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What it's about
Assuming
that the reader knows what gauge field theory
is, the story can be made short. It is well known that
massless spin 1 gauge fields describe the three known non-gravitational
interactions between elementary particles: electro-magnetic, weak and strong.
The gravitational interaction can be described as mediated by a massless spin
2 field. Then the question arises, what kinds of interactions are mediated by
massless spin 3, 4, 5, ... fields?
Non, as far
as we know at the present. This is the experimental situation. Nevertheless,
higher spin gauge fields are interesting from a theoretical point of view. As
far as non-interacting free fields go, there is nothing strange with higher
spin (spin greater than 2). The field equations are natural generalizations
of those for lower spin fields. However, free fields are not very interesting, and the theoretical
question is: Is it possible to introduce non-trivial interactions for higher
spin gauge fields? As a matter of fact, this has turned out to be an
extremely difficult question to answer.
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Why it's interesting
There are
lots of theoretical problems with interacting higher spin gauge fields, well
researched over the last 40 or so years. All the standard recipes for
introducing interactions between higher spin gauge fields and the low spin
gauge fields fail for some reason or another. It is also known that they cannot generate long range forces. These problems
might indicate that there is something wrong with them, at least as real
physics goes - perhaps nature cannot use them for anything - and the
mathematical problems, even non-consistency, is a symptom of this fact.
However, on the theoretical side, evidence is gathering that self-interacting
higher spin gauge field theories can be defined, as will be described below. Impressive progress on interactions in
anti-de Sitter space was achieved by M. Vasiliev during the 1990's when
almost no-one else was interested in the problem. This helped sustain the
area of research and it came into focus again in the early 2000's by new
groups of researchers. At the present time (2012) higher spin field theory is
an active area.
The problem is thus very intriguing. Higher spin massless particles exist as representations of the
Minkowski spacetime Poincaré group and the free field theories are natural
generalisations of spin 1 and 2 free field theory. We know that in order to
have any chance of introducing interactions we need to have an infinite tower
of fields of increasing spin. It is indeed likely that interacting theories
exist but that they are very complicated, at least when viewed in terms of
component fields.
My point of view as regards the physics of higher spin gauge fields is the
following.
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If the
mathematical problem of introducing interactions of any kind cannot be
solved, then I cannot see any way that they can exist (unless as free
felds) in nature. I realize that this explicitly assumes
a presuppostion that anything that exists in nature, at least at a
fundamental level, has a consistent mathematical description.
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If
mathematically consistent interactions can be introduced, however
complicated, then the question as to their physical existence is open, and
can only be solved experimentally or observationally. Not everything that
can be mathematically described need be physically realized (as far as
I understand).
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Lightning history
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P.A.M.
Dirac is considered to be the first (in 1936) to study relativistic field
equations for fields of higher spin.
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However,
there is a not very well known paper by Majorana from 1932 that considers
arbitrary spin particles.
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A few
years later, M. Fierz and W. Pauli, studied electromagnetic coupling for
massive higher spin fields. Inconsistencies (of the kinds that were to
plague the theory for the coming years) were encountered.
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The
theory of higher spin fields then seems to have lain dormant until the mid
1960's when it was taken up by S-J. Chang, and later on in the mid 1970's
by C.R. Hagen and L.P.S. Singh. However, in 1964, S. Weinberg showed, using
S-matrix methods, that higher spin massless particles cannot generate long
range forces. J. Schwinger studied wave equations for spin 5/2 and spin 3
in 1970.
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Then the
theory of massless higher spin fields was taken up in earnest by C.
Fronsdal and collaborators (J. Fang) in late 1970's. Fronsdal clarified the
free field theory and derived both the field equations and Lagrangians for
massless higher spin fields and clearly established their nature of being
gauge fields.
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Fronsdal's
theory was then further systematised by B. de Wit and D.Z. Freedman in
1979.
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Fronsdal
and Fang seems to have been the first to raise the question of self
interactions for higher spin gauge fields generalising the gauge field
theories for spin 1 (Yang-Mills theory) and 2 (Gravitation). They phrased the
problem as one of deforming the
free Lagrangian and the free field gauge transformations. As a parallel
track to this, Fronsdal researched an approach to higher spin gauge fields
in anti-de Sitter spacetime (AdS). Fronsdal also discerned the first hints
that a theory of interacting higher spin gauge fields would require an
infinite tower (in spin) of such fields.
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During
the 1970's and 1980's, partly in connection to the intensive research on
supersymmetry, many authors (H. A. Buchdal, K. Jonhson & E. C. G.
Sudarshan, G. Velo & D. Zwanziger, C. Aragone & S. Deser, B. deWit
& F. A. Berends & J. W. van Holten & P. van Nieuwenhuizen, N.
