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Supersymmetry in Euclidean Quantum Field Theory

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\¤.. ..,,~ "’" ·-* *` "¥-’ S cu 9)% C) 2 KL—TH—93 / 8 SUPERSYMMETRY IN EUCLIDEAN QUANTUM FIELD THEORY Z. Haba Institut of Theoretical Physics University of Wroclaw, Wroclaw, Poland J. Kupsch Fachbereich Physik Universitat Kaiserslautern, Kaiserslautern, Germany Supersymmetry is formulated within a functional approach of euclidean quantum Eeld theory. The role of the euclidean Lie superalgebra and its relation to the Poincaré Lie superalgebra are investigated in detail. As example we study the Wess—Zumino model in two dimensions. iiiilllillilliixiilll m l\\\\\\\\\\\\\\\§l\§L E,,52 OCR Output - 1 OCR Output1. Introduction Supersymmetry of a relativistic quantum field theoretical model is usually based on the supersymmetry of a classical Lagrangean (with anticommuting fermi fields) and on a (formal) representation of the Poincaré Lie superalgebra on the physi cal state space, see e.g. [37,40,43,46]. These prerequesites are sufficient to derive a perturbation expansion with supersymmetric Ward identities for the time ordered Green’s functions in a Schwinger—Symanzik type formalism of the generating functio nal or with a Berezin type of integration, see e.g. [14,37]. On the other hand for functional integration the euclidean approach is more adequate. A euclidean formulation of the Wess—Zumino (WZ) model in four dimensions has been given by Nicolai [32], and a functional version of this model in two dimensions based on euclidean supertields can be found in the lectures of Jaffe and Lesniewski [19]. But these approaches do not incorporate the role of the euclidean Lie superalgebra which has been studied e.g. in [27,28]. The significance of this algebra and the type of its representation in euclidean quantum field theory is not clarified in the literature [29]. In this paper we present a general definition of euclidean supersymmetry which incorporates the euclidean Lie superalgebra and which treats bosons and fermions with the same techniques of tensor algebras. To avoid all problems with euclidean field operators or with stochastic fields for ferrnions we use the functional point of view which goes back to Osterwalder and Schrader [34] in a formulation related to the Wightman functional of Borchers [2]. The generating functional of all Schwinger functions is then a linear functional on the free tensor algebra T(S) of test functions of dN a space S = S(|R) 0 C. The symmetry properties (either totally symmetric or totally antisymmetric) of the Schwinger functions yield an ideal J of equivalent test functions. The factor algebra A(S) = T/J is a H., graded algebra or superalgebra which is isomor phic to A"”(SB) 0 ,{_(SF). Here SB and SF are the basic test function spaces for the bosonic or fermionic fields, S = SB sn SF, and A€(’l) denotes for e = +1 the algebra of symmetric tensors and for e = -1 the algebra of antisymmetric tensors (Grassmann OCR Output - 2 algebra) of the space 'l. There is an abundand literature about graded algebras (superalgebras), graded manifolds and superanalysis, the list [4,7,15,20,%,38,39,41, 42,44] is not nearly complete. Nevertheless we shall present the mathematics of the superalgebra of test functions in some detail in Sect. 2,, because the cited literature is either too general or it elaborates on superspaces (mainly based on Grassmann algebras) which are not adequate for our construction. In Sect. 3 the generating functional of all Schwinger functions is presented as linear functional S on the test function algebra A(S): F 6 .l(S) -•<S | F> E C, such that the restriction to A", the space of all tensors of rank n, exactly yields the n—point Schwinger functions. Then Osterwalder-Schrader (OS) positivity, euclidean invariance and supersymmetry are formulated for these functionals. The investigation of super symmetry starts with a discussion of the complex euclidean Lie superalgebra which includes the real Lie algebra of the euclidean transformations. The complex euclidean Lie superalgebra 6" is represented by differential operators L on the basic test function space S. The corresponding superderivations Dr, L e é", provide then a representation of that Lie superalgebra on the whole algebra of test functions .4(S). A theory with generating functional S is supersymmetric if <S|DT_F> = 0 for all test functions F 6 A(S) and all superderivations DT, L 6 E`. The representation of the complex eucli dean Lie superalgebra is continuous in the nuclear topology of the test function alg¢+ bra. All efforts to impose a reality/hermiticity condition on the representation leads to serious difficulties which can be overcome only by modifying the algebra [29]. Hermi ticity with respect to the OS form singles out the real Poincaré Lie superalgebra and ensures supersymmetry of the reconstructed minkowskian theory. Supersymmetry is easier formulated with auxiliary bosonic degrees of freedom. Also in our approach the auxiliary degrees of freedom are necessary for a simple linear representation of the Lie superalgebra. The corresponding Schwinger functions cannot OCR Output - 3 .. be derived from a nonnegative bosonic measure. But all "unphysical" features disap pear in the nontrivial kernel of the OS form if one goes over to the reconstructed minkowskian theory. There is no need to modify the theory by introducing imaginary auxiliary fields as was done by Nicolai [32]. In the literature the euclidean quantum field theory for Majorana spinors is mainly based on the construction of field operators [10,32,3]. Here we shall follow a purely functional point of view which avoids the problems with euclidean field opera tors or with fermionic stochastic fields [10,24]. The generating functional for the Schwinger functions of massive Majorana fermions can be defined for d = 2,3,4 mod 8 dimensions, i.e. exactly when minkowskian Majorana spinors exist [25]. For these dimensions a real unitary charge conjugation matrix C can be dehned which satisfies the conditions o* =-c and oy,,o=T F (1.1) for all hermitean Dirac matrices yu, jr: 1,...,d, ·y·y+·y·y= 26. The two point uVVuW Schwinger function of the free Majorana field operator, i.e. the analytic continuation of d -1 <vac| ¢»·d:|vac>, is given by S(x,y)C with S(x,y)[] = u'=]_m — E P7 N8 (x,y) (m —3)—*(x,y). In writing formal euclidean lagrangeans we may therefore substitute the adjoint Majorana spinor %(x)aa6 by E Gg 1pH(x) as in the minkowskian case and in agreement with the operator construction in [32]. In Sect. 4 we study as example the neutral WZ model in d = 2 dimensions. The formal euclidean lagrangean of this model is, see [47,5,31] for the minkowskian ver sion, OCR Output - 4 E = [free + [mv fm., = - i A(¤=) A Mx) - A B(¤=)2 + ¤¤ A(¤¤) B(¤<) -·i ¢(¤¤)T C (¤¤ -9) 1/(X) —5<A<»=>.B<x>>[ A] m []]] _13,.+A x ,l ¢ T(X) T c on —»v>¢<»>. um Lim = »\A‘(x)B(x) — A1/:*(x) C ¢(x) A(x). (1.3) Here A and B are bosonic fields and ¢(x) is a two component anticommuting Majorana spinor. Our formalism is used to study supersymmetry of the free WZ model and of perturbation theory. Moreover we indicate how the auxiliary degrees of freedom can be eliminated, and finally the generating functional is transferred to functional integrals of a type investigated by Jaffe et al. [17] and by Arai [1] in their hamiltonian solution of the WZ model. In Appendix A we give some norm estimates for the algebra of test functions. In Appendix B we discuss the two point Schwinger functions of free charged or neutral spin—0 or spin——; particles including the problem of OS positivity. Some emphasis is laid on a construction to derive the functionals for neutral particles from those of the charged ones. Appendix C presents some formulas which are needed for the calculation of euclidean functionals for Majorana fermions. OCR Output - 5 .. 2. Graded Algebras In this section we introduce the basic H., graded algebra (superalgebra) of test functions, its topology and its dual algebra. The main references are [4,41] and, concerning the topological structure, [13]. 2.1. Algebras of Test Functions Test functions and Schwinger functions are defined on E = Q1 ¤ Il“ where *21 is a inite set with |2l| elements which label spin and field indices. The basic function space is now the complex linear space 6`(E) of rapidly decreasing C`”—functions. This space is a dense subspace of the Hilbert space 'l = £‘(E) with the inner product (fls) = ITG) s(£) dé = 2 I ¥(¤.><) s(¤=»¤¤) dx (2-1) where { = (a,x) E E, and the measure d§ includes Lebesgue integration with respect to x E Ru and summation over or 6 Ql. The norm is denoted by Finer topologies can be dened with the differential operator of the shifted harmonic oszillator Wf(a,x) = (1 + x‘ -· A) f(a,x) (2.2) 2 d 2 . with x = E x and A = E JE . This operator has been extensively used in white noise calculus [13]. This operator satisfies W gQ3 f;-and W_* is a Hilbert Schmidt operator on 1. Then ||f||“ = ||WPf]|, p 6 H is a family of norms on 6`(E) which satisfy ||f||5p ||f||+1.p We denote by 1()p = W"P’l the closure of 8 with the norm ||f\| The spaces 1)( P are ordered by inclusion (p1+1c 1,) and( 1(P)P ) is antidual to ’l(_p) with respect to the pairing (2.1). Since the embedding 1(+1)P c P1() is given by the Hilbert—Schmidt operator W—* the intersection = flpp=0 1() is a nuclear space in the projective limit topology, see e.g. [11]. The antidual space is the space of tempered distributions S’(E)pit) (_p).= 1 OCR Output To define a H., gradation the index set 21 = 210 U 21, is split into nonempty sub sets 210 and 21,, 21,, n 21, = G.
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  • Perturbation Theory in Celestial Mechanics

