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*
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 «£»«
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
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. using the following limits
<**£:' «>';*
The same argument using the limits : A** - *
allows us to prove that ^ ft ^ (ft. y and 0 *s Proved. ii) is well-known, ft^ is a direct sum of operators which are essen tially self-adjoint by Nelson's analytic vectors theorem. We omit the s A proof of the positivity of G. when A is positive. iii) From i) one sees that ((e*)É J O- * 6~* an—d Ê- A ^- & o- «2 ~*& It follows that
(**) e s e
But self-adjoint operators are maximal and consequently from ii) 0e ) e - e. By the same argument ,
Since e is self-adjoint and positive then JAI t = /« /
78/P.1016 Moreover from i)
U M/ O /AI A
The assertion iii) results now from the followingfrivlal remark : the.,, . closure of the range of C can be realized in e as € S> 6 where *5I ^) is the closure of thfl range of A and the tens-ir product is introduced as in Proposition 2 with respect to the decomposition K c K*rA*G> ft(*)
Before proving iv), first remark that, by the same argument as before, A / A\* A** e (* J D e Consequently by ii)
and
A stan .iri property of the polar decomposition gives 1**1 *JO /*/ / t/'* •e. e (e"f Therefore * * oU*' J *JfA*l ^U*^U/»* „A»// U\* e"A = e «''rA « e e * O" 0 I*) U* lAj . «''«"«"V , t'*.o*\ (.«*-)
(*">
78/P.1016 -6-
We have used
which follow from the boundedness of U . v) is easy using for example iv). This completes the proof of Proposi tion 1.1.3.
1.2 REAL SUBSPACES OFft COMPLEX HUBERT SPACE
As we shall see, in field theory, it is interesting to con sider real subspaces of H .We gather together in this section, the results we shall use in the following. These results are very simple but we shall prove them for the sake of completeness.
Definition 1.2.1 Let H be a subset of K , The symBleçtiç_çomDlernent M' of M is the set of i é H such that IK (kfb.) so if h «• M .
Proposition 1.2.2 i) H1 is a closed, real subspace of H. ii) If flcAf . then N C M . iii) M" is the closed real subspace of H generated by H. iv) (t*, + il*l) = M CI i M «the set of vectors orthogonal to M. v) (v)' s ] Ptf' C «' If one of these conditions is true, then Proof : i) and ii) are obvious. iii) follows from the fact that H" is the biorthogonal of M with respect to the real Hilbert space structure given on H by the real part of the scalar product. iv) One sees immediately that L n r (t M/ . Sobyi) (M fi M)'C M'nin' If JM f»,fc)=o and ^('^ ^)~°, then Ch, k) =. o so the set of vectors orthogonal to M contains M fli" . If h is orthogonal to M then, T-v» ( U *,+i'fej.) s O for $; é M i. =/, i and Ji &(M+iM)' and iv) is proved. v) follows 'ran iii). vi) The equivalence of the stated properties can be directly verified. Moreover (PMJ'OM'S PM1 and The converse inclusion is trivial. We now give a definition which is of interest for the proof of abstract duality. 78/P.1016 -8- Pefinition 1.2.3 A real closed subspace M in H is said to be standard if M r\ i Mal°J and if M+ifJ is dense in H . If M is a standard subspace of H • the following map defined (because M r»îM»i^ )on ^+'M °y is said to be the Ç|ngniçal_invglutign_gf H. BC9fi9§iïi9D.iiiiî Let H be a standard subspace of H. Then : 1} The canonical Involution A of M is a densely defined, closed antllinear involution. ii) M' is standard and the canonical involution of M' is the adjoint f of -j . iii) If 4 — I »J is the polar decomposition of *C , then /-I . jfr*S*j and j(K)=M\ iv) Let P be an orthonormal projection in H such that PM CM then PM (resp. (l-P)M) is standard in P^H (resp. (l-P)H). If «*j (resp. ^44 ) is the canonical involution of PH (resp. (l-P)M) then Standard subspaces and their canonical involutic.is