Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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normal to the superconducting phase in a metal. The
identification of the state space where the field
operators have to be realized is thus a physically
nontrivial problem in QFT. In this respect, the QFT
structure is drastically different from the one of
quantum mechanics (QM). The reason is the
following.
The von Neumann theorem (1955) in QM states
that for systems with a finite number of degrees of
freedom all the irreducible representations of the
canonical commutation relations are unitarily
equivalent. Therefore, in QM the physical system
can only live in one single physical phase: unitary
equivalence means indeed physical equivalence and
thus there is no room (no representations) for
physically different phases. Such a situation drasti-
cally changes in QFT where systems with infinitely
many degrees of freedom are treated. In such a case,
the von Neumann theorem does not hold and
infinitely many unitarily inequivalent representa-
tions of the canonical commutation relations do in
fact exist (Umezawa 1993, Umezawa et al. 1982). It
is such richness of QFT that allows the description
of different physical phases.
QFT as a Two-Level Theory
In the perturbative approach, any quantum experi-
ment or observation can be schematized as a
scattering process where one prepares a set of free
(noninteracting) particles (incoming particles or in-
fields) which are then made to collide at some later
time in some region of space (spacetime region of
interaction). The products of the collision are
expected to emerge out of the interaction region as
free particles (outgoing particles or out-fields).
Correspondingly, one has the in-field and the out-
field state space. The interaction region is where the
dynamics operates: given the in-fields and the in-
states, the dynamics determines the out-fields and
the out-states.
The incoming particles and the outgoing ones
(also called quasiparticles in solid state physics) are
well distinguishable and localizable particles only far
away from the interaction region, at a time much
before (t = 1) and much after (t = þ1) the
interaction time: in- and out-fields are thus said to
be asymptotic fields, and for them the interaction
forces are assumed not to operate (switched off).
The only regions accessible to observations are
those far away (in space and in time) from the
interaction region, that is, the asymptotic regions
(the in- and out-regions). It is so since, at the
quantum level, observations performed in the inter-
action region or vacuum fluctuations occurring there
may drastically interfere with the interacting objects,
thus changing their nature. Besides the asymptotic
fields, one then also introduces dynamical or
Heisenberg fields, that is, the fields in terms of
which the dynamics is given. Since the interaction
region is precluded from observation, we do not
observe Heisenberg fields. Observables are thus
solely described in terms of asymptotic fields.
Summing up, QFT is a ‘‘two-level’’ theory: one level
is the interaction level where the dynamics is specified
by assigning the equations for the Heisenberg fields.
The other level is the physical level, the one of the
asymptotic fields and of the physical state space
directly accessible to observations. The equations for
the physical fields are equations for free fields,
describing the observed incoming/outgoing particles.
To be specific, let the Heisenberg operator fields
be generically denoted by
H
(x) and the physical
operator fields by ’
in
(x). For definiteness, we choose
to work with the in-fields, although the set of out-
fields would work equally well. They are both
assumed to satisfy equal-time canonical (anti)-
commutation relations.
For brevity, we omit considerations on the renor-
malization procedure, which are not essential for the
conclusions we will reach. The Heisenberg field
equations and the free-field equations are written as
ð@Þ
H
ðxÞ¼J½
H
ðxÞ½1
ð@Þ’
in
ðxÞ¼0 ½2
where (@) is a differential operator, x (t, x) and
J is some functional of the
H
fields, describing the
interaction.
Equation [1] can be formally recast in the
following integral form (Yang–Feldman equation):
H
ðxÞ¼’
in
ðxÞþ
1
ð@ÞJ½
H
ðxÞ½3
where denotes convolution. The symbol
1
(@)
denotes formally the Green function for ’
in
(x). The
precise form of Green’s function is specifi ed by the
boundary conditions. Equation [3] can be solved by
iteration, thus giving an expression for the Heisen-
berg fields
H
(x) in terms of powers of the ’
in
(x)
fields; this is the Haag expansion in the LSZ
formalism (or ‘‘dynamical map’’ in the language of
Umezawa 1993 and Umezawa et al. 1982), which
might be formally written as
H
ðxÞ¼F½x; ’
in
½4
(A (formal) closed form for the dynamical map is
obtained in the closed time path (CTP) formalism
(Blasone and Jizba 2002). Then the Haag expansion
[4] is directly applicable to both equilibrium and
nonequilibrium situations.)
222 Quantum Fields with Topological Defects
We stress that the equality in the dynamical map
[4] is a ‘‘weak’’ equality, which means that it must
be understood as an equality among matrix elements
computed in the Hilbert space of the physical
particles.
We observe that mathematical consistency in the
above procedure requires that the set of ’
in
fields
must be an irreducible set; however, it may happen
that not all the elements of the set are known from
the beginning. For example, there might be compo-
site (bound states) fields or even elementary quanta
whose existence is ignored in a first recognition.
Then the computation of the matrix elements in
physical states will lead to the detection of unex-
pected poles in the Green’s functions, which signal
the existence of the ignored quanta. One thus
introduces the fields corresponding to these quanta
and repeats the computation. This way of proceed-
ing is called the self- consistent method (Umezawa
1993, Umezawa et al. 1982). Thus it is not necessary
to have a one-to-one correspondence between the
sets {
j
H
} and {’
i
in
}, as it happens whenever the set
{’
i
in
} includes composite particles.
The Dynamical Rearrangement of Symmetry
As already mentione d, in QFT the Fock space for
the physical states is not unique since one may have
several physical phases, for example, for a metal the
normal phase and the superconducting phase, and so
on. Fock spaces describing different phases are
unitarily inequivalent spaces and correspondingly
we have different expectation values for certain
observables and even different irreducible sets of
physical quanta. Thus, finding the dynamical map
involves singling out the Fock space where the
dynamics has to be realized.
Let us now suppose that the Heisenberg field
equations are invariant under some group G of
transformations of
H
:
H
ðxÞ!
