Spinors in Weyl Geometry

Spinors in Weyl Geometry A. H. Fariborz

∗

and D. G. C. McKeon

†

Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7,

arXiv:hep-th/9607010v1 1 Jul 1996

CANADA. (February 1, 2008)

Abstract We consider the wave equation for spinors in D-dimensional Weyl geometry. By appropriately coupling the Weyl vector φµ as well as the spin connection ωµab to the spinor field, conformal invariance can be maintained. The one loop effective action generated by the coupling of the spinor field to an external gravitational field is computed in two dimensions. It is found to be identical to the effective action for the case of a scalar field propagating in two dimensions.

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∗ e-mail:

[email protected]

† e-mail:

[email protected]

Fax: 519-661-3523 PACS No: 4.90e

1

I. INTRODUCTION

The effective action for gravity generated by the coupling of an external gravitational field to quantum matter fields is of crucial importance in string theory [1,2]. The behaviour of this induced gravitational theory under a conformal transformation is of particular interest. In refs. [3,4], the induced gravitational action in the case of a scalar matter field propagating in a background Weyl geometry is examined. In this paper, we extend these considerations to the case in which the matter field is a spinor field. In the next section we consider how to couple a spinor field to a background gravitational field in the case of Weyl geometry in D-dimensions. We then compute the first SeelyGilkey/DeWitt-Schwinger coefficient E1 (x, D), needed to determine the one-loop effective action in two dimensions when using the formalism of ref. [5].

II. SPINORS IN WEYL GEOMETRY

Weyl in 1918 [6,7] introduced the idea of a conformal transformation gµν (x) → Ω2 (x)gµν (x)

(1)

on the metric tensor gµν (x). (We use the conventions of [8], except that we employ the metric with signature ( + + + ...+) in D dimensions. ) In addition, he employed a vector field φµ (x) in order to preserve the conformal invariance of the theory; the transformation of φµ that accompanies that of (1) is φµ → φµ + Ω−1 ∂µ Ω.

(2)

As is explained in [6,8], the role of φµ is to ensure that the magnitude of a vector ξ α (l2 = gαβ ξ α ξ β ) transforms as dl = (φα dxα )l

(3)

under a conformal transformation. This in turn implies that the symmetric connection Γαβγ is given by 2

1 Γαβγ = − g αλ (gλβ,γ + gλγ,β − gβγ,λ ) + δβα φγ + δγα φβ − gβγ φα . 2

(4)

The curvature tensor Rµναβ is defined by ξ µ;α;β − ξ µ;β;α = Rµναβ ξ ν

(5)

ξ µ;ν = ξ µ,ν − Γµλν ξ λ

(6)

Rµναβ = −Γµνα,β + Γµνβ,α + Γµλβ Γλνα − Γµλα Γλνβ .

(7)

where

and hence

We also find from these equations that g µν;λ = −2g µν φλ ,

(8)

gµν;λ = +2gµν φλ .

(9)

so that

Consequently, we find that 1 √ ξ µ;µ = √ ( gξ µ ),µ − Dφµ ξ µ g = g µν (ξµ;ν ) − (D − 2)φµ ξ µ

(10)

and therefore ξα;β;γ − ξα;γ;β = Rαηβγ ξ η + 2ξα (φβ;γ − φγ;β ).

(11)

Since ψ;β;γ − ψ;γ;β = 0, we see that if ψ = gτ λ ξ τ ξ λ then ξ α ξ η Rαηβγ + ξ 2(φβ;γ − φγ;β ) = 0 and hence Rαηβγ is not anti-symmetric in the indices α and η. 3

(12)

We now consider the vierbein field eµa defined so that gµν = η ab eµa eνb

(13)

ηab = g µν eµa eνb .

(14)

eµm;λ = −eµm φλ

(15)

eµm;λ = eµm φλ .

