differential geometric and topological ideas in the present and future development of local quantum physics (please

not axiomatic QFT since Haag and I intensely disliked the terminology already since the beginning of the 60s,

because it gives a completely wrong impression of what we tried to achieve). Starting from 1972 I began to look at

conformal QFT from a "Lie-field" point of view (for a brief review of those old concepts see which are coming from

Wally Greenberg and John Lowenstein see "Two-dimensional models as testing grounds for principles and concepts of

local quantum physics" published in the February 2006 issue of Ann. of Phys.) and after realizing that the 2-dim.

energy momentum tensor is such a Lie-field, I As promised in my email from last Friday, I wanted to return to some

differences started to look together with Swieca at more general fields with anomalous scale dimensions (which

necessarily live on the covering of the Dirac-Weyl compactified Minkowski spacetime). In contrast to naive

expectations they are reducible under the center of the covering (all published in 1974/75 in Phys. Rev. D) and the

ensuing decomposition theory global fields into its central irreducible components (the global fields fulfilling the

Irvine Segal concept of global causality on the covering are being decomposed into operator field valued sections on

the compactified Minkowski spacetime, or to put it into more physical terms the fields living in the hells and

heavens (the Luescher-Mack fields) are decomposed into objects which live in our laboratory world where the non

conformal invariant hardware is localized). Only in d=1+1 where the covering space decomposes into a product of two

coverings of the circle we found nontrivial examples (the exponential Bose fields). The component fields were 10

years later called "conformal blocks" by BPZ. These component fields were strange objects because they had a central

source and range projector and if the source projector did not match the vacuum quantum numbers the application to

the vacuum was zero, not a property which Lagrangian or Wightman fields have and which is consistent with the

Osterwalder-Schrader Euclideanization. As a result of our prejudice in favor of Euclidean functional integral

representations at that time we left the exploration of that interesting decomposition theory and looked at

geometric aspects of functional integrals.

During a visit of the CERN theory group 1976/77 I looked again at the beautiful Lowenstein-Swieca operator

presentation of the 2-dim. Schwinger model. When I observed a general connection between the zero modes of the

Euclidean spinor field in a generic vector potential with the winding number of that vectorpotential I got extremely

excited (my rather late Sturm and Drang period had arrived) and thought that this must be the tip of an iceberg. I

remember when I presented my concrete observations and made some of these sweeping claims which came from my gut

feelings (but for which I had no convincing arguments at that time), Roman Jackiw rightfully challenged my

speculative statements. I roamed the CERN theory library about elliptic differential equations in topologically

interesting external fields but only after I went down to the Geneva University mathematical library I finally found

something under the name Atiyah Singer which remotely resembled what I was looking for. It took me several month

and the invaluable help of Hartmann Roemer (who came to CERN from the University of Freiburg) to get a rough

impression of what it was about. When I finally wrote a paper with N. K. Nielsen on the Euclidean functional

integral approach to the Schwinger model with special emphasis on these new topological structures I used these new

words but there were still a lot of things outside my intellectual range. When Swieca visited CERN for a short time

I infected him with my enthusiasm and soon several of his younger Brazilian collaborators were trying these ideas

on all kind of two-dimensional models. What facilitated the situation was the fact that Leo Kadanoff already had

prepared the ground by preempting many two-dimensional structures on the lattice. Actually Swieca, having had the

same thesis adviser (Rudolf Haag) but 3 years after me, knew much more about algebraic QFT than I at that time and

he once asked me whether we should not have a look at the representation theoretical operator approach of Doplicher

Haag and Roberts of how to get the charged fields by knowing only the (by definition) neutral local observables. I

told him that this may be a loss of time in view of the interesting Euclidean functional&geometry approach, a

suggestion which years later I should regret

To make a long story short, I continued this topological geometrical functional integral-based approach after

Swieca's death (end of 1980) but my confidence was waning. Around 1982/83 together with Roger Picken (who spend more

than a year at the FU Berlin) we tried to understand the Schulman paradoxon that if on does the quantum mechanical

rigid top (movement on a compact group) via functional integrals and takes into account the infinitely many saddle

points but stops at the quadratic fluctuations around them (i.e. simply ignore the nonvanishing higher fluctuation

terms) one obtains the rigorous operator QM result! There was something strange going on here, some sort of

