Quantum Fields with Topological Defects

Quantum Fields with Topological Defects
Grothaus M and Streit L (1999) Construction of relativistic
quantum fields in the framework of white noise analysis.
Journal of Mathematical Physics 40(11): 5387.
Morchio G and Strocchi F (1980) Infrared singularities, vacuum
structure and pure phases in local quantum field theory. Ann.
Inst. H. Poincare´ 33: 251.
221
Steinmann O (2000) Perturbative Quantum Electrodynamics and
Axiomatic Field Theory. Berlin: Springer.
Strocchi F (1993) Selected Topics on the General Properties of
Quantum Field Theory. Lecture Notes in Physics, vol. 51.
Singapore: World Scientific.
Quantum Fields with Topological Defects
M Blasone and G Vitiello, Universita` degli Studi di
Salerno, Baronissi (SA), Italy
P Jizba, Czech Technical University, Prague,
Czech Republic
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The ordered patterns we observe in condensed
matter and in high-energy physics are created by
the quantum dynamics. Macroscopic systems exhibiting some kind of ordering, such as superconductors, ferromagnets, and crystals, are described by the
underlying quantum dynamics. Even the large-scale
structures in the universe, as well as the ordering in
the biological systems appear to be the manifestation of the microscopic dynamics governing the
elementary components of these systems. Thus, we
talk of macroscopic quantum systems: these are
quantum systems in the sense that, although they
behave classically, some of their macroscopic features nevertheless cannot be understood without
recourse to quantum theory.
The question then arises how the quantum
dynamics generates the observed macroscopic properties. In other words, how it happens that the
macroscopic scale characterizing those systems is
dynamically generated out of the microscopic scale
of the quantum elementary components (Umezawa
1993, Umezawa et al. 1982).
Moreover, we also observe a variety of phenomena where quantum particles coexist and interact
with extended macroscopic objects which show a
classical behavior, for example, vortices in superconductors and superfluids, magnetic domains in
ferromagnets, dislocations and other topological
defects (grain boundaries, point defects, etc.) in
crystals, and so on.
We are thus also faced with the question of the
quantum origin of topological defects and their
interaction with quanta (Umezawa 1993, Umezawa
et al. 1982): this is a crucial issue for the understanding of symmetry-breaking phase transitions
and structure formation in a wide range of systems
from condensed matter to cosmology (Kibble 1976,
Zurek 1997, Volovik 2003).
Here, we will review how the generation of
ordered structures and extended objects is explained
in quantum field theory (QFT). We follow Umezawa
(1993) and Umezawa et al. (1982) in our presentation. We will consider systems in which spontaneous
symmetry breaking (SSB) occurs and show that
topological defects originate by inhomogeneous
(localized) condensation of quanta. The approach
followed here is alternative to the usual one
(Rajaraman 1982), in which one starts from the
classical soliton solutions and then ‘‘quantizes’’
them, as well as to the QFT method based on dual
(disorder) fields (Kleinert 1989).
In the next section we introduce some general
features of QFT useful for our discussion and treat
some aspects of SSB and the rearrangement of
symmetry. Next we discuss the boson transformation theorem and the topological singularities of the
boson condensate. We then present, as an example,
a model with U(1) gauge invariance in which SSB,
rearrangement of symmetry, and topological defects
are present (Matsumoto et al. 1975a, b). There we
show how macroscopic fields and currents are
obtained from the microscopic quantum dynamics.
The Nielsen–Olesen vortex solution is explicitly
obtained as an example. The final section is devoted
to conclusions.
Symmetry and Order in QFT:
A Dynamical Problem
QFT deals with systems with infinitely many degrees
of freedom. The fields used for their description are
operator fields whose mathematical significance is
fully specified only when the state space where they
operate is also assigned. This is the space of the
states, or physical phase, of the system under given
boundary conditions. A change in the boundary
conditions may result in the transition of the system
from one phase to another. For example, a change
of temperature from above to below the critical
temperature may induce the transition from the
222 Quantum Fields with Topological Defects
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 drastically 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 representations 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 experiment or observation can be schematized as a
scattering process where one prepares a set of free
(noninteracting) particles (incoming particles or infields) 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 outfield state space. The interaction region is where the
dynamics operates: given the in-fields and the instates, 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 interaction 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 outfields would work equally well. They are both
assumed to satisfy equal-time canonical (anti)commutation relations.
