[tex]\Psi[/tex]How to intepret the four components of the dirac spinor?????? the volume integral of the [tex]\Psi[/tex]^T*[tex]\Psi[/tex] give the probability of finding the releativistic electron in a given voulme of space but what exactly do the four components really mean. I have read in many Pop physics books that the 4 components mean the following --the magnitude square of the first component gives the prob that electron is spinning up with positve energy -- the magnitude square of second component gives the prob that electron is spinning down with positve energy --the magnitude square of the third component gives the prob that electron is spinning up with negative energy state -- the magnitude square of fourth component gives the prob that electron is spinning down in negative energy state But when i started reading Advanced quantum mechanics By J.J sakurai ; i realized that the above identification makes no real sense. And the author himself commented on this common misintepretation
The spinor can be constructed from very different basis, I think. The viewpoint in pop books in correct in a certain sense. Sakurai's comment is more rigorous, this is my own view. Maybe it is not correct.
In my book I'm focusing on the physical meaning of the Dirac equation and the (bi)spinors. You have to look at the Chiral (= Weyl) representation for physical insights though. Your book from Sakurai is very good but it uses an older representation. Chapter on Dirac's equation from my Book There's some material to be added at the end but the chapter is almost finished Regards, Hans
Classical or quantum? Spinors have a fairly clear classical geometrical construction, whereas their interpretation is not anywhere near so clear in quantum (field) theory. Mathematically, the Dirac algebra is a complexified Clifford algebra, and very respectable. Back in the day, when I was obsessed with these things, A.Crumeyrolle: Orthogonal and Symplectic Clifford Algebrre, (Kluwer, Dordrecht, 1990) was a fairly good mathematical reference; I.M.Benn and R.W.Tucker, An introduction to Spinors and Geometry with applications in physics, (Adam Hilger, Bristol, 1987) is a fairly elementary text; P.Budinich and A.Trautman: The spinorial chessboard, (Springer, Berlin, 1988) was in some ways a good half-way point. The original thing is E.Cartan: Lecons sur la theorie des spineurs, (Hermann, Paris, 1938), The theory of Spinors, (Hermann, Paris, 1966). Spinors are conventionally taken to be elements of a representation space of the Dirac algebra associated with a (3,1) metric. A warning is in order: spinors have a Physics health warning. The Man to follow at your peril is probably David Hestenes, who has pushed for his approach to spinors to be adopted by Physicists for perhaps 30 years. There is a small group of people who have pushed with him. His idea is that spinors, instead of being elements of a representation space simpliciter, are in fact elements of a left ideal of the Dirac algebra. This extra mathematical structure makes more interpretation possible. Despite a lot of work, Hestenes et al. (and me) has failed to show that this extra mathematical structure has Physical relevance. Although mathematical work in this area can be interesting, there can be a crossing over into partisanship. The other thing that can be problematic is a simple-minded approach to the relationship of Dirac spinors to quaternions. Again, quaternions are mathematically respectable, and amazing -- and useful, for example, in robotics -- but again there is sometimes a partisan attitude to the importance of quaternions in Physics. As so often, Wikipedia has an interesting entry on Spinors, but the mathematical level is perhaps somewhat high.
Great comment on spinor! I ever read the book by David Hestenes, It is interesting but almost useless for physicists.
A good place for an elementary geometrical approach to spinors, which I absurdly couldn't remember details of earlier, partly because I sadly, and badly, don't have my own copy, is in "Misner, Charles W.; Kip. S. Thorne & John A. Wheeler (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0". I recommend the sections of Hans de Vries' chapter, posted earlier, on the 2d Dirac equation, 11.2-11.4. Quite pretty, and something of a new way to look at spinors for me.
Thank you for your nice and stimulating words Peter. Did you see how this naturally solves the speeds of +/-c from the commutation with the Hamiltonian? I added a short section on the 4d propagation (11.5) The spinor algebra will be worked out later. in the chapter. The end of the chapter shows this in preparation. My book, chapter on the Dirac equation Regards, Hans
Hans, I just opened your PDF, so I've just started to read it...right off I see in the chapter heading, "The linearized Poison equation." Probably ought to change that to "The linearized Poisson equation"... In fact, it appears "Poisson" is misspelled in several places as "Poison".
