1988 • 239 Pages • 8.37 MB • English

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Part I N o n - l i n e a r Fie ld Trans format ions in 4 D i m e n s i o n s Transforming fields non-linearly causes problems in quantum field theory: products of fields at one and the same space-time point are singular and hence have to be made well-defined prior to any application. The ambiguities inherent in any such r~ormalization have to be understood and to be taken care of. These remarks constitute the program for the present part: First it is recalled what renormalization is about; then those examples are presented where non-linear field transformations have been mastered (in 4-dimensional space-time).

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R e n o r m a l i z a t i o n Theory~ a Shor t A c c o u n t of Resu l t s and P ro b l em s * DIETER MAISON Max-P1anck-Institut fi/r Physik und Astrophysik - Werner-Heisenberg-Institut flit Physik - P.O.Box 40 12 12, Munich (Fed. Rep. Germany) 1. H i s to ry Historically Quantum Field theory arose from the attempt to quantize charged particles coupled to the electromagnetic radiation field. Already the first calcula- tions by Dirac, Heisenberg and Pauli treating the interaction between the particles and the radiation field as a small perturbation were plagued by infinities for energies, polarizabilities e tc . . Not all of these came as a surprise since infinite self-energies resp. -stresses were already known from the classical theory of point particles cou- pled to the electromagnetic field. Although it was remarked that from a pragmatic point of view the parameters of non-interacting (bare) particles or fields are unob- servable and can therefore be made suitably infinite in order to cancel the infinities arising from the interaction, this position is quite unsatisfactory as it renders the starting point of the calculations, the Lagrangean, ill-defined. A more satisfactory attitude is to take the divergencies as an indication that the theory is incomplete and should be embedded in a theory behaving more decently at short distances resp. at large momenta. A divergent but renormalizable theory could then be considered as an 'effective' low energy approximation which is made self-consistent by the renor- malization of a finite number of parameters diverging with the high energy cut-off. *Chapters 2 and 3 have been added for the convenience of readers less familiar with the formalism of perturbation theory.

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In fact we may even learn some interesting things studying the cut-off dependence of the theory considering it as an 'effective' low-energy theory. For instance the question of 'naturalness' of super-renormalizable couplings resp. mass terms arises precisely from the self-consistency of the 'effective' theory. However, quite independently of the particular 'philosophy' favoured to cope with the undesirable presence of the divergencies, it turns out to be possible to develop calculational procedures avoiding the infinities and reducing them to an ambiguity which can be removed through a fit to the observed values of the param- eters ('Renormalization Theory'). In the early days of perturbative quarttization the main emphasis was put on finding simple calculational schemes mitigating the unwanted divergencies. How- ever it was soon recognized that subtracting infinity from infinity was not a terribly unique recipe. 'Hence there was a definite need for a structural investigation of the divergencies of QED and its consistent removal' (Dyson). In addition, beyond the one-loop approximation one was faced with a principal problem in form of the so-called overlapping divergencies. The consistent removal of these turned out to be a rather tricky entertainment leading to a satisfactory answer only after a number of erroneous steps about which A. Wightman commented: 'Renormalization Theory has a history of egregious errors by distinguished savants. It has a justified reputa- tion of perversity; a method that works up to 13 th order in the perturbation series fails in the 14t h order.' Here Wightman refers to a method of Ward to renormalize QED that works perfectly well until one meets graphs of the type X X when things go wrong.

