Rigid and local Lorentz symmetry.

74 CHAPTER 3. THE SPINNING STRING. 3A Rigid and local Lorentz syetry. For the spinning string we need a new bosonic gauge syetry: local Lorentz syetry. For readers who are unfailiar with this syetry, we give in this section an introduction to rigid and local Lorentz syetry for spinors in n diensions. One can divide syetries into rigid and local syetries, or internal and spacetie syetries. For rigid syetries the paraeters are constants, but for local syetries the paraeters are arbitrary functions of the coordinates x of spacetie. For internal syetries one does not transfor the x but for spacetie syetries one ust in general transfor the x and the fields as well. We discuss here the relation between rigid Lorentz invariance which is a rigid spacetie syetry, and local Lorentz invariance which is a local spacetie syetry. Let us begin by discussing rigid Lorentz invariance of the action of spin 0,, fields 2 in flat space Minkovski spacetie, to be definite. In this case one ust transfor both the fields except for spin 0 and the coordinates. Let us begin with the Maxwell case to show this. We define δ L A x = λ A x λ σx σ A 3A. The second ter is due to δ L x = λ x, so the δ L in δ L A acts both on A the first ter and on x the second ter. The first ter is called the spin ter it acts on the indices of the field and the second ter is the orbital ter it acts on x. The relative inus sign between the spin ter and the orbital ter has a deep explanation based on the theory of induced representations which we do not explain. We shall just be content with the stateent that coordinates transfor opposite to fields naely opposite to A and check that these transforations leave the Maxwell and other actions invariant. One can also obtain 3A. as a special case of an infinitesial Einstein transforation general coordinate transforation A x = x x A x, x = x ξ = x + λ x A x A x = δ L A x, δ L A x = ξ x A x + ξ A x = λ A x + ξ A = λ A x λ σx σ A x 3A.2 Let us work out the variation of the Maxwell action. To organize the algebra, we borrow a bit fro general relativity where a Lagrangian density transfors under a general coordinate transforation x x = x ξ into a local derivative δl = ξ L, in general relativity 3A.3 We view δx = λ σx σ as a kind of general coordinate transforation ore on this later, and anticipate that we shall find δ L 4 F = 4 ξ F F 3A.4 = 2 ξ F F Or fixed functions of x, for exaple in the case of translations in polar coordinates.

3A. RIGID AND LOCAL LORENTZ SYMMETRY. 75 where we used that ξ = 0 for ξ = λ σx σ. Using 3A. we find the following result δ L F 4 F = δ L A F 3A.5 = F λ A λ σx σ A We subtract the expected result in 3A.4 fro the result obtained in 3A.5 to see how far off we are λ F A λ F A 3A.6 λ σx σ F A 2 λ σx σ F F The first two ters cancel because they add up to λ F F, and the last two ters also cancel because F A = F 2 F. The inus sign in the second ter in 3A.6 cae fro the inus sign in 3A., so this confirs that rigid Lorentz invariance requires that coordinates transfor opposite to vector fields. Note that the action transfors into a total derivative, see 3A.4; this indicates that rigid Lorentz syetry is a spacetie syetry. Before we turn to spin, let us quickly check the rigid Lorentz invariance of the spin 2 0 action. We now have only an orbital ter but no spin ter because φ x has no indices to act on δ L φ x = ξ φ = λ σx σ φ 3A.7 We find for the variation of the Klein-Gordon action δ L 2 φ φ = φ λ σx σ φ 3A.8 = [ λ σ x σ 2 φ φ ] We used that the ter with λ σx σ = λ does not contribute as it is ultiplied by φ φ, and λ σx σ vanishes, of course. So the Klein-Gordon action also transfors into a total derivative under rigid Lorentz transforations. Now we coe to our ain subject of interest: the Dirac action. The Dirac field transfors into a spin ter and an orbital ter. δ L ψ = 4 λab γ a γ b ψ λ σx σ ψ 3A.9 The variation of the assless Dirac action in flat space under this transforation rule yields δ L L D = δ L ψγ ψ 3A.0 = ψγ 4 λab γ a γ b ψ λ σx σ ψ + ψ 4 λab γ a γ b ψ γ ψ + λ σx σ ψ γ ψ We expect again that this is equal to a total derivative δ L L D = λ σ x σ ψγ ψ 3A. The two spin ters in 3A.0 yield a coutator of the Lorentz generators λ ab γ a γ b with the Dirac field operator γ 4 ψλ ab [γ, γ a γ b ] ψ = ψλ b γb ψ 3A.2