H. Barth & S. M. Christensen, T. Curtright) investigated higher spin
interactions. Inconsistencies were encountered.
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The
first positive results on self interactions were obtained by L. Brink, I. Bengtsson
and myself (A.K.H. Bengtsson) in a lightfront
formulation in 1983.
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Soon
after this F.A. Berends, G.H.J. Burgers and H. van Dam (BBvD) obtained a
covariant gauge invariant cubic self interacting vertex for spin 3 fields.
These authors also performed a general study of the higher spin self
interaction problem in the deformation theoretic approach.
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In the
late 1980's, S. Ouvry and J. Stern, and myself
independently, discovered that all of Fronsdal’s free higher spin equations
and gauge transformations could be nicely collected in a simple BRST
formulation that was discovered as a limit of string theory (the zero
tension limit).
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In the
late 1980's, I
formulated the first steps towards an interacting theory in this BRST
framework and managed to derive the Yang-Mills cubic coupling within the
formalism. Further progress was hindered by the lack of efficient
calculational techniques.
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Then the
area of research took a new turn when E.S. Fradkin and M.A. Vasiliev in the late 1980's
started to study higher spin gauge fields in AdS. After sorting out the
free field theory, they generalised the AdS spacetime symmetry algebra to
higher spin algebras which could then be "gauged" and in that way field equations could be
obtained. This approach, lead by M. A. Vasiliev, can be said to have
dominated the area up to the turn of the millennium.
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During
the period from the late 1990's up to the present, the
BRST formulation of the free higher spin gauge theory has been rediscovered
quite few times by authors apparently unaware of the mid 1980's work.
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A
tremendous amount of work on free fields has been done by many researchers
during the last 10 years, and is still being done. This is not the place to
review that work (even if I could - which I can't). Reviews are easily
found through the arXiv http://arxiv.org/.
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In the
mid 1990's, the mathematicians R. Fulp, T. Lada and J. Stasheff showed that
the formulation of BBvD, if it actually defined an existing interacting
field theory, must be a model of a strongly homotopy algebra.
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My role and research
I came in
contact with the higher spin problem in 1983 through my PhD supervisor Lars
Brink and my friend and fellow PhD student Ingemar Bengtsson. Lightfront
methods were popular at that time in string theory, and I got my own first problem of finding the cubic supergravity
coupling in the lightfront gauge. There was a parameter λ in the theory
with the interpretation of helicity which I set to the value 2 in the case of
gravity. Actually to be completely honest, for a couple of weeks I happily
calculated away with the value 0, such was the depth of my ignorance! This
error however made it completely obvious to me that λ could take any
non-negative integer value, and I had a rough idea of what would happen if
that was the case. But I did not really understand the significance. Higher
spins were discussed, and Ingemar started to do the calculations with
arbitrary λ. Having got stuck on late night (with the Ji- generator) he called me and I knew what to do
since I had the corresponding supergravity calculations in front of me at my
desk. We wrote two papers that contained the first positive results on
massless higher spin self-interactions. We tried to do the calculations
covariantly, but our ansatz was too restrictive, and not very much came out
of it, except that I studied the covariant spin 3 gauge
algebra and managed to show that it cannot close on spin 3 fields.
This piece of work together with hints to the same effect in Fronsdal's
papers, convinced us that in order to have self interactions for gauge fields
of higher spin, you must include in infinite tower of such fields (with ever
increasing spin). Then the above mentioned paper by Berends, Burgers and van
Dam appeared with the covariant cubic spin three vertex.
The existence of cubic interaction terms is not in conflict with the infinite
tower of fields, the reason being that the non-closure of the gauge algebra
probes the quartic level of interaction.
During my first year as a post-doc at Queen Mary College in London I finally
realized (one spring afternoon while walking from Kentishtown to Highgate to fetch my son) that the free field
theory could be formulated in a very simple way using BRST methods borrowed
from string theory. All the Fronsdal free field equations could be collected
into one simple equation Q|Φ>=0, where Q
is the BRST operator and |Φ> contains all higher spin gauge
fields. During the second year, I managed to calculate enough of the cubic
vertex to show that it reproduces Yang-Mills 3-point coupling. Then progress
was halted due to the complexity of the calculations
(i.e. they were to boring to carry out, or rather, I knew before starting
that I would lose confidence half way through because the probability of at
least one error would be 1).