(16)

and

By (8) and (9), we see that

and

These equations, when combined with the definition of the covariant derivative of eµm , eµm;λ = eµm,λ − Γµκλ eκm + ωλmn eµn

(17)

show that the spin connection is given by  1 κ 1 1 e m,λ eκa − eκa,λ eκm − eκm eλa,κ + eκa eλm,κ 2 2 2   1 + gπλ eκm eπa,κ − eκa eπm,κ − φa eλm + φm eλa . 2

ωλma = −

(18)

Under the conformal transformations of (1) and (2), we find that eµa → Ωeµa

(19)

Γαβγ → Γαβγ

(20)

ωµmn → ωµmn

(21)

R → Ω−2 R

(22)

and

4

where R = g µν Rµν and Rµν = Rλµλν . A conformally invariant coupling between the gravitational field defined by φµ and gµν and a spinor field ψ is given by S=

Z

¯ cµ iγc ∂µ + 1 ωµab σ ab + D − 1 φµ dD x eψe 2 2 





ψ

(23)

where γc is a set of D-dimensional Dirac matrices satisfying {γa , γb} = 2ηab , and σab = (1/4) [γa , γb ]. The field ψ undergoes the conformal transformation ψ → Ω−(

D−1 2

)ψ

(24)

√ in conjunction with (1) and (2). A term of the form eψ¯ Rψ in (23) would also be conformally invariant, but will not be considered due to its non-polynomial dependence on R. At one-loop order, the effective action for gravity due to the propagation of the spinor field ψ in (23) is given by detD / where D−1 1 φµ . D / = ecµ γc ∂µ + ωµab σ ab + 2 2 



(25)

Since D / is linear in the derivatives, we will replace detD / by det1/2 (D / 2 ). Despite the fact that D / is not Hermitian due to the term proportional to φµ in (23), the anti-Hermitian part of D / 2 does not contribute in two dimensions as is demonstrated by the following calculation in which D / 2 is simplified. If γµ = eµc γ c and Dµ = ∂µ + 21 ωµab σ ab +

D−1 φµ , 2

then by (25),

D / 2 = γ µ ([Dµ , γ ν ] + γ ν Dµ ) Dν

(26)

i h 1 [Dµ , γ ν ] = eνa,µ γ a + ωµab eνc σ ab , γ c 2

(27)

with

where in both four and two dimensions [σab , γ c ] = γa δbc − γb δac . Together, (15), (17) and (27) yield 5

(28)

h

i

[Dµ , γ ν ] = Γνµκ eκp − eνp φµ γ p

(29)

so that h





i

D / 2 = eµq (δ qp + 2σ qp ) Γνµκ eκp − eνp φµ + (g µν + 2σ µν ) Dµ Dν .

(30)

We also can use the relation eµq eκp δ qp Γνµκ = g µκ Γνµκ which by (4) becomes 1 = g νσ,σ + g νλ g µσ gµσ,λ + (2 − D)φν . 2

(31)

√ 1 ∂ g 1 µσ g gµσ,λ = √ 2 g ∂xλ

(32)

Consequently as

we have "

#

1 √ νλ gg D / = √ + (1 − D)φν − 2σ qp eµq eνp φµ + (g µν + 2σ µν ) Dµ Dν . ,λ g 2

(33)

We now find that as 1 σ µν Dµ Dν = σ µν [Dµ , Dν ] 2  i h 1 µν D − 1 1 ab 1 ab cd = σ (34) (φν,µ − φµ,ν ) + σ (ωνab,µ − ωµab,ν ) + ωµab ωνcd σ , σ 2 2 2 4 with [σab , σcd ] = − [ηac σbd − ηad σbc + ηbd σac − ηbc σad ] , ( and, in four dimensions {σab , σcd } = 12 [−ηac ηbd + ηad ηbc + ǫabcd γ5 ]) we have 1 σ Dµ Dν = σ µν 2 µν



1 D−1 Fµν − σ ab Rµνab . 2 2 



(35)

This involves use of the relations Rµνab = −ωνab,µ + ωµab,ν − ωµam ωνmb + ωνam ωµmb with Rµνab = eαa eβb Rµναβ and Fµν = φν,µ − φµ,ν . 6

