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Lorentz Covariance in Epstein-Glaser Renormalization

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Lorentz Covariance in Epstein-Glaser Renormalization LORENTZ COVARIANCE IN EPSTEIN-GLASER RENORMALIZATION DIRK PRANGE II. INSTITUT FÜR THEORETISCHE PHYSIK UNIVERSITÄT HAMBURG LURUPER CHAUSSEE 149 D-22761 HAMBURG GERMANY EMAIL: ABSTRACT. We give explicit and inductiveformulas for the constructionofa Lorentz covariant renormalization in the EG approach. This automatically provides for a co- variant BPHZ subtraction at totally spacelike momentum useful for massless theories. INTRODUCTION Our introduction is rather brief since this paper mainly is a continuation of [BPP99]. There we a gave a formula for a Lorentz invariant renormalization in one coordinate. We use the same methods to construct a covariant solution for more variables. We review the EG subtraction and give some useful formulas in section 1. The cohomological analysis that leads to a group covariant solution is reviewed in section 2. In sections 3, 4 we give an inductive construction for these solutions in the case of tensorial and spinorial Lorentz covariance. Since the symmmetry of the one variable problem is absent in general some permutation group calculus will enter. Necessary arXiv:hep-th/9904136v2 27 Apr 1999 material is in the appendix. In the last section 5 we give a covariant BPHZ subtraction for arbitrary (totally spacelike) momentum by fourier transformation. 1. THE EG SUBTRACTION We review the extension procedure in the EG approach [EG73]. A more general introduction can be found in [Pra99], a generalization to manifolds in [BF96, BF99]. Let Dω(Rd) be the subspace of test functions vanishing to order ω at 0. As usual, any operation on distributions is defined by the corresponding action on testfunctions. Work supported by the DFG Graduiertenkolleg “Theoretische Elementarteilchenphysik” Hamburg. 1 2 DIRK PRANGE Define Rd ω Rd ϕ ϕ W(ω;w) : D( ) D ( ), W(ω;w) , → α→ ϕ ϕ x ∂ ϕ 1 W(ω;w) (x) = (x) w(x) ∑ α α w− (0), (1) − α ω ! | |≤ with w D(Rd),w(0) = 0. If 0t is a distribution on D(Rd 0 ) with singular order ∈ 6 ω \{ } ω then 0t can be defined uniquely on D (Rd), and ϕ . 0 ϕ t(ω;w), = t,W(ω;w) (2) defines an extension – called renormalization – of 0t to the whole testfunction space, d t ω D (R ). With the Leibnitz rule ( ;w) ∈ ′ ∂ γ 1 ∂ ∂ γ( fg) = ! ∑ ν µ f νg, (3) µ+ν=γ µ! ! we find α ∂ x ∂ ϕ 1 ∂ ϕ γ w ∑ α α( w− )(0) (0) = γ (0), (4) α ω ! | |≤ for γ ω, verifying the projector properties of W. The W -operation (1) is simplified | | ≤ if we require w(0) = 1 and ∂αw(0) = 0, for 0 < α ω (this was our assumption in | | ≤ 1 1 [BPP99]). It can be achieved by the following V -operation (∂µw− means ∂µ(w− )): µ d d . x 1 Vω : D(R ) D(R ), (Vωw)(x) = w(x) ∑ ∂µw− (0), (5) 7→ µ ω µ! | |≤ where w(0) = 0 is still assumed. We can write W as 6 α ϕ ϕ x ∂ ϕ W(ω;w) (x) = (x) ∑ α Vω α w α (0). (6) − α ω ! −| | | |≤ The extension (2) is not unique. We can add any polynomial in derivatives of δ up to order ω: α ϕ ϕ a ∂ ϕ t(ω;w), = t(ω;w), + ∑ α α (0), (7) α ω ! | |≤ or rearranging the coefficients α ϕ c ∂ ϕ 1 = t(ω;w), + ∑ α α w− (0) (8) α ω ! | |≤ α α α ω Since W(ω;w)(wx ) = W(ω;w)(x Vω α w) = 0 for , c resp. a are given by −| | | | ≤ α α α α a = t(ω;w),x Vω α w , c = t(ω;w),x w . (9) −| | LORENTZCOVARIANCE IN EPSTEIN-GLASERRENORMALIZATION 3 They are related through: µ µ α α c 1 α α a a = c ∑ ∂µw− (0), c = a ∑ ∂µw(0), 1 α ω, µ ω α µ! µ ω α µ! ≤| | ≤ | |≤ −| | | |≤ −| | µ µ 0 c 1 0 a a = ∑ ∂µw− (0), c = ∑ ∂µw(0). (10) µ ω µ! µ ω µ! | |≤ | |≤ The equation for a follows from the Leibnitz rule in (8), while the equation for c is derived from (9). In quantum field theory the coefficients a are called counter terms. They are not arbitrary. They have to be chosen in such a way that Lorentz covariance of 0t is pre- served in the extension. This follows in the next sections. The remaining freedom is further restricted by discrete symmetries like permutation symmetry or C,P and T symmetries. At the end gauge invariance or renormalization constraints will fix the extension uniquely. But up to now there is no local prescription of the latter. 