In SSB, why shifting a field suffices to pick a corresponding vaccum?

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In summary, the shift from \langle\phi\rangle=0 to \langle\phi\rangle\neq0 in the simplest \phi^4 with Z_2 breaking was not an ad hoc procedure, but rather picked out the true ground state. This is because the lagrangian dictates that the ground state should have a non-zero vev, and expanding around a false vacuum will not work.
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ingenue
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Consider the simplest [tex]\phi^4[/tex] with [tex]Z_2[/tex] breaking. Before the shift, [tex]\langle\phi\rangle=0[/tex] by symmetry. After the shift, the vev of the shifted field is zero, which means [tex]\langle\phi\rangle\neq0[/tex], which in turn means we have picked the corresponding vacuum out of two possibilities. However, through the calculation of path integral, shifting a field by a constant only has done nothing as to fixing the boundary condition. Then why did this happen?
 
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"The shift" isn't an ad hoc procedure. It happened to pick out a true ground state. With [tex]\langle\phi\rangle=0[/tex], you're trying to perturbatively expand around a false vacuum, and it ain't going to work.
 
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daschaich said:
"The shift" isn't an ad hoc procedure. It happened to pick out a true ground state. With [tex]\langle\phi\rangle=0[/tex], you're trying to perturbatively expand around a false vacuum, and it ain't going to work.

that's exactly why I'm asking. why did it happen to do this?
 
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Why did it happen that the ground state has [tex]\langle\phi\rangle\neq 0[/tex]? That comes from the lagrangian.
 
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No, I meant why did it happen to pick out the right vacuum?
daschaich said:
Why did it happen that the ground state has [tex]\langle\phi\rangle\neq 0[/tex]? That comes from the lagrangian.
 

Related to In SSB, why shifting a field suffices to pick a corresponding vaccum?

1. What is SSB and why is it important in physics?

SSB stands for Spontaneous Symmetry Breaking, which is a phenomenon in physics where a symmetric system will spontaneously acquire a non-zero expectation value for a field. This is important because it helps explain how particles acquire mass, and is a key concept in the Standard Model of particle physics.

2. How does shifting a field lead to SSB?

In SSB, a field is said to be shifted when its expectation value is no longer equal to zero. This can happen due to interactions with other fields or through the Higgs mechanism, resulting in a spontaneous breaking of symmetry. This shift in the field is what leads to SSB and the emergence of a non-zero expectation value.

3. Why is it sufficient to just shift a field to pick a corresponding vacuum?

In SSB, the vacuum state (lowest energy state) of a system is determined by the expectation value of the field. As mentioned earlier, shifting the field results in a non-zero expectation value, which then corresponds to a specific vacuum state. Therefore, by shifting the field, we are effectively picking a corresponding vacuum state.

4. How does SSB relate to the concept of broken symmetries?

SSB is a type of broken symmetry, where the symmetry of a system is not preserved in its ground state. This means that the system behaves differently than it would with a symmetric ground state. In SSB, the symmetry is broken due to the shift in the field, leading to a non-zero expectation value and a new ground state.

5. What are some real-world applications of SSB?

SSB has many applications in physics, including explaining how particles acquire mass in the Standard Model. It is also used in condensed matter physics to understand phase transitions and the behavior of materials at low temperatures. Additionally, SSB is important in cosmology and has been used to explain the origins of the universe and the formation of galaxies.

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