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Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes on Quantum Gravity:

In an ordinary quantum field theory without gravity, in flat spacetime, there two types of physical observables that we most often talk about are correlation functions of gauge-invariant operators $\langle O_{1}(x_{1}) \dots O_{n}(x_{n})\rangle$, and S-matrix elements. The correlators are obviously gauge-independent. S-matrix elements are also physical, even though electrons are not gauge invariant. The reason is that the states used to define the S-matrix have particles at infinity, and gauge transformations acting at infinity are true symmetries. They take one physical state to a different physical state - unlike local gauge transformations, which map a physical state to a different description of the same physical state.


  1. What does it mean for electrons to not be gauge invariant and how could this have possibly mucked up the gauge-independence of the S-matrix elements?

  2. Why are gauge transformations acting at infinity true symmetries which take take one physical state to a different physical state?

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  1. "Electrons are not gauge invariant" is a rather strange statement. The physical space of states should be thought of as having all gauge transformations quotiented out, and containing only gauge-invariant states. There is certainly an electron state in that space for e.g. QED, so it's rather puzzling what the author might mean by his statement.

  2. "Gauge transformations at infinity" can refer to two different concepts that both map physical states to distinct physical states: Either large gauge transformations (for a discussion of this somewhat murky notion, see this question and its linked questions) or global gauge transformations, which are simply the constant ones in the simple case of a $\mathrm{U}(1)$-symmetry as in electromagnetism. The contrast to "local gauge transformations" leads me to believe the author means these.

    Consider that the potential changes under a gauge transformation as $$ A\mapsto A+\mathrm{d}\Lambda$$ for some scalar function $\Lambda$, while the charged fields transform as $\psi\mapsto\mathrm{e}^{\mathrm{i}q\Lambda}\psi$. All transformations with $\mathrm{d}\Lambda\neq 0$ change the potential, which is unphysical, and must be quotiented out, they are do-nothing operations with respect to the physical state. However, if $\mathrm{d}\Lambda = 0$, then the pure gauge theory does not actually "see" this function - it does not change the potential, so it does not contribute e.g. to the overcounting of states in the path integral where we integrate over all potential configurations modulo gauge transformations - yet this symmetry does act on the charged states. We have no reason to quotient it out along with the true gauge symmetries, so it mediates between actually distinct physical states. So, on flat space where $\mathrm{d}\Lambda = 0$ just means that $\Lambda$ is constant, we have a global $\mathrm{U}(1)$-symmetry surviving after quotienting by the gauge transformations. Analogously, if the gauge symmetry is $\mathrm{SU}(N)$, there is a $\mathbb{Z}_N$ symmetry surviving, corresponding to the center of the gauge group.

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