Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
Подождите немного. Документ загружается.
equations and the operators D which give the
admissible BT. A technique to do so is provided by
the so-called Lax technique introduced by Bruschi
and Ragnisco (1980a–c). Using the Lax technique,
we can easily obtain the nonlinear partial differ-
ential equations and BT associated with the Lax
operator [16a] both in the isospectral and non-
isospectral case (when k
,t
= 0andwhenk
,t
6¼ 0)
and the corresponding evolution of the spectrum.
We have
u
;t
¼ f ðL; tÞu
x
þ gðL; tÞ½xu
x
þ 2u½33a
k
;t
¼ kgð4k
2
; tÞ
dRðk; tÞ
dt
¼ 2ikf ð4k
2
; tÞRðk; tÞ
½33b
FðÞð
~
u uÞþGðÞ 1 ¼ 0 ½33c
~
Rðk; tÞ¼
Fð4k
2
Þ2ikGð4k
2
Þ
Fð4k
2
Þþ2ikGð4k
2
Þ
Rðk; tÞ½33d
where the functions f, g, F, and G are entire
functions of their first argument and the recursive
operators L and are given by
Lf ðxÞ¼f
;xx
ðxÞ4uðx; tÞf ðxÞ
þ 2u
;x
ðx; tÞ
Z
þ1
x
f ðyÞdy ½34a
f ðxÞ¼f
;xx
ðxÞ2½
~
uðx; tÞþuðx; tÞf ðxÞ
þ
Z
þ1
x
f ðyÞdy ½34b
f ðxÞ¼½
~
u
;x
ðx; tÞþu
;x
ðx; tÞf ðxÞþ½
~
uðx; tÞuðx; t Þ
Z
þ1
x
½
~
uðy; tÞuðy; tÞf ðyÞdy ½34c
In the limit when
˜
u ! u the operator !L.ABT
is obtained by choosing the functions F and G in
eqn [33c] . The simplest BT is obtained by setting
F = and G = 1:
~
v
;x
þ v
;x
þð
~
v vÞ
1
2
ð
~
v vÞ
¼ 0 ½35
with u(x, t) = v
,x
(x, t) and is the Ba¨cklund
parameter. By combining together BT of the form
[35] with different parameters as in eqn [5], we get
the permutability theorem for the KdV BTs:
~
v
0
¼ v
ð
1
þ
2
Þ½v
0
~
v
1
2
þð1/2Þðv
0
~
vÞ
½36
Its proof is immediate from the point of view of the
spectrum.
Ba¨ cklund and Symmetries
A symmetry of the nonlinear equation [15] is given
by a flow c ommuting with it, that is, by an
equation
u
;
¼ f ðu; u
x
; u
t
; ...Þ½37
where is the group parameter, u = u(x, t; ), and the
derivative of [15] is zero on its set of solutions.
A group transformation is obtained by integrating it.
Usually this is possible only when eqn [37] is a
quasilinear partial differential equati on of the first
order. Taking into account the evolution of the
spectrum of the KdV equation [15], it is easy to
prove that its symmetries are given by
u
;
¼
X
þ1
n¼0
n
L
n
3
X
þ1
n¼0
n
tL
n
()
u
;x
þ
X
þ1
n¼0
n
L
n
()
½xu
;x
þ 2u½38
where
n
and
n
are a set of constant parameters.
For each choice of the parameters
n
and
n
,
one gets a symmetry of the KdV equation [15].
With eqn [38] one can associate the following
evolution of the reflection coefficient R(k, t; ):
dR
d
¼ 2ik
X
þ1
n¼0
n
ð4k
2
Þ
n
(
3
X
þ1
n¼0
n
tð 4k
2
Þ
nþ1
)
R ½39
and of the spectral parameter k
k
;
¼
X
þ1
n¼0
n
ð4k
2
Þ
n
k ½40
As (1/2)L 1 = xu
,x
þ 2u, one can add to the
symmetries [38] the exceptional one (which has no
spectral counterpart as u is not bounded
asymptotically):
u
;
¼ 1 þ 6tu
;x
½41
By a proper natural choice of the constant para-
meters
n
and
n
, one can define two infinite series
of symmetries. The first one is obtained by choosing
n
= 0and
n
=
n, m
with m = 1, 2, ..., 1 and can
be denoted as the isospectral series as k
,
= 0. This is
formed by commuting symmetries. The second one
is given by
n
= 0 and
n
=
n, m
with m = 1, 2, ..., 1
and can be denoted as the nonisospectral series as
k
,
6¼ 0. The nonisospectral symmetries have a
nonzero commutation relation among themselves
and with the isospectral ones.
Ba¨ cklund Transformations 245
Except for a few Lie point symmetries (given by
eqn [41] and by choosing inside the series [38] those
with different from zero only
0
or
0
or
1
) they
are all generalized symmetries (Olver 1993). By
analyzing their spectrum, it is easy to prove that the
choice [38] is such that they are all independent. For
the isospectral class, the evolution of the spectrum is
simple and can be integrated to provide the group
transformation of the spectrum
Rðk; t; Þ¼Rðk; tÞ
exp 2ik
X
þ1
n¼0
n
ð4k
2
Þ
n
()
"#
½42
Let us now consider the simplest BT obtained by
choosing, in eqn [33c], F() = and G() = 1, where
is an arbitrary parameter. In the spectral space, this
corresponds to the following change of the spectrum:
~
Rðk; tÞ¼
2ik
þ 2ik
Rðk; tÞ½43
Defining
˜
R(k, t) = R(k, t; ), eqn [42] is equal to
eqn [43] iff
n
¼
2
2nþ1
ð2n þ 1Þ
; n ¼ 0; 1; ...; 1½44
So we need an infinite number of symmetries to
be able to reconstruct the change of the spectrum
given by the BT. This shows that the existence of a BT
is strictly connected to the existence of an infinity of
symmetries which is a condition for the exact
integrability of the nonlinear partial differential
equation (Fokas 1980, Ibragimov and Shabat 1980).
