# Physical motivation behind gauging a global symmetry

Consider complex scalar field Lagrangian

$$\mathcal{L}=(\partial^\mu\psi)^\dagger(\partial_\mu\psi) - m^2\psi^\dagger\psi\tag{1}$$

Which exhibits $$U(1)$$-invariance, i.e $$\psi\mapsto e^{i\alpha}\psi$$. On the other hand, to get a locally $$U(1)$$-invariant Lagrangian, one introduces a gauge field $$A_{\mu}$$ and promotes the operator $$\partial_{\mu}\mapsto D_{\mu} = \partial_{\mu}+iA_{\mu}$$. This leads us to the gauged Lagrangian:

$$\mathcal{L}=(D^\mu\psi)^\dagger(D_\mu\psi)-m^2\psi^\dagger\psi+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\tag{2}$$

What is the physical motivation behind gauging a global symmetry?

The physical reason to introduce gauge symmetry is a combination of two facts:

1. We are working in the context of a theory that obeys special relativity (Lorentz invariance) and quantum mechanics
2. We observe massless spin-1 particles and so want to describe them in our formalism,
3. We want our formalism to encode local interactions (since, in special relativity, non-local interactions are also not causal).

Point 3 leads us to wanting to use a vector field $$A_\mu(x)$$ to describe the spin-1 field, since then it is easy to write a Lagrangian with local interactions by multiplying fields at the same spacetime point.

However, in order to describe a masslesss spin-1 particle, there should be 2 propagating degrees of freedom, yet $$A_\mu$$ has 4 components. Gauge invariance (or gauge redundancy) is introduced to project out the two "unphysical" components of $$A_\mu$$ in a Lorentz invariant way.

Once we have committed to introducing a gauge symmetry to describe our spin-1 particle, then all its interactions should also obey the gauge symmetry -- otherwise the unphysical components of $$A_\mu$$ will couple to physical particles. "Gauging a global symmetry" is a method of generating consistent couplings of a spin-1 field to other matter fields in a way that respects the gauge symmetry.

• The idea of gauging a global symmetry is just wrong. That is just a pedagogical way to introduce the concept of gauge theory, but nothing like this happens in nature. Commented Mar 21 at 12:49
• @Oбжорoв I totally agree -- I'd say it's purely a mathematical tool to construct consistent interactions. It's not itself physically meaningful. Commented Mar 21 at 16:50