0 T (L int (x 1 )...L int (x n )) = i

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1 LORENTZ INVARIANT RENORMALIZATION IN CAUSAL PERTURBATION THEORY K. BRESSER, G. PINTER AND D. PRANGE II. Institut für Theoretische Physik Universität Hamburg Luruper Chaussee Hamburg Germany Abstract. In the framework of causal perturbation theory renormalization consists of the extension of distributions. We give the explicit form of a Lorentz invariant extension of a scalar distribution, depending on one difference of space time coordinates. 1. Introduction Causal perturbation theory was founded by the work of [1], [2] and [3]. In this approach, the S-matrix is constructed as a formal functional expansion in a so-called switching function of the interaction. The coefficients in the expansion, determined by induction, are time ordered products of operator valued distributions. The advantage of this approach in comparison with the derivation of the S-matrix in the Hamilton formalism is that the time plays no distinguished role, and only a few physical properties of the S-matrix like causality, locality and unitarity are assumed. The definition of the time ordered products in the S-matrix requires an extension of their domain of definition which corresponds to renormalization. This is briefly discussed in section 2. To obtain a Lorentz invariant S-matrix, the extension has to be Lorentz invariant. The existence of such an extension was first proved in [3], later it was discussed in [4] and [5] as a cohomological problem. In spite of the knowledge of its existence the explicit form of this extension has not been calculated up to now. We fill this gap, giving a solution for a distribution in one argument in section 3. In the last section we calculate the dependence of the Lorentz invariant extension on the auxiliary function needed in the extension procedure. 2. From the S-matrix to the Extension of Distributions In causal perturbation theory the S-matrix is expanded as a functional of the testfunction g DR 4 )switching the coupling: i n S g) =1+ d 4 x 1... d 4 x n T L int x 1 )...L int x n )) g x 1 )...gx n ). 1) n! n=1 The coefficients of this expansion are the time ordered T -) products of the interaction part L int of the Lagrangian density. They are symmetric operator valued Lorentz invariant distributions on a dense domain of the Fock space of free fields. A T -product of given order n is defined by all T -products of lower order up to the total diagonal ={x 1,...,x n ) x i =x j i, j} through causality: T D R 4n \ ) = 0 T. Using Wick's theorem 0 T can be decomposed into a sum of 1

2 2 K. BRESSER, G. PINTER AND D. PRANGE products of Lorentz invariant, translation invariant numerical distributions and Wick ordered free field operators, 0 T L int x 1 )...L int x n )) = i 0 t i x 1,...,x n ):A i x 1,...,x n ):. 2) Note that for noncoincident points 0 t is a product of Feynman propagators. According to theorem 0 of [3] it is sufficient to extend the numerical part 0 t i to the diagonal to find T. This is the inductive causal construction. An extension of the numerical distribution 0 t always exists, but the uniqueness of the extension depends on the singular behaviour of 0 t at the total diagonal [6]. By choosing difference coordinates, the singularity is shifted to the origin. The singular behaviour at the origin is described by the scaling degree and the singular order. Definition 2.1. The scaling degree of a numerical distribution t at 0 is defined by { } := inf lim ɛ t ɛx) =0. 3) ɛ 0 Definition 2.2. The singular order ω of a numerical distribution t at 0 is defined by ω =[ d] 4) where d is the space time dimension and [x] denotes the largest integer n with n x. With D ω R 4n ) we denote the subspace of testfunctions vanishing to order ω at 0: D ω R 4n ) = { ψ D R 4n) D α ψ0) = 0 }, 5) where α is a multiindex. Distributions with singular order ω are well defined on D ω [6]. The extension of the numerical distribution to D is achieved with the help of a projection operator, acting on the testfunctions. Definition 2.3. To each ω N 0 and each w D0 fulfilling w0) = 1, D α w0) = 0 multiindices α with 1, we associate a projection operator W ω, w) : D R 4n) D ) ω R 4n φx) φ x) wx) x α Dα φ 0). 6) W is a modified Taylor subtraction operator. Since the function w has compact support, the result is a function with compact support and vanishes to order ω at 0. Therefore the distribution 0 t with singular order ω is defined on all W ω, w)φ. The extension of 0 t to D has the following form [6]: t, φ = 0 t, W ω, w)φ + c α Dα φ0). 7) If the singular order ω of 0 t is negative, the extension is unique, in the other case there are free constants c α in the extension procedure. The form of the c α can be restricted by demanding invariance of t under symmetry operations, e.g. under the action of the Lorentz group.

