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In this video, I show you how to use the Gupta-Bleuler technique for quantizing the free electromagnetic field. My Quantum Field Theory Lecture Series: https://www.youtube.com/playlist?list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou Superfluid Helium Resonance Experiment video: https://youtu.be/unUNQNmuvUQ Additional comments: #1: In a video this complicated, I will inevitably forget to say a few things that are worth noting. The first one is as follows. In this calculation (and in most examples of this treatment), the photons are taken to be traveling in the z-direction (with no loss of generality because the z-direction can refer to any arbitrary direction). This means that (given the usual energy momentum relation) omega=k_3. This is why the physical state condition proof works despite the fact that the omega/k_3 factor appears to be in the wrong place. Because omega=k_3, that fraction is just one, and there is no problem. I should have mentioned this in the video, but forgot to. I added this note because it is important that you know why there is no mistake. #2: Gauge Breaking And Gauge Fixing Subtleties: First, there is a slight difference between gauge breaking and gauge fixing. Gauge fixing represents actually choosing a gauge, where as gauge breaking technically just means breaking the gauge invariance. Gauge constraints like the coulomb gauge actually fix the gauge, but gauge fixing Lagrangians just break the gauge. This obviously makes the usual name of “gauge fixing Lagrangian” instead of “gauge breaking Lagrangian” quite confusing, but still, that is the way things are. Additionally, it is important to know that selecting a particular value for the alpha gauge parameter doesn’t represent actually selecting a real gauge, but confusingly, doing so is still referred to as selecting a gauge. We know that we can mess with and even eliminate the unphysical aspects of electromagnetism (temporal (scalar) and longitudinal components of the gauge field, and therefore those polarizations) by fixing the gauge with the right gauge constraint (the coulomb gauge), and that not all gauge conditions do this. However, unlike gauge fixing with actual gauge constraints, mere gauge breaking with a (confusingly named) gauge fixing Lagrangian can’t eliminate even such unphysical things, although it is enough to achieve quantizability. This is why a physical state condition is necessary in the Gupta-Bleuler quantization method. Of course, not even gauge fixing with an actual gauge constraint can eliminate the transverse polarizations. No manner of messing with the gauge can change the “physical physics” even though gauge constraints can be used to eliminate the “unphysical physics” (scalar and longitudinal polarizations). The gauge fixing Lagrangian doesn’t even eliminate the “unphysical physics”. The physical state condition is required to do that.

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