Reﬂection Positivity of Free Overlap Fermions
Yoshio Kikukawa1 and Kouta Usui2, 3
1
Institute of Physics, the University of Tokyo, Tokyo 153-8902, Japan
Department of Physics, the University of Tokyo, Tokyo 113-0033, Japan
3
Institute for the Physics and Mathematics of the Universe (IPMU), the University of Tokyo, Chiba 277-8568, Japan(Dated: October 30, 2010)2

It is shown that free lattice fermions deﬁned by overlap Dirac operator fulﬁll the Osterwalder-Schrader
reﬂection positivity condition with respect to the link-reﬂection. The proof holds true in non-gauge models
with interactions such as chiral Yukawa models.
I.

INTRODUCTION

To give the constructive deﬁnition of a relativistic quantum
ﬁeld theory through a lattice model, it is desirable for thelattice model to satisfy several fundamental requirements suchas locality, reﬂection positivity and hypercubic rotation andtranslation symmetry. Locality is believed to assure that thecontinuum limit of the lattice model is universal and the model
belongs to the same universality class of the target continuum
local ﬁeld theory. Reﬂection positivity garantees that thelattice model is a consistent quantum mechanical system satisfying unitarity. Hypercubic rotation and translation symmetry
is hopefully expected to result in the recovery of the Euclideangroup symmetry in the continuum limit. These properties ofthe lattice model should be established rigorously, if possible,
before the applications to “ﬁrst-principle” computation.
In the lattice model, it is also desirable to keep the important
symmetries of the target continuum ﬁeld theory. Gauge
symmetry can be preserved by introducing link variables.
Chiral symmetry, despite of the infamous no-go theorem[1, 2, 3], can be also preserved by adopting lattice Dirac
operators which satisfy the Ginsparg-Wilson (GW) relation[4, 5, 6, 7, 8, 9]. For the overlap Dirac operator [5, 6],
a gauge-covariant solution to the GW relation which has
been derived in the ﬁve-dimensional domain wall approach[10, 11, 12], a rigorous proof of locality has been given undera certain condition for admissible gauge-link variables [13].
Even the applications to large scale numerical simulation oflattice QCD have been attempted, obtaining clear numericalevidences for the spontaneous chiral symmetry breaking in
QCD [14, 15, 16].
However, for the GW fermions, reﬂection positivity is
not fully understood yet [17, 18, 19]. The situation should
be compared to the case of Wilson fermions, for which
the rigorous proofs of reﬂection positivity have been given[20, 21, 22]. Then, one might think that it is still somewhat
premature to refer the above numerical applications as “ﬁrstprinciple” computations[35].
In this letter, we will examine the reﬂection positivity oflattice fermions deﬁned through overlap Dirac operator. It will
be shown rigorously that free overlap Dirac fermions fulﬁll thereﬂection positivity with respect to the link-reﬂection. The
proof will be extended to the cases of Majorana and Weyl
fermions. In ref [17], Lüscher discussed the unitarity property
of free overlap Dirac fermion by investigating the positivity
through the spectral representation of free propagator andconcluded that free overlap Dirac fermion has a good unitarity
property. Our direct proof of the reﬂection positivity given
here is consistent with this observation. Our proof will bealso extended to the non-gauge models with interactions suchas chiral Yukawa models. For gauge models, however, a proofof reﬂection positivity, if any, seems to be more involved andwe will leave it for future study.

II.

