# Is it possible to derive the correct QED Lagrangian without demanding local gauge invariance?

Usually, the correct interaction term $$A_\mu \bar{\Psi} \gamma_\mu \Psi$$ in the Lagrangian is derived by demanding local gauge invariance.

Is there any other argument that fixes the form of the interaction term like, e.g., conservation of electric charge?

• Just pointing out that conservation of electric charge arises from global gauge invariance (or a global phase change), which is included by the local one. So conservation of charge would not be really a different argument. – Bruno De Souza Leão Sep 29 '18 at 12:27
• Or alternatively, you could actually think if this interaction term being motivated precisely by demanding that conservation of charge be local, and not just global. If that's what you meant in the question, then the answer is yes. But I don't really see that as a different argument from the local gauge invariance one; maybe it's just a more physically appealing one. – Bruno De Souza Leão Sep 29 '18 at 12:29
• I was thinking of "local" in the sense of "being defined point by point", in which case you could introduce a position-dependent generator, but I acknowledge that might have been ambiguous. From the top of my head I have some lecture notes from a course I took a while ago, but they are written in Portuguese; let me try to look for something else. – Bruno De Souza Leão Sep 29 '18 at 12:51
• @JakobH I don't know the answer, and I think this is a little different than what you're asking, but I'll offer it as food for thought: gauge fields and gauge invariance can be "emergent". Examples occur in condensed matter theory. This is mentioned on pages 6-7 in Witten (2017) “Symmetry and Emergence,” arxiv.org/abs/1710.01791, and on page 12 in Horowitz and Polchinski (2006) “Gauge/gravity duality,” arxiv.org/abs/gr-qc/0602037. – Chiral Anomaly Oct 28 '18 at 3:31
• @JakobH ...here's another one (last one): Section 5 in Harlow (2015) "Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture", arxiv.org/abs/1510.07911. This one goes into more detail. The basic message seems to be that models with constraints can sometimes be re-expressed as models with gauge fields/invariance. – Chiral Anomaly Oct 28 '18 at 3:47