We also have − 2σ qp eµq eνp φµ Dν = 2Aν D˙ ν

√   1 h ˙ √ =√ Dµ + Aµ gg µν D˙ ν + Aν − D˙ µ ( gg µν ) D˙ ν g i  √  √ − D˙ µ gg µν Aν − gAµ Aµ

(36)

where Aµ = σ µν φν

(37)

1 D˙ µ ≡ ∂µ + ωµab σ ab , 2

(38)

and

 h i  √  √ √ gg µν Aν + 12 gg µν ωµab σ ab , Aν ) as well as ( so that D˙ µ gg µν Aν = ,µ

"

#

1 D−1 1 √ √ √ (g µν g),µ + (1 − D)φν + g µν Dµ Dν = √ D˙ µ ( gg µν ) D˙ ν − g g 2   D − 1 1 √ µν + √ ( gg φµ ),ν . 2 g 

2

φν φν (39)

Together (30),(35) and (36) reduce (26) to "

√     √ 1 ˙ D−1 √ gg µν D˙ ν + Aν − D˙ µ gg µν Aν − gAµ Aµ − Dµ + Aµ D /2=√ g 2     1 D − 1 √ µν D−1 Fµν − σ ab Rµνab . ( gg φµ ),ν + σ µν + 2 2 2 

2

√

gφµ φµ (40)

√ We note that D / 2 is not Hermitian due to the presence of the terms − D˙ µ gg µν Aν + D−1 σµν F µν 2





in (40). However, as we shall show in the next section, these terms do not

contribute to the effective action in two dimensions.

III. THE EFFECTIVE ACTION FOR GRAVITY

In order to compute the effective action for gravity induced by the propagation of a spinor in two-dimensional Weyl geometry, we use the formalism of ref. [5]. In this approach, we consider the operator 7

∆ / = e1/2 (iD /) e−1/2

(41)

where D / is defined in (25). If we make the transforms of eqs. (1), (2), (19 - 22) so that ¯ then it is easy to see that in all dimensions ∆ /→∆ /, ¯ 1/2 ∆ / = Ω1/2 ∆ /Ω

(42)

and thus the formalism used to determine effective action for spinors propagating in a Riemannian background in ref. [5] can be used in the case of a Weyl background; the one difference being that the expansion coefficient E1 (x) given in (A21) must be used in conjunction with the explicit forms for Vµ and X that occur in (40), viz. 1 Vµ = ωµab σ ab + σµν φν 2

(43)

and     1 1√ 1 √ µν 1  ˙ √ µν  √ µ µ gφµ φ − ( gg φν ),µ − σ µν Fµν − σ ab Rµνab . X=√ Dµ gg Aν + gAµ A + g 4 2 2

(44) Since in two dimensions σµν =

1 ǫ γ, 2i µν 5

tr (σ µν ) = 0 and tr (σµν σαβ ) = − 12 (δµα δνβ − δµβ δνα )

we see that " 

1 1  ˙ √ µν  √ 1 1√ tr (E1 (x)) = gφµ φµ tr 2 ωµab σ ab + σµν φν φµ − √ Dµ gg Aν + gAµ Aµ + 4π 2 g 4    1 αβ 1 1 µν  1 √ µν ab Fµν − σ Rµνab − g φα;β − R − ( gg φν ),µ + σ 2 2 3 6 i 1 h (45) R + 2g αβ (φα );β . = 24π √  Since in D dimensions [6,8] R = R˙ + (D − 1)(D − 2)φα φα − 2(D − 1) √1g gφα where 



,α

R˙ is the Ricci scalar in the limit φα → 0, tr (E1 ) =

1 ˙ R, 24π

which is the Riemannian space

limit. (Those contributions to X in (44) that come from the non-Hermitian parts of H have vanishing trace and hence do not contribute to the effective action, as was noted at the end of the preceding section.) The result of (45) is precisely the same as the result that one obtains for a model in which a scalar propagates in a two-dimensional space with Weyl geometry [3,4]. Consequently, the 8

arguments of [5] can be used in the same way that they were in the scalar case, and we obtain Sef f

1 = 24π

Z

!

