The primary problems in the field of constructive quantum field theory are to establish in which rigorous mathematical sense the theoretical models used by quantum field theorists to understand elementary particle physics actually exist, what are their mathematical properties and what are the physical consequences of these properties.

To begin, you will find below a non-technical overview of the recent advances in algebraic quantum field theory (AQFT) in these questions. At the bottom of the page is a link to my technical survey article on constructive quantum field theory in general.

**Constructive AQFT **

The development of the tools and techniques of algebraic quantum field theory (AQFT) has reached the point where they can be turned upon the knotty question of existence of quantum field models. Although the program of constructing models via AQFT is still in its infancy and only a few researchers are working in the field, already some encouraging successes can be displayed. I personally find it stimulating that the ideas employed go well beyond the range of the semiclassical ideas which were mathematically developed by researchers in constructive quantum field theory in the 70’s and 80’s. There is no appeal to Lagrangians, actions and perturbation theory, nor does one “work in the Euclidean realm”, and one generally avoids a direct construction of strictly local quantum field operators (as these either do not exist or are prohibitively difficult to construct), preferring to construct more physically relevant quantities such as the scattering amplitudes and local “observables”. Some of the constructed models are local and free, some are local and have nontrivial S-matrices, and yet others manifest only certain remnants of locality, although these remnants suffice to enable the computation of nontrivial two-particle S-matrix elements. This includes models with nontrivial scattering in four spacetime dimensions.

I offer here a very informal and relatively non-technical overview (without equations) of the research in this direction. References are provided in case you want to follow up on any of this, though I do not attempt to give a exhaustive list of these. For your convenience, note that the paragraphs denoted by (1) concern work which admits spacetime dimension d = 2,3,4, but results only in free theories. (Nonetheless, interesting matters were learned, as I try to indicate.) In paragraphs marked with (2), I discuss work which is limited to d = 2, but which constructs a large class of quantum field models with nontrivial and asymptotically complete scattering. In paragraphs marked with (3), papers constructing models for d = 2,3,4, which have nontrivial scattering are treated. I have chosen to present the overview in a more or less historical (from the point of view of AQFT) order.

(1) In [BGL], commencing with an arbitrary positive-energy irreducible unitary representation of the Poincaré group, Brunetti, Guido and Longo construct a local net of observable algebras on a Fock space canonically associated with the initial data. The net is covariant under the associated representation of the Poincaré group on this Fock space. The net is not constructed by first constructing local quantum field operators on this Fock space, as has been normally done in the past, and the localizations of the algebras are fixed purely algebraically by the intitial representation, not by the supports of test functions of a certain class. If the initial data is a representation with strictly positive mass, then the constructed net coincides with that associated with the usual free Wightman field on the usual Fock space. But no appeal is made to such fields in the construction of the net. Indeed, if the initial data is a massless infinite spin representation, then it is known [Yng] that there is no Wightman field covariant under such a representation which also creates a corresponding one-particle state out of the vacuum; nonetheless, this algebraic construction works perfectly well also in this case. In other words, it is not possible to construct the net of observable algebras corresponding to that particular choice of representation of the Poincaré group by using what is usually called a quantum field, whereas it is possible to do so by using algebraic techniques. The approach in [BGL] appears to be limited to the construction of free quantum field theories.

(1) It is essential to note that, normally, quantum fields (and certainly Wightman fields) are point-like. (See footnote 1 below.) Although a quantum field at a point has no physical meaning, as realized early on by Bohr and Rosenfeld, point-like fields can be smeared with test functions having compact support to yield strictly localizable quantum field operators whose localizations may be made as small as one likes. In the nets constructed by Brunetti, Guido and Longo only those algebras localized in space-like cones (pick a point, a spacelike direction and an opening angle; these data determine a spacelike cone in the obvious manner) are known to contain sufficiently many operators (i.e., they create a dense set of states out of the vacuum) for any choice of initial data, whereas the algebras associated with compact regions can be trivial (consist only of multiples of the identity). This latter situation is believed to be the case for the models associated with massless infinite spin representations. Note, however, that nets of algebras associated with point-like fields have sufficiently many nontrivial operators localized in arbitrarily small compact regions (suggested by the Reeh-Schlieder Theorem – but see also [DSW] for technical details). Therefore, nets in which algebras associated with bounded localization regions are trivial cannot be constructed using standard fields.

