Spinor with Schrödinger symmetry and non-relativistic supersymmetry

Physics Letters B 702 (2011) 69–74

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Physics Letters B

www.elsevier.com/locate/physletb

Spinor with Schrödinger symmetry and non-relativistic supersymmetry

Hiroshi Yoda a,∗, Shin’ichi Nojiri a,b

a Department of Physics, Nagoya University, Nagoya 464-8602, Japanb Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya 464-8602, Japan

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 May 2011Received in revised form 8 June 2011Accepted 17 June 2011Available online 28 June 2011Editor: A. Ringwald

Keywords:Schrödinger algebraSpinor representationSuperconformal algebraNon-relativistic supersymmetry

We construct the d-dimensional “half” Schrödinger equation, which is a kind of the root of theSchrödinger equation, from the (d + 1)-dimensional free Dirac equation. The solution of the “half”Schrödinger equation also satisfies the usual free Schrödinger equation. We also find that the explicittransformation laws of the Schrödinger and the half Schrödinger fields under the Schrödinger symme-try transformation are derived by starting from the Klein–Gordon equation and the Dirac equation ind + 1 dimensions. We derive the 3- and 4-dimensional super-Schrödinger algebra from the superconfor-mal algebra in 4 and 5 dimensions. The algebra is realized by introducing two complex scalar and one(complex) spinor fields and the explicit transformation properties have been found.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Motivated with the original AdS/CFT correspondence [1–3],non-relativistic conformal symmetry [4,5], especially for the coldatom [6] has attracted some attentions as a correspondence be-tween the gravity in the anti-de Sitter (AdS) background and thenon-relativistic conformal field theory [7–12]. The non-relativisticconformal symmetry is the Galilean conformal symmetry contain-ing the Galilei symmetry and conformal symmetry.

The symmetry in the free Schrödinger equation [13–16], calledas the Schrödinger symmetry, is a kind of Galilean conformal sym-metry. The algebra of the Schrödinger symmetry in d − 1 spa-cial dimensions is the subalgebra of (d + 1)-dimensional confor-mal algebra, which is the algebra of the isometry of (d + 2)-dimensional AdS space–time. Then the d-dimensional (d − 1 spa-cial dimensional) free Schrödinger equation can be derived fromthe (d + 1)-dimensional Klein–Gordon equation by choosing thewave function to be an eigenstate of the momentum of a light-cone direction. In this Letter, we define the “half” Schrödingerequation, which is a kind of the root of the Schrödinger equa-tion, from the (d + 1)-dimensional free Dirac equation.1 We findthe solution of the “half” Schrödinger equation also satisfies thefree Schrödinger equation. Since the Klein–Gordon equation and

* Corresponding author.E-mail address: yoda@th.phys.nagoya-u.ac.jp (H. Yoda).

1 The Schrödinger symmetry for the spinor field has been investigated in [17].

0370-2693/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2011.06.048

the Dirac equation have conformal symmetry, the explicit trans-formation laws of the Schrödinger and the half Schrödinger fieldsunder the Schrödinger symmetry transformation can be derived.Since there are scalar field(s) corresponding to the Klein–Gordonfield(s) and spinor field(s) to the Dirac field(s), we may considerthe supersymmetry [18–21]. We should note that a non-relativisticsupersymmetry for the vector and spinor has been considered in[24]. Starting from the superconformal algebra in 4 and 5 dimen-sions, we derive the 3- and 4-dimensional super-Schrödinger alge-bra. The algebra is realized by introducing two complex scalar andone (complex) spinor fields.

2. Schrödinger symmetry from the Klein–Gordon and Diracequations

We now show that the d-dimensional free Schrödinger equa-tion can be derived from (d + 1)-dimensional d’Alembert equation(massless Klein–Gordon equation) by choosing the wave functionto be an eigenstate of the momentum of a light-cone direction.

