A Construction of Euclidean Invariant, Reflection Positive Measures on a Compactification of Distributions
aa r X i v : . [ m a t h . F A ] J a n A Construction of Euclidean Invariant, ReflectionPositive Measures on a Compactification ofDistributions
T. Tlas
Abstract
A simple construction is given of a class of Euclidean invariant, re-flection positive measures on a compactification of the space of distri-butions. An unusual feature is that the regularizations used are notreflection positive.
The goal of this paper is to give, under mild conditions, a very simple con-struction of a class of reflection positive, Euclidean invariant measures on acertain compactification of the space of distributions. The construction willwork in any number of dimensions and for a wide class of local actions. Wewill restrict ourselves to a single real scalar field, but it will be clear thatthe arguments below can be easily extended to other situations. Roughlyspeaking, we will give a rigorous meaning to expressions of the followingfamiliar form R e − S [ φ ] F [ φ ] dφ R e − S [ φ ] dφ , (1)where φ is supposed to live in some space of “functions”, F [ φ ] is a memberof a useful class of functionals of φ (e.g. a trigonometric polynomial), andfinally, S [ φ ] = R ( ∇ φ ) + m φ + L [ φ ], where L [ · ] is a “local” functional of φ and its derivatives.Of course, it is well-known that the expression above as it stands is a mathe-matical fiction, since there is no useful way of giving meaning to the measure dφ . Nonetheless, it is possible to proceed by combining the quadratic partof S with dφ and work with the resulting Gaussian measure. Even then,the expression above is ill-defined due to the fact that L [ φ ] is undefined onthe support of the Gaussian measure. An enormous amount of work wasexpended to try to solve these difficulties, see [1, 2, 3] and the numerousreferences therein. Roughly speaking, there are two kinds of problems oneneeds to deal with: the ones which appear at the short scales (ultraviolet)nd those appearing at the long scales (infrared). One typically proceeds byregularizing the theory in some fashion, by putting in cut-offs and removingthem in the end. We shall proceed in the same way.Let us give now the precise definitions. Fix D ∈ N . This will be the numberof dimensions in which the φ ’s will live in. It will be fixed throughout thepaper. Let us first describe the infrared regularization. This is accomplishedby essentially moving expression (1) to a sphere. Thus, let R > S R in R D × R given by the equation x + ( y − R ) = R , where x ∈ R D and y ∈ R . Let s denote the stereographic projection from S R − (0 , R ) to R D given by s ( x, y ) = 2 R R − y x. Let ∆(
D, l ) stand for the dimension of the space of spherical harmonics ofdegree l . Let { Y l,m } ∆( D,l ) m =0 be an orthonormal basis of the this space withrespect to the L product on S D , where the sphere is given the Hausdorffmeasure Ω R induced from the Lebesgue measure on R D +1 . If we denote by ∇ the Laplace-Beltrami operator on the sphere, and recalling that ∇ Y l,m = − l ( l + D − Y l,m , we see that there is a unitary isometry between theSobolev space on the sphere of order k , H k , and the set of all ‘sequences’, { f l,m : m = 0 , , . . . , ∆( D, l ); l = 0 , , . . . } satisfying X l =0 , ,... ; m =0 , ,..., ∆( D,l ) (cid:18) l ( l + D −
1) + 1 (cid:19) k | f l,m | < ∞ . It follows in turn that the dual of H k , H − k is isometric with the space ofsequences { φ l,m : m = 0 , , . . . , ∆( D, l ); l = 0 , , . . . } satisfying X l =0 , ,... ; m =0 , ,..., ∆( D,l ) (cid:18) l ( l + D −
