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\begin{document}


\title{Pascual Jordan, Glory and Demise \\and his legacy in contemporary local quantum physics}
\author{Bert Schroer\\present address: CBPF, Rua Dr. Xavier Sigaud 150, \\22290-180 Rio de Janeiro, Brazil\\email schroer@cbpf.br\\permanent address: Institut f\"{u}r Theoretische Physik\\FU-Berlin, Arnimallee 14, 14195 Berlin, Germany}
\date{March 2003}
\maketitle
\begin{abstract}
After recalling episodes from Pascual Jordan's biography including his pivotal
role in the shaping of quantum field theory and his much criticized conduct
during the NS regime, I draw attention to his presentation of the first phase
of development of quantum field theory in a talk presented at the 1929 Kharkov
conference. He starts by giving a comprehensive account of the beginnings of
the new quantum theory and then passes to his recent discovery of quantization
of ``wave fields'' and problems of gauge invariance. The most surprising
aspect of Jordan's presentation is however his strong belief that his field
quantization is a transitory not yet optimal formulation of the principles
underlying causally localizable quantum physics. The expectation of a future
more radical change coming from the main architect of field quantization
already shortly after his discovery is certainly quite startling.

I try to answer the question to what extend Jordan's 1929 expectations have
been vindicated. The larger part of the present essay consists in arguing that
Jordan's plea for a formulation without ``classical correspondence crutches'',
i.e. for an intrinsic approach without the classical parallelism called
quantization, is successfully addressed in past and recent publications on
local quantum physics.
\end{abstract}



\section{Pascual Jordan, glory and demise}

There are not many physicists in whose biography the contradictions of human
existence, the proximity of glorious scientific achievements and disturbing
human weaknesses in the face of the great catastrophies of the 20ieth century,
are as starkly reflected as in the personality of Pascual Jordan.

Born on October 18, 1902 in Hannover of mixed German-Spanish
ancestry\footnote{He owes his name to his great grandfather Pascual Jorda
(probably unrelated to the biblical river), who comes from the Alcoy branch
(Southern Spain) of the noble Jorda family whose origin can be traced back to
the 9th century. After the British-Spanish victory of Wellington over
Napoleon, the family patriarch Pascual Jorda settled in Hannover where he
continued his service to the British crown as a member of the
''Koeniglich-Grossbritannisch-Hannoverschen Garde-Husaren Regiments until
1833. Every first-born son of the Jordan (the n was added later) clan was
called Pacual.}, he became (starting at age 22) a main architect of the
conceptual and mathematical foundations for quantum theory and the protagonist
of quantum field theory. There is no doubt that he took the lead in the
formulation of the conceptual and mathematical underpinnings called ''Matrix
Mechanics'' in his first paper\cite{B-J} jointly with Max Born submitted on
27. September 1925 (3 months after the submission of Heisenberg's pivotal
paper!) entitled ''Zur Quantenmechanik''. His mathematical preparation,
particularly in the area of algebra, was superb. He had taken courses at the
G\"{o}ttingen mathematics department from Richard Courant and became his
assistant (helping in particular with the famous book); through Courant he got
to know Hilbert before he met the 20 year older Max Born, a well established
member of the physics department. By that time Jordan already had gained his
physics credentials as a co-author of a book which he was writing together
with James Franck \cite{Franck}.

After Born got a hold of Heisenberg's manuscript he tried to make sense of the
new quantum objects introduced therein. While he had the right intuition about
their relation to matrices, he felt that it would be a good idea to look for a
younger collaborator with a stronger mathematics background. After Pauli
rejected his proposal (he even expressed some reservations that Born's more
mathematically inclined program could stifle Heisenberg's powerful physical
intuition), Jordan volunteered to collaborate in this problem \cite{Pais}%
\cite{Jammer}. Within a matter of days he confirmed that Born's idea was
indeed consistent (by proving that the combination of the correspondence
principle in the form used by Heisenberg together with the calculation of the
time derivative of the p-q commutation relation permitted the derivation of
the full p-q canonical commutation relation). Whereas the imprecisely defined
correspondence principle was still part of the somewhat artistic
quasiclassical old quantum theory, the p-q commutation relation became the
cornerstone of the new quantum approach in all subsequent papers. The
Born-Jordan results made Heisenberg's ideas more accessible. Probably as a
consequence of the acoustic similarity of pq with Pascual, the younger members
of the physics department (the protagonists of the ``Knabenphysik'') in their
discussions often called it the Jordan relation. The 20 year older Max Born
became his mentor in physics. Jordan always maintained the greatest respect
which withstood all later political and ideological differences.

The year 1925 was a bright start for the 22 year old Jordan who at that time
had already written a book with James Franck in addition to his already
mentioned collaboration on the Courant-Hilbert mathematica physics project.
After the submission of the joint work with Max Born on Matrix Mechanics, in
which the p-q commutation relation appeared for the first time, there came the
famous ''Dreimaennerarbeit''\cite{B-H-J} with Born and Heisenberg in November
of the same year, only to conclude the year's harvest with a paper by him
alone on the ``Pauli statistics''. Jordan's manuscript contained what is
nowadays known as the Fermi-Dirac statistics; however it encountered an
extremely unfortunate fate after its submission because it landed on the
bottom of one of Max Born's (in his role as the editor of the Zeitschrift fuer
Physik) suitcases on the eve of an extended lecture tour to the US, where it
remained for about half a year. When Born discovered this mishap, the papers
of Dirac and Fermi were already in the process of being published. In the
words of Max Born \cite{Born}\cite{Schucking} a quarter of a century later:
''I hate Jordan's politics, but I can never undo what I did to him......When I
returned to Germany half a year later I found the paper on the bottom of my
suitcase. It contained what one calls nowadays the Fermi-Dirac statistics. In
the meantime it was independently discovered by Enrico Fermi and Paul Dirac.
But Jordan was the first''. From later writings of Born and Heisenberg we also
know that Jordan contributed the sections on the statistical mechanics
consequences to the joint papers on matrix mechanics; this led to a
(unfortunately largely lost) correspondence with Einstein.

In Jordan's subsequent papers on this subject \cite{Gas}, including those with
other authors as Eugene Wigner \cite{J-W} or Oscar Klein \cite{J-K}), it was
always referred to as ''Pauli statistics'', simply because for him it resulted
from a straightforward algebraization of Pauli's exclusion principle (in those
early years he did not refer to Dirac, neither did he quote his own
unpublished manuscript).

In 1926 he submitted two papers; the one together with Heisenberg (submitted
in March) contained the first fully quantum mechanical perturbative treatment
of the anomalous Zeeman effect \cite{H-J}, based on the spin hypothesis of
Goudsmit and Uhlenbeck. Despite the successful calculation, the authors were
quite aware that a complete understanding of half-integer spin within a
setting of relativistic dynamics was still missing (a problem which still had
to wait 3 years before Dirac finally laid it to rest). The second \cite{Trans}
paper contains Jordan's presentation of what was later called ''transformation
theory'' i.e. a more intrinsic formulation of QT which was independent of the
different choice of bases\footnote{The previous calculations in the setting of
matrix-theory either in the energy-basis (Heisenberg, Born, Jordan...) or in
Schroedinger's position eigenstate basis became special cases of a
basis-independent operator theory.} in the Matrix Mechanics formulation. This
was an important step towards a basis-independent operator theory, because not
every system of matrix coefficients (or sesquilinear forms) of a possibly
unbounded operator allows the operator representation. With this discovery he
established himself as the friendly adversary of Dirac on the continental side
of the channel and in its printed form one finds an acknowledgment of Dirac`s
manuscript\footnote{In those days the papers were presented in a factual and
very courteous style, although this was not always the case in verbal disputes
and mail correspondences as e.g. published letters of Pauli showed
\cite{Pais}.}. As an interesting sideline, one notes that in a footnote at the
beginning of the paper Jordan mentions a ``very clear and transparent
treatment'' of the same problem in a manuscript by Fritz London which he
received after completing his own work. Apparently there exists no printed
account of London's work. Most physicists are more familiar with Dirac's
notation as the result of his very influential textbook whose first edition
appeared in 1930.