are stu died in £l9i in connection with the Tomita-Takesaki modular theory of von Neumann algebras. If Wl is a von Neumann algebra with a cyclic and and separating vector ii , denote by 4W* the Hermitian part of 7JL , then_the real subspace fljj.lî. is standard, the canonical invo lution^ W^iL is the operator S of the nodular theory, 78,..1016 •9- <)*L, • In that case, the Toraita theorem says that the polar decomposition f s ° 5 s J A* satisfies :k J VlJ : W and à VL A"'"** WL EC29f-2f-ErSD95iïiSS.Ii2i3- : ^) is obvious. ii) M' is standard by Prop.' 1.2.2 (iii), iv), v )). Let f. be the canonical involution of H'. If ' A/ *• 6 M and k'fll e f*' , we have , = CAl*').-;^v;-i r*,* ;-A/*V .-c A *; *,*'-;*') Consequently fc V Conversely, «4 is a densely defined, antilinear, closed involution. Let ffs Jit €«>{•**); *+h. hi Then £)/^*J»^iV. More over, lit At//- , A é-M , then (A» = («% *; r 6M>> = (*,>>) Therefore Aé« , ff C M and <* C f . iii) Let k £M . Then Consequently if k 6 Jtf : 78/P.1016 -10- Therefore (j */ *•) is reel and JM c A»' . The following equalities are true on A3*"'M : Therefore : -4- . • fW and The density of M *• l M and the boundedness of / gives V = JT everywhere. This relation can be used to prove j />1 s M ' because iv) Using PA' /"»,) = «' ?V>*,/ , we see that ?M +i ?AJ a 7>(M+iM) is dense in PH. On the other hand. T>M a i- ?M « MAM =/«>>, P M is closed, therefore Pl*1 is standard in ?H. Moreover {1-P)l*!c M and (1-P)/^ is standard in (l-P)H. The relation 4s^,«^ fol lows from a direct computation. The following proposition will enable us to reduce most ques tions to the case of standard real subspaces. Proposition 1.2.5 Let M be a real closed subspace of H. Let P (resp. P') the ortho- 78/P.1016 -11- gonal projection on the complex subspace orthogonal to M (resp. to M') Then : i) P P' =0 and, consequently, Q = I-P-P' is an orthogonal pro jection in H; ii) PK » {0} VW - PH iii) P"H* • •{oj P'ti » F-'H iv) OM is standard in QH v) ON' - (QH)' O QH. Proof : By Proposition I.Z.Z, the range of P (resp. P") is M'/II'M* (resp. A) ni M }. Therefore i) follows from the fact that M<1lM'=/o} because <*M , i A e M' implies (h.J,)s Pc- (k.K) = 3», (t\ /,)s O. One has PW'CMOI'M'C/»'. By Proposition 1.2.2 vi), P?JcM. So ?W C M n 9 H s Mnt'M'n M's/oJ. By the same argument ?'H' - {oj . PH is a complex subspace » therefore {.rMj sQ-P)H. it follows that (M+ .- M J " = (TH)' a Q-rj H and A3*' ^7 is dense in (7-?,^ . Then, since Q £• T-T* and ?'$][-? , ?'(M+t'M)= ?M *-i'P'W is dense in P'H and Q(M+iM) =• QM+i QM 1s dense in QH. We prove now ?'/; +i ?'tf * 9'rf . If £ =. ?'h +'>'* is an element of 7>'AJ +i7>'f*) , then * P'jfc € ?''*/=» Mn lM C.M. So and l. ?'h+ip'k = T'(k*l^k) € T? /? The same argument as before allows us to show that 6?M »»" QM is dense in QH and that Pjl' - PH. 78/P.1016 -12- v) follows from Prop. 1.2.2 vi). In order to complete the proof of Prop. 1.2.5 It suffices to note that 2 M ni fi M = {*>} This follows easily from v) and Prop. 1.2.2 v). 1.3 THE VON NEUMANN ALGE8RA ASSOCIATED WITH REAL SU8SPACES Definition 1.3.1 Let X be a subset in a complex Hllbert space H. The von Neumann alge bra of operators acting in S generated by I VJ(k)•; li feUV is denoted by (ft. (*!} . In practice we only consider the case where H. is a closed real subspace in H. The following results are standard. For H= For the general case, an easy application of Proposition 1.1.2 shows that @L[H) is the set 3>(c^) f>' all bounded operators acting on e .We can now state the following theoreu [3j. Theorem 1.3.2 Let H be a complex Hilbert space, M a real subspace of H and M the closure of M. Then „ & (M) * d (iï) iij e is cyclic for (jL(M) if and only if tf+'M 1s dense in H. 78/P.1016 -13- iii) S. is separating for Ct(M) if and only if, Mfl»'«i {ù} iv, A M ' • & CM') v) (3L(»)v CL(i))" = &(»**) vi) #M « ^^;= foftinX) and consequently (Rlrfis a factor if and only if h) n M'a. { C$ • lliii '• i) follows froa the strong continuity of *• —s» Therefore, without loss of generality, we consider H closed in the fol lowing. Using Proposition 1.2.5 and Proposition 1.1.2 with respect to th» decomposition H * "P H <3 T'H © QH ? (£^ = # > * ® *^ (In the above, Ût(QM) and ^/^ act on c ). ii) Assume that /*!