20.3 The "Magnetic" Vacuum of QCD
539
ization, replacing a = 1/2 by a = 1/2  e in (20.57). The integral then splits into a
finite part (which is uninteresting in this connection) and an infinite part proportional to (gH)2
Jdx/x l  8 (gH)2B = (gH)2B/e
576
References
Di 28
Di 30a
Di 30b
Di 76
Do 71
Dy 49
Dy 67
P. A. M. Dirac: Proc. R. Soc. All7, 610 (1928)
P. A. M. Dirac: Proc. R. Soc. AI26, 360 (1930)
P. A. M. Dirac: Proc. Cambridge Philos. Soc. 26, 361 (1930)
W. Dittrich: J. Phys. A9, 1171 (1976)
H. G
20.4 Spontaneous Quark Pair Production and Fission of Quark Bags
545
only the first two terms in the bracket of (20.69) would be present. For these
terms alone, and for the chosen value of the coupling constant as = 0.385, the
potential would not become s
21.1 Dirac Particles in a Gravitational Field
SS1
which transforms like a twocomponent tensor (the partial derivative 8e IJ/8x v
does notl).
cfw_
describes both gravitational and
Because the Christoffel symbol
:v
inertial forces, it is not invariant und
20.3 The "Magnetic" Vacuum of QeD
Fig. 20.8. Schema of a hadronic quark bag
bag pressure B
perturbative
vacuum
535
true
vacuum
eq
a) its energy density must be lower than that of the perturbative vacuum;
b) it must be a perfect colour dielectric, i.e. e
16.3 SelfEnergy in Superheavy Atoms
443
The term i nRo is finite and is basically determined by the 1 s wave function
i nRo =
 a Jdr Jr dx
00
o
0
[2
1
]
x Q(r) Q(x)  P(r)P(x) ,
"2
3 r
r

with
Here L1 E(2) is
= (
) .
1
Here G 1s (p) and F 1s (p) den
16.5 The Problem of a Supercritical Point Charge
453
p = [(E  VN )2  m5] 112
T = / sgnx.
The matching condition on the nuclear surface requires
(16.68)
We set
(16.69)
and aR = Y In(2pR) + 17 and calculate the ratio of the wave functions with the
help o
17.2 Alternate Form of the KleinGordon Equation
Fig. 17.1. Spectrum of free KleinGordon equation and
signature of the norm of the plane wave solutions
particle modes
E
483
x'+J
o
Xl)
antiparticle modes
Note here the existence of two sets of solutions,
Subject Index
Metric tensor 550
Minimal coupling see Coupling
MIT bag model 540  541
Mixing between bound state and negative
energy continuum 125
Model of the vacuum
QCD 534535
QED see also Dirac's hole theory 94
Modes of a giant nuclear molecule 382
Mo
582
References
Ra 76a
Ra 76b
Ra 76c
Ra 76d
Ra 78a
Ra 78b
Ra 78c
Ra 78d
Ra 81
Ra 83
Re
Re
Re
Re
Re
70
71
73
76a
76b
Re
Re
Re
Re
77
79
80a
80b
Re
Re
Re
Re
81a
81b
83
84
Re 85
Rh 83
Ri 66
Ri 75
Ri 78a
Ri 78b
Ro60
Ro 61
Ro69
Ro 82
Ru52
Ru 76
Ru 78
Ru 79
Sa 31
19.3 Solutions of the Condensate Equations
517
We now discuss the properties of the numerical results. The free parameter in
the solutions in the "nuclear charge" Z, and all quantities will be given in units of
m n Consider arbitrarily chosen values of Z
18.3 The Fock Space and Diagonalization of the Hamiltonian
(N' n INn)
493
1
= :r:=:=:=:=(0 l(on)N'(on+)NIO)
VNIN'I
=
1
VNIN'I
(OI(On)N'l[N(o,i)Nt+(on+)NOn ] 10)
1
r== (01(N(N1) . . 1).
VNIN'I
=
cfw_1
N'=N.
(18.33)
N'>N
In view of the above remark
584
References
R. F. Streater, A. S. Wightman: PCT, Spin and Statistics, and All That (Benjamin, New
York 1964)
St 77 C. Stoller, W. WOlfli, G. Bonani, M. SWckli, M. Suter: J. Phys. B10, L347 (1977)
St 78 C. Stoller, W. WOlfli, G. Bonani, E. Morenzoni, M.