Discretization via Ba¨ cklund
BTs, apart from providing classes of exact solutions
to nonlinear equations, play a very important role in
the discretization of partial differential equations. As
noted earlier, an auto-BT is a differential relation
between two different solutions of the same non-
linear partial differential equation. If it is assumed
that the new solution
˜
u is just the old solution u
computed in a different poin t of a lattice, then the
BT becomes just a differential–difference equation
(Chiu and Ladik 1977, Levi and Benguria 1980).
This can be carried out also at the level of the
associated compatibility condition and in such a
way one is able to also obtain its Lax pair. This
demonstrates the integrability of the differential–
difference equation
vðn þ 1; tÞ
;t
þ vðn; tÞ
;t
þ½vðn þ 1; tÞvðn; tÞ
1
2
½vðn þ 1; tÞvð n; tÞ
¼ 0 ½45
which is an integrable differential–difference
approximation to the KdV equation or
wðn þ 1; tÞ
;t
¼ wðn; tÞ
;t
þ 2a sin
wðn þ 1; tÞþwðn; tÞ
2
½46
a discrete integrable differential–difference approxima-
tion to the sG equation (Hirota 1977, Orfanidis 1978).
As the nonlinear superposition formulas are
purely algebraic relations involving potentials asso-
ciated with integrable nonlinear partial differential
equations, one can interpret them as difference–
difference equations. In the case of the sG equation
from eqn [7],wehave
w
nþ1;mþ1
w
n;m
¼ 4 arctan
1
a
1
þ a
2
a
1
a
2
tan
w
n;mþ1
w
nþ1;m
4
½47
where w(x, t) = w
n, m
,
˜
w(x, t) = w
nþ1, m
, w
0
(x, t) =
w
n, mþ1
, and
˜
w
0
(x, t) = w
nþ1, mþ1
. In a similar manner,
from [36], one gets
v
nþ1;mþ1
¼ v
n;m
ð
1
þ
2
Þ½v
nþ1;m
v
n;mþ1
1
2
þ
1
2
½v
nþ1;m
v
n;mþ1
½48
The continuous limit of eqn [47], obtained by setting
x =
1
n and y =
2
m and choosing
a
1
a
2
¼
1
2
4
gives back eqn [2] (Rogers and Schief 1997). It is
worth mentioning that one can also use known
nonlinear lattice equations to construct BT for
nonlinear partial differential equations (Levi 1981).
See also: Integrable Systems and Discrete Geometry;
Integrable Systems: Overview; Painleve
´
Equations;
Solitons and Kac–Moody Lie Algebras; Toda Lattices.
Further Reading
Ba¨cklund AV (1874) Einiges u¨ ber Curven und Fla¨chentransfor-
mationen. Lund Universite¨ts Arsskrift 10: 1–12.
Bianchi L (1879) Ricerche sulle superficie a curvatura costante e sulle
elicoidi. Annali della R. Scuola normale superiore di Pisa 2: 285.
Bruschi M and Ragnisco O (1980a) Existence of a Lax pair for
any member of the class of nonlinear evolution equations
associated to the matrix Schro¨dinger spectral problem. Lettere
al Nuovo Cimento 29: 321–326.
Bruschi M and Ragnisco O (1980b) Extension of the Lax method
to solve a class of nonlinear evolution equations with
x-dependent coeffcients associated to the matrix Schro¨ dinger
spectral problem. Lettere al Nuovo Cimento 29: 327–330.
Bruschi M and Ragnisco O (1980c) Ba¨ cklund transformations
and Lax technique. Lettere al Nuovo Cimento 29: 331–334.
Chiu S-C and Ladik JF (1977) Generating exactly soluble
nonlinear discrete evolution equations by a generalized
246 Ba¨ cklund Transformations
Wronskian technique. Journal of Mathematical Physics
18: 690–700.
Clarin J (1903) Sur quelques e´quations aux de´rive´es partielles du
second ordre. Annales de la Facult des Sciences de Toulouse pour
les Sciences Mathmatiques et les Sciences Physiques. Serie 2
5: 437–458.
Coley A, Levi D, Milson R, Rogers C, and Winternitz P (eds.) (2001)
Ba¨ cklund and Darboux transformations. The Geometry of
solitons. Proceedings of the AARMS-CRM Workshop, Halifax,
NS, June 4–9, 1999. CRM Proceedings and Lecture Notes, vol.
29. Providence, RI: American Mathematical Society.
Faddeev LD and Takhtajan LA (1987) Hamiltonian Methods in
the Theory of Solitons. Berlin: Springer.
Fokas AS (1980) A symmetry approach to exactly solvable evolution
equations. Journal of Mathematical Physics 21: 1318–1325.
Gardner CS, Greene JM, Kruskal MD, and Miura RM (1967)
Method for solving the Korteweg–de Vries equation. Physical
Review Letters 19: 1095–1097.
Hirota R (1978) Nonlinear partial difference equations. III.
