Quantum Field Theory as a Lorentz Invariant Statistical Field Theory

Quantum Field Theory as a Lorentz Invariant Statistical Field Theory  We propose a reformulation of quantum field theory (QFT) as a Lorentz invariant statistical
field theory. This rewriting embeds a collapse model within an interacting QFT and thus
provides a possible solution to the measurement problem. Additionally, it relaxes structural
constraints on standard QFTs and hence might open the way to future mathematically rigorous
constructions. Finally, because it shows that collapse models can be hidden within QFTs, this
article calls for a reconsideration of the dynamical program, as a possible underpinning rather
than as a modification of quantum theory. In its orthodox acceptation, quantum mechanics is not a dynamical theory of the world. It provides
accurate predictions about the results of measurements, but leaves the reality of the microscopic
substrate supporting their emergence unspecified. The situation is no different, apart from additional
technical subtleties, in the relativistic regime. Quantum field theory (QFT) is indeed no
more about fields than non-relativistic quantum mechanics is about particles. At best these entities
are intermediary mathematical objects entering in the computation of probabilities. They
cannot, even in principle, be approximate representations of an underlying physical reality. More
precisely, a QFT (even regularized) does not a priori yield a probability measure on fields living in
space-time1
, even if this is a picture one might find intuitively appealing.
This does not mean that the very existence of tangible matter is made impossible, but rather
that the formalism remains agnostic about its specifics. It seems that most physicists would want
more and it is uncontroversial that it would sometimes be helpful to have more (if only to solve
the measurement problem [1, 2]). One would likely feel better with local beables [3] (or a primitive
ontology [4, 5]), i.e. with something in the world, some physical “stuff”, that the theory is about and
that can ultimately be used to derive the statistics of measurement results. In the non-relativistic
limit, Bohmian mechanics [6–9] has provided a viable proposition for such an underlying physical
theory of the quantum cookbook [10, 11]. It may not be the only one nor the most appealing
to all physicists, but at least it is a working proof of principle. In QFT, finding an underlying
description in terms of local beables has proved a more difficult endeavour. Bohmian mechanics
can indeed only be made Lorentz invariant in a weak sense [12] and its extension to QFT is sublte
[13, 14]. At present, there does not seem to exist a fully Lorentz invariant theory of local beables
that reproduces the statistics of QFT in a compact way (even setting aside the technicalities of
renormalization), although some ground work has been done [15]. The first objective of this article is to propose a solution to this problem and provide a reformulation (or interpretation) of QFT
as a Lorentz invariant statistical field theory (where the word “field” is understood in its standard
“classical” sense). For that matter, we shall get insights from another approach to the foundations
of quantum mechanics: the dynamical reduction program.
The idea of dynamical reduction models2
is to slightly modify the linear state equation of
quantum mechanics to get definite measurement outcomes in the macroscopic realm, while only
marginally modifying microscopic dynamics. Pioneered by Ghirardi, Rimini, and Weber [16], Diósi
[17], Pearle [18, 19], and Gisin [20] (among others), the program has blossomed to give a variety
of non-relativistic models that modify the predictions of the Standard Model in a more or less
gentle way. The models can naturally be endowed with a clear primitive ontology, made of fields
[21], particles [22, 23] or flashes [24]. Some instantiations of the program, such as the Continuous
Spontaneous Localization (CSL) model [18, 19] or the Diósi-Penrose (DP) model [17, 25, 26] are
currently being put under experimental scrutiny. These models have also been difficult to extend
to relativistic settings despite recent advances by Tumulka [27], Bedingham [28] and Pearle [29].
For subtle reasons we shall discuss later, these latter proposals, albeit crucially insightful for the
present inquiry, are difficult to handle and not yet entirely satisfactory. The second objective of
this article is thus to construct a theory that can be seen as a fully relativistic dynamical reduction
model and that has a transparent operational content.
The two aforementioned objectives –redefining a QFT in terms of a relativistic statistical field
theory and constructing a fully relativistic dynamical reduction model– shall be two sides of the
same coin. Indeed, our dynamical reduction model will have an important characteristic distinguishing
it from its predecessors: its empirical content will be the same as that of an orthodox
interacting QFT, hence providing a potential interpretation rather than a modification of the Standard
Model. This fact may be seen as a natural accomplishment of the dynamical program, yet
in some sense also as a call for its reconsideration. Surely, if a dynamical reduction model that is
arguably more symmetric and natural than its predecessors can be fully hidden within the Standard
Model, it suggests that the “collapse” manifestations currently probed in experiments are but
artifacts of retrospectively unnatural choices of non-relativistic models.
We should finally warn that the purpose of the present article should not be seen as only foundational
or metaphysical. The instrumentalist reader, who may still question the legitimacy of a
quest for ontology on positivistic grounds, might nonetheless be interested in its potential mathematical
byproducts. As we shall see, because it relaxes some strong constraints on the regularity
of QFTs, our proposal might indeed be of help for future mathematically rigorous constructions.
The article is structured as follows. We first introduce non-relativistic collapse models in
section 2 to gather the main ideas and insights needed for the extension to QFT. The core of our
new definition of QFT is provided in section 3. We show that the theory allows to understand
the localization of macroscopic objects providing a possible natural solution to the measurement
problem in section 4. Finally, we discuss in section 5 the implications for QFT and the dynamical
reduction program, as well as the limits and the relation to previous work, of our approach.