Appendix E old not used REVIEWED

Appendix E PhL 1.11.15 I decided not to use this as an Appendix. The material is all part of Section 2.11 now, but this is where things got their start. Dirac is firmly entrenched now in wedge doc. // 5.16.16 E.1 Scalar product notations for Cartesian Space A Hilbert Space has an inner product that can be written many ways. The scalar product (inner product) of two vectors belonging to an n-dimensional real Hilbert space may appear as aTb a b (a,b) <a,b> <a | b > . (E.1.1) In Cartesian space one has ai = ai and any of these notations may be regarded as a shorthand for the scalar quantity Σi=1n aibi . The leftmost notation in (E.1.1) is meant to imply a simple linear algebra combination of a row vector multiplied by a column vector. For n = 2 one would have aTb = (a1 a2) = a1b1 + a2b2 . (E.1.2) The notation aT means the Transpose of the column vector a. There are three elements here. There is a row vector aT, there is a column vector b , and there is the way they are combined to make a scalar. This threeness is clearly expressed in the notation a b where the third element is represented by the operator. It is less clearly expressed in aTb and <a | b > where the way the two vectors are combined is implied by juxtaposition. In mapping language we might write : VV → K (scalars). The form <a | b > is called the Dirac bra-ket notation and is commonly used in quantum mechanics. We find it easiest to think of <a | b > as being defined by aTb which is the combination of a vector b with a transposed vector aT . From this definition, the basic rules of bra-ket notation will be derived below. In general the vectors in a Hilbert space can have components which are complex numbers. In that general case, we have to adjust the notations in (E.1.1) as follows, (a*)Tb a* b (a,b) <a,b> <a | b > . (E.1.3) In the three notations on the right, this complex conjugation is implied by the notation. All these notations mean the same thing which is Σi=1n a*ibi where * means complex conjugation. In (E.1.3) have adopted the physics scalar product convention that <a,b> = Σi=1n a*ibi . The math convention is to say <a,b> = Σi=1n aib*i . In that case we could write these inner product notations, aTb* a b* (a,b) <a,b> <a | b > . (E.1.4) There are good arguments for each convention. In the physics convention the leftmost notation in (E.1.3) is written a†b where a† ≡ (a*)T is called the Hermitian conjugate of vector a. In either convention, we have <a,b> = <b,a>* in any of the five notations shown. For example, (b*)Ta = [(a*)Tb]* (b* a) = (a* b)* (b,a) = (a,b)* <b|a> = <a|b>* (E.1.5) In quantum mechanics, the fact that the Hilbert spaces involved have complex-valued scalar products is absolutely crucial, as we shall comment on below. But in our current document, we regard the Hilbert Spaces involved as being real, in which case we have <b|a> = <a|b> and a† = aT. E.2 Rules for the Dirac Notation (Cartesian Space) Here we use the definition <a | b> ≡ a†b to derive some standard manipulation rules for scalar products in the Dirac notation. Suppose b = Mc where M is a square nxn matrix and b and c are vectors. Then <a | b> ≡ a†b = a†(Mc) = a†(M)c = (M†a)†c = <M†a | c> (E.2.1) where we have used the fact that MTT = M and so M†† = M. The object M† is the Hermitian conjugate of the matrix M, and as with vectors, M† ≡ (M*)T = (MT)*. So we have our first rule which is <a | Mc > = <M†a | c> (E.2.2) Notice the appearance of the form a†(M)c in (E.2.1). In the Dirac notation, this is represented using two vertical bars, so <a | M | c> ≡ a†(M)c . The matrix M is referred to as an "operator" and |c> is a "state" and <a| is a "conjugate state" (or an adjoint state). So we can extend (E.2.2) to say <a | M | c> = <a | Mc > = <M†a | c> (E.2.3) for any a, c or M. Complex conjugating the equations of (E.2.3) gives <a | M | c>* = <a | Mc >* = <M†a | c>* = <Mc | a> = <c | M†a> = <c | M† |a> (E.2.4) If one has |b> = |Mc> = M |c> (E.2.5) then the implication is that the following is valid for all vectors a in vector space V, <a | b> = <a |Mc> = <a| M |c> . Complex conjugation then says <b | a>* = <Mc |a>* = <c| M† |a>* so in fact <b | a> = <Mc |a> = <c| M† |a> Since this last equation is true for all vectors |a>, we have <b | = <Mc | = <c| M† and this shows the rule for converting an expression between the ket space to the bra space: |b> = |Mc> = M |c> <b | = <Mc | = <c| M† (E.2.6) If the vectors ei form a basis for our vector space V, we can write an arbitrary