*« AI is not dense in H. Then there exists an h ^ o which is orthogonal to M . For any k e M , then : (s] yffk) e') = e * € ' , « * -(e^(U)e°) Therefore the vector e - € , which is not zero, is orthogonal to &(!*) e° and e" is not cyclic for (JL(M) . Conversely if M+i tf is dense, then ?eO and 78/P.IQ16 -14- it is sufficient to prove ii) in the case where M is standard. Actually we can prove the following more precise result. Lemma 1.3.3 '-at H , M be as in theorem 1.3.2. Assume moreover that M is closed and standard in r/ . Let si be the canonical involution of iM (which is standard). Then the set { tf (*>) t-°I A eMJ -js total in 0. with respect to the graph norm of C° This lemma is well known when H - <£ - fi\ *-• TZ-sftsTZ. Actually in that case € is the complex conjugation of functions in since 6 is a real function and e. has a real Fourier transform. So in that case the graph norm is equivalent to the ordinary norm in LZ{K) and e" is cyclic in s" for (ZftZ). Applying proposition 1.2.1 iv) to r the orthogonal projec tions on A 6 M , one reduces the general case to the preceding one. The vector can be approximated by linear combination of vectors of the form Vf(t{k) e" ol £ IS by the cyclicity of e° in the one dimensional case. The conclusion follows from the density of M-i-i'M and the continuity of A —» e . iii) is an easy consequence of ii) and iv) v) is trivial and vi) is a consequence of v) and iv). Thus it is sufficient to prove iv). For this, one can assume, without loss of generality, that M is closed and standard. In that case $ÎM) has B° as a cyclic and separating vector and the set $0C**) of linear combinations of Vf (h) } A £ A7 is a x-algebra which is x-strongly dense 1n (JtfN} therefore CSot**^^ is a core for the Tomita operator S . $0CM)& is also a core for £' as seen in lemma 1.3.3. One has : 73/P.1016 -15- therefore e-4 Hence by proposition 1.1.3and Toraita-Takesaki theory where J is the polar p.art of -d . Consequently, (R(fl) is gene rated by .£•> Yf(h) 4* A eft). But for all k. £ H : Thus is generated fay and the conclusion follows from Proposition 1.2.4 iii) applied to - ^ which is the canonical involution of /7 . This completes the proof of Theorem 1.3.2. 78/P.1016 -16- DUALITY FOR FREE FIELDS II.l Statement of the Problem Starting from a Wightman field (ft on Minkowski space, we then rapidly review the standard notation and assumptions. Let J denote the complex test functions space 3 s £ ®C£-('R'>) where £ is a finite dimensional Hilbert space and <£-(V) denotes the Schwartz space of complex CM-functions with fast decrease on 3R . The field Positivity of the spectrum, locality, cyclicity and invariance of the vacuum are also assumed. _ J on d>££)*2 $£ f) . In the decomposition jj.» C<90£^?9, J appears as J (fax) - f&J* 78/P.1016 -17- where i is an antilinear involution on £" and £ the conjugate of •$. as above. Here we shall only be concerned with free fields ; these are nightman fields which are solutions of the Klein-Gordon equation (D+»'J^ = a (or the wave equation if <*» s O ). For free fields we have : Proposition II.1.1 Let ($t $) « (aJ 4>fy Ht)**-) (two-point functions). Let K* be the completion of the quotient space of 0 by the set of vectors ). of J which satisfy (j,£) *($,{ ) = O and cV+ the canonical mapping from J into Hi * . Then, there exists an isomorphism from onto & * such that i) H s V e° ii) If I e