3 LORENTZ INVARIANT RENORMALIZATION 3 3. The Lorentz-Invariant Extension in Scalar Theories It is possible to determine the free constants c α of an extension t such that Lorentz invariance of 0 t is preserved [3, 4, 5]. We denote the action of the Lorentz group on R 4n by x D Λ) x. This leads naturally to an action on D R 4n ) and D R 4n ), respectively, given by D R 4n) φ φ Λ, φ Λ x):=φ D Λ 1) x ) D R 4n) t t Λ, t Λ,φ := t, φ Λ 1. According to equation 7), the extension has the form t, φ = 0 t, φ w x α Dα φ0) + c α Dα φ0). 8) After performing a Lorentz transformation we obtain t Λ,φ = t, φ Λ 1 = 0 x α t, φ w Λ Dα φ0) + DΛ)c) α D α φ0). 9) In order to be a Lorentz invariant extension, the difference of 8) and 9) has to be zero: D α φ0) t, w w Λ ) x α = DΛ) 1) c) α D α φ0). 10) So we have to solve t, w w Λ ) x α = DΛ) 1) c) α 11) for all α, where c α is a tensor of rank α and DΛ) is the corresponding tensor representation of the Lorentz group. From now on, we restrict ourselves to the case of distributions in one coordinate. In this case only the totally symmetric part of the c α contributes to 8). Using Lorentz indices, 11) reads t, w w Λ ) x µ 1...x µn = Λ µ 1 β1...λ µn β n µ ) 1 β 1... µn β n c β 1...β n 12) where n = α. Using infinitesimal transformations we can solve these equations for c inductively. In the case α =0, 11) is fulfilled for all choices of c, since the 1-dimensional representation of the Lorentz group is trivial. For α 1, the solution is unique up to Lorentz invariant contributions consisting only of symmetrized tensor products of the metric tensor g µν which generate Lorentz invariant counterterms like x)). We use the generators of Lorentz transformations l αβ ) ν µ = gβν α µ g αν β µ. 13) The representation of a Lorentz transformation on R 4 has the form Λ ν µ = ν µ Θ αβl αβ ) ν µ + OΘ2 ) 14) with infinitesimal parameters Θ αβ satisfying Θ αβ = Θ βα. We obtain wx) wλ 1 x)= 1 2 Θ αβl αβ ) ν µ xµ ν wx)+oθ 2 ). 15) We use the abbreviations n!! = { n for n even n for n odd 16)

4 4 K. BRESSER, G. PINTER AND D. PRANGE and b α 1...α n) = 1 b απ1)...απn) 17) n! π S n for the total symmetric part of a tensor b. Now we prove by induction over α that the symmetric part of c α for α > 0 is given by up to the aforementioned ambiguity for even α ): c α 1...α n) = n 1)!! n +2)!! [ n 1 2 ] s=0 n 2s)!! n 1 2s)!! gα 1α 2...g α 2s 1α 2s 0 t, x 2 ) s s+1...x α n 1 x 2 αn) w x αn) x β β w ). 18) At the beginning of the induction we determine the c α for α =1and α =2. 1. α =1. We obtain DΛ) 1) c) ν = 1 2 Θ αβl αβ ) ν µ cµ which yields independence of Θ αβ ) Inserting the form 13) of the l αβ ) yields 0 15)11) = t, 12 Θ αβl αβ ) ρµ xµ ρ wx ν, 19) l αβ ) ν µ cµ = 0 t, l αβ ) ρ σ xσ ρ wx ν. 20) g βν c α g να c β = 0 t, x α β w x β α w ) x ν. 21) Contracting finally with g νβ on both sides yields c α = 1 t, x 2 α w x α x β β w. 22) 3 2. α =2. We obtain from 12) t, w w Λ ) x α 1 x ) α 2 = DΛ) α 1 β1 DΛ) α 2 β2 α 1 β 1 α 2 β 2 c β 1β 2. 23) With 14) and 15) we get 0 t, l αβ ) ρ µ xµ ρ wx α 1 =l αβ ) α 1 σ c σα 2 +l αβ ) α 2 σ c σα 1. 24) Inserting the form 13) of the l αβ ) and contracting both sides with g βα1 yields 4c αα 2 g αα 2 c µ µ = 0 t, x α x σ σ w x 2 α w. 25) As in the case α =0we choose c µ µ =0and obtain by setting α = α 1 ): c α 1α 2 ) = 1 t, x α 1 x σ σ w x 2 x α 1 α2) w. 26) 4 We now assume that 18) holds for all integers smaller than n and describe the induction α = n 2 α =n. With the examples of the beginning of the induction, it is easy to see how the calculation is done for higher α. For α = n we obtain instead of 21) the following equation: n g βα i µ α g αα i ) µ β c µα 1...ˇα i...α n = i=1 = 0 t, x α β w x β α w ) x α 1...x αn. 27)