REFLECTION POSITIVITY

Reﬂection positivity is a sufﬁcient condition for reconstructing a quantum theory in the canonical formalism, i.e. theHilbert space of state vectors and the Hermitian Hamiltonian
operator acting on the state vectors, from the lattice model
deﬁned in the Euclidean space[36]. Let us formulate thereﬂection positivity condition for lattice Dirac fermions. The
cases of Majorana and Weyl fermions will be discussed later.
We assume a ﬁnite lattice Λ = [−L + 1, L]4 ⊂ Z4 in thelattice unit a = 1, and impose anti-periodic boundary condition in the time direction, and periodic boundary conditionsin the space directions. The fermionic action is deﬁned in thebilinear formX
A(ψ̄, ψ) =
ψ̄(x)DL ψ(x),(1)
x∈Λwith a lattice Dirac operator DL [37]. The kernel of the Dirac
operator should be written as
X
DL (x, y) =(−1)n0 D(x + 2nL, y), x, y ∈ Λ, (2)n∈Z4where D(x, y) is the kernel of the Dirac operator in theinﬁnite lattice Z4 . The quantum theory is then completely
characterized by the expectational functional deﬁned by thefermionic path-integration:
Z
1
D[ψ]D[ψ̄] eA(ψ̄,ψ) F (ψ̄, ψ),(3)〈F 〉 :=
Z
where the Grassmannintegration Rfor each ﬁeld variable isR
speciﬁed as dψα (x)ψα (x) = 1, dψ̄α (x)ψ̄α (x) = 1, andthe functional measure is deﬁned byY(4)
D[ψ]D[ψ̄] :={dψα (x)dψ̄α (x)}.
x∈Λ;α=1,2,3,4

2
The reﬂection positivity condition — a condition on thisexpectational functuonal — is formulated as follows: let us
deﬁne time reﬂection operator θ which acts on polynomials of
the fermionic ﬁeld variables by the relations¡¢T(5)
θ(ψ(x)) = ψ̄(θx)γ0T

θ(ψ̄(x)) = (γ0 ψ(θx))θ(αF + βG) = α∗ θ(F ) + β ∗ θ(G)θ(F G) = θ(G)θ(F ),

(6)
(7)
(8)

where we denote θ(t, x) = (−t + 1, x) and F, G are arbitrary
polynomials of fermionic ﬁelds and * means complex conjugation. Let Λ± ⊂ Λ be the sets of sites with positive or nonpositive time respectively. Let A± be the algebra of all thepolynomials of the ﬁelds on Λ± , and A on Λ. Then one says
the theory is reﬂection positive if its expectation 〈·〉 : A → C
satisﬁes〈θ(F+ )F+ 〉 ≥ 0 for ∀ F+ ∈ A+ .

(9)

A popular choice of lattice Dirac operator is the Wilson
Dirac operator,
¾
X ½1
1 †
†
Dw =
γµ (∂µ − ∂µ ) + ∂µ ∂µ ,(10)22µ=0,1,2,3
and in this case, the rigorous proofs of the reﬂection positivity
have been given [20, 21, 22]. The proofs cover the casewith gauge interaction. Therefore, the use of Wilson Diracfermions in numerical applications has a completely soundbasis[38]. Here we consider the overlap Dirac operator
Ã
!
1
1
D=
1+X√
(11)
, X = Dw − m,2
X †X
for 0 < m ≤ 1. This lattice Dirac operator describes a single
massless Dirac fermion and satisﬁes the GW relation, γ5 D +
Dγ5 = 2Dγ5 D. Although the action is necessarily non ultralocal [27], the free overlap Dirac fermion indeed satisﬁes thereﬂection positivity condition, as will be shown below.

III. PROOF OF REFLECTION POSITIVITY OF OVERLAPDIRAC FERMION

Thirdly, let us call P the set of all polynomials of the formPj θ(F+j )F+j in a ﬁnite summation, where F+j ∈ A+ .
Although the above deﬁnition of P works well for the
proof of the Wilson fermion, it is not enough for the proofof the overlap fermion. In our case of the overlap fermion,one
P needs to consider not only ﬁnite summations of the formj θ(F+j )F+j , but also inﬁnite summations or integrations
likeZ
ds θ(F (s))F (s) = lim

N →∞

N
X

θ(F (sk ))F (sk )∆sk , (14)k=1where the integration is deﬁned as a limit of a ﬁnite Riemaniansummation. (see also eq. (32) or (35)). To this end, we
consider P̄, the closure of P. The closure P̄ contains not
only elements of the original P, but also all the limit points of
conversing sequences in P. That is,F ∈ P̄

⇔∃{Fn }∞
n=1 ∈ P : lim Fn = F.
n→∞Here, the sequence {Fn }n ⊂ A is deﬁned to be convergent
to some F ∈ A, if any coefﬁcient in Fn converges to thecorresponding coefﬁcient in F as a complex number [39].
Note that with respect to this deﬁnition of convergence, thelinear operation, the product operation in A, and the linear
mappings 〈·〉0 , 〈·〉 : A → C are all continuous functions, i.e.
if Fn → F , Gn → G, thenαFn + βGn → αF + βG,〈Fn 〉 → 〈F 〉 ,Fn Gn → F G,
〈Fn 〉0 → 〈F 〉 .

