1. Introduction
One of the most impressive measurements in physics is of the electron anomalous magnetic moment of the electron, which is known with a precision of a few parts in
[
1]. On the other hand, the theoretical prediction needed to match this result requires the use of perturbation theory containing up to five loop contributions. As is well known, higher order contributions in systems containing an infinite number of degrees of freedom lead to the appearance of divergences which are taken care of by renormalization theory [
2]. The
renormalized perturbation series in enormously successful theories, such as quantum electrodynamics (qed), is, however, strongly conjectured to be a
divergent series (see Reference [
3] for the first result in this direction), and G. ’t Hooft [
4] has, among others, raised the question whether qed has a meaning in a rigorous mathematical sense.
Several years ago, Arthur S. Wightman, one of the leaders of axiomatic quantum field theory, asked the related question “Should we believe in quantum field theory?”, in a still very readable review article [
5]. After his program of constructing interacting quantum field theories in four space-time dimensions failed, the question remains in the air, and we should like to suggest in the present review that the answer to it does remain affirmative, although the open problems are very difficult.
One of the problems with quantum field theory (qft) is that it seems to have lost contact with its main object of study, viz., explaining the observed phenomena in the theory of elementary particles. One instance of this fact is that, except for a few of the lightest particles, all the remaining ones are unstable, and there exists up to the present time no single mathematically rigorous model of an unstable particle (this is reviewed in Reference [
6], and we come back to this point in
Section 4 and
Section 5). Alternative theories have not supplied any new experimentally measured numbers in elementary particle physics, such as the impressive one in qed mentioned above.
A recent, important field of applications of qft came from condensed matter physics: the structure of graphene, of which certain (possibly experimentally verifiable) phenomena have been successfully studied by Gavrilov, Gitman et al. [
7] with nonperturbative methods from the theory of strong-field qed with unstable vacuum. So far, however, the only experimental consequence of a non-perturbative quantum field theoretic model concerns the Thirring model [
8] in its lattice version, the Luttinger model. The latter’s first correct solution was due to Lieb and Mattis [
9]; the rigorous explanation of the thereby new emergent quasi-particles through fluctuation observables was provided by Verbeure and Zagrebnov in Reference [
10]. Luttinger’s model yields a well-established picture of conductivity along one-dimensional quantum wires [
11].
In
Section 2, we present a comprehensive review of divergent versus asymptotic series, with qed as background example, as well as a method due to Terence Tao which conveys mathematical sense to divergent series.
In
Section 3, we apply Tao’s method to one of the very few nonperturbative effects in qed, the Casimir effect in its simplest form, consisting of perfectly conducting parallel plates, arguing that the usual theory, which makes use of the Euler-MacLaurin formula, still contains a residual infinity, which is eliminated in our approach. The fact that Tao’s smoothing of the discrete sums eliminates the divergences is seen to be directly related to the singular nature of the quantum fields.
In
Section 4, we revisit the general theory of nonperturbative quantum fields, in the form of newly proposed (together with Christian Jaekel) Wightman axioms for interacting field theories, with applications to “dressed” electrons in a theory with massless particles (such as qed), as well as unstable particles [
12]. Here, again, the singular nature of quantum fields appears, in the form of a “singularity hypothesis”, characteristic of
interacting quantum fields, which plays a major role. It clarifies the role of the nonperturbative wave-function renormalization constant
Z, and permits to show that the condition
is not a
universal one for interacting theories, but is, rather, related to different phenomena: the “dressing” of particles in charged sectors, in the present of massless particles, considered by Buchholz [
13], as well as the existence of unstable particles. As remarked by Weinberg [
2], the characterization of unstable particles by the condition
is an intrinsically
nonperturbative phenomenon.
Section 5 discusses the latter problem in connection with some concrete models of the Lee-Friedrichs type (see Reference [
6] and references given there), both in the bound state case (atomic resonances), as well as for particles, in which case the problem remains open. The relation to Haag’s theorem [
14] is pointed out, completing the “singularity picture” of quantum fields developped in the preceding sections.