5 LORENTZ INVARIANT RENORMALIZATION 5 On the left hand side only the symmetric traceless part of the c α yields a contribution. Contracting 27) with g βα1 yields n +2)c αα 2...α n i 2 g αα i c ρ ρα 2...ˇα i...α n = = t, x α x ρ ρ w x 2 α w )...x αn. 28) The second term on the left hand side of 28) is determined by the induction hypothesis, because it is the solution of the problem for α = n 2 for the distribution x 2 t. Setting α = α 1 and symmetrizing in the indices α 1...α n, the form 18) is obtained. 4. Dependence on w If we had chosen another testfunction w DR 4 ), our result should differ only by Lorentz invariant counterterms. To use the functional derivation, we notice, that the difference of two admissible auxiliary functions has to be in D ω R 4 ). Definition 4.1. The functional derivation is defined by w Fw),ψ := d F w + λψ) dλ, λ=0 for all ψ D ω R 4 ). Applying that definition to the first term of 7), we get: t, W ω, w)φ,ψ = w Now we determine the contribution from c: w cα 1 α n w),ψ = = 1 σ x σ x α1 x αn) α 1 x α2 x αn) x 2)0 t+ n+2 + n 1 σ x σ x α1 x α n 2 η α n 1α n) x 2 + n )0 α 1 x α2 x α n 2 η α n 1α n) x 2 ) 2 t+ t, x α ψ D α φ0). 29) +. + n 1) 2 n 3 n 1) 3 n 4 σ x σ x α 1 η α 2α3 η α n 1α n) x 2 ) n α 1 η α 2α3 η α n 1α n) x 2 ) n+1 2 σ x σ x α 1 η α 3α4 η α n 1α n) x 2 ) n 2 ) 0 t, ψ n odd, 2 + α 1 η α 3α4 η α n 1α n) x 2 ) n 2 ) 0 t, ψ n even 30)

6 6 K. BRESSER, G. PINTER AND D. PRANGE Since the singular order of every term is ω n we are allowed to differentiate strongly. The first line reads σ x σ x α1 x αn)0 t α 1 x α2 x αn) x 20 t =n+2)x α1 x αn)0 t n 1)x 2 η α 1α 2 x α3 x αn)0 t + x α1 x αn) x σ 0 σ t x 2 x α1 x α n 1 αn)0 t. 31) The Lorentz invariance condition 0 t, x [α β] ψ =0, ψ D ω R d )allows to write x 2 αn 0 t = x σ x σ αn 0 t = x σ x αn σ 0 t, so the last line of 31) vanishes. The second line of 30) is proportional to σ x σ x α1 x α n 2 η α n 1α n) x 20 t α 1 x α2 x α n 2 η α n 1α n) x 2 ) 20 t =nx 2 η α 1α 2 x α3 x αn)0 t n 3)x 2 ) 2 η α 1α 2 η α 3α 4 x α5 x αn)0 t, 32) again two terms vanish because of Lorentz invariance of 0 t. Putting the n 1 in front of 32) n and adding to 31) we get n +2)x α1 x αn)0 t+ n 1 times the second term from 32). This n term cancels against the next line from 30), and so on. For n odd, the last line becomes n 1) 2 x 2 ) n 1 2 x α 1 η α 2α3 η α n 1α n)0 t, which again cancels the line before. For n even the last n 5 line is n 1) 3 ) 4x 2 ) n 2 2 x α 1 η α 3α4 η α n 1α n) x 2 ) n 2 η α 1 α2 η α n 1α n) 0 t, n 4 where again the first term is canceled by the line above but the second gives a nontrivial contribution. We end up with w cα 1 α n w),ψ =+ 0 t, x α1 x αn ψ + { 0, n odd, t, x 2 ) n 2 ψ η α 1α2 η α n 1α n), n even. Using this result and 29) we find: t, φ,ψ = w where we set d 0 =1. d n := 2n 1)!! n+2)!! ω n=0 n even 2n 1)!! n +2)!! d n n! n 2 φ0), t, x 2 ) n 2 ψ, 5. Conclusion and Outlook We give the explicit form of Lorentz invariant Epstein-Glaser renormalized distributions in one argument. In case of distributions depending on more than one argument the same calculation yields the totally symmetric coefficients c. But then the other coefficients do not vanish in general. In ϕ 4 -theory this is already sufficient for the renormalization up to third order. The general case for Lorentz invariant and Lorentz covariant distributions is treated in [7]. Acknowledgements We thank Prof. K. Fredenhagen for useful hints and Michael Dütsch for reading the manuscript.

7 LORENTZ INVARIANT RENORMALIZATION 7 References [1] E.C.G.Stueckelberg and D. Rivier, Helv. Phys. Acta, ) 215. E.C.G.Stueckelberg and J. Green, Helv. Phys. Acta, ) 153. [2] N.N. Bogoliubov and D. Shirkov, Introduction to the theory of quantized fields, John Wiley and Sons, 1976, 3rd edition. [3] H. Epstein and V. Glaser, The role of locality in perturbation theory, Ann. Inst. Henry Poincaré -Section A, Vol. XIX, n )p.211. [4] G. Popineau and R. Stora, A Pedagogical Remark on the Main Theorem of Perturbative Renormalization Theory, unpublished preprint. [5] G. Scharf, Finite Quantum Electrodynamics: The Causal Approach, Springer-Verlag, 1995, 2nd edition. [6] K. Fredenhagen, Renormierung auf Gekruemmten Raumzeiten, lecture notes R. Brunetti and K. Fredenhagen, Interacting Quantum Fields in Curved Space: Renormalizability of φ 4, gr-qc/ , Proceedings of the Conference 'Operator Algebras and Quantum Field Theory', held at Accademia Nazionale dei Lincei, Rome, July R. Brunetti and K. Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, math-ph/ [7] D. Prange, Lorentz Covariance in Epstein-Glaser Renormalization, to appear.