0
H
ðxÞ¼g
H
ðxÞ½ ½5
with g 2 G. The symmetry is spontaneously broken
when the vacuum state in the Fock space H is not
invariant under the group G but only under one of
its subgroups (Umezawa 1993, Umezawa et al.
1982).
On the other hand, eqn [4] implies that when
H
is transformed as in [5], then
’
in
ðxÞ!’
0
in
ðxÞ¼g
0
’
in
ðxÞ½ ½6
with g
0
belonging to some group of transformations
G
0
and such that
g
H
ðxÞ½¼Fg
0
’
in
ðxÞ½½ ½7
When symmetry is spontaneously broken it is
G
0
6¼ G, with G
0
the group contraction of G;when
symmetry is not broken then G
0
= G.
Since G is the invariance group of the dynamics,
eqn [4] requires that G
0
is the group under which
free fields equations are invariant, that is, also ’
0
in
is a sol ution of [2].Sinceeqn [4] is a weak equality,
G
0
depends on the choice of the Fock space H
among the physically realizable unitari ly inequiva-
lent state spaces. Thus, we see that the (same)
original invariance of the dynamics may manifest
itself in different symmetry groups for the ’
in
fields
according to different choices of t he physical state
space. Since this process is constrained by the
dynamical equations [1], it is called the dynamical
rearrangement of symmetry (Umezawa 1993,
Umezawa et al. 1982).
In conclusion, different ordering patterns appear
to be different manifestations of the same basic
dynamical invariance. The discovery of the process
of the dynamical rearrangement of symmetry leads
to a unified understanding of the dynamical genera-
tion of many observa ble ordered patterns. This is the
phenomenon of the dynamical generation of order.
The contraction of the symmetry group is the
mathematical structure controlling the dynamical
rearrangement of the symmetry. For a qualitative
presentation see Vitiello (2001).
One can now ask which ones are the carriers of
the ordering information among the system elemen-
tary constituents and how the long-range correla-
tions and the cohere nce observed in ordered patterns
are generated and sustained. The answer is in
the fact that SSB implies the appearanc e of bosons
(Goldstone 1961, Goldstone et al. 1962, Nambu
and Jona-Lasinio 1961), the so-called Nambu–
Goldstone (NG) modes or quanta. They manifest
as long-range correlations and thus they are respon-
sible of the above-mentioned change of scale, from
microscopic to macroscopic. The coherent boson
condensation of NG modes turns out to be the
mechanism by which order is generated, as we will
see in an explicit example in a later section.
The ‘‘Boson Transformation’’ Method
We now discuss the quantum origin of extended
objects (defects) and show how they naturally
emerge as macroscopic objects (inhomogeneous
condensates) from the quantum dynamics. At zero
temperature, the classical soliton solutions are then
recovered in the Born approximation. This approach
is known as the ‘‘boson transformation’’ method
(Umezawa 1993, Umezawa et al. 1982 ).
Quantum Fields with Topological Defects 223
The Boson Transformation Theorem
Let us consider, for simplicity, the case of a
dynamical model involving one scalar field
H
and
one asymptotic field ’
in
satisfying eqns [1] and [2],
respectively.
As already remarked, the dynamical map is valid
only in a weak sense, that is, as a relation among matrix
elements. This implies that eqn [4] is not unique, since
different sets of asymptotic fields and the correspond-
ing Hilbert spaces can be used in its construction. Let us
indeed consider a c–number function f (x), satisfying
the ’
in
equations of motion [2]:
ð@Þf ðxÞ¼0 ½8
The boson transformation theorem (Umezawa 1993,
Umezawa et al. 1982) states that the field
f
H
ðxÞ¼Fx; ’
in
þ f½ ½9
is also a solution of the Heisenberg equation [1].
The corresponding Yang–Feldman equation takes
the form
f
H
ðxÞ¼’
in
ðxÞþf ðxÞþ
1
ð@ÞJ½
f
H
ðxÞ½10
The difference between the two so lutions
H
and
f
H
is only in the boundary conditions. An impor-
tant point is that the expansion in [9] is obtained
from that in [4] by the spacetime-dependent
translation
’
in
ðxÞ!’
in
ðxÞþf ðxÞ½11
The essence of the boson transformation theorem is
that the dynamics embodied in eqn [1] contains an
internal freedom, represented by the possible
choices of the function f (x), satisfying the fr ee-
field equation [8].
We also observe that the transform ation [11] is a
canonical transformation since it leaves invariant the
canonical form of commutation relations.
Let j0i denote the vacuum for the free field ’
in
.
The vacuum expectation value of eqn [10] gives
f
ðxÞh0j
f
H
ðxÞj0i
¼ f ðxÞþ 0
1
ð@ÞJ½
f
H
ðxÞ
hi
0
DE
½12
The c–number field
f
(x) is the order parameter. We
remark that it is fully determined by the quantum
dynamics. In the classical or Born approximation,
which consists in taking h0jJ[
f
H
]j0i= J[
f
], that
is, neglecting all the contractions of the physical
fields, we define
f
cl
(x) lim
h!0
f
(x). In this limit,
we have
ð@Þ
f
cl
ðxÞ¼J½
f
cl
ðxÞ½13
that is,
f
cl
(x) provides the solution of the classical
Euler–Lagrange equation.
Beyond the classical level, in general, the form of
this equation changes. The Yang–Feldman equation
[10] gives not only the equation for the order
parameter, eqn [13], but also, at higher orders in
h, the dynamics of the physical quanta in the
potential generated by the ‘‘macroscopic object’’
f
(x)(Umezawa 1993, Umezawa et al. 1982).
One can show (Umezawa 1993, Umezawa et al.
1982) that the class of solutions of eqn [8] which
lead to topologically nontrivial (i.e., carrying a
nonzero topological charge) solutions of eqn [13],
are those which have some sort of singularity with
respect to Fourier transform. These can be either
divergent singularities or topological singularities.
