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Supplemental Lecture 20

The Chiral and the

Introduction to the Foundations of For Physics Students Part VII

Abstract

We consider axial symmetries and the conservation of the axial current in 1+1 and 3+1 dimensional massless . Although the axial vector current is predicted to be conserved in the classical versions of these theories as a consequence of continuous axial rotations of the field, these symmetries and conservation laws are broken in the quantum theories. We find that the response of the Dirac sea to an external electric field leads to this “axial anomaly”. The exact expressions for the divergence of the axial currents in 1+1 and 3+1 dimensions are derived. In 3+1 dimensions the degeneracy of the theory’s Landau levels plays an important role in the final result. Other approaches to understanding and calculating this and other axial anomalies are presented and discussed briefly.

Prerequisite: This lecture continues topics started in Supplementary Lecture 19.

This lecture supplements material in the textbook: , Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.

Keywords: The , , Massless , The Dirac Sea, Chiral Symmetry, Axial Anomaly, Quantum Electrodynamics in 1+1 and 3+1 Dimensions, Landau Levels.

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Contents

1. The Dirac Equation yet again...... 2

2. The Vector and Axial Vector Currents...... 5

3. The ...... 6

4. The Chiral Anomaly in 3+1 Dimensions...... 13

5. The Importance of the Chiral Anomaly...... 16

6. Appendix A. Landau Levels in 3+1 Dimensions...... 17

References ...... 20

1. The Dirac Equation yet again. We studied the Dirac equation in the previous lecture and introduced the Clifford algebra of the ,

훾훼훾훽 + 훾훽훾훼 = {훾훼, 훾훽} = 2푔훼훽 1.1

The reader should review Supplementary Lecture 19: the lecture here is a continuation of that introduction.

Our previous emphasis was on the non-relativistic limit of the Dirac equation and we used a choice of the 4 × 4 gamma matrices 훾훼 to expedite that. Here our emphasis is on massless fermions and a more convenient representation for the gamma matrices is the so-called “chiral” representation,

0 1 0 휎 훾0 = ( ), 훾푖 = ( 푖) 1.2a 1 0 −휎푖 0 where each element is a 2 × 2 matrix and the Pauli matrices are,

0 1 0 −푖 1 0 휎1 = ( ) 휎2 = ( ) 휎3 = ( ) 1.2b 1 0 푖 0 0 −1 which obey the Pauli algebra, {휎푖, 휎푗} = 2훿푖푗.

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It is useful to understand the Lorentz transformation properties of spinors in this representation. Let’s summarize calculations exploring the covariance of the Dirac equation in the chiral representation which obtain anew the 4 × 4 matrices 푆(Λ) that implement Lorentz −1 훼 훼 휌 transformations in spinor space: 푆 (Λ)훾 푆(Λ) = Λ 휌훾 that was studied in lecture 19.

Rotations and boosts were discussed separately in lecture 19. We wrote 푆푅(Λ) for rotations and † −1 saw that 푆푅(Λ) is unitary, 푆푅 = 푆푅 . However, for boosts we wrote 푆퐵(Λ) and saw that it was not unitary. This fact was traced to the fact that boosts do now conserve lengths: lengths contract under boosts. In the chiral representation Eq. 1.2 we find,

푒푖휑⃗⃗ ∙휎⃗⃗ /2 0 푒휒⃗⃗ ∙휎⃗⃗ /2 0 푆푅(Λ) = ( ) 푆퐵(Λ) = ( ) 1.4 0 푒푖휑⃗⃗ ∙휎⃗⃗ /2 0 푒−휒⃗⃗ ∙휎⃗⃗ /2 where 휑 is the angle of the rotation about a fixed axis and 휒 has the magnitude of the rapidity (see Supplementary lectures 19 and 7) and points along the boost.

The fact that 푆 is block diagonal in Eq. 1.4 means that the spinor representation of the is reducible! To make this explicit write a 4-spinor in the form of two 2-spinors,

푢 휓 = ( +) 1.5a 푢− where 푢± are called “chiral spinors”. Under rotations and boosts,

푖휑⃗⃗ ∙휎⃗⃗ /2 ±휒⃗⃗ ∙휎⃗⃗ /2 푢± → 푒 푢± 푢± → 푒 푢± 1.5b

1 1 We say that 푢 is in the ( , 0) representation of the Lorentz group and 푢 is in the (0, ) + 2 − 2 1 1 representation. Then the Dirac 4-spinor lies in the ( , 0) ⨁ (0, ) representation. We will have 2 2 more to say about these results below. They are fundamental!