1) + 1 (cid:19) − k | φ l,m | < ∞ . Now, note that the expression ∆( D, l ) = (2 l + D − l + D − D − l ! , but we are not going to need the explicit expression inwhat follows. The most convenient definition of H α for us is as the completion of the space of C ∞ functions in the norm || ( −∇ + 1) α f || L . We are slightly abusing terminology here since these are labelled by two indices, butthis should not cause any confusion. (cid:16) { f l,m } , { g l,m } (cid:17) = X l =0 , ,... ; m =0 , ,..., ∆( D,l ) f l,m g l,m l ( l + D −
1) + 1 (2)defines a trace class bilinear form on any H k for a sufficiently large k . Itthus follows by standard methods [4], that (2) is the covariance of a Gaus-sian measure µ supported on H − k for a sufficiently large k . At this pointwe select some such k and will hold it fixed in what follows. Of course, themeasure just described is a rigorous realization of the heuristic expression e − R S DR φ ( ∇ +1) φd Ω R dφ .We now move to the ultraviolet regularization. Let h be a positive, smooth,compactly supported and rotationally invariant function on R D . For anyΛ >
0, let ˜ h Λ ( x ) = h (cid:0) Λ x (cid:1) . Let h Λ ( θ ) = ˜ h Λ ◦ s ( θ ) R S D ˜ h Λ ◦ s ( θ ) d Ω R ( θ ) . It is easy to see, e.g. by considering spherical harmonics expansions, thatfor any element φ ∈ H − k , we have that φ Λ = h Λ ∗ φ ∈ C ∞ , where ∗ standsfor the convolution on the sphere. Moreover, φ Λ → φ (in the sense ofdistributions) as Λ → ∞ . Now, given any bounded measurable function L on R N , we have that the Z S R L (cid:18) φ Λ ( θ ) , ∇ φ Λ , ( ∇ ) φ Λ ( θ ) , . . . , ( ∇ ) N φ Λ ( θ ) (cid:19) d Ω R ( θ )is a well-defined, bounded function on H k , which is invariant under the or-thogonal group in D + 1 dimensions, O ( D + 1).Consider now the expression R F [ φ ] e − R S R L (cid:0) φ Λ ,..., ( ∇ ) N φ Λ (cid:1) d Ω R dµ R e − R S R L (cid:0) φ Λ ,..., ( ∇ ) N φ Λ (cid:1) d Ω R dµ . This is a well-defined version of (1) for any bounded, measurable function F on the support of µ . One would like at this stage to send R and Λto infinity. Therefore suppose { R n } ∞ n =1 and { Λ n } ∞ n =1 are two sequences ofpositive numbers with R n , Λ n → ∞ . Notice that if we replace R and Λ with R n and Λ n , then the above expression gives a bounded sequence of numbers.There are several straightforward ways to extract a number from a bounded For example, one can define f ∗ f ( θ ) = R SO ( D +1) f ( θ ) f ( gθ ) dg where dg is the Haarmeasure on the special orthogonal group in D + 1 dimensions, SO ( D + 1). equence. We’re going to do it using a Banach limit L . We are thus led toconsider the following expression L R F [ φ ] e − R S R L (cid:0) φ Λ ,..., ( ∇ ) N φ Λ (cid:1) d Ω R dµ R e − R S R L (cid:0) φ Λ ,..., ( ∇ ) N φ Λ (cid:1) d Ω R dµ ! = I ( F ) . (3)We will show momentarily that (3) can be considered an integral with respectto a certain measure. However, since we’re interested in measures which areEuclidean invariant and reflection positive, we need to restrict the classof functions F one is willing to consider simply to make these conceptsmeaningful. We shall take it to be the class of cylindrical functions given inthe following Definition.
Let D denote the space of smooth, compactly supported func-tions on R D and D ′ be its dual, the space of distributions. A function F on D ′ is said to be cylindrical if there is m ∈ N , and there are f , . . . , f k ∈ D and a bounded continuous function ˜ F on R m , such that F [ T ] = ˜ F (cid:16) f ( T ) , . . . , f m ( T ) (cid:17) . The set of all cylindrical functions will be denoted by
Cyl . It is obvious that the set of cylindrical functions is a vector space. Moreover,for any cylindrical function F [ T ] = ˜ F (cid:16) f ( T ) , . . . , f m ( T ) (cid:17) , we have that˘ F [ φ ] = ˜ F (cid:16) f ◦ s ( φ ) , . . . , f m ◦ s ( φ ) (cid:17) is a bounded continuous function on H − k .We can now state our main result in the following Theorem.