During the long interval between 1990 and 2003 I worked on the problem for a
couple of months at a time, on and off perhaps every three years but I did
not try to publish as I didn't really get any coherent results. I was teaching in a gymnasium, my
kids were small and I wanted to spend time with my family, and the time that
was left I used to read up on philosophy and history of mathematics and such
things. Inspired by
the work of Vasiliev, I set up the BRST formalism in AdS. I also tried to
look into the theory of the notorious singletons. Singletons are spin 0 and
spin 1/2 representations (named Rac and Di respectively by Fronsdal) of the
AdS group which, although they are low spin, show gauge phenomena. These
representations furthermore have the property that products of singletons
split into infinite sets of higher spin gauge fields. I tried to implement this
structure in a BRST framework but did not succeed.
Between the years 2000 and 2004 I studied computer science supported by a grant from the
Swedish Knowledge Foundation (KK-stiftelsen). In the fall of 2003 I became aware of the fact
that interest in higher spin gauge fields was increasing and that researchers
began to rediscover the old BRST free field theory. Then in early 2004 I came
across mathematical papers by Jim Stasheff on strongly homotopy algebras and
papers by Barton Zwiebach och closed string field theory and I realized that
this formalism could be applied to higher spin theory. Having studied
theoretical computer science and gotten used to thinking about structures and
systems in an abstract way in terms of syntax, semantics and interfaces, I
got the idea to treat field theory in the same way. This resulted in a new
start for me on
the higher spin problem. I won't describe this approach here, at least not
just now, but refer to my Recent papers on the subject.
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Work in various stages of progress
I'm working on the following projects
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v Based on work by Casalbuoni and
collaborators from the mid 70's, I'm trying to implement an old idea of
mine to model higher spin self interactions in terms of colliding physical
harmonic oscillators. See arXiv:0902.3915 for
preliminary results.
Update 2012: This
does not seem to work. If I get the time I should write up what the problem
seems to be.
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v
Based on
the concrete implementation in terms of Fock space vertices for the interactions,
I'm writing Haskell code to actually compute the first few terms in the
interaction.
Update
2012: This project is not active but should be pursued
eventually.
v
Active project: Returning to old work on the light-front I’m
attempting to compute quartic vertices.
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I'm sometimes thinking
about
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v Categorification of the theory
possibly employing operads to carry the structure of interactions.
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v Implementing this abstract (i.e.
categorified) approach in Haskell possibly generalizing Haskell monads to what
would be Haskell "operads".
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v Non-spacetime approaches to
higher spin gauge fields.
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v Applying the theory of
jet-bundles to the theory.
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Recent papers
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v An Abstract Interface to Higher
Spin Field Theory, J.
Math. Phys. 46:042312, 2005, arXiv:hep-th/0403267
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v
Structure
of Higher Spin Gauge Interactions, J. Math. Phys. 48:072302, 2007, arXiv:hep-th/0611067
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v Towards Unifying Structures in
Higher Spin Symmetry, SIGMA
4 (2008), 013, 23 pages, arXiv:0802.0479
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v Mechanical Models for Higher
Spin Gauge Fields, Fortsch.Phys.57:499-504,2009, arXiv:0902.3915
v Light-front
higher-helicity interactions, Fortsch.Phys. 1–6
(2012), http://dx.doi.org/10.1002/prop.201200035
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Gauge field theory
This is of course a long story in itself, but here is a very short version
that at least contains some of the right words. According to the modern view
of forces (interactions really) between elementary particles, they are always
mediated by other particles/fields. We say particles/fields since the
theoretical framework that makes the theories tick, Quantum Field
Theory, describes particles in terms of quantum fields.
There are four fundamental forces known today: Electro-Magnetic, Weak nuclear
and Strong nuclear. The electro-magnetic force is manifested on all length
scales from the sub-microscopic up to cosmic scales. The two nuclear forces
are sub-microscopic and act on the scales of atomic nuclei. Roughly speaking,
the Weak force is involved in various radioactive decays such as beta-decay.
The Strong force is responsible for making protons and neutrons out of the
more fundamental quarks, and as a secondary effect, it binds protons and
neutrons together to form stable atomic nuclei. The fourth force is the
gravitational force which is manifested from everyday scales up to cosmic
scales. It is however very weak on sub-microscopic scales, and is in practice
neglected, but in principle it is present even on extremely small scales. Due
to a fundamental relation between length scales and energy scales (small
lengths - high energy and vice versa) gravity cannot be neglected at really
small scales (much below the ones investigated today).
The mediating force-particles/fields are massless. The Electro-Magnetic, Weak
nuclear and Strong nuclear are mediated by spin 1 gauge fields. The word
"gauge" refers to a certain symmetry these field posses, and this
is a symmetry that is related to certain so-called "gauge groups"
(really Lie groups). These are (simply put), U(1)
for E-M, SU(2) for Weak and SU(3) for Strong. From here on
the story is quite complicated and does not fit into the space available
here.
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