For brevity, we omit considerations on the renormalization 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 ðxÞ
¼ J½
H ðxÞ
½1
ð@Þ’in ðxÞ ¼ 0
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 ðxÞ
¼ ’in ðxÞ þ 1 ð@Þ J ½
H ðxÞ
½3
where denotes convolution. The symbol 1 (@)
denotes formally the Green function for ’in (x). The
precise form of Green’s function is specified by the
boundary conditions. Equation [3] can be solved by
iteration, thus giving an expression for the Heisenberg 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 ðxÞ
¼ F½x; ’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.)
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 composite (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 unexpected 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 proceeding 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 { Hj } and {’iin }, as it happens whenever the set
{’iin } includes composite particles.
The Dynamical Rearrangement of Symmetry
As already mentioned, 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 ðxÞ
¼ g½
H ðxÞ
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 ! ’0in ðxÞ ¼ g0 ½’in ðxÞ
½6
with g0 belonging to some group of transformations
G0 and such that
g½
H ðxÞ
¼ F½g0 ½’in ðxÞ
½7
223
When symmetry is spontaneously broken it is
G0 6¼ G, with G0 the group contraction of G; when
symmetry is not broken then G0 = G.
Since G is the invariance group of the dynamics,
eqn [4] requires that G0 is the group under which
free fields equations are invariant, that is, also ’0in
is a solution of [2]. Since eqn [4] is a weak equality,
G0 depends on the choice of the Fock space H
among the physically realizable unitarily inequivalent 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 the 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 generation of many observable 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 elementary constituents and how the long-range correlations and the coherence observed in ordered patterns
are generated and sustained. The answer is in
the fact that SSB implies the appearance 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 responsible 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).
224 Quantum Fields with Topological Defects
f
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 corresponding 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 ðxÞ ¼ 0
½8
The boson transformation theorem (Umezawa 1993,
Umezawa et al. 1982) states that the field
f
H ðxÞ
¼ F½x; ’in þ f ½9
is also a solution of the Heisenberg equation [1].
The corresponding Yang–Feldman equation takes
the form
f
H ðxÞ
¼ ’in ðxÞ þ f ðxÞ þ 1 ð@Þ J ½
f
H ðxÞ
½10
The difference between the two solutions H and
f
H is only in the boundary conditions. An important point is that the expansion in [9] is obtained
from that in [4] by the spacetime-dependent
translation
’in ðxÞ ! ’in ðxÞ þ f ðxÞ
½11
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 freefield equation [8].
We also observe that the transformation [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 ðxÞ h0j
f
H ðxÞj0i
D h
¼ f ðxÞ þ 0 1 ð@Þ J ½
f
H ðxÞ
i E
0
½12
f
The c–number field (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 [ fH ]j0i = J [ f ], that
is, neglecting all the contractions of the physical
f
fields, we define cl (x) limh!0 f (x). In this limit,
we have
ð@Þclf ðxÞ ¼ J ½clf ðxÞ
that is, 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.
½13
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 m2 Þin ðxÞ ¼ 0
½14
and suppose that the function f (x) for the boson
transformation in (x) ! in (x) þ f (x) carries a topological singularity. It is then not single-valued and
thus path dependent:
Gþ
ðxÞ ½@ ; @ 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 m2 Þf ðxÞ ¼ 0
½16
Gþ
(x)
From the definition of
and the regularity of
@ f (x), it follows, by computing @ Gþ
(x), that
@ f ðxÞ ¼
1
@ Gþ
ðxÞ
@ þ m2
2
½17
This equation and the antisymmetric nature of
2
Gþ
(x) then lead to @ f (x) = 0, which in turn implies
m = 0. Thus, we conclude that [15] is only compatible with massless equation for in .