We denote the coordinates in (d + 1)-dimensional flat space–time by {xμ} (μ = 0, . . . ,d) and choose the metric tensor byημν := diag(−1,1, . . . ,1). We now define the light-cone coordi-nates by

x± := 1√2

(x0 ± xd). (1)

Then in terms of the light-cone coordinates, the metric tensor hasthe following form:

70 H. Yoda, S. Nojiri / Physics Letters B 702 (2011) 69–74

ημν =

⎛⎜⎜⎜⎜⎝

0 −1−1 0

1. . .

1

⎞⎟⎟⎟⎟⎠ . (2)

Now μ,ν = +,−, i = 1, . . . ,d−1. Then, for example, we have x± =−x∓ and

p± := 1√2(p0 ± pd), p± = −p∓. (3)

Now the relativistic energy–momentum relationship for themassless particle pμpμ = −2p+ p− + p2 = 0 can be rewritten ifwe write

p+ = M, p− ≡ E, (4)

as

E = 1

2Mp2, (5)

which is nothing but the expression of the kinetic energy forthe non-relativistic particle with mass M in d-dimensional space(d + 1 space–time dimensions).

Eq. (5) may suggest that the massless Klein–Gordon equationcould be rewritten as the free Schrödinger equation. By using thelight-cone coordinates, the Klein–Gordon equation has the follow-ing form:

(−2∂+∂− + ∂i∂i)φ = 0. (6)

We now choose φ to be an eigenstate of ∂+ as ∂+φ = −iMφ

and we regard a light-cone coordinate x− as a “time”-coordinatex− ≡ t . Then the Klein–Gordon equation (6) can be rewritten asthe free Schrödinger equation:

i∂tφ = − 1

2M�φ. (7)

Since we assume φ to be an eigenstate of ∂+ , φ has the followingform:

φ(x+, x−, xi) = e−iMx+

φ(x−, xi). (8)

Later we consider the conformal transformation of the Schrödingerfield φ, where the transformation of x+ can be interpreted as aphase of φ.

The free massless Dirac field also satisfies the Klein–Gordonequation. Then by rewriting the Dirac equation by choosing theDirac field ψ to be an eigenstate of ∂+ as ∂+ψ = −iMψ and re-garding light-cone coordinate x− as a “time”-coordinate x− ≡ t , weobtain the Schrödinger version of the Dirac equation. We now usethe following convention for the γ matrices:{γ μ,γ ν

} = 2ημν, γ μ† = γ 0γ μγ 0,

Σμν := 1

4

[γ μ,γ ν

], γ 5 := iγ 0γ 1γ 2γ 3,

γ ± := 1√2

(γ 0 ± γ d) = − 1√

2(γ0 ∓ γd) =: −γ∓. (9)

Then we obtain{γ +, γ −} = −2,

{γ ±, γ i} = 0 (i = 1,2, . . . ,d − 1),(

γ +)2 = (γ −)2 = 0. (10)

Then by choosing the Dirac field ψ to be an eigenstate of ∂+ as∂+ψ = −iMψ , the Dirac equation has the following form:(iγ a∂a + Mγ +)

ψ = 0 (a = −, i), (11)

which tells ψ satisfies the Schrödinger equation:

(2iM∂t + ∂i∂i)ψ = 0. (12)

Then Eq. (11) is the Schrödinger version of the Dirac equation,which we call the “half Schrödinger equation”.

The d-dimensional Schrödinger algebra is given as a subalgebraof the (d + 1)-dimensional conformal algebra SO(d + 1,2), which isgiven by

[Mμν, Mρσ ] = i(ημσ Mνρ − ημρ Mνσ − ηνσ Mμρ + ηνρ Mμσ ),

[Mμν, Pρ ] = i(ηνρ Pμ − ημρ Pν),

[Mμν, Kρ ] = i(ηνρ Kμ − ημρ Kν),

[D, Pμ] = −i Pμ, [D, Kμ] = iKμ,

[Kμ, Pν ] = −2i(ημν D + Mμν). (13)

Here Mμν generates the Lorentz boost and the rotation and Pμ

generates the translation. Furthermore the operator D generatesthe dilatation

xμ → e−λxμ, (14)

and Kμ generates the special conformal transformation given by

xμ → xμ − cμx2

1 − 2cμxμ + c2x2. (15)

The (d − 1)-dimensional Schrödinger algebra sch(d − 1) is the sub-algebra of the conformal algebra which does not change the eigen-value M of P+ , that is, the algebra given by the generators com-muting with P+ .