The year 1927 was the most fruitful in Jordan's career. The first paper
submitted in February 1927 deals with a question which remained from the
transformation theory \cite{Trans2}, namely a certain invariance property
which expressed the independence on the scale which is used to describe the
spectrum of an observable; Jordan attributes this problem to conversations
with Robert Oppenheimer. The second paper submitted in July 1927 was inspired
by Dirac's field theoretic transcription of the quantum mechanical
multiparticle configuration space for Schroedinger's formalism (``high
dimensional abstract space'') to the quantization of Schroedinger waves in
ordinary space \cite{Gas}. Jordan sets out to do something analog for
''Fermi's instead of Einstein's gas''. He \ develops what he refers to as the
``Pauli-statistics'' (probably using material from his ill-fated 1925
manuscript which ended in Born's suitcase) and uses the quantized spacetime
field formulation to compute the density fluctuations (the problem set by
Einstein which was successfully calculated in the famous
Born-Heisenberg-Jordan ''Dreimaennerarbeit'' 2 years before, but at that time
the quantized wave approach was not available) in a Fermi gas. He returned to
this subject in a joint work with Wigner (submitted in Januaray 1928) which
contains significant extensions and clarifications \cite{J-W}. This paper is
not only quoted as an alternative approach to Dirac's presentation of
anti-commutation relations, the Jordan-Wigner method received particular
attention in connection with nonlocal transformations which are capable to
change commutation relations. In this paper Jordan and Wigner discovered an
abstract (but highly ambiguous) method to write Fermions in terms of
''Paulions'' i.e Pauli matrices (which however as a result of Jordan's
lifelong love for quaternions appear in a quaternionic camouflage). The
ordering prescription which they need in order to write concrete Fermion
formulas in the Hilbert space of a discrete array of Pauli spin matrices
becomes physically unique in case of the presence of a natural spatial
ordering as in the transfer matrix formalism of the 2-dim. Lenz-Ising model
\cite{Lieb}. This kind of nonlocal formula involving a line integral (a sum in
the case of a lattice model) became the prototype of statistics changing
transformations (bosonization/fermionization) in d=!+1 models of quantum field
theory \cite{Thirring}\cite{Rothe} and condensed matter physics \cite{Lut}.

We are used to the fact that in publications in modern times the relation of
names to new concepts and formulas should be taken with a grain of salt e.g.
if we look at the original publication of Virasoro we are not terribly
surprized that the algebra has no central term and consists only of
frequencies of one sign (i.e. is only half the Witt algebra). But we trust
that what the textbooks say about the beginnings of quantum mechanics can be
taken literally. When we look at \cite{Trans} page 811 we note that the
relation between Born and the probability interpretation is much more indirect
than we hithertoo believed. Born in his 1926 paper was calculating the Born
approximation of scattering theory and his proposal to associate a probability
with scattering in modern terminology concerns the interpretation of the cross
section. The generalization to the probability interpretation of the absolute
square of the x-space Schr\"{o}dinger wave function according to Jordan was
done by Pauli\footnote{This may partially explain why Pauli in his 1933
Handbuch article on wave mechanics introduced the spatial localization
probability density without reference to Born.} who was of course strongly
influenced by Born. As in most cases ideas enter the market without a persons
name and the permanent label which is attached to them years later usually
represents a mode of thinking and rarely can be taken prime facie.

His increasing detachment (which started during the 30's) from the ongoing
conceptual development of QFT, and his concentration on more mathematical and
conceptual problems of quantum theory whose investigation proceed in a slower
pace (and can be done without being instantly connected to the stream of new
informations) is certainly related to his political activities, as he lets
himself to be sucked more and more into the mud of the rising NS-regime. In
trying to understand some of his increasing nationalistic and bellicose
behavior in the midst of the more cosmopolitan atmosphere of his academic
peers in G\"{o}ttingen it appears of some help to look at his background and
upbringing, although a complete understanding would probably escape us did not
experience those times of great post-war turmoil.

Pascual Jordan was brought up in a traditional religious surrounding. At the
age of 12 he apparently went through a soul-searching fundamentalist period
(not uncommon for a bright youngster who tries to come to terms with rigid
traditions) in which he wanted to uphold a literal interpretation of the bible
against the materialistic Darwinism (which he experienced as a ''qu\"{a}lendes
Aergernis'', a painful calamity), but his more progressive teacher of religion
convinced him that there is basically no contradiction between religion and
the sciences. This then became a theme which accompanied him through his life;
he wrote many articles and presented innumerable talks on the subject of
religion and sciences.

At the times of the great discoveries in QT many of his colleagues thought
that the treaty of Versailles was unjust and endangered the young Weimar
Republic, but Jordan's political inclination went far beyond and became
increasingly nationalistic and right-wing. These were of cause not very good
prerequisites for resisting the temptation of the NS movement in particular
since the conservative wing of the protestant church (to which he
adhered\footnote{The oldest son of the family patriarch Pascual Jorda was
brought up in the Lutheran faith of his forster mother, whereas all the other
children born within that marriage were raised as Catholics.}) started to
support Hitler in the 30's; in fact the behavior of both of the traditional
churches during the NS regime belongs to their darkest chapters. Hitler
presented his war of agression as a divine mission and considered himself as
an instrument of God's predestination (g\"{o}ttliche Vorsehung) while almost
all Christian churches were silent or even supportive.

In the 20s Jordan published articles (under a pseudonym) of an agressive and
bellicose stance in journals dedicated to the spirit of German Heritage; a
characteristic ideology of right-wing people up to this day if one looks at
the present days Heritage foundations and their political power in the US. He
considered the October revolution and the foundation of the Soviet Union as
extremely worrisome developments. Jordan succumbed to the NS-lure; his most
bizarre project was to join the Nazi movement in order to convince them that
modern physics as represented by Einstein and especially the new Copenhagen
brand of Quantum Theory was the best antidote against the ``materialism of the
Bolsheviks''. This explains perhaps why he joint NS organisations at an early
date when there was yet no pressure \cite{Wise}. He of course failed in his
attempts to convince the Nazis of his philosophical points; despite his verbal
support\footnote{Different from Heisenberg he did not directly work on any
armament project but rather did his military service as a meteriologist.} he
gave to their nationalistic and bellicose propaganda and even despite their
very strong anti-communist and anti-Soviet stance with which he fully agreed,
the Anti-Semitism of the Nazis did not permit such a viewpoint since they
considered Einsteins Relativity and the modern Quantum Theory with its
Copenhagen interpretations as incompatible with their Anti-Semitic propaganda;
one can also safely assume that the intense collaboration with his Jewish
colleagues made him appear less than trustworthy in the eyes of the regime.

Jordan's carrier during the NS times ended in scientific isolation at the
small university of Rostock; he never received any benefits for his pro NS
convictions and the sympathy remained one-sided. Unlike the mathematician
Teichmueller, whose rabid anti-Semitism led to the emptying of the
G\"{o}ttingen mathematics department, Jordan inflicted the damage mainly on
himself. The Nazi's welcomed his verbal support but he always remained a
somewhat suspicious character to them. As a result he was not called upon to
participate in war-related projects and spend most of those years in
intellectual isolation. This is somewhat surprising in view of the fact that
Jordan like nobody else tried to convince the NS regime that fundamental
research should receive more support because of its potential weapon-related
applications; in these attempts he came closer to a ``star wars'' propagandist
of the NS regime than Heisenberg who headed the German Uranium program but
never joint the NS party.

Jordan's party membership and his radical verbal support in several articles
published in journals devoted to ``German Heritage'' got him into deep trouble
after the war. For two years he was without any work and even after his
re-installment as a university professor his full rights (e.g. to advise PhD
candidates) were only reinstated in 1955. When his friend and colleague
Wolfgang Pauli asked him after the war: ``Jordan, how could you write such
things?'' Jordan retorted: ``Pauli, how could you read such a thing?'' Without
Pauli's help he would not have been able to pass through the process of
de-nazification (where Heisenberg helped him) and afterwards to be
re-installed as a university professor. In Pauli's acerbic way of dealing with
such problems: ``Jordan is in the possession of a pocket spectrometer by which
he is able to distinguish intense brown from a deep red''. Only after
receiving Jordan's promise to keep away from politics he wrote a
recommendation (in the jargon of those days a ``Persilschein'', i.e. a
whitewash paper) which greatly facilitated his re-entry into university.
Jordan did not keep his promise for long; during the time of Konrad Adenauer
and the big debates about the re-armament of West Germany he became a CDU
member of parliament. His speech problem (he sometimes fell into a stuttering
mode which was quite painful for people who were not accustomed to him)
prevented him from becoming a scientific figurehead of the CDU party. At that
time of the re-armament issue there was a manifesto by the ``G\"{o}ttingen
18'' which was signed by all the famous names of the early days of the
university of G\"{o}ttingen quantum theory, including Max Born. Jordan wrote
immediately a counter article with the CDU parties blessing, in which he
severely criticized the 18 and claimed that by their action they endangered
world peace and stability. Max Born felt irritated by Jordan's article, but he
did not react in public against Jordan's opinion. What annoyed him especially
were Jordan's attempts to disclaim full responsibility of his article by
argueing that some of the caused misunderstandings resulted from the fact that
it was written in a hurry. But Born's wife Hedwig exposed her anger in a long
letter to Jordan in which she blamed him of a ``deep misunderstanding of
fundamental issues''. She quoted excerpts from Jordan's books and writes:
''Reines Entsetzen packt mich, wenn ich in Ihren B\"{u}chern lese, wie da
menschliches Leid abgetan wird'' (pure horror overcomes me when I read in your
books how human suffering is done away with). Immediately after this episode
she collected all of Jordan's political writings and published them under the
title: ``Pascual Jordan, Propagandist im Sold der CDU'' (Pascual Jordan,
Propagator payed by the CDU) in the Deutsche Volkszeitung.