2. THE G-COVARIANT EXTENSION We will first define the notion of a G-covariant distribution. So let G be a linear transformation group on Rd i.e. x gx, g G. Then 7→ ∈ α α β α x g βx =(gx) (11) 7→ denotes the corresponding tensor representation. G acts on functions in the following way: . 1 (gϕ)(x) = ϕ(g− x), (12) so that D is made a G-module. We further have g(ϕψ)=(gϕ)(gψ), (13) α 1 α 1 x ∂α(g− ϕ)=(gx) g− (∂αϕ), (14) α 1 α x ∂α(g− ϕ)(0)=(gx) ∂αϕ(0). (15) 0 d Now assume we have a distribution t D′(R 0 ) that transforms covariantly under the Group G as a density, i.e. ∈ \{ } 0t(gx) detg = D(g)0t(x), (16) | | where D is the corresponding representation. That means: . 0t,gψ = D(g)0t,ψ = D(g) 0t,ψ . (17) 4 DIRK PRANGE We will now investigate what happens to the covariance in the extension process. We compute: 1 D(g) t ω ,g− ϕ t ω ,ϕ (18) ( ;w) − ( ;w) 0 1 0 = D(g) t,W ω g− ϕ t,W ω ϕ (19) ( ;w ) − ( ;w) α (6) 0 1ϕ x ∂ 1ϕ 0 ϕ = D(g) t,g− ∑ α Vω α w α(g− )(0) t,W(ω;w) (20) * − α ω ! −| | + − | |≤ α (13,15) 0 1 ϕ x ∂ ϕ 0 ϕ = D(g) t,g− ∑ α gVω α w α (0) t,W(ω;w) (21) * − α ω ! −| | + − | |≤ ∂ ϕ (17) 0 α I α (0) = ∑ t,x ( g)(Vω α w) α (22) α ω − −| | ! | |≤ . α ∂αϕ(0) = ∑ b (g) α . (23) α ω ! | |≤ Then (23) defines a map from G to a finite dimensional complex vectorspace. Now we follow [SP82], [Sch95][chapter 4.5]: Applying two transformations α 0 α b (g1g2) = t,x (I g1g2)(Vω α w) (24) − −| | 0 α = t,x ((I g1) + g1(I g2 ))(Vω α w) (25) − − −| | α 0 α = b (g1) + detg1 t(g1x),(g1x) (I g2 )(Vω α w) , (26) | | − −| | and omitting the indices we see b(g1g2) = b(g1)+D(g1)g1b(g2), which is a 1-cocycle for b(g). Its trivial solutions are the 1-coboundaries b(g)=(I D(g)g)a, (27) − and these are the only ones if the first cohomology group of G is zero. In that case we can restore G-covariance by adding the following counter terms: G cov . 1 α t ,ϕ = t ω ,ϕ + ∑ a (w)∂αϕ(0). (28) (ω−;w) ( ;w) α α ω ! D E | |≤ The task is to determine a from (27) and (22): 0 α α t,x (I g)(Vω α w) = (I D(g)g)a (29) − −| | − 3. TENSORIAL LORENTZ COVARIANCE The first cohomology group of L+↑ vanishes [Sch95][chapter 4.5 and references there]. We determine a from the last equation. The most simple solution appears in the case of Lorentz invarinance in one coordinate. This situation was completely 0 analyzed in [BPP99] for ∂αw(0) = δα. The following two subsections generalize the results to arbitrary w, w(0) = 0. 6 LORENTZCOVARIANCE IN EPSTEIN-GLASERRENORMALIZATION 5 4 3.1. Lorentz invariance in R . If we expand the index α into Lorentz indices µ1,...,µn, (29) is symmetric in µ1,...,µn and therefore a will be, too. We just state our result from [BPP99] which is modified through the generalization for the choice of w: n 1 [ − ] n 1 !! 2 n 2s !! (µ1...µn) ( ) ( ) (µ1µ2 µ2s 1µ2s a = − ∑ − η ...η − (n + 2)!! (n 2s 1)!! × s=0 − − β 0 2 s µ2s+1 µn 1 2∂µn) µn) ∂ t,(x ) x ...x − x x x β Vω nw , (30) × − − D E if we choose the fully contracted part of a to be zero in case of n being even. We used the notation 1 1 b(µ1...µn) = ∑ bµπ(1)...µπ(n) , b[µ1...µn] = ∑ sgn(π)bµπ(1)...µπ(n) , n! n! π Sn π Sn ∈ ∈ for the totally symmetric resp. antisymmetric part of a tensor. 3.2. Dependence on w. Performing a functional derivation of the Lorentz invariant extension with respect to w, only Lorentz invariant counter terms appear. Definition. The functional derivation is given by: δ . d F(g),ψ = F(g + λψ) . δg dλ λ=0 This definition implies the following functional derivativ es: δ ϕ ψ 1 αψ ∂ ϕ 1 δ t(ω;w)( ), = ∑ α t(ω;w),x α w− (0), (31) w − α ω ! | |≤ δ 1 α 1 S,Vωw ,ψ = ∑ S,W ω (x ψ) ∂αw− (0), (32) δw h i α! ( ;w) α ω | |≤ for any distribution S. Proof. We show how to derive the first relation. Inserting the definition we find: α α d 0 x 1 x 2 t ω λψ (ϕ) = t, ψ ∑ ∂α ϕw− (0) + w ∑ ∂α ϕψw− (0) , dλ ( ;w+ ) − α! α! λ=0 * α ω α ω + | |≤ | |≤ 6 DIRK PRANGE using Leibnitz rule and rearranging the summation in the second term, 1 0 ψ α∂ ϕ 1 = ∑ α t, x α w− (0) α ω !* − | |≤ ν α∂ ϕ 1 x ∂ ψ 1 +wx α w− (0) ∑ ν ν w− (0) ν ω α ! + | |≤ −| | 1 0 α ψ ∂ ϕ 1 = ∑ α t,x W(ω α ;w) α w− (0) − α ω ! −| | | |≤ 1 αψ ∂ ϕ 1 = ∑ α t(ω;w),x α w− (0), − α ω ! | |≤ α ϕ αϕ where we used the relation x W(ω α ;w) = W(ω;w)(x ) on the last line.
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