Dirac Operators REVIEWED

The Dirac Hilbert Space Operators PhL 3.31.16 5.16.16: This was my development of wedge Section 2.11 (g) on Dirac notation where we have M and M . The covariant transpose and its clear meaning and role got clearer here and I even updated tensor doc with it. I never really needed fancy things like RVR-1 = R-1 V, but I do have M' = RMRT for matrix. That is, I never had to "rotate a vector Dirac operator". So I don't really have all of QM covered in wedge doc and that is fine. Motivation: 1 1. What did I do with those rotations? 1 2. Then "confusion about rotation operators" has a lot: 1 3. Then in "rotation matrices" doc I show 2 4. What does Messiah have to say? 2 5. What does Schiff have to say? 2 6. I will now roll my own presentation 2 7. Proposed rewrite of wedge doc (g) with new section (h) added 4 (g) The Covariant Transpose 4 (h) Linear Dirac Space Operators 4 Motivation: 1) Dirac is quite important in my wedge doc, and I feel fuzzy about this concept. 2) I remember once before making a distinction between a rotation operator R and R, so at one time it seemed to me there was a distinction work noting. 3) I am curious about this issue in general. 1. What did I do with those rotations? D:\Work\My Interests\Physics\Quantum Mechanics\angmom_spin math stuff Here are some clips of interest from "comparison of Cartesian" RVR-1 = R-1 V R = R() exp(-iJ) R = R() exp(-iJ) [Ji]jk = -i ijk // see "confusion about rotation operators.doc " Rz() Ry() Rz() = U D(1)(,,) U-1 = <1m'|k><k| Ry()|j><j|1m> = U†m'k <k| Ry()|j> Ujm = U†m'k Ry()kj Ujm = [ U† Ry() U ] m'm 2. Then "confusion about rotation operators" has a lot: Fact: I am using <k| Ry()|j> = Ry()kj so in my notation R must refer to a matrix, whereas R is a diract space operator. It is the summary at the end of this confusion doc that matters. M' = <| I' |> = <| R I R-1 |> = <| R-1 I |> = R-1 <| I |> = R-1M. Here I is a vector operator, and I have rotated the apparatus but not the observer frame, so that rotated apparatus then has I' as its vector operator. I use R I R-1 = R-1 I and get M' = R-1M for the expectation value. Now in this discussion, I and R and R-1 seem to be operators in the Hilbert Space and R does NOT seem to be a matrix. So there is definitely "something going on here" but I think the basics are missing. 3. Then in "rotation matrices" doc I show the matrix 3D ones as R and the Pauli 2D ones as R. I say R J R-1 = R-1 J . R V R-1 = R-1 V So that is it for this folder on ang mom. Again, there is something going on. 4. What does Messiah have to say? He has a good section in Vol I starting page 245. First he does states, and notes and bras are the dual space, fine. Operators starts on page 250. Linear operators only, He comments that operators generally don't commute, also true of matrices. Hermitian operators. Unitary operators p 257. He is just using normal font for operators. Operators have a spectrum of eigenvalues, so don't forget the eigenvalue equations. Completeness mentioned. Finally on page 283 we get <n|A|m> = A*nm and so on. Page 285 discusses "matrix transformations" where A' = TAT-1. But this is just pure matrix stuff. So Messiah really has no special notations for operators in Hilbert spaces. 5. What does Schiff have to say? He starts on page 165 with bras and kets. Shows on that page operators Ω and H in regular font. I swee nothing really strange here in terms of notation. 6. I will now roll my own presentation I first write this v' = (Mv) = (M)v (v')T = (Mv)T = vT MT |v'> = |Mv> = M|v> <v'| = <Mv| = <v|MT . (2.11.g.4) Now consider just this much |v'> = |Mv> = M|v> In the non-Dirac notation I make no distinction between (Mv) = and (M)v except possibly this. Mv is a vector, whereas (M)v is a matrix times a vector. But nobody EVER makes that distinction. The basis change stuff is maybe important. In wedge I have ui as the axis-aligned basis vectors and I would say Mij = <ui| M | uj> Then I would have [M(e)]ij = <ei| M | ej> so I denote the default case with no superscript. So I think that much is under control. Now back to Mij = <ui| M | uj> The thing on the left is the matrix element of the matrix M in the (u) basis. But what is M on the right? In one sense it is a linear operator