Spontaneous Symmetry Breaking And The Higgs Mechanism | Quantum Field Theory

In this video, I use a set of example theories to explain and demonstrate spontaneous symmetry breaking and the Higgs mechanism. Quantum Field Theory Lecture Series:    • Quantum Field Theory Lecture Series   Superfluid Helium Resonance Experiment video:    • Superfluid Helium Resonance Experiment | C...   A clarification on tachyon condensation: Tachyon condensation handles the tachyon problem at high energy in theories that are used for phenomenology. The first two theories discussed in this video only contain the scalar field used for breaking symmetry. There is therefore nothing for the tachyons present at high energy to decay into. Tachyon condensation therefore doesn’t take care of the problem for these theories. This is ok, however, because they aren’t used for phenomenology. In theories like the third example and ones like the standard model, which are used for phenomenology, tachyon condensation is free to take place on account of there being many particles that the tachyons can decay into. So when I say that tachyons don’t imply unphysicality at high energy, I mean that they don’t guarantee it. It ultimately depends on the theory. Generalizing Gladstones Theorem To Complex Fields: The way that I stated the Goldstone theorem in this video was that the number of broken generators (and Nambu-Goldstone bosons) is equal to the number of unbroken directions in isospin space, resulting from the zero isospin components in the new vacuum about which we were expanding the theory. This makes perfect sense when the scalar field is real, and there is therefore one field per component. How this could be correct for a complex scalar field, where there are two fields per component is a little less obvious. It turns out that the immediate generalization is to say that there are as many broken generators (and Nambu-Goldstone bosons) as there are fields set equal to zero in the vacuum about which we are expanding. However, we can always recast an N component complex scalar field as a 2N component real scalar field, in which case the original statement would apply again. So in a round about way, that statement of the Goldstone theorem is generally valid, even for a complex scalar field. Take the U(1) example in this video. It includes a one component complex scalar field. The new vacuum about which we are expanding the theory is real. One field has therefore been set equal to zero in that vacuum, namely the imaginary component. From this and the Goldstone theorem, we would expect one broken generator, one Nambu-Goldstone boson, and one massive gauge field. This is exactly what we see. We could always recast the U(1) theory as an SO(2) theory with a two component real scalar field, instead of a one component complex scalar field. The new vacuum would then have one zero component, and one real constant component, and the earlier statement of the Goldstone theorem would apply and produce the same answer. One final note on the U(1)/SO(2) problem. One is used to seeing residual symmetry associated with zero isospin components in the scalar field, but in U(1)/SO(2) theory, there is only one generator to be broken, so there is no residual symmetry.