta(a, -b)

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_name='L')
>>> Lorentz.metric
metric(Lorentz,Lorentz)

Attributes

metric	(the metric tensor)
delta	(Kronecker delta)
epsilon	(the Levi-Civita epsilon tensor)
data	((deprecated) a property to add ndarray values, to work in a specified basis.)

class sympy.tensor.tensor.TensorIndex(name, tensor_index_type, is_up=True)

Represents a tensor index

Parameters:

name : name of the index, or True if you want it to be automatically assigned

tensor_index_type : TensorIndexType of the index

is_up : flag for contravariant index (is_up=True by default)

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. Adding - to a covariant (is_up=False) index makes it contravariant.

Dummy indices have a name with head given by tensor_inde_type.dummy_name with underscore and a number.

Similar to symbols multiple contravariant indices can be created at once using tensor_indices(s, typ), where s is a string of names.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, TensorIndex, TensorHead, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> mu = TensorIndex('mu', Lorentz, is_up=False)
>>> nu, rho = tensor_indices('nu, rho', Lorentz)
>>> A = TensorHead('A', [Lorentz, Lorentz])
>>> A(mu, nu)
A(-mu, nu)
>>> A(-mu, -rho)
A(mu, -rho)
>>> A(mu, -mu)
A(-L_0, L_0)

Attributes

name
tensor_index_type
is_up

class sympy.tensor.tensor.TensorHead(

name,

index_types,

symmetry=None,

comm=0,

)

Tensor head of the tensor.

Parameters:

name : name of the tensor

index_types : list of TensorIndexType

symmetry : TensorSymmetry of the tensor

comm : commutation group number

Notes

Similar to symbols multiple TensorHeads can be created using tensorhead(s, typ, sym=None, comm=0) function, where s is the string of names and sym is the monoterm tensor symmetry (see tensorsymmetry).

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

Define a fully antisymmetric tensor of rank 2:

>>> from sympy.tensor.tensor import TensorIndexType, TensorHead, TensorSymmetry
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> asym2 = TensorSymmetry.fully_symmetric(-2)
>>> A = TensorHead('A', [Lorentz, Lorentz], asym2)

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
>>> from sympy import diag
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> 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:

>>> F = TensorHead('F', [Lorentz, Lorentz], asym2)

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], TensorSymmetry.no_symmetry(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	(total number of indices)
symmetry
comm	(commutation group)

commutes_with(other)

Returns 0 if self and other commute, 1 if they anticommute.

Returns None if self and other neither commute nor anticommute.

sympy.tensor.tensor.tensor_heads(

s,

index_types,

symmetry=None,

comm=0,

)

Returns a sequence of TensorHeads from a string \(s\)

class sympy.tensor.tensor.TensExpr(*args)

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.

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.

get_matrix()

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=None,

)

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. The original index order will be used if no value is passed.

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])
>>> A(i).replace_with_arrays({A(i): [1, 2]}, [i])
[1, 2]

Since ‘indices’ is optional, we can also call replace_with_arrays by this way if no specific index order is needed:

>>> A(i).replace_with_arrays({A(i): [1, 2]})
[1, 2]

>>> expr = A(i)*A(j)
>>> expr.replace_with_arrays({A(i): [1, 2]})
[[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])
>>> 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]]})
[[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)
[[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]]

class sympy.tensor.tensor.TensAdd(*args, **kw_args)

Sum of tensors.

Parameters:

free_args : list of the free indices

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_heads, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> p, q = tensor_heads('p,q', [Lorentz])
>>> t = p(a) + q(a); t
p(a) + q(a)

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()

Canonicalize using the Butler-Portugal algorithm for canonicalization under monoterm symmetries.

contract_metric(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

class sympy.tensor.tensor.TensMul(*args, **kw_args)

Product of tensors.

Parameters:

coeff : SymPy coefficient of the tensor

args

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.

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()

Canonicalize using the Butler-Portugal algorithm for canonicalization under monoterm symmetries.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, TensorSymmetry
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> A = TensorHead('A', [Lorentz]*2, TensorSymmetry.fully_symmetric(-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)

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, tensor_heads
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensor_heads('p,q', [Lorentz])
>>> 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() → list[TensorIndex]

Returns the list of free indices of the tensor.