Discrete sine-Gordon equation. Journal of the Physical Society
of Japan 43: 2079–2086.
Ibragimov NH and Shabat AB (1980) Infinite Lie–Bcklund algebras
(in Russian). Funktsional. Anal. i Prilozhen 14: 79–80.
Lamb GL (1976) Ba¨cklund transformations at the turn of
the century. In: Miura RM (ed.) Ba¨ cklund Transformations,
pp. 69–79. Berlin: Springer.
Lax PD (1968) Integrals of nonlinear equations of evolution and
solitary waves. Communications in Pure and Applied Mathe-
matics 21: 647–690.
Levi D (1981) Nonlinear differential difference equations as
Backlund transformations. Journal of Physics A: Mathematical
and General 14: 1083–1098.
Levi D (1988) On a new Darboux transformation for the
construction of exact solutions of the Schroedinger equation.
Inverse Problems 4: 165–172.
Levi D and Benguria R (1980) Ba¨cklund transformations and
nonlinear differential difference equations. Proceedings of the
National Academy of Science USA 77: 5025–5027.
Levi D and Winternitz P (1989) Non-classical symmetry reduction:
example of the Boussinesq equation. Journal of Physics A:
Mathematical and General 22: 2915–2924.
Levi D, Ragnisco O, and Sym A (1984) Dressing method vs. classical
Darboux transformation. Il Nuovo Cimento 83B: 34–42.
Lie S (1888, 1890, 1893) Theorie der Transformationgruppen.
Leipzig: B.G. Teubner.
Matveev VB and Salle LA (1991) Darboux Transformations and
Solitons. Berlin: Springer.
Moutard Th-F (1878) Sur la construction des e´quations de la forme
(1=z)(d
2
z=dxdy) = (x,y), qui admettent une integrale ge´ne´ral
explicite. Journal de l’Ecole Polytechnique, Paris 28: 1–11.
Olver PJ (1993) Applications of Lie Groups to Differential
Equations. New York: Springer.
Orfanidis SJ (1978) Discrete sine-Gordon equations. Physical
Review D 18: 3822–3827.
Rogers C and Schief WK (1997) The classical Ba¨ cklund
transformation and integrable discretization of characteristic
equations. Physics Letters A 232: 217–223.
Rogers C and Shadwick WF (1982) Ba¨cklund Transformations
and Their Applications. New York: Academic Press.
Toda M (1989) Theory of Nonlinear Lattices. Berlin: Springer.
Wojciechowski S (1982) The analogue of the Ba¨cklund transforma-
tion for integrable many-body systems. Journal of Physics A:
Mathematical and General 15: L653–L657.
Zaharov VE and Shabat AB (1979) Integration of the nonlinear
equations of mathematical physics by the method of the
inverse scattering problem. II (Russian). Funktsional Analiz
i ego Prilozheniya 13: 13–22. (English translation: Functional
Analysis and Applications 13: 166–173 (1980)).
Batalin–Vilkovisky Quantization
A C Hirshfeld, Universita
¨
t Dortmund,
Dortmund, Germany
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The Batalin–Vilkovisky formalism for quantizing
gauge theories has a long history of development. It
begins with the Faddeev–Popov procedure for
quantizing Yang–Mills theory, involving the Faddeev–
Popov ghost fields (Faddeev and Popov 1967). It
continued with the discovery of BRST symmetry by
Becchi et al. (1976). Then Zinn-Justin (1975)
introduced sources for these transformations, and
a symmetric structure in the space of fields and
sources in his study of renormalizability of these
theories. Finally, Batalin and Vilkovisky (1981)
systematized and generalized these developments.
A more detailed account of this history can be
found in Gomis et al. (1994), where many worked
examples of the Batalin–Vilkovisky formalism are
given. At the present time, it is the most general
treatment available. Alexandrov, Kontsevich, Schwarz,
and Zabaronsky (AKSZ 1997)havepresenteda
geometric interpretation for the case in which the
action is topologically invariant.
Structure of the Set of Gauge
Transformations
Consider a system whose dynamics is governed by
a classical action S[
i
] which depends on the
fields
i
(x), i = 1, ..., n. We employ a c ompact
notation in which the multi-index i may denote
the various fields involved, the discrete indices on
which they depend, and the dependence on the
spacetime variables as well. The generalized
summation convention then means that a
repeated index may denote not only a sum over
discrete variables, but also integration over
the spacetime variables.
i
= (
i
) d enotes the
Batalin–Vilkovisky Quantization 247
Grassmann parity of the fields. Fields with
i
= 0
are called bosonic, with
i
= 1 fermionic. The
graded commutation rule is
i
ðxÞ
j
ðyÞ¼ð1Þ
i
j
j
ðyÞ
i
ðxÞ½1
For a gauge theory the action is invariant under a set
of gauge transformations with infinitesimal form
i
¼ R
i
"
;¼ 1or2or...m ½2
The "
are the infinitesimal gauge parameters and
R
i
the generators of the gauge transformations.
When
= ("
) = 0 we have an ordinary symmetry,
when
= 1 the equation is characteristic of a
supersymmetry. The Grassmann parity of R
i
is
(R
i
) =
i
þ
(mod 2).
