In studying about the Higgs field and related, I find little mention of the equilibrium point at -V. I would like help conceptualizing what a negative vacuum expectation value is, ideally with respect to the Higgs field. Do you have any hint on how to observe or create it?
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Unfortunately I cannot comment to ask you for more details.
What exactly do you mean 'equilibrium point at -V'?
Is this the potential, $V(\phi^* \phi)$, or the VEV, $v$ ?
Is it the fact that we put $$ \mu^2 < 0$$ where $$ V(\phi^* \phi) = \mu^2 (\phi^* \phi) + \frac{\lambda}{4}(\phi^* \phi )^2 $$ that is bothering you?
The Vacuum Expectation Value (VEV) is, after spontaneous symmetry breaking, $$ \langle 0 \vert \phi \vert 0 \rangle = \phi_0 = \sqrt{ \frac{v^2}{2}} = \sqrt{\frac{-\mu^2}{2 \lambda}} $$
This is a positive quantity, since we must set $$ \mu^2 < 0$$ to spontaneously break the Mexican Hat potential $ V(\phi^* \phi)$ above.
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$\begingroup$ I'm talking about the VEV. I thought the VEV could be stable (at equilibrium) at v or -v, No? I must be misunderstanding a statement by Matt Strassler from link: "Now, although I’ve set things up so that H could be either v or -v, it doesn’t matter whether the value of the Higgs field is positive or negative; the world comes out looking the same, with the same physics, because nothing depends on the overall sign of H." $\endgroup$ Commented Apr 11, 2014 at 8:15
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1$\begingroup$ I always found it better to view the higgs potential as a literal "mexican hat/wine bottle" potential, that is as a 3D bowl shaped object. In this way one can see that the dip on either side simply becomes a trough instead that causes there to be essentially one minimum value all around the middle at a location v from the center. (Of course this does then move into the realm of complex valued functions when we make the potential 3D and because of this technically the field has a phase which represents the infinitely many solutions around the circle, this is the subject of gauge symmetries) $\endgroup$ Commented Apr 11, 2014 at 9:03
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$\begingroup$ Yes, I quite like that conceptual view of the trough of the wine bottle, except for the left-hand (negative VEV) side of the bottle! Perhaps this will elucidate what bothers me: In this link, Flip Tanedo puts it this way "A loose interpretation for the Higgs vev is a background probability for there to be a Higgs boson at any given point in spacetime." I'm bothered by a negative probability, and more so by a stable negative probability at -v. Obviously there is a hole in my understanding...what am I missing? $\endgroup$ Commented Apr 11, 2014 at 15:24
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2$\begingroup$ In the mexican hat potential there are minima at $re^{i\theta}$ for some $r$, and every $\theta$. Suppose then that the VEV is $\langle H \rangle = re^{i\theta_0}$. Then you can redefine the field $H$ as $H' = e^{-i\theta_0}H$, and $\langle H' \rangle = r$ is real and positive. Since the Lagrangian is real, this redefinition does nothing to it. $\endgroup$ Commented May 15, 2014 at 0:05