In the middle of the twenties the authors of the ``Dreimaennerarbeit'' were
proposed twice for the Nobel prize by Einstein, but understandably the support
for Jordan dwindled after the war. Nevertheless, in 1979 it was his former
colleage and meanwhile Nobel prize laureate Eugen Wigner who proposed him. But
at that time the Nobel committee was already considering second generation
candidates associated with the second phase of QFT which started after the war
with perturbative renormalization theory; there was hardly any topic left of
the first pioneering phase which was not already taken into account in
previous awards.

Jordan did not only have to cope with political satirical remarks as that from
Pauli, but as a result of his neutrino theory of the photon he also received
some carnavalesc good-humored criticism as in the following song (the melody
according to Mack the knife):

``Und Herr Jordan \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ``Mr. Jordan

Nimmt Neutrinos \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ takes Neutrinos

Und daraus baut \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ and
from those he

Er das Licht
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ builts
the light.

Und sie fahren
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ And in pairs they

Stets in Paaren
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ always travel.

Ein Neutrino sieht man nicht.'' \ \ \ \ \ \ \ \ One Neutrino's out of sight''.

Actually Jordan's idea is less ``crazy'' as it appears at first sight. In a
fictitious mass-less world in d=1+1 spacetime dimensions it is literally
correct \cite{Lut} (analog to the ``Jordan-Wigner transformations'' in the
previous section), and in d$\geq$1+3 it is correct in the sense that from even
composite neutrino field operators one can easily find one which communicates
via its non-vanishing matrix-elements between the vacuum and a one-photon
state vector (i.e. as a result of the presence of electro-weak interactions
not all the matrixelements of neutrino-antineutrino field polynomials can
vanish); but this implies via scattering theory (which in turn results from
QFT) that the components of the generated vector beyond a one-photon state
vanish asymptotically i.e. the chosen composite field from neutrinos is a
valid interpolating field for a photon\footnote{This is of course just one
manifestation of the impressive ability of the vacuum to couple (via vacuum
polarization) to every state vector with given super-selected charges which
can be reached by applying localized operators which carry that ``charge'' to
the vacuum. These structural consequences of the causal locality principle in
local quantum physics at the time of Jordan were only known in very special
cases.}. The reason why Jordan could not be correct in a literal sense has to
do with the subtle modification of the general bound state concept in the
presence of vacuum polarization in passing from quantum mechanics to causally
local quantum field theory and nothing with the specific neutrino-photon
setting. Vacuum polarization ``democratizes'' the relations between different
particles (''Nuclear Democracy''), the elementary-bound hierarchy is shifted
to charges and their fusion laws. When we nowadays say that, e.g., a meson is
made-up of two quarks, we are using the magic cover of the word
''confinement'' which is expected to legitimize an ''effective'' return of the
quantum mechanical bound state picture as a substitute for a proof. Part of
the problem is of course the conceptual vagueness of the particle-field
relation. Without the 1939 Wigner theory of particles combined with its
multi-particle extension through the quantum field theoretical scattering
theory of the 50s it is not possible to attribute an unambiguous meaning to
the particle-field dichotomy. Particles in those days were the quanta of free
wave fields.

Although Jordan took (with the majority of German physicists) a strong
position against those behind ``German Physics''\footnote{Jordan thought that
nationalistic and racist views had no place in science; in his own bellicose
style of ridicule (in this case especially directed against nationalistic and
racist stance of the mathematician Biberbach): ``The differences among German
and French mathematics are not any more essential than the differences between
German and French machine guns''.} and in this way contributed to their
downfall, he defended bellicose and nationalistic positions outside science
and he certainly supported Hitlers war of aggression against the ``Bolshewik
peril''. The fact that he was a traditional religious person and that the
majority of bishops in the protestant church were pro Hitler had evidently a
stronger spell on him than his friendship with his Jewish colleagues who by
that time had mostly left Germany (in some cases he tried to maintain a link
through mail correspondence).

Unlike the majority of the German population for which the early Allied
re-education effort (which was abandoned after a few years) to rid society of
aggressive bellicist and racist ideas was a huge success so that the
subsequent change of US policy in favor of re-armament of West Germany run
into serious opposition during the Adenauer period, Jordan joined the CDU
party where he had to undergo the least amount of change concerning his
relation to the issue of military and arms \footnote{The leadership of the CDU
recently supported Bush's war (against the majority of its voters).}, thus
breaking his promise of political abstinence to Pauli.

Jordan died in 1980 while working on his pet theory of gravitation with a
time-dependent gravitational coupling, but this work never reached the level
of the papers from those glorious years 1925-1930 or his subsequent pre-war
mathematical physics contributions. In the words of Silvan Schweber in his
history of quantum electrodynamics, Jordan became the ``unsung hero'' of a
glorious epoch of physics which led to the demise of one of its main architects.

It is however fair to note that with the exception of Max Born Jordan's other
collaborators, especially von Neumann and Wigner shared the bellicose kind of
anti-communism; Wigner later became an ardent defendor of the Vietnam war.
Since both of them came from a cosmopolitan Jewish family background, their
anti-communist fervor probably had its roots in their experience with the
radical post world war I Bela Kuhn regime in the Hungarian part of the
decaying Habsburg empire.

The cultural and scientific achievements in a destroyed and humiliated Germany
of the post world war I Weimar republic within a short period of 15 years
belong to the more impressive parts of mankind's evolution and Jordan, despite
his nationalistic and destructive political viewpoints is nevertheless part of
that heritage.

\section{Jordans role in the first phase of quantum field theoretical development.}

In times of stagnation and crisis as the one we presently face in the post
standard model era of particle physics, it is helpful and encouraging to look
at the better times. There are of course various motives for revisiting the
past. One is to preserve the historical continuity\ and the stock of
knowledge; a particularly strong motivation at a time when fundamental physics
is being threatened by the loss of conceptual abilities and the increasing
superficiality of a calculatory monoculture from inside its own community.
Another reason is that the past, if looked upon with care and hindsight, may
teach us where we possibly took a wrong turn and could point towards an
alternative path.

In this connection Jordan's plenary talk\footnote{The papers commented upon in
the previous section are not exhaustive, but since our present aim is more on
the side of relating Jordan's work and his remarkable views about open
questions with meanwhile solved or still open fundamental problems of QFT (and
less with standard type of centennial commemorations), we will not review the
content of all of his contributions.} at the 1929 conference in
Kharkov\cite{Kharkov} is particularly interesting. In a way it marks the
culmination of the first pioneering phase of QFT; it also already raised some
of the questions which were partially solved almost 20 years later in the
second phase of development (i.e. through renormalized perturbation theory).
In his talk Jordan reviews in a very profound and at the same time simple
fashion the revolutionary steps from the days of matrix mechanics to the
subsequent formulation of basis-independent abstract operators (the
transformation theory which he shares with Dirac) and steers then right into
the presentation of the most important and characteristic of all properties
which set QFT apart from QM: Locality and Causality as well as the inexorably
related Vacuum Polarization. He ends by emphasizing that even with all the
progress already achieved and that expected to clarify some remaining
unsatisfactory features of gauge invariance (\textit{Die noch bestehenden
Unvollkommenheiten, betreffs Eichinvarianz, Integrationstechnik usw., duerften
bald erledigt sein)}, one still has to confront the following
problem\footnote{Here we have actualized in brackets the content of this
sentence since ``QED'' (the only existing QFT in those days) was used in the
same way as ``QFT'' in present days.}:

\textit{Man wird wohl in Zukunft den Aufbau in zwei getrennten Schritten ganz
vermeiden muessen , und in einem Zuge, ohne klassisch-korrespondenzmaessige
Kruecken, eine reine Quantentheorie der Elektrizitaet zu formulieren
versuchen. Aber das ist Zukunftsmusik.} (In future constructions one perhaps
will have to avoid the construction in two separated steps and rather have to
formulate in one swoop, without the crutches of classical correspondences, a
pure quantum theory of electricity \textit{(}QFT\textit{). }But this is part
of a future tune.)

He returns on this point several times, using slightly different formulations
(....\textit{muss aus sich selbst heraus neue Wege finden) }for a plea towards
a future intrinsic formulation of QFT which does not have to take recourse to
quantization which requires to start with a classical analog.