A subscript after a comma denotes the right
derivative with respect to the corresponding field,
that is, the field is to be commutated to the far right
and then dropped. The field equations may then be
written as
S
0;i
¼ 0 ½3
where S
0
is the classical action. Let denote the
surface in the space of solutions where the field
equations are satisfied:
S
0;i
j
¼ 0 ½4
If the gauge transformations are ‘‘independent’’
on-shell, that is,
rank R
i
j
¼ m ½5
the gauge theory is said to be ‘‘irreducible.’’ We
assume here that this is the case. When it is not, the
theory is ‘‘reducible.’’ For details of the treatment in
that case, see Gomis, Paris, and Samuel. The
classical solutions are
0
2 .
The Noether identities are
S
0;i
R
i
¼ 0 ½6
The general solution to the Noether identity is
i
¼ R
i
T
þ S
0;j
E
ji
½7
The commutator of two gauge transformations is
½
1
;
2
i
¼ R
i
;j
R
j
ð1Þ
R
i
;j
R
j
"
1
"
2
½8
Since this commutat or is a symmetry of the action, it
satisfies the Noether identity
S
0;i
R
i
;j
R
j
ð1Þ
R
i
;j
R
j
¼ 0 ½9
which by eqn [7] implies that
R
i
;j
R
j
ð1Þ
R
i
;j
R
j
¼ R
i
T
þ S
0;j
E
ji
½10
Equations [8] and [10 ] lead to the following
condition:
½
1
;
2
i
¼ R
i
T
S
0;j
E
ji
"
1
"
2
½11
The tensors T
are called the structure constants of the
gauge algebra, although they depend, in general, on
the fields of the theory. When E
ij
= 0, the gauge
algebra is said to be ‘ ‘clo sed,’’ otherwise it is ‘ ‘open.’’
Equation [11] defines a Lie algebra if the algebra is
closed and the T
are independent of the fields.
The gau ge tensors have the follo wing graded
symmetry properties:
T
¼ ð 1Þ
T
E
ij
¼ ð 1Þ
E
ji
¼ ð 1Þ
E
ij
½12
The Grassmann parities are
ðT
Þ¼
þ
þ
ðmod 2Þ½13
and
ðE
ij
Þ¼
i
þ
j
þ
þ
ðmod 2Þ½14
Various restrictions are imposed by the Jacobi
identity
X
cyclicð123Þ
½
1
; ½
2
;
3
¼ 0 ½15
These restrictions are
X
cyclicð123Þ
R
i
A
S
0;j
B
ji
"
"
"
¼ 0 ½16
where
3A
T
k
R
k
T
T
þð1Þ
ð
þ
Þ
T
k
R
k
T
T
þð1Þ
ð
þ
Þ
T
k
R
k
T
T
and
3B
ji
E
ji
k
R
k
E
ji
T
ð1Þ
i
R
j
;k
E
ki
þð1Þ
j
ð
i
þ
Þ
R
i
;k
E
kj
þð1Þ
ð
þ
Þ
ð ! ! Þþð1Þ
ð
þ
Þ
ð ! ! Þ
As in the familiar Faddeev–Popov procedure, it is
useful to introduce ghost fields C
with opposite
Grassmann parities to the gauge parameters "
:
ðC
Þ¼
þ 1 ðmod 2Þ½17
248 Batalin–Vilkovisky Quantization
and to replace the gauge parameters by ghost fields.
One must then modify the graded symmetry proper-
ties of the gauge structure tensors according to
T
1
2
3
4
...
!ð1Þ
2
þ
4
þ
T
1
2
3
4
...
½18
The Noether identities then take the form
S
0;i
R
i
C
¼ 0 ½ 19
and the structure relations [10] become
ð2R
i
;j
R
j
R
i
T
þ S
0;j
E
ji
ÞC
C
¼ 0 ½20
Introducing the Antifields
We incorporate the ghost fields into the field set
A
= {
i
, C
}, where i = 1, ..., n and = 1, ..., m.
Clearly A = 1, ..., N, where N = n þ m. One then
further increases the set by introducing an antifield
A
for each field
A
. The Grassmann parity of the
antifields is
A
¼ ð
A
Þþ1 ðmod 2Þ½21
Each field is assigned a ghost number, with
gh½
i
¼0
gh½C
¼1
gh
A
¼gh½
A
1
½22
In the space of fields and antifields, the antibracket
is defined by
ðX; YÞ¼
@
r
X
@
A
@
l
Y
@
A
@
r
X
@
A
@
l
Y
@
A
½23
where @
r
denotes the right, @
l
the left derivative. The
antibracket is graded antisymmetric:
ðX; YÞ ¼ ð1Þ
ð
X
þ1Þð
Y
þ1Þ
ðY; XÞ½24
It satisfies a graded Jacobi identity
ððX;YÞ; ZÞþð1Þ
ð
X
þ1Þð
Y
þ1Þ
ððY; ZÞ; XÞþð1Þ
ð
Z
þ1Þð
X
þ
Y
Þ
ððZ;XÞ;YÞ¼0 ½25
It is a graded derivation
ðX; YZÞ¼ðX; YÞZ þð1Þ
X
Y
ðX; ZÞY
ðXY; ZÞ¼XðY; ZÞþð1Þ
X
Y
YðX; ZÞ
½26
It has ghost number
gh½ðX; YÞ ¼ gh½Xþgh½Yþ1 ½27
and Grassmann parity
ððX; YÞÞ ¼ ðXÞþðYÞþ1 ðmod 2Þ½28
For bosonic fields
ðB; BÞ¼2
@B
@
A
@B
@
A
½29
for fermionic fields
ðF; FÞ¼0 ½30
and for any X
ððX; XÞ; XÞ¼0 ½31
If one groups the fields and the antifields together
into the set
z
a
¼f
A
;
A
g; a ¼ 1; ...; 2N ½32
then the antibracket is seen to define a symplectic
structure on the space of fields and antifields
ðX; YÞ¼
@
r
X
@z
a
!