"

2 √ λ  d x g −σ R + √ + σ2σ gφ ,λ g 2

√

#

(46)

(where σ = ln (Ω−1 ) ). The integrand in (46) reduces to what occurs in Reimannian space in two dimensions as was argued above. Hence in two dimensions, the effective action for gravity induced by the propagation of a spinor in a background with Weyl geometry is identical to what one obtains with Riemannian background.

IV. DISCUSSION

We have analyzed in some detail the formalism appropriate for a spinor field propagating in a space-time with Weyl geometry. Despite the fact that the action for the spinor field is not Hermitian, we have shown that the anti-Hermitian contribution does not contribute to the effective action for gravity in two dimensions that is induced by the propagation of the spinor field. Indeed, the spinor contribution turns out to be precisely equal to that of the scalar field in two dimensions. Currently we are considering expanding our considerations to include a version of supergravity that involves Weyl geometry in the bosonic sector.

ACKNOWLEDGMENTS

We would like to thank the Natural Sience and Engineering Research Council of Canada for financial support. R. and D. Mackenzie posed the questions that stimulated this investigation.

APPENDIX A:

In this appendix we consider the matrix element 9

Mxy = hx|e−Ht |yi

(A1)

1 √ H = − √ (∂µ + Vµ ) gg µν (∂ν + Vν ) + X. g

(A2)

for the operator

In order to compute the coefficient E1 (x) in the expansion [9] Mxx =

i 1 h 2 E (x) + E (x)t + E (x)t + ... 0 1 2 tD/2

(A3)

we employ the approach of [10,11] in conjunction with a normal coordinate expansion, as was employed in the non-linear sigma model [12]. We begin by expanding the metric tensor in normal coordinate about a point x to the order that will eventually be required to obtain E1 (x), gµν (x + π(ξ)) = gµν

 1 κ 1 gµν;λσ + R λµσ gκν + Rκλνσ gκµ ξ λξ σ + ... + gµν;λ ξ + 2 3 λ





(A4)

which, by (8), becomes  1 κ 1 4gµν φλ φσ + gµν (φλ;σ + φσ;λ ) + R λµσ gκν + Rκλνσ gκµ ξ λ ξ σ + ... + 2gµν φλ ξ + 2 3 λ

= gµν





(A5) Consequently, we find that 1 g (x + π(ξ)) = exp tr [ln(gµν )] 2 # " D 1 D2 √ 2 = g 1 + Dφ.ξ + (φ.ξ) + φλ;σ ξ λξ σ + Rλσ ξ λξ σ + ... 2 2 6 

q



(A6)

and so 1 D2 D 1 q = √ 1 − Dφ.ξ + (φ.ξ)2 − φλ;σ ξ λ ξ σ − Rλσ ξ λ ξ σ + ... . g 2 2 6 g (x + π(ξ)) 1

"

#

(A7)

We also will have occasion to use the expansions Vµ (x + π(ξ)) = Vµ + Vµ;λ ξ λ +

1 1 Vµ;λ;σ + Rκλµσ Vκ ξ λ ξ σ + ... 2! 3 

10



(A8)

and X (x + π(ξ)) = X + X;λ ξ λ +

1 X;λ;σ ξ λξ σ + ... 2!

(A9)

For purpose of illustration, we now let X = Vµ = 0 in (A2), so that 1 √ H = − √ ∂µ gg µν ∂ν g 







µν λ σ = 1 + Yλ ξ λ + Yλσ ξ λ ξ σ pµ δ µν + Zλµν ξ λ + Zλσ ξ ξ pν

(A10)

to the order which we need to compute E1 (x). (Here we have defined p to be −i∂ and have assumed gµν (x) = δµν .) In order to expand Mxx in (A1) in powers of t as in (A3), we follow the approach of [10,11]. This involves first using the Schwinger expansion [13] e−(H0 +H1 )t = e−H0 t + (−t) +(−t)

2

Z

1

0

Z

1

0

du e−(1−u)H0 t H1 e−uH0 t

Z

du u

0

1

dv e−(1−u)H0 t H1 e−u(1−v)H0 t H1 e−uvH0 t + ...