It does not only mark a high point in presentations of developments of QFT and
in particular his own participation in the shaping of physical concepts, but
in a way it also can be viewed as rounding off the pioneering stage of QFT; in
the nearly 20 years up to the beginnings of the second stage of perturbative
renormalization theory due to Feynmann, Schwinger and Dyson (with important
conceptual methodical and computational contributions by Kramers, Tomonaga,
Bethe and many others), there was a kind of conceptional lull apart from some
isolated but important contributions whose potential was only appreciated much
later as e.g. Wigner's famous 1939 group representation-theoretical approach
to particles and their classification (an illustration of what is meant by
intrinsic, without quantization), and the identification of the scattering
operator as the most important invariant observable by Heisenberg and Wheeler.

In contrast to Pauli who contributed to the second post war phase of QFT
(renormalized perturbation theory) and always followed the flow of ideas in
QFT up to his early death, the Kharkov talk was in some sense Jordan's
swan-song as far as his active participation in the elaboration of quantum
field theory is concerned . Afterwards he turned his attention to more
mathematical and conceptual problems as well as to biology \cite{Beyler} and
psychology. His enduring interest in psychology was presumably related to the
psychological origins of his stuttering handicap which prevented him to use
his elegant writing style in discussions with his colleagues and
communications with a wider audience, which perhaps explains why even in the
physics community his contributions are not as well known as they merit to be.
During his visit of Niels Bohr's institute in Copenhagen he was offered some
money from Bohr in order to get in touch with Adler in Vienna and receive some treatment.

His withdrawal from the mainstream of particle physics may have partially been
the result of his frustration that his influence on the NS regime was not what
he had expected and (after the defeat of Germany) his attempts to account for
his membership in the Nazi party as well as the difficult task to make a
living with the weight of his past NS sympathies (which cost him his position
as a university professor for the first two years after the war). In this way
he missed to take notice of the contribution of his former colleague and
collaborator Eugene Wigner from the glorious early days of QFT who by 1939
elaborated a purely intrinsic (i.e. without invoking any classical parallelism
of quantization) approach to the modern particle concept based on
representation theory of the covering of the Poincar\'{e} group \cite{Wigner}
and in this way accomplished at least in a more limited context what Jordan
wanted to be achieved in full QFT (see next section). This together with the
later realization that scattering theory \cite{LSZ} already follows
\cite{Haag} from the causal locality and the assumption about the energy
momentum spectrum then clarified the particle-field relation; in fact it
allowed to infer that with every Wigner particle in the energy momentum
spectrum the locality (via derived strong spacelike cluster properties of
correlation functions) automatically secures the existence of multiparticle
vectors which are symmetrized/antisymmetrized) tensor products of Wigner
particle states without having to make any explicit assumptions about the kind
of interaction (e.g. the form of the Hamiltonian), as it would be necessary in
Schroedinger quantum mechanics. Together with asymptotic completeness the
Hilbert space arena of QFT becomes that of a Fock multiparticle space without
any assumption about a canonical commutation structure (which would
immediately generate conflicts with the presence of interactions). But this
chain of arguments is limited to Lorentz frames and there is no reason to
believe that e.g. an LSZ asymptotic behavior holds in the natural time in the
Rindler world of uniformly accelerated Unruh \cite{Unruh}\cite{Halvor1}
observers\footnote{Therefore it may be somewhat misleading to use the
terminology of ``particles'' in connection with the Rindler-Unruh situation
(as one find it in the literature \cite{Wald}) since there is no reason to
believe that the Hilbert space analyzed in terms of thermal Unruh excitations
has a Fock space structure (apart from atypical case of non-interacting
quantum matter).}. Many of these pivotal conceptual conquests which were
achieved in the 50s-70s are presently being lost at an alarming rate and this
is not only due to the fact that they belong to the stock of pre-electronic
knowledge but also the result of an extreme particle physics monoculture and
unilateralism stemming from the globalization of certain trends and fashions.

Finally the causal locality structure in a new powerful algebraic setting also
led to a complete de-mystification of the notion of inner symmetry in
fundamental papers of Doplicher Haag and Roberts\cite{Haag}\cite{D-R}. For
this these authors has to go far beyond Lagrangian QFT\footnote{In the
Lagrangian setting this question is meaningless because the inner symmetry is
part of the very formalism of coupling multiplets of fields.} and develop new
mathematical concepts in operator algebra theory. Such a derivation was out of
the question within the setting of QFT in the 30's, but one may say that
Jordan found the best possible formulation within his momentum space creation
and annihilation operator formalism of linear systems (where the emphasis on
the particle-field dichotomy is unnecessary) for connecting the
representations of the symmetric group of particle statistics with inner SU(N)
unitary symmetries \cite{Sym}. This was before Heisenberg in 1936 promoted his
ideas about nuclear symmetries to the SU(2) isospin and years before Schwinger
rediscovered this connection in a presentation which became widely known
(especially through a book written by Biedenharn and van Dam \cite{Schwinger}).

In his Kharkov talk Jordan also addressed the problem of gauge invariance to
which he attributed special importance. He pointed out that the Fermi
treatment in the canonical setting was not satisfactory in the new local and
covariant setting and he expressed his opinion that a more satisfactory
solution should be expected to be a forthcoming technical step. When finally a
better understanding of some of those points \cite{GP} was reached after 20
years, Jordan probably never noticed it, since he had left the area of gauge
theory; in fact QFT in those days was exclusively QED, with simpler scalar
interactions only serving to test new ideas in a simpler context. It is
interesting to note that the potential use of non-abelian gauge in particle
physics was pointed out the first time in a talk which Oscar Klein (one of his
collaborators in Copenhagen during the 30's) gave at the eve of the beginning
of the 2nd world war in Warsaw \cite{Klein} long before the meson-nucleon
models of the 50's and the Yang-Mills proposal. The interruption of this
fruitful line of gauge-theoretic thinking in particle physics as a result of
the war was cemented by the change of emphasis towards meson-nucleon physics
in the post-war period and the new start in this direction had to wait for
three decades.

We do not know what Jordan's reaction immediately after the war to the
impressive breakthrough in QED may have been, but reading the text of his
Kharkov talk one gets the impression that he may have been a bit disappointed
by the very conservative nature of renormalized perturbation theory since he
believed that only very revolutionary conceptual changes and additions could
resolve the problems described in his talk. In strange contrast to his
political stance, in physics Jordan was a true revolutionary.

\section{Jordan's radical expectations and the present development}

In this section we will address the question to what extend Jordan's
expectations concerning a more intrinsic formulation of the physical
principles underlying his ``quantized wave field'' setting (i.e. the standard
standard Lagrangian setting) have become reality. We will argue that local
quantum physics or algebraic quantum field theory (for the development up to
1990 see \cite{Haag}) has all the radical aspects which he hoped for, although
there exists no continuous line between his 1929 Kharkov manifesto and modern
algebraic QFT. In this respect there is some similarity with the fate of the
old gauge theory which culminated in Oscar Klein's 1939 Warsaw conference
talk, which was forgotten thereafter and only found again many years after the
sucessful application of the new non-abelian gauge theory.

The radical aspect of algebraic QFT, namely the absence of any reference to
quantization of classical structures explains why it developed much more
slowly. Lacking a quantization dictionary (e.g. the Lagrangian formalism) one
has to create a new conceptual framework and one also needs new mathematics to
implement it. Field quantization and renormalization theory (even including
gauge theory) uses classical intuition and geometry in its mathematical
formulation with the quantum aspects entering in the form of recipes whereas
algebraic QFT uses more abstract concepts and operator algebras. Geometric
properties are often obtained as consequences but are generally not part of
the input. Although the new algebraic approach is faithful to the same
physical principles as its Lagrangian predecessor pointlike covariant fields
are not the actors on the center of the stage but rather become generators of
nets of operator (von Neumann) algebras. As coordinatizations of algebras they
loose their distinguished position (apart from Noether generators of
symmetries) and in analogy to coordinates in geometry there are infinitely
many different fields which (different from fields in classical physics have
their own directly measurable life) all describe the same physical reality and
constitute just one local equivalence class. It is not possible to describe
all new concepts and result of this new operator algebra-based local quantum
physics. Our intention in the sequel is to highlight those aspects which are
directly related to the problems which led Jordan to keep a critical distance
to his own brainchild of ``quantized wave fields'' which have to do with
vacuum polarization, causal locality, the particle versus fields issue,
symmetry, scattering theory and how to introduce interactions.

In the historical \ retrospective there were two important reasons why the
local quantum physical approach i.e. the formulation in terms of nets of
observable algebras and a dichotomy between algebras and states is
conceptually preferable to that based on point-like fields in terms of their
vacuum expectation values. First it incorporated the observation that the
S-matrix does not depend on which local field one chooses from the local
equivalence class; instead the S-matrix is being directly related to the net
structure of the local algebras without the intervention of ``field
coordinatizations''. The second equally important reason is that the net
structure of observable algebras allows for a comprehensive solution of the
appearance of (particle and field) statistics and internal symmetries, with
both properties being inexorably linked to each other (as already mentioned in
the previous section \cite{Haag}). This is at least in part reminiscent of the
intrinsic spirit (i.e. no use of classical relations through Lagrangian
formalism or other forms of quantization) underlying Wigner's 1939 particle theory.

