A four dimensional fermion determinant is presented as a path integral of the
exponent of a local five dimensional action describing constrained bosonic
system. The construction is carried out both in the continuum theory and in the
lattice model.Comment: 9 pages, latex, no figure

A.A. Slavnov

Coleman

Faddeev

Furuya

Gamboa-Saravi

Huerta

Lusher

Marino

Naon

Polyakov

Semenoff

Slavnov

Witten

Physics Letters B

English

arXiv

A four dimensional fermion determinant is presented as a path integral of the exponent of a local five dimensional action describing constrained bosonic system. The construction is carried out both in the continuum theory and in the lattice model

Slavnov, A A

CERN Document Server

SMI-3-95
Bosonization of fermion determinants
A.A.Slavnov 
Steklov Mathematical Institute, Russian Academy of Sciences,
Vavilov st.42, GSP-1,117966, Moscow, Russia
July 28, 1995
Abstract
A four dimensional fermion determinant is presented as a path integral of
the exponent of a local ve dimensional action describing constrained bosonic
system. The construction is carried out both in the continuum theory and in
the lattice model.
1 Introduction
The problem of bosonization of fermionic theories was studied by many authors (see
e.g. [1], [2], [3], [4], [5], [6]), mainly in the context of two-dimensional models. There
were also some attempts to extend this procedure to higher dimensions [7], [8], [9],
[10], but at present they lead only to partial success.
The problem of bosonization is of a particular importance for lattice gauge theori-
es as lattice QCD, because the fermionic degrees of freedom complicate numerical
simulations enormously.
Recently M.Lusher [11] proposed the algorithm for the approximate inversion of
the QCD fermion determinant replacing it by an innite series of bosonic determi-
nants. In the present paper I will describe an alternative approach which allows
to write the exact expression for the fermion determinant as a path integral of the
exponent of a local bosonic action. In my approach a four dimensional fermionic
system is replaced by a ve dimensional constrained bosonic one.
In the second section I present the construction for the continuum theory. In this
case the proof of equivalence will be given in the framework of perturbation theory
as the problem of nonperturbative denition of a continuum path integral is at
present beyond our possibilities. In the third section the corresponding procedure
will be given for the lattice QCD where the equivalence may be proven at the
nonperturbative level as well.
E-mail: slavnov@class.mian.su
1
2 Perturbative bosonization in the continuum the-
ory
We consider the following fermion determinant
det(D +m)n =
Z
expf−
nX
i=1
Z
d4x[  i(x)(D +m) 
i
(x)]gd  
i
d 
i
 (1)
where
D = (γ@ + igγA) (2)
A belongs to the Lie algebra of some compact group (SU(3) for QCD). Indices 
denote spinorial and colour components. The eq.( 1) describes n degenerate fermion
flavours interacting vectorialy with the Yang-Mills eld. To make contact with the
discussion of lattice models we assume that the four dimensional space is Euclidean
and the γ matrices ( = 1; 2 : : : 5) are Hermitean. As all the discussion in this
section will be restricted to perturbation theory all the equations are understood
as a series over g, and the Gaussian path integrals are dened axiomatically as in
ref. [12], [13]. Therefore we need not to bother about the convergence of the path
integrals and all the conclusions are valid (perturbatively) for any number of fermion
flavours. However in the next section where we deal with the nonperturbative proof
for lattice models we have to be more careful at this point and to make all the
integrals convergent we restrict our discussion to the case of even number of flavours.
For simplicity we take n = 2.
We present the determinant of the Dirac operator as the determinant of the
Hermitean operator by using the identity
det(D +m) = det[γ5(D +m)] (3)
Therefore
det(D +m)2 = det(γ5(D +m)γ5(D+m)) = det(−D
2 +m2) (4)
Being the square of the Hermitean operator the operator −D2 + m2 is positive
denite.
Let us introduce the following action depending on ve dimensional bosonic elds
(t; x)
S =
Z
dtd4xf(t; x)@t(t; x) + 