vector in V as v = Σi vi ei or | v > = Σi vi | ei> (E.2.7) Then v† = Σi vi* (ei)† or < v | = Σi vi* < ei| (E.2.8) If the basis vectors are orthonormal, we have <ei|ej> = δi,j or ei† ej = δi,j (E.2.9) If the basis vectors are complete for V, we have this strange-looking notation, Σi=1n | ei><ei| = 1 or eiei† = diag(1,1...1) = "1" (E.2.10) Notice that eiei† = | ei><ei| is an nxn matrix, where for n = 2 one would have eiei† = ((ei)1* (ei)2*) = = . In more conventional notation, one writes orthonormality and completeness of a basis as Σi=1n (ei)n*(ej)n = δi,j // orthonormality Σi=1n (ei)n*(ei)m = δn,m // completeness (E.2.11) Completeness is always a bit more opaque than orthogonality in the student (author) mind. A simple example is to imagine that e1 = (1,0.0...), e2 = (0,1,0...) and so on so that (ei)n = δi,n. Then the above completeness statement is obviously true. If (e'i) = Σa=1n Uia(ea) where U is any unitary matrix (UU† = U†U = 1), we find that Σi=1n (e'i)n*(e'i)m = Σi=1n [Σa Uia* (ea)n*] [Σb Uib (eb)m] = Σi=1n [Σa Uia* δa,n] [Σb Uib δb,m] = Σi=1n Uin*Uim = Σi=1n U†niUim = (U†U)nm = (1)n,m = δn,m (E.2.12) The Dirac notation provides perhaps a more intuitive view of completeness. Consider, | v > = 1 | v > = Σi=1n | ei><ei| v > = Σi=1n | ei> vi = Σi=1n vi | ei> . (E.2.13) This says that if the basis ei is complete, then any vector v can be expanded on that basis as shown. If R is a unitary nxn matrix in complex Cartesian space Vn = Cn , we can express the idea that the inner product of two vectors is a scalar under such rotations in this manner: <a' | b'> = <Ra | Rb> = <a | R†R | b> = <a | 1 | b> = <a | b> (E.2.14) In real Cartesian space the requirement R†R = 1 becomes RTR = 1 or RT = R-1 and R is then a rotation matrix (or a parity reflection type matrix). Notice that we replaced <Ra | by <a|R† using (E.2.6). E.3 Rules for the Dirac notation (non-Cartesian Space): covariant notation In the previous section we assumed Cartesian space with its metric tensor gij = δi,j and we expressed results for both real and complex Hilbert spaces. Here, we consider only real Hilbert Spaces, but we allow for a general metric tensor. In this case, the inner product is a b = Σi,j=1ngijaibj. Conveniently, the notions of contravariant and covariant vectors as discussed in Chapter 2 allow this to be written in the simpler form a b = Σi=1n aibi = Σi=1n aibi where mention of the metric tensor can be avoided. In this more general world, every basis {en} has a dual basis {en} where en em = δnm as in ***. How then are the results of the previous sections stated for this non-Cartesian but real space V? First of all, the notations for an inner product are still those of (E.1.1) aTb a b (a,b) <a,b> <a | b > . (E.1.1) Most Dirac notation rules are unchanged, in particular. <a | M | c> = <a | Mc > = <MTa | c> (E.2.3) |b> = |Mc> = M |c> <b | = <Mc | = <c| MT (E.2.6) Differences arise with respect to the orthogonality and completeness relations. We now have <ei|ej> = δij = δi,j = δij = <ei|ej> = orthogonality Σi=1n | ei><ei| = 1 = Σi=1n | ei><ei| completeness (E.3.1) The notion of a matrix M and MT needs clarification. We have for example |b> = M |c> bi = Mij cj or bi = Mijcj (E.3.2) where the ket notation could imply either the contravariant or covariant matrix sense. The notion of a transpose matrix is somewhat vague, but we argue in Tensor that the most reasonable and consistent definition of the transpose of a rank-2 tensor is this: (MT)ab = Mba (RT)ab = Rba (MT)ab = Mba (RT)ab = Rba (MT)ab = Mba (RT)ab = Rba (MT)ab = Mba (RT)ab = Rba Tensor (7.9.3) (E.3.3) where we state this for a general rank-2 tensor M and also for the R-matrix Rij of Chapter 2 which is the linearization (or differential) of a general non-linear transformation x' = F(x). The main idea in the above four lines appears in the middle two lines which show that the transpose of an up-tilt or a down-tilt matrix is obtained by reflecting the indices in a vertical line between the indices (and not by just swapping the indices). Each of the above equations is "covariant" and either index can be taken up or down to generate a valid new equation. Recall that for the R-matrix there are four orthogonality rules (2.1.9). 