Another verbal attempt (for the proofs the reader is referred to the
literature in the reference list at the end of this essay) to reveal some of
its deep content would be to say that the dichotomy between causal observables
and (representation-) sector generating fields (here ``field'' in the sense of
charge-carrying operators and not necessarily in that of pointlike covariant
objects) and the reconstruction of charged fields from observables is the
analog of the geometric reconstruction in Marc Kac's famous phrase ''how to
hear the shape of a drum''. \ In the context of local quantum physics this
includes not only the reconstruction of the internal symmetry groups from the
local observables (neutral fields), but also the derivation of the particle
statistics (and natural commutation relations of charge-carrying fields) from
the structure of observables\footnote{In the aforementioned (previous section)
work of Jordan and Schwinger the investigation of the relation between inner
symmetries and permutation group statistics representations was done in the
interaction-free global setting of observables as bilinear expressions in
(momentum space) creation/annihilation operators. Without the powerful causal
locality structure of the observables it is not possible to \textit{deduce}
the statistics of the charge-carrying operators (particles) from their
observable shadow.}. The DHR theory and the DR theorems \cite{D-R} constitute
the finishing touch for the internal symmetry problem, a problem started by
Jordan, Heisenberg and Schwinger.

There is a third powerful pragmatic as well as conceptual reason why this
algebraic operator-algebra-based setting of QFT may be crucial for resolving
once and for all the (only partially solved) short distance problem. The
Tomonaga-Schwinger-Feynman-Dyson renormalized perturbation theory of quantized
fields is limited by the power-counting rule i.e. only interaction densities
of maximal operator short distance dimension d=spacetime dimension (coupling
parameters with non-positive ``engineering'' dimensions) lead to
renormalizable theories. This rule excludes the coupling of all higher spin
fields with s%
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%BeginExpansion
$>$%
%EndExpansion
1 (since the dimension is
%TCIMACRO{\TEXTsymbol{>}}%
%BeginExpansion
$>$%
%EndExpansion
1 and the resulting power-counting exceeds 4) and if one does not use the
``gauge magic'' (see later remarks), it even excludes massive vector mesons.
More precisely the physical vector meson field consistent with the Wigner
particle concept has operator dimension 2 which renders any physical coupling
among its own kind and as well as with other fields of operator dimension
$\geq5$ i.e. also vectormesons would violate the powercounting
renormalizabilty test. There is no statement or argument that this definition
of renormalizability sets the natural frontiers allowed by the underlying
principles, rather it appears to be a technical borderline set by the use of
singular point-like field coordinatizations which is an undesired but
unavoidable collateral aspect of the classical parallelism of Lagrangian
quantization\footnote{The most general standard formulation in the presence of
interactions is ``causal perturbation theory'' which does not require the
chosen free field representation to be characterized in terms of an equation
of motion (i.e. a free Lagrangian is not needed).
\par
{}}. If one does implement interactions by locally coupling free fields (there
is no compelling reason for implementing interactions in terms of these
non-intrinsic and singular field coordinatizations!), then a violation of
power-counting would wreck the stabilization of the short-distance dimensions
of the interacting fields around their free field values (modulo logarithmic
corrections); the short-distance degree together with the number of parameters
would keep growing in every order.

Renormalization theory resulted from the recognition that part of the
problems, for whose solution Jordan in his 1929 Kharkov talk deemed new
revolutionary steps to be necessary, turned out to be soluble essentially in
the standard framework known at that time without any radical conceptual
change. As a result of Kramers's historically pivotal analogy with the
Lorentz-Poincar\'{e} infinities (which these authors encountered in their
attempts to combine classical fields with particles of classical mechanics),
renormalization was initially viewed as the dumping of infinities into a
finite number of physical parameters. But after the dust settled, a slightly
different picture emerged. Since particles according to Wigner are part of
Poincar\'{e} group positive energy representation theory, which in turn is
completely fixed by the action of the fields on the vacuum and the structure
of the resulting Hilbert space, the quantum field theoretic situation is
different from the classical counter part and conceptually even simpler. If
one formulates perturbation as the iterative construction of time-ordered
functions which fulfill the correct locality and spectral property (the
LSZ-setting) there are no infinities.

The feeling, that despite the overwhelming success of this post Jordan second
phase of progess in ``quantization of wave fields'' and the later extension of
that success through the ``standard model'' within the renormalization
formalism, one eventually will need a new conceptual and mathematical basis,
started to take a more concrete form in the framework of Wightman \cite{S-W}
and the addition of scattering theory (the LSZ framework and its derivation
from the Wightman theory in the presence of a spectral gap by Haag and Ruelle
\cite{Haag}). These formulations still used (what we called) ``singular field
coordinatizations'' (covariant point-like fields), although their analogy with
classical fields was considerably weakened by stripping off some of their
individual physical significance (apart from currents associated with
symmetries) and combining them into local equivalence classes which describe
the same physics and in particular have the same S-matrix as a class
invariant. The full liberation from these field coordinatization required
large conceptual and mathematical investments which, as a result of their
distance to quantization approaches had to pass through uncharted territories
and for this reason took a rather long time in terms of the speed of progress
with which the first (Jordan's quantization of wave fields ) and second phase
(renormalization) unfolded. An account of most of the structural results
obtained in this third phase up to the beginning of the 90ies can be found in
Haag's book on ''Local Quantum Physics'' \cite{Haag}.

These results were often criticized (and mostly ignored) by some particle
physicists as a result of what they conceived to be an unbalanced relation
between conceptual and mathematical investments and the scarcity of physically
testable results and interesting models. Since several years there are clear
signs that the situation is beginning to develop more favorably towards the
constructive side\footnote{In order to avoid misunderstandings it should be
stressed that the caution in using pointlike fields for the coordinatization
of nets of algebras refers to the process of their perturbative construction
(integrals over their intermediate vacuum fluctuations) and not to their use
in presenting the final result (i.e. the interacting net).}. On the one hand
the algebraic method leads to a decomposition theory of fields on the
conformal covering (with respect to the center of the covering) in terms of
operator-valued sections on the Dirac-Weyl compactification of Minkowski
space, a fact which was already noted at the beginning of the 70ies
\cite{S-S}. Combined with the increase in mathematical knowledge about
algebras related to loop groups (Kac-Moody) and diffeomorphisms of the circle
(Witt-Gelfand-Fuchs-Virasoro) and together with the experience of physicists
from the representation theory of abelian (Thirring model) and non-abelian
current algebras (nonabelian generalization of the Thirring model), the
discovery of exactly soluable d=1+1 conformal models started with the
construction of exact 4-point functions of minimal models (which contain the
field theories of the Ising- and Potts- model) in an extremely seminal paper
\cite{BPZ}. A significant boost came also from the identification of braid
group statistics \cite{R-S}\cite{Fro} as the determining property for the
commutation properties between these component (``block'') fields\footnote{In
higher dimensional conformal theories the appearance of the braid group is
limited to the commutation relation of the conformal component (block) fields
in the timelike Huygens region which sets the spectrum of anomalous dimensions
but is not related to statistics \cite{S1}.} (in case of d=1+1 one has
field-valued sections on the circle). It turned out that pointlike fields and
nets of algebras in the conformal setting are closely related in that the
latter always permit such coordinatizations \cite{Jrss} i.e. the net point of
view is equivalent to the Wightman formulation, a fact which one also believes
to hold in general.

Interacting conformal theories do not permit a Wigner particle interpretation
\cite{S2} and the derivation of an S-matrix; for this reason are not of
\textit{direct} physical relevance to particle physics. In addition their
explicit construction is based on the representation theory of infinite
dimensional Lie algebras which limit the conformal field theories to those
whose building blocks are chiral fields on the one-dimensional lightray.
Barring subtle connections between such chiral theories and higher dimensional
quantum field theories (see the later discussion of algebraic lightfront
holography), the crucial question for a better understanding of the short
distance problem in bona fide particle physics is: can one find a constructive
approach which does not use correlation functions of pointlike fields (i.e.
those singular coordinatizations of algebraic nets which are the potential
carriers of the ``short distance germs'' whose successful cure within the
standard Lagrangian setting leads us back to the limitations set by
powercounting) from the outset but rather employs on-shell quantities as the
S-matrix and formfactors \footnote{These are matrix elements of operators (or
even pointlike fields) between Wigner multiparticle states; or in a more
graphical terminology: objects which apart from one (composite) leg have all
other legs ``on-shell''. Such ``sesquilinear forms'' do not reveal explicit
short distance fluctuation. The Feynman theory cannot be formulated in terms
of formfactors only.} of localized operators? This is the kind of guiding idea
which was behind Heisenberg's 1943 unsucessful attempt to formulate
relativistic particle physics in a pure S-matrix setting \cite{Hei}. The
subsequence appearance of renormalization theory took some of the original
motivation which was based on the believe that QFT was beset by an incurable
short-distance disease, but basically it only shifted the short-distance
problem to another conceptual level. 

