Analyticity m Axiomatic Field Theory

V. Glaser

CERN, Geneva

Half an hour is certainly not much time really to discuss a subject characterized largely by con­ siderable technical intricacies. So I shall necessarily have to be partial. In keeping with Einstein's proverb, ''Eigener Dreck stinkt nicht" I shall naturally favour my own contributions to the field.

My exposition will try to be a little more than the transcription of the summaries of three papers: 1) The work of K. Hepp1 and independently R. Haag and D. Robinson (unpublished, private com­ munication by D. Robinson) by which the authors manage to bridge almost completely the gap between the LSZ and the Wightman formalisms. 2) A paper by J. Bros. H. Epstein and myself2 in which we prove within the framework of the LSZ formalism the "crossing analyticity" for the 2-particle scattering amplitudes for stable particles with arbitrary but strictly positive masses and 3) A paper by A. Martin3 containing a proof of the Pomeranchuk theorem which uses only analyticity contained in (2) and starts from assumptions which are in a sense optimal.

It seems that analyticity properties of scattering amplitudes - judging if nothing only by the amount of papers on the subject - has turned out to be quite a useful tool in the study of strongly interacting particles. The attractiveness of the method stems from its model independence - in fact all the proofs4 of the dispersion-relations are based essentially on such simple assumptions as the spectral condition and micro-causality. On the other hand most of the work is done these days assuming the validity of the Mandelstam representation, an assumption that to my knowledge has not been proved even in all orders of perturbation theory so far. It is therefore a challenge for the theoretical physicist to try to understand better the basis on which the analyticity properties rest (this is the moral background of the first contribution) and to derive new results.

There are several significantly different sets of axiomatic formalisms that have been proposed in 5 order to study the contents of relativistic quantum mechanics • We shall be interested only in those that involve "the principle of micro-causality" as a basic postulate: the LSZ field theory, the Wight­ man theory and the local-ring theory of Haag and Araki. Besides the Lorentz covariance they all three assume also the positivity of the energy-momentum spectrum. While the Wightman theory postulates the existence of Lorentz-covariant fields in the sense of operator-valued tempered dis­ tributions, the Haag-Araki axioms admit only the existence of bounded operators corresponding to physical measurements in finite space-time regions. The LSZ theory includes all the Wightman axioms plus a complete particle interpretation: the S-matrix is assumed to be interpolated by a set of local fields satisfying a weak convergence condition for t _, ± 00 • Furthermore it is assumed that the fields are such that the Greens functions, i.e. the vacuum expectation value of time-ordered and retarded functions' can be defined as tempered distributions sufficiently regular in momentum space around the mass-shell in order to give sense to the reduction formulae. If it is the LSZ reduction formulae, as is well known, that form the natural starting point for the search of analy­ ticity properties of physical quantities. A natural question to be asked is whether the LSZ formalism could not be derived from the Wight­ man axioms using some simpler additional assumptions. Haag and Ruelle5 have indeed succeeded in developing a scattering theory within the framework of the Wightman theory under more detailed assumptions on the energy-momentum spectrum. If the spectrum of the mass operator p2 contains besides the vacuum state a discrete particle state of strictly positive mass m 0 and if there is a local Wightman field A (x) such that< ljA(x)JO> " O, then the operator A*(f, t), where

P = (Po, p) w = +{[I + m2 27 4 creates a 1-particle state lf(p)> from the vacuum if the test function is in S(G), i.e. is in S(R ) and has its support in a sufficiently small neighbourhood of the "forward ma~s shell" G = 2 2 { p : I p - m 1 :S c, p0 > 0 }. Here A (p) is the Fourier transform of A(x) and f (p) = f(c..i, p) the (c.t', p) restriction off to the mass shell. As a consequence of a fast decrease of the (truncated) Wightman functions in space -like directions the limits

n (1) lim n A* (f 1 , t) I 0 > = I f1 , • •• , f. > ex, ex = in or out, t_, ± 00 1 f1 E S(G) as shown by Haag and Ruelle, exist in the strong topology in the Hilbert space H and the limits t;.hus defip.ed have all the properties of a state of n free particles characterised by the wave packets f11 ••• , fn. There exist also the limits m m (2) <