The first are associated to a divergence of f (x) for
jxj= 1, at least in some direction. Topological
singularities are instead present when f (x) is not
single-valued, that is, it is path dependent. In both
cases, the macroscopic object described by the
order parameter, carries a nonzero topological
charge.
Topological Singularities and Massless Bosons
An important result is that the boson transformation
functions carrying topological singularities are only
allowed for massless bosons (Umezawa 1993,
Umezawa et al. 1982).
Consider a generic boson field
in
satisfying the
equation
ð@
2
þ m
2
Þ
in
ðxÞ¼0 ½14
and suppose that the function f (x) for the boson
transformation
in
(x) !
in
(x) þ f (x) carries a topo-
logical singularity. It is then not single-valued and
thus path dependent:
G
þ
ðxÞ½@
;@
f ðxÞ 6¼ 0; for certain ; ; x ½15
On the other hand, @
f (x), which is related with
observables, is single-valued, that is, [@
, @
]
@
f (x) = 0. Recall that f (x) is solution of the
in
equation:
ð@
2
þ m
2
Þf ðxÞ¼0 ½16
From the definition of G
þ
(x) and the regularity of
@
f (x), it follows, by computing @
G
þ
(x), that
@
f ðxÞ¼
1
@
2
þ m
2
@
G
þ
ðxÞ½17
This equation and the antisymmetric nature of
G
þ
(x) then lead to @
2
f (x) = 0, which in turn implies
m = 0. Thus, we conclude that [15] is only compa-
tible with massless equation for
in
.
224 Quantum Fields with Topological Defects
The topological charge is defined as
N
T
¼
Z
C
dl
@
f ¼
Z
S
dS
@
@
f
¼
1
2
Z
S
dS
G
þ
½18
Here C is a contour enclosing the singularity and S a
surface with C as boundary. N
T
does not depend on
the path C provided this does not cross the
singularity. The dual tensor G
(x)is
G
ðxÞ
1
2
G
þ
ðxÞ½19
and satisfies the continuity equation
@
G
ðxÞ¼0
, @
G
þ
ðxÞþ@
G
þ
ðxÞþ@
G
þ
ðxÞ¼0 ½20
Equation [20] completely characterizes the topolo-
gical singularity (Umezawa 1993, Umezawa et al.
1982).
An Example: The Anderson–Higgs–Kibble
Mechanism and the Vortex Solution
We consider a model of a complex scalar field (x)
interacting with a gauge field A
(x)(Anderson 1958,
Higgs 1960, Kibble 1967). The lagrangian density
L[(x),
(x), A
(x)] is invariant under the global
and the local U(1) gauge transformations (we do not
assume a particular form for the Lagrangian density,
so the following results are quite gen eral):
ðxÞ!e
i
ðxÞ; A
ðxÞ!A
ðxÞ½21
ðxÞ!e
ie
0
ðxÞ
ðxÞ; A
ðxÞ!A
ðxÞþ@
ðxÞ½22
respectively, where (x) !0 for jx
0
j!1 and/or
jxj!1 and e
0
is the coupling constant. We work
in the Lorentz gauge @
A
(x) = 0. The generating
functional, including the gauge constraint, is
(Matsumoto et al. 1975a, b)
Z½J; K¼
1
N
Z
½dA
½d½d
½dB
exp i S½A
; B;
½23
S¼
Z
d
4
x
h
LðxÞþBðxÞ@
A
ðxÞ
þ K
ðxÞðxÞþKðxÞ
ðxÞ
þ J
ðxÞA
ðxÞþijðxÞvj
2
i
N¼
Z
½dA
½d½d
½dB
exp i
Z
d
4
x LðxÞþijðx Þvj
2
B(x) is an auxiliary field which implements the
gauge-fixing condition (Matsumoto et al. 1975a, b).
Notice the -term where v is a complex number ; its
roˆ le is to speci fy the condition of symmetry breaking
under which we want to compute the functional
integral and it may be given the physical meaning of
a small external field triggering the symmetry
breaking (Matsumoto et al. 1975a, b). The limit
!0 must be made at the end of the computations.
We will use the notation
hF½i
;J;K
1
N
Z
½dA
½d½d
½dBF½
exp iS½A
;B;
½24
with hF[]i
hF[]i
,J = K =0
and hF[]ilim
!0
hF[]i
.
The fields , A
,andB appearing in the generating
functional are c-number fields. In the following, the
Heisenberg operator fields corresponding to them
will be denoted by
H
, A
H
,andB
H
,respectively.
Thus, the spontaneous symmetry breaking condition
is expressed by h0j
H
(x)j0i
~
v 6¼ 0, with
~
v constant.
Since in the functional integral formalism the
functional average of a given c-number field gives
the vacuum expectation value of the corresponding
operator field, for example, hF[]ih0jF[
H
]j0i,we
have lim
!0
h(x)i
h0j
H
(x)j0i=
~
v.
Let us introduce the following decompositions:
ðxÞ¼
1
ffiffiffi
2
p
ðxÞþiðxÞ½
KðxÞ¼
1
ffiffiffi
2
p
K
1
ðxÞþiK
2
ðxÞ½
ðxÞ ðxÞh ðxÞi
Note that h( x)i
= 0 because of the invariance
under !.