Quantum Fields with Topological Defects
The topological charge is defined as
Z
Z
NT ¼
dl @ f ¼
dS @ @ f
C
S
Z
1
þ
dS G
¼
2 S
½18
Here C is a contour enclosing the singularity and S a
surface with C as boundary. NT does not depend on
the path C provided this does not cross the
singularity. The dual tensor G (x) is
G ðxÞ 12 Gþ
ðxÞ
½19
and satisfies the continuity equation
@ G ðxÞ ¼ 0
þ
þ
, @ Gþ
ðxÞ þ @ G ðxÞ þ @ G ðxÞ ¼ 0
½20
Equation [20] completely characterizes the topological 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 general):
ðxÞ ! ei
ðxÞ;
A ðxÞ ! A ðxÞ
½21
ðxÞ ! eie0 ðxÞ ðxÞ; A ðxÞ ! A ðxÞ þ @ ðxÞ ½22
respectively, where (x) ! 0 for jx0 j ! 1 and/or
jxj ! 1 and e0 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)
Z
1
½dA ½d½d ½dB
Z½ J; K ¼
N
exp i S½A ; B; ½23
S¼
Z
h
d4 x LðxÞ þ BðxÞ@ A ðxÞ
þ K ðxÞðxÞ þ KðxÞ ðxÞ
2
i
þ J ðxÞA ðxÞ þ ijðxÞ vj
Z
N ¼ ½dA ½d½d ½dB
Z
4
2
exp i d x LðxÞ þ ijðxÞ vj
225
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
roˆle is to specify 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
Z
1
½dA ½d½d ½dBF½
hF½i;J;K N
½24
exp i S½A ; B;
with hF[]i hF[]i, J =K =0 and hF[]i lim!0
hF[]i .
The fields , A , and B appearing in the generating
functional are c-number fields. In the following, the
Heisenberg operator fields corresponding to them
will be denoted by H , AH , and BH , respectively.
Thus, the spontaneous symmetry breaking condition
is expressed by h0jH (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[]i h0jF[H ]j0i, we
have lim ! 0 h(x)i h0jH (x)j0i = ~v.
Let us introduce the following decompositions:
1
ðxÞ ¼ pﬃﬃﬃ ½ ðxÞ þ iðxÞ
2
1
KðxÞ ¼ pﬃﬃﬃ ½K1 ðxÞ þ iK2 ðxÞ
2
ðxÞ ðxÞ h 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 K1 and K2 lead to
pﬃﬃﬃ Z
h ðxÞi ¼ 2v d4 yhðxÞðyÞi
pﬃﬃﬃ
¼ 2v ð; 0Þ
½25
In momentum space the propagator for the field has the general form
Z
ð0; pÞ ¼ lim 2
!0 p m2
þ ia
þ (continuum contributions) ½26
226 Quantum Fields with Topological Defects
Here Z and a are renormalization constants. The
integration in eqn [25] picks up the pole contribution at p2 = 0, and leads to
~
v¼
pﬃﬃﬃ Z
v , m ¼ 0;
2
a
~
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 massless, it manifests as a longrange 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 example, the magnon in
(anti)ferromagnets.) Note that
pﬃﬃﬃ Z 4
@
h ðxÞi ¼ 2
d yhðxÞðyÞi
½28
@v
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
ð2Þ
!0
hBðxÞA ðyÞi ¼ @x
(
hBðxÞBðyÞi ¼ lim
!0
Z
i
d4 p eipðxyÞ
4
i
ð2Þ4
i
ð2Þ4
Z
Z
e0 ~
v
p2 þ ia
d4 p eipðxyÞ
d4 p eipðxyÞ
1
1
p2 þ ia p2
)
1
p2
½29
½30
ðe0 ~
vÞ 2
Z
(, 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)
(
)
Z1=2
H ðxÞ ¼ :exp i
in ðxÞ ~v þ Z1=2
in ðxÞ
~v
þF ½in ; Uin
; @ðin bin Þ :
½32
Z1=2
@ bin ðxÞ
e0 ~v
þ : F ½in ; Uin
; @ðin bin Þ:
1=2
ðxÞ þ
AH ðxÞ ¼Z3 Uin
BH ðxÞ ¼
e0 ~v
1=2
Z
½bin ðxÞ in ðxÞ þ c
The absence of branch-cut singularities in propagators [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 Uin
also exists in the theory.
Note that because of the invariance (, A , B) !