This 〈·〉0 deﬁnes a linear function from A into C. Second, we
decompose the lattice action A into the following three parts :A = A+ + A− + ∆A

(13)

where A+ ∈ A+ , A− ∈ A− , and ∆A is the part of theaction which contain both positive and negative time ﬁelds.

(16)
(17)

Now, we note the fact that the following four statements(i)-(iv) imply the reﬂection positivity:(i) If F, G belong to P̄ then F G also belongs to P̄.
(ii) For all F ∈ P̄, 〈F 〉0 ≥ 0.
(iii) θ(A+ ) = A− .
(iv) ∆A ∈ P̄.
In fact, from these statements, it follows that
­ A
®
­
®
e θ(F+ )F+ 0 = eA+ +A− +∆A θ(F+ )F+ 0
DE= eA+ +θ(A+ )+∆A θ(F+ )F+
0
*
+
=

≥0

θ(eA+ )eA+ e∆A θ(F+ )F+|{z} | {z }∈P̄ (by (i),(iv))

To prove the reﬂection positivity, we need some additionaldeﬁnitions and notations. First, let us denote
Z〈F 〉0 := D[ψ]D[ψ̄]F (ψ̄, ψ).
(12)

(15)

∈P̄

0

(18)
for arbitrary F+ ∈ A+ . ­Considering
the special case where®
F+ = 1 ∈ A+ , we have eA 0 ≥ 0. Hence, we obtain­ A®
e θ(F+ )F+ 0〈θ(F+ )F+ 〉 =≥ 0.
〈eA 〉0(19)

Therefore the proof is reduced to showing these four statements (i)-(iv).
Next, we will give the proofs of the statements (i)-(iv). The
statement (i) follows from the similar statement with P, which

3
has been proved for the Wilson case [20] . In fact, let F, G ∈P̄. Then, there exist sequences {Fn }n and {Gn }n in P suchthat
Fn → F,Gn → G.

(20)

From the continuity of the product operation in A (see (16)),
we getFn Gn → F G.

(21)

Since Fn , Gn ∈ P, Fn Gn ∈ P. Therefore F G is the limit of
the sequence {Fn Gn }n ⊂ P, which means that F G ∈ P̄.
To show the statement (ii), one should refer to the deﬁnitionof fermionic integration measure. With the deﬁnition (4), it is
sufﬁcient to consider F+ ∈ A+ of the formYF+ ={ψ̄α (x)ψα (x)} ∈ P,

whether x0 − y0 > 0 or x0 − y0 < 0, we choose the contours[iE1 , i∞] or [−iE1 , −i∞], respectively, to obtain¯¯
D(x, y)¯
x0 −y0 >0
Z
Z ∞
d3 p
dE −E(x0 −y0 ) ip·(x−y)
X(iE, p)p=ee3(2π) E1 2π−X † X(iE, p)(27)¯¯
D(x, y)¯
x0 −y0 <0
Z
Z ∞
d3 p
dE E(x0 −y0 ) ip·(x−y) X(−iE, p)
p
=
e
e
.
(2π)3 E1 2π−X † X(iE, p)(28)
where E1 is the edge of the cut coming from the square root,
and is determined by the relations(22)

X † X(iE1 , p) = 0,x∈Λ+ ;α=1,2,3,4

for which one can see
Z
4
D[ψ]D[ψ̄] θ(F+ )F+ = {det(γ02 )}16L = 1 ≥ 0.

