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 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()[source]

Clear the TensorManager.

comm_i2symbol(i)[source]

Returns the symbol corresponding to the commutation group number.

comm_symbols2i(i)[source]

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)[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 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 corrispondence 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)
set_comms(*args)[source]

set the commutation group numbers c for symbols i, j

Parameters :args : sequence of (i, j, c)
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 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)

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    
dim_eps    
dummy_fmt    
class sympy.tensor.tensor.TensorIndex[source]

Represents an abstract tensor index.

Parameters :

name : name of the index

tensortype : TensorIndexType of the index

is_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)

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 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, 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  
class sympy.tensor.tensor.TensorType[source]

Class of tensor types.

Parameters :

index_types : list of TensorIndexType of the tensor indices

symmetry : 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 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, tensorsymmetry, TensorType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1]*2)
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> A = S2('A')

Attributes

name    
index_types    
rank    
types equal to typ.types  
symmetry equal to typ.symmetry  
comm commutation group  
commutes_with(other)[source]

Returns 0 (1) if self and other (anti)commute.

Returns None if self and other do not (anti)commute.

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 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.

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)

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, contract_all=False)[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, -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 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_args()[source]

Returns (g, dummies, msym, v), the entries of canonicalize

see canonicalize in tensor_can.py

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, contract_all=False)[source]

Raise or lower indices with the metric g

g metric

contract_all if True, eliminate all g which are contracted

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)
static from_indices(*indices)[source]

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, TensMul
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> TensMul.from_indices(m0, m1, -m1, m3)
([(m0, 0, 0), (m3, 3, 0)], [(1, 2, 0, 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

g permutation corrisponding to the tensor in the representation used in canonicalization

canon_bp if True, then g is the permutation corresponding to the canonical form of the tensor

sorted_components()[source]

Returns a tensor with sorted components

The sorting is done taking into account the commutation group of the component tensors.

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)]
substitute_indices(*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.substitute_indices((i,j), (j, k))
A(j, L_0)*B(-L_0, -k)
sympy.tensor.tensor.canon_bp(p)[source]

Butler-Portugal canonicalization

sympy.tensor.tensor.tensor_mul(*a)[source]

product of tensors

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

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