Wolfhart Zimmermann: life and work - MPP Theory Group€¦ · Wolfhart Zimmermann: life and work Klaus Sibold Universität Leipzig Colloquium Munich, MPP Talk K. Sibold (Leipzig)

Wolfhart Zimmermann: life and work

Klaus Sibold

Universität Leipzig

Colloquium Munich, MPP

Talk K. Sibold (Leipzig) Wolfhart Zimmermann May 23, 2017 1 / 35

Outline

1 The beginning

2 LSZ – 1rst highlight

3 Intermediate years

4 Renormalization theory – 2nd highlightFinite diagrams, equations of motion, symmetriesOperator product expansion

5 Reduction of couplings – 3rd highlight

6 The man behind the scientist

7 Summary


The beginning

childhood, study

born February 17, 1928 in Freiburg im Breisgau (Germany)father: medical doctorolder sister: theater, “Giganisch”1946: entering university in Freiburg, study of mathematics & physics.lectures/seminars:“Either they were too fast or too slow for me. Either I had to think aboutthe new content – then I was too slow. Or I understood it instantly, thenthe lecture was boring.”measure: in 1950 doctoral degree in mathematicsthesis devoted to topologyearlier dissertation: but abandoned, because he found out that themain result could be proven in a much simpler way, hence consideredthis work as inadequate for a doctoral degreea further article on topology (1952)papers written in style and spirit of BOURBAKIhis comment: “I can read BOURBAKI like the newspaper.”


LSZ – 1rst highlight

general remark

1952 WZ: research associate (group of Werner Heisenberg)

Max-Planck-Institut f. Physik in Göttingen

first physics paper (1952): on thermodynamics of a Fermi gas

first QFT paper (1953): on the bound state problem in field theory

with co-author Vladimir Glaser

entry ticket to “der Feldverein”



LSZ papers

truely famous: three papers (1955, 1955, 1957)with Harry Lehmann and Kurt Symanzikthe “LSZ formalism” of quantum field theory

principles: Lorentz covariance, unitarity, causalityrealized on Green functions and S-matrixfirst 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 expansionshowever: greatly sucessful in perturbative realizationextremely powerful in practiceuntil the present day the most efficient description of scatteringamplitudes in particle physics.



asymptotic condition, reduction formula

key idea: in remote past and future scattering experiment deals withfree particles

interaction only in a finite region of spacetimerespective fields related by asymptotic condition:

φ(x) −→x0→±∞

√zφ out

in(x),

z a number, φ outin

free fields(� + m2

)φ out

in(x) = 0,

φ(x) is an interacting fieldlimit: in the weak sense, i.e. it is valid only for matrix elements



reduction formula

scattering experiment: ni particles in intial stateinto nf particles in final state.transition by S-operator, matrix elements Sfi : LSZ-reduction formula

Sfi = 〈f |i〉 = 〈p1 . . . pnf |q1 . . . qni 〉

=

(−1√

z

)nf +ni

limnf ,ni∏k ,j

(p2k −m2)(q2

j −m2)G(−p1, ...,−pn,q1, ...,ql)

(with lim : p2k → m2, q2

j → m2, p0k > 0, q0

k > 0)Here G denotes the FT of the Green functions

G(y1, . . . , ynf , x1, . . . , xni ) = 〈Tφ(y1)...φ(xni )〉,

vacuum expectation value of time ordered product of field operatorsdetermined 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 andclarification of the role of fundamental fields have been given by Haag(1958, 1959) and in particular by Ruelle (1962).



perturbative treatment

for Green functions, S-matrix: Feynman diagramsfor every elementary interaction: vertexfor particles propagating in spacetime: linex1 x2

x3x4 scattering process: vertices linked by linesmathematical prescription for vertices, lines: “Feynman rules”ordering of diagrams: by numbers of verticesperturbation series: power series of coupling constantsconsistent algorithm required



