# What is the difference between many body theory and quantum field theory methods in condensed matter?

I am starting to studying condensed matter theory and I do not understand if Many-Body Quantum Mechanics and Quantum Field Theory are just synonyms or are two different methods.

It seems to me that the first is used when the starting point is a many-body wavefunction and then the occupation numbers formalism is adopted. In the second case, the starting point is a classical field which is quantized to describe some features of a material. However, in the end, it seems to me that the types of techniques are the same (Green's functions, Path integrals, Faymann diagrams and so on).

Are just two synonyms and QFT is just the cooler, newer name of the same topic or are they different methods?

• Many-body theory and quantum field theory are just synonyms. They're both quantum mechanical theories of many particles. Any difference in connotation is just due to history. – knzhou Dec 21 '18 at 14:35
• Usually the term "many body theory" is used for non-relativistic theories of many particles, while the term "QFT" is mostly used for relativistic quantum theories. The main difference is that in many body theories the total number of particles is conserved (at least in closed systems), and thus the tools of second quantozation, fock spaces, etc. are a mathematical convenience that allows to do more efficient computations. On the other hand, in relativistic theories the number of particles is variable due to the possibility of particle creation and annihilation. – yuggib Dec 22 '18 at 6:15
• In this context the aforementioned tools are necessary to describe the theory in a proper way. – yuggib Dec 22 '18 at 6:16
• @knzhou but in QFT particle number is not conserved whereas in many-body theory (assuming we're both talking about the occupation number representation of non-relativistic many-body quantum mechanics) particle number is conserved. Am I wrong in this? – TheQuantumMan Jan 1 at 10:46

They overlap at least, and they may even be synonymous, depending on who you ask. Specialists in different fields often end up borrowing each other's ideas and methods, and the lines between them often become blurred or found to be nonexistent in the first place. Refs 1, 2, 3, and 4 are just a few of the many examples. And by the way, the author of ref 1 is a Physics SE user.

To explain why the meanings of "many body theory" and "QFT methods" are so fluid, I'll give a broader perspective that includes relativistic QFT.

The broad class of models that includes models called "many body theory" and models called "QFT" can be characterized like this:$$^\dagger$$ instead of assigning observables to individual particles, observables are assigned to regions of space (in the Schrödinger picture) or spacetime (in the Heisenberg picture). In such models, individual particles don't carry their own observables, but we can have observables that detect particles in a given region of space(time). This approach is important even if the total number of particles is conserved, because particles (like fermions or bosons) are often indistinguishable — which is really just a cryptic way of saying that the model's observables are tied to space(time) instead of to individual particles.

$$^\dagger$$ In topological QFT (TQFT), "spacetime" is a misnomer because such models don't have a metric field, and observables may not be assigned to local (contractible) regions. TQFT is still a rich subject because we can consider the model on a whole class of different spacetimes simultaneously. This is also an interesting thing to do in non-topological QFT, but I won't go into that here.

Here are a few comments to highlight the diversity within this broad class of models:

• It includes both strictly nonrelativistic and strictly relativistic models, and also models that are neither strictly nonrelativistic nor strictly relativistic. One example of the "neither" type is a variant of quantum electrodynamics with strictly nonrelativistic electrons: the number of electrons is conserved, but the number of photons is not.

• It includes models where the total number of particles is conserved, and models where it is not conserved. (More generally, within a single model, the total number of a given species of particles may be conserved for some species and not for others.) This is different than the distinction between relativistic and nonrelativistic models. The total number of particles is conserved in relativistic models that don't have any interactions at all, and it's not conserved in many nonrelativistic models. Even in nonrelativistic models where the number of "fundamental" particles is conserved, we still often have emergent quasiparticles (like spinons and holons) whose number may not be conserved.

• It includes models that assign observables to regions of continuous space and also models that assign observables only to discrete lattice points. The latter is not limited to condensed matter, and it's not limited to numerical calculations. In fact, lattice QFT is currently the most broadly applicable method we have for constructing relativistic QFTs nonperturbatively. A strict continuum limit may not always exist (unless we forfeit the interactions that make the model interesting), but we can still take the lattice spacing to be much smaller than the Planck scale, and that's close-enough-to-continuous for all practical purposes.

• Many of the models that were originally conceived for condensed matter turn out to define relativistic QFTs when the parameters are tuned to make the correlation length much larger than the lattice spacing, and this situation has enriched both subfields. We can use techniques originally developed for relativistic QFT to study critical points and phase transitions in condensed matter, and we can use techniques originally developed for condensed matter to help construct new examples of relativistic QFTs — or sometimes just new and enlightening ways of constructing familiar examples.

Whatever we call them, interesting models whose observables are tied to space(time) are so diverse that any attempt to neatly classify them is doomed. The more we learn about the subject, the more apparent this becomes. Words like "many body theory" and "QFT methods" are little more than vague clues about some of the kinds of things the author/speaker might be referring to. We just can't rely on words like that to convey anything precise, unless the author/speaker tells us exactly how we should interpret them in that particular article/lecture.

References:

1. X.G. Wen (2004), Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons, Oxford

2. Zinn-Justin (1996), Quantum Field Theory and Critical Phenomena, Oxford

3. Abrikosov, Gorkov, and Dzyaloshinski (1963), Methods of Quantum Field Theory in Statistical physics, Dover

4. McGreevy (2016), "Where do quantum field theories come from?" (https://mcgreevy.physics.ucsd.edu/s14/239a-lectures.pdf)

• Excellent answer, thank you. In my mind, many body theory always meant the occupation number representation of non-relativistic many-body quantum mechanics". When I then heard people saying it's equivalent to QFT, I got confused. I guess my question to you is, with this less broad definition of many body theory, it is not equivalent to QFT, right? (even though I do understand that the point of your answer is that it is a bit ill-defined for the many reasons you have outlined) – TheQuantumMan Jan 2 at 7:41
• @TheQuantumMan Right. Here's an example from Fetter and Walecka's Quantum Theory of Many-Particle Systems (1971), in the start of ch 1: "In principle, the $N$-body wave function ... [has] all possible information, but a direct solution... is impractical. [We will] resort to other techniques, [such as]... quantum-field theory... In a relativistic theory, the concept of second quantization is essential to describe the creation and destruction of particles. Even in a nonrelativistic theory, however, second quantization greatly simplifies the discussion of many identical interacting particles." – Chiral Anomaly Jan 2 at 14:52
• @TheQuantumMan I don't know if Fetter and Walecka's book uses the exact words "many body theory" anywhere, but they do say "$N$-body" and "many particle," so I think that's an example of what you're talking about. So that's a case where "many body theory" refers to a subset of QFT, at least if we define QFT as "observables are tied to space" as their book seems to do, instead of being a synonym for QFT. And since QFT is often used in the narrower sense of relativistic QFT, some might even say that they don't overlap at all. – Chiral Anomaly Jan 2 at 14:52
• This clears up a lot of confusion. At the end of the day it boild down to definitions and what matters is knowing the tools that you want to use, but thank you for clearing up some of my confusion. – TheQuantumMan Jan 2 at 19:16
• Really nice answer. – Árpád Szendrei Jan 3 at 17:16