NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
234
9. Second Quantization of the Dirac Field and Definition of the Vacuum
The field operator is quantized according to
/iI(x, t)
=
L 6n tJJn (x) e  iEn t +
En> me
L
En< me
qJn (X)
e  iEn t
(9.99)
with the specifications
6n 10, qo) = 0,
anlO, qo)
= 0,
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
324
12. The Dynamics of HeavyIon Collisions
various channels, because for R + 00 (V AB + 0), Xn(R) become eigenfunctions of
i.e. plane waves with good momentum P:
p2/2/l,
Xn(R)
R+oo
i
) .
exp ( ;P.R
(12.45)
Physically, however, the asymptotic relati
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
228
9. Second Quantization of the Dirac Field and Definition of the Vacuum
We shall now prove that
If exp [
i: I
d'x([Y;, litl 
(Y; Vi' +.po
\Ii)]
(9.72)
has the desired properties. The sign in the exponent can be selected freely,
because <j is idempoten
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
10.5 The Vacuum in a Constant Electromagnetic Field
285
where (10.131) was used. We now solve (10.134) for e/lVwhich appears on both
sides:
(10.135)
where 1VA = gVA is the Minkowski metric (1, 1, 1, 1). Through the antisymmetry of FVA we have
(10.136)
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
9.7 Charge and Energy of the Vacuum (I)
237
where F characterizes the boundary between particle and antiparticle states with
the possible choices Fo or F_ m Clearly, the above expressions for qvac and Evac
make little sense as they stand, because they inv
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
9.5 The Vacuum State (II)
231
that satisfies our requirements and does not contradict (9.87). The Fock space
expression for the charge operator is
Q
= e Jr d 3X[VI t(x), VI x)] = e I: [otno 0n] + e I:
A
A
(
2
2
2
n>Fo
n<Fo
at
:J
[Un,
n],
(9.89)
where
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
222
9. Second Quantization of the Dirac Field and Definition of the Vacuum
From Taylor's theorem the lhs is
(9.42)
if we retain only terms linear in LI
gives
(j/'(x) = (j/(x)
e
Il
+
ax v
2
a(j/(x)
axil
and ell. Combining the last two equations
axil
(j/(x
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
12.5 Collisions with Nuclear Interaction
333
Fig. 12.9. Various trajectories R(t) in
a heavyion collision: (a) Rutherford
trajectory, (b) Rutherford trajectory
with finite contact time T, (c) trajectory with friction and energy loss
R
T
t
parameters owin
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
246
9. Second Quantization of the Dirac Field and Definition of the Vacuum
emission of two positrons. Upon further approach of the nuclei the energy of the
vacuum continues to grow, but less rapidly than before Rcr as a result of the binding effect of the
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
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
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
10.5 The Vacuum in a Constant Electromagnetic Field
279
It is possible to start directly from the expression (9.114) for the vacuum
energy:
Evac
= Sd 3xtr[yO
at
.
X'+X
However, this expression does not form a useful basis for a covariant treatment,
becau
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
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
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
9.1 Canonical Quantization of the Dirac Field
213
9.1 Canonical Quantization of the Dirac Field
The transition from the classical theory of the Dirac field to the associated quantum theory is most easily made in the framework of Hamiltonian formalism, in
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
252
9. Second Quantization of the Dirac Field and Definition of the Vacuum
stationary states at t =  00, and an out vacuum, defined according to stationary
states at t = + 00. This is reflected in different possible definitions of the
Feynman propagator,
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
10.4 Decay of the Neutral Vacuum  Schrodinger Picture
273
We finally evaluate the time evolution operator in the supercritical potential
(10.49) in terms of the projected resonance basis, because this result will be
needed in Chap. 12. With the help of (
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
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
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
11.2 The TwoCentre Dirac Equation
303
asymptotic ion velocity estimated before. From (11.2) follows that Ref must be
considerably smaller than 135 fm (for U + U the true value is about 30 fm, see
below), hence the collision time 'eoll is less than 10 21
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
9.7 Charge and Energy of the Vacuum (I)
oJ
243
Fig. 9.5. (a) Feynman diagram for the lowest order vacuum
polarization function n(1); (b) graphical representation of the
vacuum energy
in secondorder perturbation theory
are attracted. However, the result i
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
216
9. Second Quantization of the Dirac Field and Definition of the Vacuum
On applying the operator i(a/at)  Ho(x, t) to this equation, it follows from
(9.9) that the matrix function S satisfies the homogeneous Dirac equation
[i :t  tl]
HD(x,
S(x, t;x',
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
9.8 Charge and Energy of the Vacuum (II)
249
where ro, ro [V] are the radii where the square roots vanish, respectively (the
classical turning points). The square root expressions occurring in (9.151) are just
the classical radial momenta with and without
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
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
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
10.2 Evolution of the Vacuum State
261
Fig. 10.3. At to the subcritical potential is suddenly rendered
supercritical
FUn)
Vcr
F'(out)
Vcr+V'
(10.14)
whereas the backward propagating functions after to are stationary in the supercritical basis
)(x, t)
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
9.2 Fock Space and the Vacuum State (I)
(0, bare 10, bare) = 1 ,
219
(9.28)
then every scalar product of the type
(ni, . , nk, bare Into . , nk> bare)
(9.29)
can be readily evaluated by the commutation relations and definition (9.25). The
scalar product v
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
9.4 Gauge Invariance and Discrete Symmetries
225
commute with the Hamiltonian (in the Fock space of localized states) and,
being Hermitian operators, correspond to observable constants of motion.
In the presence of an external potential, the energy moment
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
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
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
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
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
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
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
240
9. Second Quantization of the Dirac Field and Definition of the Vacuum
tr [yO
ot
OA
Sp(x, X' 1 AA p)]
= Ap(O IJp(x) 10) 
X'+X
i tr [yO HO[AAp]
1AAp)l
.
Jx'+x
OA
(9.122)
The interpretation of the first term is obvious: it describes the interaction o
NED University of Engineering & Technology, Karachi
Project Management
INDUSTRIAL IM 419

Spring 2014
318
12. The Dynamics of HeavyIon Collisions
Fig. 12.2. Schema of how the supercritical resonance
moves through the discretized continuum. Strong
and very much localized couplings occur between
neighbouring continuum states
E
me 2
E,s
t cr
+tcr
me 2
imp