Wolfhart Zimmermann: life and work

Abstract In this report, I briefly describe the life and work of Wolfhart Zimmermann. The highlights of his scientific achievements are sketched and some considerations are devoted to the man behind the scientist. The report is understood as being very personal: at various instances I shall illustrate facets of work and person by anecdotes.

LSZ papers truely famous: three papers (1955,1955,1957) with Harry Lehmann and Kurt Symanzik the "LSZ formalism" of quantum field theory principles: Lorentz covariance, unitarity, causality realized on Green functions and S-matrix first axiomatic formulation of quantum field theory conversely: Lehmann, Glaser and Zimmermann (1957) suffient conditions on functions → a field theory LSZ does not refer to perturbative expansions however: greatly sucessful in perturbative realization extremely powerful in practice until the present day the most efficient description of scattering amplitudes in particle physics. asymptotic condition, reduction formula key idea: in remote past and future scattering experiment deals with free particles interaction only in a finite region of spacetime respective fields related by asymptotic condition: z a number, φ out in free fields φ(x) is an interacting field limit: in the weak sense, i.e. it is valid only for matrix elements reduction formula scattering experiment: n i particles in intial state into n f particles in final state. transition by S-operator, matrix elements S fi : LSZ-reduction formula HereG denotes the FT of the Green functions G(y 1 , . . . , y n f , x 1 , . . . , x n i ) = T φ(y 1 )...φ(x n i ) , vacuum expectation value of time ordered product of field operators determined by equations of motion historical remark Historical remark: Another axiomatic formulation of QFT has been initiated by Wightman (1956). The relation of the LSZ-scattering theory to those axioms and clarification of the role of fundamental fields have been given by Haag (1958Haag ( , 1959 and in particular by Ruelle (1962 x 3 x 4 scattering process: vertices linked by lines mathematical prescription for vertices, lines: "Feynman rules" ordering of diagrams: by numbers of vertices perturbation series: power series of coupling constants consistent algorithm required Dyson, Gell'man-Low Green functions: Gell'man-Low  IHES (Bures-sur-Yvette, France) noteworthy: contribution to "relativistic" SU(6)-symmetry (in hindsight: prepares the way to supersymmetry, anticommutators → Jordan algebras (Hironari Miyazawa, (1967)) general remark next absolute landmark work: renormalization theory Bogoliubov & Parasiuk, Hepp (BPH): finite diagrams via recursive prescription WZ: first step explicit solution of recursion -"forest formula" second step: subtractions in momentum space → integrals absolutely convergent (BPH: conditional convergence) "BPHZ renormalization scheme" (1968,1969) → S-matrix elements → Green functions involving arbitrary composite operators → equations of motions, currents, symmetries → precise notion of anomalies → link to mathematics → truely QFT effects pivotal tool: "Zimmermann identities" between different normal products (meaning even beyond perturbation theory) vertex correction: logarithmically divergent integral subtract first Taylor term at p = 0 introduce Zimmermann's ε Z = ε(m 2 + p 2 ) integral is absolutely convergent limit ε → 0: integral Lorentz covariant function.
no series problem for non-overlapping diagrams like Here one can remove the divergences by subsequently removing in an analogous way first those of the subdiagrams and thereafter that of the entire diagram. The result does in particular not depend upon in which order the subdiagrams have been subtracted sum: over all families of non-overlapping diagrams ("forests") in Γ t: Taylor subtractions at p = 0 S: relabels the momentum variables appropriately.
forest formula: (I Γ − · · · subtractions) theorem: the integral over the internal momenta of the closed loops is absolutely convergent and yields in the limit ε → 0 a Lorentz covariant vertex function or (for general Green functions) a Lorentz covariant distribution.

Renormalization theory -2nd highlight
Finite diagrams, equations of motion, symmetries normal products, Zimmermann identity "obvious" extension: standard vertices → composite operator via Green functions with composite operator as a special vertex and use of the respective reduction formula harbours all fundamental deviations of quantum field theory from classical field theory action principle, equation of motion define functional differential operators which represent field for a massive scalar field with the action principle reads (non-integrated transformation) replace δ X ϕ by 1: → well-defined operator field equation via LSZ-reduction symmetries, anomalies suppose: variations δ X satify an algebra implies algebraic restrictions on the insertions Q X if Q X (x) = variation modify Γ eff : symmetry can be implemented if not: anomaly Note: method is constructive; insertion Q X (x) in action principle is determined uniquely, can be characterized by covariance and power counting; extremely powerful tool operator product expansion arrive at normal products by merging external lines isolate singularities, capture them as coefficients of operators find: the operator product expansion (as introduced by K. Wilson) provides existence proof for OPE in perturbation theory (1973) ...
x study limit ξ → 0 for x = (x 1 + x 2 )/2 and ξ = (x 1 − x 2 )/2. for a bilinear product of a scalar field A eliminate scale paramenter t, find ode's, singular at vanishing couplings, case by case study power series solutions → initial value condition, no free parameter general solution: n free parameters, say, they replace λ i possible symmetries: solutions in the reduced model (1) massless theory, pseudo-scalar field B, spinor field ψ, interaction igψγ 5 Bψ − λ 4! B 4 for λ positive, g sufficiently small embedded into general solution with d 11 g 2 5 √ 145+2 + higher orders d 11 arbitrary.