Section 6 is reserved to the conclusion and brief comments on alternative approaches.
3. Application of Tao’S Method to the Casimir Effect for Perfectly Conducting Parallel Plates
In this section, we demonstrate the applicability of Tao’s method to an important effect in nonperturbative qed, the Casimir effect [
24,
25,
26]. For a review, see Reference [
27]; for recent advances, see Reference [
28]; and, for a pedagogical treatment, see Reference [
29]. We shall follow Reference [
30]; also see Reference [
25], Section 2.7. Only the very simplest case, that of perfectly conducting parallel plates, will be considered.
Consider an empty cubic box
with perfectly conducting boundaries. It has thickness
d and lateral sizes of surface
:
The electric field in the box is the solution of Maxwell’s equations, with proper boundary conditions (see References [
25,
30], Section 2.7). It yields a classical energy, whose quantization provides the expression for the total Hamiltonian
:
Above, the modes
are described by a wave-number
and a polarization index
,
are creation and annihilation operators satisfying commutation relations
,
is the photon energy. and the prime in
means that only one polarization is possible when one of the wave-numbers
equals zero. The wave-numbers
and the eigenfrequencies are
where
c denotes the speed of light. The boundary condition on the electric field, i.e., that its tangential components vanish on the boundary of
, lead to the property of its
y and
z components to be proportional to
, with
, as above. The (infinite) last term in the Hamiltonian represents the zero-point energy of the electromagnetic field. As in Reference [
30], we adopt Casimir’s point of view that the vacuum energy
represents mean square fluctuations of the fields in the box
that exist even in the absence of photons but may produce physically observable effects because they depend on the geometry (shape, size) of the spatial domain containing the field. Accordingly, let
denote the force per unit surface induced by these vacuum fluctuations between two faces of the metallic box at distance
d. Since a real metal is characterized by a frequency-dependent dielectric function
such that
as
, but which tends to the vacuum value
as
, namely, when
, with
a characteristic atomic frequency, high enough frequencies should not contribute to the force. For this reason, we introduce a cutoff function
in the above formula for the zero point energy, such that
as well as
This means to consider
above as given by
The cutoff function will be removed at the end by letting
.
Extending the plates to infinity in the
directions yields the energy per unit surface
where
Above,
is the two-dimensional wave-number vector in the
-plane,
, and we have introduced polar coordinates in the
plane, with angular sector
. The prefactor 2 in
is due to the two polarization states of the photon an the prime in the sum means that the term
must have an additional factor
. With the change of variable
,
may be written, finally,
therefore,
where
A possibility for a cutoff function
g satisfying (
80) and (
81) is to consider, for each finite integer
N (this is no restriction because one may always consider for any given sequence of numbers
the largest integers
), the function
, where
denotes the characteristic function of the interval
, i.e.,
if
, and
otherwise. Recalling, however, the problems mentioned after (
79), we are led to adopt the following
definition for
g instead:
with
defined as in (
66), (
67). Since
implies
, we may write (
83) and (
84) in the form
where
The force between the plates, assuming that the electromagnetic field is entirely enclosed in the cavity, is
, with
given by (
83) and
F replaced by
. There is, however, also a field in the space external to the cavity: the external face of the plate at
d will be subject to a force in the opposite direction due to the vacuum fluctuations in the semi-infinite space to its right, namely
which yields
By the formula for
just preceding (
83), the corresponding energy per unit surface is
as
, up to negligible corrections which are discarded. Note that this Ansatz amounts to normalizing the force in such a way that a single plate in infinite space feels no resulting force. Define, thus, the total energy
per unit surface by
By (
83) and (
89), it is given by
where
Our main result is the following theorem, which is an application of Tao’s method to a less trivial case than (
65), i.e., in which the relevant function
f is not a polynomial:
Theorem 1. where is a Bernoulli number (13). Proof. Let
By (
86),
Above, as before, the superscripts denote the order of the derivatives. From the above formulas, we find that
,
. Inserting these results into (
74)–(
76), and taking into account (
92) and (
93), we obtain
where, from the above explicit formula for
,
The explicit formula for
yields
Inserting (
96) and (
97) into (
95), and taking into account the normalization (
67), we obtain
from which, together with (
92), (
94) follows.