ab
@
l
Y
@z
b
½33
with
!
ab
¼
0
A
B
A
B
0
½34
The antifields can be thought of as conjugate
variables to the fields, since
A
;
B
¼
A
B
½35
The Classical Master Equation
Let S[
A
,
A
] be a functional of the fields and
antifields with the dimension of an action, vanishing
ghost number and even Grassmann parity. The
equation
ðS; SÞ¼2
@S
@
A
@S
@
A
¼ 0 ½36
is the classical master equation. Solutions of the
classical master equation with suitable boundary
conditions turn out to be generating functionals for
the gauge structure of the theory. S is also the
starting point for the quantization. One denotes by
the subspace of stationary points of the action in
the space of fields and antifields:
¼ z
a
@S
@z
a
¼ 0
½37
Given a classical solution
0
of S
0
one stationary
point is
i
¼
i
0
; C
a
¼ 0;
A
¼ 0 ½38
Batalin–Vilkovisky Quantization 249
An action which satisfies the classical master
equation has its own set of invariances:
@S
@z
a
R
a
b
¼ 0 ½39
with
R
a
b
¼ !
ac
@
l
@
r
S
@z
c
@z
b
½40
This equation implies
R
a
c
R
a
b
¼ 0 ½41
One says that R
a
b
is invariant on-shel l. A nilpotent
2N 2N matrix has rank N. Let r be the rank of
the hessian of S at the stationary point:
r ¼ rank
@
l
@
r
S
@z
a
@z
b
½42
We then have r N. The relevant solutions of the
classical master equation are those for which r = N.
In this case the number of independent gauge
invariances of the type in eqn [39] equ als the number
of antifields. When at a later stage the gauge is fixed,
the nonphysical antifields are eliminated.
To ensure the correct classical limit, the proper
solution must contain the classical action S
0
in the
sense that
S
A
;
A
A
¼0
¼ S
0
½
i
½43
The action S[
A
,
A
] can be expanded in a series in
the anti fields, while maintaining vanishing ghost
number and even Grassmann parity:
S½;
¼S
0
þ
i
R
i
C
þ C
a
1
2
T
ð1Þ
C
C
þ
i
j
ð1Þ
i
1
4
E
ji
ð1Þ
C
C
þ ½44
When this is inserted into the classical master
equation, one finds that this equation implies the
gauge structure of the classical theory.
Gauge Fixing and Quantization
Equation [39] shows that the action S still possesses
gauge invariances, and hence is not yet suitable for
quantization via the path integral approach: a
gauge-fixing procedure is necessary. In the Batalin–
Vilkovisky approach the gauge is fixed, and the
antifields eliminated, by use of a gauge-fixing
fermion which has Grassmann parity () = 1
and gh[] = 1. It is a functional of the fields
A
only; its relation to the antifields is
A
¼
@
@
A
½45
We define a surface in functional space
¼
A
;
A
j
A
¼
@
@
A
½46
so that for any functional X[,
]
Xj
¼ X ;
@
@
½47
To construct a gauge-fixing fermion of ghost
number 1, one must again introduce additional
fields. The simplest choice utilizes a trivial pair
C
,
with
ð
C
Þ¼
þ 1;ð
Þ¼
gh½
C
¼1; gh½
¼0
½48
The fields
C
are the Faddeev–Popov antighosts.
Along with these fields we include the corresponding
antifields
C
,
. Adding the term
C
to the
action S does not spoil its properties as a proper
solution to the classical master equati on, and one
gets the nonminimal action
S
non
¼ S þ
C
½49
The simplest possibility for is
¼
C
ðÞ½50
where
are the gauge-fixing conditions for the
fields . The gauge-fixed action is denoted by
S
¼ S
non
j
½51
Quantization is performed using the path inte gral
to calculate a correlation function X, with the
constraint [45] implemented by a -function:
I
ðXÞ¼
Z
DD
A
@
@
A
exp
i
h
W½;
X½;
½52
Here W is the quantum action, which reduces to S in
the limit h !0. An admissible leads to well-
defined propagators when the path integral is
expressed as a perturbation series expansion.
The results of a calculation should be independent
of the gauge fixing. Consider the integrand in eqn
[52],
I½;
¼exp
i
h
W½;
X½;
½53
Under an infinitesimal change in
I
þ
ðXÞI
ðXÞ
Z
DI ½54
250 Batalin–Vilkovisky Quantization
where the Laplacian is
¼ð1Þ
A
þ1
@
@
A
@
@
A
½55
Obviously, the integral I
(X) is independent of if
I = 0. For X = 1 one gets the requirement
exp
i
h
W
¼exp
i
h
W
i
h
W
1
2h
2
ðW; WÞ
¼ 0 ½56
The formula
1
2
ðW; WÞ¼ihW ½57
is the quantum master equation. A gauge-invariant
correlation function satisfies
ðX; WÞ¼ihX ½58
The terms of higher order in h by which the
quantum action W may differ from the solution of
the classical master equation S correspond to the
counter-terms of the renormalizable gauge theory if
S ¼ 0 ½59
One must, of course, use a regularization scheme
which respects the symmetries of the theory. For
W = S þ O(h) the quantum master equati on [57]
reduces in this case to the classical master equation
ðS; SÞ¼0 ½60
Hence, up to possible counter-terms, one may
simply choose W = S.