(A11)

where H0 is independent of ξ λ , and then converting this into a power series in t by insertion of complete sets of momentum states and then performing the appropriate momentum integrals. Explicitly, we find that to the order that contributes to E1 , 

Mxx ≈ hx| (−t) +(−t) 

2

Z

Z

1

0 1

0

du e−(1−u)p

du u

Z

0

1

2t



dv e−(1−u)p 

β 2



µν α β Yαβ ξ α ξ β p2 + pµ Zαβ ξ ξ pν + Yα ξ α pµ Zβµν ξ β pν e−up 2t





Yα ξ α p2 + pµ Zαµν ξ αpν e−u(1−v)p i

2

× Yβ ξ p + pλ Zβλσ ξ β pσ e−uvp t + ... |xi.

2t

2t

(A12)

If we now insert complete sets of position and momentum states into (A12) we obtain Mxx

  dpdq µν α β α β 2 −i(p−q).z −[(1−u)p2 +uq 2 ]t Y z z q + p Z z z q dz e e ≈ (−t) du αβ µ ν αβ 0 (2π)2D Z Z 1 Z   dpdqdr µν β α −i[(p−q).z1 +(q−r).z2 ] −[(1−u)p2 +ur 2 ]t du +(−t) Y z q Z z r dz dz e e α 1 µ β 2 ν 1 2 (2π)3D 0 Z Z Z 1 Z 1 2 2 2 dpdqdr dz1 dz2 e−i[(p−q).z1 +(q−r).z2 ] e−[(1−u)p +u(1−v)q +uvr ]t dv du u +(−t)2 3D (2π) 0 0 Z



1

Z

Z

× Yα z1α q 2 + pµ Zαµν z1α qν





Yβ z2β r 2 + qλ Zβλσ z2β rσ . 11

(A13)

It is now a straightforward exercise to first integrate over zi in (A13), and then after utilizing the resulting delta functions, evaluate the momentum integrals over p,q and r (all of which are Gaussian). We finally arrive the result Mxx ≈

1 µµ 1 µν D 2 t + − Zνν Yαα − + − Zµν D/2 (4πt) 6 3 6 3       D−4 D D−4 1 +Yα Yα (D + 2) + Yµ Zνµν − + + Yµ Zµνν 48 12 3 24   1 µν µλ 1 µµ νν 1 µλ νν 1 µν µν + . Z Z + Zλ Zν + Zλ Zλ − Zµ Zλ 24 α α 12 48 12 









(A14)

If in (A10) we were to include X and Vµ as in (A2), then by employing the expansions of (A8) and (A9), we find that in order to determine E1 (x), (A14) must be supplemented by Mxx ≈

t (−Vν Y ν − X) . (4πt)D/2

(A15)

µν It is now possible to determine E1 for the particular case in which Yα , Yαβ , Zαµν and Zαβ are

fixed by the expansions (A5 - A9). We find that these imply that Yα = −Dφα

(A16)

2

Yαβ =

D D 1 φα φβ − (φα;β + φβ;α ) − Rαβ 2 4 6

(A17)

Zαµν = (D − 2)g µν φα

(A18)

and µν Zαβ

D2 D 1 (φα;β + φβ;α ) = g − 2D + 2 φα φβ + g µν − 2 4 2  1 µν 1 µν ν µ + g Rαβ − R α β +R α β . 6 6 "

!

µν





(A19)

Together, (A14 - A18) imply that when D = 2, Mxx

1 1 t 2Vµ φµ − X − g αβ φα;β − R . ≈ (4πt) 3 6 



(A20)

By (A20) we see that in two dimensions 1 1 4πE1 (x) = 2Vµ φµ − X − g αβ φα;β − R 3 6 when considering the operator H in (A2). 12

(A21)

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