= < cp •• I n Ae x (f;) jgi, ... 'g,> ex i = 1 i = i in the sense of weak convergences for all <{J ex > E Hex = linear closure of all the states jf 1, •• fn > provided all the g1 E S(G) but with no Sl:lPPO.rt conditions on the f 1 E S(R*). The expression on the right hand side defines the fields Aex (f) = J Aex (x)f(x)d-ix, which can be shown to have all the required properties of the usual free in- or out-fields. If one adds, as in LSZ, the condition Hin (or Houtl = H, it follows from the CPT theorem that Hin = Hout = H and that Aout is unitarily equivalent to A1,., which in turn defines the S-matrix by Aout = s- 1A1nS. Now the characteristic feature of these formulae is that in all the limits the times of the different operators involved have to be kept equal, whereas in the derivation of the LSZ reduction formulae the time limits have to be taken independenlly. So in the Wightman theory the matrix element 1 < cp ex IA (f 1 ) ••• A (fn) I <{Jex> has a priori no meaning, since one does not know whether the m-particle states are in the domain of definition of the polynomials in the Wightman fields. The authors in (1) were able to show that this is nevertheless true for the set D 0 ex of collision states (dense in He x!) describing divergoot symptotic particle configurations, i.e. states I f 1 , ••• fm>, where the f 1 (p) E S(G) have mutually disjoint support in momentum space. The crucial point of the method is to prove that the limit in (1) is attained in faster than any inverse power of time if the f 1 (p) have the above property. The next step consists in proving that the limit (1) exists also if the 0 vacuum state is replaced by any

P~ = J p~ + mz

The restriction of these distributions to the energy shell is therefore proved to be well defined for momentum configurations describing diverging particle beams, as expected, the scattering ampli­ tudes turn out to be independent of the regularization, and so this justifies the LSZ reduction for­ mulae also in the case when sharp Green's functions can be defined. As indicated the definition of the sharp Green's functions starting from the Wightman functions remains still an open question. It is a generalisation of the "renormalisation problem" posed by perturbation theory in a given order. Steinman7 has shown that in x-space it reduces to the well defined and solved distribution - theoretical problem of decomposing a distribution with support in a set K into a sum of distribution having their supports in given sets K 1 such that U 1K1 = K. But in field theory the difficulty consists in showing that the ambiguities which arise in the boundary points of two contguous sets (in the case of the n-point function the indeterminacy points are the j set of all planes x 1-x1 = O, i, = 1, ... , n) have to be determined in such a way, that the coinci­ dence regions in momentum - space given by the spectral condition be respected by the different members of the decomposition. It is a problem akin to the Dyson representation, where a distri­ bution with given support properties in x and p space has to be completely characterised. To my

28 knowledge this devilishly complicated problem has been completely resolved only in the case of the 3-point function by R. Stora8 just by the use of the Dyson fotmula. (The properly (1) below was obtained by the authors already earlier15110. A special case of it, namely a na.lyticlty in a neighbour­ hood of the physical cut in s for fixed negative twas independently proved by Minguzzi14 .) As far as the second paper is concerned, I shall be able only to state its results in terms of the usual Mandelstam variables. By a partial analytic completion of the primitive domain of holomorphy of the 4-point function arising from the properties of the retarded, advanced and the Steinman7 functions in p-space it was shown by the authors that the scattering amplitude is analytic: (1) in an open (complex) neighbourhood (in all the variables) of all the physical points of the three channels except for the usual s, t and u cuts. (2) the 6 regions defined in 1) are connected (the crossing property) as in the Haag-Araki scat­ tering theory, the only restriction on the validity of the above theorem is that all the four particles involved have strictly positive masses (and hence satisfy the stability conditions). The word "open" in the statement (1) means that also the physical thresholds are, except for the cut, within the region of analyticity*). The "crossing region" stated in (2) is such that it contains for any fixed negative momentum transfer (t = Ret < o) the whole upper or lower half plane in the center of mass energy(s) outside a finite circle not touching the physical thresholds as visualized in figure 1.

physical region I

t • physical region II or ID

Figure 1

This is an improvement of the results obtained so far. The classical proofs of the dispersion re1a.­ tions4 yielded only analyticity either in the momentum transfer for fixed physical energies (the Lehman ellipses) or the whole cut p1a.ne in the center of mass energy for a fixed sufficiently small physical momentum transfer provided the threshold masses were fulfilling certain rather restrictive conditions. So e.g. the fig. 2 indicates the new domain of analyticity for the forward pp scattering

PP pp

Figure 2 amplitude, whereas so far nothing has been proved on the subject. I shall be able to add only two 9 remarks: the first is that the proof was obtained without the exploitation of the Steinman identities • Their inclusion (as is implicitely done in the c1a.ssical proofs) permits en1a.rgement of the above domain, but not in general to the point of obtaining the whole cut plane in fig. 1 within the linear 10 problems (i.e. without the use of unitarity), as can be shown by a counter example a la Jost • The second is, that the above result is equally valid within the LSZ, the Wightman (thanks to reference 11 13) and the Haag-Araki formalisms* (due to the reduction formulae developed by H. Araki .)