(t; x)[−D
2(x) +m2](t; x)g (5)
Here  is a small positive parameter which eventually will be put equal to zero. The
elds  have the same spinorial and colour structure as the elds  .
We claim that the determinant (1) (for n = 2) may be presented as the following
path integral:
det(D +m)2 = lim
!0
Z
expf−S + i
Z 1
0
dt
Z
d4x(x; t)(x) +
+i
Z 1
0
dt
Z
d4x(x)(x; t)gd

dd

d (6)
2
The elds (x) play a role of Lagrange multipliers imposing the constraintsZ 1
0
dt(t; x) =
Z 1
0
dt(t; x) = 0 (7)
The Green functions of the elds  dened by the action(5) are retarded:
~G(k; p) =
1
ik + p2 +m2
(8)
G(t; p) =
1
2
Z
expfiktg ~G(k; p)dk; G(t) = 0; t < 0 (9)
Integrating in the eq.(6) over ;  one gets
I = det(@t −D
2 +m2)−1 (10)Z
expf−
Z 1
0
dtds
Z
d4xd4y(x)[@t −D
2 +m2]−1(x; y; s− t)(y)gd
d
Due to the fact that the Green functions of the elds  are retarded, for any  dif-
ferent from zero the rst factor in the eq.( 10) is equal to one. Indeed a perturbative
expansion of this determinant generates the diagrams of the type
n 
Z
dk
F (k; p1 : : : pn)
(ik + p21 +m
2) : : : (ik + p2n +m
2)
(11)
where F is some polynomial over pi; k. Note that the elds A do not depend on s,
therefore the diagrams generated by eq.(10) are the cycles with zero fth component
of momenta. The integral over k is convergent and all the poles are in the upper
complex half plane. Closing the integration contour in the lower half plane one sees
that the integral is equal to zero. In fact one has to be slightly more careful at this
point, as due to momentum conservation all these diagrams are multiplied by the
singular constant factor  (0). To give a precise meaning to this expression some
infrared cut-o must be introduced to the eq. ( 5). In the next section we shall
show that a nite lattice may serve this purpose.
The second factor is not singular in the limit ! 0 and one can calculate it by
putting  = 0. Let us prove it in some more details.
Expansion of the exponent in the eq.(10) over g generates the terms of the formZ 1
0
ds
Z 1
0
dt
Z
dk
expfik(s− t)gP (k; p1 : : : pn)
(ik + p21 +m2) : : : (ik + p2n +m
2)
(12)
where P is some polynomial over k; pi. Integrating over s and t one getsZ
dk
(2− e−ik − eik)P (k; pi)
(k − i)2(ik + p21 +m2) : : : (ik + p2n +m
2)
(13)
The pole at k = 0 is spurious. It is compensated by the zero of the nominator.
So we are free to dene the roundabout of this pole arbitrarily. We choose the
prescription k ! k−i. With this prescription the part of the integrand proportional
3
to (2−expf−ikg) gives zero contribution as in this case we may close the integration
contour in the lower half plane and all the poles are situated in the upper half plane.
The last term, proportional to expfikg is calculated by closing the contour in the
upper half plane. The integral is equal to the sum of the residues at the poles at
k = 0; k = i−1(p2i + m
2). The contribution of the poles at k = i−1(p2i + m
2)
vanishes at ! 0. The contribution of the pole at k = 0 gives
P (k; pi)
(ik + p21 +m
2) : : : (ik + p2n +m
2)
jk=0 (14)
Therefore the integral (10) acquires a form
I =
Z
expf−
Z
(x)[−D
2 +m2]−1(x; y)(y)dxdy +O()gd