[ J i j, Jkl] = i(δil J jk − δik J jl − δ jl J ik + δ jk J il),

[ J i j, H] = 0, [ J i j, E] = 0, [ J i j, F ] = 0,

[Pi, P j] = 0, [Pi, H] = 0, [Pi, E] = i P i, [Pi, F ] = 0,

[ J i j, Pk] = i(δ jk P i − δik P j),

[Gi, G j] = 0, [Gi, H] = i P i, [Gi, E] = iGi,

[Gi, F ] = 0, [ J i j, Gk] = i(δ jkGi − δikG j),

[Pi, G j] = iMδi j, [E, H] = −2iH, [E, F ] = 2i F ,

[F , H] = iE. (16)

Here

H = P−, Pi = Pi, P+ = M, J i j = Mij,

Gi = M+i, E = D + M−+, F = 1

2K+. (17)

The generators H , Pi , J i j , Gi , E , and F generate the translation oftime, the translation of the spacial coordinates, the spacial rotation,the Galilei boost, the dilatation, and the special conformal transfor-mation, which are explicitly given as the following transformationof the coordinates:

H : t → t + a, Pi : xi → xi + bi,

J i j : xi → Rijx j(

Rij ∈ SO(d − 1)), Gi : xi → xi + vit,

E : t → λ2t and xi → λxi,

F : t → 1

1 + f tt and xi → 1

1 + f txi . (18)

In (16), since M can be regarded as a c number, the commuta-tor of Pi and G j becomes a central extension. In fact, this algebrais one of the central extended of the Galilean conformal algebra

H. Yoda, S. Nojiri / Physics Letters B 702 (2011) 69–74 71

which is constructed by non-relativistic limit (c → ∞) of confor-mal algebra [22].

As well known, the Klein–Gordon equation (6) and the Diracequation are invariant under the conformal transformation andtherefore the Schrödinger equation (7) and the half Schrödingerequation (11) is invariant under the Schrödinger transformation.The invariance of the Klein–Gordon and the Dirac equations andthe transformation law of the Klein–Gordon and the Dirac fields ingeneral D dimensions are explicitly given in Appendix A.

3. Super-Schrödinger symmetry from superconformal symmetry

Since we have found that both of the Klein–Gordon equa-tion and the Dirac equation have the Schrödinger symmetry andfound the explicit transformation laws of the Klein–Gordon andthe Dirac fields, we now like to find the transformation connect-ing the Klein–Gordon field(s) and the Dirac field(s), which is thesuper-Schrödinger symmetry transformation.

The supersymmetric extension of the conformal symmetry isknown as the superconformal symmetry. In this section, for sim-plicity, we construct (2 + 1)- and (3 + 1)-dimensional super-Schrödinger algebra starting with the D = 4, N = 1 and D = 5,N = 2 superconformal algebra.

3.1. Construction of (2 + 1)-dimensional super-Schrödinger algebrafrom the reduction of D = 4, N = 1 superconformal algebra

D = 4, N = 1 superconformal algebra [23], which is given by,in addition to the conformal algebra SO(4,2) (13),

{Q α, Q β} = −1

2iγ μ

αβ Pμ, {Sα, Sβ} = 1

2iγ μ

αβ Kμ,

{Q α, Sβ} = −1

2iδαβ D + 1

2iΣμν

αβ Mμν + γ 5αβ A,

[Q α, Mμν ] = i(Σμν Q )α, [Sα, Mμν ] = i(Σμν S)α,

[Q α, D] = 1

2i Q α, [Sα, D] = −1

2i Sα,

[Q α, Kμ] = −i(γμS)α, [Sα, Pμ] = i(γμ Q )α,

[Q α, A] = −3

4

(γ 5 Q

)α, [Sα, A] = 3

4

(γ 5 S

)α. (19)

In order to obtain the closed algebra, we define projection opera-tors P and P as follows

