Today I read a bit of Ryder’s QFT (which was $20 when I bought it on Amazon; a day after I bought it, it dropped to $14 (!), but it is now again $53?), starting over with Chapter 2.  Ryder approaches relativistic wave equations from symmetry arguments, which I like.  Below are some notes to myself on his introduction of the Dirac equation.

   First, he demonstrates the problems with the Klein-Gordon equation for scalar particles,

     (\Box + m^2)\psi = 0, \ \ \Box \equiv \eta^{\mu\nu}\partial_{\mu}\partial_{\nu} = \frac{\partial^2}{\partial t^2}-\bold{\nabla}^2.

   The first problem is obvious; this equation is obtained by the substitution of differential operators for E \rightarrow \imath \hbar \frac{\partial}{\partial t} and \bold{p} \rightarrow - \imath \hbar \bold{\nabla}, as usual in quantum theory, into the relativistic energy-momentum relation p^{\mu}p_{\mu} = E^2 - \bold{p}^2 = m^2.  But this relation gives both positive and negative energy solutions, E = \pm \sqrt{m^2 + \bold{p}^2}.  These negative energy solutions are the first problem.

   The second problem stems from attempting to modify the probabilistic interpretation of the wave function into one consistent with relativity.  Taking the usual non-relativistic probability density, \rho = \psi^{\ast} \psi, and attemping to make it the time component of a 4-vector

     j^{\mu} = (\rho, \bold{j}),

where

     \bold{j} = -\frac{\imath \hbar}{2m}(\psi^{\ast} \bold{\nabla} \psi - \psi \bold{\nabla} \psi^{\ast}) 

is the probability current, gives

     \rho = \frac{\imath \hbar}{2m} (\psi^{\ast} \frac{\partial \psi}{\partial t} - \psi \frac{\partial \psi^{\ast}}{\partial t}).

   However, since the Klein-Gordon equation is second-order in time, initial conditions for \psi and \frac{\partial \psi}{\partial t} can be chosen independently, and thus \rho is not guaranteed to be positive-definite – which makes a probabilistic interpretation difficult!

   To fix these problems, Dirac introduced his first-order equation.  Now, some texts, like Griffiths’s Elementary Particles book, attempt to introduce the Dirac equation by “factoring” the Klein-Gordon equation, and then showing that these factors must necessarily imply a first-order matrix equation – i.e., the Dirac equation.  Ryder does this too, but first derives the Dirac equation by considering the transformation of spinors under groups isomorphic to the Lorentz group.  In this way, the Dirac equation is given by symmetry principles.

   Even before showing this, as an analogy Ryder demonstrates that SU(2) is a double cover of SO(3).  I also found a good discussion of this here (there is also more detailed discussion of group representations and the Lorentz group in Maggiore and in Aitchison’s notes).  Basically, one can show that any element of SO(3), the group of orientation-preserving rotations of a 3-dimensional vector space, can be mapped to two elements of SU(2), the group of unitary transformations of a 2-dimensional complex spinor space.  This two-to-one mapping derives from the similar-but-not-identical Lie algebras of the two Lie groups, given by the commutator relations obeyed by the generators of the groups: \left[J_i,J_j\right] = \imath \epsilon_{ijk} J_k for the angular momentum operators \bold{J} that generate SO(3) vs. \left[\frac{\sigma_i}{2},\frac{\sigma_j}{2}\right] = \imath \epsilon_{ijk} \frac{\sigma_k}{2} for the Pauli matrices \bold{\sigma} that generate SU(2).

   Likewise, the generators of the Lorentz group are similar to those of other groups; however, in this case, the similarity must be made explicit.  To see this, first notice that the set of pure Lorentz transformations, the generators of which we shall call \bold{K}, is not closed; e.g., the group commutator of boosts in the x and y directions gives a rotation in the z direction.  Thus, the true Lorentz group includes both boosts and rotations.  That is, the Lie algebra relations of the Lorentz group intertwine both \bold{J} and \bold{K}, like so:

     \left[K_i,K_j\right] = -\imath \epsilon_{ijk} J_k,
     \left[J_i,K_j\right] = \imath \epsilon_{ijk} K_k,

along with the usual commutator for rotations.

