Direct Proof of Reﬂection Positivity of Free Overlap
Dirac Fermion

Yoshio Kikukawa
Institute of Physics, the University of Tokyo, Tokyo 153-8902, Japan
E-mail: kikukawa@hep1.c.u-tokyo.ac.jp

Kouta Usui∗
Department of Physics, the University of Tokyo 113-0033, Japan
Institute for the Physics and Mathematics of the Universe (IPMU), the University of Tokyo,
Chiba 277-8568, Japan
E-mail: kouta@hep-th.phys.s.u-tokyo.ac.jp
It is shown that free lattice Dirac fermions deﬁned by overlap Dirac operator fulﬁll theOsterwalder-Schrader reﬂection positivity condition with respect to the link-reﬂection.

The XXVIII International Symposium on Lattice Field Theory, Lattice2010
June 14-19, 2010
Villasimius, Italy∗ Speaker.

c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.
⃝

http://pos.sissa.it/Direct Proof of Reﬂection Positivity of Free Overlap Dirac Fermion

Kouta Usui

1. Introduction
In this short article, we will examine the reﬂection positivity [1, 2, 3] of lattice fermions deﬁned
through overlap Dirac operator [4, 5], a gauge-covariant solution to the GW relation [4, 5, 6, 7, 8, 9],which has been derived in the ﬁve-dimensional domain wall approach [10, 11, 12]. The reﬂection
positivity of the GW fermions is not fully understood yet [13, 14, 15], while that of Wilson fermionshas been rigorously proved in various ways including the gauge interacting case [3, 16, 17]. In
the following, It will be shown rigorously that free overlap Dirac fermion fulﬁlls the reﬂection
positivity with respect to the link-reﬂection. In ref [13], Lüscher discussed the unitarity propertyof free overlap Dirac fermion by investigating the positivity through the spectral representation offree propagator and concluded 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 be also extended to the non-gauge models with interactions such as chiral Yukawa models. For
gauge models, however, a proof of reﬂection positivity, if any, seems to be more involved and wewill leave it for future study.

2. Reﬂection positivity
Reﬂection positivity is a sufﬁcient condition for reconstructing a quantum theory in the canonical formalism, i.e. the Hilbert space of state vectors and the Hermitian Hamiltonian operator
acting on the state vectors, from the lattice model deﬁned in the Euclidean spaceLet us formulate
the reﬂection positivity condition for lattice Dirac fermions.
We assume a ﬁnite lattice Λ = [−L + 1, L]4 ⊂ Z4 in the lattice unit a = 1, and impose antiperiodic boundary condition in the time direction, and periodic boundary conditions in the spacedirections. The fermionic action is deﬁned in the bilinear formA(ψ̄, ψ) = ∑ ψ̄(x)DL ψ(x),(2.1)

x∈Λwith a lattice Dirac operator DL 1 . The kernel of the Dirac operator should be written as
DL (x, y) = ∑ (−1)n0 D(x + 2nL, y),n∈Z4x, y ∈ Λ,(2.2)

where D(x, y) is the kernel of the Dirac operator in the inﬁnite lattice Z4 . The quantum theoryis then completely characterized by the expectational functional deﬁned by the fermionic pathintegration:〈F〉 :=1ZZ

D[ψ]D[ψ̄] eA(ψ̄,ψ) F(ψ̄, ψ),(2.3)

1 The reader might prefer the sign convention where A = −ψ̄D ψ in stead of (2.1). These two sign convention inL
the fermionic 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 [3]. As we will see in section 3, this sign
convention is suitable to prove the statement (iv) ∆A ∈ P̄.

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Direct Proof of Reﬂection Positivity of Free Overlap Dirac Fermion

Kouta Usui

where the Grassmann integration for each ﬁeld variable is speciﬁed asZZd ψ̄α (x)ψ̄α (x) = 1,(2.4){dψα (x)d ψ̄α (x)}.

(2.5)

dψα (x)ψα (x) = 1,and the functional measure is deﬁned by∏

D[ψ]D[ψ̄] :=x∈Λ;α=1,2,3,4

The reﬂection positivity condition — a condition on this expectational functuonal — is formulated as follows: let us deﬁne time reﬂection operator θ which acts on polynomials of the fermionicﬁeld variables by the relations
θ (ψ(x)) = (ψ̄(θ x)γ0 )T ,θ (αF + β G) = α ∗ θ (F) + β ∗ θ (G),θ (ψ̄(x)) = (γ0 ψ(θ x))T

(2.6)

θ (FG) = θ (G)θ (F),(2.7)

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 non-positive timerespectively. Let A± be the algebra of all the polynomials of the ﬁelds on Λ± , and A on Λ. Then
one says the theory is reﬂection positive if its expectation 〈·〉 : A → C satisﬁesfor ∀ F+ ∈ A+ .