d (15)
Integrating over ;  we get
I = det(−D2 +m2) +O() (16)
Therefore
lim
!0
I = det(D +m)2 (17)
3 Bosonization on the lattice
In this section we generalize the construction given above to the case of lattice
models, where the proof of equivalence can be given without any references to per-
turbation theory.
We again present the determinant of the Dirac operator as the determinant of
the Hermitean operator by using the identity
det(D^ +m) = det[γ5(D^ +m)]; D^ = γD (18)
In eq.( 18) D is the lattice covariant derivative
D (x) =
1
2a
[U+ (x) (x+ a)− U(x) (x− a] (19)
U is a lattice gauge eld. We consider a nite lattice with periodic boundary
conditions.
As it has been already mentioned, to provide the positivity of the determinant
we consider the case of two degenerate fermion flavours interacting vectorialy with
the Yang-Mills eld. We shall not deal explicitely with the problem of fermion
doubling. All the reasonings are trivially extended to the model improved by adding
for example the Wilson term.
It is convinient to present the fermion determinant in the following form
det[γ5(D^ +m)]
2 =
Z
expfa4
X
x
 (x)(D^2 −m2) (x)gd  d (20)
4
We shall prove that in analogy with the continuum case the integral over fermionic
elds  can be replaced by an integral of a ve dimensional lattice bosonic action.
The spatial components x are dened as above. The fth component t to be dened
on the one dimensional lattice of the length L with the lattice spacing b:
L = 2Nb; −N < n  N (21)
We choose b in such a way that b << a and in the continuum limit
2Nb2 = Lb! 0 (22)
The key ingredient of the construction presented in the preceeding section was the
using of retarded Green functions for the elds  providing triviality of the deter-
minant in the eq.(10). This phenomenon is a continuum analog of the well known
property of a triangular matrix: the determinant of a triangular matrix is equal to
the product of the diagonal elements. Therefore to achieve the same goal in the
lattice case we choose the lattice derivative with respect to the fth coordinate t in
the form of a triangular matrix. More precisely we are going to prove the following
equality Z
expfa4
X
x
 (x)(D^2 −m2) (x)gd  d = (23)
= lim
!0;b!0
Z
expfa4b
NX
n=−N+1
X
x
[
n+1(x)− 

n(x)
b
n(x) + 

n(x)(D^
2 −m2)n(x)−
−
i
p
L
(n(x)(x) + 
(x)n(x))]gd

ndnd
d:
Here n(x) and (x) are bosonic elds which carry the same spinorial and colour
indices as the elds  (x). The elds n(x) satisfy free boundary conditions i.e.
n = 0; n  −N; n > N (24)
The operator −D^2 +m2 is Hermitean and can be diagonalized by a unitary trans-
formation. As it does not depend on t this transformation will make the exponent
in the r.h.s. of eq.( 23) diagonal with respect to all variables except for t. After this
transformation the r.h.s. of eq.( 23) acquires the form
I = lim
!0;b!0
I(; b)
I(; b) =
Z
expfb
NX
n=−N+1
X

[
n+1 − 

n
b
n − 

n B
n
−
i
p
L
(n 
 + n 

n)]gd

ndnd
d; N+1 = 0 (25)
Index  refers now to the eigenstates of the operator −D^2 + m2, B being the
corresponding eigenvalues. Obviously B > 0.
5
The integral ( 25) is convergent as the real part of the action in the exponent is
positive. It is easy to see by rewriting the action in terms of Fourier components:
S =
1
2
Z 
b
−
b
f~(k)[−(e−ikb − 1)b−1 +B] ~(k)] +
+
i
2
p
L
(e−iknb ~(k) + eiknb ~(k))gdk (26)
The real part is
ReS =
1
2
Z 
b
−
b
f~(k)[−(coskb− 1)b−1 +B] ~(k)]dk (27)
As  < 0 and B > 0, ReS > 0.
Performing in the eq.( 25) the integration over n one gets
I(; b) = det(C−1)
Z
expf
−b
L
X

NX
n;m=−N+1
(C)−1mn
gdd (28)
Where C(k) is the kernel of the quadratic form in the eq.( 25).
As follows from the eq.( 25)
det(C) =
Y

det(C) (29)
In the coordinate space C is a triangular matrix with the diagonal elements
 (+Bb) (30)
Therefore
det(C) = expf
X

ln( +Bb)
2Ng (31)
Separating the constant term one gets
det(C) = expf2Nb−1
X

B +O(Lb)g = (32)
= expf−1LTr [−D^2 +m2] +O(Lb)g
Using the explicit form of the operator D one can easily verify that Tr [−D^2 +m2]
is a nonessential constant. It follows also from the fact that the trace of a local
operator is local. It has to be gauge invariant and a polynomial of the second order
in the elds A. The only possible solution is a constant. So we can include det(C−1)
into normalization constant.
To get an explicit form of the remaining terms it is sucient to nd the stationary
point of the exponent in the eq.( 25). The corresponding classical equations look as
follows
−(n+1 − 

