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17.1 Introduction

- We have seen how the non-Abelian symmetries SU(2) and SU(3) can be used to describe typical physical phenomena as p article multiplets and m assless gauge.
- In two cases, the physical phenomena appear to b e ve r y d iff er.
- The answers to these questions all involve the same fundamental idea, which is a crucial component of the Standard Model.
- The idea is that a symmetry can be hidden.
- By contrast, the symmetries co n sid er to m a y b e ter m e d'm a n if e st sy m m e tr ies'

- In the global and local cases, the consequences of symmetry breaking are different.
- The essentials for a theoretical understanding of the phenomenon are contained in the simpler global case.
- Spontaneous broken lo cal symmetry will be discussed in chapter 19 and applied in chapter 22.

- In response to question, what could go wrong with the argument, we gave in chapter 12.
- To understand, we have a field and a sectio.
- The states are created by operators acting on the vacuum.

- If the vacuum is left invariant, a multiplet structure will emerge.

- The argument for the existence of multiplets of mass-degenerate particles breaks down and this will not manifest in the form of multiplets of mass-degenerate particles.

- There is another way of thinking about what is meant by a broken symmetry in field theory, but it is less rigorous.

- We will see in a moment why this notion is not rigorous.

- 0 does not exist in the space.

- It was suggested before.

- 0 has an infinite norm.

- In a sense, the argument can be reversed.

- We might just use the application for the symmetry.
- We believe that the concept of s sy mmetry is so important to particle physics that a more extended discussion is justified.

- There are crucial sights to the phenomenon of b e g ained b y consid erin g. We will describe the model for the ground state of a superfluid after a brief look at the ferromagnet.
- The 'Goldstone model' is the simplest example of a broken global U(1) symmetry.
- The direction of the Standard Model will be drawn from the generalization of this to the non-Abelian case.
- The ground state for a superconductor is introduced in a way that builds on the model of a superfluid.
- We are prepared for the application in chapter 18.
- In chapter 19 we will see how a different aspect of superconductivity provides a model for the answer to question.

- Everything is dependent on the properties of the vacuum state.

- The Hamiltonian states that the equilibrium state is the lowest energy.
- It is possible that the ground state of a complicated system has unsuspected properties, which may be very hard to predict from the Hamiltonian.
- The properties of the quantum field theory vacuum are similar to those of the ground states of many physically interesting many-body systems.

- In quantum mechanics, the ground state of any system described by a Hamiltonian is non-degenerate.
- Sometimes we meet systems in which more than one state has the lowest energy eigenvalue.
- The true ground state will be a unique linear superposition of the various states, and tunnelling will take place between them.
- In practice, a state which is not the true ground state may have an extremely long lifetime.

- In the case of fields, there is an alternative possibility that is often described as a 'degenerate ground state'.
- The ground state is unique if the theorem is true.

- All of them are the same if the charge destroys a ground state.
- We have the possibility of many ground states.
- One can verify that the alternative ground states are related in the infinite volume limit with simple models.
- All the members of the other towers are also part of the Copyright 2004 IOP Publishing.
- Any tower must be a complete space of states.

- The spins should be fully aligned.

- The Hamiltonian depends on the dot product of the spin operators.

- This ground state is not good.
- The ground state is not invariant under the symmetry of the Hamiltonian and the ground state is degenerate.
- In due course, we will explore this for the superfluid and the superconductor.

- There are some useful insights from the ferromagnet.
- Two ground states are different by a spin rotation.

- The two su ch 'rotated g round states' are indeed rthogonal.

- The spins have all three components, but the magnet is one-dimensional.

- When a symmetry is spontaneously broken, we should expect massless particles when quantized.

- The ferromagnet gives us more information.
- Copyright 2004 IOP Publishing was 'chosen'.

- Thus'spontaneously' breaking the symmetry.

- An 'order parameter' is C.

- The basis of the second-order phase transitions theory is based on this concept.

- The superfluid is similar to the particle physics applications.

- There is symmetry broken in the superfluid ground state.
- Since this is an Abelian, the physics will have symmetry.
- The U(1) case will serve as a physical model for non-.
- We will always be at zero temperature.

- I re 17 I act on er m.

- Ground B is not expected to be an eigenstate of the number operator.

- There is an infinite amount of particles, with which the ground state can be exchanged.
- A number-non-conserving ground state may appear more reasonable from this point of view.
- The ultimate test is whether such a state is a good approximation to the true ground state for a large but finite system.

- The transformation is said to be 'canonical' because they have the same commutation relations.

- The function of the equation is a 'dispersio n relation'.
- We will have massless 'phonon-like' modes.
- The 'plasma Frequency' is just that.

- The topic of chapter 19 will be this.

- Let's focus on the ground state in this model after discussing the spectrum of quasiparticles.

- B case but we don't need the dtailed r esult: an analogous result for the ground state is discussed more fully in section 17.7.

- The following is important.
- In our discussion of the ferromagnet, this situation is mentioned in the paragraph before equation.

- Spontaneous symmetry is breaking in field theory.

- We expect a non-zero vacuum expectation value for an operator to be the key requirement for symmetry breaking in field.

- In the next section, we show how this requirement necessitates at least one massless mode.

- We examine (17.37) and (17.52) in another way, which is only rigorous for a finite system, but is suggestive.

- We have to choose one, thus breaking the symmetry, because no physical consequence follows from choosing one rather than another.
- ground B is in the real direction.
- A definite phase is what B has.

- We return to quantum field theory proper and show how massless particles can be present.
- Whether these particles will actually be observable is one of the questions contained in the theory.

- The vanishing of (17.64) would seem to be unproblematic because the commutator in (17.64) involves local operators separated by a large space-like interval.