15 years later the S-matrix setting was enriched by the introduction of
analytic properties related to the adaptation of Kramers-Kronig dispersion
theory to relativistic particle physics. Together with the use of Feynman
rules it became clear that the on-shell program in Heisenberg's formulation
still lacked a very important property: the crossing property which links a
scattering process analytically via the complexified mass shells to its
crossed (''dual'') counterpart involving antiparticles. Although the
dispersion relation technology permitted a relatively easy (in certain cases
even nonperturbative \cite{Bros}) derivation for the elastic scattering part
of the amplitude, its conceptual understanding remained somewhat of a mystery,
apart from the general impression that crossing seemed to be a kind of
on-shell imprint of causal locality\footnote{The reader is advised to be
careful about the literature on crossing. In most articles the crossing
property is established by connecting the crossed channel to the original
process via an analytic path which leaves the mass shell. The true crossing
property in the sense used here requires to find a path which remains on the
(complex) mass shell \cite{Bros}. Its physical interpretation has remained
somewhat mysterious even after the observation that it is closely linked with
the KMS thermal properties of Unruh wedge algebras.}. A new construction
program, the so-called ``bootstrap program'' was formulated \cite{Chew} and
some physicists had high hopes about an ultraviolet-finite passage into
particle physics in which all the physical principles which on-shell objects
inherited from the off-shell setting were to be maintained. Despite many
ingenious supportive inventions (Mandelstam representation, Regge poles,..),
the program did not produce physically interesting results and fell out of
favor. There was one obvious explanation had been advertized as an alternative
to QFT, instead of a complementing tool. Even worse, in some of the old papers
and conference reports one notices a kind of cleansing rage against good old
QFT. But around that time QFT had an impressive comeback; it re-appeared as a
physically very healthy object bringing fruitful progress via renewed
interests and substantial progress especially in gauge theories. Although the
historical continuity with Oscar Klein's gauge theory had been lost, the
powerful renormalization technology (combined with the progress in the
experimental situation in weak and strong interactions) elevated gauge
theories to one of the most valuable instrument of particle physics which the
protagonists of the pre-war gauge theory could not even dream of.

Another more intrinsic reason for the demise of the S-matrix bootstrap program
was that its protagonist never succeeded to implement an operator formulation
of crossing in order to be able to use it as a computational tool e.g. in
perturbative iterations. This shortcoming is of course connected with the lack
of complete understanding of how it is conceptually related to the underlying
principles of QFT.

As it happens often in theoretical physics, if a problem appears insoluble
with the help of the available methods, physicist invent another similar
problem of similar appearance. In the present case Veneziano was able to come
across an ingenious construction of a nonunitary model (to be iteratively
unitarized later on, similar to perturbation theory) of elastic scattering
which fulfills crossing as a result of functional relations for $\Gamma
$-functions. Even more, it turned out that these properties allowed for a
mathematical interpretation in terms of space-time commutation relations
(duality relations) of a 4-point-function of an underlying auxiliary d=1+1
conformally covariant free field theory. Its main unusual property was the
existence of infinite particle towers (in zero order unitarization) and its
consistency requiring specific high space-time dimensions. This p-space
spectrum which resembles that of a quantized string then led to the name
string theory \cite{GSW}. As a result of its somewhat ad hoc construction its
position to the causal localization aspects and related to this its physical
interpretation remained mysterious. Its unclear status with respect to
stringlike causal localizablity (or for that matter any kind of localization)
has plagued it ever since and even after more than 30 years is a serious
liability, since localization is the most important property for any kind of
physical interpretation and for the derivation of scattering theory (which in
string theory is imposed as a kind of cooking recipe).

On the other hand the string theory turned out to give rise to interesting new
mathematical structures; in fact one of the remaining unexplained surprising
aspects for field theorists is how such a wealth of mathematics can come from
ideas suggested by physicists without having any tangible success in particle physics.

Recently one has been able to combine some aspects of the old mass shell
bootstrap approach with localization properties of local quantum physics and
in this way one has made significant progress in setting up equations for
generalized formfactors which bypass the ultraviolet problems of the
Lagrangian approach. This progress was made possible by using a very powerful
mathematical tool of operator algebras: the (Tomita-Takesaki\footnote{This
discovery of this theory goes back to the mathematician Tomita and was refined
and extended by Takesaki. In a more special setting it was independently found
by the physicists Haag Hugenholtz and Winnik \cite{Haag}. Its significance for
the problem of causal localization was first pointed out by Bisognano and
Wichmann \cite{Haag}.}) modular theory \cite{Borchers}. It turns out that for
the vacuum state restricted to the operator algebra localized in a (Rindler)
wedge region of Minkowski spacetime the associated modular objects have
geometrical-physical significance in terms of the L-boost and the antilinear
TCP operator. The spatial adaptation of this theory to the Wigner one particle
representations (irreducible positive energy representations of the covering
of the Poincar\'{e} group) leads to a direct construction of interaction free
nets of algebras without having to pass through pointlike fields
\cite{BGL}\cite{F-S1}. A surprising new result is that the hitherto somewhat
mysterious zero mass spin towers (which in Wigner's classification were called
``continuous spin representations'') turn out to be free zero mass strings,
but this time not only in the sense of an infinite spin tower structure but
also as objects with a well-defined semiinfinite stringlike
localization\footnote{The compactly modular subspaces are empty and the
non-trivial subspaces with the smallest modular localization are associated
with arbitrarily thin space-like cones (semiinfinite string cores).}. These
string localizability on the one-particle level leads to ``string fields'' in
a Fock space \cite{M-S}. The fact that for these string-localized Wigner zero
mass spin towers the Poincar\'{e} invariance does not lead to a restriction of
spacetime dimensionality (d=26 for massive strings of string theory) suggests
that there may be an important and subtle point which still needs to be
settled. 

The d=1+2 Poincar\'{e} group turns out to be a special case in that the
universal covering group is infinite-dimensional and the value of the spin is
a priori not quantized (``anyons''). Here the surprise is that the more subtle
spin-statistics connection is already preempted by the one-particle properties
\cite{Mund1}. The best possible localization also turns out to be also
semiinfinite string like. But in contrast to the spin tower representations
any stringlike localized operator applied to the vacuum with a one-particle
component is always accompanied \cite{Mund2} by vacuum polarization
clouds\footnote{Usually one expects these clouds to be present in interacting
theories; here it is the sustension of braid group statistics in an otherwise
interaction-free situation which requires their presence.}; the smallest
localization region which allows a polarization-free particle creation is a
wedge region. In d%
%TCIMACRO{\TEXTsymbol{>}}%
%BeginExpansion
$>$%
%EndExpansion
1+2 modular theory allows to define ``interacting'' in a completely intrinsic
way without any reference to Lagrangians only in terms of net properties.

The re-investigation of the Wigner particle spaces in the new modular setting
has also shed some new light onto a very old problem from the
Dirac-Jordan-Klein days. The interpretation of the c-number solutions of
relativistic wave equation had exposed the Klein paradox as a kind of warning
against interpreting such waves too naively in the setting of quantum
mechanics. The modular approach shows that this problem has its origin already
in a certain behavior of the positive energy wave function alone (without
coupling to external fields and without explicitly introducing the negative
energy solutions). The Wigner one-particle theory is ``heterotic'' with
respect to localization. Besides the Newton-Wigner localization which results
from adapting the Born localization probability of the Schr\"{o}dinger theory
there is a second localization based on modular defined real subspaces of the
complex Wigner representation space \cite{F-S1}\cite{Halvor2}. Wheras the N-W
localization is only asymptotically covariant and causal\footnote{The
derivation of time dependent scattering theory uses the N-W localization and
the Poincar\'{e} invariance and cluster properties of the S-matrix only
require its asymptotic (large time-like distances) invariance of the N-W
localization.}, the modular localization is covariant and causal and must be
used in problems of microcausal causal propagation in spacetime (no
superluminal signals in QFT). The most surprising aspect of modular
localization is that it prepares the analytic properties which link positive
with negative frequency solutions, the virtual presence of antiparticles and
the prerequisites for crossing properties from particles to anti-particles.
Even more surprising: it preempts the connection between spin\&statistics in
the setting of one-particle spaces!

This raises the question whether in the presence of interactions there exists
an approach which generalizes this modular construction from particle data to
a full-fledged LQP. Knowing the mere presence of interactions (via the vacuum
polarization behavior of subwedge-localized dense subspaces of state vectors
in the aforementioned sense) is however not enough; one also has to be able to
classify interacting nets and give a road map of their concrete construction.
One expects that the modular data of local algebras (especially the modular
groups) will lead to more detailed properties of the vacuum polarization
clouds which can be used for classifying interactions in a completely
intrinsic way.