n )b
−1 +Bn + iL
−1
2 = 0; n 6= N (33)
(n − 

n−1)b
−1 +Bn + iL
−1
2 = 0; n 6= −N + 1
6
−N b
−1− N B
 − iL−
1
2 = 0 (34)
−N+1b
−1 + −N+1B
 + iL−
1
2 = 0
for small b eq.s( 33) may be approximated by the dierential equations
−@t
 +B + iL−
1
2 = 0 (35)
@t
 +B + iL−
1
2 = 0
whereas eq.s( 34) play the role of boundary conditions:
(
L
2
) = 0; (−
L
2
) = 0 (36)
The solution of these eq.s is
(t) = −
i
p
LB
(1− expfB−1(t−
L
2
)g) (37)
(t) = −
i
p
LB
(1− expf−B−1(t+
L
2
)g)
Substituting these solutions to the eq.( 25) we get in the limit b! 0
lim
b!0
I(; b) =
Z
expf−
Z L
2
−L
2
X


BL
[1− expf
B

(t−
L
2
)g]dtgdd = (38)
=
Z
expf−
X

(B)−1[1−

BL
[1 + expf−B−1Lg]]gdd
In the limit ! 0 this equation reduces to
I = lim
!0;b!0
I(; b) = expf−
X

(B)−1gdd = det(−D^2 +m2) (39)
The equality (23) is proven. The nal recipe is the following: Any gauge invariant
observable may be expressed as a purely bosonic path integral of the form
Z = lim
!0;b!0
Z
expfa4b
NX
n=−N+1
X
x
[
n+1(x)− 

n(x)
b
n(x) +
+n(x)(D^
2 −m2)n(x)−
i
p
L
(n(x)(x) + 
(x)n(x))] +
+L(U) + s:t:gdndnd
d (40)
where L(U) is the lattice Yang-Mills action and s.t. stands for the source term
depending on the elds U; n; n. All the elds are bosonic and the integration
goes over the elds n; n dened on the one-dimensional lattice with length L and
free boundary conditions.
7
4 Discussion
In this paper we showed that a four dimensional fermion determinant can be written
as a path integral of the exponent of a ve dimensional local bosonic action. In the
same way one can present a two dimensional fermion determinant as a path integral
for three-dimensional bosonic theory. In the case of lattice models this procedure
leads to a well dened bosonic path integral. No numerical simulations in this
approach have been tried so far and it would be very important to see how the
method works in practical calculations. In this case one need not of course to take
the limit ! 0; b! 0, but the following conditions b << a; b << ; b << L−1
must be respected.
Acknowledgements.
This work was started while the author was visiting Universite Libre de Bruxelles
and The Institute for Theoretical Physics of University of California at Santa Bar-
bara, participating in the Workshop on Chiral symmetry on the lattice organised
by M.Creutz. I am grateful to Marc Henneaux for hospitality at ULB and useful
discussions. I also wish to thank the sta of ITP for hospitality and all the partici-
pants of the chiral workshop for numerous stimulating discussions. I am indebted to
S.A.Frolov for helpful comments. This researsh was supported in part by Interna-
tional Science Foundation and Russian Governement under grant MNB000, Russian
Basic Research Fund under grant 94-01-00300a and the National Science Foundation
under Grant PHY89-04035.
References
[1] S.Coleman, Phys.Rev.D11 (1975) 2088.
[2] A.Polyakov and P.Wigman, Phys.Lett.B131 (1983) 121; 141 (1984) 223.
[3] E.Witten, Comm.Math.Phys.92 (1984) 455.
[4] R.Gamboa-Saravi, F.Shaposnik and J.Solomin, Nucl.Phys.B185 (1981) 239.
[5] K.Furuya, R.Gamboa-Saravi and F.Shaposnik, Nucl.Phys.B208 (1982) 159.
[6] C.Naon, Phys.Rev. D31 (1985) 2035.
[7] G.W.Semeno, Phys.Rev.Lett.61 (1988) 817.
[8] M.Lusher, Nucl.Phys. B326 (1989) 557.
[9] E.C.Marino, Phys.Lett. B263 (1991) 63.
[10] L.Huerta,F.Zanelli, Phys.Rev.Lett. 71 (1993) 3622.
[11] M.Lusher Nucl.Phys. B418 (1994) 637.
[12] A.A.Slavnov, Theor.Math.Phys. 22 (1975) 177.
8
[13] L.D.Faddeev, A.A.Slavnov,Gauge Fields. Introduction to Quantum Theory. Second
edition. Moscow, Nauka, 1988. English translation ed.Addison-Wesley 1990
9


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