260
10. Evolution of the Vacuum State in Supercritical Potentials
particle amplitudes Smn. This is a consequence of neglecting true twobody interactions between Dirac particles arising from electromagnetic interactions of electrons with other electrons o
9.9 Appendix: Feynman Propagator for TimeDependent Fields
255
Each of these two sets of equations determines the coefficients I3nm and Pnm, but
they cannot be explicitly resolved. Still, they suffice to derive a relation between
the StuckelbergFeynman p
286
10. Evolution of the Vacuum State in Supercritical Potentials
1m cfw_s
Cft!
I
IRe
cfw_s
Fig. 10.11. Rotation of the integration contour in the complex s
plane. The contribution of the quarter circle at infinity vanishes
due to the exponential factor e
10.1 The In/Out Formalism
259
the operators for in particles to those for out particles by projecting (10.1,2) with
With the singleparticle Smatrix elements Smn defined in (S.4S),
EC;:ut) = Jd 3x
d<out)
=(
m
)(x, t) t Qt(x, t) = (
E(inlt
+
n
n>P
I:
n>P
268
10. Evolution of the Vacuum State in Supercritical Potentials
Resonance Wave Function
U,
10
Fig. 10.8. Negative energy continuum wave
functions in the resonance region for
Z = 184. Upper part: continuum eigenstate
lower part: projected continuum wave
to.5 The Vacuum in a Constant Electromagnetic Field
=
=
tr
2
m;]
0
I
281
(m e  ie)2)]
ds exp[ i(m e ie)2s]
(10.110)
s
In the step from the first to the second line we used the property that the trace of
any product of an odd number of y matrices vanis
10.3 Decay of a Supercriticai K Vacancy  Projection Formalism
271
To go further, we must know the overlap matrix elements between the true
eigenstates tPp, and the projected basis functions
i.e. we must diagonalize
the full Hamiltonian Ho in the modified
to.7 Klein's Paradox Revisited
297
To find the relation between the in and out operators the singleparticle Smatrix
elements according to (10.5) are needed:
11
SE'
E=
<If/r,E'
() IIf/r,E
(+
+1
OOs
4no
o
Y
S d Z If/r,E'
refl ()t
()
(1)2
Z If/r,E Z
=
n
10.3 Decay of a Supercritical K Vacancy  Projection Formalism
267
to the subspace projected by P = J  Q is a Hermitian operator, because J, Q and
therefore P are Hermitian:
(PHDP)t
= ptHbPt = PHDP.
(10.38)
Its eigenfunctions for E <  me are therefore o
290
10. Evolution of the Vacuum State in Supercritical Potentials
Fig. 10.13. Measurement showing the spectrum of the
Hercules XI pulses. The peak at S8 keV is by the Larmor
transition of electrons in a magnetic field of S x 108 T.
Indications of the sec
280
10. Evolution of the Vacuum State in Supercritical Potentials
(10.104)
This relation enables us to express the vacuum amplitude (to. 100) solely in terms
of the Feynman propagator through the infinitesimal equation
(to.l05)
This equation is the starti
264
10. Evolution of the Vacuum State in Supercritical Potentials
but for small switchon times T the distribution is much broader than the
resonance width (Fig. 10.6):
N.
Ii
_ IV. 12 T2 (sin tT(c; Er)2
Ii
+T(c;Er)
(10.30)
The width in this case is appr
278
10. Evolution of the Vacuum State in Supercritical Potentials
y(t) = exp[ i(EvacER)t ttt]
,
(10.97a)
(10.97b)
The amplitude y(t), belonging to the Kshell component of the state 1.Q(t,
propagates like an excited state of energy (ER) = IERI above t
282
10. Evolution of the Vacuum State in Supercritical Potentials
(10.114)
The trace is obviously divergent, if taken over all of spacetime, but for the contribution of a finite volume to the vacuum amplitude the expression is well defined.
The spinor tr
10.4 Decay of the Neutral Vacuum  SchrOdinger Picture
00
0= WECO)
277
00
= J dp Weep) = J dpj(p)hE(p)*
00

(10.90b)
00
for the unknown function j(p).