1: Rba Rbc = δac 2: Rba Rbc = δac // sum is on 1st index 3: Rab Rcb = δac 4: Rab Rcb = δac . // sum is on 2nd index (7.6.4)' (2.1.9) Using the transpose definition just stated, these rules can be written as 1: (RT)ab Rbc = δac 2: (RT)ab Rbc = δac 3: Rab (RT)bc = δac 4: Rab (RT)bc = δac (E.3.4) We can summarize these rules as saying RTR = RRT = 1 or for either up-tilt or down-tilt matrix multiplication (E.3.5) RT = R-1 . In Cartesian space the statement RT = R-1 (real orthogonal) implies that R must be a rotation matrix (or parity/reflection matrix), whereas in the non-Cartesian space RT = R-1 as defined above is valid for any transformation differential matrix R, whether or not it is a rotation matrix. That is to say, R could contain both rotation and stretch components. If one were to smoothly morph from a non-Cartesian space gij to a Cartesian space δi,j one would find that RT = R-1, while originally being true for a general R matrix, would end up being true only for a rotation R matrix. In this notation, we can restate our notion of the scalarity of an inner product in Dirac notation, <a' | b' > = <Ra | Rb > = <a | RTRb > = <a | RTR| b > = <a | 1| b > = <a| b > where b' = Rb means either b'i = Rijbj or b'i = Rijbj. E.4 Transformations and Experiments Three transformations. Think of three transformations A,B and C applied to a given starting position. We shall use the word rotate and rotation as conceptual aids, but we imply more general transformations as indicated by the matrix R above in covariant notation. The "starting position" is an "experiment". There is an apparatus (the experiment) and there is an observer (the basis vectors). In transformation A, the apparatus is rotated while the basis vectors stay put. In transformation B, the basis vectors are rotated while the apparatus stays put. In transformation C, the apparatus and basis vectors are rotated together by the same amount. The apparatus is described by a set of tensors. In transformation A we find for rank-1 and rank-2 tensors, v'i = Rijvj T'ij = RiaRjbTab ei, eiej etc all stay put Thus, vector v is "actively" rotated into vector v', and tensor Tij is actively rotated into T'ij, and similarly for all higher rank tensors. Transformation A is our transformation of interest. Rank-1 tensor. Consider in more detail what happens to a vector v under transformation A. vi = ei v starting position vi' = ei v' ending position after the action of transformation A In transformation A, the basis vectors ei do not move. Although (ei v) is the dot product of two vectors, it does not transform as a scalar under transformation A. It looks like a scalar, but in fact it transforms under transformation A as the component of a vector. It would transform as a scalar under transformation C since then one finds e'i v' = <e'i | v'> = <Rei | Rv> = <ei | RTR | v> = <ei| 1 | v> = <ei| v> = ei v , but we are not interested in transformation C. Our experiment uses transformation A. The expansions going with the above lines are these v = Σiviei starting position v' = Σiv'iei ending position after the action of transformation A Rank-2 tensor. We can generalize these expansions to a rank-2 tensor. As for the vector case, the basis vectors stay put in transformation A : T = ΣabTab eaeb starting position T' = ΣabT'ab eaeb ending position after the action of transformation A The projections corresponding to ** are then Tab = (eaeb ) T starting position T'ab = (eaeb ) T' ending position after the action of transformation A Rank-1 tensor function Now we return to the vector case we define a function fi(v) as follows, fi(v) ≡ vi = ei v = <ei | v > Recall our discussion in Section E.1 above that the scalar product has three elements: two vectors and an instruction for combining them into the scalar product. In the notation ei v that instruction is encoded into the operator, whereas in the Dirac notation <ei | v > it is implicit in the juxtaposition of the bra and the ket. In either case the instruction is that ei v = <ei | v > = Σj=1n (ei)jvj = Σj=1n (ei)jvj . We wish to talk about a linear functional here, and to clarify the notation used to represent it. As a mapping, one can define the linear functional fi : V → K where V is our vector space and K is the space of scalars. One could write this as fi : V1 → V0. Evaluating at a vector v in V one gets fi(v) ϵ K. In the notation fi(v) = ei v, we really want to denote the functional by fi = ei fi ϵ V* fi(v) = ei v = vi fi(v) ϵ K In the Dirac notation, things are perhaps less specific , fi = < ei | fi ϵ V* fi = < ei | v> = vi fi(v) ϵ K The question of interest here is this: is fi a scalar, or is it a vector, or is it neither? We know it is a vector of the dual space V*, but we want to know in what sense fi transforms under transformation A discussed above.