In the previous lecture we wrote the Dirac equation in covariant form,

휕 (푖ℏ훾휇 − 푚푐) 휓 = 0 1.6 휕푥휇 which could be obtained from the Lorentz invariant ,

휕 푆 = ∫ 푑4푥 휓̅ (푖ℏ훾휇 − 푚푐) 휓 1.7 휕푥휇

3 using the Euler-Lagrange equations. We can write the Lagrangian density in terms of chiral spinors,

̅ 휇 † 휇 † 휇 † † ℒ = 휓(푖ℏ훾 휕휇 − 푚푐)휓 = 푖푢−휎 휕휇푢− + 푖푢+휎̅ 휕휇푢+ − 푚(푢+푢− + 푢−푢+) = 0 1.8a where we used the notation from the previous lecture,

휎휇 = (1, 휎푖) 휎̅휇 = (1, −휎푖) 1.8b

푎nd we also used

0 1 0 휎 −휎 0 훾0 훾푖 = ( ) ( 푖) = ( 푖 ) 1 0 −휎푖 0 0 휎푖 in simplifying the right-hand side of Eq. 1.8a and we set ℏ = 푐 = 1,. Note that for massless fermions 푢+ and 푢− decouple and Eq. 1.8b becomes two independent differential equations, called the “Weyl” equations,

휇 휇 푖휎̅ 휕휇푢+ = 0 푖휎 휕휇푢− = 0 1.9

So far, these manipulations depend on the explicit chiral representation Eq. 1.2. But we need an invariant way to do these calculations. The 훾5 gamma matrix is crucial in achieving this. We define,

훾5 = −푖훾0훾1훾2훾3 1.10a which differs from the definition used in the previous lecture by a phase choice. (the −푖 will prove convenient for chiral spinors). Recall the properties of 훾5,

{훾5, 훾휇} = 0 (훾5)2 = 1 1.10b

훾5 allows us to define a Lorentz invariant chiral projection operator,

1 푃 = (1 ± 훾5) 1.11 ± 2

2 The operators 푃± are idempotent: 푃± = 푃± , 푃+푃_ = 0 and 푃+ + 푃− = 1. These points are particularly trivial in the chiral representation Eq. 1.2 where

1 0 훾5 = ( ) 1.12a 0 −1

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So in the chiral representation

1 0 0 0 푃 = ( ) 푃 = ( ) 1.12b + 0 0 − 0 1

But the real value of 푃± is to define chiral spinors in any representation,

휓± = 푃±휓 1.13 which form the irreducible representations of the Lorentz group. 휓+ is called “left-handed” and

휓− is called “right-handed”. This terminology will make more sense when we have considered the spin carried by a massless fermion. In addition, the chiral spinors 휓± are related to each other by parity, 푃

푃: 푥0 → 푥0 , 푥푖 → −푥푖 1.14a

So, P changes the sign of the boost’s velocity, 푣 → −푣 , so from Eq. 1.5b,

푃: 휓±(푥 , 푡) → 휓∓(−푥 , 푡) 1.14b

2. The Vector and Axial Vector Currents. We have already seen that the vector current 푗휇(푥) = 휓̅(푥)훾휇휓(푥) is conserved, 휇 휕 푗휇(푥) = 0, for free Dirac fermions. We can check this critical directly from Eq. 1.6,

휇 휇 휇 휕휇푗 = (휕휇휓̅)훾 휓 + 휓̅훾 (휕휇휓) = 푖푚휓̅휓 − 푖푚휓̅휓 = 0 2.1

This continuity equation leads to the conservation of the total charge 푄 = ∫ 푑푥 휓̅훾0휓 = ∫ 푑푥 휓†휓. Local and global conservation of charge is a crucial ingredient in quantum electrodynamics.