There is a unique (up to homeomorphism) compactification ˚ D ′ of D ′ such that every cylindrical function F on D ′ has a unique continuousextension ˚ F to ˚ D ′ . Also, there is a unique, rotationally invariant probabilitymeasure µ such that Z ˚ F d ˚ µ = I ( ˘ F ) . Moreover, one can choose the sequences { R n } ∞ n =1 and { Λ n } ∞ n =1 such that ˚ µ is reflection positive. If, additionally, one has that I (cid:0) || φ || H − k (cid:1) < ∞ , then ˚ µ is invariant under translations as well. Another intuitively appealing procedure would be to use nonstandard analysis, bytaking R and Λ unlimited, and then extracting the standard part of the limited expressionabove. ote that usually [1], Euclidean invariance and reflection positivity are de-fined for measures supported on D ′ . However, in view of our choice of theclass of functions which we’re interested in integrating, we can use essentiallythe same definitions, which are: • Euclidean invariance: For any element E of the Euclidean group, andany cylindrical F , we have that Z ˚ F E d ˚ µ = Z ˚ F d ˚ µ, where F E [ φ ] = ˜ F (cid:16) Ef ( φ ) , . . . , Ef m ( φ ) (cid:17) and Ef j ( x ) = f j ( E − x ). • Reflection positivity: Let V + stand for the subspace cylindrical func-tions such that the supports of f , . . . , f m are contained in the subset( x , . . . , x D ) ∈ R D with x D >
0. Let Θ : R D → R D be the reflectionin the x D coordinate. Then, for any F ∈ V + , we have that Z (cid:16) ˚ F ˚ F Θ (cid:17) d ˚ µ ≥ . Proof.
Recalling the definition of the topology on D ′ , it follows at once thatthe set of cylindrical functions separates points from closed sets in D ′ . Thisimplies that one can imitate the standard arguments (see e.g. [5]) used toshow the existence and the properties of the Stone-Cˇech compactification,but with the algebra of all continuous functions being replaced with the al-gebra of cylindrical ones. This shows the existence and uniqueness of thecompactification ˚ D ′ that we want, as well as the unique extension propertyfor cylindrical functions.Let A = { ˚ F : F ∈ Cyl } . It is obvious that A is a subalgebra of continuousfunctions on ˚ D ′ . Note that if { F n } ∞ n =1 is a sequence in Cyl which convergesuniformly on D ′ , then { ˚ F n } ∞ n =1 is also a uniformly convergent sequence on˚ D ′ . It is to check that A vanishes nowhere and separates points on ˚ D ′ .Thus, by Stone-Weierstrass, the uniform closure of A coincides with C ( ˚ D ′ ),the algebra of all continuous functions on ˚ D ′ .Now, it is obvious that ˚ F → I ( ˘ F ) is a linear positive functional on A .Moreover, as was mentioned above, we trivially have that (cid:12)(cid:12)(cid:12) I ( ˘ F ) (cid:12)(cid:12)(cid:12) ≤ || F || L ∞ .It follows that the functional above extends uniquely to a linear positivefunctional on C ( ˚ D ′ ). By Riesz-Markov, we have that there is a probabilitymeasure ˚ µ such that this functional coincides with the integral with respecto ˚ µ . Uniqueness of ˚ µ follows from the fact that A is dense in C ( ˚ D ′ ). Now, let ˘ F [ φ ] = ˜ F (cid:16) g ( φ ) , . . . , g m ( φ ) (cid:17) . If O ∈ O ( D + 1), let( ˘ F ) O [ φ ] = ˜ F (cid:16) O ( g )( φ ) , . . . , O ( g m )( φ ) (cid:17) , where, as usual, O ( g )( · ) = g ( O − ( · )).Now using the fact that µ and R S R L (cid:16) φ Λ , . . . (cid:17) d Ω