76 CHAPTER 3. THE SPINNING STRING. where we used that [γ, γ a γ b ] = 2δ aγ b 2δ b γ a 3A.3 The two orbital ters in 3A.0 yield two ters with λ σx σ and one ter with λ λ σx σ ψγ ψ + ψγ ψ + λ ψγ ψ 3A.4 It is clear that the last ter in this expression cancels 3A.2, while the two orbital ters indeed produce the total derivative in 3A.. So also the Dirac action is invariant under rigid Lorentz transforations. Note that given either the spin ter or the orbital ter in 3A.9, the other ter is fixed by requiring the action to be invariant. As an exercise one ay prove the rigid Lorentz invariance of the free spin 3 action in 2 flat space L3/2 = ψ γ σ ψ σ 3A.5 Hint: use the generalization of 3A.3, λ ab [γ σ, γ a γ b ] = 4λ σ σ γσ + 4λ γ σ + 4λ γ σ 3A.6 After this rather explicit review of rigid Lorentz invariance in flat space we turn to local Lorentz invariance in curved space. There are now four crucial differences: ˆ the paraeters becoe x-dependent: λ x ˆ coordinates no longer transfor; there are only spin ters δ ll ψ = 4 λab x γ a γ b ψ 3A.7 As we insist that every spacetie syetry ust transfor the coordinates, we conclude that local Lorentz syetry is an internal syetry. ˆ gravitational fields are present. For exaple, in curved space L D = e ψγ e D ψ D ψ = ψ + ω n 4 γ γ n ψ 3A.8 where e = det e, and the vielbein fields e are square roots of the etric in the sense that g = e e n η n. The field ω n x is called the spin connection, and it can be obtained fro the requireent that the total covariant derivative of the vielbein vanishes, e Γ e + ω n e n = 0 3A.9 ˆ Vector indices are divided into flat indices, n,... tangent indices, anholonoic indices, and curved indices,... base anifold indices, holonoic indices. Fields with flat vector indices transfor under the spin part of Lorentz syetry. One uses the vielbein fields e to transfor flat vector indices to curved vector indices, and back. For exaple, δ ll A x = λ n x A n x δ ll A x = 0 } A = e A A = e A 3A.20

3A. RIGID AND LOCAL LORENTZ SYMMETRY. 77 ˆ Spinor indices are always flat. Thus ψ transfors as in 3A.7. 2 Let us now analyze the invariance of the spin 0,, actions under general coordinate 2 transforations Einstein transforations, and local Lorentz transforations. Just as one needs a gauge field the vielbein field, or the etric if one generalizes translations to general coordinate transforations local translations, one ay anticipate that one needs a new gauge field if one generalizes rigid Lorentz syetry to local Lorentz syetry. This gauge field should transfor into the derivative of paraeters, λ n, plus ore, so it ust have the index structure ω n. The Einstein invariance of the Dirac action is rather easy to prove. Recall that g x = e x e n x η n, so g is equal to e = det e. Thus in L D = e ψγ e ψ 3A.2 the cobination e ψ is an Einstein scalar because e for fixed is a contravariant vector, and ψ is a covariant vector, assuing, as we did, that ψ is an Einstein scalar. Thus also ψγ e ψ is an Einstein scalar. It follows that e ψγ e ψ is an Einstein scalar density, hence δ E e ψγ e ψ = ξ L D 3A.22 under Einstein transforations x x ξ. Under local Lorentz transforations the action in 3A.2 is not invariant: if one varies the field ψ in ψ one obtains ters with λ n x which we shall try to reove by introducing a gauge field ω n for local Lorentz syetry, replacing with a covariant derivative D. Direct evaluation of the variation of 3A.2 under the following local Lorentz transforation rules yields δ ll ψ = 4 λpq x γ p γ q ψ 3A.23 δ ll e = λ n x e n δ ll L D = e ψγ e 4 λpq x γ p γ q ψ e ψγ λ n x e n ψ + e ψ 4 λpq x γ p γ q γ e ψ We used that δ ll det e = 0, which follows fro We also used that ψ transfor as 3A.24 δ ll det e = ee δ ll e = ee λ ne n = eλ = 0 3A.25 δ ll ψ = ψ 4 λpq γ p γ q 3A.26 2 One can construct a so-called superspace with couting coordinates x and anti-couting coordinates θ A, θȧ. Then there are supervielbein fields VΛ M where Λ =, A, A are curved and M =, A, A are flat superindices. One can then define spinor fields with flat spinor indices which transfor under local Lorentz transforations not local super-lorentz transforations because this leads to a field theory with ghosts, and spinor fields with curved spinor indices which transfor under general super-coordinate transforations super-einstein transforations. We do not discuss superspace in this note.

