spaghetti3451
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Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) 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.
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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?
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.
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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?