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The main philosophy behind the application of field theory methods in condensed matter is that, since solid state systems are usually hopelessly too complicated to allow an accurate microscopic description, we will not attempt to write down an "ab-initio" model that describes any particular system/material. Instead, we attempt to write down a (usually) continuum quantum field theory that captures the long-wavelength, low-energy properties of a larger class of systems.

These effective field theoretic models have a reduced number of parameters, which are usually fit to experiments (or to numerics). We ignore many microscopic details, including e.g. the underlying atomic crystalline structure. When is such a description justified?

Typically, we will see that the field theory description is valid in the vicinity of a critical point, where there is a correlation length, that becomes much larger than any microscopic length in the theory (e.g., the lattice spacing). So the use of quantum (or classical) field theories in condensed matter physics is intimately related to the study of quantum (or classical) phase transitions, in particular, continuous (second order) transitions, in which the correlation length diverges and many physical properties become universal. If we are not close to any critical point, the correlation length is typically of the same order of the microscopic length scales. The use of field theory is then invalid, and one has to use other methods, e.g., mean-field theory. So in principle, the use of field theory is mostly suited to studying the properties of phase transitions rather than of phases. There are exceptions to this rule, which we will discuss below.

There is a close relation between the field theory techniques that are used in condensed matter physics and the one used in high-energy theory, despite the different language used. Indeed, historically, there have been a lot of useful exchanges of ideas between condensed matter and particle physics. The main difference is in the philosophy. In both fields, one typically has to introduce a cutoff in the theory in order to control it. In particle physics, however, this cutoff is not a physical quantity, but rather represents our ignorance about the correct theory at high energies. Therefore, ultimately, the physical results have to be independent of the cutoff; the cutoff dependence is removed by a procedure called renormalization, by which we fix certain physical quantities (such as the mass of the electron) to their experimentally measured values, and then tune the parameters of the field theory such that these quantities do not change when we send the cutoff to infinity. In condensed matter physics, the cutoff has a physical meaning of a microscopic length, such as the lattice spacing or the distance between particles. Therefore, in principle, we will not be so opposed to the cutoff appearing in our final answers. Nevertheless, it is often convenient to pretend that we are doing high energy theory, and to perform a renormalization procedure such that the cutoff dependence is replaced by physically measurable quantities at low energies and long wavelengths.

Another difference is the use of the renormalization group (RG) technique. In quantum field theory, this is a practical way to perform the renormalization procedure described above; it is often described as a technique to eliminate the "infinities" that arise when perturbation theory is applied naively. In condensed matter, the renormalization group is often interpreted in the "Wilsonian" way, as integrating out high energy modes and finding an effective lower-energy theory, in an attempt to reveal the properties of the low-energy behavior. Despite this difference in viewpoint, the techniques used are actually equivalent. The Wilsonian approach is much more physically transparent (to our taste) and conceptually simpler. However, for the purpose of performing complicated calculations (e.g., ones that involve going beyond "one- loop" order), the field theoretic formulation of the renormalization group is often more practical. We will mostly use the Wilsonian approach here, but may see an application of the field theoretical approach at some point.

Finally, let us mention some cases in which a field theoretical description is actually valid even though we are not in the vicinity of any continuous phase transition. These are cases where we are in a gapless phase, whose low-energy properties are (to some degree) universal by some reason other than the proximity to a second-order transition. Examples include:

1. Fermi liquids and Luttinger liquids, which will be the subject of the first few lectures.
2. Systems with continuous broken symmetries, which exhibit gapless Goldstone modes.
3. Systems with emergent continuous gauge fields, which may support a phase with gapless gauge excitations (much like photons in quantum electrodynamics).