R are O ( D + 1) invariant,we have Z ( ˘ F ) O [ φ ] e − R S R L (cid:0) φ Λ ,... (cid:1) d Ω R dµ = Z ˘ F [ φ ] e − R S R L (cid:0) φ Λ ,... (cid:1) d Ω R dµ. Now, if O belongs to the O ( D ) subgroup preserving the y axis, it followsat once that I ( ˘ F O ) = I (( ˘ F ) O ) = I ( ˘ F ), and thus ˚ µ is rotationally invariant.What remains is to deal with translations and with reflection positivity.We shall handle reflection positivity first. The proof will, in effect, use theMarkov property of the free quantum field [9]. The fact that a free quantumfield on a Riemannian manifold with a reflection is reflection positive sinceit’s Markovian was shown in [10]. We shall, along the way, show essentiallythe same thing by a somewhat different route which is more convenient toour setting.First, note that all of the discussion above works for any two sequences { R n } ∞ n =1 and { Λ } ∞ n =1 . In order to show reflection positivity, we need toassume that the sequences satisfy R Dn Λ n → . (4)Now, let δ >
0. Let S + δR = S R ∩ { ( x , . . . , x D − , x D , y ) ∈ R D +1 : x D > δ } , with S − δR having the same definition with the replacement x D < − δ . Also,let S δR = S R − ( S + δR ∪ S − δR ). Let H k + be the closed subspace of H k which isthe closure of C ∞ functions supported in S + δR with H k − the analogous spacefor S − δR , and let H k denote the orthogonal complement of H k + ⊕ H k − . It istrivial to see that the support of every element of H k is contained in S δR .Finally, let P ± denote the orthogonal projections of H k onto H k ± , with P This procedure of defining a measure on a space by going to the compactification wasused in a different, simpler context in [6]. Also, a similar idea is utilized in the constructionof the celebrated Ashtekar-Lewandowski measure, see e.g. [7, 8]. The same fact was shown by different methods in [11, 12] as well. eing the projection onto H k .We extend now Θ to R D +1 in the obvious way, by keeping y fixed, i.e.Θ( x , . . . , x D − , x D , y ) = ( x , . . . , x D − , − x D , y ). It should be clear that Θinduces a unitary map, f ( · ) → f (Θ · ), from H k + onto H k − .Now, observe that B (cid:16) f, g (cid:17) = B (cid:16) ( P + + P + P − ) f, ( P + + P + P − ) g (cid:17) = B (cid:16) P + f, P + g (cid:17) + B (cid:16) P f, P g (cid:17) + B (cid:16) P − f, P − g (cid:17) = B + (cid:16) f, g (cid:17) + B (cid:16) f, g (cid:17) + B − (cid:16) f, g (cid:17) . (5)To see that there are no cross-terms above, consider e.g. B (cid:16) P f, P + g (cid:17) .From (2), we see that it is equal to h P f, ( −∇ + 1) − P + g i L . We wantto show that this expression vanishes. To do that, it is enough to showthat h P f, ( −∇ + 1) − h i L = 0 for any C ∞ function h which is supportedin S + δR , as such functions are dense in H k + . Now, notice that there is asmooth function ˇ h such that h = ( −∇ + 1)ˇ h . Moreover, the support of ˇ h is contained in S + δR (in fact in the support of h ). Probably the easiest wayto see this is to use the fact that0 = Z S R − supp( h ) h ˇ hd Ω R = Z S R − supp( h ) (cid:16) |∇ ˇ h | + | ˇ h | (cid:17) d Ω R ≥ Z S R − supp( h ) | ˇ h | d Ω R . Therefore, since the support of P f is disjoint from that of ˇ h , we have that h P f, ( −∇ + 1) − h i L = h P f, ˇ h i L = 0 . The other cross terms are dealt with