(23)

Therefore, one concludes that for arbitrary F ∈ P, 〈F 〉0 ≥0. Take arbitrary F ∈ P̄. Then there exists a converging
sequence {Fn }n such that Fn → F . From the continuity of〈·〉0 (see (17)), we obtain〈F 〉0 =

DE
lim Fnn→∞0= lim 〈Fn 〉0 ≥ 0.
n→∞

(24)

∓γ0 X(±iE, p) = Y±† Y± (E, p)

X X

x∈Λ+ y∈Λ+=X

Y± (E, p) = −3Xl(E, p) sin pk

where W (E, p) ="l(E, p) =†ψ̄(θy) γ0 DL(y, x)γ0 ψ(θx)
|
{z
}

X

(E ≥ E1 ).

(30)

In fact, it is not difﬁcult to check that Y± (E, p) are given byx∈Λ+ y∈Λ+=(29)

In this spectrum representaion of D, it is very crucial tonotice the fact that ∓γ0 X(±iE, p) (E ≥ E1 ) are positivedeﬁnite matrices and there exist matrices Y± (E, p) such that

W (E, p)k=1

The statement (iii) can be shown by using the property of†the overlap Dirac kernel: DL(x, y) = γ0 DL (θx, θy)γ0 . In
fact, one gets´
X X ³θ(A+ ) =θ ψ̄(x)DL (x, y)ψ(y)E1 > 0.

γk ∓ iW (E, p)γ0 + il(E, p),2l(E, p)P3

k=1 (1 − cos pk ) + 1 − cosh E − m and1sinh E×P3
2
2 k=1 sin pk /W (E, p)2 + 1
s
Ã
! 12
P3
2
2
sin
p
+
W
(E,
p)
k
 .
1 + 1 − k=1
sinh2 E=D(θy,θx)′ψ̄(x )DL (x′ , y ′ )ψ(y ′ )From the equations (2), (27) and (28), we ﬁnd the spectrumx′ ∈Λ− y ′ ∈Λ−= A− .

(25)

To show the statement (iv) ∆A ∈ P̄, we use a spectral representation of DL (x, y). To derive the spectral representationof DL , we ﬁrst Fourier transform the overlap Dirac operator
kernel D(x, y) in the inﬁnite volume:¯¯D(x, y)¯Z
x0 ̸=y0=X(p0 , p)
d4 p ip·(x−y)
pe, (26)(2π)4
2 X † X(p0 , p)PP
where X(p0 , p) =µ iγµ sin pµ +µ (1 − cos pµ ) − m.
Then, we change the p0 integration region, [−π, π], to thecontours along the imaginary axis in the complex p0 plane by
Cauchy’s integration theorem, as shown in FIG. 1. Depending

Im

6
?
r iE1
−π

0

Im

↑∞

6
π

- Re

6
−π

π

0

r −iE
6

1

- Re

?
↓ −∞

FIG. 1: Complex integration contours

4
representation of DL (x, y) as follows: putting V = 1/(2L)3 ,¯
X Z ∞ dE
11¯DL (x, y)¯=−2ELVx0 ̸=y0E1 2π 1 + ep−E|x0 −y0 | ip·(x−y)

×e
+

e

XZ ∞

p

X(ϵiE, p)−X † X(iE, p)

−2EL

dE e
1
−2EL V
2π
1
+
e
E1

p

where ϵ is deﬁned as the sign of x0 − y0 , and the spacial
momentum pk runs over pk = nk π/L, (−L ≤ n ≤ L) in
the above summation. In (31), the ﬁrst term becomes D(x, y)in the limit L → ∞, and the second term represents a ‘ﬁnite
lattice effect’ which vanishes in the limit L → ∞. The latteris the contribution of the wrong-sign-energy modes and theminus sign appearing in front of X(−ϵiE, p) comes from theanti-periodicity in the time direction, which is required for the
positivity, as will be seen.
From these observations, now we can show that ∆A ∈ P̄:for the term with x0 > 0, y0 ≤ 0 (in this case ϵ = 1), we
obtainX X
ψ̄(x)DL (x, y)ψ(y)=−X Z ∞ dE 1
p

E1

"

