In physics, contraction is a dilation with coefficient λ<1\lambda\lt 1. This notion is used in fixed point theory, theory of topological vector spaces etc. There is also a notion of contraction from metric space theory; see short map. Finally, the contraction rule is a structural rule in logic and type theory. This entry will be predominantly about another notion of a contraction.

Contraction of tensors

This entry will be predominantly about contraction of tensors, where by tensor we mean a vector in some tensor power V nV^{\otimes n} of a vector kk-space VV (or a projective kk-module if kk is only a commutative ring). Let V *=Hom k(V,k)V^* = Hom_k(V,k) be the dual vector space and (V *) m(V^*)^{\otimes m} be some tensor of V *V^*. Then one may define (l,s)(l,s)-contraction

(V *) mV n(V *) (m1)V (n1)(V^*)^{\otimes m}\otimes V^{\otimes n}\to (V^*)^{\otimes (m-1)}\otimes V^{\otimes (n-1)}

by pairing by the evaluation map the ll-th tensor factor of (V *) r(V^*)^{\otimes r} and ss-th tensor factor of V nV^{\otimes n}. In fact as a map written, one can contract also elements of (V *) mV n(V^*)^{\otimes m}\otimes V^{\otimes n} which did not come from a product of a pair of element (i.e. which are not decomposable tensors).

Let the rank rr of VV be finite. If SV nS\in V^{\otimes n} is given in some basis by components S i 1,,i nS^{i_1,\ldots, i_n} and T(V *) rT\in (V^*)^{\otimes r} is given in the dual basis by components T j 1,,j rT_{j_1,\ldots,j_r}, then the components of the contraction will be

contr l,s(T,S) j 1,,j l1,j l+1,,j m i 1,,i s1,i s+1,,i n= u=1 rT j 1,,j l1,u,j l+1,,j mS i 1,,i s1,u,i s+1,,i ncontr_{l,s}(T,S)^{i_1,\ldots, i_{s-1},i_{s+1},\ldots,i_n}_{j_1,\ldots, j_{l-1},j_{l+1},\ldots,j_m} = \sum_{u = 1}^r T_{j_1,\ldots, j_{l-1},u,j_{l+1},\ldots,j_m} S^{i_1,\ldots, i_{s-1},u,i_{s+1},\ldots,i_n}

More generally one can contract mixed tensors and do several contractions simultaneously. In fact it is better to think of a contraction as a tensor multiplication of two tensors and then doing the genuine contraction of one upper and one lower index of the same tensor:

contr l,s(A) j 1,,j l1,j l+1,,j m i 1,,i s1,i s+1,,i n):= u=1 rA j 1,,j l1,u,j l+1,,j m i 1,,i s1,u,i s+1,,i n contr_{l,s}(A)^{i_1,\ldots, i_{s-1},i_{s+1},\ldots,i_n}_{j_1,\ldots,j_{l-1},j_{l+1},\ldots, j_m}) := \sum_{u = 1}^r A^{i_1,\ldots, i_{s-1},u,i_{s+1},\ldots,i_n}_{j_1,\ldots,j_{l-1},u,j_{l+1},\ldots, j_m}

The simplest case is the trace of a (1,1)(1,1)-tensor: trA= i=1 rA i itr A = \sum_{i=1}^r A^i_i.

These operations can be symmetrized or antisymmetrised appropriately to make sense on symmetric or antisymmetric powers.

For example, there is a contraction of a vector XVX\in V and a nn-form ωΛV *\omega\in \Lambda V^*:

(X,ω)ι X(ω)(X,\omega)\mapsto \iota_X(\omega)

and ι X:ωι X(ω)\iota_X: \omega\mapsto \iota_X(\omega) is a graded derivation of the exterior algebra of degree 1-1. This is also done for the tangent bundle which is a C (M)C^\infty(M)-module V=TMV = T M, then one gets the contraction of vector fields and differential forms. It can also be done in vector spaces, fibrewise.


Last revised on March 28, 2021 at 00:20:58. See the history of this page for a list of all contributions to it.