Dyson, Gell’man-Low

S-matrix: DysonS = 〈Tei

∫Lint〉

Green functions: Gell’man-Low

G(x1, ..., xn) = 〈T (φ(x1)...φ(xn))〉

=

⟨T(φ(0)(x1)...φ(0)(xn)ei

∫L(0)

int

)⟩⟨

ei∫L(0)

int

⟩evaluation: Wick’s theorem with 〈T (φ(x1)(0)φ(x2)(0)〉 = ∆c(x1 − x2)

fundamental axioms:Lorentz covariance, unitarity, causality: satisfied



observe: propagator

∆c(p) =i

p2 −m2 + iε

distribution, not a functioncalculate: ∆c(x − y)∆c(x − y)find: infinite ! meaningless !many diagrams with closed loops not well-definedproblem: give mathematical meaning to such expressions

do not violate the fundamental axiomsSchwinger, DysonBogoliubov & Parasiuk, Hepp (BPH) first satisfactory solutionWolfhart Zimmermann (BPHZ), (s.b.)


Intermediate years

various problems

1957 WZ leaves Göttingen

positions in: Instit. for Advanced Study in Princeton, Univ. of Hamburgvisitor at: Physics Dep. of UCB (Berkeley), CERN, Univ. of Viennaproblems studied: bound states, one-particle singularities of Green’sfunctions, analyticity structure of scattering amplitudes

1962 appointed professor of physics at New York Universityvisitor at: Enrico Fermi Institute (Chicago)

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))


Renormalization theory – 2nd highlight

general remark

next absolute landmark work: renormalization theoryBogoliubov & Parasiuk, Hepp (BPH): finite diagrams via recursive

prescriptionWZ: 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 normalproducts

(meaning even beyond perturbation theory)Talk K. Sibold (Leipzig) Wolfhart Zimmermann May 23, 2017 13 / 35

Renormalization theory – 2nd highlight Finite diagrams, equations of motion, symmetries

finite diagrams

propagator decreases in p-space for large p like 1/p2; implication

∼ λ2∫

d4k1

(p − k)2 −m2 + iε1

k2 −m2 + iε

integrand ∼ (k)0 integral '(ln( Λ

m ))

vertex correction: logarithmically divergent integralsubtract first Taylor term at p = 0introduce Zimmermann’s εZ = ε(m2 + p2)integral is absolutely convergentlimit ε→ 0: integral Lorentz covariant function.



no series problem for non-overlapping diagrams like

Here one can remove the divergences by subsequently removing in ananalogous way first those of the subdiagrams and thereafter that of theentire diagram. The result does in particular not depend upon in whichorder the subdiagrams have been subtracted



However, in diagrams like

γ λ1

λ2 λ3

removal of divergences in a subdiagram λ interfers with those of theothers and the removal of the overall divergence (i.e. of γ): thediagram γ contains “overlapping divergences”



WZ: “forest formula”• explicit solution of the recursion problem involved• deals properly with the overlaps

RΓ(p, k) =∑

U∈FΓ

SΓ

∏γ∈U

(−td(γ)pγ Sγ)IΓ(U)

sum: over all families of non-overlapping diagrams (“forests”) in Γt : Taylor subtractions at p = 0S: relabels the momentum variables appropriately.

forest formula:∫

(IΓ − · · · subtractions)

theorem: the integral over the internal momenta of the closed loops isabsolutely convergent and yields in the limit ε→ 0 a Lorentz covariantvertex function or (for general Green functions) a Lorentz covariantdistribution.



normal products, Zimmermann identity

“obvious” extension: standard vertices→ composite operatorvia Green functions with composite operator as a special vertexand use of the respective reduction formula

〈T (Q(x)ϕ(y1)...ϕ(yk ))〉 =⟨

T(

Nd [Q(0)(x)]ϕ(0)(y1)...ϕ(yk )(0)ei∫L(0)

int

)⟩(0)

d : naive dimension of Q.find: δ = d + c, c ∈ N possibleresult: Zimmermann identity

Nδ[Q] · Γ = Nϕ[Q] · Γ +∑

i

u(Q)i Nϕ[Qi ] · Γ

with ϕ > δ ≥ dim(Q)harbours all fundamental deviations of quantum field theory fromclassical field theory



action principle, equation of motion

define functional differential operators which represent fieldtransformations δX

on Γ

W X Γ ≡ i∫

d4x δXϕ(x)δ

δϕ(x)Γ (1)

for a massive scalar field with

Γeff =

∫d4x (

12

(∂ϕ∂ϕ−m2ϕ2)− λ

4!ϕ4) + Γcounter (2)

the action principle reads

δXϕ(x)δ

δϕ(x)Γ =

[δXϕ(x)

