Re: Contraction of Tensors in Mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg112537] Re: Contraction of Tensors in Mathematica*From*: Peter Breitfeld <phbrf at t-online.de>*Date*: Sun, 19 Sep 2010 05:37:55 -0400 (EDT)*References*: <i727in$etb$1@smc.vnet.net>

Sam Takoy wrote: > Hi, > > I this message I will suppress the covariant/contravariant nature of > tensors. > > Suppose I have two tensors A_ijkl and B_rstu (denoted by a and b in > Mathematica). Each tensor is presented for a 4-deep list. Now, I want to > form a new tensor: > > C_ijklrstu = A_ijkl*B_rstu > > and I want C to be represented by an 8-deep list. I've gathered is that > what I need to do is > > c = Outer[Times, a, b]; > > Great! > > Now supposed I want to do some contraction to define a new tensor > > D_jklrsu = C_ijklrsiu? > > > How do I do that? > > Many thanks in advance! > > Sam > I use the following functions, which seem to work: This function sums over the first two indices of the tensor: Spur12[T_]:=Tr[T,Plus,2]//Simplify The following function adds the missing indices of the list up to the maximal one given: padindexlist[indices_]:= Join[indices,Complement[Range[Max[indices]],indices]] the following function IndicesVorn[T,{i,j,...}] transposes the tensor so that the index i becomes the first index, the index j the second etc.: IndicesVorn[T_,indices_]:= Transpose[T,Ordering[padindexlist[indices]]] This makes the contraction: contract[T,{i,j,m,n,...}] contracts over the indices i and j, m and m, etc.: contract[T_,indices_]:= Nest[Spur12[#]&, IndicesVorn[T,indices],Length[indices]/2] Example with a smaller Tensor: T=Array[a[##]&,{2,2,2,2,2}] contract[T,{1,2}] Out={{{a[1, 1, 1, 1, 1] + a[2, 2, 1, 1, 1], a[1, 1, 1, 1, 2] + a[2, 2, 1, 1, 2]}, {a[1, 1, 1, 2, 1] + a[2, 2, 1, 2, 1], a[1, 1, 1, 2, 2] + a[2, 2, 1, 2, 2]}}, {{a[1, 1, 2, 1, 1] + a[2, 2, 2, 1, 1], a[1, 1, 2, 1, 2] + a[2, 2, 2, 1, 2]}, {a[1, 1, 2, 2, 1] + a[2, 2, 2, 2, 1], a[1, 1, 2, 2, 2] + a[2, 2, 2, 2, 2]}}}out contract[T,{1,2,4,5}] Out= {a[1, 1, 1, 1, 1] + a[1, 1, 1, 2, 2] + a[2, 2, 1, 1, 1] + a[2, 2, 1, 2, 2], a[1, 1, 2, 1, 1] + a[1, 1, 2, 2, 2] + a[2, 2, 2, 1, 1] + a[2, 2, 2, 2, 2]} Hope this helps //Peter -- _________________________________________________________________ Peter Breitfeld, Bad Saulgau, Germany -- http://www.pBreitfeld.de