□
We have shown that Tao’s method of smoothed sums yields the correct formula (
94) for the total energy density for the simplest Casimir effect, of perfectly conducting parallel plates. Most proofs of this effect use the Euler-MacLaurin formula, and write (
94) in the form
(see, e.g., Reference [
31], p. 171, (13.121), for a recent reference). If
were a
known function, the r.h.s of (
99) would be its usual asymptotic expansion, which would determine it with any degree of precision. If, however, nothing is known about the l.h.s., as is the case with the Casimir effect, the r.h.s. of (
99) must be taken as the
definition of
, and then Wightman’s Observation A applies. Indeed, by this definition,
, since the
term in (
99) indicates that one is supposed to sum the series, which, however, diverges, whatever (nonzero) value of the small parameter is filled in. Due to (
75) and (
80), every finite approximation to
is independent of the cutoff
g, but the rest diverges as
. Taking a sufficiently large number of terms, even the sign of
eventually changes from negative to positive, in analogy to (
72) and (
73).
It is to be remarked that the aforementioned “residual divergence” is not removed by any process of renormalization, and is, in this respect, quite analogous to the situation in perturbative qed (
1), which refers to the
renormalized perturbation series (for the gyromagnetic ratio of the electron). It is present in all approaches which use the Euler-MacLaurin series. This does not, of course, mean that these approaches are “wrong”: it means that they are not
mathematically precise, the issue being one of striving towards a higher level of understanding.
We would like to expand slightly on this important issue because it touches on the philosophy of science. As Jaffe observes in Reference [
32], lesson III, p.7: “Arthur (Wightman) insisted: A great physical theory is not mature until it has been put in precise mathematical form”. As discussed in
Section 2, perturbative qed is also, in a similar way, not mathematically precise, in spite of remaining one of the greatest successes of physics, but, as Lieb observes in Reference [
33], “it is as much an enigma as it is a success”. One important point in this connection is that perturbation theory provides a wrong picture of the photon cloud which surrounds an electron; see Reference [
33] and references given there, as well as References [
34,
35].
What can be said, in analogy, about the Casimir effect? It is certainly one of the very few
nonperturbative effects of qed. As remarked in Reference [
31], p. 170, together with blackbody radiation, it provides the most direct (experimental) evidence for the quantum nature of the Maxwell field (to which one might add the phenomenon of spontaneous emission; see Reference [
15]). On the theoretical side, there are strong
conceptual arguments which require that the electromagnetic field be quantized [
36]. In this same paper, Bohr and Rosenfeld point out that the square of the fields at a single point, such as in the first expression for the vacuum energy
, are ill-defined. In fact, as discussed in Reference [
15], p. 33, what we measure by a test body is the field strength averaged over some small region about a point: fields are what is termed operator-valued distributions, a notion which is basic to the axiomatic (or general) theory of quantized fields, according to which only the so-called Wick dots
exist [
37]. Building on this notion, a few different approaches to the Casimir effect, which do not use the Euler-MacLaurin series (see References [
38,
39,
40]), arrive at the same result independently. Of particular interest is Reference [
39], which uses the image method in a field theoretic context and arrives at (
94) by summing a convergent series. These different conceptual formulations are free of infinities, even of these mentioned in connection with (
99), but they are rather special, in contrast to Tao’s formulation, which is quite general: there, the would-be rest in (
99) disappears in the limit
(see Theorem 3.1).