similarly.It is obvious that B + , B , and B − in the decomposition (5) are symmetric,positive, and trace class. Moreover, they are supported on H k + , H k , and H k − respectively. It follows that there are three Gaussian measures µ + , µ , and µ − , and a decomposition of the support of µ of the form φ = φ + + φ + φ − ,with the corresponding supports of the Sobolev functions being in S + δR , S δR and S − δR .Now, let F ∈ V + with F [ T ] = ˜ F (cid:16) f ( T ) , . . . , f m ( T ) (cid:17) . If R and Λ aresufficiently large compared with δ , such that the supports of f ◦ s , . . . , f m ◦ s are contained in S + δR , we have that ˘ F [ φ ] ˘ F Θ [ φ ] e − R S +( αδ ) R ∪ S − ( αδ ) R L (cid:0) φ Λ ,... (cid:1) d Ω R dµ [ φ ] = Z ˘ F [ φ + ] ˘ F Θ [ φ − ] e − R S +( αδ ) R L (cid:0) φ +Λ ,... (cid:1) d Ω R e − R S − ( αδ ) R L (cid:0) φ − Λ ,... (cid:1) d Ω R dµ + [ φ + ] dµ − [ φ − ]= Z ˘ F Θ [ φ − ] e − R S − ( αδ ) R L (cid:0) φ − Λ ,... (cid:1) d Ω R dµ − [ φ − ] ! × . . . · · · × Z ˘ F [ φ + ] e − R S +( αδ ) R L (cid:0) φ +Λ ,... (cid:1) d Ω R dµ + [ φ + ] ! == Z ˘ F [ φ + ] e − R S +( αδ ) R L (cid:0) φ +Λ ,... (cid:1) d Ω R dµ + [ φ + ] ! , where α > φ ± Λ = h Λ ∗ φ ± . The first equality above is aconsequence of the decomposition of the measure µ just described, the factthat the integrand is independent of φ , and that the Z S +( αδ ) R ∪ S − ( αδ ) R L (cid:0) φ Λ , . . . (cid:1) d Ω R = Z S +( αδ ) R L (cid:0) φ +Λ , . . . (cid:1) d Ω R + Z S − ( αδ ) R L (cid:0) φ − Λ , . . . (cid:1) d Ω R The final line is a consequence of the change of variables φ − = Θ φ + in thefirst term. To see that this is so, note that by a direct calculation, for any f ∈ H k + we have that Z e iφ + ( f ) dµ [ φ + ] = e − B + ( f,f ) = e − B − (Θ( f ) , Θ( f )) = Z e iφ − (Θ( f )) dµ [ φ − ] . From this, it follows by taking limits that for any element F ∈ V + , we havethat Z ˘ F [ φ + ] dµ + [ φ + ] = Z ˘ F Θ [ φ − ] dµ − [ φ − ] . If we now use that Θ φ +Λ ( x ) = φ − Λ (Θ( x )), then approximating L by a C ∞ function, and then taking limits of Riemann sums and using dominated con-vergence we have what we want.Now, notice that (3) is invariant under L → L + constant. This meansthat without loss of generality, we can assume that
L ≥
0. Let M =sup x ∈ R N L ( x ). We then have that (cid:12)(cid:12)(cid:12) e − R S +( αδ ) R ∪ S − ( αδ ) R L (cid:0) φ Λ ,... (cid:1) d Ω R − e − R S R L (cid:0) φ Λ ,... (cid:1) d Ω R (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − R S R L (cid:0) φ Λ ,... (cid:1) d Ω R (cid:12)(cid:12)(cid:12)(cid:12) e R S αδ ) L (cid:0) φ Λ ,... (cid:1) d Ω R − (cid:12)(cid:12)(cid:12)(cid:12) . e − R S R L (cid:0) φ Λ ,... (cid:1) d Ω R Z S αδ ) L (cid:0) φ Λ , . . . (cid:1) d Ω R . e − R S R L (cid:0) φ Λ ,... (cid:1) d Ω R (cid:16) M δR D (cid:17) , where the harmonic analysis notation . above stands for “less or equal thanan irrelevant constant multiple of”. We thus have, in view of (4), that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R ˘ F [ φ ] ˘ F Θ [ φ ] (cid:16) e − R S Rn L (cid:0) φ Λ n ,... (cid:1) d Ω Rn − e − R S +( α Λ n ) Rn ∪ S − ( α Λ n ) Rn L (cid:0) φ Λ n ,... (cid:1) d Ω Rn (cid:17) dµ R e − R S Rn L (cid:0) φ Λ n ,... (cid:1) d Ω Rn dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . || F || L ∞ M R Dn Λ n → . (6)Putting it all together, we get that I ( ˘ F ˘ F Θ ) ≥ t is a trans-lation by a vector t . Let t ⊥ stand for the subspace orthogonal to t in R D .For every R >