The Goldstone Theorem
Since the functional integral [23] is invariant under
the global transformation [21], we have that
@Z[ J, K]=@ = 0 and subsequent derivatives with
respect to K
1
and K
2
lead to
h ðxÞi
¼
ffiffiffi
2
p
v
Z
d
4
yhðxÞðyÞi
¼
ffiffiffi
2
p
v
ð; 0Þ½25
In momentum space the propagator for the field
has the general form
ð0; pÞ¼lim
!0
Z
p
2
m
2
þ ia
þ (continuum contributions)
½26
Quantum Fields with Topological Defects 225
Here Z
and a
are renormalization constants. The
integration in eqn [25] picks up the pole contribu-
tion at p
2
= 0, and leads to
~
v ¼
ffiffiffi
2
p
Z
a
v , m
¼ 0;
~
v ¼ 0 , m
6¼ 0 ½27
The Goldstone theorem (Goldstone 1961, Goldstone
et al. 1962) is thus proved: if the symmetry is
spontaneously broken (
~
v 6¼ 0), a massless mode must
exist, whose field is (x), that is, the NG boson
mode. Since it is massle ss, it manifests as a long-
range correlation mode. (Notice that in the present
case of a complex scalar field model , the NG mode
is an elementary field. In other models, it may
appear as a bound state, for exampl e, the magnon in
(anti)ferromagnets.) Note that
@
@v
h ðxÞi
¼
ffiffiffi
2
p
Z
d
4
yhðxÞðyÞi
½28
and because m
6¼ 0, the right-hand side of this
equation vanishes in the limit !0; therefore,
~
v is
independent of jvj, although the phase of jvj
determines the one of
~
v (from eqn [25]): as in
ferromagnets, once an external magnetic field is
switched on, the system is magnetized independently
of the strength of the external field.
The Dynamical Map and the Field Equations
Observing that the change of variables [21] (and/or
[22]) does not affect the generating functional, we may
obtain the Ward–Takahashi identities. Also, using
B(x) ! B(x) þ (x)in[23] gives h@
A
(x)i
, J, K
= 0.
One then finds the following two-point function pole
structures (Matsumoto et al. 1975a, b):
hBðxÞðyÞi ¼ lim
!0
i
ð2Þ
4
Z
d
4
p e
ipðxyÞ
e
0
~
v
p
2
þ ia
()
½29
hBðxÞA
ðyÞi ¼ @
x
i
ð2Þ
4
Z
d
4
p e
ipðxyÞ
1
p
2
½30
hBðxÞBðyÞi ¼lim
!0
i
ð2Þ
4
Z
d
4
p e
ipðxyÞ
ðe
0
~
vÞ
2
Z
(
1
p
2
þ ia
1
p
2
½31
The absence of branch-cut singularities in propaga-
tors [29]–[31] suggests that B(x) obeys a free-field
equation. In addition, eqn [31] indicates that the
model contains a massless negative-norm state
(ghost) besides the NG massless mode . Moreover,
it can be shown (Matsumoto et al. 1975a, b) that a
massive vector field U
in
also exists in the theory.
Note that because of the invariance (, A
, B) !
(, A
, B), all the other two-point functions
must vanish.
The dynamical maps expressing the Heisenberg
operator fields in terms of the asymptotic operator
fields are found to be (Matsumoto et al. 1975a , b)
H
ðxÞ¼:exp i
Z
1=2
~
v
in
ðxÞ
()
~
v þ Z
1=2
in
ðxÞ
þF½
in
; U
in
;@ð
in
b
in
Þ
: ½32
A
H
ðxÞ¼Z
1=2
3
U
in
ðxÞþ
Z
1=2
e
0
~
v
@
b
in
ðxÞ
þ : F
½
in
; U
in
;@ð
in
b
in
Þ: ½33
B
H
ðxÞ¼
e
0
~
v
Z
1=2
½b
in
ðxÞ
in
ðxÞ þ c ½34
where : ...: denotes the normal ordering and the
functionals F and F
are to be determined within a
particular model. In eqns [32]–[34],
in
denotes the
NG mode, b
in
the ghost mode, U
in
the massive
vector field, and
in
the massive matter field. In eqn
[34] c is a c-numb er constant, whose value is
irrelevant sinc e only derivatives of B appear in the
field equations (see below). Z
3
represents the wave
function renormalization for U
in
. The corresponding
field equations are
@
2
in
ðxÞ¼0;@
2
b
in
ðxÞ¼0
ð@
2
þ m
2
Þ
in
ðxÞ¼0
½35
ð@
2
þ m
2
V
ÞU
in
ðxÞ¼0;@
U
in
ðxÞ¼0 ½36
with m
V
2
= (Z
3
=Z
)(e
0
~
v)
2
. The field equations for
B
H
and A
H
read (Matsumoto et al. 1975a, b)
@
2
B
H
ðxÞ¼0; @
2
A
H
ðxÞ¼j
H
ðxÞ@
B
H
ðxÞ½37
with j
H
(x)= L(x)=A
H
(x). One may then require
that the current j
H
is the only source of the gauge
field A
H
in any observable process. This amounts to
impose the condition:
p
hbj@
B
H
(x)jai
p
= 0, that is,
ð@
2
Þ
p
hbjA
0
H
ðxÞjai
p
¼
p
hbjj
H
ðxÞjai
p
½38
where jai
p
and jbi
p
denote two generic physical
states and A
0
H
(x) A
H
(x) e
0
~
v : @
b
in
(x):. Equa-
tions [38] are the classical Maxwell equations. The
condition
p
hbj@
B
H
(x)jai
p
= 0 leads to the Gupta–
Bleuler–like condition
½
ðÞ
in
ðxÞb
ðÞ
in
ðxÞjai
p
¼ 0 ½39
where
()
in
and b
()
in
are the positive-frequency parts
of the corresponding fields. Thus, we see that
in
and
b
in
cannot participate in any observable reaction.
226 Quantum Fields with Topological Defects
This is confirmed by the fact that they are present
in the S-matrix in the combination (
in
b
in
)
(Matsumoto et al. 1975a, b). It is to be remarked,
however, that the NG boson does not disappear from
the theory: we shall see below that there are situations
in which the NG fields do have observable effects.