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
21.4 Event Horizon and Thermal Particle Spectrum
567
We now have a complete set of solutions of the Dirac equation in Rindler
coordinates. In the following section these solutions become a basis for a second
quantized theory of fermions in Rindler space,
574
Be
Be
Be
Be
Be
21
34
47
63
71
Be 74
Be 75
Be
Be
Be
Be
76a
76b
77
78
Be 79
Be 80a
Be 80b
Be 81
Be 82
Bi 70
Bi 75
Bi 82
Bj 64
Bj 65
BI 82
Bo 51
Bo 59
Bo 67
Bo 78
Bo 80
Bo 81
Bo 82
Bo 83a
Bo 83b
Bo 84
Br 59a
Br 59b
References
G. Bertrand: Compo Rend. 172
l
20.1 Quantum Chromodynamics
523
Urr
'I' . U 'I' =
(20.6)
U gr
U br
Any unitary matrix can be written as the imaginary exponential of a Hermitian
matrix: U = exp(iL). We have U+ U = 1 and L + = L, and det(U) = 1 means
tr(L) = O. All traceless Hermitian
21.1 Dirac Particles in a Gravitational Field
/p'
Elm
1.0
Fig. 21.1. The energy eigenvalues of the
Dirac equation in the field of an extended
gravitating source. The energies are shown
as a function of the source radius ro (in
units of the Schwarzschild r
19. Overcritical Potential for Bose Fields
For a discussion of overcritical behaviour of bosons a full understanding of the
single particle energy spectrum is required. The solutions of the KleinGordon
equation can exhibit two distinct types of behaviour
16.5 The Problem of a Supercritical Point Charge
457
In our case this leads to
(16.85)
This model implies that the screening effects are described by the function
y(r) = [Zz(r) a Z xZ] 112, which has the important feature that it contains the
electron an
17.1 The KleinGordon Field
473
where the covariant derivatives have been introduced. Here, as before, X O =
Xo = c t and Oil = O/OXIl, and hence
0ll
= (_0_
cot
, ) .
V
(17.13)
Recall that XII = (ct, x), XII = (ct, x).
The Lagrangian density is a functio
16.5 The Problem of a Supercritical Point Charge
459
and take into account that VN is much deeper than the energy of the deepest
bound state resonance, so that E and mo can be neglected. Thence
2 () _ V2
x(x 1)
(x t)ZNa
3 (ZNar)2
r  Nr2
R 3 VN
 4 R 3 V
19.3 Solutions of the Condensate Equations
515
approximate, coherent ground state (19.48) is in fact an approximate eigenstate
of the operator
Ii' = Ii  /.lL ,
L = Sxt !tXd3X
where
(19.59a)
(19.59b)
is proportional to the charge operator.
The coherent st
16.3 SelfEnergy in Superheavy Atoms
441
The double line indicates the exact electron propagator and wave function in the
Coulomb field of a nucleus. According to this diagram the level shift follows
from the Gel/Mann  Low theorem [Oe 51J
LIE
= 4 ni a J
19.2 The True Ground State and Bose Condensation
S07
which vanishes in view of (19.27c). Thus the paradox (19.24) is resolved. We have
found two complexconjugate bound state solutions with ER < mc 2 of
vanishing norm. Appearance of such solutions beyond
16.4 Supercharged Vacuum
fiE leV]
II
IS
445
Fig. 16.3. The selfenergy shift LlE of Kshell electrons is plotted versus the nuclear charge number Z.
(e e e) = numerical results of Mohr [M074] for is
electrons in the Coulomb field of pointlike nuclei;
(00
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_
16.5 The Problem of a Supercritical Point Charge
461
scribed in the previous section. From the formulae obtained it is obvious that
screening does not destroy this shell structure; its effect is only to shift the outer
shells of the vacuum charge further
18.3 The Fock Space and Diagonalization of the Hamiltonian
491
a;
and the adjoint equations for
and Ck' The Hamiltonian Ii (18.2) is a bilinear
form in these operators. Our aim is to find a space in which the action of these
operators can be better unders
20. t Quantum Chromodynamics
527
ization, and so on, can be summed into a geometric series for the full propagator
(see Sect. 14.4):
.
D(q2)
= Do(q2)+ i2D o(q2)Jl(q2)Do(q2)
+ i4Do(q2) Jl( q2)Do(q2) Jl(q2)Do(q2) + .
= DO(q2) [1 Jl(q2)Do(q2)]1
= 1/q2 [1 +
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
19.1 The Critical Potentials
O.l."r"tr;,..,
0.2
0.3
0.
503
Fig. 19.4. Energy of the lowest bound state
in the Coulomb potential of the nucleus
with V=  VQI1(rIR)2/3), R = tOhi
moc. Insert is an enlargement of the critical
area, showing (. ) the
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,
20.2 Gluon Condensates in Strong Colour Fields
531
contain a source term, but instead the thirdorder term g\ W* x W) x W. Except
jor this term, it is identical to the usual wave equationjor a charged vector boson
in an external Coulomb jield . Hence the