휇 ̅ 휇 5 We can also investigate the conservation of the axial vector current 푗퐴 = 휓훾 훾 휓. We can 휇 calculate the four divergence of 푗퐴 using the free Dirac equation,

휇 ̅ 휇 5 ̅ 휇 5 ̅ 5 휕휇푗퐴 = (휕휇휓)훾 훾 휓 + 휓훾 훾 (휕휇휓) = 2푖푚휓훾 휓 2.2

휇 So, only if 푚 = 0 will 푗퐴 be conserved. We can also relate this conservation law to a symmetry operation just like we have done for the conservation of the vector current. In that case the

5 symmetry 휓 → 푒푖훼휓 is a symmetry of the Lagrangian Eq. 1.8a and Noether’s Theorem, which was introduced in Supplementary lecture 14, leads to the identification and conservation of the vector current. For the axial current, the associated symmetry is

5 5 휓 → 푒푖훼훾 휓 휓̅ → 휓̅ 푒푖훼훾 2.3

휇 The kinetic piece of the Lagrangian density, 휓̅ 푖훾 휕휇휓, is invariant to this transformation but the term, 푚휓̅휓, is not. This analysis agrees with the explicit calculation Eq. 2.2.

Now for the point of this discussion. What happens to these symmetries when interactions 휇 are turned on? In the case of the vector current, we strongly suspect that 휕휇푗 = 0 should survive the introduction of the interactions of quantum electrodynamics because current conservation is so critical to the properties and consistency of the theory. By contrast, we will see that axial symmetries may not survive the introduction of such interactions. In the case of quantum 휇 electrodynamics, we will find that the conservation law 휕휇푗퐴 = 0 of the massless theory is violated by quantum interactions and there is a “chiral anomaly” which plays an important role in several theories of elementary particles and condensed matter phenomena.

3. The Chiral Anomaly. When massless Dirac fermions couple to an , the wavefunction satisfies the equation,

휕 푒 훾휇 (푖ℏ − 퐴 ) 휓 = 0 3.1 휕푥휇 푐 휇

The electromagnetic field 퐴휇 may be quantized and dynamical or might be an external specified field. Eq. 3.1 leads to the conservation of the vector current and the axial current if we accept classical manipulations as discussed in Sec. 2. However, explicit calculations of quantized fluctuations show that in quantum electrodynamics, axial current conservation suffers from a “chiral anomaly” and is replaced by,

푒2 휕 푗휇 = 퐹 ∙ 퐹̃ 3.2 휇 퐴 16휋2

훼훽훾훿 where 퐹 ∙ 퐹̃ = 휖 퐹훼훽퐹훾훿. The source of the non-vanishing right-hand side to Eq. 3.2 is the “triangle diagram” shown in Fig. 1,

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Fig.1 The Triangle Anomaly in 3+1 dimensional QED

The anomaly, credited to theorists S. Adler, J. Bell and R. Jackiw in 1969, was actually discovered by an experimentalist J. Steinberger in 1949 who computed the lifetime for a neutral to decay to two in a nuclear physics model [1]. These early calculations of Feynman diagrams did not thoroughly elucidate the source of the violation of the conservation of the axial charge. For that reason, let’s consider the axial anomaly in a simpler setting, quantum electrodynamics in 1+1 dimensions, where the relation to counting axial charges carried by fermions is clear.

Let’s begin with the Dirac equation for free fermions in one time-one space dimensions. We will understand the violation of the conservation of axial charge by quantum effects here before turning to more realistic theories. The Clifford algebra in 1+1 dimensions reads,

{훾훼, 훾훽} = 2푔훼훽 3.3a for 훼, 훽 = 0,1. We can write down 2 × 2 matrices which satisfy the Clifford algebra using Pauli matrices. For example

0 1 0 1 훾0 = 휎1 = ( ), 훾1 = 푖휎2 = ( ) 3.3b 1 0 −1 0

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These matrices act on 휓 which is a two component spinor. Since there is no real “spin” in 1+1 dimensions, a two component spinor is sufficient to describe relativistic fermions, particles and anti-particles. Of course, fermions must satisfy Fermi statistics and the Pauli exclusion principle: Fermions and bosons are distinguished by their statistics in 1+1 dimensions. We will learn more about fermions and bosons in 1+1 dimensions in the next lecture where we will introduce “bosonization” and explicitly solve quantum electrodynamics in 1+1 dimensions.

