Appendix D v3

written 11/22/15 Installed on 1.26.16, do not edit here. Appendix D: A Unified View of Tensors and Tensor Functions 1 D.1 Tensor functions in Dirac notation 1 D.2 Basis change matrix 2 D.3 Transformations of tensors and tensor functions 4 D.4 Tensor Functions and Quantum Mechanics 6 Appendix D: A Unified View of Tensors and Tensor Functions In this section multiindex notations are shown in red to the right. D.1 Tensor functions in Dirac notation The vector space V has dimension n, and k ≤ n. The vector space is real, so <a|b> = <b|a>. In the bra-ket notation (Paul Dirac, 1939), a rank-k tensor functional T is represented by the bra <T| which is an element of the dual space V*k. Meanwhile, elements of the space Vk are written as kets which are a tensor product of smaller kets, | vi,vi, .... vi > = |vi> |vi> ..... |vi> . | vI> (D.1.1) Here the ir are labels, not components. Each vi is a vector in V having n components (vi)j . The tensor function T(vi,vi, .... vi) is then represented by the application of the functional <T| to vectors in Vk so that, <T | vi,vi, .... vi > = T(vi,vi, .... vi) . T(vI) = <T | vI > (D.1.2) Due to the tensor product (of vector spaces) construction of the "ket" shown in (D.1.1), the function shown in (D.1.2) is manifestly k-multilinear. The bra-ket notation represents an inner product (scalar product) so the spaces here are Hilbert spaces, not just vector spaces. The covariant tensor Tii....i we claim is this (each label ir ranges from 1 to n ), Tii....i = <T | ei,ei, .... ei > . TI = <T | eI > (D.1.5) The n vectors |ei> for i=1 to n form a basis for V, and the n*k kets | ei,ei, .... ei > form a basis for Vk. From this one would conclude from (D.1.2) that Tii....i = T(ei,ei, .... ei) ir = 1 to n TI = T(eI) (D.1.6) in agreement with our established fact (6.2.5). The contravariant form is then Tii....i = T(ei,ei, .... ei) . TI = T(eI) (D.1.7) Let us now assume that the n vectors |vi> for i=1 to n form some alternative basis for V, and then the n*k kets | vi,vi, .... vi > form an alternative basis for Vk. The dual basis is {vi} where vi vj = δij as in (2.11.2) for the ei basis and its dual ej. Looking at our two equations from above, <T | vi,vi, .... vi > = T(vi,vi, .... vi) ir = 1 to n (D.1.2) <T | ei,ei, .... ei > = Tii....i ir = 1 to n (D.1.5) one can say that the tensor Tii....i and the tensor function T(vi,vi, .... vi) are both representations of the same abstract tensor <T| in two different Vk bases, |vI> and |eI>. Both bases have dimension n*k. Recall T = ΣI TIeI ϵ Vk = a tensor |T> = ΣI TI |eI > T = ΣI TIλI ϵ V*k = a tensor functional <T| = ΣI TI <eI| T ~ T by the isomorphism V*k ~ Vk [see circa (2.11.12)] . (D.1.8) Notice that for the basis {vi}, <vI|vJ> = < vi|vj>< vi|vj> .... < vi|vj> = (vi vj)(vi vj) .... (vi vj) = δijδij...δij // see (2.3.2) for basis {vr} with dual basis {vr} = δIJ . // orthonormal basis in the multiindex notation (D.1.9) This result applies as well to the basis |eI>, so <eI|eJ> = <vI|vJ> = δIJ . (D.1.10) D.2 Basis change matrix The basis-change transformation matrix between the |vI> and |eI> bases is given by, MIJ ≡ < ei,ei, .... ei | vj,vj, .... vj > MIJ = <eI|vJ> (D.2.1) = < ei|vj>< ei|vj> .... < ei|vj> // see (2.9.17) = (ei vj)(ei vj) .... (ei vj) = λi(vj)λi(vj) .... λi(vj) // see (2.11.3) = (vj)i (vj)i ...