"topological protection?". We found that this problem has a finite dimensional counterpart: the Duistermaat-Heckmann

theory and of course the formal application to the infinite dimensional Euclidean Feynman would confirm this; but

there does not exist a D-H theory in infinite dimension unless one finds a regularization (approximation by a finite

integral) which maintains the D-H situation. We could not, and I think nobody did find such a thing after us. So how

could you think of doing group-valued sigma models by functional integration if not even this quantum mechanical

trivial example fits into this setting? It that time Luis Alvarez Gaume wrote a nice little paper of a "derivation"

of the Atiyah-Singer index theorem in terms of a quantum mechanical supersymmetric path integral but all this was

formal; it does not amount to a proof of the A-S theorem inasmuch as the work with Picken did not resolve the

Schulman paradox (i.e. no proof of an infinite dim. extension of Duistermaat-Heckmann). This was the end of my 8

year old love affair with differential geometry&Euclidean functional integrals. I cleaned up my desk and started to

go back to those interesting algebraic structures I found from the central decoposition theory with Swieca. This was

around 1983/84. I had the good luck to have a collaborator of phantastic scientific qualities, Karl Henning Rehren.

He came from Pohlmeyer with an excellent background on integrable systems, in particular with algebraic structures

arising from infinitely many conserved charges (in particular the ones associated with the Nambu-Goto model) and was

enthusiastically looking at those new structures in the FQS and the BPZ papers on 2-dim. conformal QFT whereas I

told him about the old algebraic structures with Swieca. There was no doubt in our mind that the two things belong

together, but the details turned out to be tricky since the conceptual and mathematical setting was quite different.

I remember that I got excited when it became clear to us that the structures which appeared in our old conformal

decomposition theory where new fields with source and range projections which formed "exchange algebras" with braid

group commutation relation. I am still proud of a little paper where we constructed explicitly these exchange fields

in the conformal Ising field theory and computed with their help explicit analytic expressions for n-point

functions. Our collaboration culminated in a paper which gave a setting for chiral quantum field theory in which

the new exchange algebra structures are simply new realizations of the old causality principles which also underly

the Lagrangian quantization- or the Wightman- setting and in order to highlight the fact that all these new

structures still obey the same principles we chose the title "Einstein causality and Artin braids". This was

published at approximately the same time as the Moore-Seiberg paper which was based on a categorical analysis of the

braid group statistics setting and gained particular popularity by mathematicians for who causality and localization

properties are outside their conceptual radar screen. with the exception of a group in Rome and of the Japanese

mathematician Kawahigashi who got most of their powerful new results from localization (spacetime-indexed nets of

observable algebras). Geometry has a role in local quantum physics but it is totally subject to the local quantum

principles. The most important concept which is capable to convert abstract algebraic properties (domain and range

properties of local operator algebras) into geometry (inner symmetries, spacetime symmetries) is the Tomita-Takesaki

modular theory of operator algebras and I am convinced that in will gain importance in the near future. My

constructive results which I obtained over the last couple of years on factorizing models, lightfront and double

cone holography and results on localization entropy (including a very interesting result which I will post on hep-th

on wednesday) all depend on the use of this amazingly powerful theory.

It is often said by outsiders that the algebraic approach is weak if it comes to computation. This is a

misunderstanding. It can do all the computations (by causal perturbation theory) one does with Feynman rules but it

avoids functional integrals for two simple reasons. First in functional integral representations one is limited to

those special covariant representations which allow an Euler Lagrange description and as already Weinberg has shown

in the 60s most covariantizations of the unique (m,s) Wigner representations do not have this property (causal

perturbation does not depend on it) and second the correlation functions of strictly renormalizable theories after

renormalization are simply not representable in the Feynman-Kac functional integral form; so a functional integral

is only a formal starter of a computation, afterwards you follow the intrinsic logic and do not give a damn for F-K

representability (renormalization cannot be done functionally). This is similar to the old Bohr-Sommerfeld quantum

theory which was more on the artistic and less on the mathematical-conceptual side. Scattering theory and cross

sections are not only a part of AQFT but AQFT was born together with LSZ scattering theory being at its cradle. That

algebraic field theorist do not do much computing is another story; violin builders like Strativari were mostly not

good fiddlers. AQFT extends standard QFT; example: for doing perturbation theory in curved spacetime the usual

rules are useless, in particular there is no functional integral representation. One has to separate the algebraic

structure from the structure of states on algebras and the standard approach is incapable of doing this. But in the

algebraic approach this is very natural and one can find the appropriate formalism in recent works of Wald,

Hollands, Brunetti-Fredenhagen...There are two interesting results. On the one hand one understands why the

expectations in individual states cannot be diffeomorphism-invariant, rather this invariance is a property of the

particular folium of states to which the given state belongs. The second result is perhaps the most recent really

great achievement: for the first time we are able to formulate Einstein's classical local covariance principle in

the setting of QFT in curved spacetime. This leads to a completely new view of what a QFT is: it is an object which

lives simultaneously on all hyperbolic spacetimes, i.e. an abstract algebraic substrate which (not unlike

stem-cells) can be spacetime-indexed by any globally hyperbolic manfold (this corresponding to the spatially

organized organs you can grow from stem cells).

