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I recently read "Fifty Years of Yang-Mills Theory and my Contribution to it" by R. Jackiw, and stumbled upon an intriguing sentence:

Usually in quantum field theory tunneling is suppressed by infinite energy barriers between degenerate vacua – this leads to spontaneous symmetry breaking.

  1. In what sense is there an infinite energy barrier? How can we see this explicitly?
  2. What does this have to do with spontaneous symmetry breaking?
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  • $\begingroup$ There is a more or less intuitive discussion of this in section 2.11 of arxiv.org/abs/1703.05448 $\endgroup$ – AccidentalFourierTransform Mar 27 '17 at 13:21
  • $\begingroup$ @AccidentalFourierTransform Thanks! I'll have a look at it $\endgroup$ – jak Mar 27 '17 at 13:27
  • $\begingroup$ @AccidentalFourierTransform If I'm not missing sth crucial, I think his argument only holds for an infinite volume system, which our universe is not. In addition, from his argument it seems very well possible that transitions take place. The only problem seems to be that it can not happen everywhere in an infinite volume system, because that would need, well an infinite amount of energy. For a finite volume system the energy would be finite. However, these lecture notes were incredibly helpful for some other problems I was currently struggling with. $\endgroup$ – jak Mar 27 '17 at 14:32
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Suppose you have a quantum field theory with one scalar field $\phi$, and the equation of motion for this field tells you that the lowest energy configuration is obtained when $\phi$ is constant throughout spacetime, with $\phi^2 = \phi_0^2$ (here $\phi_0$ is a constant determined by the parameters of the Lagrangian, for instance).

Now let's address your questions :

  1. There are two classical vacua, characterized by $\langle \phi \rangle = \pm \phi_0$. If you want to tunnel from one to the other, the energy barrier will be some constant integrated on all of space. Because of the infinite volume of space, this energy barrier is infinite, independently of how small the energy density is.
  2. As a consequence, the quantum states which are superpositions of the two classical vacua are completely suppressed. This means that there are two quantum vacuum, which are identical to the classical ones. Hence the $\mathbb{Z}_2$ symmetry of the equation $\phi^2 = \phi_0^2$ is broken in any given vacuum by $\langle \phi \rangle = \pm \phi_0$. This is spontaneous breaking of the $\mathbb{Z}_2$ symmetry.

In summary, tunneling in QFT in strictly more than one dimension (as opposed to quantum mechanics, that you can see as QFT in $0+1$ dimension) costs an infinite amount of energy because of the integration over space. This suppresses superposition states, and entails spontaneous symmetry breaking.

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  • $\begingroup$ Thanks a lot for your answer! Two small questions: 1). The crucial point seems to be that spacetime has an infinite volume. How is this assumption justified? Isn't the volume of spacetime, in reality, finite? 2.) Couldn't we, with the same line of thought, compute that energy barrier that the scalar field takes the other VEV in some "bubble", i.e. a finite volume? This barrier should be finite and therefore local tunneling to other vacuum states seem possible... $\endgroup$ – jak Mar 27 '17 at 14:45
  • $\begingroup$ If space is compact (eg a sphere or a torus), then indeed the energy is finite and you can have tunneling. But in general, space is not assumed to have finite volume. Why do you think so ? $\endgroup$ – Antoine Mar 27 '17 at 16:26
  • $\begingroup$ For your other remark, yes you can have a bubble where the scalar takes the other VEV, but in that case the bubble is surrounded by a domain wall, which has some tension. In other words, this configuration does not have minimal energy, because at the boundary of the bubble the field $\phi$ is varying. Because of the tension of the wall, the bubble will ultimately disappear. $\endgroup$ – Antoine Mar 27 '17 at 16:29
  • $\begingroup$ I always thought that the big bang scenario implies that spacetime is finite. It started at zero and now becomes bigger. In addition, at a minimum, we have a causal horizon and anything farther away shouldn't matter. Thus instead of integrating "all of space" we could equally integrate to the causal horizon. This again would mean there is only a finite energy barrier between degenerate vacua. But then, there could be tunneling (instantons) between them and the correct vacuum would be a superposition of all possible degenerate vacua. (Analogous to the QCD vacuum). $\endgroup$ – jak Mar 28 '17 at 4:39
  • $\begingroup$ I started a new question for this: physics.stackexchange.com/q/321857 $\endgroup$ – jak Mar 28 '17 at 5:14

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