The Dynamical Rearrangement of Symmetry
and the Classical Fields and Currents
From eqns [32]–[33] we see that the local gauge
transformations of the Heisenberg fields
H
ðxÞ!e
ie
0
ðxÞ
H
ðxÞ
A
H
ðxÞ!A
H
ðxÞþ@
ðxÞ; B
H
ðxÞ!B
H
ðxÞ
½40
with @
2
(x) = 0, are induced by the in-field
transformations
in
ðxÞ!
in
ðxÞþ
e
0
~
v
Z
1=2
ðxÞ
b
in
ðxÞ!b
in
ðxÞþ
e
0
~
v
Z
1=2
ðxÞ
in
ðxÞ!
in
ðxÞ; U
in
ðxÞ!U
in
ðxÞ
½41
On the other hand, the global phase transformation
H
(x) !e
i
H
(x) is induced by
in
ðxÞ!
in
ðxÞþ
~
v
Z
1=2
f ðxÞ; b
in
ðxÞ!b
in
ðxÞ
in
ðxÞ!
in
ðxÞ; U
in
ðxÞ!U
in
ðxÞ½42
with @
2
f (x) = 0andthelimitf (x) !1 to be performed
at the end of computations. Note that under the above
transformations, the in-field equations and the
S-matrix are invariant and that B
H
is changed by an
irrelevant c-number (in the limit f !1).
Consider now the boson transformation
in
(x) !
in
(x) þ (x): in local gauge theories the
boson transformation must be compatible with the
Heisenberg field equations but also with the physical
state condition [39]. Under the boson transforma-
tion with (x) =
~
vZ
1=2
f (x) and @
2
f (x) = 0, B
H
changes as
B
H
ðxÞ!B
H
ðxÞ
e
0
~
v
2
Z
f ðxÞ½43
eqn [38] is thus violated when the Gupta–Bleuler-
like condition is imposed. In order to restore it, the
shift in B
H
must be compensate d by means of the
following transformation on U
in
:
U
in
ðxÞ!U
in
ðxÞþZ
1=2
3
a
ðxÞ;@
a
ðxÞ¼0 ½44
with a convenient c-number funct ion a
(x). The
dynamical maps of the various Heisenberg operators
are not affected by [44] since they contain U
in
and
B
H
in a combination such that the changes of B
H
and of U
in
compensate each other provided
ð@
2
þ m
2
V
Þa
ðxÞ¼
m
2
V
e
0
@
f ðxÞ½45
Equation [45] thus obtained is the Maxwell equa-
tion for the massive potential vector a
(Matsumoto
et al. 1975a, b). The classical ground state current j
turns out to be
j
ðxÞh0jj
H
ðxÞj0i¼m
2
V
a
ðxÞ
1
e
0
@
f ðxÞ
½46
The term m
2
V
a
(x) is the Meissner current, while
(m
2
V
=e
0
)@
f (x) is the boson current. The key point
here is that both the macroscopic field and current
are given in terms of the boson condensation
function f (x).
Two remarks are in order: first, note that the
terms proportional to @
f (x) are related to obser-
vable effects, for example, the boson current which
acts as the source of the classical field. Second, note
that the macroscopic ground state effects do not
occur for regular f (x)(G
þ
(x) = 0). In fact, from [45]
we obtain a
(x) = (1=e
0
)@
f (x) for regular f (x)
which implies zero classical current (j
= 0) and
zero classical field (F
= @
a
@
a
), since the
Meissner and the boson current cancel each other.
In con clusion, the vacuum current appears only
when f (x) has topological singularities and these can
be created only by condensation of massless bosons,
that is, when SSB occurs. This explains why
topological defects appear in the process of phase
transitions, where NG modes are present and
gradients in their condensate densities are nonzero
(Kibble 1976, Zurek 1997).
On the other hand, the appearance of spacetime
order parameter is no guarantee that persistent
ground state currents (and fields) will exist: if f (x)
is a regular function, the spacetime dependence of
~
v
can be gauged away by an appropriate gauge
transformation.
Since, as already mentioned, the boson transfor-
mation with regular f (x) does not affe ct obs ervable
quantities, the S-matrix is actual ly given by
S ¼: S
in
; U
in
1
m
V
@ð
in
b
in
Þ
: ½47
This is indeed independent of the boson transforma-
tion with regular f (x):
S ! S
0
¼:S
in
; U
in
1
m
V
@ð
in
b
in
Þ
þZ
1=2
3
ða
1
e
0
@
f Þ
: ½48
Quantum Fields with Topological Defects 227
since a
(x) = (1=e
0
)@
f (x) for regular f (x). However,
S
0
6¼ S for singular f (x): S
0
includes the interaction of
the quanta U
in
and
in
with the classically behaving
macroscopic defects (Umezawa 1993, Umezawa
et al. 1982).
The Vortex Solution
Below we consider the example of the Nielsen–
Olesen vortex string solution. We show which one is
the boson function f (x) controlling the nonhomoge-
neous NG boson condensation in terms of which the
string solution is described. For brevity, we only
report the results of the computations. The detailed
derivation as well as the discussion of further
examples can be found in (Umezawa 1993,
Umezawa et al. 1982).
In the present U(1) problem, the electromagnetic
tensor and the vacuum current are (Umezawa 1993,
Umezawa et al. 1982, Matsumoto et al. 1975a , b)
F
ðxÞ¼@
a
ðxÞ@
a
ðxÞ
¼ 2
m
2
V
e
0
Z
d
4
x
0
c
ðx x
0
ÞG
þ
ðx
0
Þ½49
j
ðxÞ¼2
m
2
V
e
0
Z
d
4
x
0
c
ðx x
0
Þ@
x
0
G
þ
ðx
0
Þ½50
respectively, and satisfy @
F
(x) = j
(x). In these
equations,
c
ðx x
0
Þ¼
1
ð2Þ
4
Z
d
4
pe
ipðxx
0
Þ
1
p
2
m
2
V
þi
½51
The line singularity for the vortex (or string)
solution can be param etrized by a single line
parameter and by the time parameter . A static
vortex solution is obtain ed by setting y
0
(,)= and
y(,)= y(), with y denoting the line coordinate.
