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with x' = Ax
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(2-9b) (2-9c)
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iyJl ,",0, Jl l/J'(x') - ml/J'(x') = 0 ux
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There must be a local relation between l/J and l/J', so that the observer in the second frame may reconstruct l/J' when l/J is given. We assume that this relation is linear:
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l/J'(x') = S(A)l/J(x)
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(2-14)
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where S(A) is a nonsingular 4 x 4 matrix. Equation (2-9c) now reads
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~ ~ S(A)l/J(x) Jl
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ux uX
- mS(A)l/J(x) = 0
In order that this equation be a consequence of (2-9b) for any l/J, and since
ox"/ox,Jl = (A -l)"Jl' we must have
(2-15) Let us first construct S(A) for an infinitesimal proper transformation A, which
52 QUANTUM FIELD THEORY
may be written as (A -l)1'v = g''v - wi'.
+ ...
where the infinitesimal matrix wl'V is antisymmetric. We write
S(A) = I -
al'vwl'V
+ ...
(2-16)
where the matrices yields
al'V
are antisymmetric in IlV. To first order
III W,
Eq. (2-15) (2-17)
A set of matrices a ap satisfying this relation is given by (2-18) A finite transformation is of the form
S(A)
= r(i/4)(J,poJ"P
(2-19)
where w aP is now finite. For spatial rotations S is unitary, whereas it is hermitian for Lorentz boosts.
The form of the finite transformations is most easily derived in the chiral representation for
l' matrices:
1'0 =
f3 =
( 0 -1)
-1 0
i . ((Ji (Joi=2[YO,y;]=-ux i =i 0
-(Ji
(2-20)
In tbis representation, tbe two Pauli spinors of tbe decomposition of tbe bispinor !/J transform independently under rotations and boosts. Tbe representation of tbe Lorentz group [more exactly, of its covering group SL(2, C)] is reducible into a sum of two inequivalent representations: (t, 0) + (0, i). However, we sball see tbat tbe representation is irreducible if we include tbe transformation under parity (space reflection).
We recall that the representations of the Poincare group are classified according to the values of two Casimir operators p 2 and W2; PI' is the momentum energy operator, which is the infinitesimal generator of translations, whereas WI' is constructed from the angular momentum operator JI'", the infinitesimal generator of Lorentz transformations, as (2-21)
THE DIRAC EQUATION
If M2 denotes the eigenvalue of'p2, W 2 takes'only values of the form
W 2 = -M 2 S(S
+ 1)
where the spin S is integer or half integer. For the solutions of the Dirac equation, and therefore of the Klein-Gordon equation, p2 = - 02 takes the value m2, while J JlV is given by
l/J'(X)
(I - ~ = (I - ~ (I - ~
JJlVWJlv)l/J(X) (TJlVW Jlv ) l/J(x P- wPvX V ) (TJlvW JlV
+ XJlWJlVOv)l/J(X)
(2-22)
which yields
Let us then compute
from Eq, (2-21):
(2-21a)
The orbital contribution has disappeared, justifying that intrinsic angular momentum. We then use the identity
eJlaPyf,ila'P'y' =
corresponds to
-det (grr.)
L (-It gap",gPPP,gyP,
(2-23)
where in the first expression r (or r' respectively) takes the values rx, f3, y (rx', f3', y/), and in the second one, the sum runs over the permutations P of (rx', f3', y/). After some algebra using the Dirac equation, this leads to (2-24) Thus the equation describes spin i particles. Finally, we derive the transformation law of the spinor l/J under parity. We have again to find S(A) satisfying (2-15), where A denotes the matrix
(2-25)
It is easy to see that
(2-26)
QUANTUM FIELD THEORY
is the desired transformation. Here rfp is an arbitrary, unobservable phase. The important point is that the positive and negative energy solutions have relative opposite parities corresponding to the two opposite eigenvalues of yO. After the reinterpretation of negative energy solutions, this will mean opposite intrinsic parities for particle and antiparticle. The various bilinear forms constructed from 1/1 and Ij/ play an important role in the sequel. The remainder of this section is devoted to the study of their transformation properties under Lorentz transformations. From Eq. (2-14), we deduce that
Ij/'(x') = lj/(x)yOS(A)tyo
lj/(x)S-l(A)
where the second expression is verified using the explicit expressions (2-19) of S(A) [and (2-26) for parity]. Thus a bilinear product Ij/(x) A 1/1 (x) transforms according to
1j/'(x')AI/I'(x') = lj/(x)S-l(A)AS(A)I/I(x)
(2-27)
For instance, from (2-15), we learn that Ij/ylll/l transforms as a four-vector:
(2-28)
whereas Ij/(x) I/I(x) is a (nonpositive definite) scalar density. More generally, any 4 x 4 matrix may be expanded on a basis of 16 matrices. It may be shown that the algebra generated by the y matrices-a Clifford algebra for mathematicians-is nothing but the complete algebra of these 4 x 4 matrices. Let us introduce the notation~ (2-29) In the representation (2-10)
Y The matrix yS satisfies
(0I I)
(2-30)
(2-31)
We now consider the 16 matrices:
['S =' I
['VIl=' Yll [' Ilv=CJllv =2 YIl'Yv]
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