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The difficulties pertaining to the proper treatment of overlapping divergencies were finally resolved by a systematic approach based on general postulates like locality, unitarity and Poincar~ invariance. This 'axiomatic' approach emerging from ideas of Stueckelberg was fully formalized by Bogoliubov and resulted in a rigorous construction of the renormalized perturbation expansion to all orders due to the penetrating work of Hepp. A particularly powerful formulation was given by Zimmermann, who succeeded to resolve the result of the recursive addition of counter-terms to the Lagrangean resp. subtractions of vertex functions on Feynman amplitudes into a closed expression called the 'forest formula'. Many of the further developments of renormalization theory used this particularly lucid formalism. Characterizing the renormalized theory by abstract principles instead of defining it through a particular subtraction scheme has the advantage that one can study its properties in a scheme independent way. It only remains to show that there exists some method leading to the desired result, whatever method is used in any particular case turns into a matter of convenience. Some renormalization schemes~ as e.g. Zimmermann's have simple formal properties making them ideally suited for general considerations, whereas others like dimensional renormalization are more suitable for actual calculations. An approach staying as closely as possible to the 'axioms' of renormalization theory was given by Epstein and Glaser 1. Using the x-space support properties of advanced and retarded Green functions they can avoid undefined quantities al- together. The recursive construction of the perturbation series is reduced to the problem of 'cutting' distributions. At this point the usual ambiguities of the result emerge, which can be removed as usual by suitable normalization conditions. The 'axiomatic' approach also turns out to be a fiducial guide on the treach- erous field of theories with local invariance groups. As in the early days of renor- malization theory also in this case the situation was and still is characterized by misinterpreted calculational results: and inconsistent assumptions leading often to paradoxical conclusions. What we can, however, learn from an excursion into the history of perturbative renormalization is the fact that - as frequently in science - progress is stimulated by these paradoxical results which can only be resolved by clarifying the basic physical requirements masked by complicated calculational pro-

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cedures erroneously taken to be a substitute for the latter. Clearly that does not mean that we should underestimate the value of intelligent calculational methods which after all make the renormalized perturbation expansion more than an exercise in mathematics. The overwhelming success of perturbative QED in cases like the higher order corrections to the anomalous magnetic moment of the electron or the muon is an impressive example. In fact, for calculations beyond one loop in the Weinberg-Salam theory it may be vital to find a renormalization scheme minimizing the calculational effort exhausting easily the capacities of even the biggest existing computers. The renormalized perturbation expansion has also been a powerful guide for non-perturbative considerations. Much of the work of LSZ on quantum field theory has been abstracted from the perturbative series. Of particular importance is the development of 'Constructive QFT' emerging from the attempt to use renormaliza- tions as suggested by perturbation theory, but otherwise proceed non-perturbatively. Its recent development is strongly influenced by the close connection between 'eu- clideanized' relativistic quantum field theories and and the theory of phase transi- tions in statistical mechanics. The essential conceptual tool is the 'renormalization group' of Wilson, which also provides a new understanding of the concepts of renor- malization theory. Renormalizable theories turn out to be related to the fixed points of the renormalization group transformation. This viewpoint supersedes the con- ventional perturbative classification and may also allow consistent theories which are perturbatively non-renormalizable. 2. The Free Field The n-dimensional scalar free field ~(x) of mass m >_ 0 is a Wightman field 2 acting on a Hilbert space of free particles, the Fock space 9v . ~" has the structure of a direct sum ~" = ( ~ = 0 "T'N of N-particle spaces ~'N which are symmetric tensor products of the one-particle space ~'1 = L2(d#) with dp = 8(p2 _ m2)O(pO)dnp. ~'0 = C~ is the (no-particle) vacuum sector. ~'0 and ~'1 carry irreducible unitary representations of the Poincar~ group through U(h,a)f~ = ~2 U(A, a)¢(p) = eipa¢(A-lp)

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inducing a unitary representation on ~'. The free field ~(x) may be defined through its truncated Wightman functions 2 1 = iA+(xl -- x2, m 2) w T = 0 for k # 2 It obeys the field equation (a 2 + ~2)~,(x) = o derived from the Lagrangean = / + Similarly one may define a generalized free field £pp(x) replacing the measure d#(p) by a superposition dgp(p) = f 6(p 2 - ~2)e (p° )dp(~2) with some (signed) measure dp(x 2) leading to the two-point function (a, ~p(x)~.(o)~) = i X A+(x, ~2)dp(~2) If the moments Kj = f x2Jdp(a2) vanish for 0 < j < J (for J sufficiently big) the two-point function of ~p(X) becomes differentiable. Hence generalized free fields £pp(x) may be used as regularized versions of ~(x).