G
þ
(x) is nonzero only on the line at y (we can
consider more lines but let us limit to only one line,
for simplicity). Thus, we have
G
0i
ðxÞ¼
Z
d
dy
i
ðÞ
d
3
½x yðÞ G
ij
ðxÞ¼0
G
þ
ij
ðxÞ¼
ijk
G
0k
ðxÞ; G
þ
0i
ðxÞ¼0
½52
Equation [49] shows that these vortices are purely
magnetic. We obtain
@
0
f ðxÞ¼0
@
i
f ðxÞ¼
1
ð2Þ
2
Z
d
ijk
dy
k
ðÞ
d
@
x
j
Z
d
3
p
e
ipðxyðÞÞ
p
2
½53
that is, by using the identity (2)
2
R
d
3
p(e
i px
=p
2
) =
1=2jxj,
rf ðx Þ¼
1
2
Z
d
dy
k
ðÞ
d
^r
x
1
jx yðÞj
½54
Note that r
2
f (x) = 0 is satisfied.
A straight infinitely long vortex is specified by
y
i
() =
i3
with 1 <<1. The only nonvanish-
ing component of G
(x)areG
03
(x) = G
þ
12
(x) =
(x
1
)(x
2
). Equation [54] gives (Umezawa 1993,
Umezawa et al. 1982, Matsumoto 1975a, b)
@
@x
1
f ðxÞ¼
1
2
Z
d
@
@x
2
½x
2
1
þx
2
2
þðx
3
Þ
2
1=2
¼
x
2
x
2
1
þx
2
2
@
@x
2
f ðxÞ¼
x
1
x
2
1
þx
2
2
;
@
@x
3
f ðxÞ¼0
½55
and then
f ðxÞ¼tan
1
x
2
x
1
¼ ðxÞ½56
We have thus determined the boson transformation
function corresponding to a particular vortex solu-
tion. The vector potential is
a
1
ðxÞ¼
m
2
V
2e
0
Z
d
4
x
0
c
ðx x
0
Þ
x
0
2
x
02
1
þ x
02
2
a
2
ðxÞ¼
m
2
V
2e
0
Z
d
4
x
0
c
ðx x
0
Þ
x
0
1
x
02
1
þ x
02
2
a
3
ðxÞ¼a
0
ðxÞ¼0
½57
and the only nonvanishing component of F
:
F
12
ðxÞ¼2
m
2
V
e
0
Z
d
4
x
0
c
ðx x
0
Þðx
0
1
Þðx
0
2
Þ
¼
m
2
V
e
0
K
0
m
V
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
1
þ x
2
2
q
½58
Finally, the vacuum current eqn [50] is given by
j
1
ðxÞ¼
m
3
V
e
0
x
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
1
þ x
2
2
q
K
1
m
V
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
1
þ x
2
2
q
j
2
ðxÞ¼
m
3
V
e
0
x
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
1
þ x
2
2
q
K
1
m
V
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
1
þ x
2
2
q
j
3
ðxÞ¼j
0
ðxÞ¼0
½59
We observe that these results are the same of the
Nielsen–Olesen vortex solution. Notice that we did
not specify the potential in our model but only the
invariance properties. Thus, the invariance proper-
ties of the dynamics determine the characteristics of
the topological solut ions. The vortex solution
228 Quantum Fields with Topological Defects
manifests the original U(1) symmetry through the
cylindrical angle which is the parameter of the
U(1) representation in the coordinate space.
Conclusions
We have discussed how topological defects arise as
inhomogeneous condensates in QFT. Topological
defects are shown to have a genuine quantum
nature. The approach reviewed here goes under the
name of ‘‘boson transformation method’’ and relies
on the existence of unitarily inequivalent representa-
tions of the field algebra in QFT.
Describing quantum fields with topological
defects amounts then to properly choose the physical
Fock space for representing the Heisenberg field
operators. Once the boundary conditions corre-
sponding to a particular soliton sector are found,
the Heisenberg field operators embodied with such
conditions contain the full information about the
defects, the quanta and their mutual interaction.
One can thu s calculate Green’s functions for
particles in the presence of defects. The extension
to finite temperature is discussed in Blasone and
Jizba (2002) and Manka and Vitiello (1990).
As an example we have discussed a model with
U(1) gauge invariance and SSB and we have obtained
the Nielsen–Olesen vortex solution in terms of
localized condensation of Goldstone bosons. These
thus appear to play a physical role, although, in the
presence of gauge fields, they do not show up in the
physical spectrum as excitation quanta. The function
f (x) controlling the condensation of the NG bosons
must be singular in order to produce observable
effects. Boson transformati ons with regular f (x) only
amount to gauge transformations. For the treatment
of topological defects in nonabelian gauge theories,
see Manka and Vitiel lo (1990) .
Finally, when there are no NG modes, as in the
case of the kink solution or the sine-Gordon
solution, the boson transformation function has to
carry divergence singularity at spatial infinity
(Umezawa 1993, Umezawa et al. 1982, Blasone
and Jizba 2002). The boson transformation has also
been discussed in connection with the Ba¨ klund
transformation at a classical level and the confine-
ment of the constituent quanta in the coherent
condensation domain.
For further reading on quantum fields with
topological defects, see Blasone et al. (2006).
Acknowledgments
The authors thank MIUR, INFN, INFM, and the
ESF network COSLAB for partial financial support.
See also: Abelian Higgs Vortices; Algebraic Approach to
Quantum Field Theory; Quantum Field Theory: A Brief
Introduction; Quantum Field Theory in Curved
Spacetime; Symmetries in Quantum Field Theory:
Algebraic Aspects; Symmetries in Quantum Field Theory
of Lower Spacetime Dimensions; Topological Defects
and their Homotopy Classification.
Further Reading
Anderson PW (1958) Coherent excited states in the theory of
superconductivity: gauge invariance and the Meissner effect.
Physical Review 110: 827–835.
Blasone M and Jizba P (2002) Topological defects as
inhomogeneous c ondensates in quantum field theory: kinks
in (1 þ 1) dime nsional
4
theory. Annals of Physic s 295:
230–260.
Blasone M, Jizba P, and Vitiello G (2006) Spontaneous Break-
down of Symmetry and Topological Defects, London: Imper-
ial College Press. (in preparation).
