G LO SSARY 2 85
threestage model of group decision making. According to Bales and Strodtbeck (1951),
groups proceed through three stages before eventually arriving at a decision: orientation,
discussion, and decision making.
traits. Personality charact
17.2 Alternate Form of the KleinGordon Equation
479
where the powers of (moc 2 ) have been chosen that the dimension of qJ(l), qJ(2) is
[L  312). A trivial consequence of (17.44) is a relation between the components
qJ(l) and qJ(2)
(ih
:t 
eAO) qJ(l)
=
498
18. Subcritical External Potentials
Fig. 18.1. 1sand 2p energies of the KleinGordon equation in units of m"c2 , as a function of the nuclear charge Z. V  Zir
1s
1r
allows us to identify
n=n'+1+1,
(18.56)
where n = 1,2,3 . (where, of course 1= 0,1,2
18.4 The Coulomb Problem for SpinO Particles
497
which has the form of Whittaker's differential equation [Ab 65, Eq. (13.1.31)]
and hence the solution regular at origin is
1
1
U(s) = NWx,.u(s) = NeTSsT+.u lFl (t+ /1 x, 1 + 2/1; s) .
(18.49a)
Here lFda,
17.2 Alternate Form of the KleinGordon Equation
477
+ hCVx*(x) (X') [ft(X'), ihcVx$(x)]
+ (mOc 2)2*(X) (X')[ft(X'), (X)] + h.C. ,
(17.35)
which is obviously an antiHermitian operator.
Aside from the usual commutators (17.31), one also needs here
[ft(x')
448
16. ManyBody Effects in QED of Strong Fields
5.0
I
7
4.0
'e
.
3.0
Cf
>
5

0
.
0
_
I
Fig. 16.5

6
_
10
fm
100
_
1000
I. 0
V
100
10
Fig. 16.6
If
1000
1m
Fig. 16.5. Solutions for the selfconsistent potential Vof the relativistic ThomasFermi equati
18.4 The Coulomb Problem for SpinO Particles
495
The same approach must be employed to determine the form of B(s) (18.35)
in terms of the mode operators. But the (FeshbachVillars) field equations must
be used for the timeindependent potential (17.60) s
18. Subcritical External Potentials
To prepare for the discussion of overcritical phenomena in the case of spinO
particles, in this chapter the quantized theory of bosons in external potentials will
be treated. Starting from the Hamiltonian field equatio
16.5 The Problem of a Supercritical Point Charge
451
The transformation (3.113), which is
= 2( Emo)1I2Recfw_l/>d
U2 = 2(  E + mo)112lm cfw_I/>d,
U1
(16.56)
gives the components U1 and U2 of the Dirac spinor
Ul
= 2Ne rr y12( 
E  mo)112 . [(1
+ Recfw_
17.1 The KleinGordon Field
471
The SchrOdinger wave equation is easily generalized to describe the relativistic
motion. Recall the relativistic energymomentum relation
(17.1)
and the quantization prescription
(17.2a)
at
pihV.
(17.2b)
Inserting (17.2a,
16.5 The Problem of a Supercritical Point Charge
449
radius of the background charge distribution is taken to zero. Naturally, the selfconsistent potential that emerges must be just subcritical, that is, no further state
can become supercritical. This mea
16.4 Supercharged Vacuum
447
We now proceed to discuss the solution of (16.41). Since the charge density of
the vacuum must be confined to the vicinity of the external charge, we require a
solution such that
e VCr)
ya
4 _
r+
(16.42a)
r
00
(a is the fine
19.2 The True Ground State and Bose Condensation
(q+ 11Jilq+ 1)
= W(q) + tL.
511
(19.44)
Thus tL is the energy of the last particle added to the condensate. Therefore we
must seek solutions of (19.43) for a particular choice of (a real) tL within the
boun
450
16. ManyBody Effects in QED of Strong Fields
The results described above are altered when one takes into account that the
nuclear charge is screened by the vacuum charge located close to the nucleus. A
similar effect occurs for a charged sphere in an
476
17. Bosons Bound in Strong Potentials
We can now turn to the canonical quantization of the KleinGordon field.
In this conventional method to quantize a Bose field, assume that the fields
and their conjugate momentum fields (17.19) satisfy the followi
462
Z
200
16. ManyBody Effects in QED of Strong Fields
Fig. 16.12. Screening function Z(r) =
 Vr/afor a nucleus with charge ZN = 200
and radius R = 10  5 fm. (   ) solution
of the Poisson equation; (    ) approximate Zeff
,
I
I
IL _ _
,
I
190
I
482
17. Bosons Bound in Strong Potentials
where en is + 1 for particles (negatively charged, observe that e is the electron
charge), and 1 for antiparticles. In component form (17.60) becomes [cf. (17.45)]
(En eA o)qJ(1) = moc 2qJ(2)
(17.62a)
(En eA o)
492
18. Subcritical External Potentials
(18.27)
Hence, as an annihilates the vacuum state,
anlNn> = VNI(Nl)n>.
(18.28a)
Thus an' in general, reduces the number of particles in the mode" n" . Contrary to
this, an+ increases the number of particles in the
454
16. ManyBody Effects in QED of Strong Fields
tanaR
=
1 sgnx(y+Za) URllxl
URsgnx(y+Za)/lxl
tanqJ= rno (
E
Ixl
2(y+Za)
_ Y+3Za) sgnx.
21xl
To investigate the dependence on the sign of x, we introduce the notation
aR(x = 1) and qJ for qJ(x = 1). Becau
16.6 Klein's Paradox Revisited
467
charge cannot be observed and we must thus conclude that, in this case, the real
vacuum charge would have to be renormalized. This would then also be true for
the shielding charge of the nucleus. In addition, the "real"
500
19. Overcritical Potential for Bose Fields
We now solve this equation first for a shortrange potential, taking here the
example of the square well potential
VCr)
=
YoO(R  r) .
(19.4)
Then
(En 1m + Yo)2  (mo C2 )2 > 0
for the (bound) s states (I =
17. Bosons Bound in Strong Potentials
This book would remain incomplete without a discussion of the behaviour of
Bose particles in supercritical external fields. Indeed, the physics explored in the
following sections differs fundamentally from the conclus
16.5 The Problem of a Supercritical Point Charge
465
100
50
1
ue
Fig. 16.16
Fig. 16.15
Fig. 16.15. The function q2 r2 at the energy E = 0.5 mo. (    ): computed without approximations
in the singleparticle case for ZN = 136 and 200. (   ) : with sc
488
18. Sub critical External Potentials
(18.11a)
which should vanish in a consistent quantization. Note that only if
[ap , a;]
= o\pp')
(18.11b)
[Cp,C;]
= 03(p_p')
(18.11c)
will we get the desired answer, that is (18.4b). Assuming (18.11),
[X(x,O),X+(x,
18.2 (Quasi) Particle Representation of the Operators
x(x,O) =
I:
I: XJc>(X)ct
k
n
487
(18.9)
,
introducing the socalled singleparticle operators On, ct. Their commutation
properties are determined by attempting to satisfy (18.8). Equation (18.9) is
wr
468
16. ManyBody Effects in QED of Strong Fields
m
L
L
m

