# What are Quantum Field Theories?

Every time I read about quantum field theories, I wrongly assume and associate the theory to the Standard Model, that is, our current theory of particles and interactions.

However, it seems that the Standard Model is just a type of Quantum Field Theory, of many existent. Unfortunately, everywhere I search about QFT, them are talking about the Standard Model itself (QED,Weak,...) (for example, Wikipedia says "In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics"). So, my question is:

What exactly is a Quantum Field Theory? For example, CFT is a QFT, why? Is that because we imply commutators/anti-commutators algebra, while we impose that the whole theory is invariant under the Poincaré group?

• Commented Aug 17, 2023 at 14:09

A field theory is a mathematical model where the "basic ingredients" are fields. Maxwell's theory of electromagnetic fields and continuum mechanics are prominent examples of classical field theories.

A quantum field theory is a quantum theory, where you promote fields to operators, roughly speaking. The standard model is a specific instance of a relativistic quantum field theory.

There are also non-relativistic quantum field theories, used e.g. in (non-relativistic) condensed matter physics, with a plethora of applications, such as the description of phonons in solids, superfluids and superconductors as well as the fractional quantum Hall effect, to name a few.

Two books which discuss quantum field theories in both mentioned domains are:

1. Quantum Field Theory for the Gifted Amateur. Tom Lancaster, Stephen Blundell. OUP Oxford, 2014.
2. Quantum Field Theory. An Integrated Approach. Eduardo Fradkin. Princeton University Press, 2021.

A quantum field theory is a field theory (i.e. a theory in which each and every point in spacetime obtains a value, according to a function) which is also consistent with the (anti) commutation relations.

The function can be a scalar function (i.e. $$\mathbb{R}^n\rightarrow\mathbb{R}$$), or a vector function (i.e. $$\mathbb{R}^n\rightarrow\mathbb{R}^m$$, with $$n\ne m$$ in general), or even a tensor function or a spinor. The first three obey commutation relations and they are used to describe particles that obey Bose statistics, whereas the spinor fields are used to describe particles that obey Fermi statistics.

The standard model is a theory in which there exist several of the above examples of functions/fields, one for each type of particle known to us to be real. It describes all the fundamental forces in nature, with gravity being the only exception.

The simplest quantum field theory one can study is the scalar field theory. In the scalar field theory, the scalar fields, that are present in the Lagrangian/action each carry one and only degree of freedom. They obey Bose-Einstein statistics. When promoted to operators, the fields obey the commutation relations and upon acting on the vacuum, they create a state containing one particle localised at a single spacetime point.

A conformal field theory (CFT) is a quantum field theory with a very specific symmetry bult in: the conformal symmetry. Namely, upon performing the "so-called" conformal transformations, the theory remains invariant.

Quantum field theory is, very broadly, the area of physics that combines classical field theory and quantum mechanics. Its a generalization of Quantum Mechanics that enables us to quantize and study "richer" objects.

Some would say that within this framework, it also combines special relativity, but this corresponds to only a subset of quantum field theories, such as the Standard model. Actually, the Standard model is a model within the framework that is called Quantum field theory. In condensed matter physics, special relativity is not needed in general (although this might have some exceptions, such as in the case of some effective theories). Actually, the fields we quantise, and indeed our whole theory, can be invariant under different transformations which would distinguish the typical QFTs found in high-energy physics which are Lorentz invariant. One such QFT is conformal field theory, which is just a QFT that's invariant under the so-called conformal transformations.

So, the explanation here is clear: quantum field theory is a marriage between classical field theory and quantum mechanics. Any extra ingredient makes it more specific. If you alo combine special relativity, you get a Lorentz invariant QFT, something that is still pretty general; there is still the action, which defines a model out of the framework. You choose the fields and their interactions. A specific set of fields and interactions, along with the appropriate symmetries, such as Lorentz invariance, gets you the Standard Model.

Now, if you want conformal invariance (which includes things such as scale invariance) instead, you get a conformal field theory. This is still not a rigid model, it is still a framework in that lots of fields and their interactions can be conformally invariant.

The same happens in condensed matter physics. There you do not generally have Lorentz invariance (hence, neither Poincare invarince), so your framework is non-relativistic. Once you figure out your fields and their interactions etc, then you get a model. That is why there is such a broad set of theories in condensed matter (models describing superconductors, others describing Hall effects, etc).

Quantum field theories are defined by their field content (i.e., the number of different types of fields in the theory), their masses and strengths of various kinds of interactions possible among these various different fields. The different kinds of fields differ by what kind of transformations done to them leave the theory invariant. So a scalar field like the Higgs and a vector field like the photon transform differently under Lorentz transformations. You could also have varying numbers of copies of the same type of field to get different theories. For example, the standard model has three generations of quarks, each with same transformation properties, but differing in mass.

The standard model of particle physics is a theory with a very specific combination of these fields with 19 "free" parameters which determine the interaction strengths and masses of the theory. These parameters are determined by experiments; they're free because they are not constrained by theoretical considerations of symmetries, well-definedness, consistency, etc.

So, you could imagine a theory exactly like the standard model but with values different from those found in the experiments. That would be a QFT that is not the standard model. You could also write down theories whose field content is different, whose interactions look different, the symmetries obeyed are different. All these would be QFTs distinct from the standard model.