2π V

#
CE,p θ(CE,p ) + DE,p θ(DE,p ) ,(32)

where CE,p and DE,p are deﬁned by
CE,p
r
=

X
1
ψ̄(x)γ0 Y˜+ (E, p)† e−Ex0 eip·x , (33)
1 + e−2EL
x∈Λ+

DE,p
s
=

X
e−2EL
ψ̄(x)γ0 Y˜− (E, p)† eEx0 eip·x ,
−2EL
1+e

(34)

x∈Λ+

with Y˜+ (E, p) = Y+ (E, p)/(−X † X(iE, p)) 4 . The overallminus sign in the r.h.s. of (32) results from (31) by using (30).
This minus sign is canceled after exchanging the order of theGrassmann products in (32), and we see that this term belongs
to P̄. Similarly, for the term with x0 ≤ 0, y0 > 0 (in this caseϵ = −1), we obtain
X X
ψ̄(x)DL (x, y)ψ(y)1x∈Λ− y∈Λ+=

X Z ∞ dE 1
p

E1

2π V

′
CE,p
r

=

"

#
′
′
′
′
θ(CE,p)CE,p+ θ(DE,p)DE,p, (35)X
1
Y˜− (E, p)ψ(y)e−Ey0 e−ip·y ,
−2EL
1+e

(36)

X
e−2EL
Y˜+ (E, p)ψ(y)eEy0 e−ip·y .
−2EL
1+e

(37)

y∈Λ+

′
DE,p
s

=

−X(−ϵiE, p),
× eE|x0 −y0 | eip·(x−y) p−X † X(iE, p)(31)x∈Λ+ y∈Λ−′′
where CE,pand DE,p
are deﬁned byy∈Λ+

In this case we do not need to exchange the order of the
product, and we immediately see that this term belongs toP̄. Thus we obtain ∆A ∈ P̄ and complete the proof of thereﬂection positivity in the Dirac case.

IV. MAJORANA AND WEYL FERMIONSThe above proof for Dirac fermion can be extended for Majorana fermions[28, 29] and Weyl fermions[30, 31, 32, 33].
In the case of Majorana fermion, the anti-ﬁeld ψ̄ is the charge
conjugation of the ﬁeld ψ : ψ̄ ≡ ψ T C, where C is the charge
conjugation matrix satisfying Cγµ C −1 = −γµT , Cγ5 C −1 =the path-integral
γ5T , C † C = 1, C T = −C. Accordingly,Q
measure reduces to D[ψ] := s x∈Λ;α {dψα (x)}, where asign factor s(= ±1) is introduced for later convenience. This
Majorana-reduction does not contradict with the deﬁnition oftime reﬂection θ, because both (5) and (6) imply θ(ψ(x)) =Cγ0 ψ(θx). Then, the conditions (iii) A− = θ(A+ ) and (iv)
∆A ∈ P̄ follow immediately. The propery (ii) of P̄ also
holds ture by theR fact that one can always choose the sign
4
factor s so thatD[ψ]θ(F+ )F+ = {det(Cγ0 )}8L > 0
Q
for F+ =x∈Λ+ ;α=1,2,3,4 {ψα (x)}. Thus the reﬂection
positivity (9) follows from the conditions (i), (ii), (iii) and (iv)also for the Majorana case.
For Weyl fermion, we deﬁne the chiral components by¶µ¶µ1 ∓ γ51 ± γ̂5
ψ(x), ψ̄± (x) = ψ̄(x),ψ± (x) =2
2(38)
where γ̂5 = γ5 (1 − 2D). We adopt, for simplicity, the chiralbasis for gamma matrices in which γ5 = σ3 ⊗ 1, γ0 = σ1 ⊗ 1,
and denote the spinor indices as α+ = {1, 2}, α− = {3, 4}.
Then ψ̄±α∓ (x) = ψ̄α∓ (x). The action of the Weyl fermion isP
given by A(±) = x ψ̄± (x)DL ψ± (x), and the Dirac fermion
action (1) decomposes as A = A(+) + A(−) . To deﬁne thepath-integral for the Weyl fermion, we introduce the chiral
bases
i
i
i
{v±
(x) | γ̂5 v±(x) = ±v±(x); i = 1, · · · , n± },

(39)

4
where
P i n± i= 2(2L) , and expand the ﬁelds as ψ± (x) =
v
(x)c
.
The
path-integral
measure is then deﬁned by±i ±D[ψ± ]D[ψ̄± ] =

Y
i

dci±

Y
x∈Λ;α∓dψ̄α∓ (x),

(40)