〈θ (F+ )F+ 〉 ≥ 0A popular choice of lattice Dirac operator is the Wilson Dirac operator,
½¾
11 †
†
Dw = ∑γµ (∂µ − ∂µ ) + ∂µ ∂µ . ,2µ=0,1,2,3 2
Here we consider the overlap Dirac operator
Ã!
1
1
D=
1+X √,2X †X

X = Dw − m,

(2.8)

(2.9)

(2.10)

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 ultra-local [18], thefree overlap Dirac fermion indeed satisﬁes the reﬂection positivity condition, as will be shown
below.

3. Proof of Reﬂection Positivity of overlap Dirac fermion
To prove the reﬂection positivity, we need some additional deﬁnitions and notations. First, let
us denote〈F〉0 :=

Z

D[ψ]D[ψ̄]F(ψ̄, ψ).

3(3.1)

Direct Proof of Reﬂection Positivity of Free Overlap Dirac Fermion

Kouta Usui

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(3.2)

where A+ ∈ A+ , A− ∈ A− , and ∆A is the part of the action which contain both positive and negative
time ﬁelds. Thirdly, let us call P the set of all polynomials of the form ∑ j 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 isnot enough for the proof of the overlap fermion. In our case of the overlap fermion, one needs
to consider not only ﬁnite summations of the form ∑ j θ (F+ j )F+ j , but also inﬁnite summations orintegrations likeZNds θ (F(s))F(s) = lim ∑ θ (F(sk ))F(sk )∆sk ,N→∞

(3.3)

k=1

where the integration is deﬁned as a limit of a ﬁnite Riemanian summation (see also eq. (3.18)).
To this end, we consider P̄, the closure of P. The closure P̄ contains not only elements of theoriginal P, but also all the limit points of conversing sequences in P. That is,F ∈ P̄⇔∃{Fn }∞
n=1 ∈ P : lim Fn = F.
n→∞

(3.4)

Here, the sequence {Fn }n ⊂ A is deﬁned to be convergent to some F ∈ A , if any coefﬁcient in
Fn converges to the corresponding coefﬁcient in F as a complex number 2 . Note that with respect
to this deﬁnition of convergence, the linear 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 + β Gn → αF + β G,

Fn Gn → FG,(3.5)〈Fn 〉 → 〈F〉 ,〈Fn 〉0 → 〈F〉 .

(3.6)

Now, we note the fact that the following four statements (i)-(iv) imply the reﬂection positivity:(i) If F, G belong to P̄ then FG 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 thatDE­®
­ A®
e θ (F+ )F+ 0 = eA+ +A− +∆A θ (F+ )F+ 0 = eA+ +θ (A+ )+∆A θ (F+ )F+0*+=θ (eA+ )eA+ e∆A θ (F+ )F+|{z} | {z }∈P̄ (by (i),(iv))∈P̄≥0(3.7)0

2 This deﬁnition of convergence in A is equivalent to the norm convergence of the Grassmann algebra induced fromthe metric of the underlying vector space, where the fermionic ﬁelds form an orthonomal basis.

4

Direct Proof of Reﬂection Positivity of Free Overlap Dirac Fermion

Kouta Usui­ ®
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.
(3.8)
〈eA 〉0
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 thesimilar statement with P, which has been proved for the Wilson case [3] .
To show the statement (ii), one should refer to the deﬁnition of fermionic integration measure.
With the deﬁnition (2.5), it is sufﬁcient to consider F+ ∈ A+ of the form

∏F+ ={ψ̄α (x)ψα (x)} ∈ P,(3.9)

x∈Λ+ ;α=1,2,3,4for which one can seeZ4D[ψ]D[ψ̄] θ (F+ )F+ = {det(γ02 )}16L = 1 ≥ 0.

(3.10)

Therefore, one concludes that for arbitrary F ∈ P, 〈F〉0 ≥ 0. Take arbitrary F ∈ P̄. Then thereexists a converging sequence {Fn }n such that Fn → F. From the continuity of 〈·〉0 (see (3.6)), weobtain
DE〈F〉0 = lim Fn = lim 〈Fn 〉0 ≥ 0.
(3.11)n→∞0n→∞