    Perturbation Theory in Celestial Mechanics

    Perturbation Theory in Celestial Mechanics Alessandra Celletti Dipartimento di Matematica Universit`adi Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) ([email protected]) December 8, 2007 Contents 1 Glossary 2 2 Definition 2 3 Introduction 2 4 Classical perturbation theory 4 4.1 The classical theory . 4 4.2 The precession of the perihelion of Mercury . 6 4.2.1 Delaunay action–angle variables . 6 4.2.2 The restricted, planar, circular, three–body problem . 7 4.2.3 Expansion of the perturbing function . 7 4.2.4 Computation of the precession of the perihelion . 8 5 Resonant perturbation theory 9 5.1 The resonant theory . 9 5.2 Three–body resonance . 10 5.3 Degenerate perturbation theory . 11 5.4 The precession of the equinoxes . 12 6 Invariant tori 14 6.1 Invariant KAM surfaces . 14 6.2 Rotational tori for the spin–orbit problem . 15 6.3 Librational tori for the spin–orbit problem . 16 6.4 Rotational tori for the restricted three–body problem . 17 6.5 Planetary problem . 18 7 Periodic orbits 18 7.1 Construction of periodic orbits . 18 7.2 The libration in longitude of the Moon . 20 1 8 Future directions 20 9 Bibliography 21 9.1 Books and Reviews . 21 9.2 Primary Literature . 22 1 Glossary KAM theory: it provides the persistence of quasi–periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.e. a non– degeneracy of the integrable system and a diophantine condition of the frequency of motion.