0, there is a unique SO ( D + 1) rotation O t,R of S R such that s ( O t,R (0)) = t and s ( O t,R ( s − ( t ⊥ ))) is orthogonal to t at t . Now, if F [ T ] = ˜ F (cid:16) f ( T ) , . . . , f m ( T ) (cid:17) is cylindrical, such that ˜ F is C on R m , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z (cid:18) ( ˘ F ) O t,R [ φ ] − ˘ F t [ φ ] (cid:19) e − R S R L (cid:0) φ Λ ,... (cid:1) d Ω R dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤|| ˜ F ′ || L ∞ Z (cid:18) max j =1 ,...,m (cid:12)(cid:12)(cid:12) φ (cid:16) ( tf j ) ◦ s − O t,R ( f j ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:19) e − R S R L (cid:0) φ Λ ,... (cid:1) d Ω R dµ ≤|| ˜ F ′ || L ∞ max j =1 ,...,m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( tf j ) ◦ s − O t,R ( f j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H k Z || φ || H − k e − R S R L (cid:0) φ Λ ,... (cid:1) d Ω R dµ. Now, notice that (cid:12)(cid:12)(cid:12)(cid:12) ( tf j ) ◦ s − O t,R ( f j ) (cid:12)(cid:12)(cid:12)(cid:12) H k goes to zero as R → ∞ . Then,if I (cid:0) || φ || H − k (cid:1) < ∞ , and ˜ F is C , we have that I ( ˘ F t ) = I ( ˘ F ). Since thisequation holds on a dense subset of Cyl , it in fact holds everywhere, whichconcludes the proof. Here, the reader should perhaps draw the case when D = 2. e would like to point out that the condition I (cid:0) || φ || H − k (cid:1) < ∞ can be con-sidered a very mild version of the analyticity axiom [1], which in effect,would require that I ( e zφ ( f ) ) is a holomorphic function of z . Intuitively,this corresponds to the constructed measure having an exponential fall off‘at infinity’ as opposed to the linear one required in the theorem. This, in-cidentally, would also guarantee that the support of ˚ µ is contained within D ′ .Of course, from renormalization group arguments, one in general would not expect that the measure constructed above would be useful. This is because,as is familiar, one needs to adjust the bare parameters in the Lagrangian asthe cutoffs are removed. This would correspond to making the function L above dependent on the cutoff, i.e. dependent on n . Moreover, one shouldalso allow (e.g. by considering the case D = 2) for L to be unbounded,at least in the limit [2]. Therefore, suppose that {L n } ∞ n =1 is a sequence ofbounded functions. It is now easy to demonstrate the following strongerversion of the theorem just proven:
Theorem.
The theorem above remains valid if we replace L everywhere with L n .Proof. Notice that the only place in the proof above where the constancyof L was used is in (6). Again, assuming without loss of generality that all L n ’s are nonnegative, let M n = sup x ∈ R Nn L n . Then, provided we choose thesequences { R n } ∞ n =1 and { Λ n } ∞ n =1 such that M n R Dn Λ n → , the entire argument above goes through, and we have what we want. Acknowledgments:
The author would like to thank J. Merhej for readinga preliminary version of this paper and for the numerous comments whichgreatly improved its readability.
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Department of Mathematics, American University of Beirut, Beirut, Lebanon.Email address ::