When I read about the spontaneous symmetry breaking (SSB) in the simple scalar field theory, I was wondering many things. In SSB, what happens is that, the ground state of the scalar potential breaks the symmetry but the overall Lagrangian still preserves it. I was wondering if the reverse is possible and if yes, then how will we describe it. My curiosity was natural in the sense that we see more conserved symmetries at low energy like color (strong interaction), electromagnetism, baryon number, lepton number etc than spontaneously broken symmetries like electroweak symmetry. Although we know theories where Lepton number and Baryon number can be violated by the decay of very heavy particles, at GUT (Grand Unified Theory) scale for example, but we always take it for granted that the conserved gauge symmetries like color and electromagnetism at low energy are conserved at high energy as well. However, recently I found some papers which try to investigate such issues in details, the high energy behavior of symmetries. And it can be possible that finite temperature effects can:
1. Restores a symmetry (global or local) at high temperature which is broken at zero temperature and
2. Violates a symmetry (global or local) at high temperature which is preserved at zero temperatures.
Although the first effect of finite temperature is what most of us would naively expect (like in the case of SSB), but the second effect is something counter-intuitive as well unexpected. The second effect is known as Anti Restoration of symmetries at high temperatures. Langacker and Pi had shown in their work that the U(1) symmetry of electromagnetism which is perfectly conserved at zero temperature can be broken at very high temperatures provided your theory has sufficiently rich to account for that. Of course, with just the standard model field content, the second effect would be difficult to show. Although such anti-restoration at high temperatures wont be observed at present universe which is in a supercooled state, but such effects can have very important implications in cosmology. There might be multiple phase transition in the early universe because of this restoring and anti-restoring effects. There are models where the electroweak symmetry is broken at very high scale (~GUT scale for example), and as the Universe cools down, this symmetry gets restored after some critical temperatures and upon further cooling, it gets broken at the electroweak scale. This means that the standard model quarks and leptons were ultra-heavy at very early stage of the Universe which can have various implications. Although this field was very active in the eighties, I do not see anybody working on such models seriously nowadays except the models of supersymmetry breaking. But I find these ideas very interesting although they are kind of out-dated from current research trends point of view.
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