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.

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.

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.