The action for massless Dirac fermions reads,

휇 푆 = ∫ 푑푥 푑푡 ℒ = ∫ 푑푥 푑푡 휓̅ 푖훾 휕휇 휓 3.4

Let’s write out the inner product,

휇 † 0 0 1 † 5 ℒ = 휓̅ 푖훾 휕휇 휓 = 푖휓 훾 (훾 휕0 + 훾 휕1)휓 = 푖휓 (휕0 − 훾 휕1)휓 3.5 where we introduced 훾5 and used the 1+1dimensional chiral representation Eq. 3.3b,

1 0 훾5 = −훾0훾1 = −푖휎1휎2 = 휎3 = ( ) 3.6 0 −1

It is convenient to decompose 휓 into chiral spinors using the projection operators introduced in 1 the previous lecture, 푃 = (1 ± 훾5), ± 2

휓± = 푃±휓 1.13

In the chiral representation where 훾5 = 휎3, we have simply,

휒+ 0 휓+ = ( ) 휓− = ( ) 3.7 0 휒− and 휒+ has positive chirality and 휒− has negative chirality. Further, let’s introduce the light cone variables,

휕± = 휕0 ± 휕1 3.8 so the Lagrangian Eq. 3.5 becomes,

† † ℒ = 푖휒+휕−휒+ + 푖휒−휕+휒− 3.9

We learn that the chiral spinors decouple and their free wave equations read,

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휕−휒+ = 0 휕+휒− = 0 3.10a

So

휒+ = 휒+(푡 + 푥) 휒− = 휒−(푡 − 푥) 3.10b

So 휒+ describes left-moving fermions and 휒− describes right-moving fermions. Note that this decoupling only occurs for massless fermions: a mass term in ℒ would couple 휒+ and 휒− ̅ † † because 푚휓휓 = 푚(휒+휒− + 휒−휒+).

If we consider free fermions with well-defined energy-momentum, then

1 휒(퐸)(푡 ± 푥) = 푒−푖퐸(푡±푥) with 푝 = ∓퐸 ± √퐿

(퐸) For 휒+ , 퐸 = −푝, and the fermions with positive chirality with positive energy have negative momentum, while the solutions with positive chirality have positive momentum. For negative chirality, the situation is reversed. See Fig. 2.

Building on the ideas of Dirac Theory presented in Supplementary lecture 19, we realize that the of the 1+1 dimensional theory must have its negative energy states filled as shown in Fig. 2.

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Fig. 2. Dirac vacuum of 1+1 dim fermions: The red states are occupied, the blue are not.

So, there are unoccupied positive energy right moving states ( 푝 > 0), unoccupied positive energy left moving states (푝 < 0), occupied right moving negative energy states and occupied left moving negative energy states.

What are the symmetries of the fermions described by the Lagrangian Eq. 3.9? As discussed in the previous lecture, there is phase symmetry 휓 → 푒푖훼휓 which does not distinguish 휇 휇 between 휒+and 휒−, and leads to the conservation of the vector current 푗 = 휓̅훾 휓. Classically, 5 there is also the chiral symmetry 휓 → 푒푖훼훾 휓 which leads to the conservation of the axial vector 휇 ̅ 휇 5 current 푗퐴 = 휓훾 훾 휓. To better understand the conservation laws, let’s write the currents in 휇 휇 terms of 휒+and 휒−. First, 푗 = 휓̅훾 휓,

0 † † † 1 † 0 1 † 5 † † 푗 = 휓 휓 = 휒+휒+ + 휒−휒− 푗 = 휓 훾 훾 휓 = −휓 훾 휓 = −휒+휒+ + 휒−휒− 3.11

휇 ̅ 휇 5 And next, 푗퐴 = 휓훾 훾 휓,

0 † 5 † † 1 † 0 1 5 † † † 푗퐴 = 휓 훾 휓 = 휒+휒+ − 휒−휒− 푗퐴 = 휓 훾 훾 훾 휓 = −휓 휓 = −휒+휒+ − 휒−휒− 3.12

0 Let’s restate these conservation laws in terms of the charges 푄 = ∫ 푑푥 푗 (푡, 푥) and 푄퐴 = 0 푖훼 ∫ 푑푥 푗퐴 (푡, 푥). We see from Eq. 3.11 that the first symmetry 휓 → 푒 휓 implies that the sum of the right moving fermions 푛+ and left moving fermions 푛− , 푛+ + 푛− , is conserved. (Fermion number counts fermions minus anti-fermions, the “net” number of fermions.) And, reading from

푖훼훾5 Eq. 3.12, the chiral symmetry 휓 → 푒 휓 leads to the conservation of the difference 푛+ − 푛− . So, if both symmetries are unbroken the total number of right movers and left mover fermions are separately conserved.