Goldstone J (1961) Field theories with ‘‘superconductor’’ solu-
tions. Nuovo Cimento 19: 154–164.
Goldstone J, Salam A, and Weinberg S (1962) Broken symmetries.
Physical Review 127: 965–970.
Higgs P (1960) Spontaneous symmetry breakdown without
massless bosons. Physical Review 145: 1156–1163.
Kibble TWB (1967) Symmetry breaking in non-abelian gauge
theories. Physical Review 155: 1554–1561.
Kibble TWB (1976) Topology of cosmic domains and strings.
Journal of Physics A 9: 1387–1398.
Kibble TWD (1980) Some implications of a cosmological phase
transition. Physics Reports 67: 183–199.
Kleinert H (1989) Gauge Fields in Condensed Matter, vols. I & II
Singapore: World Scientific.
Manka R and Vitiello G (1990) Topological solitons and tempera-
ture effects in gauge field theory. Annals of Physics 199: 61–83.
Matsumoto H, Papastamatiou NJ, Umezawa H, and Vitiello G
(1975a) Dynamical rearrangement in Anderson–Higgs–Kibble
mechanism. Nuclear Physics B 97: 61–89.
Matsumoto H, Papastamatiou NJ, and Umezawa H (1975b) The
boson transformation and the vortex solutions. Nuclear
Physics B 97: 90–124.
Nambu Y and Jona-Lasinio G (1961) Dynamical model of
elementary particles based on an analogy with superconduc-
tivity. I. Physical Review 122: 345–358.
Nambu Y and Jona-Lasinio G (1961) Dynamical model of
elementary particles based on an analogy with superconduc-
tivity. II. Physical Review 124: 246–254.
Rajaraman R (1982) Solitons and Instantons: An Introduction to
Solitons and Instantons in Quantum Field Theory. Amsterdam:
North-Holland.
Umezawa H (1993) Advanced Field Theory: Micro, Macro and
Thermal Physics. New York: American Institute of Physics.
Umezawa H, Matsumoto H, and Tachiki M (1982) Thermo
Field Dynamics and Condensed States. Amsterdam: North-
Holland.
Vitiello G (2001) My Double Unveiled. Amsterdam: John
Benjamins.
Volovik GE (2003) The Universe in a Helium Droplet. Oxford:
Clarendon.
von Neumann J (1955) Mathematical Foundation of Quantum
Mechanics. Princeton: Princeton University Press.
Zurek WH (1997) Cosmological experiments in condensed matter
systems. Physics Reports 276: 177–221.
Quantum Fields with Topological Defects 229
Quantum Geometry and Its Applications
A Ashtekar, Pennsylvania State University, University
Park, PA, USA
J Lewandowski, Uniwersyte Warszawski, Warsaw,
Poland
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
In general relativity, the gravitational field is
encoded in the Riemannian geometry of spacetime.
Much of the conceptual compactness and mathema-
tical elegance of the theory can be traced back to
this central idea. The encoding is also directly
responsible for the most dramatic ramifications of
the theory: the big bang, black holes, and gravita-
tional waves. However, it also leads one to the
conclusion that spacetime itself must end and
physics must come to a halt at the big bang and
inside black holes, where the gravitational field
becomes singular. But this reasoning ignores quan-
tum physics entirely. When the curvature becomes
large, of the order of 1=‘
2
Pl
= c
3
=Gh, quantum effects
dominate and predictions of general relativity can
no longer be trusted. In this ‘‘Planck regime,’’ one
must use an appropriate synthesis of general
relativity and quantum physics, that is, a quantum
gravity theory. The predictions of this theory are
likely to be quite different from those of general
relativity. In the real, quantum world, evolution may
be completely nonsingular. Physics may not come to
a halt and quantum theory could extend classical
spacetime.
There are a number of different approaches to
quantum gravity. One natural avenue is to retain the
interplay between gravity and geometry but now use
‘‘quantum’’ Riemannian geometry in place of the
standard, classical one. This is the key idea under-
lying loop quantum gravity. There are several
calculations which indicate that the well-known
failure of the standard perturbative approach to
quantum gravity may be primarily due to its basic
assumption that spacetime can be modeled as a
smooth continuum at all scales. In loop quantum
gravity, one adopts a nonperturbative approach.
There is no smooth metric in the background.
Geometry is not only dynamical but quantum
mechanical from ‘‘birth.’’ Its fundamental excita-
tions turn out to be one dimensional and polymer-
like. The smooth continuum is only a coarse-grained
approximation. While a fully satisfactory quantum
gravity theory still awaits us (in any approach),
detailed investigations have been carried out to
completion in simplified models – called mini- and
midi-superspaces. They show that quantum space-
time does not end at singularities. Rather, quantum
geometry serves as a ‘‘bridge’’ to another large
classical spacetime.
This article will focus on structural issues from
the perspective of mathematical physics. For com-
plementary perspectives and further details, see
Loop Quantum Gravity, Canonical General Relativity,
Quantum Cosmology, Black Hole Mechanics, and
Spin Foams in this Encyclopedia.
Basic Framework
The starting point is a Hamiltonian formulation of
general relativity based on spin connections
(Ashtekar 1987). Here, the phase space G consists
of canonically conjugate pairs (A, P), where A is a
connection on a 3-manifold M and P a 2-form, both
of which take values in the Lie algebra su(2). Since G
can also be thought of as the phase space of the
SU(2) Yang–Mills theory, in this approach there is a
unified kinematic framework for general relativity
that describes gravity and the gauge theories which
describe the other three basic forces of nature. The
connection A enables one to parallel transport chiral
spinors (such as the left-handed fermions of the
standard electroweak model) along curves in M. Its
curvature is directly related to the electric and
magnetic parts of the spacetime ‘‘Riemann tensor.’’
The dual P of P plays a double role (the dual is
defined via
R
M
P ^ ! =
R
M
P
y
! for any 1-form !
on M). Being the momentum canonically conjugate
to A, it is analogous to the Yang–Mills electric field.