*The speaker regrets very much that due to lack of time he could not include in his talk a discussion of the implications of the very important paper "Local Rings and the Connection of Spin with Statis­ tics" by H.J. Borchers (preprint) for the theory of scatt~ring.

29 Martin3 has given a particularly simple proof of the Pomeranchuk theorem. If one makes in addition to the analyticity properties proved in reference 2 the following three assumptions: 1. Inside the analyticity domain the forward scattering amplitude F(s, 0) is bounded by C(E) x exp E I s I for arbitrarily small E > 0. *

2. For physical energies I~ co I F(s, O) I I Isl In Isl = 0 s ±

3. The difference between the particle-particle and particle-anti-particle cross-sections a - /j has a limit for s __, co (including ± "") then the Pomeranchuk theorem, i.e. lim (a - a) = 0 follows. s __, co The set of these three assumptions is optimal in the following sense: if any one of them is released, counter examples can be found. Neiman12 has given proof of the Pomeranchuk theorem in which Martin's conditions 1) and 2) were replaced by I F(s, O)I < C Isl in the whole upper half plane, so the improvement is obvious. It is also shown in the paper - but using now much deeper properties of the theory of functions of one complex variable - that under the analyticity assumptions proved in reference 2 for fixed t > 0 the assumptions: 1. IF(s, t)I < C(E) exp E Isl for all E > 0 ands in the domain of analyticity.

2. for s real IF(s, t)j is bounded by Isl H as s ± co.

3. Im F(s, t) has a constant sign for s > s 0 ands< - s 0 • 4. [F(s, t)/ F(-s, t)I has a limit for s - > 00 (including 0 or co) lead to lim IF(s, t)/F(-s, t)I =1 as s - > co . This theorem strengthens considerably the result obtained recently by Biatas and Czyzewski13 and contains interesting implications for the pion-proton scattering. As a concluding remark, I would like to say the following. One would like of course to extend to the n-point function the analyticity properties proved sci far only for the 4-point function. Haag and Robinson1 have proved by using rather poor methods of analytic completion that the intersection of the holomorphy envelope of the 5-point function and the mass shell is not empty. Unfortunately not all the physical points could be reached so far, in particular not those where the three outgoing momenta get too close to each ot~r (which incidentally reminds one of the "diverging particle condition" used in reference 13). If it turned ()Ut that this is in principle impossible (a counter­ example satisfying all the Steinmann-relations would suffice) within the linear problem starting from a finite n, then the speaker believes that the whole idea of using analyticity as an approach to scattering is in danger since he does not see how then the "crank of unitarity" could get started.

References 1. K. Hepp, Comm. Math. Phys. 1 (1965). 2. J. Bros, H. Epstein, V. Glaser, to be published in Com. Math. Phys. J. Bros, Lectures at the Seminar on High Energy Physics, Trieste (1965). 3. A. Martin, CERN preprint to be published. 4. See e.g. R. Omnes: ''Relations de Dispersion et Particules Elementaire" (Hermann, Paris, 1960). 5. See R. Jost, Proceedings of the Sienna Conference, Bologna, {1963). 6. R. Haag, Phys. Rev. 112, 669 (1958). D. Ruelle, Helv. Phys. Acta 35, 147 {1962). 7. 0. Steinman, Helv. Phys. Acta 36, 90 (1963). 8. R. Stora, private communication. 9. 0. Steinman, Helv. Acta 33, 257, 347 (1960). D. Ruelle: Nuovo Cimento 19, 356 (1961). H. Araki: Journ. Math. Phys. 2, 163 (1961) . H. Araki and N. Burgoyne: Nuovo Cimento 18, 342 (1960). 10. R. Jost, Helv. Phys. Acta 31, 263 (1958). 11. H. Araki, Vorlesungen ilber axiomatische Quantenfeld theorie, ETH, Zurich (1961/62). 12. N. V. Meimann, J. Exptl. Theoret. Phys (USSR) 43, 2277 (1962). 13. A. Biatas and 0. Czyzewski; Phsy. Letters 13, 337 (1964). 14. A. Minguzzi, Nuovo Cimento 32, 198 (1964). 15. J. Bros, H. Epstein, V. Glaser, Nuovo Cimento 31, 1265 (1964).

* Within the LSZ formalism Bros et al.2 (unpublished) were able to prove this assumption. It is believed to be true also within the Wightman and the Haag-Araki theories but not yet proved.

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