δ

δϕ(x)Γeff

]· Γ ≡

[QX (x)

]· Γ (3)

(non-integrated transformation)replace δXϕ by 1:→ well-defined operator field equation via LSZ-reduction



symmetries, anomalies

suppose: variations δX satify an algebra[W X ,W Y

]= iW Z , (4)

implies algebraic restrictions on the insertions QX

if QX (x) = variationmodify Γeff : symmetry can be implementedif not: anomalyNote: method is constructive; insertion QX (x) in action principle isdetermined uniquely, can be characterized by covariance and powercounting; extremely powerful tool


Renormalization theory – 2nd highlight Operator product expansion

operator product expansion

arrive at normal products by merging external linesisolate singularities, capture them as coefficients of operatorsfind: the operator product expansion (as introduced by K. Wilson)provides existence proof for OPE in perturbation theory (1973)

...

x1 x2

→...

x

study limit ξ → 0 for x = (x1 + x2)/2 and ξ = (x1 − x2)/2.Talk K. Sibold (Leipzig) Wolfhart Zimmermann May 23, 2017 21 / 35

Renormalization theory – 2nd highlight Operator product expansion

for a bilinear product of a scalar field A

T A(x + ξ)A(x − ξ) = E0(ξ)1 + E1(ξ)A(x)− iEµ2 (ξ)∂µA(x)

+12

E3(ξ)N[A(x)2] + R(x , ξ)

directional dependence of composite operatorsunderstood (1971)→ lightlike and spacelike operator product expansions

application:strong sector (QCD) of standard model of particle physicsdeep inelastic scattering of ν’s and e’s off hadronscomposite structure of hadrons confirmed


Reduction of couplings – 3rd highlight

general remark

• 1974 WZ scientific member of the Max-Planck Societydirector at MPP, Munich, Germany• 1977 honorary professor at Technical University of Munich• visitor at:Centre de Investigación y de Estudios Avanzados del IPN, Mexico City,MexicoPurdue University West Lafayette, IN, USA.• prime subject of his group: formulation of gauge and supersymmetric

models to all orders, possible only with BPHZ•WZ & Reinhard Oehme study asymptotically free theories like

QCD and analyze the Renormalization Group in models with severaleffective couplings (1984)



generalization of symmetry

starting point: a perturbatively renormalizable model has a primarycoupling g and n secondary couplings λi , i = 1, ...,n.effective couplings satisfy the renormalization group equations

ddt

g(t) = βg(g, λi)ddtλi(t) = βλi (g, λi) (5)

eliminate scale paramenter t , find

βg(g, λ(g))d

dgλi(g) = βλi

(g, λ(g)). (6)

ode’s, singular at vanishing couplings, case by case studypower series solutions→ initial value condition, no free parametergeneral solution: n free parameters, say, they replace λipossible symmetries: solutions in the reduced model



simple examples

Simple examples (1984):(1) massless theory, pseudo-scalar field B, spinor field ψ, interaction

igψγ5Bψ − λ4!B

4

for λ positive, g sufficiently small

∃1 power series λ = 13(1 +

√145)g2 + o(g4)

embedded into general solution with d11g25

√145+2 + higher orders

d11 arbitrary.(2) massless Wess-Zumino model, couplings g, λ, interaction

gψ(A + iγ5B)ψ − λ2 (A2 + B2)2

supersymmetric solution λ = g2 embedded into a non-supersymmetricgeneral solution λ = g2 + ρ3g8 +

∑ρjg2j+2, ρ3 arbitrary

third solution λ = −45g2 +

∑ρjg2j+2 not related to supersymmetry



further notable examples

• non-supersymmetric embeddings of models which can haveN = 2,4 supersymmetry

• non-abelian gauge symmetry as unique solution, if embeddingtheory has rigid invariance

