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(2-77)
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(2-78) with <p a Dirac spinor and P a four-vector which is not orthogonal to n. By adding to P some quantity of the form An, we may realize the condition
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The interpretation of this four-vector is the following. In the frame where eO = 0, and thus A O = 0, E and A along Ox, B along Oy, and n along Oz, the operators PI = ia b P2 = ia 2, and i(a o + a3 ) = Po + P3 commute with the Dirac hamiltonian. The substitution of (2-78) into Eq. (2-77) leads to the condition
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2eA P -
ieU')<p(~) =
<pW= (m)1/2 exp {e 4 _ if< d~'[eA(~') P _ 2n P 0 P
e2A2(~')J}u
2n . P
(2-79)
where u is a constant bispinor. Since ( 4)2 = - n2A2 = 0, we may write
t/!p(x)
-e )(m)1/2 ue iI ( 1 + 2n P 4 Po
where I stands for the action of a classical particle in a plane wave (damped at infinity), with P = mu(oo) = p(oo) [compare Eq. (1-84)]:
1= -p x - fn.x d~[~ A(~) p - ~ A2(~)J o n p 2n P
For t/! to satisfy the original Dirac equation and not only the squared equation (2-73), u has to obey some auxiliary condition. After some algebra, we find that
(i~ - e/1- m)t/!p(x) =
(m)1/2 (
1 + 2p. n U
e)(p -
m)u e'l
Therefore
(p - m)u =
THE DIRAC EQUA nON
and u = u(p) is a solution of the free Dirac equation. The reader will verify that the solution (2-79) has the correct normalization
and that the associated current reads
r(x) = iJ!p(x)y"I/Ip(x) =
~ [p" - eA" + n" (e ~ - eA2)] pO n' p 2n' p
If A(~) is a quasiperiodic function of ~ (slowly damped at infinity), the average value ofr is
showing the same phenomenon as its classical counterpart. These expressions were originally derived by Volkow in 1935.
2-2-4 Foldy-Wouthuysen Transformation
In the last subsections, the physical meaning of the Dirac equation has been investigated using a nonrelativistic approximation. It is worth while showing that this may be pursued in a systematic way. This is the purpose of the FoldyWouthuysen transformation. To be more explicit, we want to find a unitary transformation
e-iSI/I '
which decouples the small and large components and where S may be time dependent. Let us call odd the operators such as IX which couple large and small components, even those which do not (for example, I, /3, ... ). Since 1/1 ' satisfies the equation .
(2-80)
our problem is to find S so as to get rid of the odd operators in H' at a given order in 11m. In practice, we shall do it up to terms of order (kinetic energy1m or (kinetic energy x potential energy/m 2 ). This will lead to a further insight in relativistic corrections entailed by the Dirac equation. In the free case, we can construct S exactly. It is time independent and can be chosen as
S= -z/3-8= - z - 8
IX'P
Since (y'
p/ipJ)2 =
I, we may compute e iS and H' in a closed form:
e's = e(Y'p/!p!l8
cos 8 + /3 -
IX'P
SIll
QUANTUM FIELD THEORY
(COS 8 + f3 O(I~I sin 8}0(. P + f3m) (cos 8 - f3 O(I~I sin 8)
0(. p(cos 28 -
1:1 sin 28) + f3(m cos 28 + Ipi sin 28)
The choice of 8 such that sin 28
cos 28 =]f
eliminates the odd operator 0( . p and leaves
(p2 + m E
f3(m 2
+ p2)1/2
as we would expect. In other words, we have decomposed H into a direct sum 2 of two nonlocal hamiltonians + p2. It is clear that these square roots cannot be represented in configuration space by a finite set of differential operators. In the interacting case, we therefore expect S to be of order m-l, and we expand H' to the desired order:
+ i[S, H] - ~ [S, [S, H]] - ~ [S, [S, [S, H]]]
+ 24 [S, [S, [S, [S, H]]]] - S - 2 [S, S] + "6 [S, [S, S]] + ...
Here use has been made of the general identity
eA Be- A
L.., n=O
,[A,[A,[ [A,B]]- ]]
(2-81)
where, in the generic term of the sum, A appears n times. This identity is easily derived by computing the successive derivatives of esA Be-sA at s = 0, and using them in the (formal) Taylor expansion at s = 1.
We start from H = f3m + (!) + $ where (!) denotes the odd operator (!) = 0( . (p - eA) and $ the even one eAo. The solution of the free case suggests that we take S = - if3(!)/2 to first order. Then according to the above expression for H', we calculate
+ (!)' + $'
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