C/y ^s- Jf , then 4>(f) is essentially self- adjoint on and w 1^(4-)) * V exp(L*lt))v" 78/P.1016 -18- This result shows that the local von Neumann algebras gene rated by a free nightman field are algebras of the form (R(M) where M is a real subspace of the one particle space T* . Thus we can use the abstract duality theorem in Section I to study the duali ty properties of the free field algebras. However this task is facilitated by choosing a different representation of the one particle space /f^. . To this end we follow Araki and let /C be the Hilbert sum of K+ and /f. . /C is the completion of the quotient space of \j by the set of vectors JL of J with zero length with respect to the scalar product ( ?,$) s C4'S)+ *"$'$)- ini tne canonical mapping a( from J into /f is «(f)* *+li) +«-(*) /*? Let *P be the projection of K onto /f+ and /"* the unique anti- unitary involution on /{* defined by : Let «C % be the real Hilbert space of P -'nvariant vectors of K : Then, since PPs Q-7*) f* , « -9 'fïrk, is an isomorphism of Ke.K with K+ considered as a real Hilbert space. 'We will regard A£ K as our one particle space using this isomorphism to put a complex Hilbert space structure on Ke. \" . Thus multiplication by t is given by ! and the complex scalar product by (k,k>) * *(*,*) This formalism may be found in £23}. 78/P.1016 . — -19- Kere "C denotes the Kerraitian form on /C defined by Note that 0 corresponds to the comnutator function : Other useful aspects of the one-particle space are now mentioned. Le.Tf.ia 11.1.2 (Relativistic invariance) The capping d if) -* « (*>A*f) j * f , fa *} * %' defines a strongly continuous unitary representation (A. of the restricted Poincaré group on A* which commutes with the operators "P and P (and thus, its restriction to ne/f is also a strongly continuous unitary representation of this group in n-e K }. With the same notations as in Section I and Proposition II.1.1, we have Ufa*) ml e = U u («,*) V Lemma II.1.3 Let p be a real positive function p e- such that Jj%(rfc)c 2"A - .£&*; © f we define on \7^ J^tt**) » /• 78/P.1016 -20- Then i) given Fe J4 , the following limits exist in H" and are independent of p : .&*- tf (F® 4) (fc(*) ii) For 5? S ft* , F €r "Xt where rfim - rcf-z) iii) If (A* is a bounded measure on 1R and then Lfr) pi?) « £ (p"*r) L (IL) p(p) - {& ^=*-r; Proof By the temperedness of the field, the scalar products (£ a V can be written as follows : 78/P.1016 where £ and B are Fourier transforms of spin components of £ and q and Mt" are polynomially bounded measures with support in Then, putting and F^/t^j) « fj/t; F^ ««eh., , /[«fcj- «ft';//-- £ Jj*jtr)(?(£)-f'(Ç))*%iï)W9 * ' A/ Using the properties of fl , polynomial boundedness of U-2' and the Lebesgue dominated convergence theorem, we deduce that <* ^^/ is a Cauchy sequence in tÇ whose limit is independent of p . The same method applies for &(?) and the rest of the Lemma is standard p2J . We turn now to describe the local von Neumann algebras associated with a free field. If 0" is a closed double cone, i.e. the Poincaré transform of a set of the form {x. , /*/ +!*•'! £ ccj for some a- > O , then we define 0\ ["} to be the von Neumann algebra generated by : If F is a more general subset of Minkowski space, we define Œ(^/ to be the von Keumann algebra generated by \ tH. [&) J & C-F^fr&JcJ where JQ denotes the set of closed double cones in Minkowski cpace. 78/?.1016 -22- If F denotes the spacelike complement of F : then (H(F)c In the above, we have followed algebraic field theory in assigning a special role to the set •/£ of closed double cones. However if duality holds for F , then $(F) is essentially the only von Neumann algebra we can reasonably associate with F . More precisely if F —*> (A (F) is some local net of voji Neumann algebras defined for F € \S say with go^ and {%(&)?