To implement the gauge fixing, one uses for the
action W = S
non
. For the path integral Z = I
(X = 1),
the integration over the antifields in eqn [52] is
performed by using the -function. The result is
Z ¼
Z
D exp
i
h
S
½61
Geometrical Interpretation of Topological
Field Theories
The Batalin–Vilkovisky formalism for topological
field theories has been given a geometrical inter-
pretation by AKSZ (1997).
A supermanifold equippe d with an odd vector
field satisfying Q
2
= 0 is called a Q-manifold. A
Q-manifold provided with an odd symplectic struc-
ture ! (P-structure) is called a QP-manifold if the
odd symplectic structure is Q-invariant, that is,
L
Q
! = 0. Every solution to the classical master
equation determines a QP-structure on M and vice
versa. The geometric object corresponding to a
classical mechanical system in the Batalin–Vilkovisky
formalism is a QP-manifold.
The nondegenerate closed 2-form ! is written as
! ¼ dz
a
ab
dz
b
½62
where z
a
are local coordinates in the supermanifold
M. For functions on M, an (odd) Poisson bracket is
defined as in eqn [33], where !
ab
stands for the
inverse matrix of !
ab
. An even function S on M
satisfies the classical master equation if (S, S) = 0.
The correspond ence between vector fields and
functions on M is given by K
F
G = (G, F), wher e K
F
is the vector field, F the given function, and G an
arbitrary function. The function F is called the
Hamiltonian of the vector field K
F
.
Geometrically, equivalent QP-manifolds describe
the same physics. In particular, one can consider
an even Hamiltonian vector field K
F
corresponding
to an odd function F. T his vector field determines
an infinitesimal transformation preserving P-structure.
It transforms a solution S to the classical master
equation into the physically equivalent solution
S þ (S, F), where is an infinitesimally small
parameter.
A submanifold L of a P-manifold M is called a
Lagrangian submanifold if the restriction of the
form ! to L vanishes. In the particular case when
M =T
N (the cotangent bundle to N with reversed
parity of fibres) with standard P-structure, one can
construct many exampl es of Lagrangian submani-
folds in the following way. Fix an odd function on
N, the gauge fermion. The submanifold L
2 M
determined by the equation
a
¼
@
@x
a
½63
where {x
a
,
a
} are coordinates corresponding to the
identification of M, will be a Lagrangian submani-
fold of M.
The P-manifold M in the neighborhood of L can
be identified with T
L. In other words, one can
find such a neighborhood U of L in M and a
neighborhood V of L in T
L that there exists an
isomorphism of P-manifolds U and V leaving L
intact. Using this isomorphism a function defined
on a Lagrangian submanifold L M determines
another Lagrangian submanifold L
M.
Consider a solution S to the classical master
equation on M. In the Batalin–Vilkovisky formalism
we have to restrict S to a Lagrangian submanifold
L 2 M, then the quantization of S can be performed
by integration of exp (iS=h) over L. One may
construct an odd vector field Q on L in such a
Batalin–Vilkovisky Quantization 251
way that the functional S restricted to L is
Q-invariant. This invariance is BRST invariance.
AKSZ apply these geometric constructions to obtain
in a natural way the action functionals of two-
dimensional sigma-models (Witten 1998) and to
show that the Chern–Simons theory (Axelrod and
Singer 1991) in Batalin–Vilkovisky formalism arises as
a sigma-model with target space G,whereG stands
for a Lie algebra and denotes parity inversion.
The Poisson-Sigma Model
The quantization of the Poisson-sigma model was
performed by Hirshfeld and Schwarzweller (2000)
and by Cattaneo and Felder (2001). The Poisson-
sigma model is the simplest topological field theory
in two dimensions. It is a field theory on a two-
dimensional world sheet without boundary (Schaller
and Strobl 1994). It involves a set of bosonic scalar
fields, which can be seen as a set of maps
X
i
: M !N, where N is a Poisson manifold. In
addition, one has a 1-form A on the world sheet M
which takes values in T
(N), for x coordi nates on M
we have A = A
i
dx
i
^ dX
i
. Its action is
S
0
½X; A¼
Z
M
ðA
i
@
X
i
þ P
ij
ðXÞA
i
A
j
½64
where
is the antisymmetric tensor and is the
volume form on M. The gauge transformations of
the model are
X
i
¼ P
ij
ðXÞ"
j
;A
i
¼ D
j
i
"
j
½65
where D
j
i
= @
j
i
þ P
kj
,i
A
k
. The equations of motion
are
D
j
i
A
j
¼ 0 ½66
and
ð@
X
i
þ P
ij
A
j
Þ¼
D
X
i
¼ 0 ½67
The gauge algebra is given by
ð "
1
Þ;ð"
2
Þ½X
i
¼ P
ji
ðP
mn
;j
"
1n
"
2m
Þ
ð"
1
Þ;ð"
2
Þ½A
i
¼ D
j
i
ðP
mn
;j
"
1n
"
2m
Þ
ð
D
X
j
Þ
P
mn
;ji
"
1n
"
2m
½68
In our general notation the generators of the gauge
transformations R are here P
ij
and D
j
i
. The gauge
tensors T and E are P
ij
,k
and
P
mn
,ji
. The higher-
order gauge tensors A and B vanish.