Explanation

The indices are listed in the order in which they appear in the component tensors.

Examples

>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensor_heads
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensor_heads('p,q', [Lorentz])
>>> 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()

Returns the list of indices of the tensor.

Explanation

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, tensor_heads
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensor_heads('p,q', [Lorentz])
>>> 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)

Returns the tensor corresponding to the permutation g

For further details, see the method in TIDS with the same name.

sorted_components()

Returns a tensor product with sorted components.

split()

Returns a list of tensors, whose product is self.

Explanation

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, tensor_heads, TensorSymmetry
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
>>> A, B = tensor_heads('A,B', [Lorentz]*2, TensorSymmetry.fully_symmetric(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.canon_bp(p)

Butler-Portugal canonicalization. See tensor_can.py from the combinatorics module for the details.

sympy.tensor.tensor.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)

sympy.tensor.tensor.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, TensorSymmetry
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> R = TensorHead('R', [Lorentz]*4, TensorSymmetry.riemann())
>>> t = R(i,j,k,l)*(R(-i,-j,-k,-l) - 2*R(-i,-k,-j,-l))
>>> riemann_cyclic(t)
0

class sympy.tensor.tensor.TensorSymmetry(*args, **kw_args)

Monoterm symmetry of a tensor (i.e. any symmetric or anti-symmetric index permutation). For the relevant terminology see tensor_can.py section of the combinatorics module.

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 (i.e., Bianchi identity), are not covered. See combinatorics module for information on how to generate BSGS for a general index permutation group. Simple symmetries can be generated using built-in methods.

Examples

Define a symmetric tensor of rank 2

>>> from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, TensorHead
>>> Lorentz = TensorIndexType('Lorentz', dummy_name='L')
>>> sym = TensorSymmetry(get_symmetric_group_sgs(2))
>>> T = TensorHead('T', [Lorentz]*2, sym)

Note, that the same can also be done using built-in TensorSymmetry methods

>>> sym2 = TensorSymmetry.fully_symmetric(2)
>>> sym == sym2
True

See also

sympy.combinatorics.tensor_can.get_symmetric_group_sgs

Attributes

base	(base of the BSGS)
generators	(generators of the BSGS)
rank	(rank of the tensor)

classmethod direct_product(*args)

Returns a TensorSymmetry object that is being a direct product of fully (anti-)symmetric index permutation groups.

Notes

Some examples for different values of (*args): (1) vector, equivalent to TensorSymmetry.fully_symmetric(1) (2) tensor with 2 symmetric indices, equivalent to .fully_symmetric(2) (-2) tensor with 2 antisymmetric indices, equivalent to .fully_symmetric(-2) (2, -2) tensor with the first 2 indices commuting and the last 2 anticommuting (1, 1, 1) tensor with 3 indices without any symmetry

classmethod fully_symmetric(rank)

Returns a fully symmetric (antisymmetric if rank``<0) TensorSymmetry object for ``abs(rank) indices.

classmethod no_symmetry(rank)

TensorSymmetry object for rank indices with no symmetry

classmethod riemann()

Returns a monotorem symmetry of the Riemann tensor

sympy.tensor.tensor.tensorsymmetry(*args)

Returns a TensorSymmetry object. This method is deprecated, use TensorSymmetry.direct_product() or .riemann() instead.

Explanation

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.

class sympy.tensor.tensor.TensorType(*args, **kwargs)

Class of tensor types. Deprecated, use tensor_heads() instead.

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)

class sympy.tensor.tensor._TensorManager

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.

clear()

Clear the TensorManager.

comm_i2symbol(i)

Returns the symbol corresponding to the commutation group number.

comm_symbols2i(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.

get_comm(i, j)

Return the commutation parameter for commutation group numbers i, j

see _TensorManager.set_comm

set_comm(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, TensorSymmetry
>>> Lorentz = TensorIndexType('Lorentz')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> A = TensorHead('A', [Lorentz])
>>> G = TensorHead('G', [Lorentz], TensorSymmetry.no_symmetry(1), 'Gcomm')
>>> GH = TensorHead('GH', [Lorentz], TensorSymmetry.no_symmetry(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)

set_comms(*args)

Set the commutation group numbers c for symbols i, j.

Parameters:

args : sequence of (i, j, c)