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S O/RS. OPT-??/P. I^Lй s O/RS. OPT-??/P. I^lé \1^3OO3T>2/ ' DUALITY FOR QUANTUM FREE FIELDS John ROBERTS ** Daniel TESTARD1*** 78/P.1016 JULY 1978 " Centre de Physique Théorique, CNRS Marseille "" Universitat Osnabriick, Fachbereich 5, C-45 Osnabrilck, R.F.A. """Centre Universitaire d'Avignon and Centre de Physique Théorique, CNRS, Marseille POSTAL ADDRESS : C.N.R.S. - LUMINY - Case 907 Centre de Physique Théorique F - 13288 MARSEILLE CEDEX 2 (France) INTRODUCTION nti==siwn Duality is one of the most intriguing concepts in algebraic field theory. First proposed by R. Haag [l,2] , it can be interpreted as a converse to the principle of local commutativity according to which ob­ servables Aj and A2 localized in spacelike separated regions 8f and 6£ must commute since the measurement of A, cannot interfer with the measurement of A., . Duality strenghtens this by claiming that, conversely, an observable A commuting with all observables localized spacelike to ff must Itself be localized in & . Expressing this in terms of the von Neumann algebra 9H.9") generated by the observables localized in & , we have where & denotes the spacelike complement of v . Araki succeeded in verifying duality for the free neutral scalar Boson field. His proof consisted of two parts. In the first part [3j, fcp examined the algebraic aspects of second quantization and characterized the commutant of the von Neumann algebra R(M) generated by tne Weyl operators based en a closed real subspace H of the 1-particle space as R(M') , where M1 denotes the symplectic complement of M. Thus R(M)' » R(H'). This theorem, which is of interest in its own right, is referred to here «s abstract duality. Eckmann and Osterwalder [4] had the good idea of simpli­ fying this proof by using Tomita-Takesaki theory. They identified the anti- unitary involution J associated with the faithful vacuum state of R{M) as the second quantization of an antiunitary involution j of the 1-particle space so that the proof could be summarized : R(M)' - 0 R(H)J • eJR(M)e^ - R(jH) - R(M'). 78/P.1016 -b- However the form of j is relatively complicated and,the operator J of Tomita-Takesaki theory is best described as the polar part of the antilinear involution S : j? » T A ** .The second-named author, hearing of [4J attempted to reconstruct the proof for himself and began by writing S » «• ft • Since where Vi(h) denotes the Weyl operator associated with h e M , it is clear that we should define Shs -Ji ,. h & M (see the proof of Theorem 1.3.2). Mow second quantization preserves polar decompositions (Proposition 1.1.3) , so if is 3* '» is the polar decomposition of S , we deduce J s <*•* and û1'0- » *< . These unpublished results art- the basis of Section I. An alternative proof of abstract duality has been given by Rieffel [5] using an abstract commutation theorem and implictly by Dell'Antonio [6] using infinite tensor products. Abstract duality reduces the proof of duality for free Bose fields to a discussion of the real subspaces M t&) of the 1-particle space associated with the space-time regions & . The case of the free neutral scalar field of mass -»w>0 was dealt with by Araki in [7j and his proof was extended by Dell'Antonio [6 J to charged Bose fields and Bose fields of higher spin. Kis discussion of the mass-zero case is however Incomplete. We are primarily interested in the free electromagnetic field where the cons­ traints on the Cauchy data mean that some topological restriction on is necessary if duality is to hold. Our duality theorem (Theorem II.2.7) applies to contractible regions 0" and is complemented by a counterexample in Section II.3 for a region that is not simply connected. Partial results for the free electromagnetic field have been obtained independently by Benfatto and Nicolo [B]. For contractible regions, the proof of duality can anyway be simplified by using dilatations and this applies even if the theory is not itself dilatation invariant as we illustrate by treating the free neutral scalar field of mass m^.0 in Section II.4. Another proof of duality which deserves mention is that of Landau [9j. Ke Bakes effective use of an Idea of Osterwalder [lO] that Araki's ori- 78/P.1016 -c- ginal proof [3] using an expansion of an operator in terms of annihilation and creation operators should be replaced by a manifestly local expansion in terms of Wick polynomials in the time-zero field and its conjugate momentum. For physicists familiar with the techniques of Mick ordering, Landau's proof has the merits of conciseness and intelligibility. However it neither contains the