Thursday, June 15, 2017

vacuum - What is the difference between quantum fluctuations and thermal fluctuations?


Start with a simple scalar field Lagrangian $\mathcal{L}(\phi)$ at zero temperature $T = 0$, which has a hidden symmetry and spontaneously break it. By the standard procedure a field $\phi$ is redefined


$$\phi \rightarrow \langle \phi \rangle + \phi',$$


where $\phi'$ is a quantum fluctuation around some constant value $\langle\phi\rangle$. The constant value $\langle \phi \rangle$ is called a condensate (or vacuum expectation value) of the field $\phi$. (For example, in the case of pions and sigma mesons ($\mathcal{L}$ is a linear sigma model Lagrangian) fluctuations $\phi'$ are physical pions and sigma mesons, with pion condensate equal to zero, and sigma meson condensate equal to $\langle \sigma \rangle = f_\pi$.)


The spontaneus symmetry breaking looks the same for $T \neq 0$ scalar field theory. Again, we redefine the field $\phi \rightarrow \langle \phi \rangle + \phi'$ and obtain physical particles $\phi'$ as a fluctuations around the condensate, which is now temperature dependent variable; and it can serve as an order parameter of the theory. (For example, in the case of sigma mesons and pions, the condensate $\langle \sigma \rangle $ will vanish at the chiral temperature point, displaying the existance of the chiral phase transition.)


So my question is, are the quantum fluctuations $\phi'$ (i.e. the physical particles) the same in $T = 0$ and $T\neq0$ field theory? Or are they somehow 'mixed', so they are both thermal and quantum fluctuations? In addition, the diagram here http://upload.wikimedia.org/wikipedia/commons/0/06/QuantumPhaseTransition.png basically says that quantum and classical (critical) behaviour is the same thing, which adds up to my confusion.


Of course, if I completely missed the point, I hope that someone can explain in a better way what is the concept of the symmetry breaking and emergence of a condensate (and physical particles).





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