P := −1

2γ −γ +, P := −1

2γ +γ −, (20)

which satisfy the following equations:

P + P = −1

2

{γ −, γ +} = 1,

P 2 = 1

4γ −γ +γ −γ + = −1

2γ −γ + = P,

Pγ − = γ −P = γ −, γ +P = Pγ − = γ +,

Pγ + = γ +P = γ −P = Pγ − = 0,

ηP = −1

2η† P † = −1

2ηγ +γ −, P = −P † := −1

2γ +γ −,

Pγ a P = 0 (a = −,1, . . . ,d − 1), Pγ +P = γ +. (21)

If we define

Tα := Pαβ Sβ, Tα := Sβ Pβα = Pαβ Sβ, (22)

we obtain(γ +T

) = (Tγ −) = 0. (23)

α α

Then a closed algebra, which we call the super-Schrödinger alge-bra, is given by

{Q α, Q β} = −1

2iγ −

αβ H − 1

2iγ i

αβ Pi − 1

2iγ +

αβ M,

{Tα, Tβ} = iγ +αβ F ,

{Q α, Tβ} = 1

4i(γ −γ +)

αβE − 1

4i(Σi jγ

−γ +)αβ

J i j

+ 1

2i(γ +γi

)αβ

Gi − 1

2

(γ 5γ −γ +)

αβA,

[Q α, J i j] = i(Σi j Q )α, [Tα, J i j] = i(Σi j T )α,

[Q α, Gi] = −1

2i(γ −γi Q

)α, [Tα, Gi] = 0,

[Q α, E] = −1

2i(γ −γ + Q

)α, [Tα, E] = −iTα,

[Tα, H] = −i(γ + Q

)α, [Tα, Pi] = −1

2i(γ +γ −γi Q

)α,

[Tα, F ] = 0, [Q α, F ] = 1

2i(γ −T

)α,

[Q α, A] = −3

4

(γ 5 Q

)α, [Tα, A] = 3

4

(γ 5T

)α. (24)

We now consider the following Lagrangian

L = iψγ μ∂μψ − 1

2

∑i=i,2

∂μφ∗i ∂μφi . (25)

The Lagrangian is invariant under the following superconformaltransformation:

δQ φ1 = −1

2i(ξψ), δQ φ2 = −1

2i(ξγ 5ψ

),

δQ ψ = 1

2

(iγ −∂t + iγ i∂i + Mγ +)(

ξφ1 − γ 5ξφ2),

δT φ1 = −1

2i(ηP

(tγ + − xiγi

)ψ

),

δT φ2 = 1

2i(ηPγ 5(tγ + − xiγi

)ψ

),

δT ψ = 1

2i[(2t∂t + xi∂i + 2) + γ +γ i(t∂i − iMxi)

− Σi j(xi∂ j − x j∂i)](

φ1 + γ 5φ2)

Pη,

δAφ1 = −1

2θφ2, δAφ2 = 1

2θφ1, δAψ = −1

4θγ 5ψ. (26)

Here the ξ , η and θ are parameters of the transformation. Thetransformation (26) is closed on shell.

As we are considering the field theory in 4 dimensions, we maydefine γ 5 and the chiral projection operators PL,R by

γ 5 := iγ 0γ 1γ 2γ 3,

PL := 1

2

(1 − γ 5), P R := 1

2

(1 + γ 5). (27)

We also define the left and right parts for the Dirac field and alsothe scalar field by

ψL := PLψ, ψR := P Rψ,

φL := φ1 − φ2, φR := φ1 + φ2. (28)

Then the transformation laws (26) can be rewritten as

72 H. Yoda, S. Nojiri / Physics Letters B 702 (2011) 69–74

δQ L φL = −i(ξψL), δQ R φR = −i(ξψR),

δQ R ψL = 1

2

(iγ −∂t + iγi∂i + Mγ +)

P RξφL,

δQ L ψR = 1

2

(iγ −∂t + iγi∂i + Mγ +)