(vj)i // see (2.11.7) = (vJ)I . // using a multiindex notation shown below (7.8.2) (D.2.2) There are n*k values for I and n*k values for J, so matrix M has dimension nk x nk. Entirely in multiindex notation, MIJ = <eI|vJ> = (vJ)I // mixed, see (2.1.6) line 2 or (D.2.3) MIJ = <eI|vJ> = (vJ)I . // pure covariant, see (2.1.6) line 4 The transpose is then, (MT)JI = MIJ = <eI|vJ> = <vJ| eI> = (vJ)I // Hilbert Space is real (MT)JI = MIJ = <eI|vJ> = <vJ| eI> = (vJ)I . (D.2.4) In the bra-ket notation completeness of an orthonormal basis is expressed this way: 1 = ΣJ |eJ><eJ| = ΣJ |eJ><eJ| = ΣJ |vJ><vJ| = ΣJ |vJ><vJ| . (D.2.5) Proof: (example) Consider a general Vk tensor T : (1) |T> = 1|T> = ΣJ |eJ><eJ| T> = ΣJ TJ |eJ> // so basis |eJ> must be complete (2) |eI> = 1|eI> = ΣJ |eJ><eJ|eI> = ΣJ |eJ>δJI = |eI> // why orthonormal is needed Therefore the up-tilt basis-change matrix M is real orthogonal, meaning MMT = 1 or MT = M-1 : (MMT)IK = ΣJ MIJ(MT)JK = ΣJ <eI|vJ><vJ| eK> = <eI| (ΣJ|vJ><vJ| )eK> = <eI | 1 | eK> = <eI | eK> = eI eK = δIK // see (2.11.2) or (D.2.6) MMT = 1 . // real orthogonal in the multi-index sense In quantum mechanics with complex Vk, one gets instead MM† = 1 (M is unitary) . The connection then between the tensors and tensor functions is given by, TI = <eI| T> = <eI| 1 | T> = <eI| ΣJ |vJ><vJ| T> = ΣJ <eI|vJ><vJ| T> = ΣJ <eI|vJ><T| vJ> = ΣJ MIJ T(vJ) . // MIJ = (vJ)I (D.2.7) Going the other direction, T(vI) = <vI | T > = <vI | 1 | T > = <vI | ΣJ |eJ><eJ| T > = ΣJ <vI|eJ> <eJ|T > = ΣJ (MT)IJ TJ . // (MT)IJ = (vI)J (D.2.8) Example of (D.2.7): Tii = Σj,j=1n (vj)i(vj)i T(vj,vj) ir = 1 to n or Tij = Σa,b=1n (va)i(vb)j T(va,vb) . // n2 terms in the sum (D.2.7a) Example of (D.2.8): T(vi,vi) = Σj,j=1n (vi)j (vi)j Tjj ir = 1 to n or T(vi,vj) = Σa,b=1n (vi)a (vj)bTab . // n2 terms in the sum (D.2.8a) Comment: These examples can be compared to a simple quantum mechanics case. Let |x> be a basis vector describing a 1D particle at location x (coordinate representation), and let |p> be a basis vector describing a plane-wave particle having momentum p (momentum representation). Then it turns out that the basis change matrix is <x|p> = ψp(x) = C eipx where C is a normalization constant. So the basis change "matrix" (continuous matrix subscripts p and x) is a function of p, just as the basis change matrix in (D.2.8a) is a function of vi and vj. D.3 Transformations of tensors and tensor functions In this section we write vectors in bold font. Consider two sets of n vectors vi and v'i where vi form a basis for V. One can then write, v'i = Qijvj i = 1,2...n implied sum on j (D.3.1) where Qij is a matrix describing the linear combinations of the vi that make up the v'i. Since the tensor function T(vi,vi, .... vi) is k-multilinear, one can certainly write T(v'i,v'i, .... v'i) = QijQij .... Qij T(vj,vj, .... vj) (D.3.2) or just showing the ket part, | v'i,v'i, .... v'i > = QijQij .... Qij | vj,vj, .... vj > . (D.3.3) Equation (D.3.2) vaguely resembles the Chapter 2 transformation of a covariant tensor field, T'ii...i (x') = RijRij .... Rij Tjj...j (x) (D.3.4) where x' = F(x) and dx'  = R dx. // R is the differential of F. The resemblance is perhaps closer if we restrict to x' = F(x) to be a linear transformation, so then x' = R x or x'i = Rijxj . (D.3.5) The resemblance between (D.3.2) and (D.3.4) we claim is really superficial and misleading, which is the main reason for bringing it up. We just make a few comments on this matter. The linearized transformations of Chapter 2 like v'i = Rijvj for a vector vi are component transformations. The j on vj is a component index, and v'i = Rijvj (v' = Rv) is an instruction for creating a new vector v' by linearly combining the components of v. Transformation (D.3.5) is such a component transformation. In contrast, the transformation (D.3.1) that v'i = Qijvj is not a component transformation. It constructs n new vectors v'i by linearly combining the n vectors