78 CHAPTER 3. THE SPINNING STRING. which follows fro ψ = ψ iγ 0 and δψ = 4 λpq γ p γ q ψ = 4 ψ γ qγ pλ pq = 4 ψ γ pγ qλ pq 3A.27 Note that γ pγ qiγ 0 = iγ 0 γ p γ q for exaple γ 0γ iγ 0 = γ 0 γ iγ 0 = iγ 0 γ 0 γ. So we get δ ll L D = 4 λ pq e ψγ e γ p γ q ψ 4 eλpq e [γ, γ p γ q ] ψ e ψγ λ n x e n ψ 3A.28 The last two ters cancel. To show this, it is helpful to introduce the notation ψ = e ψ. With 3A.3 one gets eλ pq ψγq p ψ e ψγ λ n n ψ which indeed cancels. So, as announced, we are left with a ter with λ n δ ll L D = 4 λ pq e ψγ e γ p γ q ψ 3A.29 We now odify the Dirac action by adding a ter with ω n which replaces by D L D = e ψγ e D ψ D ψ = + 4 ω n γ γ n ψ 3A.30 We shall require that D ψ transfor in the sae way as ψ under a local Lorentz transforation thus without λ n ter. This will fix how ω n transfors δ ll D ψ = 4 λn γ γ n D ψ 3A.3 If this is the case, the local Lorentz invariance of 3A.8 is clear: it follows fro the cancellation of the two last ters in 3A.28 but this tie with D ψ instead of ψ. Writing 3A.3 out gives the following equation for δ ll ω pq pq in ters of ω δll ω pq γp γ q ψ + 4 ω pq γ p γ q 4 λn γ γ n ψ 4 λpq γ p γ q ψ + 4 = 4 λn γ γ n ψ + ω pq 4 γ p γ q ψ 3A.32 Rearranging ters yields an equation for δ ll ω pq δll ω pq 4 γ pγ q ψ = λ pq 4 γ pγ q ψ + 6 λn ω pq [γ γ n, γ p γ q ] ψ 3A.33 The coutator of two Lorentz generators yields again Lorentz generators the Lorentz algebra closes [γ 2 n, γ pq ] = η np γ q η nq γ p η p γ nq + η q γ np 3A.34 where γ n = γ 2 γ n γ n γ. So δll ω pq γ 4 pγ q ψ = λ q + 2λ pω pq γ 4 qψ 3A.35 We can reove the factor 4 γ pγ q ψ and find then how ω pq should transfor under local Lorentz transforations: δ ll ω pq = λ pq ω p p λ p q ω q q λ pq = D λ pq 3A.36 We have thus obtained a satisfactory result as far as local Lorentz invariance is concerned

3A. RIGID AND LOCAL LORENTZ SYMMETRY. 79 ˆ fields transfor only with a local spin ter ˆ ordinary derivatives are replaced by local-lorentz-covariant derivatives ˆ the gauge field ω pq transfor into the local-lorentz-covariant derivative of the local-lorentz paraeter. However, we odified the action when we added the ter containing ω n, so we ust go back to Einstein syetry, and check that we did not loose it. Under Einstein transforations the covariant derivative of ψ D ψ = ψ + 4 ω n γ γ n ψ 3A.37 transfor as a covariant vector, provided we define that ω n which is a new and independent field, so we have the freedo to define how it transfors transfors like a covariant vector δ E ω n = ξ ω n + ξ ω n 3A.38 Recalling δ E ψ = ξ ψ because ψ is an Einstein scalar, yields δ E D ψ = ξ ψ + ω n 4 γ γ n ξ ψ 3A.39 ξ ω n + ξ ω n γ γ n ψ + 4 In order that D ψ is a covariant vector, it should transfor as δ E D ψ = ξ D ψ + ξ D ψ 3A.40 If one stares for a oent of both expressions, one sees that they are equal. So, luckily, the Dirac action in curved space L D = e ψγ e D ψ D ψ = ψ + ω n 4 γ γ n ψ 3A.4 is both local-lorentz and Einstein invariant. So far the field ω n was taken as a new independent field first-order foralis, but one can also construct a coposite field ω n e fro the vielbein field e which transfors the sae way under both local syetries as the independent field ω n. This construction is discussed in the next appendix, but the final result is that one iposes the vielbein postulate e + ω ne n Γ e = 0 3A.42 Taking for Γ Christoffel sybol { }, one can solve for ω n, and one finds in this way ω n e. Another way is to require that the field equation of the independent field ω n be satisfied. This field equation is algebraic it contains no derivatives of ω n, and, in fact, it is equal to the vielbein postulate. To see this we begin with L = er n ω e e n 3A.43 where R n ω is the Rieann curvature in ters of the spin connection R n = ω n + ω pω pn 3A.44