- In less formal terms, we treat the spontaneously broken case in chapter 19.

- The necessity of having a massless particle, or particles, in the theory is caused by the existence of a non-vanishing vacuum expectation value for a field.
- The result is the Goldstone.

- The now expected massless mode emerged from the ground.
- A simpler model in which the symmetry breaking is brought about by hand is discussed.

- We will see how this symmetry may be broken.

- The vacuum of the quantum field theory is the nature of the ground state of this field system.

- It reduces the.

- The usual quantized modes are expected to be followed by small oscillations of the field about this minimum.

- 2 is a maximum rather than a minimum.

- The system must choose one direction.

- This sy mmetry is still a classical analogue, though it has been broken spontaneously.
- The model suggests that we should think of the'symmetric' and 'broken symmetry' as different phases of the same stem.

- In contrast to 1776), 0 B does not disappear.
- The condition for the existence of massless (Goldstone) modes is clear, as is the fact that this is exactly the situation met in the superfluid.
- We can see how they emerge in this model.

- Particles are thought of as coming from a ground state in quantum field theory.
- The vacua have no restorin g force and are massless.

- In the superfluid case, the ansatz and the non-zero vev may be compared with the other two.

- Goldstone's model contains a non-zero vacuum value of a field which is not an invariant under the symmetry group and zero mass bosons.
- The Goldstone model is phenomenological.

- In the 'broken symmetry' case, it is interesting to find out what happens to the symmetry current.

- We are going to generalize the U(1) model to the non-Abelian case.

- We can show the essential features by looking at a particular example, which forms part of the Standard Model's Higgs sector.

- neutral anti-particles are created when 0 destroys neutral particles.
- The Lagrangian has an additional U(1) symmetry so that the full symmetry is SU(2)xU(1).

- We can see what happens in the broken symmetry case.

- The stable ground state (17.98) is about a point.
- expand about it, as in (17.84).

- It is not obvious what an appropriate generalization of (17.84) and (17.85) might be.

- This would mean that this particular choice of the vacuum state respected the subset of symmetries, which would not be'spontaneously broken' after all.
- We would get fewer of the Goldstone bosons than we expected since each broken symmetry is associated with a massless goldstone.
- This happens in the present case.

- We would expect four massless fields if we broke the SU(2)xU(1) symmetry completely.

- It is not possible to make such a choice.
- This point may be made clearer by an analogy.

- It is easy to look at infinitesimal transformations if you consider what symmetries are respected or broken by.

- When we look at the spectrum of oscillations about the vacuum, we expect to find three massless bosons, not four.

- The number o f d eg rees of freedom is the same in each case.

- The SU( 2 )xU(1) sy mmetry will be 'gauged' in th e Standard Model.
- Replacing ordinary derivatives with suitable covariant ones is easy.

- The subject of chapter 19 will be exactly how this happens.

- We end this chapter by considering a second example of s sy mmetry b reakin g in s sy mmetry, as a p relimin ary to our discussion of s sy mmetry b reakin g in s sy mmetry.

- The existence of a gap is a fundamental ingredient of the theory of superconductivity.
- We emphasize at the beginning of the chapter that we will not treat the interactions in the superconducting state.
- We work at zero temperature again.

- Ity to th at o f sectio n 17.2 is the Copyright of 2004.
- We will be dealing with electrons instead of the bosons of a superfluid.
- We all see the same phenomenon in the superconducto.

- It can only happen for bosons.
- It is essential that an ism wh e r e b y p a tificatio n o f a m ech an ism.
- A p air o f electrons is repulsive and it remains so in a so lid.
- Positively charged ion can be used as a source of attractio n for electrons in certain circumstances.
- The value of F is the electron d ensity.
- The Debye Frequency is associated with lattice vibrations.
- Cooper was the first to observe that the Fermi'sea' was unstable with respect to the formation of bound pairs.
- The instability modifies the sea in a fundamental way and we need a formalism to handle the situation.

- We all see the same thing, that the ground state of the BCS does not correspond to the symmetry of the superfluid.

- I am the last c o n d itio.

- We will make a crucial number-non-conserving approximation soon.

- The assumption is only valid if the ground state does not have a definitive number of particles.

- The fundamental result at this stage is Equation 17.

- If we consider experimental probes which do not inject or remove electrons, we must be careful to reckon energies for an excited state as relative to a BCS state having the same number of pairs.

- The by now anticipated form for a spontaneously broken U(1) symmetry is in the condition (17.128).
- The massless photon field will enter at the same time.
- Remarkably, we learn in chapter 19 that the expected massless ( Goldstone) m ode is, in this case, not observed: instead, that degree of freedom is incorporated into the gauge field, rendering it massive.
- This is the physics of the Meissner effect in a superconductor and the Higgs mechanism in the Standard Model.

- The electron-electron attraction operates over D. The ensity of states is called F.

- No perturbative treatment starting from a normal g round state could reach this resu lt because F cannot be expanded as a power series in this quantity.
- The estimate is in agreement with the experiment.

- The method used to find the ground state in this model is similar to the method used to find the superfluid.

- Great simplifications occur when the vacuum state is expanded out.

- The superfluid (17.140) is a coherent superposition of correlated pairs with no restraint on the particle number.

- The barest outline of a simple version of the theory has been omitted.
- The Fermi momentum is F.

- The Bohr radius is 0.
- The right-hand side of conventional superconductors is of order 10-3.
- As many as 106 pairs may have their centres of mass within one coherence length of each other, as the pairs are not really bound, only correlated.
- The simple theory presented here contains essential features which attempt to understand the occurrence of symmetry breaking in fermionic systems.

- We are going to apply in particle physics.

- If 17.
- 112 holds, the required anti-commutation relations will be satisfied.

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