The ghost fields are again denoted by C
i
. The
Noether identities are then
Z
M
D
j
i
A
j
P
ki
þð
D
X
i
ÞD
k
i
C
k
¼ 0 ½69
Considering the commutator of two gauge transfor-
mations leads to (see eqns [8]–[11])
Z
M
2P
mi
;j
P
nj
P
ji
P
mn
;j
C
m
C
n
¼ 0
Z
M
2ðP
jk
i
D
l
j
þ P
mk
;ij
A
m
P
jl
Þ
D
m
i
P
kl
;m
þð
D
X
j
Þ
P
kl
;ji
C
l
C
k
¼ 0
½70
The Jacobi identity is
P
ij
;m
P
mk
C
i
C
j
C
k
¼ 0 ½71
The fields and antifields of the model are
A
¼fA
i
; X
i
; C
i
g and
A
¼ A
i
; X
i
; C
i
½72
The extended action is
S¼
Z
M
ðA
i
@
X
i
þ P
ij
ðXÞA
i
A
j
Þ
þ A
i
D
j
i
C
j
þ X
i
P
ji
ðXÞC
j
þ
1
2
C
i
P
jk
;i
ðXÞC
j
C
k
þ
1
4
A
i
A
j
P
kl
;ij
ðXÞC
k
C
l
½73
The gauge-fixing conditions are taken to be of the
form
i
(A, X), so that the gauge fermion [50] becomes
=
C
i
i
(A, X). The antifields are then fixed to be
A
i
¼
C
j
@
j
ðA; XÞ
@A
i
X
i
¼
C
j
@
j
ðA; XÞ
@X
i
C
i
¼ 0
C
i
¼
i
ðA; XÞ
½74
The gauge-fixed action is
S
¼
Z
M
ðA
i
@
X
i
þ P
ij
ðXÞA
i
A
j
Þ
þ
C
k
@
k
ðA; XÞ
@A
i
D
j
i
C
j
þ
C
k
@
k
ðA; XÞ
@X
i
P
ij
C
j
þ
1
4
C
m
@
m
ðA; XÞ
@A
i
C
n
@
n
ðA; XÞ
@A
j
P
kl
;ij
ðXÞ
C
k
C
l
þ
i
i
ðA; XÞ
½75
Now consider different gauge conditions:
1. First, the Landau gauge for the gauge potential
i
= @
A
i
, so that the gauge fermion becomes
=
C
i
@
A
i
. The antifields are fixed to be
A
i
¼ @
C
i
X
i
¼ C
i
¼ 0
C
i
¼ @
A
i
½76
252 Batalin–Vilkovisky Quantization
for this gauge choice the gauge-fixed action is
S
¼
Z
M
ðA
i
@
X
i
þ P
ij
ðXÞA
i
A
j
Þþ
C
i
@
D
j
i
C
j
þ
1
4
ð@
C
i
Þð@
C
j
Þ
P
kl
;ij
ðXÞ
C
k
C
l
i
ð@
A
i
Þ
½77
Translating this action into the notation of Cattaneo
and Felder, one sees that it is exactly the expression
they use to derive the perturbation series.
2. Now consider the temporal gauge
i
= A
0i
. The
gauge fermion is given by =
C
i
A
0i
. The anti-
fields are fixed to
A
0i
¼
C
i
A
1i
¼ 0
X
i
¼ C
i
¼ 0
C
i
¼ A
0i
½78
The gauge-fixed action is
S
¼
Z
M
ðA
i
@
X
i
þ P
ij
ðXÞA
i
A
j
Þ
þ
C
i
D
j
0i
C
j
i
ðA
0i
Þ
½79
3. Finally consider the Schwinger–Fock gauge
i
= x
A
i
. Then the antifields are fixed to be
A
i
¼ x
C
i
X
i
¼ C
i
¼ 0
C
i
¼ x
A
i
½80
for this gauge choice the gauge-fixed action is
S
¼
Z
M
ðA
i
@
X
i
þ P
ij
ðXÞA
i
A
j
Þ
þ
C
i
x
D
j
i
C
j
i
ð@
A
i
Þ
½81
Notice that in the noncovariant gauges 2 and 3 the
action simplifies, in that the term which arose
because of the nonclosed nature of the gauge algebra
vanishes.
See also: BF Theories; BRST Quantization; Constrained
Systems; Graded Poisson Algebras; Operads;
Perturbative Renormalization Theory and BRST;
Supermanifolds; Topological Sigma Models.
Further Reading
Alexandrov M, Kontsevich M, Schwarz A, and Zaboronsky O
(1997) Geometry of the Master Equation. International
Journal of Modern Physics A12: 1405–1430.
Axelrod S and Singer IM (1991) Chern–Simons Perturbation
Theory, Proceedings of the XXth Conference on Differential
Geometric Methods in Physics, Baruch College/CUNY, NY.
(hep-th/9110056).
Batalin IA and Vilkovisky GA (1977) Gauge algebra and
quantization. Physics Letters 69B: 309–312.
Becchi C, Rouet A, and Stora R (1976) Renormalization of gauge
theories. Annals of Physics (NY) 98: 287–321.
Cattaneo AS and Felder G (2001) On the AKSZ formulation of
the Poisson–Sigma model. Letters of Mathematical Physics
56: 163–179.
Faddeev LD and Popov VN (1967) Feynman diagrams for the
Yang–Mills field. Physics Letters 25B: 29–30.
Gomis J, Paris J, and Samuel S (1994) Antibracket Antifields and
gauge-theory quantization. Physics Reports 269: 1–145.