which acts on the space V so can say M|v> ≡ |Mv> . That seems a pretty good definition of M: it is defined by its action on all vectors. But M can also "act to the left" on something in the dual space. I need to explain how it is that something like M can act either to the right or to the left. What does that mean to a reader? v' = (Mv) = (M)v (v')T = (Mv)T = vT MT |v'> = |Mv> = M|v> <v'| = <Mv| = <v|MT . (2.11.g.4) I first define operator M by writing M|v> ≡ |Mv> and this is then only defined acting to the right on a vector. I can of course get the dual vector <Mv| . I guess I could make these two definitions: M|v> ≡ |Mv> M acting to the right. M in Mv is a matrix <u| N ≡ <NTu| N acting to the left. N in NTu is the covariant transpose matrix of N. Now we can close either one from the other side in a totally logical fashion <w| M |v> ≡ <w | Mv> = an ordinary scalar product. In the other case <u| N | s> = <NTu| s> where here we have taken M as acting to the left. In regular matrix notation you write aTMb and you can think of matrix M as acting either to the left or the right aT (Mb) M acts to the right (aTM)b M acts to the left (aTM)b = (MTa) b aTM b can think of M acting either to the right or to the left. So here "what is M" ? By default, it is a matrix in the axis-aligned basis or M(u). 7. Proposed rewrite of wedge doc (g) with new section (h) added (g) The Covariant Transpose Whereas the matrix transpose of a matrix Mab would be (MT)ab = Mba (swap the rows and columns), it is the covariant transpose (MT)ab = Mba that is significant in covariant notation. We quote from Tensor where M is a general rank-2 tensor while R and S are the "differentials" of (2.1.2), (MT)ab = Mba (RT)ab = Rba (ST)ab = Sba (MT)ab = Mba (RT)ab = Rba (ST)ab = Sba (MT)ab = Mba (RT)ab = Rba (ST)ab = Sba (MT)ab = Mba (RT)ab = Rba (ST)ab = Sba . (7.9.3)' (2.11.g.1) Equations in any column can be obtained by lowering one or both indices in the top equation, so that the covariant transform MT is a rank-2 tensor if M is a rank-2 tensor. For all-up or all-down indices, the two kinds of transposes are the same: (MT)ab = (MT)ab = Mba. The transpose always has the indices reflected in a vertical line between the indices. This subject is discussed in Tensor Section 7.9 where all claims are proved. We quote some of the conclusions: det(M) = det(MT) = det(MT) (7.9.7)' (2.11.g.2) RRT = RTR = 1 SST = STS = 1 RS = SR = 1 RT = R-1 = S ST = S-1 = R . (7.9.8)' (2.11.g.3) (h) Linear Dirac Space Operators Consider these three ways of writing the same real number, where M is a matrix sandwiched between vector b on the right and transpose vector a on the left, aT (Mb) M acts to the right ( * * *) [ ] (aTM)b M acts to the left, and note that (aTM) = (MTa)T [ (* * *) ] aTM b can think of M acting either to the right or to the left. (2.11.h.1) In writing these equations, one normally thinks of M as being a matrix Mij = (M[u])ij or M = M[u] . By default, the matrix elements are taken in the axis-aligned ui basis on both left and right, so that (ui)TM (uj) = Σa,b (ui)a Mab (uj)b = Σa,b δia Mab δjb = Mij // = (M[u])ij. (2.11.h.2) One could, however, do this in some other basis, for example, (ei) TM (ej) = Σa,b (ei)a Mab (ej)b = Σa,b Ria Mab Rjb = (RMRT)ij // = (M[e])ij (2.11.h.3) and the result is a completely different matrix. In this case the matrices are related by a covariant similarity transformation by R M[e] = R M[u]RT . (2.11.h.4) It is useful to think of the object M as being a basis-independent abstract linear operator which, when sandwiched between certain basis vectors, has certain matrix elements. Different types of basis vectors yield different matrices. We could even have mixed basis elements, (ui) TM (ej) = Σa,b (ui)aMab(ej)b = Σa,b δiaMabRjb = (MRT)ij // = (M[u,e])ij (2.11.h.5) so in this case we get M[u,e] = MRT . (2.11.h.6) The abstract operator M only becomes a matrix when it is properly sandwiched. This notion of thinking of the object M as a basis-independent linear operator becomes more pronounced in the Dirac notation. We restate the above equations as follows, all of which evaluate to the same real number, <a| ( M |b>) M acts to the right = <a | Mb > (<a|M ) |b> M acts to the left, and note that <a|M = <MTa | = <MTa | b > <a| M |b> can think of M acting either to the right or to the left. (2.11.h.7) The space between the vertical bars is inhabited by abstract linear operators like M. The matrix elements shown above are then <ui | M | uj> = (M[u])ij = Mij <ei | M | ej> = (M[e])ij = (RMRT)ij <ui | M | ej> = (M[u,e])ij = (MRT)ij (2.11.h.8) To emphasize this notion of abstract operator, one could write the operator in a different font, so M is a matrix and M is a Dirac space operator, in which case <a| M |b> = <a| ( M |b>) = <a |M b> = a scalar product of two vectors <a| M |b> = (<a|M ) |b> = <MTa | b > = a scalar product of two vectors <ui | M | uj> = (M[u])ij = Mij etc . (2.11.h.9) Here then is a review of the matrix and Dirac notations, a' = (Ma) = (M)a (b')T = (Mb)T = bT MT matrix notation |a'> = |Ma> = M|a> <b'| = <Mb| = <b|MT . Dirac notation (2.11.h.10) Then consider the following claim Fact: <a | M | b> = <b| MT | a> (2.11.h.11) where both M and MT are the names of abstract linear operators. Proof: <a | M | b> ≡ <a | M b> = a (Mb) = ai(Mb)i = ai[ Mijbj] = ai Mij bj = bj Mij ai = bj (MT)ji ai = bj [MTa]j = b (MTa) = <b| MTa> = <b| MT |a> . Operator M is defined by its action on an arbitrary ket vector M | b> = | M b> Operator MT is defined by its action on an arbitrary ket vector MT | b> = | MT b> Notice in the proof that the covariant transpose MT is the correct transpose to use since Mij = (MT)ji. Exercise: Show that wv is a scalar under any transformation x' = F(x) : w'v' = <w' | v'> = <Rw|Rv> = <w|RTR|v> = <w| 1 |v> = <w|v> = wv . (2.11.h.12) In this example R is a matrix, whereas R is the corresponding Dirac space operator. The statement RTR = 1 (2.11.h.13) is the operator version of our matrix statement RTR = 1 (2.11.h.14) which we verify as follows, (RTR)ac = (RT)abRbc = Rba Rbc = δac // (2.1.9) #1 (2.11.h.15) and which is valid for any transformation differential matrix Rij. One may represent a Dirac operator M in various ways M = Σij | ui> Mij <uj| = Σij | ei> [M[e]] ij <ej| = Σij | ui> [M[u,e]] ij <ej| (2.11.h.16) as can be verified by closing with the appropriate basis vectors. For example, for the last line above, <ua | M | eb> = <ua | { Σij | ui> [M[u,e]] ij <ej|} | eb> = Σij <ua | ui> [M[u,e]] ij <ej| eb> = Σij δai [M[u,e]] ij δjb = [M[u,e]] ab . (2.11.h.17) When M = 1 we find 1 = Σij | ui> δij <uj| = Σi | ui><ui| (2.11.h.18) which is just a statement that the | ui> basis is complete. Finally, we make a comparison between the abstract Dirac operator M and the abstract rank-2 "vector" M M = Σij | ui> Mij <uj| M = Σij Mij ui uj // (2.8.10) or |M> = Σij Mij | ui> | uj> . (2.11.h.19) The first object M is an operator in the Dirac Hilbert Space V. The second object M or |M> is a vector in the tensor product space V V. These are completely different objects, though they both involve the same matrix elements Mij. In each case, we can project out those matrix elements in an appropriate fashion: <ua | M | ub> = <ua | { Σij | ui> Mij <uj|} | ub> = Mab [< ua| < ub| ] | M > = [< ua| < ub| ] Σij Mij | ui> | uj> = Mab . (2.11.h.20) The above discussion is presented implicitly for a square matrix M, but only small adjustments are needed for it to apply to a non-square matrix. In this case, in aTM b one thinks of vectors a and b as having different dimensions. Perhaps b lies in x-space which is Rn while a lies in x'-space which is Rm with m > n, and then Mij is an m x n matrix. The x'-space V' = has n basis vectors |u'i> while the x-space V has m basis vectors |ui>. Then one would have, for example, <ui | M | u'j> = Mij M = Σi=1m Σj=1n | ui> Mij <u'j| |M> = Σi=1m Σj=1n Mij | ui> | u'j> 1' = Σi | u'i><u'i| completeness in V' 1 = Σi | ui><ui| completeness in V (2.11.h.21) This is exactly the situation we shall encounter in Chapter 9 where the matrix R is an m x n matrix. We shall not use a script font to represent Dirac space operators, but one should keep in mind that any non-scalar object which is seen to be operating to the left on a bra, or to the right on a ket, or which is found between the two vertical bars in < | | > is a Dirac space operator, regardless of the font used to represent it. Such operator objects are not matrices, but their matrix elements form matrices.