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, bin the ghost mode, Uin
the massive
vector field, and in the massive matter field. In eqn
[34] c is a c-number constant, whose value is
irrelevant since only derivatives of B appear in the
field equations (see below). Z3 represents the wave
function renormalization for Uin
. The corresponding
field equations are
@ 2 in ðxÞ ¼ 0;
@ 2 bin ðxÞ ¼ 0
ð@ 2 þ m2 Þin ðxÞ ¼ 0
ðxÞ ¼ 0;
ð@ 2 þ m2V ÞUin
@ Uin
ðxÞ ¼ 0
½35
½36
with mV 2 = (Z3 =Z )(e0 ~v)2 . The field equations for
BH and AH read (Matsumoto et al. 1975a, b)
@ 2 BH ðxÞ ¼ 0; @ 2 AH ðxÞ ¼ jH ðxÞ @ BH ðxÞ ½37
with jH (x) = L(x)=AH (x). One may then require
that the current jH is the only source of the gauge
field AH in any observable process. This amounts to
impose the condition: p hbj@ BH (x)jaip = 0, that is,
ð@ 2 Þp hbjA0H ðxÞjaip ¼ phbj jH ðxÞjaip
½31
½33
½38
where jaip and jbip denote two generic physical
states and A0
v : @ bin (x):. EquaH (x) AH (x) e0 ~
tions [38] are the classical Maxwell equations. The
condition p hbj@ BH (x)jaip = 0 leads to the Gupta–
Bleuler–like condition
ðÞ
ðÞ
½in ðxÞ bin ðxÞjaip ¼ 0
½39
()
where ()
in and bin are the positive-frequency parts
of the corresponding fields. Thus, we see that in and
bin cannot participate in any observable reaction.
Quantum Fields with Topological Defects
This is confirmed by the fact that they are present
in the S-matrix in the combination (in bin )
(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 ! eie0 ðxÞ H ðxÞ
AH ðxÞ ! AH ðxÞ þ @ ðxÞ;
BH ðxÞ ! BH ðxÞ
½40
with @ 2 (x) = 0, are induced by the in-field
transformations
in ðxÞ ! in ðxÞ þ
bin ðxÞ ! bin ðxÞ þ
in ðxÞ ! in ðxÞ;
e0 ~
v
1=2
Z
e0 ~
v
1=2
Z
ðxÞ
ðxÞ
½41
Uin
ðxÞ ! Uin
ðxÞ
On the other hand, the global phase transformation
H (x) ! ei
H (x) is induced by
in ðxÞ ! in ðxÞ þ
in ðxÞ ! in ðxÞ;
~
v
1=2
Z
f ðxÞ;
bin ðxÞ ! bin ðxÞ
Uin
ðxÞ ! Uin
ðxÞ
½42
with @ 2 f (x) = 0 and the limit f (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 BH 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 transformation with (x) = ~
vZ1=2
f (x) and @ 2 f (x) = 0, BH
changes as
BH ðxÞ ! BH ðxÞ e0 ~
v2
f ðxÞ
Z
½43
eqn [38] is thus violated when the Gupta–Bleulerlike condition is imposed. In order to restore it, the
shift in BH must be compensated by means of the
following transformation on Uin
:
1=2 Uin
ðxÞ ! Uin
ðxÞ þ Z3
a ðxÞ;
@ a ðxÞ ¼ 0 ½44
with a convenient c-number function a (x). The
dynamical maps of the various Heisenberg operators
are not affected by [44] since they contain Uin
and
227
BH in a combination such that the changes of BH
and of Uin
compensate each other provided
ð@ 2 þ m2V Þa ðxÞ ¼
m2V
@ f ðxÞ
e0
½45
Equation [45] thus obtained is the Maxwell equation for the massive potential vector a (Matsumoto
et al. 1975a, b). The classical ground state current j
turns out to be
1
j ðxÞ h0jjH ðxÞj0i ¼ m2V a ðxÞ @ f ðxÞ ½46
e0
The term m2V a (x) is the Meissner current, while
(m2V =e0 )@ 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 observable 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)(Gþ
(x) = 0). In fact, from [45]
we obtain a (x) = (1=e0 )@ 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 conclusion, 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 transformation with regular f (x) does not affect observable
quantities, the S-matrix is actually given by
1
S ¼ : S in ; Uin @ðin bin Þ :
½47
mV
This is indeed independent of the boson transformation with regular f (x):
1
0
@ðin bin Þ
S ! S ¼ :S in ; Uin
mV
1
1=2
þZ3 ða @ f Þ :
½48
e0
228 Quantum Fields with Topological Defects
since a (x) = (1=e0 )@ f (x) for regular f (x). However,
S0 6¼ S for singular f (x): S0 includes the interaction of
the quanta Uin
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 nonhomogeneous 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 ðxÞ ¼ @ a ðxÞ @ a ðxÞ
Z
m2
0
d4 x0 c ðx x0 ÞGþ
¼ 2 V
ðx Þ ½49
e0
j ðxÞ ¼ 2
m2V
e0
Z
0
d4 x0 c ðx x0 Þ@x0 Gþ
ðx Þ
½50
respectively, and satisfy @ F (x) = j (x). In these
equations,
Z
1
1
0
d4 p eipðxx Þ 2
c ðx x0 Þ ¼
½51
p m2V þ i
ð2Þ4
The line singularity for the vortex (or string)
solution can be parametrized by a single line
parameter and by the time parameter . A static
vortex solution is obtained by setting y0 (, )= and
y(, )= y(), with y denoting the line coordinate.