• “finite” models existfinite: β-functions vanish to all orders

superconformal symmetry is realized as in the classical theory( mainly of theoretical interest)



phenomenological implications

reduction in the standard model?problem:coupling of abelian subgroup asymptotically free in the infrared... of non-abelian subgroups asymptotically free in the ultraviolet→ generalize reduction principlefind: bounds on Higgs and top mass (1991)including two-loop corrections

mt = 89.6± 9.2 GEV , mh = 64.5± 1.5 GEV

values already overruled by precision experimentsmodel must be extendedwithin supersymmetric extensions of the standard model (2008):

mh = 121 . . . 126 GeV (uncertainty: 3 GeV)

Wolfhart Zimmermann was pleased by thisTalk K. Sibold (Leipzig) Wolfhart Zimmermann May 23, 2017 27 / 35

The man behind the scientist

general remark

1991 WZ was awarded the Max-Planck-Medal1996 retirement; WZ kept ties to the institute until his endup to now: the scientist and his workthe man:

enjoyed eating and drinking wellloved having company for dinner in his househis wife graceful & competent hostgenerous towards members & guests of the institutecared very much about his three daughtersloved music, theater, the flowers in his terrace garden

Let’s have another look at the person via anecdotes.



anecdotes

Why did WZ never do refereeing work for journals?

answer (comes close to comment at lectures & seminars):

“If the problem addressed in the paper is interesting I am attracted tosolve it myself. If I don’t find it interesting I can not press myself to readit further and just do nothing but criticising. In any case it distracts metoo long from my own work.”



He simply hated committee meetings, in particular those of thedirectorate of the institute. There was just too much of trouble andstrife and bad behaviour for him. In the breaks of directorate meetingshe used to come to my office to discuss physics as a kind ofrecreation. At some time there was a “chance” that he had to becomeexecutive director (Geschäftsführender Direktor). His comment: “Achwissen Sie, ich habe einen Zettel in meiner Jackentasche. Daraufsteht: mir kann ja nichts passieren.” Indeed, nothing happened to him;a colleague of his was very eager to get this job.



WZ as “boss”?two remarks

first: he quoted a well-known mathematician:“ Mr. X at university Y said once in public: ‘Ich bin ein Bonze undmöchte als solcher behandelt werden.’ I would never say this.”

second: he himself filled in and kept the list of vacation days for themembers of the theory group and not the administration of the institute.reason: a scientist is most effectivley controlled by his work ad not byadministrative measures like presence in the institute.

no abuse of this freedom in the theory group, people there quite wellunderstood that their rank is being fixed by their scientific reputation.

It is obvious which sort of atmosphere is being created on such abackground.



In the same spirit he supervised the guest program of the theory group.The only relevant criterion for admission was the expected scientificoutcome and its quality. Mainstream arguments were not consideredto be sufficient. And, of course, the program was international. Noarguments like “Germany first” have ever been heard. This wasseemingly trivial at that time. But it has to be outspoken today.


Summary

Summary

When looking at the highlights a clear pattern emerges:• LSZ clarify basic notions in their fundamental papers. Those have

been used over and over again and have become textbook knowledge.•WZ improves the basis of renormalization theory. A wealth of papers

tackles successfully the structure of models: equations of motion,symmetries, anomalies.

•WZ proves operator product expansion in Minkowski space. Measurablequantities in QCD become available; they confirm the theory.

•WZ formulates the principle of reduction of couplings. Withinsupersymmetric extensions of the standard model the Higgs mass can bepredicted to quite some level of accuracy.

“Wenn Könige bauen, haben die Kärrner zu tun!”(F. Schiller in den “Xenien” (1798) über Kant)


Summary

Wolfhart Zimmermann has ended a journey in which he not onlydevoted his gifts to mathematics and physics but above all of thisto his family, his friends and his collaborators. We will miss him.


Summary

Thank you for your attention


Documents

Wolfhart Zimmermann: life and work - MPP Theory Group€¦ · Wolfhart Zimmermann: life and work Klaus Sibold Universität Leipzig Colloquium Munich, MPP Talk K. Sibold (Leipzig)