> fc(F) c t(F) c $(F')' C We now pose the problem of duality in terms of closed real subspaces of the one particle space ftcfç by making the following definitions. For &C JC , let M for) be the norm closure i;, TUk of and let, for a general subset F in Minkowski space, f*)(F) be the norm closed subspace of A£^" generated by £ M(O-) i & c-F & ejrj Then by Theorem 1.3.2. (Z (F)s (£. (M(F)J and duality holds for F 'f M (F)- M IF')' where M (F')'s {*€?** .• rft&JLo +Ï£ M(F'Jj 78/P.1016 -23- Or.e of the technical problems in deciding whether duality holds for F is to check the property of outer regularity. This property expressed in term of closed subspaces reads where JC0 denotes the set of closed double cones centered at the origin of Minkowski space. It is easy to see that duality implies outer regularity. Our exposition differs from that of p J in that we use dila tations to prove outer regularity rather than cutting down supports using multipliers for ^7 . The advantage is that the Fourier transform of a dilatation is again a dilatation. However this procedure only works for a special class of sets F" . These sets are contractible in a rather strong sense. Definition 11.1,4 ft set 25 in ^ Is said to be contractible about O if given ^y I>& >o , there exists an open neighbourhood "\F of 0 such that 7\ ( B + \J) C "& .3 is said to be is contractible if B-*o contractible about O for some *D . If S is contractible, so are Int 8 and S and we have Int S s S . If 3, and 8a are contractible so is 3, •*• 34 . Any contractible set 3 is star-shaped with respect to some interior point and is simply connected. If 3 is bounded and •i > Z. then the complement of S is also simply connected. To see that there are many contractible sets, we note : Proposition II.1.5 Let 8 be a compact set in 5Ç , star-shaped with respect to an interior point "o and such that the mapping 78/P.10Ï6 •f-l u.tS —* A(u)~ J"bÙ>o'-tt**Aue Bt is continuous, then p is contractible. In particular any set which is compact» convex with non empty interior is contractible. This proposition will not be used in the sequel and we omit the proof. II,g The Free Electromagnetic Field The free electromagnetic field rip is an irreducible Uightman free field which is an antisymmetric second rank tenser satis fying Maxwell's equations in the sense of operator-valued tempered dis tributions. As is well known, Fui cannot be derived from a local electromagnetic potential flu. .Its relation to the local electro magnetic potential flu. defined on the indefinite-metric Hilbert space jt,. of the Gupta-Bleuler formalism is as follows. The posi tive semi-défini te subspace *fe(r.li defined by the condition "a^Ap Instead, we begin with the two-point function for \y\J which in accordance with the conventions of II.1, induces the following positive semi-definite sesquilinear form on the test function space of antisymmetric tensors (*) See F. Strocchi, A. Wightmann, Journ. Math. Phys. 15. (12) 2138-224 (1974). 78/P.1016 -25- where ,.. £2n) In addition to the usual axioms for a nightman free field which can be deduced from this formula if desired, the free electroma gnetic field is also dilatation «variant : lemrj lt.?.l Let /A (X) = y*f(f*)s /fJ,>>° ' Then, there is a strongly continuous unitary representation A —y 2>W of the multiplicative group of the positive real numbers on A" such that J)(l) LL(a.,/\) s U (*«, A) D(A) and 2(>) M(F) * M (IF) One checks immediately that (f^.