Why information field theory *is* a field theory

Ewan Cameron wrote a blog titled “Why information field theory is *not* a field theory”. In the following debate I had with him, it became clear that information field theory (IFT) is indeed a field theory, only that the understanding of what a field theory is and what it is not had been confused. As the blog and the following debate are long and the solution to this confusion only appears at the end, here a short summary of the insights.

What is a field theory?

A field is a physical quantity that has a value at each point in space and time [from Wikipedia: fields in the physical sense]. Physical quantities can be measured by definition. Fields are described by mathematical functions of space-time locations. However, this does not mean that every function represents a different field, as functions can differ on immeasurably small sets of locations without changing any measurement (an integral of the function over a finite domain). From a physical point of view, such functions are equivalent (and physics actually deals with the equivalence class of functions with identical measures).

A field theory describes properties of and knowledge on a given field. Field theories come in various flavors.

Classical field theory (CFT, e.g. Maxwell equations, Newtonian gravity, and general relativity) describes the state of classical fields like electromagnetic or gravitational fields with unique macroscopic field values given at each location. A classical field has a single state, a function providing the field value at each location in space-time. Thus, there exist well defined field states and CFT describes how those evolve in time.

Statistical field theory (SFT) describes the statistics of a (fluctuating) classical field. How often does the field have a certain value and what is its average state? SFT uses probabilities of field states to describe for instance how often the field can be found in an individual state. SFT investigates ensembles of field states owing to the inherent problem of ignorance of the exact micro-physical state of a classical field. SFT is successfully applied in solid state physics and in other areas.

Quantum field theory (QFT) deals very successfully with the dynamics of quantum fields, where quantum uncertainty forbids the specification of unique field values. In QFT an ensemble of field states (= classical fields) are considered where the uncertainty in the field state is inherent to the theory; each state is qualified by a complex-valued probability amplitude. The probability that a measurement provides a certain result is then the absolute square value of this amplitude after integrating all possible field states weighted by a test function, which is defined by the measurement.

Thermal field theory (TFT) combines SFT and QFT to describe thermally excited quantum fields.

Information field theory (IFT) takes an intermediate position, in that IFT deals with classical fields whose state is not fully specified by the information provided. In IFT, the field has definite values at all locations, those are, however, simply not known precisely. This uncertainty in field values is described by probabilities over the space of field states. Probabilities are therefore used in IFT to describe knowledge and not necessarily describe how often the field is in a certain state as in SFT and TFT. Although the IFT formalism is mathematically equivalent to the SFT formalism, IFT keeps book of a broader class of uncertainties like the information loss of measurement processes due to imperfection and noise. IFT can take into account prior knowledge derived from some theoretical model (like the CFT the field is obeying). Also, it's aim is not necessarily to describe the evolution of the field but to best quantify the knowledge one has of the state of a field given a measurement, data model, and prior knowledge.

IFT can be regarded as a certain limit of QFT (the complex valued amplitudes being marginalized out) or simply as a Bayesian summary of knowledge on field configurations. In real-world signal reconstruction applications, IFT is typically used as an effective field theory (EFT) for the knowledge on a macroscopic quantity like a density of matter or an intensity of a material property.

What is a field theory *not*?

A physical field theory (SFT, QFT, TFT, or IFT) is not probability theory for the space of all mathematical functions. This space is too rich by containing many functions which differ on sets of measure zero and therefore are physically equivalent. Building well defined measures over general functional spaces is difficult, and it can be shown that a so called Lebesgue measure as conveniently used for integration in finite dimensional spaces does not exist in the infinite dimensional case (thanks to any volume being infinite there). This does not mean that field theories are inconsistent, they just use probability measures over functional spaces that are restricted in some way. They deal with infinitely many degrees of freedom, which are, however, not infinitely free (e.g. constrained by certain continuity conditions). It is not fully clear at this moment what is the best way to specify the canonical mathematical measure on function spaces field theories should use. However, it is already clear that field theories describe nevertheless our world very successfully, despite open mathematical questions.

Does IFT deal with a finite or an infinite number of degrees of freedom?

In practical calculation - no matter which field theory one is dealing with - one always has to discretize the field; in CFT, SFT, QFT, TFT (e.g. in lattice QCD) as well as in IFT. Thus, practically applied field theories deal with finitely many degrees of freedom. However, if the space is sufficiently finely pixelized, the details of the pixelization become irrelevant and one gets the same results as in the continuum limit (modulo renormalization subtleties). A finite data set can constrain an infinite number of degrees of freedom, however, not to complete knowledge.

Whether the number of degrees of freedom in applied IFT examples, where often the field is just an effective description of a huge number of atoms, is really infinite, can be argued. In this context, “infinite” might just be read as a metaphor for “a very large number of which its precise value or even its order of magnitude is irrelevant to the problem at hand”. Probabilities over vast, but still finite-dimensional spaces, as IFT applications building on the effective field theory perspective only require, are perfectly well defined.

Information field theory is a field theory!

IFT is as much a field theory as CFT, SFT, and QFT are. Despite the mathematical difficulties to deal with probabilities on function spaces, those field theories, and in particular QFT, were successfully confronted with experimental data (e.g. the large hadron collider confirmed QFT predictions of the standard model of particle physics with breathtakingly high accuracy). Also, IFT has successfully been applied to real world problems (see IFT resource page)

Torsten Enßlin in May 2015.