abstract duality theorem nor does it prove the full concrete duality theorem since the question of outerregularity is not treated. There have been many other results on duality which are not treated here. Thus twisted duality, the form of duality appropriate to Fermi fields, was proved for the free Fermi fields in [11, Appendix] using the structural results of [6]. The relationship between duality for the field algebra and duality for the observable algebra is also discussed in [llj. Undoubtedly the most important development is due to Bisognano and Wickmann [l2,13J who proved duality for a special case of regions, the wedges, i.e. the Poincaré transforms of the region f*<r ("*• '• * * &• **•* j Their arguments are of a general nature and only require the algebra of ojservables to have been suitably derived from a field thf.ory. These as­ sumptions have been verified by Driessler and Frohlich [11J for a class of 2-dimensional models. Bisognano and Hickmann also attempted to prove duality for double cones but arrived instead at a property ca1led essential duality in [l5], namely that the net (& of algebras defined by ©C6>> s 6lL&')' satisfies duality. However the most important application of duality, the theory of superselection structure [l6,17j, can be derived under the weaker assumption of essential dualicy [15]. Furthermore, as shown in [18], duality for double cones is in fact incompatible with spontaneously broken gauge symmetries of the first kind. 78/P.1016 I. ABSTRACT DUALITY This section is devoted to giving a proof of results appearing in £3} . We first recall the basic definitions of the exponential of an Hilbert space. For more details, see e.g. [,20 3* (21J 1.1. Exponentiation of a complex Hilbert space Let H be a complex Kilbert space with k.k -^ Cf>. R ) as scalar product. The Bose-Einstein-Fock-space constructed from H is denoted by ©•" in the following (*). The scalar product in fr 1s denoted by 41, ^-^L+i v) • Let k fc- U and e-^ be the vector in ê.** whose component on the n-particle subspace 1s (*\!) *• 1\ © » - • €>k (n times). It is well known that the vectors e^ , U e- H are linearly independent and satisfy tt,ll) (1) (*U . ** ) - Moreover the set \t^ \ KfH|iS total in e" . It follows from (1) that k.->«.k- is continuous from H to <," . The formula defines a linear operator on the linear subspace generated by W(h) is an isoraetry and therefore extends to a unitary operator on C also denoted by W(h), called a Weyl operator on * " - The Keyl ope­ rators satisfy the following canonical commutation relations: (x) Another widely used notation is P {,Vl) 78/P.1016 -2- If dim H • 1 , this structure is described as follows J2o): Proposition I.1.1 S may be identified with L (JR.) in such a way that for ^ft. eût *. V/^) is multiplication by ô' and W^fcJ is translation by —t /I . For «£»« <V > / , the following proposition exhibits the tensorial nature of e . Proposition 1.1.2 Assume H s .&? *•* . There exists a Hilbert isomorphism from _, a onto the Infinite tensor product QP C * containing the . O- n *** .1 vector® £ * where Ui is the null vector in H; . «'«X , This isomorphism transforms S into .Qv c and yf(£ h;) into <S iVYV) '/(All these infi- nite tensor products are convergent in the appropriate sense). We omit the proof of this Lemma.(See poj). a An interesting class of operators on & is discussed in the following. Proposition 1.1.3 Let A be a linear (resp. antilinear) densely defined, closed operator from H to K . Let us define two -a priori unbounded- linear (resp. if tf antilinear) operators from fi to C as follows : 78/P.1016 -3- i) ïhe domain of e, is the subspxegenorated by £ A f'*^ and ii) The restriction of St to the subspace of e generated by tensors of the form his-—» A (n-times) ^*»5tJ is ACS--- CBA (n-times). These operators satisfy the following properties : i) <2, and £2 are densely defined, closable and have the same closure denoted by É '"'. 11) If H = K , A 1s linear and self-adjoint (resp. positive) then <£ is self-adjoint (resp. positive). iii) If A = U |A/ is the polar decomposition of A, then the polar A decomposition of £. is A U JAI e r £. e iv) (eA)* , eA* v) If H = tf, © //», and A = A, <&At then BC92f : ^rom tne Hilbert space structure on A* , it is sufficient to consider the case of linear operators. A A i) G is densely defined and so is C2 by a polarization ar­ gument. We prove that ( £t j ^ g™ . It is sufficient to see that ^ £d{(e,V**) and A .A (e, )** 4*"" = #*/*' (") e 1s often denoted by P(A) . 78/P.1016 -4- for /j g- SVJ . We can proceed inductively with respect to n.
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