1 Introduction

Given the very restrictive no-go results that surround the theory of higher spin interactions—to be reviewed in Chapter 2—one may very well question the meaningfulness of pursuing the subject at all. The common view is to see the no-go results as something positive—almost as a prediction of general quantum field theory—that delimit the spin spectrum that one needs to consider regarding fundamental theory. This has been the mainstream view since the supergravity days in the late 1970s, although the experimentally verified spin spectrum of fundamental fields conspicuously does not contain spin 3/2.

For those who anyway have chosen to pursue the subject, one may be curious as to their motivations. In an answer to that, one could then argue that it is an interesting problem of mathematical physics that need not have anything to do with reality. However, given that theoretical physics is ultimately about understanding fundamental physics as it presents itself to us through observation and experimentation, the hope would be that there is some role for higher spin gauge fields to play in nature.

Let me begin by stating my own point of view regarding these questions. I do think that higher spin gauge field theory constitutes a very interesting and challenging area of mathematical physics. But I also sincerely hope that the theory has some role to play in the mechanics of observed nature.

In our physical world, fundamental matter is described by spin 1/2 fermionic fields while the fundamental forces of electromagnetism and weak and strong nuclear interactions are described by spin-1 bosonic gauge fields. The universal force of gravity may be thought of as being mediated by a spin-2 gauge field. It is well understood macroscopically, but far from it microscopically. Fortunately, the weakness of gravity makes it possible to ignore it at the energies attainable in present day experimental high energy physics.

Beginning in the 1950s, higher spin matter particles were discovered in the accelerator laboratories. Fundamental or composite, they prompted research into higher spin wave equations for massive fields. Massless fields were mentioned in some papers, but only for “completeness of treatment” it seems. They had no inherent interest. Wigner and Bargmann–Wigner of course treated them, as did Weinberg. Except for Weinberg’s early 1960s papers, it is not clear from the literature that I have seen, if there was a recognition that massless higher spin fields would be force fields rather than matter fields. It might have been too obvious to point out.

In this dualistic picture of fundamental reality as being constituted of matter and interactions, one may be more tantalized by the interaction side of the picture. A new matter constituent may be interesting, but a new kind of interaction is really intriguing!

1.1 Setting the stage: What is the problem?

What then is the higher spin problem? Vague as the question is, the mathematical physics problem may stated as follows:

Constructing interacting field theories for classical and quantum gauge fields of spin higher than 2, possibly together with lower spin fields.

This problem may be considered to have received a particular solution by the Vasiliev theory. The theory was developed almost single-handedly—if not in isolation—by Mikhail Vasiliev from 1987 to the end of the 1990s, when it started to attract interest from other theoreticians. The Vasiliev theory then received a lot of attention during the early 2000s, largely in the context of the AdS/CFT-dualities. This does not mean that the subject of higher spin interactions is anywhere near being closed. There are still conceptual and technical questions left in a state of confusion. In actual fact, it could be said, at least within the confines of local field theory, that the theory has failed its objectives in that it suffers from serious problems with locality. These problems were actually unearthed as a consequence of the interest the theory attracted after the millennium. This issue has become contentious.

The question of interactions in Minkowski space-time has also received renewed attention in the last 10 years or so, in particular, in the light-front formulation. The light-front cubic interactions found in 1983, when set in the context of quartic investigations of 1990 and 1991, has produced what is called the cubic chiral theory, until very recently only known in the light-front formulation. Inroads to a covariant theory have been made during the last couple of years using twistor theory. Since nonlocality is almost bound to appear in higher spin theories at the quartic level, the cubic chiral theory escapes this fate, at the price of nonunitarity. One may debate which is the worse, nonlocality or nonunitarity, but the conventional view is to deem both alternatives ample reasons to scrap a theory. Normally, nonunitary time evolution, leading to the nonpreservation of probabilities in quantum physics, is seen as something bad, indeed as unacceptable. A nonconventional view would be to instead try to make a virtue out of a vice, and ask what purpose a nonlocal or a nonunitary theory could serve. Let us pause this line of inquiry, and instead simply formulate a more phenomenological aspect of the higher spin problem.

The theoretical physics problem is ultimately related to phenomenology, to what is observed or could possibly be observed in the future of physics. The problem may be stated as follows, allowing for a wider mindset than a narrow unification of forces narrative:

What is the role played by higher spin gauge fields in the fundamental structure of the world? Where are they and what purpose do they serve?

From the point of view of experimental physics, it is clear that there are sub-microscopic phenomena that can be well described by the concept of spin and the concomitant quantum theory. As already alluded to, the fundamental particles and fields are all described by very low values of spin: 1, 1/2 and 0. This—in the form of the Standard Model—seems to cover all presently known phenomena in the microworld. The gravitational interaction stands apart. At the submicroscopic scales probed today, gravity plays no role. It is a theory of the macroscopic world. It sets the stage of modern cosmology. Therefore, gravity—in its still largely unknown quantum version—is also the force of the very early universe and of black holes and the very late universe.

There is a dichotomy of description here. Gravity seems to lend itself naturally to a geometrical picture of the phenomena. This was the way the theory was developed by Einstein and the way most subsequent work on it has followed. But it can also be viewed as a classical field theory on flat space-time. It becomes highly nonlinear when expressed in terms of a spin-2 field. It may seem unnatural to view the theory in that way, but it can be done.1

On the other hand, the gauge theories of electromagnetic, weak and strong interactions can be quite easily formulated and manipulated with no particular reference to geometry. This is so, even though gauge theory can be rephrased in terms of the geometry of certain fiber bundles, and in this way the theory can be made to conform closely to generalized geometrical ideas. Doing this offers useful techniques for unraveling deeper properties of the theories, as well as investigating topological phenomena.

From the existing approaches to the description of the fundamental forces of nature, we can extract at least three aspects of the theories of massless low spin fields, i. e., spin 2 and spin 1:

(i)

Gauging. Both theories can be seen as “nongeometrical” gauge theories where a global symmetry group is made local. For spin 1, the procedure is a standard textbook exercise. However, while technically possible for spin 2, it is conceptually not straightforward.

(ii)

Deforming. Both theories can be seen as nonlinear deformations of a local gauge symmetry. Again, for spin 1, the procedure is fairly straightforward, although seldom performed in textbooks. Performing it for spin 2 is not very easy, and authors are reluctant to write out the details.

(iii)

Differential geometric. Both theories can be seen as generalized differential geometric theories.

These three aspects are not exclusive, although at low levels of mathematical sophistication, the first two aspects appear rather different in their philosophy and implementation.2 As the sophistication is increased, they look more and more as different ways to follow the third aspect. On a high enough level of abstraction and using sufficiently powerful mathematics, the three viewpoints may be merged into one.

This seems to be a general phenomena. Given that a certain theory of physics exists in the sense of being internally mathematically consistent and consistent with accepted basic tenets of theoretical physics (these can change of course), it would seem that...