^X. = ^Ti3^3 arui it follows from the Lebesgue dominated convergence theorem that A —» (T> â )/-i is continuous. This shows the existence of the strongly continuous unitary representation ^ . The commuta tion relations with Poincaré transformations are checked by direct computation on 0 .Finally 3> (*) M(F) - M(*>F) since 0" £ JC implies <* 0" € JC and the result obviously holds for double cones. Dilatation invariance automatically implies out*:- regula rity for contractible sets in 'K . Lemma II.2.2 Let / be contractible in . Then V(F)S C\ Mir**) dreJt 78/P.I016 -26 Proof Without loss of generality, we may suppose that F is contraction about 0 .Let To every element A €• , we pay associate a distribution solution of Maxwell's equations by setting (fej) The physical interpretation of is that it is the expec tation value of the electromagnetic field in the coherent state created from the vacuum by W (à.) : F,(.VM = (&> *Ck) fyW Vf^Sl) It is easy to see. using partitions of the unity that if F" is a closed subset of "ÏRV then : fl(F'/ = {keffeio F^l*)k)*° f"-**?'} We shall not need this result but it suggests a strategy for proving duality : first regularize & and then prove that it is of the form ei (£) where £ has its support in FT & for Ô" arbitra rily chosen in JC„ . In essence we adopt this strategy but we need to work in terms of Cauchy data. 78/P.1016 -27- Given f &3 . we first define &., £_«<£"© Oe(&) by and write o1 (4) B ^illj * &'l (TJ.) . This corresponds to writing r^t in terms of the electric and magnetic fields : p/«» _ "g" /J*) + (7 [6) . With these variables we have SÏ»F*-AA -*V*M* where and A s 3*4) T>* 1 We next pass to the- time-zero fields and define continuous tr.appling s &, h > Lerroi 11.2.3 Given fs % G ^tr^)*3 C3 and f>„ as in lemma II.1.3. then « it) = ^ *, />„«/) and k@)*tz, ** fa • J7 exist in ft" . F.-thermore if f and 7 are real-valued and Suftf, **ftf -S B then e^f), fifj; are in fl M (s£f&) + 0') where &Î-&) 1s the d1amond generated by Q ; 78/P.1016 -28- Proof The first statement is proved in lemma Ji.1.3. The second follows from the support properties of PttO T and P tf j* using a parti tion of the identity argument. We list the basic formulae in terms of these variables. Lemma 11.2.4 If f,f * /WO Z?J (where For the proof of duality it is important to note Lemma 11•2•S / 4 / 4 = ô//> A£7, £f *<##>**/ is dense in £e * Proof This is a routine computation : one checks that 78/P.1016 -29- 4.t£) *-*(*-*(**&)' h (**<**&) where the variables in e. and A are understood to be evaluated at U.* = ° and We now associate with en element >i of distributions E (h)(2) and WA;^; by setting (?,$ & 3C&)& &) Z(Vl?) - * *(*,*(?)) H(V(f) - i r(k,k £(k) and H (à) are just the Cauchy data of the distribution solution of Maxwell's equation fû^ (A) defined previously. As such they satisfy the constraint condition The main properties of these distributions are summed up in Lemma II.2.S i) £(e(f) **(f))* -X cut? ii) tin) = »{*) = o 1mplies 4 » o 78/P.1016 -30- 111, lenfea/Jif and only If and Su.JfH(k)<. Q. Proof 1} 1s a direct consequence of lemma 11.2.4 1i) follows if one also uses lerona II.2.5 and 1ii) if one also uses lemma II.2.3. He now state the main result of this section. Theorem II.Z.7 (Quality) Let S be a bounded contractible subset of Z> (in the sense of Definition 11.1.4} then M (Z (*)')' = M (CCB)) Proof Given k & M(eCB)') and £ >o , pick f 6 C^C^*) 5 with S<*}j>f C 3£ = /a'e!? :)/^ & £} and define A routine computation shows that c (**(>) - ^J^J^'fi1 so £ (A »») is a C -function with support in G v- S£ and that Aiy}" E (kna)s O . Since -B is contractible, so is S f Be and B •»- 3t has a simply connected complement. Hence, by the cohomological lemma of the Appendix, there is a C -function 9 such tl-,at with J^pjf « S*- 3a£ and similarly a C*°-function £ with ^"Jfi ^ £-*-£?,£ such that -* /i \ / />?" 