Higgs mechanism is defined to operate (give spontaneous symmetry breaking) below some critical temperature. As mentioned, for ex. in Wikipedia:

Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass...

So what is meant by critical temperature quantum mechanically? Is this referring to a 'real' temperature as in a real gas, or some QM (virtual) temperature? Please explain.

  • $\begingroup$ I believe (hence comment, not answer) that the temperature in this case is equivalent to the background energy of the universe. The same way as $k_B T = \langle E \rangle$ for a gas. The early universe was much more energetic than the one we live in right now due to inflation. At certain energies, the symmetry breaking stops playing any role in the scattering amplitudes. $\endgroup$ – Darkseid Jul 20 '19 at 20:52

It's a real temperature.

In statistical mechanics, temperature is an indirect measure of energy: the higher the temperature, the higher the energy. Here's one way to express the relationship: when we say that the system (the universe, in this case) is in equilibrium with temperature $T$, we mean that as far as any local observations are concerned, we might as well use a weighted average over all possible states, weighted by $e^{-E/kT}$ where $E$ is the energy of the state and $k$ is Boltzmann's constnat. The higher the value of $T$, the less rapidly these weights fall off with increasing $E$, so higher-energy states contribute more and more as $T$ increases, especially because the number of high-$E$ states is much greater than the number of low-$E$ states. This way of thinking about temperature is just as valid in quantum field theory as it is for an ideal gas. In both cases, this $T$ is a real temperature: with minor qualifications, it's basically what temperature means.

The Higgs mechanism has a characteristic energy scale, which in the standard model is $\sim 200$ GeV. If we consider only states that have low energies compared to that scale, then the consequences of the broken symmetry (experts would say "non-linearly realized" symmetry) are prominent. But when we're averaging over states most of which have energy $E$ far larger than the $W$ or $Z$ boson mass (which are both within a factor of $\sim 3$ of the scale $\sim 200$ GeV), the lower-energy states where those masses are important are overwhelmed by the far-more-numerous higher-energy states. As a result, the familiar low-energy consequences of electroweak symmetry breaking are less prominent at such high temperatures.

  • $\begingroup$ THank you Chiral ; so we are talking about the temperature (or avg. energy state of the Higgs field? So my question now is , what does that mean in a practical way....for ex., what raises (or lowers) the energy (temp.) of the Higgs field?? How can it be raised ? $\endgroup$ – Gary Jul 21 '19 at 0:18
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    $\begingroup$ @Gary The temperature was (presumably) high enough in the very early universe. Not just the temperature of the Higgs field, though; t's not clear to me how one would even define, much less manipulate, the temperature for an individual field in quantum field theory, and that's not what people mean when they talk about the electroweak symmetry breaking transition at high temperature. They're talking about the temperature of the whole system, Higgs field and matter fields and gauge fields. $\endgroup$ – Chiral Anomaly Jul 21 '19 at 1:30
  • $\begingroup$ OK; In other words, since Higgs is supposedly ubiquitous, does a strong electric field in the vacuum, for example, raise the energy density of the vacuum and thus the Higgs field? And if so how do we determine how much the energy scale of the whole system is raised in terms of GeV?? $\endgroup$ – Gary Jul 21 '19 at 1:34
  • $\begingroup$ ...Furthermore, your interesting user name brought to mind another question about Higgs field....In Chiral anomaly do the Dirac fermions acquire their mass from the Weyl fermion's coupling to the Higgs field, and if so, how does that work in the Chiral anomaly? IOW, is C.A. a result of a differential coupling between left and right handed Weyl fermions to Higgs? Can you please explain. $\endgroup$ – Gary Jul 21 '19 at 17:16
  • $\begingroup$ @Gary You're asking some interesting questions, and in keeping with the design principles of Physics SE, I'd recommend posting them as separate questions. That way, you're likely to get more complete answers -- and from other users who might know more about these things than I do. I'll just briefly address the one question about C.A.: it's a result of a different coupling between left and right handed Weyl fermions to the gauge fields. Their couplings to the Higgs are also different, but that's not the source of the C.A. $\endgroup$ – Chiral Anomaly Jul 21 '19 at 23:33

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