PLξφR ,

δT R φL = −ix(ηP

(tγ + − xiγi

)ψL

),

δT L φR = −ix(ηP

(tγ + − xiγi

)ψR

),

δT L ψL = 1

2i[(2t∂t + xi∂i + 2) + γ +γi(t∂i − iMxi)

− Σi j(xi∂ j − x j∂i)]

P P LηφL,

δT R ψR = 1

2i[(2t∂t + xi∂i + 2) + γ +γ i(t∂i − iMxi)

− Σi j(xi∂ j − x j∂i)]

P P RηφR ,

δ AφL = −θφR , δ AφR = θφL,

δ AψL = 1

2θψL, δ AψR = −1

2θψR . (29)

Here A := 2A. Except the transformation generated by A, thetransformations are closed on the left and right parts, respectively.

3.2. Construction of (3 + 1)-dimensional super-Schrödinger algebrafrom the reduction of D = 5, N = 2 superconformal algebra

D = 5, N = 2 superconformal algebra [25,26] is given by

{Q Aα, Q B

β

} = −1

2iδB

Aγμαβ Pμ,

{S Aα, S B

β

} = 1

2iδB

Aγμαβ Kμ,

{Q Aα, S B

β

} = δBA

(1

2iδαβ D − 1

2iΣμν

αβ Mμν

)+ δαβ R B

A,

[Q Aα, Mμν ] = i(Σμν Q A)α, [S Aα, Mμν ] = i(Σμν S A)α,

[Q Aα, D] = 1

2i Q Aα, [S Aα, D] = −1

2i S Aα,

[Q Aα, Kμ] = −i(γμS A)α, [S Aα, Pμ] = i(γμ Q A)α,

[Q Aα, R BC ] = 3

4(εAB Q kα + εAC Q jα),

[S Aα, R BC ] = 3

4(εAB Skα + εAC S jα),

[R AB , RC D ] = 3

4(εAC R B D + εB D R AC + εAD R BC + εBC R AD) (30)

and conformal algebra SO(5,2) (13).Here fermionic generators with extra indices, A, B, C, . . . = 1,2,

is SU(2) Majorana spinors and R AB = R B A is SU(2) R-symmetrygenerators. These indices are raised and lowered with totally anti-symmetric tensor εAB and εAB (ε12 = ε12 = 1), as

ζ A := εABζB , ζA := ζ BεB A, (31)

and SU(2) R invariant inner product is defined by contraction ofraised and lowered indices,

ζ A1 ζ2A = εAB ζ1Bζ2A = εAB ζ 1Aζ 2B . (32)

In the same way as in the (2 + 1)-dimensional case (22), wereduce this algebra using by projection operators (20), such as

{Q Aα, Q B

β

} = −1

2iδB

A

[γ −αβ H + γiαβ Pi + γ +

αβ M],

{S Aα, T B

β

} = iδBAγ + F ,

{Q Aα, T B

β

} = −δBA

[1

4i(γ −γ +)

αβE − 1

4i(Σi jγ

−γ +)αβ

J i j

+ 1

2i(γ +γi

)αβ

Gi

]− 1

2

(γ −γ +)

αβR B

A,

[Q Aα, J i j] = i(Σi j Q A)α, [T Aα, J i j] = i(Σi j T A)α,

[Q Aα, Gi] = −1

2i(γ −γi Q A

)α,

[T Aα, Gi] = 0, [Q Aα, E] = −1

2i(γ −γ + Q A

)α,

[T Aα, E] = −iT Aα, [T Aα, H] = −i(γ + Q A

)α,

[T Aα, Pi] = −1

2i(γ +γ −γi Q A

)α,

[T Aα, F ] = 0, [Q Aα, F ] = 1

2i(γ −T A

)α,

[Q Aα, R BC ] = 3

4(εAB Q Cα + εAC Q Bα),

[T Aα, R BC ] = 3

4(εAB TCα + εAC T Bα),

[R AB , RC D ] = 3

4(εAC R B D + εB D R AC + εAD R BC + εBC R AD). (33)

Thus, the on-shell transformation of hypermultiplet includingtwo scalar fields φA and one spinor field ψα is