5
and the path-integral measure of Dirac fermion (4) is factorized as D[ψ]D[ψ̄] = J D[ψ+ ]D[ψ̄+ ]D[ψ− ]D[ψ̄− ] where theJacobinJ[ψα (x); ci+ , cj− ]= | v+α (x)1 · · · v+α (x)n+ v−α (x)1 · · · v−α (x)n− |(41)
can be set to unity by choosing the chiral basis vectors appropriately. The expectational functional for the left-handed
Weyl fermion is deﬁned by
Z
(−)
1
(−)
〈F 〉
:= (−)
D[ψ− ]D[ψ̄− ] eA (ψ̄− ,ψ− ) F (ψ̄− , ψ− ).
Z(42)Because of the factorization properties of the action and thepath-integral measure, we note the identity­®(−)­®(−) (+)
F (ψ̄− , ψ− )= F (ψ̄− , ψ− )〈1〉­®
= F (ψ̄− , ψ− ) .
(43)
In this setup, we can formulate the reﬂection positivity
condition for the Weyl fermion as follows. We deﬁne the time
reﬂection operator θ for the left-handed ﬁelds asθ(ψ−α− (x)) = {ψ̄− (θx)γ0 }α− = ψ̄−α+ (θx),(44)θ(ψ̄−α+ (x)) = {γ0 ψ− (θx)}α+ = ψ−α− (θx),(45)

where α+ = α− − 2. And let A± (−) be the algebra of all
the polynomials of the left-handed ﬁeld components ψ−α− (x)and ψ̄−α+ (x) on Λ± . Then one can show〈θ(F+ )F+ 〉

(−)

≥0

for

∀

(−)
F+ ∈ A + .

(46)

Note that, in this formulation of the reﬂection positivity, theﬁeld components ψ−α+ (x) are completely excluded fromobservables.
To prove (46), we note the fact that the expectational
functional for the left-handed Weyl fermion (42) is simply
related to the expectational functional for the Dirac fermion
(−)
(−)
(+)
(3) by 〈F 〉= 〈F 〉 〈1〉= 〈F 〉 for F (ψ̄− , ψ− ).
Moreover, since
½µ
¶
¾
½µ
¶
¾
1 − γ̂5
1 − γ5
D−1
=
D−1
,
2
2
α− ,α+
α− ,α+(47)
one can show®
­®(−) ­= F (ψα− , ψ̄α+ ) ,F (ψ−α− , ψ̄−α+ )(48)
by performing the Wick contructions explicitly. Then, (46)follows immediately from the reﬂection positivity condition(9) for the overlap Dirac fermion.
V. REFLECTION POSITIVITY OF CHIRAL YUKAWA
THEORYIn this section, we consider the case with an interaction —chiral Yukawa model. Chiral Yukawa model is deﬁneded by

the action[9]Xnλ0A=ψ̄Dψ − φ∗ ∂µ† ∂µ φ − m20 φ∗ φ − (φ∗ φ)2 − 2χ̄χ
2
x∈Λ
o
©1
ª
1
+ g0 (ψ̄ + χ̄) (1 − γ5 )φ + (1 + γ5 )φ∗ (ψ + χ) ,2
2
(49)
where ψ is a Dirac ﬁeld, χ is an auxiliary Dirac ﬁeld, andφ is a complex scalar ﬁeld. In this case, we deﬁne the ﬁeld
algebra A(Y) of the chiral Yukawa theory as the set of all thepolynomials F (ψ, χ, φ) of fermionic ﬁelds ψ and χ whose
coefﬁcients are complex valued continuous (not necessarily
holomorphic) functions of bosonic ﬁeld conﬁguration φ, with(Y)
converging expectation value 〈F 〉 deﬁned through the path
integration as usual:
Z
1
(Y)
〈F 〉
:= (Y)
D[path] eA(ψ,χ,φ) F (ψ, χ, φ) < ∞.
Z
(50)
Here, D[path] stands for the path integration measure,D[path] = D[ψ]D[ψ̄]D[χ]D[χ̄]D[φ]D[φ∗ ].

