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Edited Question

The absence of certain terms can make a field theory conformally invariant. For example, the absence of a mass term in the Maxwell action makes it conformally invariant. Here is a nice question and answer.

But in general, what is the physical significance of the conformal invariance of any conformally invariant field theory (Maxwell action being a special case only)? Does it lead to some sort of conservation, restriction, etc?

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    $\begingroup$ One consequence I could think of is that it has a traceless stress energy (see: physics.stackexchange.com/q/505466). Is your question restricted to only Maxwell's equation or is it also for any conformally invariant field equation? $\endgroup$
    – KP99
    Feb 19, 2022 at 14:11
  • $\begingroup$ You are right. I would like to know what conformal invariance of field equations would mean, in general, and not why the field equations are conformally invariant. $\endgroup$
    – Solidification
    Feb 19, 2022 at 14:14
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    $\begingroup$ I have an explanation, not very well structured, so I'm commenting here: The trajectory of center of mass for system of massless particles in flat space-time is unique only when helicity = 0, for non-zero helicity (like photons) , all points on null hypersurface $x^ap_a$=const are equivalent (see section 1.3 of doi.org/10.1016/0370-1573(73)90008-2). Conformal rescaling preserves light cone structure, so it should preserve the dynamics of photon as well. This consequence is also true in field description. In qft , fields $\iff$ particles, so it roughly makes sense $\endgroup$
    – KP99
    Feb 19, 2022 at 14:33

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