(1) In [MSY], Mund, Schroer and Yngvason apply this technique to representations of the Poincaré group for three and four dimensional Minkowski space. They then try to find the most general cone-localized (but not necessarily point-like) field which generates this representation when applied to the vacuum and is completely determined by its vacuum two-point function. They show that such fields exist for all irreducible bosonic representations for which the representation of the “little group” is either faithful or trivial. This includes all representations of integer spin associated with strictly positive mass, the massless scalar representations, and the massless infinite spin representations. They show that in the first two cases, these string-localized fields (see footnote 2 below) can actually be expressed as a suitable line integral over point-like fields, and they show in the third case that this is not possible. They also explain how these results generalize to massless representations with finite helicity and to representations associated with half-odd integer spin. In other words, they establish the degree to which the theories constructed by Brunetti, Guido and Longo can be interpreted as arising from point-like quantum fields.

(2) In [S1,S2] Schroer made some suggestions how aspects of this approach (called “modular localization”) could allow one to construct the so-called “quantum field models with factorizing S-matrix”. A little background discussion might be useful here.

(2) A classic problem in dynamical theories like quantum theory is the inverse scattering problem, which, in the case of quantum field theory, takes the following form: given the results of scattering experiments on a suitable quantum system in the form of (all possible) scattering amplitudes, thereby determining the full S-matrix, construct a quantum field model whose associated scattering theory reproduces the specified amplitudes. Though mathematically rigorous progress has been made with this problem in classical mechanics and quantum mechanics, it has so far proven to be particularly formidable in quantum field theory. In order to make the problem more manageable, workers in the field have focussed their attention on the special case of factorizing S-matrices in two spacetime dimensions; this is the relatively simple situation where all scattering processes reduce to (suitable combinations of) two-body scattering, so that specification of the two-body scattering amplitude completely determines the S-matrix.

(2) Heretofore, the primary effort in this direction has been made in the context of the form factor program [KW,ZZ,Sm], in which local quantum field operators associated with the quantum field purportedly having the prescribed scattering behavior are expressed in terms of a certain algebra. Rigorous formulas for matrix elements of local operators between scattering states have been obtained, but the computation of products of local operators at different spacetime points is not under mathematical control, because infinite sums over intermediate states are involved. So the Wightman axioms have not yet been checked in such models, apart from simple cases. An apparent lesson of this work is that a direct (rigorous) computation of the required local field is impracticable. However, algebraic techniques developed in AQFT have reached the point where a mathematically rigorous construction of the desired quantum field model having the prescribed scattering behavior has been carried out.

(2) An intriguing approach to this form factor program in integrable quantum field theories which uses certain wedge-localized (see footnote 3 below) but nonlocal quantum field operators called polarization-free generators (which create covariant one-particle states out of the vacuum) was proposed by Schroer [S1,S2]. Unfortunately, it was subsequently shown by Borchers, Buchholz and Schroer [BBS] that in more than two spacetime dimensions the existence of (tempered) polarization-free generators which are defined on a translation-invariant, common dense domain (tacitly assumed in [S1,S2]) entails the triviality of the associated S-matrix. However, the door was left open for their possible employment in quantum field models with nontrivial scattering in two dimensional Minkowski space.

(2) So in [L1] Lechner considered two spacetime dimensional theories. Given a function S2 (which has the properties of an elastic scattering amplitude), he constructs such (tempered) polarization-free generators. In subsequent papers [L2,L3] he showed that (for an even larger class of two-body scattering amplitudes) algebraic techniques employed in [SW,BL,BS1] allow the construction of a net of strictly localized (in arbitrarily small regions) observable algebras, which admits an asymptotically complete scattering theory whose two-body scattering coincides with the initially given function S2. The S-matrix elements factor into suitable products of two-body amplitudes. The local algebras are obtained by suitable intersections of the algebras generated by the wedge-localized polarization-free generators, and not by finding strictly localized quantum fields. This work is a satisfying solution of the inverse scattering problem for the class of theories with factorizing S-matrix in two spacetime dimensions.

(3) In [BS2] Buchholz and Summers studied a model of a nonlocal quantum field on Minkowski space of any spacetime dimension d greater than or equal to 2, which can be viewed for d = 2 as a special case of the models Lechner examined. Using algebraic techniques they showed that for d > 2 the algebras of observables associated with bounded spacetime regions are small, but that there are sufficiently many observables localized in certain unbounded spacetime regions (intersections of wedges) to permit the definition of a scattering theory. This scattering turns out to be nontrivial. Note that although the quantum field is nonlocal, the constructed algebras of observables are local in the usual sense that observables localized in spacelike separated regions must commute.