Are both conservation laws true in 1+1 dimensional quantum electrodynamics? In fact, they are not! This was discovered by J. Schwinger in the early days of the subject, 1962 [2]. Schwinger showed that the theory can be solved as a free but massive(!) scalar field. Where does the mass come from? In 1+1 dimensions, the electric charge e has dimension of [mass] and the mass of the scalar particle is exactly 푚2 = 푒2/휋. The dynamical breaking of the model’s chiral

10 symmetry leads to mass generation! There is much to understand in this simple model (it was popularized in the 1970’s as a soluble model of confinement [3], became a model for the 휂′ problem in hadron spectroscopy [4] and became a model for 휃-vacua in non-abelian gauge theories [5]).

휇 We can see that 푗퐴 is not conserved when electromagnetic interactions are incorporated into the Dirac equation in 1+1 dimensions. To prove this let’s begin with the simplest case: place the fermions into an external electric field E. First, we use minimal substitution to make the coupling,

휇 ℒ = 휓̅훾 (푖휕휇 − 푒퐴휇)휓 3.13

5 where we have taken ℏ = 푐 = 1. The fermion symmetries, 휓 → 푒푖훼휓 and 휓 → 푒푖훼훾 휓, are apparently not effected in Eq. 3.13 by the presence of 퐴휇. However, there is more to the quantum theory than just Eq. 3.13: there is the quantum vacuum and the quantum fluctuations in the fields

휓 and 퐴휇!

Imagine turning on the electric field adiabatically [6,7], then holding it constant for a time t which should be very long compared to other time scales in the problem. Charged particles, all the filled states in the Dirac sea, will feel the force 푒퐸 and after time t will have picked up an extra momentum of,

∆푝 = 푒퐸푡 3.14

So, Fig. 2 turns into Fig. 3,

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Fig. 3 The Dirac sea after the application of an electric field E for time t.

So, the electric field has created left-moving anti-particles and right-moving particles. This 0 violates the conservation of the axial charge 푄퐴 = ∫ 푑푥 푗퐴 (푡, 푥)! Let’s add up the charges carefully. Recall that the density of states in one spatial dimension is 푑푥 푑푝⁄2휋 . Call the density of right moving fermions 휌+ and the density of left moving fermions 휌−. The shift in momentum Eq. 3.14 then leads to shifts in these densities,

푒퐸푡 푒퐸푡 휌 = 휌 = − 3.15 + 2휋 − 2휋

The total fermion number (particles minus anti-particles) is conserved as the figure indicates.

The total density of fermions is 휌 = 휌+ + 휌− and Eq. 3.15 gives,

푑 휌̇ = 0 ∫ 푑푥 푗0(푡, 푥) = 0 3.16a 푑푡

However, the difference between fermion numbers is not conserved, 휌퐴 = 휌+ − 휌−, and Eq. 3.15 gives,

푒퐸 푑 푒퐸 휌̇ = ∫ 푑푥 푗0(푡, 푥) = 3.16b 퐴 휋 푑푡 퐴 휋

This is the chiral anomaly! Collecting these results and writing them as local conservation laws,

푒 휕 푗휇 = 0 휕 푗휇 = 휖훼훽퐹 3.17 휇 휇 퐴 2휋 훼훽

12 where we wrote the chiral anomaly in Lorentz invariant form using the fact that 퐸 = 퐹01 and 1 1 1 휖훼훽퐹 = (퐹 − 퐹 ) = (2 퐹 ) = 퐸. The covariance and generality of this exercise will be 2 훼훽 2 01 10 2 01 discussed further below.

How did the symmetry violation occur? Clearly the quantum vacuum of Dirac was the culprit! This aspect of the problem has no classical analog and would be missed in a classical argument based on Noether’s Theorem! Still, we can ask where the extra fermions came from? They came from infinity [6,7]! View Fig. 3 and compare it to Fig. 2. In Fig. 2 the occupied states at negative energy on the left (and right) extend to infinity, so when the external electric field is applied, fermions are available to populate the states at positive energy and positive momentum.