78/P.1016 -31- Applying Lerasa II.2.6 we have * * f = e. (f ) + ^(tf) 50 that k*f £ A? (€(B -r SJ£)^ . Using a sequence P» to approximate the £ -function, A*»-A, —^ k in fteJÇ ana we deduce that * 78/P.1316 -32- II.3 COUNTEREXAMPLE TO DUALITY FOR THE FREE ELECTROMAGNETIC FIELD Me show that duality does not hold for the diamond generated by a torus. To see this one considers a pair of disjoint but interlocking tori t0 and \n and constructs elements n(j)ÉM(e(C)')' •*» *e?,)*M(C&)')' «"h r(€f£), kff,))+o •This w *>&) 4 M(*i To make the counterexample as explicit as possible ù 4 a let 3>0 and I>, be the discs £ •? eR •- «,* "» =* * / "*&- } J and {ite R ." vt s o (v^t) •> */• *; -4 J- oriented so that thsir normals are in the positive Mi and »( directions respectively. Let 'Si denote the torus d-Dc *• "^s; where , and àhl denotes the boundary ofJh.i'Ofd . Ï, and '€, are disjoint if Sa +£, < 1 Let c£ denote the^ve^tor-valued^dist^ibution corresponding to the "flux" through T>; : H^lÇ) - J t'^Dc and we regularize this with a real C -function pt- with'support contained in &g^ and let , jf s. $c*fr . By lemma II.2.6, e{%) whire, using polar coordinates in the appropriate planes 78/P.1016 -33- Thus 1 III ZT kn / tU(f), ^(j^^frdrjeltLsoi.V *•(-{*», Urt*+-o*9yr*i.f) ° • o where 0"= f0*P and f(/?j= Pil-%) is a C*°-function with support in g . We only get a non-zero contribution to this integral if l Cu?& * (if ffift-oLe) y- riM*i- S (£0+e,) < 1 giving ^r*^r«w(*j < 2 +i>8 (1 + ' *o#) Thus we can restrict the range of the 0 -integration to l0,™! • For each fixed Ô in 1.0 JTJ the integrand vanishes unless Irl £ 1 *° /* / r r Thus we get and this is non-zero for a suitable choice of f> and p . This counterexample to duality has a rather simple inter pretation in field theory : the field-theoretical conmutator we have just computed is 78/P.1016 -31- Mow £(£) and M (%>) are Just tne flux of tne regularized electric and magnetic fields 1Ê * f, and ff * pv through 2>, and 2>„ respectively. Since dif n*P,~o , the flux of n*p0 through Z>a depends only on ?J0 , the boundary of 2), . Thus H (<}e) can be localized about any surface spanning 1) J>0 . However, the torus Z0 is not simply connected ; there is no surface spanning *3J0 contai ned in Ç, and, as we have seen A//j ) is not localized in xa . This also explains why there is no local electromagnetic po tential for the free electromagnetic field and the right way to looi at this argument and associated computation with its unexpectedly sinple final result is in terms of the local 2-cohoaiology ^15 J • *hc salr,a ideas may be applied to the local 1-cohoraology , here one has quantities which may be localized about any path with given endpoints but which are not necessarily localized about the endpoints. This yields exaaples where duality, while holding for double cones, fails for diamonds based on disconnected sets. This is the case for the observable algebra which is the gauge-invariant part of the algebra generated by the free charged scalar field. We content ourselves with sketching the arguaient. The operators violating duality are those which transfer charge from a point in one connected component of the base of the diamond to a point in another component. These operators do not belong to the algebra of the diamond because they do not commute with operators which measure the total charge in one connected component of the base. 78/P.1016 •35' 11.4 DUALITY FOR THE FREE SCALAR FIELD OF ANY MASS For completeness, we present a proof rf duality for the free scalar field. This is simple and less interesting than in the case of the free electromagnetic field because the Cauchy data are no longer subject to constraints. Nevertheless there is one technical difficulty which must be overcome in adapting the proof of the last section because one can no longer use dilatation invariance to prove outer regularity as in lemma II.2.2. In accordance *."'th the conventions of II.1, the test func tion space 3 is now jL ("R?) with a positive sesquilinear form where and H is the associated Hilbert space with a canonical map et : J —» fc. The first step is to describe H 1n terms of the Cauchy data using leirma II.1.3 to give mappings tfh. from ^4 ( £ j£. (&*)) into ^C whose properties Proposition II.4.1 9 i For fe^CR ), F/&€^c(-R) one has: 15 (?