(3) Common to the work done in [S1,S2,SW,L1,L2,L3,BS1,BS2] is the use of auxiliary nonlocal quantum fields, which are relatively easy to construct, to obtain the local observables and the subsequent scattering theory. The new idea here is that it might be overly ambitious to try to directly construct the local field operators, since they might be unmanageable (or nonexistent), even though the model may exist in the sense of AQFT. (See footnote 4 below.) This is illustrated most readily by the model discussed in [BS2] with d = 2, which is equivalent to a certain scaling limit of the two-dimensional Ising model. The complicated nature of any expression for the associated local fields is suggested in [MTW].

(3) More recently, Grosse and Lechner [GL] begin with a free quantum field (on Minkowski space with spacetime dimension greater than or equal to 2) and perform a deformation (motivated by the desire to produce models on noncommutative Minkowski space, which we can ignore for now and just consider the models as living in classical Minkowski space) on that field to produce a family of fields which, taken together, is covariant under the initial representation and relatively wedge-local but not relatively local. Although there are indications that the algebras associated with compact localizations are small, once again there is enough remaining locality to define two-body scattering theory, which turns out to be nontrivial.

(3) The arguments which Grosse and Lechner employ to obtain their results restrict their choice of initial quantum field to the class of free quantum fields. In [BS3] Buchholz and Summers generalize Grosse and Lechner’s procedure to (nearly) any Minkowski space quantum field theory in any number of dimensions. This is done by using what we call warped convolutions: given a quantum field theory, either in the sense of Wightman (where the field operators must additionally satisfy polynomial H-bounds) or in the sense of AQFT (where no additional restrictions are required), this construction provides wedge-localized operators which commute at spacelike distances, transform covariantly under the underlying representation of the Poincaré group and admit a scattering theory. The corresponding scattering matrix is nontrivial but breaks the Lorentz symmetry, in spite of the covariance and wedge-locality properties of the deformed operators. This is perhaps not a surprise, since the deformed theory can also be interpreted as living on noncommutative Minkowski space. When taking the free quantum field as the initial model, our deformation coincides with that of Grosse and Lechner.

(3) In a subsequent paper [GL2] Grosse and Lechner study the above-mentioned deformations in the context of Wightman field theory. In particular, they view the deformation of the (arbitrary) initial Wightman theory as a deformation of the underlying Borchers-Uhlmann tensor algebra of test functions, which is, however, endowed in this case with a twisted (Moyal) tensor product instead of the usual tensor product. They explain how these deformed fields relate to vacuum representations of Weyl-Wigner deformed fields. Again they find nontrivial two-particle scattering in the deformed theory which breaks Lorentz symmetry.

(3) More recently still, in [BLS] Buchholz, Lechner and Summers have filled in the technical gaps in the proofs of [BS3] and have established a number of new results, including a proof that the modular objects associated by Tomita-Takesaki theory [Ta] (see [Su2] for a brief overview) with the wedge algebras and the vacuum do not change under the warping deformation. Unexpectedly, they also established a close connection between the warped convolutions and the strict deformations of Rieffel [Ri]. In addition, Dappiaggi, Lechner and Morfa-Morales [DLM] have generalized some of the results of [BLS] to the class of globally hyperbolic four-dimensional spacetimes admitting two linearly independent, spacelike, complete, commuting smooth Killing vector fields. Of course, in these spacetimes one does not (yet) have a scattering theory, so there is no corresponding result about nontrivial scattering of the warped theory. However, they do consider the free Dirac field on such spacetimes and show that the four-point functions of the deformed fields do depend on the choice of deformation parameter, establishing that for this model the deformed theories are not equivalent (in that sense) to the initial theory.

Work is currently under way to generalize these results in a number of directions. It is likely that the deformations studied so far are merely examples of a much larger class of deformations (there are suggestive indications of this in [GL2]), which may themselves include models of more immediate physical relevance. Even though a dynamical principle is still lacking, it is already encouraging that the first examples of (algebraic) quantum field models in four spacetime dimensions having nontrivial scattering have been constructed by these new methods.