Gþ
(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
Z
dyi ðÞ 3
½x yðÞ Gij ðxÞ ¼ 0
G0i ðxÞ ¼
d
d
½52
Gþ
Gþ
ðxÞ
¼
0
ij ðxÞ ¼ ijk G0k ðxÞ;
0i
Equation [49] shows that these vortices are purely
magnetic. We obtain
@0 f ðxÞ ¼ 0
@i f ðxÞ ¼
Z
1
2
dijk
dyk ðÞ x
@j
d
ð2Þ
Z
eipðxyðÞÞ
d3 p
p2
½53
R
that is, by using the identity (2)2 d3 p(ei px =p2 ) =
1=2jxj,
Z
1
dy ðÞ
1
d k
^ rx
½54
rf ðxÞ ¼ 2
d
jx yðÞj
Note that r2 f (x) = 0 is satisfied.
A straight infinitely long vortex is specified by
yi () = i3 with 1 < < 1. The only nonvanishing component of G (x) are G03 (x) = Gþ
12 (x) =
(x1 )(x2 ). Equation [54] gives (Umezawa 1993,
Umezawa et al. 1982, Matsumoto 1975a, b)
Z
@
1
@ 2
d
f ðxÞ ¼
½x þ x22 þ ðx3 Þ2 1=2
@x1
2
@x2 1
x2
¼ 2
½55
x1 þ x22
@
x1
@
f ðxÞ ¼ 2
;
f ðxÞ ¼ 0
2
@x2
x1 þ x2 @x3
and then
f ðxÞ ¼ tan1
x2
¼ ðxÞ
x1
½56
We have thus determined the boson transformation
function corresponding to a particular vortex solution. The vector potential is
Z
m2
x0
a1 ðxÞ ¼ V d4 x0 c ðx x0 Þ 02 2 02
2e0
x1 þ x2
2 Z
½57
m
x0
a2 ðxÞ ¼ V d4 x0 c ðx x0 Þ 02 1 02
2e0
x1 þ x2
a3 ðxÞ ¼ a0 ðxÞ ¼ 0
and the only nonvanishing component of F :
Z
m2
d4 x0 c ðx x0 Þðx01 Þðx02 Þ
F12 ðxÞ ¼ 2 V
e0
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
m2
¼ V K0 mV x21 þ x22
e0
½58
Finally, the vacuum current eqn [50] is given by
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
m3
x2
j1 ðxÞ ¼ V qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ K1 mV x21 þ x22
e0 x2 þ x2
1
2
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
3
mV
½59
x1
q
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ
j2 ðxÞ ¼
K1 mV x21 þ x22
e0 x2 þ x2
1
2
j3 ðxÞ ¼ j0 ðxÞ ¼ 0
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 properties of the dynamics determine the characteristics of
the topological solutions. The vortex solution
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 representations 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 corresponding 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 thus 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 transformations with regular f (x) only
amount to gauge transformations. For the treatment
of topological defects in nonabelian gauge theories,
see Manka and Vitiello (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 confinement 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.
229
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 condensates in quantum field theory: kinks
in (1 þ 1) dimensional 4 theory. Annals of Physics 295:
230–260.
Blasone M, Jizba P, and Vitiello G (2006) Spontaneous Breakdown of Symmetry and Topological Defects, London: Imperial College Press. (in preparation).
Goldstone J (1961) Field theories with ‘‘superconductor’’ solutions. 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 temperature 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 superconductivity. I. Physical Review 122: 345–358.
Nambu Y and Jona-Lasinio G (1961) Dynamical model of
elementary particles based on an analogy with superconductivity. 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: NorthHolland.
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.

Download Report