In conclusion, the previously mentioned references, as well as Theorem 1, are mathematically precise statements of the Casimir effect. In different ways, they introduce a “smoothing”, which has its roots in the previously discussed singular nature of the quantum fields. This “smoothing” is familiar from distribution theory, which is one of the basic mathematical pillars of classical mathematical physics [
41]. Coming back to sequences and series of functions, as in
Section 2, consider the sequence of infinitely differentiable functions
It certainly has no limit in the sense of functions, but let
be an infinitely differentiable function, equal to zero outside the set
. A partial integration shows that
where the prime indicates differentiation. Thus,
. Similarly (see, e.g., Reference [
42], Chapter 3, p. 36), it may be shown that the series
is such that
that is, “
” is a “delta-sequence”: the r.h.s of (
100) becomes more and more concentrated around the point
as
j grows large, but the “distributional limit” (
101) does exist. That is, a smoothing around the singular point
enables the limit to exist. In analogy, with (
95) and (
97),
This means that a smoothing of the “steps” at each integer
N also enables the limit
in Theorem 3.1 to exist. Since the derivative of the step function is a “delta function” in the sense of distributions (see, e.g., Reference [
41], p. 82), the two notions are related.
We now come to more general nonperturbative approaches, in which the singular nature of quantum fields also plays a major role.
5. A Proposal for the Meaning of the Condition : The Presence of Massless and Unstable Particles
In the presence of massless particles (photons), Buchholz [
13] used Gauss’ law to show that the discrete spectrum of the mass operator
is empty. Above,
is the generator of time translations in the physical representation, i.e., the physical hamiltonian
H, and
is the physical momentum. This fact is interpreted as a confirmation of the phenomenon that particles carrying an electric charge are accompanied by clouds of soft photons.
Buchholz formulates adequate assumptions which must be valid in order that one may determine the electric charge of a physical state
with the help of Gauss’ law:
denotes the electromagnetic field observable, and (
119) is assumed to hold in the sense of distributions on
.
When endeavoring to apply Buchholz’s theorem to concrete models, such as
, problems similar to those occurring in connection with the charge superselection rule [
50] arise. The most obvious one is that Gauss’ law (
119) is only expected to be valid (as an operator equation in the distributional sense) in non-covariant gauges, the Coulomb gauge in the case of
, but not in covariant gauges [
50]. If we adopt the present framework, our option is to use the Coulomb gauge and to define the theory in terms of the
n-point Wightman functions of observable fields, i.e., gauge-invariant fields, thus maintaining Hilbert-space positivity. The hypotheses of Buchholz’s theorem are then in consonance with the requirements of Wightman’s theory [
14], and are applicable to
. In a charged electron sector, denoting spinor indices by
, we have
with
positive, measures, and
satisfying certain bounds with respect to
(Reference [
51], p. 350).
with
the renormalized electron mass, according to conventional notation. Recalling (
120), we have the following immediate corollary of Buchholz’s theorem:
Corollary 3. For in the Coulomb gauge, assuming it exists in the sense of the framework of this section and satisfies the assumptions of Buchholz’s theorem, the following condition holds: Above,
denote observable fermion fields, which we assume to exist as a generalization of those constructed by Lowenstein and Swieca [
52] in
; also see Reference [
53] for a similar attempt in perturbative
. In the words of Lieb and Loss [
34], who were the first to observe this phenomenon in a relativistic model of qed, “the electron Hilbert space is linked to the photon Hilbert space in an inextricable way”. Thereby, in this way, “dressed photons” and “dressed electrons” arise as new entities.
We now come back to Weinberg’s suggestion that the condition describes unstable particles.
Turning to scalar fields for simplicity, we consider the case of a scalar particle
C, of mass
, which may decay into a set of two (for simplicity) stable particles, each of mass
m. We have energy conservation in the rest frame of
C, i.e.,
with
, and
the momenta of the two particles in the rest frame of
C:
In order to stress the practical effects of the confusion of the two definitions of
Z in the literature, when the ETCR is assumed (including [
2]), it should be remarked that the first reference given by Weinberg on a model for unstable particles [
54] assumes (what ammounts to)
instead of (
123) (that is, we are in the stability range!), and finally obtains
! This is solely due to the “double definition” of the non-perturbative wave function renormalization constant!