In this sense the derivation shows the “reality” of the Dirac sea. We will see, however, that there are alternative derivations of the chiral anomaly which do not explicitly manipulate or even refer to the vacuum. Those derivations find subtleties in the sum over quantum fluctuations (the measure of the Feynman path integral proves to be non-chiral invariant) or they compute Feynman diagrams ( in 1+1 dimensions and triangle graphs in 3+1 dimensions) and one must define the integrals over loop momenta in the Feynman diagrams carefully to preserve ordinary charge conservation. These views of the chiral anomaly will be discussed in later lectures.

4. The Chiral Anomaly in 3+1 Dimensions. It is important to see that the chiral anomaly, Eq. 3.17, has a generalization to 3+1 dimensions. In that case we will find,

푒2 푒2 휕 푗휇 = 휖훼훽훾훿퐹 퐹 = 퐹 퐹̃훼훽 4.1a 휇 퐴 16휋2 훼훽 훾훿 16휋2 훼훽

훼훽 훼훽훾훿 where we identified the dual of 퐹훼훽, 퐹̃ = 휖 퐹훾훿. Writing out Eq. 4.1a in terms of 퐸⃗ and 퐵⃗ , we find

푒2 휕 푗휇 = 퐸⃗ ∙ 퐵⃗ 4.1b 휇 퐴 4휋2

So, the anomaly arises from the pseudo-scalar 퐸⃗ ∙ 퐵⃗ and both electric and magnetic fields are required in 3+1 dimensions. So, a generalization of the argument in Sec. 3 will require parallel

13 electric and magnetic fields. We will see how this works below. We will have to understand the states of charged particles in an external magnetic field 퐵⃗ , the so-called Landau levels that the reader has probably seen before in discussions of electrodynamics, accelerator science, or condensed matter physics. We will include a discussion of Landau levels in Appendix A for completeness and the convenience of the reader.

We begin with our discussion in Sec. 1 of the massless Dirac equation in 3+1 dimensions. We use the chiral representation Eq. 1.2 and write

휓 휓 = ( +) 4.2 휓−

5 where the chiral spinors 휓± are eigenstates of 훾 . Beginning with Eq. 1.8a, we couple the fermions to an external electromagnetic field 퐴휇 [7],

† 휇 † 휇 ℒ = 푖휓+휎̅ 퐷휇휓+ + 푖휓−휎 퐷휇휓− 4.3

휇 푖 휇 푖 where 퐷휇 = 휕휇 − 푖푒퐴휇 and 휎 = (1, 휎 ) and 휎̅ = (1, −휎 ). To describe an external magnetic field, we take 퐴0 = 0 and 퐴 ≠ 0. Then the equation of motion that follows from the Lagrangian

Eq. 4.3 for 휓+ is,

푖 푖휕0휓+ = 푖휎 퐷푖휓+ 4.4a and a similar equation for 휓− that we will discuss after focusing on 휓+. We identify the Hamiltonian in Eq. 4.4a,

푖 퐻 = −푖휎 퐷푖 = (푝 − 푒퐴 ) ∙ 휎 4.4b

1 The spin of the fermion is 푆 = 휎 (we are taking ℏ = 푐 = 1 again), and we can compute 퐻2 2 using algebra illustrated in the previous lecture, Sec. 2, Eq. 2.4-2.6b, and find

퐻2 = 휋⃗ ∙ 휎 휋⃗ ∙ 휎 = 휋⃗ 2 − 2푒퐵⃗ ∙ 푆 4.4c where 휋⃗ = 푝 − 푒퐴 . This Hamiltonian has some familiar features. The first term in Eq. 4.4c is the Hamiltonian for a non-relativistic, structureless, charged particle in a magnetic field. The second term expresses Zeeman splitting between spin states. Choose the 퐵⃗ field in the z-direction, 퐵⃗ =

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(0,0, 퐵). If we choose the Landau gauge where 퐴 = (0, 푥퐵, 0), then 퐵⃗ = ∇⃗⃗ × 퐴 produces 퐵⃗ = (0,0, 퐵) as required. Now the Hamiltonian reduces to,

2 2 2 2 퐻 = 푝푥 + (푝푦 − 푒퐵푥) + 푝푧 − 2푒퐵푆푧 4.5

The first two terms lead to Landau levels. We review the relevant aspects of Landau levels in Appendix A and use those results here. In particular, in Appendix A we solve the physics in the transverse plane (푥, 푦) and the energy levels of Eq. 4.5 prove to be,