vi. The j on vj is a label, not a component index. In (D.3.4), the left-side object T'ii...i(x') has a prime on T. It is a tensor different from Tjj...j (x), and this would be true even if there were no x dependence of the field. In (D.3.2), the left-side object T(v'i,v'i, .... v'i) has no prime, it is the same T as on the right. In fact, as was shown in (2.11.f.8), the object T(v1, v2, ...vk) under any Chapter 2 component transformation transforms as a scalar, so there are no Rij or Qij matrices involved, T'(v'1, v'2, ...v'k) = T(v1, v2, ...vk) . T'(v'Z) = T(vZ) (2.11.f.8) where (v'r)i = Rij (vr)j r = 1...k implied sum on j i = 1...n T(v1, v2, ...vk) is a scalar because it is the scalar product of a rank-k tensor functional T = <T| with a rank-k tensor | v1, v2, ...vk >, just as <a | b> = a b is a scalar. (D.3.2) is nothing more than a statement that the tensor function T(v1, v2, ...vk) is k-multilinear. D.4 Tensor Functions and Quantum Mechanics Eq. (D.1.2) defining a tensor function as a bra-ket combination <T | vi,vi, .... vi > = T(vi,vi, .... vi) (D.1.2) has the following quantum mechanics incarnation, which was the original use Dirac intended for his bra-ket notation, <ψ| r1, r2...rk> = ψ(r1, r2...rk) . (D.1.3) Here object <ψ| plays the role of the abstract tensor <T|, and the generic arguments vi become the physical positions ri of k particles. The object ψ is a functional in V*k which gets applied to |r1,r2....rk> = |r1>|r2>....|rk> and the resulting function ψ(r1,r2...rk) is called a "wavefunction" which describes the "probability amplitude" that the k particles are near spatial locations r1, r2, ....rk. The probability that the k particles are near these spatial locations is given by |ψ(r1,r2...rk)|2dnr1dnr2 ...dnrk. "Near" means that ri lies somewhere in the range ri to ri+dnri , It happens that in quantum mechanics literature it is the ket that is the functional in V*k and the bra which is the element of Vk. So in a physics text one always sees equations like, ψ(r1, r2...rk) = <r1, r2...rk| ψ>. (D.1.4) This is a long-standing convention difference between the physics and math worlds. When talking about a functional f applied to a vector x, it seems natural to have f(x) = <f | x>, which is the math convention. The physics person writes <x|ψ> = ψ(x) and says that the state vector |ψ> is being projected onto the coordinate representation basis element <x|. Usually ψ is not called a "functional". A ket is thought of as a vector v, while the bra is a transpose vector vT and then <v1|v2> = v1Tv2 in a matrix notation sense, so here it seems logical to put "the vector", whether v, v2 or ψ, on the right. Our functional T maps elements of Vk to the real numbers, and <a|b> = <b|a> = a b, so one can "for free" switch the role of which is the functional, and which is the ket acted upon by the functional. In quantum mechanics the functional maps to complex numbers, and <a|b> = a* b where * is complex conjugation. Then <b|a> = b* a = (a b*)* = a* b = <a|b>*. And <v1|v2> = v1T* v2 = v1†v2 . It is a crucial element of quantum mechanics that the space Vk is complex and not real. In the math world, one usually sees instead <b|a> = b a* . If the k particles are electrons or other half-integral spin particles which are in a "symmetric spin state", then the wavefunction (D.1.4) must be replaced by [Alt(ψ)](r1, r2...rk) in order to make it be totally antisymmetric in the coordinates ri, as required by "Fermi statistics" for half-integral spin particles. We mention this just to show that the Alt operator and the permutation group in general have important applications in quantum mechanics.