In order to check that
when (
123) holds, while
is valid in the stable case
, in a model, we are beset with the difficulty to obtain information on the two-point function.
There exists a quantum model of Lee type of a composite (unstable) particle, satisfying (
123), where (
113) was indeed found, that of Araki et al. [
55]. Unfortunately, however, the (heuristic) results in Reference [
55] have one major defect: their model contains “ghosts”. A very good review of the existent (nonrigorous) results on unstable particles is the article by Landsman [
56], to which we refer for further references and hints on the intuition behind the criterion (
113).
In the next section, we come back to a set of models for atomic resonances and particles, which might support the suggested picture of quantum field theory in terms of “dressed” and unstable particles, and, in the last section, we discuss the crucial conceptual issues and difficulties associated with this program, comparing it with alternative approaches.
6. Models for Atomic Resonances, Unstable and “Dressed” Particles: What Distinguishes Quantum Field Theory from Many-Body Systems?
In Reference [
6] the model below—the Lee-Friedrichs model of atomic resonances—was revisited. Its Hamiltonian may be written
with
and
The operators act on the Hilbert space
where
denotes symmetric (Boson) Fock space on
(see, e.g., Reference [
57]), which describes the photons. We shall denote by
the scalar product in
. Formally,
, and
k denotes a three-dimensional vector. The † denotes adjoint,
, and
are the usual Pauli matrices. The operator
commutes with
H. We write
and introduce the notation
is the restriction of
H to the subspace
. Let, in addition,
Then, the one-dimensional subspace
consists of the ground state vector
with energy zero, where
denote the upper
and lower
atomic levels, and
denotes the zero-photon state in
. Note that
is also eigenstate of the free Hamiltonian
, with energy zero, and we say, therefore, that the model has a persistent zero particle state.
We shall refer to the model described by (
124) as
Model 1. Replace, now, in Model 1,
by
We shall refer to the ensuing model as
Model 2. Let, now,
satisfy
anticommutation relations, i.e.,
together with all other anticommutators equal to zero, i.e.,
, etc. By means of Schwinger’s representation
with the further correspondences
where
denotes the fermion no-particle state, Model 1 becomes the usual Lee model for particles, and Model 2 a “refined” Lee model for particles; we refer to them as
Model 3 and
Model 4, respectively. There is one big difference, however, between the atomic resonance case and the particle case. In the former case, the function
g in (
126) or (
134) is square-integrable, and, indeed, the physical dipole-moment matrix elements provide natural cutoffs, so that neither infrared nor ultraviolet problems arise. In the latter case, however, one aims at the pointwise limit
, which should lead to an euclidean-invariant theory. This step is delicate and requires mass and wave function renormalizations. To the particle versions, Model 3 and Model 4, assuming they are well-defined, the forthcoming framework is applicable.
Let a theory of a scalar field of mass
be invariant under the euclidean group, that is, the group of translations and rotations of euclidean space
, where
R denotes a rotation. By Haag’s theorem (Reference [
46], p. 249), in a euclidean field theory which uses the Fock representation, the no-particle state
is euclidean invariant, i.e.,
We have the following:
Theorem 3. Let the Hamiltonian be of the formwhere satisfiesThen, if Ψ
is any state invariant under , i.e.,then Ψ
belongs to the domain of H only if . Proof. We have that
depends only on
, so that
according to whether
or
, that is,
H must annihilate any translation-invariant state to which it is applicable. □
Choosing
in the Fock representation, it follows from Theorem 3, together with the consequence of Haag’s theorem (
135), that
H can be applied to the no-particle state only if it annihilates it. The above was extracted from Reference [
46], p. 250, and, to make it entirely rigorous, (
136) should be written as a limit of the “smearing” of
with smooth functions. This result may be applied to Model 3 (assuming the renormalizations performed such that the pointwise limit can be taken, leading to an euclidean invariant quantum field theory), with the no-particle state identified to the state given by
, in and
the no-particle photon state:
. We say that there is no
vacuum polarization. On the other hand, for Model 4,
does not belong to the domain of
H due to the term
in (
134), which, in terms of fermion operators equals,
, and
. In the case there is vacuum polarization, Theorem 3 implies that there exists an “infinite energy barrier” between the Fock no-particle state and the true vacuum: non-Fock representations are required, that is, the “physical” Hilbert space is not unitarily equivalent to Fock space.