(51)

Note that all the polynomials of bosonic ﬁeld conﬁguration
belong to A(Y) .
The θ operation for the fermionic ﬁelds ψ, χ is the same asin the free case (5)(6). For the bosonic ﬁeld φ, the θ reﬂection
is deﬁned as
θφ(x) := φ(θx).

(52)

For F ∈ A(Y) of the form F (ψ, χ, φ) = f (φ)M (ψ, χ) with fbeing a continuous function of {φ(x)}x∈Λ and M (ψ, χ) somemonomial of {ψα (x), ψ̄α (x), χα (x), χ̄α (x)}x∈Λ , we deﬁneθ(F )(ψ, χ, φ) = f ∗ (θφ)M † (θψ, θχ),(53)

where M † means the monomial whose order of the Grassmann product is reversed in the original M . We extend the θoperation for arbitrary F ∈ A(Y) by anti-linearity. Then, thereﬂection positivity of this chiral Yukawa model is deﬁned in
the same way as in the free overlap fermion case :〈θ(F+ )F+ 〉

(Y)

≥ 0,

for ∀ F+ ∈ A+ .
(Y)

(54)

We will prove the reﬂection positivity of the chiral Yukawa
model in the same manner as in the free fermion case, basedon the statements (i)-(iv). The statement (i) clearly holds true.
(Y)
In the statement (ii), we deﬁne the expectation 〈·〉0 for the
Yukawa model as
Z
(Y)
(55)
〈F 〉0 := D[path] F (ψ, χ, φ).
For this deﬁnition to make sense, F should be a special element in A(Y) so that the right hand side of (55) is convergent.
Let B be the subset of A(Y) whose elements are integrablewith respect to the above 〈·〉0 -measure. Note that if F belongs6
to A(Y) , eA F belongs to B because of the rapidly decreasing
property of the bosonic weight. Then, the statement (ii) should
be rephrased as : (ii) For all F ∈ P̄ ∩ B, 〈F 〉0 ≥ 0. To show(Y)this, it is sufﬁcient to consider F+ ∈ A+ of the form
Y

F+ = f (φ+ )χ̄α (x)χα (x)ψ̄α (x)ψα (x),(56)

x∈Λ+ ;αwhere φ+ = {φ(x)}x∈Λ+ . For such an F+ , we obtain(Y)〈θ(F+ )F+ 〉0

¯Z
¯2
¯
¯
= ¯¯ D[φ+ ]D[φ∗+ ] f (φ+ )¯¯ ≥ 0.

(57)

Therefore, for arbitrary F ∈ P ∩ B the statement (ii) holds.
This result can be extended for F ∈ P̄ ∩ B by a similarargument in the case of free fermion given above [40]. The
statement (iii) can be checked by noting that the interaction
terms are strictly local and belongs to either A+ or A− ,depending on the time coordinate, and they are mapped toeach other by the θ transformation. To show the statement(iv), one should note that only the ﬁrst two terms in the action(49) contribute to ∆A. The ﬁrst fermionic part belongs to P̄
as shown in the above proof of the free Dirac fermion. As tothe second bosonic part, it is a well-known fact.
From these statements (i)-(iv), the reﬂection positivity of
the chiral Yukawa model follows immediately. By (18) and(Y)
the fact that eA θ(F+ )F+ ∈ B for arbitrary F+ ∈ A+ , weobtain®(Y)­ A ®(Y) ­ A­ A®(Y)
e 0 = e θ(1)1 0 ≥ 0.
e θ(F+ )F+ 0 ≥ 0,(58)This implies〈θ(F+ )F+ 〉

(Y)

=

­ A
®(Y)
e θ(F+ )F+ 0(Y)〈eA 〉0

≥ 0,

∀

(Y)