Perhaps the greatest conceptual achievement of AQFT is the derivation of the origin of inner (charged) symmetries

from the (neutral) observable. These phantastic contributions of Doplicher, Haag and Roberts are like Mark Kac's

"hearing the shape of a drum" i.e. reconstructing the whole structure of QFT from its observable shadow. In this

case the problem cannot even be formulated in the lagrangian setting, less solved.

I could continue the list with other results of the AQFT setting which you cannot hope to obtain in other way, but I

think the relevant question to be asked here is:

What is the explanation why such a rich setting is suffering from so much prejudice?

This question I asked already a long time ago and one of my reputable colleagues gave me an answer which I find very

convincing. Sometime in the early 60s somewhere at an international conference somebody (whose name I will not

mention sincs he is still alive) made a very stupid remark after a talk by Weinberg and he claimed that what he was

telling Weinberg is a rigorous result of AQFT (which in that time was called axiomatic, to the distast of Haag and

myself). This caused an enormous damage and I don't blame Weinberg for considering a theory which produces such a

nonsense as not worthwhile. Weinberg's opinion carry a considerable weight in the community and string theory would

never enjoy its present status without his support.

There is also a positive aspect to this prejudice. Only mature and very qualified physicists who were able to make a

qualified judgement entered this area. In all papers from AQFT people exactly deliver what they promise and mediocre

papers are virtually nonexistent.

The reason why I wrote my essay is that times really have worsened from tolerating a small dedicated group to ethnic

cleansing of everything which is not stringy.

I cannot make an attempt to talk Weinberg out of his prejudice, but maybe it is possible to try this in the case of

a younger not totally polarized person as Aaron Bergman.

To Aaron Bergman

I have the following suggestion: take some time, maybe 2 weeks and look at math-ph/ 0511042 (not because I am a

co-author but because this is probably the simplest presentation of modular theory in its spatial form). Afterwards

follow some of the reference e.g. a more mathematical presentation by Brunetti, Guido and Longo or go directly to

the operator version of modular theory and its applications to factorizing models. There is no proselyting on my

part (to enter research in this area one needs anyhow several years of investment), I only want to let you know that

there are small time investments for getting rid of prejudices.

To Thomas Larsson

There is a conceptual error right at the beginning of your paper; Lorentz covariance, quantum theory and the cluster

property do not lead to QFT. You find the counterexamples in my work "An Anthology of non-local Quantum Field Theory

and Quantum Field Theory on non-commutative Spacetime" (published in AOP towards the end of last year). These

so-called "direct particle interaction" models fulfill the cluster-decomposition property yet they do not have a

representation in terms of a second quantization setting i.e. there is no universal (n-independent) operator which

applied to the n-particle states maps to n+1 or n-1 particle states. With other words this claim by Weinberg (based

on your 3 properties) on which there are many papers (including philosophical treatises is incorrect, but nobody

checks what Weinberg says). I have not red your paper and perhaps you later arguments and perhaps they do not depend

on what you write at the beginning. In any case considerable progress to find analoga of those structures which one

has in chiral theories (energy-momentum virasoro structure, current algebras) are well on their way in recent

articles of Todorov et al.

To Peter Woit

Coming back to Peter's viewpoint about the use of differential geometry in quantum physics, I think it does not go

significantly beyond free quantum fields coupled to external potentials. In no sense is the intrinsicness of diff.

geometry (coordinate independence) related to the intrinsicness problem of QFT (independence of the use of

coordinatization by quantum fields). AQFT addresses the latter problem and this is considerably more subtle (in

particular this is where modular operator theory comes in) than the former. In fact I do not believe in the use of

any outside mathematics unless the principles of QFT ask for it (which definitely is the case with modular theory,

in fact large parts of it were discovered independently by physicists.)

Isn't it ironic that string theorists accuse the AQFT of the use of gratitious mathematics when they themselves are

using structures like gerbes and algebraic geometry? I cannot think of more useless mathematical structures for

quantum theory (which deals with operators and states) than those. Hypocricy? Hubris? Double standards?