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3. W i c k P r o d u c t s The Wick products :qo(X): = :T(Xl). . . qO(Xk): can again be conveniently defined through their vacuum expectation values (fl, :~(Xl):... :~(xk):a)= ~ 1-IwT(zi) ~ z~=~ x ~ where the sum runs over all possible ordered pairs Zi = (xa 1, xa2) where the xa i are elements of different Xj's. Example: ( ~ , :~( Xl)~P( X2 )::~P(x3)~( x4):~ ) = -- A + ( X l -- x 3 ) A + ( x 2 - - x4 ) - - A + ( X l - - x 4 ) A ÷ ( x 2 - - x3) The Wick product :qo(X): remains well-defined even if all the elements of X = {Xl , . . . , xr} coincide, leading to the Wick power ~ . The definition of Wick products can be generalized to derivatives of qo(x) intro- ducing a suitable multi-index notation 1 :~ r : ( x ) l_~( Oa ~( x) ~r(a). r! -- : 1-I r ( . ) ! ' r , . • where only a finite number of r(a) are different from zero. The vacuum expectation values (~, ~ . . . :grk ' -~f l ) can be evaluated with r l l ~'k! / the formula given above. To each term A+(Z1) . . . A+(Zk) corresponds a graph G whose vertices are the x i and whose lines connect the vertices given by the Zj's. Example: A + ( x 1 - - x 2 ) A + ( x 1 - - x 3 ) A + ( x 2 - - x 3 ) A + ( x 2 -- x3 ) contributing to (a, :~2:(~):~3:(~2 ) :~3:(~3)a ) 2! 3! 3~

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gives the graph 2 3 Wick's theorem allows to expand multiple Wick products into simple ones: :T(X1):... :¢P(Xk): ---- (a, :~(X 1 \ Y1):..- :~(Xk \ Yk) :~) :v(Y1) .. . ~(Yk): l~cx~ From this formula one easily derives the corresponding expansion for products of Wick powers rl ! "'" rk ! :~(rl--Sl):(Xl) :~p(rk--Sk):(Xk)O,~S l (Xl) ' - s k ) ! - ' : Sl ! "'" sk ! Analogous formulae hold for generalized free fields. Sufficiently regularized free fields y)p(X) allow for the definition of the time-ordered functions resp. products (~,T:~p(X1):. . . :~p(Xk):~) obtained by replacing the A+ functions by (regular- ized) Feynman propagators 1 e ipz AF, p (2~r)n J p2 _ g2 + io ampdp(~2)" The corresponding graphs are called Feynman graphs. The generalization of Wick's theorem to time-ordered products (well-defined only for regularized fields) is T:~(X1): . . . :~(Xk): = (a, T:T(X 1 \ Y1):.-. :T(Xk \ Vk):fl):T(Yl)..- ~(Yk): ~CX~

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10 4. The Sca t te r ing ope ra to r The scattering operator S (S-matrix) providing a unitary map between the Fock spaces of in- and outgoing asymptotic particles can be characterized by 'axioms' derived from its physical interpretation. Following Bogoliubov 3 one considers the scattering operator S(g) in the presence of 'external' classical fields g(x) assumed to be smooth and localized (e.g. of compact support) which are coupled to suitable quantum fields. The corresponding interacting quantum fields can then be defined by O(x) = S(g,O)-ii6h(x) S(g,h)h: 0 where we have distinguished the particular field O by its external field h(x). In order to avoid problems with interactions of infinite duration resp. spatial extension it is convenient to replace also the coupling constants by such localized functions. The adiabatic limit g(x) -+ const, can then be studied separately. Hence we shall for the moment not distinguish between external fields and coupling con- stants. The required properties of S(g) are: i) S(O)= 1 (Normalization) ii) U(A,a)S(g)U(A,a) -1 = S(D(A)g(A-I (x -a) ) ) (Poincax~ invariance) where D(A) is the finite dimensional representation of the homogeneous Lorentz group corresponding to the covaxiance of g(x) iii) S(g)S+(g) = S+(g)S(g) = 1 (Unitarity) iv) If the support of g lies outside the causal past of the support of h, i.e. supp g VI (supp h + l / - ) = ~), then S(g)- lS(g + h) = S(h) (Causality) The perturbation expansion of S(g) is a power series in the coupling constants resp. external fields g(x) c¢ i k / S(g) = 1 + ~ k.' T k ( X l , ' " , x k / g ( x l ) ' " g ( x k / I I d x i k=l In order to avoid questions of convergence of the series it is usuMly interpreted as a formal power series in g. Since the individual terms of the expansion axe in

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