Continuum (short‑distance) limit of effective theories with physical cutoffs

Determine whether the short‑distance (UV) limit of effective quantum field theories that possess physical cutoffs, as often encountered in condensed matter physics, exists and yields a well‑defined quantum field theory in the continuum sense.

Background

The notes discuss the Wilsonian view of quantum field theory as an effective description that depends on a reference scale. For a theory to be well‑defined in the continuum, one expects the UV cutoff to be removable, producing a scale‑invariant (conformal) fixed point. However, many condensed matter systems have intrinsic (physical) cutoffs, and it is not guaranteed that a continuum QFT exists when taking the short‑distance limit.

Clarifying whether a continuum limit exists in such systems impacts both mathematical foundations of QFT and practical interpretations of effective theories used in condensed matter physics.

References

In fact, some QFTs as effective theories that appear in condensed matter physics have physical cutoffs. In such theories, it is unknown whether the short-distance limit leads to a QFT, but this does not diminish their value as effective theories.

— Modern Approach to 2D Conformal Field Theory (2412.18307 - Kusuki, 24 Dec 2024) in Section 1.1 (Why Consider CFT?), footnote