2 2 퐸 = 푒퐵(2푛 + 1) + 푝푧 − 2푒퐵푆푧 4.6 where n labels the Landau levels 푛 = 0,1,2, … We will also need the degeneracy of each Landau level. This is also calculated in Appendix A: the degeneracy per unit transverse area is 푒퐵/2휋. The interplay between the harmonic oscillator levels and the Zeeman splitting is important. Note that for 푝푧 = 0, states with 푆푧 = 1/2 (aligned along 퐵⃗ ) in the 푛 = 0 Landau level have vanishing energy 퐸 = 0. They are zero modes! In addition, the next lowest level energy states consist of a doublet: the 푛 = 0 , 푆푧 = −1/2 state has the same energy as the 푛 = 1 , 푆푧 = 1/2 state. So, there is a gap ~푒퐵 between these states and the zero mode, so let’s concentrate on the zero 1 mode. These states have spin so the spinors describing them have the form, 2

휒+(푥, 푦; 푧, 푡) 0 휓+(푥, 푦; 푧, 푡) = ( ) and 휓−(푥, 푦; 푧, 푡) = ( ) 4.7 0 휒−(푥, 푦; 푧, 푡) where we have separated the 푥, 푦 dependence because of the factorization observed in Eq. A.7 of Appendix A. In particular, they are zero energy solutions of the chiral equation Eq. 4.4a. One can substitute this form into the Lagrangian density Eq. 4.3 and form the Action by integrating over 푑푥푑푦푑푧푑푡. The (푥, 푦) dependence of the wavefunction has been determined and recorded in Eq. A.7, but the t and the 푧 dependences have not yet been addressed. We obtain from the first term in Eq. 4.3,

푆+ = 푁 ∫ 푑푧 푑푡 푖 휒+̅ (휕0 − 휕3)휒+ 4.8 where 푁 is a normalization factor coming from the integration over the transverse coordinates and we have not yet included an external electric field in the action. However, Eq. 4.8 shows us that 휒+ describes right movers in the 푧 direction.

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Now we should repeat the entire discussion for 휓−. This gives another zero mode contribution, 푆−, which reads 푁 ∫ 푑푧 푑푡 푖 휒−̅ (휕0 + 휕3)휒− where we have accounted for the change in minus signs in Eq. 4.3. Taking this result and Eq. 4.8, we have the full Action describing the zero modes,

푆 = 푆+ + 푆− = 푁 ∫ 푑푧푑푡 (푖휒+̅ 퐷−휒+ + 푖휒−̅ 퐷+휒−) 4.9 where we have re-inserted the coupling of the fermion to the external field in the covariant derivatives 퐷±, 퐷± = 푖휕± − 푖푒퐴± where 휕± = 휕0 ± 휕3 and 퐴± = 퐴0 ± 퐴3. But now we recognize Eq. 4.9: the result is proportional to the 1+1dimensional Action for massless fermions coupled to an external electromagnetic field!

Now we can generalize the previous discussion of the non-conservation of the axial charge from 1+1 to 3+1 dimensions. That discussion implies the violation of the four dimensional axial charge. To get the quantitative result, we must recall from Appendix A that the lowest Landau level has a degeneracy per transverse area of 푒퐵/2휋. So, the total rate of violation of the axial charge density, 휌̇퐴, in 3+1 dimensions is the 1+1 dimensional result times the density of states factor 푒퐵/2휋,

푒퐸 푒퐵 푒2 휌̇ = ( ) ( ) = 퐸⃗ ∙ 퐵⃗ 4.10 퐴 휋 2휋 2휋2 where we have written the electric and magnetic field dependence in the final result as 퐸⃗ ∙ 퐵⃗ : it is only the component of the electric field along the 퐵⃗ that contributes to the 1+1 dimensional effect. As noted earlier, this result means that the local non-conservation of the chiral vector current is,

푒2 휕 푗휇 = 퐹 ∙ 퐹̃ 3.2 휇 퐴 16휋2

5. The Importance of the Chiral Anomaly. The chiral anomaly is important in high energy physics, mathematical physics and condensed matter physics as well. We learned above that quantum effects can violate classical conservation laws. The counting arguments presented here illustrate the relevance and importance of the Dirac sea. The reader should be aware of other more rigorous derivations of the chiral anomaly. This work includes perturbation theory, explicit solutions of model field theories, the non-invariance 16 of the measure in the path integral approach of field theories to chiral rotations, and connections to mathematical physics topics, such as topology, of non-abelian gauge theories, index theorems of mathematics, etc. It’s quite an impressive list and it indicates the wide ranging influence of the chiral anomaly in condensed matter physics, in the standard model of high energy physics, models beyond the standard model, string and M-theory, etc. We will see the chiral anomaly again when we study 1+1 model field theories in the next lecture.

