How does spontaneous breaking symmetry occur in quantum mechanics?

[reggepole]

Does anybody have an enough simple example and explanation of spontaneous breaking symmetry in quantum mechanics? When does the process of breaking take place? After or before some small perturbation appears?

 

[marlon]

ok, since i do not know how deep you are into QFT, i suggest you first read this intro text i wrote on dynamical symmetry breaking in my journal.
Just scroll down to the last entry on the page :
https://www.physicsforums.com/journal.php?s=&action=view&journalid=13790&perpage=10&page=9 -entry


regards
marlon

 

[Norman]

an example- but not a quantum one or a very good one- but it none-the-less is somewhat illustrative is this:

Take a piece of paperboard (stiff paper) and cut a strip a couple inches long. Place this piece of paperboard between your thumb and forefinger. Apply a little pressure and the board will bend one way or the other. Before the pressure was applied (the perturbation) the symmetry between the two sides of the paper was evident. After the pressure was applied, the symmetry is broken. Is this spontaneous... not really. But you can see the idea I hope.

An example in classical physics, is the massive bead on a circular wire which is rotating about an axis through the loop itself (not through the center- I hope I have explained this clearly). If the loop rotates with an angular velocity omega, there will be a specific angular velocity omega_0 that breaks the reflection symmetry of the loop. You should be able to find something on the internet about this. Try google.

edit:

Here are some links that might help:

Wiki on SSB:
http://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking

And a paper (a bad one anyways) that has the loop argument in it:
http://philsci-archive.pitt.edu/archive/00000563/00/SSB.pittarchive.mss.pdf

p.s. I wouldn't trust anything in that paper I put in here. But the gist of the argument is there.

 

[reggepole]

I am enough deep in QFT to know Goldstone model and Higgs phenomenon. I like to see an example from quantum mechanics with square well or something like it not field theory or classical mechanics . I've read some in Sakurai book but not sure I understood right. Seems I've understood math but not the idea. When does the process of breaking take place? After or before some small perturbation appears? BTW I read somewhere that this effect does not appear in quantum mechanics.

 

[marlon]

When does the process of breaking take place? After or before some small perturbation appears? BTW I read somewhere that this effect does not appear in quantum mechanics.

Let me answer the second question first. Suppose you have a potential $$V( \phi)$$ that reaches minimal value for two distinct non zero field values.
If you replace the field by its opposite value (reflection symmetry), the Lagrangian stays the same but once you calculated the two minima, you go from one to the other by replacing the field by its opposite.

In QM, the particle will tunnel between the two minima and the probability of being in one of the two vacua is equal because of reflection symmetry of the hamiltonian. In QFT, the tunneling barrier is infinite (this is proven in QFT) and extends over the entire volume of the system. Thus, one of the two minima must be chosen. By chosing one of them, you are breaking the symmetry because if you replace the field by its opposite you will no longer be able to go to the other potential minimum, hence reflection symmetry is lost. This answers your first question : the breakdown takes place once nature has chosen one out of all possible vacuum values (this happens prior to the excitations).

once you have one minimum, you study the excitations from this vacuum configuration and you will see that extra terms will arise in the equations of motion that express the interaction of an elementary particle with a certain boson (ie the Higgs particle). It is this interaction that gives mass to elementary particles.

In order to make sure this will work, you need to adapt the potential in the Lagrangian in such a way that you are sure the associated vacuum value will be degenerate so that breakdown can take place.
regards
marlon

PS in QFT you can go from one minimum to another by going from one gauge configuration at negative spatial infinity to another at positive spatial infinity. Particles that do this are called instantons.

 

[tavi_boada]

My teacher gave us the following example:

Imagine you have a donkey with two buckets of food. One at his right and one at his left. You have specular symmetry and you are happy, because you're a physicist and you like symmetries.

After a while the animal gets hungry and eats one of the plates, thus breaking the symmetry.How could he chose one of the plates if they where equal? Answer: He's not a physicist.

Kinda stupid now that I see it written out.

 

[selfAdjoint]

My teacher gave us the following example:

Imagine you have a donkey with two buckets of food. One at his right and one at his left. You have specular symmetry and you are happy, because you're a physicist and you like symmetries.

After a while the animal gets hungry and eats one of the plates, thus breaking the symmetry.How could he chose one of the plates if they where equal? Answer: He's not a physicist.

Kinda stupid now that I see it written out.

This is the famous thought experiment called "Buridan's Ass". The medieval philosopher Jean Buridan taught that animals, unlike humans, were entirely causal mechanisms. His critics brought up the donkey, or ass. If entirely determiistic how could it ever decide between two identical hay bales equally distant from it?

 

[tavi_boada]

Ha ha!

He didn't tell us it wasn't his. Thank god symmetries in real life are approximate!