Analyticity M Axiomatic Field Theory V. Glaser
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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the transcription of the summaries of three papers: 1) The work of K. Hepp1 and independently R. Haag and D. Robinson (unpublished, private com munication by D. Robinson) by which the authors manage to bridge almost completely the gap between the LSZ and the Wightman formalisms. 2) A paper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) < <P .. I n A* (g,, t) I 0 > = < 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 ex E Do ". In order to a void all existence questions of the Green's functions (i.e. of vacuum expectation values of T - products or R - products) in connection with the reduction formulae Hepp uses regularized step functions, that is essentially ea (t) = e * a (t) with a (t) ED and f a (t) dt = A instead of the sharp step functions e (t). This procedure yields well defined tempered distribution with obviously the same asymptotic properties in the time variables as required from the "sharp" Green's functions, but with support in somewhat displaced cones (in the case of the R-functions). This entails no change in the analyticity in p-space as compared to the sharp R-functions but introduces an exponential growth in purely imaginary time directions in the tubes (the coefficient o in the exponential bounds eO 2: I ti! is equal to supp a, as can be easily verified on the example of the 2-point function). This last remark is of capital importance for analyticity. The asymptotic behaviour in x-space proved before, which is a convergence property at infinity of the (regularised) Green's functions in the variables dual to p~ - / (m2 + p~ ), yields then regularity in p-space for the amputated Green's functions around the energy shell 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.