All this being said, Model 3 turns out, very unexpectedly, to be afflicted by “ghosts”, i.e., states of negative norm (see Reference [
43], Chapter 12). We say unexpectedly because the occurrence of “ghosts” in relativistic quantum field theory is known to be a consequence of the use of manifestly covariant gauges (see Reference [
44]), and the Lee model is not relativistically invariant. Note, however, that, in the Schwinger representation, the “fermions”
V and
N are two states of an
infinitely heavy (spinless) fermion, i.e.,
there is no recoil! Of course, this is a highly unphysical assumption. One may consider, however, the model with recoil, describing the interaction between the photon field and
two particles
V and
N, with energies
and
, and interaction energy
, with
proportional to the charge, and
,
, the latter representing the (eventually massive) “photon” energy. This model is well-defined and free of “ghosts” in the pointwise limit
, the latter taken in a careful way, according to a renormalization prescription. This was proved by Yndurain [
58] in a seldom cited, but very important, paper. Thus, the pathologies associated to the original Lee model just have to do with neglecting recoil!
We henceforth refer to the Yndurain versions of models 3 and 4 as Model 3Y and Model 4Y.
Model 2 (for zero temperature, as a model of atomic resonances) may be the simplest prototype of a model with vacuum polarization. Model 3Y should be suitable to study the criterion
for unstable particles. The subtlest point in this connection is the fact that the term proportional to the delta measure in (
106) has coefficient
Z, and, in its absence, due to the condition
, the (renormalized) mass seems to remain undetermined. One must, therefore, be able to determine the renormalized mass uniquely from
alternative general conditions on the Hamiltonian, such as the requirement that it be self-adjoint and bounded below.
We close this section with some remarks of what distinguishes quantum field theory from many-body systems. The latter are characterized by an interaction Hamiltonian
, where
V denotes the interaction potential,
and
spin indices, and
are boson or fermion quantized operators (see, e.g., Reference [
57], p. 110). The interaction term, and consequently the whole Hamiltonian, annihilates, therefore, the no-particle state (Fock vacuum). In contrast, quantum field theories display in general vacuum polarization, as typified by Model 4Y, which is an approximation of the usual trilinear coupling terms which occur in qed and also in quantum chromodynamics (qcd), as a consequence of relativistic invariance and the gauge principle (see References [
2,
44]). Coupled with Haag’s theorem and Theorem 3, this implies non-Fock representations, as previously discussed. This is a further manifestation of the singular nature of quantum fields. Of course, the above reference to many-body systems concerns a finite number of particles. For nonzero density, i.e., an infinite number of degrees of freedom, non Fock representations arise even in the case of free systems, by a well-known mechanism (Reference [
59], Section 2.3). What we wish to emphasize is that, in the case of quantum field theories, such representations arise due to a particular reason, namely vacuum polarization.
7. Conclusions
In this review, we focused on the foundations of quantum field theory, which is still believed to be the most fundamental theory, describing in principle all phenomena observed in atomic and particle physics. Unlike quantum mechanics, however, its foundations are still not cleared up. We attempted to describe how some novel approaches lead to a unified picture, in spite of the fact that several difficult open problems remain. The solution of some of them seems to be nontrivial, but feasible. Many other problems demand new ideas, however.
