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4-3-2 Emission Rate to Lowest Order
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It is possible to get an expansion of W in powers of rL = e Z /4n by expanding both the logarithm and .:J(A) in Eq. (4-97). To lowest order, we find
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(4-100)
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But if q = Pl
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+ Pz, we have
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m) ( - q)]
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! tr [(h + m) (q)(pz = [ -
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~ a(q) a( - q) + Pl a(q)pz a( - q) + Pl a( -
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q)pz a(q) ]
QUANTUM FIELD THEORY
We then compute the integrals [q2 = (PI
+ P2f ~ 4m 2]:
(4-103a)
(4-103b)
Indeed, lilY has the general form lilY = I 1 glly + izqllqy and successive contractions with qllqy and gllY lead to (4-103b). This yields for W(I):
W(I)
fd;;
(1 _ 4;2 y/2 &(qO)e(l _ 4;2 ) (2q2
+ 4m2)
(4-104)
x {q. a(q)q a( - q) - q2a(q) a( - q)}
This result will be made more transparent if we introduce the Fourier transform of the electromagnetic field
FIl.(q) = - i[qllay(q) - qyall(q)]
i Flly(q)PY( -
q) = J B(q)J2 =
E(q)J2
q2a(q) a( - q) - q. a(q)q a( - q)
Finally, since the integrand in (4-104) is even, we may forget the restriction
qO> 2m:
W(I)
= ~ d4 &(q2 - 4m 2) [JE(qW -JB(qW] (1 _ 4;2 )1/2 (1 + q
2;2)
(4-105)
This is a nontrivial result offield theory, but it could have been obtained through a careful interpretation of the hole theory. Observe that for q2 > 0, there exists a frame where q = 0 and thus B(q) = - iq x a(q) vanishes; this implies that JE(q)J2 - JB(q)J2 ~ 0, that is, pair creation is an electric effect. However, electromagnetic modes with q2 > 4m 2 are not that easy to generate by the current technology! We quote also the corresponding rate for the pair creation of spin less charged bosons, to lo)"est order (4-106)
INTERACTION WITH AN EXTERNAL FIELD
4-3-3 Pair Creation in a Constant Uniform Electric Field
In the case of a constant uniform field, we are able to derive an exact, nonperturbative result. We return to Eq. (4-88) and write In So (A)
Tr In
{er -
eJ(X) - m + ie]
r -m+ Ie } 1 .
(4-107)
Since the trace of an operator is invariant under transposition and since the charge conjugation matrix C satisfies
CYpC- 1
we may write In So (A)
+m ie]
Tr In
{er -
eJ(X)
r +m-18 1 .}
12 -m
(4-108)
and the sum of (4-107) and (4-108) reads 2ln So(A) The useful identity In
Tr In
({er
eJ(X)]2 - m 2 + ie} p 2
+ 18
(4-109)
~ = foo
ds (eis(b+i ) _ eis(a+i ) s
(4-110)
enables us to write for the desired probability
w(x) = Re
foo ds e-is(m'-i ) tr xl eis(f-e.tI(x))'lx) - <xl eiSP'lx )
Re tr
ds . .) - e -IS (m , -1
(4-111)
For a constant field, this probability should not depend on x. Moreover, (JpyFPY commutes with all the other operators, and we may compute its exponential. From now on, we assume that the electromagnetic field is purely electric (we have seen in the last section that the pair creation is an electric effect) and that the electric field is along the z axis. We also choose a gauge such that only A 3 (x) = -Et(t == xo) is nonvanishing. Then tr e ise,,",P'/2
4 cosh (seE)
(4-112)
and using the commutation relation eXo, Po] = -i, we get (P - eA)2 - m 2 = P5 - p} - (P 3 + eEXO)2
e-iPOp3/eE(P5 _ p} _ e 2E2 X 02 ) eiPOp3/eE
(4-113)
QUANTUM FIELD THEORY
Therefore, tr
<xl e is [(P-eA)2+,,",P'/2] Ix)
4 cosh (seE)
d p dw dw' ei(w'-w)(t+p'/eE)-isPf <wi eis(P5-e2E2X5) Iw') (2n)4
(4-114)
= 2e;. cosh (eEs) foo dw <wi eis(P5-e2E2x5)lw)
(2n)
The last integral may be considered as the trace of the evolution operator of an harmonic oscillator with a purely imaginary frequency. This results from the correspondence Po -+ P, - X 0 -+ Q, 2ieE -+ Wo, i -+ mo, P5 - e 2E2 X5 -+ p2/2mO + imow5Q2. The energy levels of such a system are well known and it follows that
Trexp[is(2~0 +mo;5 Q2 )J= n~o eXP[is(n+DwoJ
2 sin swo/2 so that
(4-115)
OO 00
dw <wi eis(P5-e2E2X5) Iw) =
. 1 2 smh (seE)
(4-116)
Collecting all the terms, we find
W=---
(2n)2
1 foo ds [ eEcoth(eEs)--s Re(ie- ( 1J. S2
,Sm 2
le) .
(4-117)
Potential energy
----Vex)
Vo/lelE
Figure 4-1 Potential energy (solid line) of an electron submitted to the binding potential V(x) (dashed line) and to the electric potential eEx.
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