80 CHAPTER 3. THE SPINNING STRING. and vary the field ω δ ω L = e D ω δω n D ω δω n e e n 3A.45 The structure of this result should be failiar: curvatures always vary into the covariant derivative of the variation of the gauge field. To solve this equation one ay use a trick which siplifies the algebra a lot: we decopose without loss of generality ω into ω e + ω, where ω is still arbitrary, but we still denote the arbitrary variation of ωe+ ω by δω. We also add for free Christoffel ters in 3A.45 because they cancel due to the antisyetry in,. δ ω L = e ˆD δω n ˆD δω n e e n e ω pδω pn + ω n pδω p e e n where ˆD is the fully covariant Einstein and local Lorentz covariant derivative 3A.46 ˆD δω n = δω n { } δω n + ω p e δω pn + ω n p e δω p 3A.47 Partial integration of the first ter in 3A.46 yields ters with ˆD e, ˆD e and ˆD e which vanish due to the vielbein postulate ˆD e n = 0. We are only left with the ters with ω p. Since δω pq is arbitrary, we obtain ω p δ n q ω n pδ q n, e e n = 0 3A.48 We should have added the ters due to, but antisyetrizing e e n in, is equivalent to antisyetrizing in, n, and the two ters with ω are already antisyetric in, n. It is easy to show that 3A.48 iplies that ω n = 0. If one adds atter actions to the gauge action in 3A.43, and if these atter actions contain spin connections ω n, then one obtains a nonvanishing result for ω; the ters with ω are torsion ters, and in supergravity theories they are quadratic in the ferionic fields. Thus we find that the solution of the Euler-Lagrange field equation for the independent field ω n in 3A.43 is just the field ω n e which satisfies ˆD e = 0. This is the sae ωe as one obtains by solving the vielbein postulate ˆD e ˆD e = D ωe D ωe = 0 3A.49 We leave as an exercise to check that also the Klein-Gordon and Maxwell actions are in curved space invariant under Einstein and local-lorentz syetry. In the Maxwell action with A, there is nothing to prove for local Lorentz syetry because A does not transfor, but if we write it in ters of A = e A, one obtains F n = e e n F = D A n D n A D A n = A n + e ω n q ea q, = e 3A.50 We end with soe coents.

3A. RIGID AND LOCAL LORENTZ SYMMETRY. 8 ˆ the rigid Lorentz transforations of A in 3A. are a special case of Einstein transforations } δ ll A = λ A λ σx σ A ξ = λ σx σ. 3A.5 δ E A = ξ A + ξ A but this is not true for Dirac fields. ˆ the local Lorentz transforations have no relation to Einstein transforations, for exaple they do not contain the transport ter ξ φ. ˆ we leave again as an exercise the proof that the spin 3 2 local Lorentz and Einstein invariant. action in curved space is ˆ For vielbein fields, we can write Einstein transforations as a derivative of the paraeter ξ = ξ e plus ore δ E e = ξ e + ξ e = ξ e + ξ e e = ξ + ξ ω nee n + ω nee n = ξ + ω neξ n ξ ω ne e n = D ωe ξ λ ne n with λ n = ξ n ω 3A.52 We used in the third line the vielbein postulate D ωe e D ωe e = 0. So, an Einstein transforation of e can be written as a generalized gauge transforation paraeters + ore plus a local Lorentz transforation with paraeter λ n = ξ ω ne. Often one considers the difference of an Einstein transforation and this local Lorentz transforation as the basic syetry; it reads δe = D ωξ and is then for obvious reasons called a local covariant translation.