This set of equations can be solved with the help of the orthogonality relation
for the negative conti
10.3 Decay of a Supercritical K Vacancy  Projection Formalism
1t
269
Phase shift
10)
Fig.tO.9
1R, taHIR,t o10)
Fig. to. to
Fig. to.9. Energy dependence of the phase shift in the negative energy continuum for Z = 184.
(   ) shows the phase shift of the
258
10. Evolution of the Vacuum State in Supercritical Potentials
F
=blin)
I in
lout)
All variabLe
in
Fig. 10.1
F'
b1out)
=
=
out
.t
Fig.tO.2
Fig. 10.1. Stationary in and out regions are connected by an intermediate region where the potential
changes. M
256
9. Second Quantization of the Dirac Field and Definition of the Vacuum
(9.180)
where we renamed m to n in the second sum. This new relation enables the coefficients ikn to be expressed by the coefficients !3nm in (9.179) or vice versa:
(9.181)
i.e. th
10.7 Klein's Paradox Revisited
293
magnetic field strength (eB). As shown in Fig. 10.15, the K shell becomes supercritically bound when the magnetic field strength reaches 1012 T. Due to the magnetic spin interaction, only the magnetic substate of the K s
9.9 Appendix: Feynman Propagator for TimeDependent Fields
=
+ e(t t') L [ L
kl
. lfIi+ )(x) Vi + )(X')
 e(t'  t)
n>F'
253
Sn*kSnl Okl e(k > F)]
L [ L Sn*kSnl Ok1e(k<F)]lfIi+)(x) Vi+)(X') ,
k, I
n<F'
where e(k F) means that the sum over k is restrict
10.2 Evolution of the Vacuum State
dP
dE
dP
dE
Fig. 10.5a, b. Shape of the spontaneously emitted positron spectrum for different times of duration of supercriticality
rT1
b)
Width
.
263
I<
Width
1/T
EFWHM
rEFWHM
10
5.561'i
r"'r=r
0.1
10
rT 1ft
100
Fig
11.1 HeavyIon Collisions: General Remarks
301
Fig. 11.1. Definition of the coordinates in the multipole expansion of the potential
z
V(r, cos 0)
=cfw_
 2Ze2
R12 [(2r)2
1 + Ii P2(COSO) + .] ,
2
_
2
J'
r:5R12 .
(11.1)
.
Clearly, at distances r > R the pot
254
9. Second Quantization of the Dirac Field and Definition of the Vacuum
iSSF(x,X ' )
= 8(t t') I:
I: Pnm
n>F'm>F
8(t'/)
I:
I:
(9.171)
n<F m<F'
It makes states above the Fermi surface propagate into the future and those
below into the past. It also e
10. Evolution of the Vacuum State in Supercritical
Potentials
Three types of experiments probing a supercritical external field can be imagined,
in principle. First, experiments where a sub critical but strong field is made supercritical for a certain fin
262
10. Evolution of the Vacuum State in Supercritical Potentials
We have thus shown in the framework of field theory that the neutral vacuum
state decays in a supercritical external potential, producing a positron distribution centred around the supercri
274
10. Evolution of the Vacuum State in Supercritical Potentials
of an initially prepared state. The decay of the neutral vacuum in the static electric field of a supercritical nucleus is a sufficiently simple process to enable it to
be treated in the Sc
276
10. Evolution of the Vacuum State in Supercritical Potentials
Projecting with the states (0 laR and (0 IdE, respectively, a set of coupled differential equations for the functions y(t) and WE(t) arises:
(10.84a)
(10.84b)
According to (10.81) the initi
292
10. Evolution of the Vacuum State in Supercritical Potentials
Fig. 10.15. Energy of a Kshell electron in Pb as function of the surrounding magnetic field strength.
The energy is measured in the magnetic field in units of 4.4 x 109 T.
The curve labell
10.3 Decay of a Supercriticai K Vacancy  Projection Formalism
265
Fig. 10.7. An approximate wave function
for the supercriticai K shell can be
obtained by cutting the oscillating tail off
a negative continuum wave function in
the 1s resonance
r
charge of