As a mathematician, holomorphicity is an extremely good property that provides rigidity, finite dimensionality, algebraicity. etc to whatever theory that's considered. I'm curious about why (anti-)holomorphicity is considered in physics.

As an example, apparently holomorphic representations of $SL(2,\mathbb{C})$ apply to physics, while weaker kinds of representations are of interest in math as well.

As another example, 2D conformal field theory models fields as meromorphic functions. Though conformality almost equals holomorphicity in 2D, I'm still curious why picking "conformal" in the beginning?

Q. Do people consider them because they have well-developed mathematical background and thus allow us to say something interesting, or there are deeper reasons behind it?

  • $\begingroup$ This post (v2) seems somewhat broad and opinion-based. Complex function theory is used in almost all areas of physics basically because it is very powerful. $\endgroup$
    – Qmechanic
    May 17 '20 at 19:56
  • $\begingroup$ That's it? And why you think this thread can have several valid answers? $\endgroup$
    – Student
    May 17 '20 at 19:58

Scale invariance is common in physical systems at phase transitions. If the characteristic length of a system is small in one phase (disordered) and infinite in another phase (ordered), then typically the system is scale invariant at the transition between the two phases. A non-trivial physical observation is that in many cases scale invariance (together with invariance under rotations and translations) implies conformal invariance. And in two dimensions, conformal transformations are described by holomorphic functions.

Therefore, in two-dimensional conformal field theory, the appearance of holomorphic functions is the mathematical manifestation of physical symmetries: ultimately, scale invariance.


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