The strategy towards construction based on modular theory is suggested by the
previously mentioned special case of Wigner particles in the absence of
interactions. One starts from the easily established fact that the true
on-shell S-matrix (not the off-shell ``S-matrix'' in the sense of a generation
functional) is a relative modular invariant of a wedge algebra in the vacuum
reference state\footnote{As the phenomenon of vacuum polarization this modular
aspect of the S-matrix is characteristic of LQP (and related to the the vacuum
fluctuation properties).}. This characterizes the position of the dense subset
of wedge-localized states (which by using an adapted thermal topology becomes
a full Hilbert space), but it does not yet allow to compute the position of
the wedge subalgebras inside the total algebra. The aim of the modular
approach is precisely to understand this relative position since then the net
of all causally localized subalgebras can be determined with the help of
Poincar\'{e} covariance by modular inclusions and modular intersections from
the net of wedge algebras. Having achieved this, one may of course ask for
covariant pointlike generators of this net. 

It is very encouraging that this scenario, even without looking at its
detailed mathematical implementation, allows to prove the uniqueness of a
local net associated with a given crossing symmetric S-matrix \cite{S3} (but
not its existence!). There was no chance for the solution of the inverse
scattering problem to lead to QFT in terms of a distinguished pointlike field
(since the local equivalence class of fields has the same S-matrix), but at
least the QFT in the intrinsic net formulation is unique, if it exists at all.
A modest attempt towards an explicit construction would consist in testing the
present scenario for a family of d=1+1 dimensional crossing symmetric
factorizing elastic S-matrices for which investigations carried out in the
70's showed that the old bootstrap program works (albeit not to the
satisfaction of those who wanted the bootstrap approach to select a unique
``theory of everything'', one rather obtained scores of families of
factorizing crossing symmetric S-matrices). My algebraic attempt was rewarded
by an immediate success: the generators of the wedge algebras associated with
factorizable S-matrices fulfill a Zamolodchikov-Faddeev-like algebra
\cite{schroer}\cite{Lechner} without vacuum polarization and the sharpening of
the localization leads to structural equations for the formfactors
(matrixelements of these sharper localized operators between in and out
multiparticle scattering states) which describe the shape of the vacuum
polarization clouds\footnote{Factorizing models have no real (on-shell)
particle creation but exhibit the full virtual vacuum polarization structure
characteristic of causal QFT.}. Factorizing models certainly cannot exist in
d=1+3; according to an old theorem higher dimensional theories inexorably
relate the virtual to the real particle creation structure \cite{BBS}. Indeed
the generalization of these modular constructions to higher dimensional
theories pose new problems which are presently being studied \cite{on-shell}.
The best one can presently expect from such an on-shell approach in terms of
the S-matrix and generalized formfactors is a classification of all local
theories which permit a perturbative version (with the tree S-matrix and the
associated formfactors as input) which is of course expected to include all
renormalizable theories in the setting of the standard Lagrangian approach. In
this way one would expect to learn something about interactions with higher
spins. 

It is interesting to illustrate the precarious conceptual aspect of the
standard approach by looking at interacting (massive) vectormesons. The
physical vectormeson field with the smallest operator dimension is a
transverse vectorfield. Since it has d=2 instead of the classical value d=1
any interaction coupling (which must at least be of third degree in the free
fields) has inevitably operator dimension $\geq5$ which would make it
nonrenormalizable i.e. the dimension of the interacting field would keep
increasing with the perturbative order and these higher polynomials would
cause an ever increasing number of independent parameters. The standard trick
to make renormalizable vectormesons is to invoke the classical notion of gauge
theory in conjunction with the mass-providing Higgs mechanism. However there
is a completely different way to obtain the same observables without Higgs condensates:

(1) Start with free massive vectormesons in zero order and represent their
associated Wigner one-particle state space as a cohomology of a larger space
involving ghost components (indefinite metric Wigner representation space)
i.e. a kind of BRS trick on the level of one-particle spaces.

(2) Construct the larger Fock space over this Wigner space in which the
physical vectormeson field is now replaced by an unphysical multiplet of
fields of operator dimensions 1 which brings the dimension of the interaction
density into the renormalizable range $\leq4$

(3) Note that beyond zero order one runs into consistency problems unless one
introduces additional physical degrees of freedom whose simplest (and probably
unique) realization in a renormalizable context is a scalar field namely the
Higgs stripped off its one point vacuum expectation value (the vectormeson
already had a mass in zero order!)

(4) Check that the cohomological representation really stabilizes the
situation in the sense that a physical descent is possible after having done
all the perturbative calculations including the renormalization.

The derivation of these statements can be found in \cite{D-S} and the idea
especially concerning the avoidance of the imposition of the Higgs mechanism
and obtaining the presence of a scalar particle from consistency can be traced
back to \cite{Scharf}. But in the present context our main aim is not to
question the intrinsic physical meaning of the Higgs mechanism (and convince
the reader that it is not more than a mnemonic device) but rather to turn the
whole gauge formulation from top to bottom. Since there is only one
renormalizable coupling involving vectormesons there is obviously no need for
a ``gauge principle''; the local quantum situation is radically different from
the classical one where there are many ways of coupling vector fields and
where one therefore needs a (gauge) selection principle. The logic should
therefore be%
\[
renormalizability\overset{semiclassically}{\longrightarrow}%
classical\,\,gauge\text{ }principle
\]
i.e. the local quantum theory explains the classical gauge
principle\footnote{Strictly speaking the classical gauge concept arose only
for massless photons (and their non-abelian gluon counterparts). In the
context of the present vectormeson setting one would have to construct the
massless limit by a careful pertubative decoupling of the scalar particles and
an accompanying emergence of abelian charges (in the dual language: the loss
of screening).} (in the semiclassical approximation of the model) in terms of
the renormalizability requirement. One mystery has been explained in terms of
another one; but since local quantum physics is more fundamental, there is a
better chance of a future de-mystification on the renormalizability side. 

The power counting rule is a well-defined, albeit somewhat formal borderline,
but the present example shows that in case of spin 1 there are cohomological
tricks which can extend the power counting rule to include ``ghost
catalyzers'' (ghosts which are only used in intermediate steps). Since for
higher spin there may be similar or different tricks, one should admit that
one really does not know the true intrinsic borderline for models allowed by
the underlying physical principles. This brings us back to Jordan's 1929
critical assessment of the situation and the need for a radically different
approach as compared to the standard quantization formalism even after the
discovery of renormalization. 

There have been attempts to improve the short distance behavior in the
conventional point-like field setting by enforcing a partial
cancellation\footnote{The wave function normalization constant can not be
finite unless the theory is trivial.} of divergencies between Fermions and
Bosons via the so-called supersymmetry. But whereas the spontaneous breaking
of internal- and Lorentz- symmetries (e.g. by a heat bath) is a welcome
physical enrichment, the supersymmetry in a heat bath environment shows the
pathological behavior of a ''collapse'' \cite{Bu} i.e. it has some aspects of
a Pyrrhus victory. This means that there is no enlargement of the Hilbert
space by combining different values of the breaking parameter (e.g. the
preferred rotational direction in the case of a Ferromagnet) such that the
symmetry can be recovered in the larger reducible representation space. In
addition to this theoretical peculiarity, nature has refused for more than 30
years to jump over those sticks which supersymmetrist prepared for this
occasion and it is questionable that this will happen in the future.

A very important new instrument for a non-perturbative constructive analysis
base on modular ideas has recently emerged: the algebraic lightfront
holography (LFH). Its main aim is to look first at the simpler algebra on the
causal horizon of a wedge \cite{LHF} (which is half of a lightfront). In d=1+3
this algebra has a 7-parametric subgroup as a symmetry group, but as a result
of the surprising absence of transverse vacuum polarization the LFH is
identical to an extended conformal chiral theory. Apart from free fields in
which case the LFH can be obtained simply by a process of restriction to the
lightfront, the LFH process for interacting theories has to pass through
purely algebraic intermediate steps. In contrast to equal time algebras, the
existence of d-1 dimensional generating fields is not limited by short
distance behaviour since unlike canonical fields they turn out to be
gerneralized chiral fields. The very mathematical nature of the wedge- and
horizon- localized algebras (they are hyperfinite type III$_{1}$ von Neumann
factors) on which the prerequisite for density matrices namely the existence
of tracial states is not met, and hence a notion of entropy cannot be defined.
The physical origin of this phenomenon are the uncontrollably big vacuum
fluctuations which accompany the creation of sharply defined causal
localization boundaries. The remedy is to make a fuzzy surface which leaves
the vacuum polarization to reorganize themselves in a halo surrounding the
original localization region. This is done with the help of the so-called
split inclusion property, which, different from a cutoff, maintains the
original local theory while only reorganizing some local degrees of freedom in
the finitely extended halo (whose extension is a control parameter for the
size of vacuum fluctuations). The vacuum state restricted to the split
inclusion becomes a thermal density matrix (with the Hawking temperature) in
an appropriately defined tensor factorization of the total Hilbert space. This
sequence of density matrices for decreasing halo size converges (as expected)
against the dilation operator (the holographic image of the wedge-associated
boost) which is a non-trace class operator which sets the thermal KMS
properties of the Unruh-Rindler effect; thus if one would be able to associate
a split-loaclization entropy with the finite halo situation, one can be sure
that this is naturally associated with the Hawking-Unruh temperature aspect.
The transverse symmetry of the horizon (whose linear extension is the
lightfront) forces the concept of an area (the dimension of the transverse
edge of the wedge) density of split-localized entropy. This shows that the
prerequisites for a Bekenstein-like quantum area law for localization entropy
are met in a surprisingly generic manner. But for the control of a limiting
halo-independent area density two more properties remain to be established:

(1) The increase of the area density with decreasing halo size $\varepsilon$
is universal (model calculations indicate that it goes like $ln\varepsilon)$
so that it possible to have a finite relative area density between systems
with different quantum matter content.