&,*&) « J- I?») (1(f) *£(»)£• 78/P.1016 -36- I») (ft*), to)* ± J ?(?> && ^£,3.-* vi) r(%ce), {>(*))* -L(F(?) &(?) <*3? viii) «-//; = p f*; * /sfv; Moreover, if ^" is real valued (i.e. belongs to then &(P,' and fc(F) are in Ke/r - Corollary II.4.2 *> ££F) t 0^6-) =0 if and c'y ^ F- & - «* H) J fi(F) + &(*)] F, 6- e^C**)} is dense in &*. 78/P.1016 •37- The dense set in ii) is exploited to define a continuous action of the group of dilatations on RcK • In contrast to the situa tion in lemma II.2.1, this action is neither unitary, nor covariant, norcan it be defined in terms of an action on the space $ of test functions. Lemma II.4.3 Let &&) = *-Z§W $*&) of the multiplicative group of positive real numbers by bounded linear trans formations on »?e /f such that Er82f llfift) **p(<**)S1"* fi? (VU1 * Al WW*" -if lirf*>iK # * (ttpfjF** d3r\ If rn so &x is unitary. If »yn>o th?n A-y^w * y^vA* « * fa*»*• for A > / and the reverse inequalities hold for A < / . Thus The Lebesgue dominated convergence theorem shows that 1\(D(X)~Z) ($,[?)+• U&)) t-*° as -*-»* . and, since DU) is uniformly bounded in a neighbourhood of ^»7 . <* —* &(*)• is strongly continuous. 78/P.1Û16 -38- Proposition II.4.4 Let J5 be contractible in JJS • Then Proof Let "{£>) denote the closed real subspace of ftt JC spanned by {Ç(p),P (p> ' S"I/"r c S J .Now tf(&) C H M (C( 8)* Cr) (compare with lenuna II.2.3) and A? fig(Ë)).C IT(&) t"°!'e • by Proposition II.3.1 (viii). Without loss of generality we may suppose that "E> is contractible about O . then : U HI STB) <= MfV(B)) Now by lemma II.4.3 AT{À£) s Vf*) tf (&) so if k C. rt AT (A 8) and u After this proof of the outer regularity, the remaining steps are just as in section 2 except that we do not need to deal with constraints. We first associate to each fe. in xe/( distributions 4>(k)(£) and 4>(k)(j£) by setting ( F e #£**)) Lemma II.4.5 implies A -O 78/P.1016 -39- "^ ke M ("€(&)')' if and only if Sujf> The proof follows that of lemma II.2.6 Theorem II.4,6 (Duality) Let 3 be a bounded icontractibl e subset in TR (in the sense of Definition II.1.4), then M (£(8)') = A? f*f«jj Proof Let p be as in theorem II.?.7, then if A € M ( <£(t>) J are C**-functions with support in S -t- &g . Hence, by lemma II.4.5 where F" and Gr are (^functions with support in Thus k#f> 6 M (*€ (S+ Sis)) . Using a sequence fK to approximate the J" -function, ^*A, -» A in /Ce/f and Proposition 11.4.4. ACKNOWLEDGEKc.NTS This is the written version of seminars held in Marseille in Oecemtier 1976; We should like to thank all participants of these seminars, in particular J. Bellissard whose seminar influenced the fora of Section II.1. 78/P.1016 •40- APPENDIX : COHOM0LOGICAL LEMMA Lemma Let B be a closed set in 3? whose cosiplenient B is simply connected. Let f be a C -function on TRi with values in JRs , with support in S and such that eU#£ s o . Then, given £ >o there exists a C*"-function 9 on Z? with values in &\3 such that f » o-'/ and whose support is contained in Of" Oj where %-- /****•• t z i * ej Proof fl* eo —» Let ^ be as in the lemma. Then there exists a C -function 5 defined in |R" with values in !5? such that ••*? cun.? 9 * It follows that €***£ 9, = o on 3 and one can define on o a C -real valued function ¥ such that Taking a C real valued function %• such that "P~o on 2 and i Xz I an(8tBt) *• the function /= %f is defined on 7i? , c C and such that on (3 + 3£ J (which is an open set) : f, - r* s' then, it is easy to see that Q s. q — *«/ J satisfies all the stated properties. 78/P.1016 11 REFERENCES [l] R. 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