v+m
Fig. 16.18. Scattering of a positron of a
negative potential step in the Klein framework

.
X
+0
V
which as indicated can hardly be more than the electron Compton wavelength.
But most real
452
16. ManyBody Effects in QED of Strong Fields
We are interested mainly in the charge density of the supercritical electrons
imbedded in the negative energy continuum. This charge density is given by
(6.181) and can be written as
(16.64)
where LJ E is
458
16. ManyBody Effects in QED of Strong Fields
2
11
!I
r_ !I
o
1
2

./
/
III
ro
1'102
Fig. 16.10. Quasiclassical momentum of a
nucleus with Z = 200 at energy E =
 1.5 me for the Ut wave function with
x = + 1. ro denotes the point where
E  V + m =
264 INTRODUCTION TO POLITICAL PSYCHOLOGY
across existing case lists; it forces the researcher to specify the precise time frame and
exact sequence within which the appropriately designated threats, counterthreats and
responses are made. (p. 13)
In adop
266 INTRODUCTION TO POLITICAL PSYCHOLOGY
the way of your oncoming vehicle), while you continue to drive straight down the highway.
Thus, both drivers (assuming they were not suicidal maniacs) would need to not only
demonstrate their capability of causin
258 INTRODUCTION TO POLITICAL PSYCHOLOGY
Although an ancient Greek historian, Thucydides has often been described as the rst realist,
because of his attention to the anarchic, selfhelp nature of the ancient Greek international sys
tem; his emphasis up
IO. INTERNATIONAL SECURITY AND CONFLICT 269
only deterrence failures that led to war, not successful examples of deterrence that maintained
the peace. Despite decades of research and debate on the subject of deterrence, scholars still
greatly dispute wh
IO. INTERNATIONAL SECURITY AND CONFLICT 259
be automatically perceived to be offensive by their opponents). In 1914, one needed a mobi
lized army at the front line to either defend yourself from attack or invade your neighbor. Of
fensive and defensive