The first issues concern
relativistic quantum field theories, such as qed or qcd. They have been discussed in
Section 3 and
Section 4, after a pedagogic review of asymptotic and divergent series in
Section 2.
The fact that divergent series, even if asymptotic, do not define a theory mathematically, was emphasized in the case of qed; see, in particular, (
24)–(
26). From the latter, we see that even the assumption that renormalized perturbation series, such as (
1), is the asymptotic series of an unknown function, which is probably true and explains the dazzling success of perturbative qed, cannot hold for qcd, where a rough dimensionless measure of the coupling constant is of order
. The improved perturbation theory deriving from renormalization group arguments and asymptotic freedom in qcd (Reference [
44], Ch. 18.7) has no comparable experimental consequences.
We, therefore, turned our attention to
nonperturbative phenomena, one of the very few (in qed) being the Casimir effect, for simplicity in its simplest form, that of perfectly conducting parallel plates in the vacuum. We showed in
Section 3 that Tao’s method of smoothed sums [
17] eliminates the “residual infinity” present in the Euler-maclaurin series for the energy density of the field (Theorem 1). This smoothing of the series is a new method of accounting for the singular nature of quantum fields, which are not pointwise defined.
Going beyond specific models or effects, we reviewed in
Section 4 a recent novel criterion to characterize interacting theories in the Wightman framework [
12]. Here, a different aspect of the singular nature of quantum fields is touched upon: the nature of the singularity of the two-point Wightman function at small distances. Using the Källén- Lehmann representation of the two-point function
and the Steinmann scaling degree [
45]
, we propose to characterize interacting Wightman theories by the requirement
. We were then able to prove a Theorem 2 which states that, in an interacting Wightman theory, the total spectral mass which occurs in the Källén- Lehmann representation is infinite. This allows us, in turn, to state that, contrary to previous belief, the
nonperturbative wave-function renormalization constant
Z is not universally equal to zero. We then interpret the condition
in a two-fold way: it is either due to the nonexistence of a pure point part in the mass spectrum in charged sectors in theories, such as qed, due to infraparticles (electrons with their photon clouds), or to the existence of unstable particles, as conjectured by Weinberg (Reference [
2], p. 461. We believe that this new approach is connected with the theories of “dressed” photons and electrons in References [
34,
35].
In qcd, the gluons being massless, a similar phenomenon as discussed in the last paragraph should be expected: an interacting theory of “dressed” quarks and gluons. It has been suggested by Casher, Kogut and Susskind [
60], and Swieca [
61] that massless
contains what is desired of a theory of quark confinement, in the sense that under short distance probing the theory behaves as if it contained particles which do not manifest themselves as physical states: in that limit one recovers a theory of free (massless) electrons and photons. For large distances, the electrons completely disappear from the picture, giving rise to massive photons, by “Bosonization” (an extreme form of the “dressing” phenomenon). In Reference [
12], we formulated a precise criterion for confinement which applies to
and which, generalized to qcd (under assumptions on the leading infrared singularity of the gluon propagator), yields a surprisingly realistic picture of confinement and asymptotic freedom.
We should like to mention a recent alternative approach, due to Buchholz and Fredenhagen (Reference [
62] and references given there). There the assumed existence of a time-arrow yields a novel approach, centered on an interesting, non-commutative structure of dynamical algebras inspired by scattering theory. An early reference which treats unstable particles in the spirit of open systems, associated to a dynamical group (the Poincaré semigroup), is the paper by Alicki, Fannes, and Verbeure [
63]. Sewell’s book [
64] also deals with the emergent macrophysics originating from quantum mechanics, albeit not in the realm of quantum field theory.
More directly aimed at gauge theories is the theory of string-localised fields, due to Mund, Schroer and Yngvason, which has some points of contact with the present approach, specifically the insistence on positivity, i.e., “ghostless” theories. The theory has recently progressed considerably through a deep analysis of Gauss’ law in that connection, see [
65] and references given there to the previous literature.
