Understanding Quantum Symmetry Breaking

In summary, the conversation discusses the concept of symmetry breaking in classical and quantum physics. It is stated that symmetry is broken when the associated charge of the symmetry, Q, does not equal 0 when acting on the vacuum state |0>. The definition of symmetry breaking is further explained as a subgroup H remaining invariant while the larger symmetry group G is broken. The conversation also explores the use of unitary and antiunitary operators in representing symmetry transformations and the role of the associated charge, Q, in maintaining symmetry. The question of whether symmetries represented by antihermitian operators also follow this pattern is raised.
  • #1
alphaone
46
0
Hi,
I have done classical symmetry breaking and now want to understand the quantum one. I have seen the statement that the symmetry is broken if and only if Q|0> not 0. Where |0> is the vacuum and Q is the associated charge of the broken symmetry. Why does this imply symmetry breaking? The way I know it a symmetry group G is broken to a subgroup H if the theory is invariant under G whereas the vacuum is invariant only under H, meaning h|0>=|0> for any h element H and g|0> not equal |0> for any g not in H. Is this the right definition? And if so does this imply Q|0> not 0 in the case of a broken symmetry?
 
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  • #2
I think I figured it out myself:
Any symmetry transformation can be represented by either a unitary operator U or an antiunitary operator A acting on the space of states, i.e. U|phi> is the transformed state(for the case of a unitary operator) now if I am not mistaken we can write U = exp(i*x*Q) (x being a parameter here and not spacetime position) where Q is the associated charge of the symmetry. Then for the symmetry to be mainfest we require U|0>=|0> which is equivalent to saying Q|0>=0 when expanding the exponential. A question I have about this is: Does this also hold for symetries being represented by antihermitian operators, i.e. if the states transform under the symmetry as A|phi> can I still write A=exp(i*x*Q) where Q is the conserved charge? In this case the charge would then be antihermitian instead of hermitian, I guess.
 
  • #3


Hello! It sounds like you have a good understanding of classical symmetry breaking, and now you are looking to understand the quantum version. In quantum mechanics, symmetry breaking occurs when the ground state (or vacuum state) of a system is not invariant under the symmetry group of that system. This means that the vacuum state is not unchanged when a symmetry transformation is applied to it.

In the case of quantum symmetry breaking, the symmetry group is represented by an operator Q, which is associated with the broken symmetry. This operator acts on the vacuum state, and if the result is not equal to the vacuum state (Q|0> ≠ |0>), then the symmetry is said to be broken. This implies that the vacuum state is not invariant under Q, and therefore not invariant under the full symmetry group.

Your understanding of the definition of symmetry breaking for a symmetry group G being broken to a subgroup H is correct. In this case, the vacuum state is invariant under the subgroup H, but not under the full group G. This is equivalent to saying that h|0> = |0> for any h element of H, but g|0> ≠ |0> for any g not in H.

In conclusion, the statement Q|0> ≠ |0> is a way to mathematically represent the idea that the vacuum state is not invariant under the symmetry group, and therefore the symmetry is broken. I hope this helps clarify the concept of quantum symmetry breaking for you. Keep exploring and asking questions!
 

1. What is quantum symmetry breaking?

Quantum symmetry breaking is a phenomenon in quantum field theory where the symmetry of a physical system is spontaneously broken at the quantum level, resulting in a state of the system that is different from the initial symmetrical state.

2. How does quantum symmetry breaking occur?

Quantum symmetry breaking occurs when the vacuum state, or ground state, of a quantum system is not invariant under the symmetry transformations of the system. This leads to the formation of a new state that breaks the original symmetries of the system.

3. What are the implications of quantum symmetry breaking?

The implications of quantum symmetry breaking are significant in the study of fundamental particles and the behavior of matter at the smallest scales. It can also impact the understanding of phase transitions and the formation of new states of matter.

4. Can quantum symmetry breaking be observed experimentally?

Yes, quantum symmetry breaking has been observed in various experimental settings, such as in condensed matter systems and in the behavior of subatomic particles. It can also be studied through theoretical calculations and simulations.

5. How does quantum symmetry breaking relate to the Higgs mechanism?

The Higgs mechanism is a theory that explains how particles acquire mass in the Standard Model of particle physics. It is closely related to quantum symmetry breaking, as the Higgs field is responsible for breaking the symmetry of the electroweak force and giving mass to particles. This connection is a crucial aspect of understanding the behavior of fundamental particles.

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