Field Theory
        Georgia Tech PHYS-7147              spring 2005
        Tue, Thu 12:05-13:25                 Howey S107

				Homework Assignment #8
				----------------------
				due Thu, Mar 31

Dirac equation
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Reading: 

   W. Greiner and J. Reinhardt: Dirac equation
   excerpt on ChaosBook.org/library  (click on Field Theory link)
   covers Dirac's original derivation of his equation.

   Exercises: none, please work through the steps of
   derivation of the Dirac equation, in particular the matrix algebra,
   and verify the probability current conservation.

   Bonus points: 
   Exercise B (ambitious, requires a bit of a literature search): Sketch 
              Wigner's derivation of the Dirac equation. The idea is
	      that the energy of a particle at rest (of any spin) is
	      component of a Minkowsky 4-vector. Boost the Pauli 1/2
	      spinor into 4-d Minkowsky space.

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Predrag's comments:

There are at least three different ways to derive the Dirac equation. 

1) The most elegant is the one Dirac gave in his book [1], with Dirac
propagator a "square root" of the Klein-Gordon propagator.

2) The derivation based on Wigner's analysis of the irreducible unitary
representation of the Poincare group (the covering group of the
inhomogeneous proper othochronous Lorentz group) is very
important for the development of 20th century QFT [2].

3) There is an intriguing derivation which Feynman gave (for the SO(1, 1)
case with one spatial dimension and one time dimension) in his class, and
which was given as a problem in his book with Hibbs [3].

Wigner's derivation:
 In 1939, Wigner [2] observed that internal space-time symmetries of
relativistic particles are dictated by their respective little groups.
The little group is the maximal subgroup of the Lorentz group which
leaves the four-momentum of the particle invariant. He showed that the
little groups for massive and massless particles are isomorphic to the
three-dimensional rotation group and the two-dimensional Euclidean group
respectively. Wigner's 1939 paper gives a covariant picture massive
particles with spins, and connects the helicity of massless particle with
the rotational degree of freedom in the group E(2).

His main motivation was to
understand in intrinsic physical terms the ever increasing zoo of linear
relativistic (higher spin) field equations of those days, which were
proposed in the aftermath of the Dirac equation.


 [1] P.A.M. Dirac, Quantum Mechanics, 4th ed. (Oxford University Press,
London, 1958). 

 [2] E. P. Wigner, Ann. Math. 40, 149 (1939). See, e.g., A.S. Wightman, in
Dispersion Relations and Elementary Particles, edited by C. DeWitt and R.
Omnes (Wiley and Sons, New York, N.Y., 1960).

 [3] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals
(McGraw-Hill, New York, 1965). Also see T. Jacobson, J. Phys. A17 (1984)
2433, ibid. A17 (1984) 375; I. Bialynicki-Birula, Phys. Rev. D49 (1994)
6920; L.H. Kau man and H.P. Noyes, Phys. Lett. A218 (1996) 139.


 scholar.google.com will give you many derivations of the Dirac equation.
 possibly of interest:
http://arxiv.org/hep-th/9512151
http://arxiv.org/hep-th/9608092
http://arxiv.org/hep-th/0102067
http://arxiv.org/hep-th/0108067, and so on...