The statement (iii) can be shown by using the property of the overlap Dirac kernel: D†L (x, y) =γ0 DL (θ x, θ y)γ0 .
To show the statement (iv) ∆A ∈ P̄, we use a spectral representation of DL (x, y). To derive thespectral representation of DL , we ﬁrst Fourier transform the overlap Dirac operator kernel D(x, y)in the inﬁnite volume:Z¯d 4 p ip·(x−y) X(p0 , p )¯p,(3.12)e
D(x, y)¯=(2π)4x0 ̸=y02 X † X(p0 , p)
where X(p0 , p ) = ∑µ iγµ sin pµ + ∑µ (1 − cos pµ ) − m. Then, we change the p0 integration region,[−π, π], to the contours along the imaginary axis in the complex p0 plane by Cauchy’s integration
theorem, as shown in FIG. 1. Depending 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¯¯D(x, y)¯

x0 −y0 <0Z=

d3 p(2π)3Z=Z ∞dE −E(x0 −y0 ) ipp·(xx−yy)
X(iE, p )pee
E1 2π−X † X(iE, p)

d3 p(2π)3(3.13)

Z ∞dE E(x0 −y0 ) ipp·(xx−yy) X(−iE, p )pee.
E1 2π−X † X(iE, p )(3.14)

where E1 is the edge of the cut coming from the square root, and is determined by the relations
X † X(iE1 , p ) = 0,5

E1 > 0.

(3.15)

Direct Proof of Reﬂection Positivity of Free Overlap Dirac Fermion

Kouta UsuiIn this spectrum representaion of D, it is very crucial to notice the fact that ∓γ0 X(±iE, p )(E ≥ E1 ) are positive deﬁnite matrices and there exist matrices Y± (E, p ) such that
∓γ0 X(±iE, p ) = Y±†Y± (E, p ) (E ≥ E1 ).

(3.16)

In fact, it is not difﬁcult to check that Y± (E, p ) are given by
3l(E, p ) sin pkW (E, p )γk ∓ iγ0 + il(E, p ),2l(E, p )k=1 W (E, p )Y± (E, p ) = − ∑where W (E, p ) = ∑3k=1 (1 − cos pk ) + 1 − cosh E − m and
sÃ
! 12
23
2
1sinh E∑ sin pk +W (E, p ) l(E, p ) = 
1 + 1 − k=1.
2
322 ∑k=1 sin pk /W (E, p ) + 1
sinh2 E
From the equations (2.2), (3.13) and (3.14), we ﬁnd the spectrum representation of DL (x, y) as
follows: putting V = 1/(2L)3 ,Z ∞¯dE
11 −E|x0 −y0 | ipp·(xx−yy) X(εiE, p)¯
pDL (x, y)¯=∑ee−2ELVx0 ̸=y0−X † X(iE, p )p E1 2π 1 + e+∑

Z ∞pdE e−2EL 1 E|x0 −y0 | ipp·(xx−yy) −X(−εiE, p )pee,−2EL V
E1 2π 1 + e−X † X(iE, p )(3.17)

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 (3.17), the ﬁrst term becomes D(x, y) in the limit L → ∞, and thesecond term represents a ‘ﬁnite lattice effect’ which vanishes in the limit L → ∞. The latter is thecontribution of the wrong-sign-energy modes and the minus sign appearing in front of X(−εiE, p )comes from the anti-periodicity in the time direction, which is required for the positivity, as will beseen.
From these observations, now we can show that ∆A ∈ P̄: for the term with x0 > 0, y0 ≤ 0 (inthis case ε = 1), we obtain"#Z ∞dE 1(3.18)∑ ∑ ψ̄(x)DL (x, y)ψ(y) = − ∑ E1 2π V CE,pp θ (CE,pp ) + DE,pp θ (DE,pp ) ,px∈Λ+ y∈Λ−Im
6

↑∞

r iE1

6- Re?

−π0Im
6−ππ

60r

π Re−iE1?

↓ −∞Figure 1: Complex integration contours

6Direct Proof of Reﬂection Positivity of Free Overlap Dirac Fermion

Kouta Usui

where CE,pp and DE,pp are deﬁned byrCE,pp =sDE,pp =1∑ ψ̄(x)γ0Y˜+ (E, p )† e−Ex0 eipp·xx ,1 + e−2EL x∈Λ+(3.19)

e−2EL∑ ψ̄(x)γ0Y˜− (E, p )† eEx0 eipp·xx ,1 + e−2EL x∈Λ+(3.20)

1with Y˜+ (E, p ) = Y+ (E, p )/(−X † X(iE, p )) 4 . The overall minus sign in the r.h.s. of (3.18) resultsfrom (3.17) by using (3.16). This minus sign is canceled after exchanging the order of the Grassmann products in (3.18), and we see that this term belongs to P̄. One can show similarly that theterm with x0 ≤ 0, y0 > 0 (in this case ε = −1) also belongs to P̄. Thus we obtain ∆A ∈ P̄ andcomplete the proof of the reﬂection positivity.

Acknowledgements
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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