δQ φA = −1

2i(ξ Aψ

),

δQ ψ = (iγ −∂t + iγi∂i + Mγ +)

ξAφA,

δT φA = −1

2i(ηA P

(tγ + − xiγi

)ψ

),

δT ψ = −i[(2t∂t + xi∂i + 3) + γ +γi(t∂i − iMxi)

− Σi j(xi∂ j − x j∂i)]

PηAφA,

δRφA = −3

2θ A

BΦB , δRψ = 0. (34)

Here the SU(2) Majorana spinor ξ A, ηA and 2 × 2 (anti-)Hermitiantraceless matrix θ A

B are parameters of the transformation. In con-trast to the 3-dimensional algebra (except the transformation gen-erated by A), because of R-symmetry invariance of spinor field ψ ,the transformations of two scalar fields φA are not closed.

4. Summary

In this Letter, we have found the “half” Schrödinger equa-tion, which is a kind of the root of the Schrödinger equation,from the (d + 1)-dimensional free Dirac equation. Starting fromthe Klein–Gordon equation and the Dirac equation in d + 1 di-mensions, the explicit transformation laws of the Schrödinger andthe half Schrödinger fields under the Schrödinger symmetry trans-formation have been derived. We have also obtained the 3- and4-dimensional super-Schrödinger algebra from the superconformalalgebra in 4 and 5 dimensions. The algebra is realized by intro-ducing two complex scalar and one (complex) spinor fields andthe explicit transformation properties have been found. It could beinteresting to find any system which has this kind of supersymme-try.

Acknowledgements

This research has been supported by Global COE Program ofNagoya University (G07) provided by the Ministry of Education,Culture, Sports, Science & Technology and by the JSPS Grant-in-Aidfor Scientific Research (S) # 22224003 (S.N.).

H. Yoda, S. Nojiri / Physics Letters B 702 (2011) 69–74 73

Appendix A. Conformal transformations of scalar and spinorfields

A.1. Scalar

Here we review on the conformal symmetry in D-dimensionalspace–time and show the invariance of the Klein–Gordon equationand the transformation law of the Klein–Gordon field under theconformal transformation.

We consider the coordinate transformation:

xμ = f μ(xν

). (A.1)

Under the coordinate transformation, the flat metric in D dimen-sions

ds2 = ημν dxμ dxν, (A.2)

is changed as

ds2 = ημν∂ f μ

∂ xρ

∂ f ν

∂ xσdxρ dxσ . (A.3)

When the transformed metric is proportional to ημν

ηρσ∂ f ρ

∂ xμ

∂ f σ

∂ xν= C

(xτ

)ημν, (A.4)

the transformation (A.4) is called conformal transformation. HereC(xτ ) is a function of the coordinate and determined by multiply-ing Eq. (A.4) with ημν as

C(xτ

) = 1

Dημνηρσ

∂ f ρ

∂ xμ

∂ f σ

∂ xν. (A.5)

Then substituting (A.5) into (A.4), we obtain

ηρσ∂ f ρ

∂ xμ

∂ f σ

∂ xν= 1

Dημνηρσ ηαβ

∂ f ρ

∂ xα

∂ f σ

∂ xβ. (A.6)

We call f μ(xν) satisfying (A.6) as the conformal Killing vector. Wenow define a matrix Mμ

ν by

Mμν ≡ ∂ f μ(xρ)

∂ xν, (A.7)

Eq. (A.6) can be rewritten as

ημν Mμρ Mν

σ = 1

Dηρσ ηαβημν Mμ

α Mνβ . (A.8)

By regarding the quantity in (A.8) as a matrix with indexes ρand σ and considering the determinant, we obtain

ηαβημν Mμα Mν

β = DM2D . (A.9)

Here M is the determinant of Mμν : M = det Mμ

ν .We now consider the coordinate transformation (A.1) for the

massless scalar field φ:

S = 1

2

∫dD xημν∂μφ∂νφ. (A.10)

Then we obtain

S = 1

2

∫dD x Mημν

(M−1)

μρ(M−1)

νσ

× ∂ρ

(M

2−D2 φ

)∂σ

(M

2−D2 φ

). (A.11)