   Now, we can write complex linear combinations of these 6 generators to form 6 new generators for the complexified Lorentz group:

     \bold{A_{\pm}} = \frac{1}{2}(\bold{J}\pm\imath\bold{K}). 

   Then, \bold{A_{\pm}} separately satisfy the commutation relation of the SU(2) group, so the Lorentz group SO(3,1) is roughly analogous to SU(2)\otimesSU(2).  The isomorphism is not exact because of the complexification; however, we can use this similarity to study the transformation effects of boosts on the two different types of spinors associated with each SU(2) group (defining the spinor space on which SU(2)\otimesSU(2) is represented as (j=\frac{1}{2},j=0)\oplus(j=0,j=\frac{1}{2})).  It can also be shown that these two spin-\frac{1}{2} spaces switch under parity (basically because \bold{J} is an axial vector, while \bold{K} is a true vector).  If we wish to include parity in our equations, it becomes necessary to consider both types of 2-component spinors, which transform differently under boosts, combined as one 4-component spinor.

   By studying how this 4-component Dirac spinor transforms under boosts, one finds the Dirac equation.  Essentially, one boosts a rest frame Dirac spinor, and then uses the equivalence of both rest frame 2-component spinors to write relations between boosted 2-component spinors of both types.  This relation can be cast back into an equation involving a set of 4×4 matrices \gamma^{\mu}  and the Dirac spinor \psi, which is the Dirac equation:

     (\gamma^{\mu}p_{\mu}-m)\psi=(\imath\gamma^{\mu}\partial_{\mu}-m)\psi=0. 

   The \gamma^{\mu} obey the anti-commutation relation \{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}\bold{I}.  Of course, one could have found this relation by “factoring” the Klein-Gordon equation,

     p^{\mu}p_{\mu} - m^2 = (\alpha^{\kappa}p_{\kappa}+m)(\beta^{\lambda}p_{\lambda}-m) = 0,

finding that the \alpha^{\mu} and \beta^{\mu} are the \gamma^{\mu} matrices, but this is far less revealing of the rich structure of the Dirac spinor representation of the Lorentz group.

   Now the Dirac equation is first-order, and a positive-definite probability density can be found.  Negative energy solutions still exist, but because of the fermionic nature of the spin-\frac{1}{2} particles described by the equation the Pauli exclusion principle can be invoked to prevent a particle from falling to infinitely negative energy.  However, for the exclusion principle to fix this problem requires that the negative energy states be filled by an infinity of particles – the so-called Dirac sea.  Dirac postulated that antiparticles would appear as “holes” in this sea, predicting the existence of the positron.  Despite the success of this prediction, this somewhat awkward picture of the Dirac sea is voided by the full QFT formulation (which allows bosonic antiparticles, despite the lack of a bosonic exclusion principle, as well); however, even QFT requires its own sea of infinite energy – i.e., the zero-point energy of the vacuum.

   Thus, by extending the Lorentz group by parity and finding how the Dirac spinors on which the group is represented transform under boosts, one can derive the Dirac equation from group theoretic and symmetry arguments!

   Kind of sketchy for the first post of real substance, but this was mostly to check out the \LaTeX capabilities.  I wrote this post bit-by-bit over a few days, and have since then branched off from Ryder to peruse Maggiore and Aitchison and Hey.  I actually think I will make Aitchison and Hey my text of choice; the presentation is clearer than Ryder, if less interesting, and there are problems given in the book and solutions online at Aitchison’s website (see above for relevant links).  Nevertheless, I hope to find a clearer, mathematically rigorous presentation of the isomorphisms and relations between the various Lie groups in the future.  This looks like a good overview.