I finalized the updates for the 4d case now, you can find them in section 11.20. The chapter is streamlined and has grown quite a bit. My book, Chapter on the Dirac equation Regards, Hans
With a Foldy-Wouthuysen unitary transformation, U = SQRT((E+m)/2E) {1 + γ∙p/(E+m)} both electrons and positrons are described by two-component spinors -- p is the three momentum while γ is the spatial vector of Dirac matrices. Electrons are then described by the adjoint spinor of the form, (x1,x2,0,0), and by (0,0,y1,y2), x2=0 for spin up,while y1=0 for positron spin up --- if you use the spinor definitions and conventions from Franz Gross's Relativistic Quantum Mechanics. But, as Gross points out there's a huge price to pay for such a simple representation. Interactions become hopelessly complex, as in the two component Dirac eq. with E&M interactions requires seven terms to describe the interaction, and who needs that? So the interpretation of the components of a Dirac spinor is, of course, dependent on the representation. Regards, Reilly Atkinson
It is possible to interpret the classical Dirac spinor geometrically in a Lorentz covariant way that is independent of the representation of the Dirac matrices; a homomorphism from Dirac spinors to tetrads can be constructed, for example, as (1)[tex]\bar\psi\gamma^\mu\psi[/tex], the real (2) and complex (3) parts of [tex]\bar\psi\gamma^\mu\psi^C[/tex], and (4)[tex]\bar\psi\gamma^5\gamma^\mu\psi[/tex], where [tex]\psi^C[/tex] is the charge conjugate of [tex]\psi[/tex]. These four 4-vectors are orthogonal (!!, although perhaps unsurprising that such a thing is somehow possible, considering the construction of the Dirac algebra in terms of the metric), and present 7 degrees of freedom of the Dirac spinor. The final independent degree of freedom can be given by the phase of [tex]\bar\psi\psi+i\bar\psi\gamma^5\psi[/tex]. The arbitrary choice of complex phase in the definition of charge conjugation means that these are not natural homomorphisms, and we can also construct a homomorphism into bivectors, [tex]\bar\psi\gamma^\mu\gamma^\nu\psi[/tex], so there is a panoply of possible geometrical interpretations to choose from, with no natural way to choose in sight. I would say that the dependence of the first construction on the charge conjugated spinor should not be considered a strong objection to that particular interpretation, but I imagine some Physicists would discard it immediately. Personally, I wouldn't dream of writing equations in terms of such a tetrad presentation, but a Lorentz covariant and representation independent geometrical interpretation of the Dirac spinor is possible. However, there's much more to do, since there is more than just a single Dirac spinor to interpret in the standard model of particle physics. There's the neutrino spinor wave function to interpret as well, and the quark wave functions, and the muon and tau families as well. Then there's second quantization, minimal coupling to the electromagnetic field, renormalization, ... . All in all, a bit of a mess if we do anything but use Dirac wave functions in any way except instrumentally.
I haven't thought about this stuff in few years. An old post of mine on sci.physics talked parity and the four components of a Dirac spinor. Not sure if its relevant.
Dirac spinors form 4-dimensional complex vector bundle DM over the space-time manifold M. Once you are given a spinor [tex]\psi[/tex] at some point p of M, it is a vector in the special vector space [tex]D_p(M)[/tex], which is called the fiber of DM at that point p. This space has no direct relation to spacial vectors at the point p. Please, look my paper http://arxiv.org/abs/0802.1491. I would be pleased if you find it helpful for understanding Dirac spinors.
Losely speaking the 2 upper spinors "Psi" have position-representation (x) meaning and the lower "chi" has momentum-representation (p) meaning. You could think of Dirac's equation as a quantum version of Hamiltons classical equations: p'=-dH/dx x'=dH/dp instead of Newtons equation which is of second order in derivative mx''=F. Classically to solve it you reformulate it as 2 coupled ODE's which is easy to solve by software, and to solve Hamiltons equations are one of these ways. 2) The boundary conditions are different for these wave-functions. For "Psi" we use Dirichlet BC Psi=0. For example particle in a box (1D) Psi=sin(pi*x/L) so Psi=0 at x=0,L. But for the lower spinor: dChi/dx=0, which is a Neumann BC. This is a characteristics of momentum p=-i*hbar*dPsi/dx applied on sin function gives cos, exactly as the solution for the dirac equation. Note the eigenvalue E is non-linear in the energy here: lam=E(1+E/2mc^2), where lam=the "classical" eigenvalue ~(n*pi/L)^2, giving: E=lam-lam^2/2mc^2+... As a consequence, chi cannot be treated on equal foot as psi, that is to intermix x- and p-space! There is no meaning of the expression x+p. At an infinite barrier Psi=0 but the momentum has it maximum there -> dChi/dx=0 (particle is reflected at the wall). Best, Per