(2) The validity of thermodynamic basic laws for causal localization-caused
thermal behavior which parallels those of the standard heat-bath thermal setting.

These problems are presently being investigated.

This brief excursion into contemporary problems of QFT is to indicate that the
process of its radical reformulation in completely intrinsic terms i.e. away
from quantization (in the sense of a classical parallelism) as envisaged in
Jordans plenary talk at the 1929 Kharkov conference has begun some time ago,
and new interesting ideas and a completely fresh look at QFT have been coming
in at an accelerated pace. The picture which is revealed is quite different
from the string theorists ``old field theory'' i.e. Lagrangian perturbation
theory enriched with instantons and supersymmetry and what string theorist
call the low energy ``effective'' QFT from their scale-sliding on strings.

In order to get away from the present unilateralism (in terms of areas and
ideas, not countries) in particle physics and to decrease the gap between
physics and what physicists are doing\footnote{The recent ``Bogdanov
scandal'', in which a PhD in particle physics was awarded for manipulating a
pseudo-physical language (which to a large part originated from string
theory), shows how confused and fragil the situation already has become.}, it
is hepful to look at the work of the old masters which reached such a perfect
match between conceptual depth, physical relevance and mathematical skill. In
the continental part of Europe of the 20's all these qualities were
represented in the trio Wigner, Jordan and John von Neumann. Wigner who
originally studied chemistry under Polanyi at the TU in Berlin-Charlottenburg
(where he got his degree in 1925 and joined the physics group in G\"{o}ttingen
some years later) developed an excellent strength in the area of conceptual
and calculational problems. The more mathematical side of the new quantum
physics was superbly represented by the brilliance of John von Neumann whereas
Jordan's abilities was somewhere in the middle of the two in that he was very
powerful in both areas. Whereas Dirac's more geometric and analytic way of
doing quantum theory may be viewed as the pivotal link between Schr\"{o}dinger
and and the pioneers of renormalized Lagrangian field theory (Tomonaga,
Schwinger, Feynman, Dyson) one can argue that Wigner Jordan and von Neumann
with their emphasis on a more intrinsic approach which leans less on
quantization and more on algebraic concepts prepared the ground for what is
presently referred to as algebraic QFT or loal quantum physics. All the
ingredients of this new setting already existed in parts: Wigner's group
theory based approach to particles, Jordan's emphasis on 4-dimensional
formulation based on causal locality and von Neumann's idea of operator
algebras and their state space preduals. But it was not possible to amalgamate
them into a new framework without additional very large amount of progress on
a physical conceptual as well as mathematical level. It is very interesting
and profitable to look with present hindsight at what they were able to
achieve and what problems they encountered on their way towards a more
intrinsic framework for QFT and particle physics. After Wigner found his
representation theoretical method to particle physics (which was not
appreciated by his contemporaries who were still hung-up on quantizing field
equations) he tried to obtain an access to QFT by adding the idea of
localization. He found what is nowadays referred to as the Newton-Wigner
localization which amounted practically to an adaptation of the
nonrelativistic $\overrightarrow{x}$-space probability amplitude
interpretation for his group-representation approach. The non-covariant and
acausal propagation aspects of this localization bothered him very much and
since a causal localization was an indispensible prerequisite of local quantum
physics Wigner began to have doubts about QFT or rather its usefulness for
particle physics\footnote{Private communication from R. Haag.
\par
{}}. On the other hand, Jordan attributed crucial significance to causal
locality ever since he elaborated with Pauli the P-J commutator function, but
he did not know how these properties can be used for the localization of
states and how particle states can be introduced at all into an interacting
field formalism. Apart from questions of locality and localization of states
he also had a strong interest in the foundations of (global) quantum theory.
Coming from Matrix Mechanics he had some conceptual difficulties with the
suggestiveness of the superposition principle in Schr\"{o}dinger's wave
formulation. As a result he elaborated a conceptually more palatable setting
by changing the multiplication structure on the algebraic side \cite{J}. He
convinced his colleagues von Neumann and Wigner that this was a fruitful idea
and as a result they wrote up some joint work on what was later called the
Jordan algebras \cite{J-N-W}. But what could have been the beginning of a
fruitful long-lasting collaboration became a victim of the turmoil caused by
the NS regime's rise to power.

The problem of finding physically more acceptable assumptions from which the
superposition principle and the operator formalism of quantum physics could be
derived has attracted interest up to this date. In more recent times the
attention shifted somewhat from the algebraic level to the dual level of
states i.e. to the question what additional physically natural properties must
the convex space of states have in order to be identifiable with a state space
on an operator algebra \cite{Connes}, but in most of these more recent
investigations the Jordan algebras continue to play an important role
\cite{Alfsen}\cite{Bel}. Wigner's problem to find the correct localization
which links his particle theory with the causal locality of QFT has meanwhile
been solved by the aforementioned modular theory \cite{BGL}\cite{F-S1}. This
modular theory has also enriched the von Neumann operator algebra setting in
such a way that the causal locality principle has been naturally incorporated
in the net (of operator algebras) approach of local quantum physics
\cite{Borchers}\cite{Haag}. As also mentioned before, this new setting
promises to lead to a revolutionary new nonperturbative exploration
(classification and construction) of the same physical principles which
underlie the standard Lagrangian approach. Whereas the standard approach has
had many successes and appears to be played out, the new setting is very much
in its initial stage and as a result of its large distance from any
quantization setting it is expected to require more time for the revelation of
its full content and power. It should be clear from these arguments that with
a similar stretch of imagination which connects Schr\"{o}dinger and Dirac with
the protagonists of the standard approach one is able to see in Jordan von
Neumann and Wigner the originators of some alternative ideas which point into
the direction of local quantum physics.

All the protagonists of those pioneering days of quantum physics have been
commemorated in centennials except Pascual Jordan who, as the result of the
history we have described, appearantly remained a ''sticky'' problem despite
Pauli's intercession by stating ``It would be incorrect for West Germany to
ignore a person like P. Jordan''. His postwar activities consisted mainly in
creating and arranging material support (by grants from Academies and
Industry) for a very successful group of highly motivated and very talented
young researchers in the area of General Relativity who became internationally
known (Engelbert Schuecking, Juergen Ehlers,..) and attracted famous visitors
especially from Peter Bergmann (Rainer Sachs,....). In this indirect way there
is a connection between Jordan's post war activities in general relativity and
the new Albert Einstein Institute in Golm (Potsdam). This path leads from
Jordan's Seminar in Hamburg through universities in Texas (where most of its
members got positions when Jordan's chair at the university of Hamburg was
lost for general relativity group), and then via the astrophysics in Garching
(where Ehlers took up a position in 1971) to the AEI for Gravitational Physics
of the MPI where Ehlers became the founding director in 1994.

Acknowledgments: I owe thanks to Engelbert Sch\"{u}cking for informations
concerning previous article and books related to Jordan. I am particularly
indebted to Anita Ehlers for making her unpublished biographical notes
available to me. Finally I thank J\"{u}rgen Ehlers for several interesting
discussions on some problems of interpretations and chronological orders in
Jordan's papers.

{\small Epilogue}

{\small Since this essay was written during the war preparations of the Bush
administration, I cannot help thinking about parallels to the issue of
``preemptive war'' in Jordan's time. In all the sadness and bitterness about
the impunity with which the international law can be broken in our times there
is one small consolation in that at least most of the Christian churches have
learned the message from their terrible failure in the second world war. This
time the misuse of religion for the justification of violation of
international law and human rights is limited to where it already was
instrumentalized 60 years ago, namely to warmongerers who believe in their
heavenly mission of being God-chosen instruments of divine destination.}

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\end{document}