Here φ is defined by

φ = MD−2

2 φ. (A.12)

By using (A.8) and (A.9), we can rewrite the action (A.11) as

S = 1

2

∫dD x

{ημν∂μφ∂νφ + 2

(2 − D

2D

)ημν

(M−1∂μM

)φ∂ν φ

+(

2 − D

2D

)2

ημν(M−1∂μM

)(M−1∂ν M

)φ2

}. (A.13)

By using the partial integration, we can further rewriting (A.13) as

S = 1

2

∫dD x

{ημν∂μφ∂νφ − (D − 2)

4(D − 1)φ2

(−(D − 1)ημν∂μ∂νζ

− (D − 2)(D − 1)

4ημν∂μζ∂νζ

)}. (A.14)

Here ζ is defined by

ζ ≡ 2

Dln M. (A.15)

Then Eqs. (A.3), (A.6), and (A.7) tell that the metric tensor gμν isgiven by

gμν = eζ ημν. (A.16)

For the metric (A.16), the scalar curvature R is given by

R =(

−(D − 1)ημν∂μ∂νζ − (D − 2)(D − 1)

4ημν∂μζ∂νζ

)eζ .

(A.17)

Since the metric (A.16) is obtained from the flat metric (A.2),where the scalar curvature vanishes, the scalar curvature (A.17)also vanishes. Then the second term in (A.14) vanishes and we find

S = 1

2

∫dD xημν∂μφ∂νφ = 1

2

∫dD xημν∂μφ∂νφ, (A.18)

that is, the action (A.10) of massless free scalar is invariant underthe coordinate transformation (A.1) and (A.12) where f μ is theconformal Killing vector satisfying (A.6).

A.2. Spinor

Here show the invariance of the Dirac equation and the trans-formation law of the Dirac field, that is, spinor field, under theconformal transformation.

In order to investigate the conformal transformation in thespinor field, we consider the infinitesimal transformation

xμ = xμ + δxμ. (A.19)

Then by defining

mμν ≡ ∂δxμ

∂ xν, (A.20)

we can rewrite the condition (A.8) for the conformal transforma-tion as

1

2(mμν + mνμ) = 1

Dημνm, m ≡ mμ

μ. (A.21)

The action of the Dirac fermion is given by

S = i

∫dD x ψγ μ∂μψ. (A.22)

We now consider the following infinitesimal transformation:

ψ =(

1 − D − 1

2D+ 1

2mμνΣμν

)ψ, (A.23)

which gives

ψ =(

1 − D − 1 − 1mμνΣμν

)˜ψ, (A.24)

2D 2

74 H. Yoda, S. Nojiri / Physics Letters B 702 (2011) 69–74

Then we find

S = i

∫dD x

(˜ψγ μ∂μψ − mν

μ˜ψγ μ∂νψ − m ˜ψγ μ∂μψ

− D − 1

2D˜ψγ μψ∂μm + 1

2˜ψγ ρΣμνψ∂ρmμν

). (A.25)

Since

dD x(1 − m) = dD x, ∂μ ≡ ∂

∂ xμ= ∂

∂xμ− mν

μ∂

∂xν, (A.26)

we find

S = i

∫dD x

(˜ψγ μ∂μψ − D − 1

2D˜ψγ μψ∂μm

+ 1

2˜ψγ ρΣμνψ∂ρmμν

). (A.27)

In case of rotation (including the Lorentz transformation), transla-tion, and dilatation, mμν is a constant or vanishes and therefore∂μm = ∂ρmμν , which tells that the action (A.22) is invariant un-der the rotation (including the Lorentz transformation), translation,and dilatation. In case of the special conformal transformation witha infinitesimal parameter cμ ,

δx = cμ(x)2 − 2xμxρcρ, (A.28)

we find

∂μm = −2Dcμ, γ ρΣμν∂ρmμν = −2(D − 1)cμγ μ, (A.29)

and therefore the second and the third terms in (A.25) cancel witheach other and the action (A.22) is invariant under the special con-formal transformation (A.26).

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