To conclude, in my view the heroic efforts of constructive quantum field theorists in the 70’s and 80’s took semiclassical ideas to their technical limits. Since they did not succeed in fully constructing interacting quantum field models in four spacetime dimensions, it would appear that a different approach to constructing models must be found. Although workers in the field are far from reaching the holy grail of constructing, say, the Standard Model (if it even exists), what is heartening about the results outlined above is the fact that most of these new models cannot be constructed using the older ideas, either because the technical effort to do so is prohibitive or because it is quite simply excluded mathematically. Indeed, most of these models do not seem to have an associated Lagrangian. Therefore, one is truly treading upon new ground.

Readers of this summary may be interested in the slides of two talks I gave on the subject in Leipzig in June, 2009. The first, an invited address at the Workshop on Local Quantum Physics, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, June 26, 2009, is less detailed (for comparative lack of time) than the second, a followup seminar given at the Institute for Theoretical Physics, University of Leipzig, Leipzig, Germany, June 30, 2009.

In addition, I have produced a survey of all forms of constructive quantum field theory: A perspective on constructive quantum field theory PS- and PDF-Files here , which I shall try to update occasionally. A somewhat condensed version has been published [Su3].

Footnotes:

1. When we speak of a field being “pointlike”, we mean that it is an operator-valued distribution defined over a test function space admitting functions having arbitrarily small support.

2. We call such fields string-like, because a string with one finite end and the other stretching out to infinity can be localized in such a cone, and if the string is straight, then it can be localized in a cone with arbitrarily small opening angle.

3. A wedge is the image under an arbitrary Poincaré group transformation of the set of all points in Minkowski space whose first spatial coordinate is greater than the absolute value of its temporal coordinate; if you visualize such a set, you see why the term “wedge” is natural.

4. It is already understood how, once the existence of the covariant net of observables algebras is established, one may pass to the associated quantum fields [FH,RW,Su1] (if they exist). Perhaps it is worthwhile mentioning that we regard quantum fields as auxiliary objects for a number of reasons. One is that nets of observable algebras exist which have no associated Wightman fields. Another is that many different quantum fields generate the same net of observable algebras. For example, it has been proven that every field in the Borchers class of the free massive scalar field (and there are infinitely many such fields) is associated with the net generated by the free field [DSW]. We view quantum fields in a manner similar to the way modern differential geometers view coordinate systems – one is chosen for convenience of computation. This point of view is no longer unfamiliar to theoretical physicists (buzzword: duality).

References:

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[BBS] H.-J. Borchers, D. Buchholz and B. Schroer, Polarization-free generators and the S-matrix, Commun. Math. Phys., 219, 125-140 (2001).

[BC] H. Bostelmann and D. Cadamuro, An operator expansion for integrable quantum field theories, arXiv:1208.4763v1 .

[BL] D. Buchholz and G. Lechner, Modular nuclearity and localization, Ann. Henri Poincaré, 5, 1065-1080 (2004).

[BS1] D. Buchholz and S.J. Summers, Stable quantum systems in Anti-de Sitter space: Causality, independence and spectral properties, J. Math. Phys., 45, 4810-4831 (2004).

[BS2] D. Buchholz and S.J. Summers, String- and brane-localized causal fields in a strongly nonlocal model, J. Phys. A, 40, 2147-2163 (2007).

[BS3] D. Buchholz and S.J. Summers, Warped convolutions: A novel tool in the construction of quantum field theories, in: Quantum Field Theory and Beyond, edited by E. Seiler and K. Sibold (World Scientific, Singapore), pp. 107–121, 2008.

[BLS] D. Buchholz, G. Lechner and S.J. Summers, Warped convolutions, Rieffel deformations and the construction of quantum field theories, Commun. Math. Phys., 304, 95-123 (2011).

[DLM] C. Dappiaggi, G. Lechner and E. Morfa-Morales, Deformations of quantum field theories on spacetimes with Killing vector fields, to appear in Commun. Math. Phys.

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[Su2] S.J. Summers, Tomita-Takesaki modular theory, in: Volume 5 of the Encyclopedia of Mathematical Physics, edited by J.-P. Francoise, G. Naber and T.S. Tsun, pp. 251–257, 2006.

[Su3] S.J. Summers, A perspective on constructive quantum field theory, in: Fundamentals of Physics (Eolss Publishers, Oxford), 2012; this is a volume of UNESCO’s scientific encyclopedia, Encyclopedia of Life Support Systems , available online here and in hardcopy in the near future.

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