A theory of quantum fields in de Sitter space has been developed in recent years by Jaekel, Mund and Barata. It is particularly interesting because of its connection to thermal aspects of quantum fields, as early realized by Narnhofer, Peter and Thirring [
66]. We refer to [
49] for some recent references.
In addition, an important perturbative algebraic quantum field theory has been developed in the last twenty years. Its recent application to the sine-Gordon model [
67] actually yielded a convergent series, which marks the success of this approach: we refer to [
67] to references to the previous extensive literature. An earlier treatment of integrable two-dimensional models, with several novel structural features, both conceptual and technical, is due to Lechner and collaborators, see the review by Alazzawi and Lechner [
68] and references given there.
In spite of the fact that the convergence of the perturbation series for models such as the massive Thirring-Schwinger model (
) is well-known since the early days of constructive quantum field theory (1976) [
69], it is important to emphasize that no singular point fields appear in the functional analytic approach of [
67], so that it may be expected that this new method, invented by Fredenhagen, may be able to cope with physically relevant models in the future, such as gauge theories in four space-time dimensions.
Concerning the latter, since the gluons are, as the photon, massless, we are proposing a picture of interacting gauge theories as composed of “dressed” quarks and gluons, or dressed electrons and photons, for qed. The latter has been suggested by Lieb and Loss [
34] and Jaekel and myself [
35], and we suggested further that this picture is characterized by the condition
[
12]. Unfortunately, even in
this involves a construction of fermion observable fields which has not even been achieved in perturbation theory [
53]. This project is therefore very difficult for qed, and the self-interactions of the gluons in qcd make it even much more difficult there. We feel, however, that it contains a “grain of truth”: in particular, Lieb and Loss provide convincing arguments to the effect that description by “dressed” photons is the more natural one in their relativistic model of qed, while we relate this phenomenon to the non-Fock structure of the representations (when the ultraviolet cutoff is removed). The latter appear in connection with a generalization of the transformations analized by Wightman and Schweber [
70] which occur in
in the Coulomb gauge [
35]. Therefore, from our point of view, a first step would be a study of a simpler model displaying vacuum polarization. The simplest such is Model 2.
Although the Lee-Friedrichs model is very old (see [
6] and references given there), it is one of the few approximations to relativistic quantum field models which preserve some basic physical features. This may be due to the fact that its space dimension is three, and that it retains the trilinear coupling familiar from relativistic theories. We proposed in
Section 5 that the main feature distinguishing quantum field theory from many-body systems is
vacuum polarization. When the theories are euclidean invariant, Haag’s theorem, together with theorem 3 provides a way to distinguish Model 3Y from Model 4Y (the letter Y refers to the important work of Yndurain [
58], who showed that the “ghosts” in the Lee model were solely due to neglecting recoil). We believe that Model 3Y is suitable to verify the prediction
for unstable particles. Model 4Y seems at present beyond any control, but understanding Model 2, which is relevant to atomic resonances, would be an enormous advance towards grasping the main features of vacuum polarization. It should be noted that Herbert Fröhlich’s electron-phonon theory [
71] is a true quantum field theory with vacuum polarization, and the failure of being able to handle it may well be the “penalty” for the problems found in BCS theory [
64]. The study of Model 1 in [
6] shows clearly that the field-theory interaction through emission and absorption of (virtual) particles is of
qualitatively different nature from potential theory: in particular, the regeneration of the unstable state from the decay products is a virtual quantum phenomenon analogous to tunneling in potential theory which is, however, not present in potential theory. In the case of atomic resonances, Model 1 even accounts even for subtle physical aspects, such as the fact that the natural line width of Lamb states is much smaller than the corresponding (Lamb) shift, essential for the observability of the latter [
6]: this is an example of the physical features mentioned in the beginning of this paragraph.