If quantum field theory unites special relativity and quantum mechanics, why is it used in condensed matter physics?

Question

There are many successful theories in condensed matter physics in the framework of quantum field theories, such as Fermi liquid theory.

However, condensed matter physics is low-energy and non-relativistic by nature. If so, why is quantum field theory, which unites special relativity and quantum mechanics, used in condensed matter physics? Is it because quantum field theory can describe systems with infinitely many degrees of freedom? Or, is it because that it can deal with interacting multi-particle system much better than quantum mechanics?

Or, is there some other reason that I do not know yet?

Answer 1

The crucial thing that QFT brings to the table is that it allows us to naturally model systems with an indefinite number of particles. For a relativistic system this is a necessity because particle number is not conserved at relativistic energies; for non-relativistic systems with a thermodynamically large number of particles, QFT is a very convenient tool through which we may implement the grand canonical ensemble, in which the particle number is indefinite not because of relativistic pair production and annihilation but rather because the system is in chemical contact with a particle reservoir.

Even for non-relativistic systems with definite particle number, QFT is a very convenient language for describing the dynamics of quasiparticles. When one considers the low-energy excitations of electrons in a solid, for example, the vast majority of the electrons are "frozen" below the Fermi level and do not participate in any dynamics. Near the Fermi level, electrons may be excited into higher energy states which do participate in dynamics (e.g. they can conduct current and scatter with phonons). Interaction with the surrounding lattice gives these excitations effective properties (mass, charge, etc) which differ from those of bare electrons in free space, and so the excitations are regarded as quasiparticles. Crucially, these quasiparticles are not conserved (since e.g. the absorption of a photon "creates" a quasiparticle excitation), and the formalism of QFT provides a convenient language in which we might describe quasiparticles as excitations above the ground state (filled) Fermi sea - in quite a precise analogy with how QFT allows us to describe actual elementary particles as excitations above the ground state vacuum in high energy physics.

Answer 2

The name of the classical text, Methods of Quantum Field Theory in Statistical Physics by Abrikosov Gor'kov and Dzialoshinskii (often referred to as AGD) is quite telling in this respect: it is the techniques of the QFT that find themselves very useful in treatment of the condensed matter problems, even though non-relativistic ones.

Perhaps, the essential is being able to treat many degrees of freedom, principally via the second quiantization formalism. AGD focused mainly on the Feynman-Dyson expansion, but later methods, notably path integrals and renormalization group have found very wide use as well. One could even argue that some of them received greater development in condensed matter than in the QFT (but I will not defend this claim). What is more certain is that the methods are often well mastered by people without special training in QFT, and there are many texts describing them in exclusively non-relativistic setting. Thus, referring to them as "Quantum field theory methods" is mostly a matter of history/tradition.

Remark: the distinction drawn in the OP between Quantum mechanics and QFT is a matter of personal respective. E.g., Dirac equation, second quantization and path integrals are often included in a basic QM course/text.

Answer 3

If quantum field theory unites special relativity and quantum mechanics,

This is a wrong assumption, that special relativity is necessary for a consistent quantum field theory to be applied to a problem. I find this unnecessary constraint in the wiki article too.

why is it used in condensed matter physics?

Because it works in fitting data and predicting new states.

Quantum Field Theory (QFT) is founded on the postulates of quantum mechanics and does not need special relativity to work. My answer here might help.

There exist QFT models for nuclear physics, which were my first introduction to QFT back in 1961 . As long as the basic model for a system is quantum mechanical one can construct a field theory for it given certain assumptions. It is the quantum mechanical postulates that are necessary.

After all, the quantum harmonic oscillator is used to introduce students to the creation and annihilation operators of QFT , and there is nothing relativistic about it.

Answer 4

Well, if you look at Newtons law for gravitation and Coulombs law in electromagnetism, they look similar and hence we could ask why are two such different phenomena governed by the same law?

One answer that is certainly correct but not very satisfying is to say that physics uses mathematical models and its quite possible that the same mathematical model is used in very different models.

A more satisfying answer, is to suggest that the inverse square law is due to the dimensionality of space.

Another satisfying answer which opens up many other questions amd not always many answers is that they have a common root. This is the way of unification which has been successful in unifying three of the fundamental forces. But only after two centuries of continual effort.

Given that QFTs are used in both condensed matter and fundamental physics we can propose two possibilities in analogy to the ones above. Either we say that QFT is merely a mathematical model useful in two very different physical domains and this does not point to anything deeper; or in the contrary, we can say that this does. Obviously the latter suggestion is more intriguing.

First, in applications of QFT to condensed matter its important to note that the quanta here do not describe elementary particles but quasi-particles, or collective excitations. I think it is also accurate - but since I'm not a condensed matter theorist, take this with a pinch of salt - to say that QFT here is done in Euclidean signature rather than the Minkowski signature used in fundamental QFTs.

Personally, I think that this indicates that QFT is not a fundamental theory but an effective theory - only valid upto a certain range of energies. And that a deeper theory is required.

One suggestion is string theory, on which the jury is still out on (not least because it predicts an infinite tower of massive particles none of which have been seen). Here we see elementary particles as collective excitations of the underlying stringy world just as we see condensed matter quasi-particles as collective excitations of an underlying atomic world.