F ∈ A+ ,(59)

by a ﬁve-dimensional Wilson fermion plus a ﬁve-dimensionalbosonic spinor ﬁeld with the action deﬁned by the 5-dim.
Wilson-Dirac operator square |Dw(5dim.) −m0 |2 , both subject
to the anti-periodic condition in the ﬁfth-direction. (m0 is
the domain wall height in the lattice unit.) Then one cansafely take the limit of the inﬁnite extent of the ﬁfth dimension
and the reﬂection positivity of overlap Dirac fermions follows
indeed.
In the case with gauge interaction, however, the positivity
condition is not satisﬁed for the Pauli-Villars bosonic ﬁeld. In
fact, for the action of the Pauli-Villars bosonic ﬁeld,
Xφ∗ (x)|Dw(5dim.) − m0 |2 φ(x),APV(b) = −(60)x

one hasθ(APV(b) ) ̸= APV(b)(61)

under the anti-linear θ operation for the bosonic spinor ﬁeld
φα (x) and the link variable U (x, y),θφα (x) := φα (θx),θU (x, y) = U (θx, θy),and for an observable F of the form F (U, φ, ψ)f (φ, U )M (ψ),θ(F )(U, φ, ψ) = f ∗ (θφ, θU )M † (θψ),

(62)
=
(63)

as before. This is due to the non-vanishing commutators of
the covariant difference operators [∇0 , ∇k ] ̸= 0 (k = 1, 2, 3).
And this causes a difﬁculty in completing the proof. Of
course, it does not exclude the possibility that the overlap
fermion itself, which is deﬁned in the limit of the inﬁnite extent of the ﬁfth dimension, satisﬁes the reﬂection positivity. In
this approach with domain wall fermion, it would be possible
to trace the effects of the violation of the reﬂection positivity
in the limit of the inﬁnite extent of the ﬁfth dimension. Workin this direction is in progress.

completing the proof of (54).
Acknowledgements
VI. DISCUSSIONS

There is another route to the proof: it is through the connection to domain wall fermion[28, 34]. Since domain wall
fermion is deﬁned by the ﬁve-dimensional Wilson fermion,
it fulﬁlls the reﬂection positivity by itself. In the free case,fortunately, the positivity condition is also satisﬁed for thePauli-Villars ﬁelds, where the Pauli-Villars ﬁelds are deﬁned

K.U. would like to thank Tsutomu T. Yanagida for valuable
encouragements. K.U. is supported by Global COE Program“the Physical Science Frontier”, MEXT, Japan. This work
was supported by World Premier International Center Initiative (WPI Program), MEXT, Japan. Y.K. would like tothank S. Hashimoto and H. Fukaya for discussions. Y.K.
is supported in part by Grant-in-Aid for Scientiﬁc Research
No. 21540258, 21105503.

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[35] Strictly speaking, locality is not also established for the gauge
action adopted in these works and the phase structure of thegauge model is not fully understood. See the arguments in [23].
It would be desirable to clarify these points.
[36] Reﬂection positivity was ﬁrst proposed in the continuum theory,
in analyzing the relation between the Minkowski and Euclidean
versions of quantum ﬁeld theory [24][25].
[37] The reader might prefer the sign convention where A =−ψ̄DL ψ in stead of (1). These two sign convention in thefermionic action are connected to each other by the transformation ψ̄ ′ = iψ̄ and ψ ′ = iψ. We have chosen this sign
convention for the sake of the proof of the reﬂection positivity
following [20]. As we will see in section 3, this sign convention
is suitable to prove the statement (iv) ∆A ∈ P̄.
[38] One should note, however, that the reﬂection positivity is lostfor the improved gauge and fermion actions. See [26] for thetreatment for the case of the Symanzik improvement of Wilson
gauge action.
[39] This deﬁnition of convergence in A is equivalent to the norm
convergence of the Grassmann algebra induced from the metricof the underlying vector space, where the fermionic ﬁelds form
an orthonomal basis.
[40] To give a precise proof, we need to clarify what we mean bysaying that a sequence of bosonic functions {fn }∞n=1 converges
to some f . Here, we use uniform convergence, i.e. a sequence{fn }n is deﬁned to converge to some f iff
lim sup |fn (φ) − f (φ)| = 0.

n→∞φ

As is well known in the calculus, with this deﬁnition of convergence, the limit function f is guaranteed to be continuous,and the order of the integration and the limit operation can beexchanged, implying the continuity of 〈·〉0 and 〈·〉. In particular,
if {fn }n ⊂ B converges to some f , 〈fn 〉0 also converges to〈f 〉0 , meaning that f ∈ B.