Hirshfeld AC and Schwarzweller T (2000) Path integral quantiza-
tion of the Poisson–Sigma model. Annals of Physics (Leipzig)
9: 83–101.
Schaller P and Strobl T (1994) Poisson structure induced
(topological) field theories. Modern Physics Letters A9:
3129–3136.
Witten E (1988) Topological sigma models. Communications in
Mathematical Physics 118: 411–449.
Zinn-Justin J (1975) Renormalizati on of gauge theories. In:
Rollnik H and Dietz K (eds.) Trends in Elementary
Particle Physics, Lecture Notes in Physics, vol. 37. Berlin:
Springer.
Bethe Ansatz
M T Batchelor, Australian National University,
Canberra, ACT, Australia
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The Bethe ansatz is a particular form of wave function
introduced in the diagonalization of the Heisenberg
spin chain. It underpins the majority of exactly solved
models in statistical mechanics and quantum field
theory. At the heart of the Bethe ansatz is the way in
which multibody interactions factor into two-body
interactions. The Bethe ansatz is thus intimately
entwined with the theory of integrability.
The way in which the Bethe ansatz works is best
understood by working through an explicit hands-on
example. The canonical example is the isotropic
antiferromagnetic Heisenberg Hamiltonian
H ¼
X
L1
i¼1
h
i;iþ1
þ h
L;1
; h
ij
¼
1
2
ðs
i
s
j
þ 1Þ½1
Bethe Ansatz 253
where s = (
x
,
y
,
z
) are Pauli matrices and L is the
length of the chain. Periodic boundary conditions are
imposed. However, open boundary conditions may
also be treated, along with the addition of magnetic
bulk and boundary fields. The z-components of each
of the spins are either up or down. Since the
z-component of the total spin commutes with the
Hamiltonian, the total number n of up spins serves as a
good quantum number. A state of the system can
therefore be conveniently described in terms of the
coordinates of all the up spins. Denote these coordi-
nates by x
i
,with1 x
i
L. The quantum number n
ensures that the Hamiltonian decomposes into L þ 1
sectors, each of size L choose n. The antiferromagnetic
ground state occurs in the largest sector.
The normalization of the Hamiltonian [1] is such
that its action is that of the permutation operator:
hji ¼ ji; hjþþi ¼ jþþi
hjþi ¼ jþi; hjþi ¼ jþi
½2
Diagonalization of Sectors
One can address the diagonalization of the sectors
for various cases.
Case 1: n = 0
Consider the case with all spins down. The
eigenstate is =ji, with H=L and,
thus, E = L is the trivial solution.
Case 2: n = 1
There are L states, with
¼
X
L
x¼1
aðxÞj ðxÞi ½3
where j (x)i is the state with an up spin at site x.
The aim is to find the amplitudes a(x). It is clear
that
Hj ðxÞi ¼ðL 2Þj ðxÞiþj ðx 1Þi
þj ðx þ 1Þi ½ 4
in the bulk (away from either boundary). Insertion
of [3] into H=E gives
EaðxÞ¼ðL 2Það xÞþaðx 1Þþaðx þ 1Þ½5
Substitution of spin waves a(x) = e
ikx
gives
E ¼ L 2 þ 2 cos k ½6
The boundary conditions are such that a(0) = a(L)
and a(L þ 1) = a(1); either gives e
ikL
= 1, from which
the L values of k follow.
Case 3: n = 2
Here the wave function can be written in terms of
the two flipped spins as
¼
X
x<y
aðx; yÞj ðx; yÞi ½7
It is to be emphasized that one is working in the
region with x < y. There are two cases to consider:
(1) y > x þ 1and(2) y = x þ 1. Consider the
interactions in the bulk. For (1) the action of the
Hamiltonian implies
Eaðx; yÞ¼ðL 4Þaðx; yÞþaðx 1; yÞþaðx þ 1; yÞ
þ aðx; y 1Þþaðx; y þ 1Þ½8
and for (2)
Eaðx; x þ 1Þ¼ðL 2Þ
aðx; x þ 1Þ
þ aðx 1; x þ 1Þþaðx; x þ 2Þ½9
The compatibility of these two equations requires that
2aðx; x þ 1Þ¼aðx; xÞþaðx þ 1; x þ 1Þ½10
which is known as the ‘‘collision’’ or ‘‘meeting’’
condition.
Some adjustments need to be made for spins
which get flipped at the boundaries. Looking at
[8] and [9] with x = 1andx = L, it is evident that
one can take
aðy; x þ LÞ¼aðx; yÞ½11
to restore the original ordering. The terms which
arise involve up spins at sites 0 and L þ 1. This
illustrates the periodic boundary condition.
We now assume (the Bethe ansatz) that
aðx; yÞ¼A
12
e
ik
1
x
e
ik
2
y
þ A
21
e
ik
2
x
e
ik
1
y
½12
Substitution of the ansatz [12] into [8] gives
E ¼ L 4 þ 2 cos k
1
þ 2 cos k
2
½13
Substitution of [12] into [10] gives
A
12
A
21
¼
1 2e
ik
1
þ e
iðk
1
þk
2
Þ
1 2e
ik
2
þ e
iðk
1
þk
2
Þ
½14
The three relations [11], [12], and [14] give the
Bethe equations
e
ik
1
L
¼
A
12
A
21
and e
ik
2
L
¼
A
21
A
12
½15
254 Bethe Ansatz