But (apart from a constant), it is also an orthonor-
mal triad (with density weight 1) on M and
therefore determines the positive-definite (‘‘spatial’’)
3-metric, and hence the Riemannian geometry of M.
This dual role of P is a reflection of the fact that
now SU(2) is the (double cover of the) group of
rotations of the orthonormal spatial triads on M
itself rather than of rotations in an ‘‘internal’’ space
associated with M.
To pass to quantum theory, one first constructs an
algebra of ‘‘elementary’’ functions on G (analogous
to the phase-space functions x and p in the case of a
particle) which are to have unambiguous operator
analogs. The holonomies
h
e
ðAÞ :¼Pexp
Z
e
A ½1
associated with a curve/edge e on M are (SU(2)-
valued) configuration functions on G. Similarly,
230 Quantum Geometry and Its Applications
given a 2-surface S on M, and an su(2)-valued (test)
function f on M,
P
S; f
:¼
Z
S
trðf PÞ½2
is a momentum function on G, where tr is over the
su(2) indices. (For simplicity of presentation, all
fields are assumed to be smooth and curves/edges e
and surfaces S, finite and piecewise analytic in a
specific sense. The extension to smooth curves and
surfaces was carried out by Bacz and Sawin,
Lewandowski and Thiemann, and Fleischhack. It is
technically more involved but the final results are
qualitatively the same.) The symplectic structure on
G enables one to calculate the Poisson brackets
{h
e
, P
S, f
}. The result is a linear combination of
holonomies and can be written as a Lie derivative,
fh
e
; P
S; f
g¼L
X
S; f
h
e
½3
where X
S, f
is a derivation on the ring generated by
holonomy functions, and can therefore be regarded
as a vector field on the configuration space A of
connections. This is a familiar situation in classical
mechanics of systems whose configuration space is a
finite-dimensional manifold. Functions h
e
and vector
fields X
S, f
generate a Lie algebra. As in quantum
mechanics on manifolds, the first step is to promote
this algebra to a quantum algebra by demanding
that the commutator be given by ih times the Lie
bracket. The result is a ?-algebra a , analogous to the
algebra generated by operators exp i
^
x and
^
p in
quantum mechanics. By exponentiating the momen-
tum operators
^
P
S, f
one obtains W , the analog of the
quantum-mechanical Weyl algebra generated by
exp i
^
x and exp i
^
p.
The main task is to obtain the appropriate
representation of these algebras. In that representa-
tion, quantum Riemannian geometry can be probed
through the momentum operators
^
P
S, f
, which
stem from classical orthonormal triads. As in
quantum mechanics on manifolds or simple field
theories in flat space, it is convenient to divide the
task into two parts. In the first, one focuses on the
algebra C generated by the configuration operators
^
h
c
and finds all its representations, and in the second
one considers the momentum operators
^
P
S, f
to
restrict the freedom.
C is called the holonomy algebra. It is naturally
endowed with the structure of an abelian C
?
algebra
(with identity), whence one can apply the powerful
machinery made available by the Gel’fand theory.
This theory tells us that C determines a unique
compact, Hausdorff space
A such that the C
?
algebra
of all continuous functions on A is naturally
isomorphic to C .
A is called the Gel’fand spectrum
of C . It has been shown to consist of ‘‘generalized
connections’’
A defined as follows:
A assigns to any
oriented edge e in M an element
A(e)ofSU(2)
(a ‘‘holonomy’’) such that
A(e
1
) = [
A(e)]
1
;and,if
the endpoint of e
1
is the starting point of e
2
,then
A(e
1
e
2
) =
A(e
1
)
A(e
2
). Clearly, every smooth con-
nection A is a generalized connection. In fact, the
space A of smooth connections has been shown to be
dense in
A (with respect to the natural Gel’fand
topology thereon). But
A has many more ‘‘distribu-
tional elements.’’ The Gel’fand theory guarantees that
every representation of the C
?
algebra C is a direct
sum of representations of the following type: the
underlying Hilbert space is H= L
2
(
A,d) for some
measure on
A and (regarded as functions on
A)
elements of C act by multiplication. Since there are
many inequivalent measures on
A, there is a multi-
tude of representations of C . A key question is how
many of them can be extended to representations of
the full algebra a (or W) without having to introduce
any ‘‘background fields’’ which would compromise
diffeomorphism covariance. Quite surprisingly, the
requirement that the representation be cyclic with
respect to a state which is invariant under the action
of the (appropriately defined) group Diff M of
piecewise-analytic diffeomorphisms on M singles out
a unique irreducible representation. This result was
established for a by Lewandowski, Około´w, Sahl-
mann and Thiemann, and for W by Fleischhack. It is
the quantum geometry analog to the seminal results
by Segal and others that characterized the Fock
vacuum in Minkowskian field theories. However,
while that result assumes not only Poincare´invar-
iance but also specific (namely free) dynamics, it is
striking that the present uniqueness theorems make
no such restriction on dynamics. The requirement of
diffeomorphism invariance is surprisingly strong and
makes the ‘‘background-independent’’ quantum geo-
metry framework surprisingly tight.
This representation had been constructed by
Ashtekar, Baez, and Lewandowski some ten years
before its uniqueness was established. The under-
lying Hilbert space is given by H= L
2
(
A,d
o
) where
o
is a diffeomorphism-invariant, faithful, regular
Borel measure on
A, constructed from the normal-
ized Haar measure on SU(2). Typical quantum states
can be visualized as follows. Fix: (1) a graph on M
(by a graph on M we mean a set of a finite number
of embedded, oriented intervals called edges; if two
edges intersect, they do so only at one or both ends,
called vertices), and (2) a smooth function on
[SU(2)]
n
. Then, the function
ð
AÞ :¼ ð
Aðe
1
Þ; ...;
Aðe
n
ÞÞ ½4
Quantum Geometry and Its Applications 231