One of the impacts of the chiral anomaly in high energy physics is its prediction of the electromagnetic decay of the neutral pion to two photons [1]. To get the rate correct one finds the need for the three colors of quarks in quantum chromodynamics! Because the chiral anomaly is an exact result(!) and because it involves the counting of fermions, it provides compelling evidence for the very successful standard model of high energy physics. The violation of the conservation of the axial current in quantum chromodynamics is also an ingredient in the calculation of the mass of the particle 휂′ and the problem of the ninth axial current [5,8]. That discussion goes well beyond the illustrations here; important ingredients include the color- carrying elementary constituents of matter, the gluons of quantum chromodynamics, non-abelian dynamics and its vacuum state. The reader is encouraged to look into these topics but they require more background than provided in this introductory lecture.

In the next lecture we will turn to related physical phenomena in 1+1 dimensional model field theories. We will look at connections between bosons and fermions in 1+1 dimensions, confinement, and more in 1+1 dimensional quantum electrodynamics.

6. Appendix A. Landau Levels in 3+1 Dimensions. Let’s review the topic of Landau levels [9] and extract results we need to establish the chiral anomaly in 3+1 dimensions in Sec. 4. We begin with the non-relativistic quantum Hamiltonian,

1 2 퐻 = (푝 − 푞퐴 ) A.1 2푚

Let the external magnetic field point in the 푧-direction, 퐵⃗ = (0,0, 퐵), so there is translation and rotational symmetry in the (푥, 푦) plane. However, when we choose a vector potential 퐴 so that 퐵⃗ = ∇⃗⃗ × 퐴 these symmetries will not be apparent in the intermediate steps of the calculations. In particular, we choose 퐴푦 = 푥퐵 with 퐴푥 = 퐴푧 = 0. Now the Hamiltonian reads,

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1 2 퐻 = (푝2 + (푝 − 푞퐵푥) + 푝2) A.2 2푚 푥 푦 푧

Clearly [푝푦, 퐻] = [푝푧, 퐻] = 0 but [푝푥, 퐻] ≠ 0. So, we can choose energy eigenstates to also be eigenstates of 푝푦 and 푝푧,

휓(푥) = 푒푖푘푦푦+푖푘푧푧휒(푥) A.3

Substituting this form into the eigenfunction equation,

1 2 퐻휓(푥) = (푝2 + (푝 − 푞퐵푥) + 푝2) 휓(푥) = 퐸휓(푥) A.4 2푚 푥 푦 푧 we find the eigenvalue equation for 휒(푥),

1 푚휔2 2 ℏ2푘2 ( 푝2 + 퐵 (푥 − 푘 푙2 ) ) 휒(푥) = (퐸 − 푧 ) 휒(푥) A.5 2푚 푥 2 푦 퐵 2푚 where 휔퐵 = 푞퐵/푚 is the familiar cyclotron frequency and 푙퐵 is the magnetic length, 푙퐵 =

√ℏ⁄푞퐵. We will see directly that ℏ휔퐵 sets the scale of the energy splitting between energy eigenstates of Eq. A.5 and 푙퐵 = √ℏ⁄푞퐵 sets the scale of the size of the wavefunction in the 푥 direction. In particular, identify Eq. A.5 as the differential equation for a 1-dimensional harmonic 2 oscillator centered at 푥푐 = 푘푦푙퐵. We read off the energy eigenvalues and eigenstates of Eq. A.5 from our knowledge of the 1-dimensional harmonic oscillator,

1 ℏ2푘2 퐸 = ℏ휔 (푛 + ) + 푧 , 푛 = 0,1,2, …. A.6 퐵 2 2푚

And the energy eigenfunctions are also familiar,

2 2 2 푖푘푦푦+푖푘푧푧 2 −(푥−푘푦푙퐵) ⁄2푙퐵 휓푛,푘(푥, 푦, 푧) = 푁푒 퐻푛(푥 − 푘푦푙퐵) 푒 A.7 where 퐻푛 is a Hermite polynomial which is multiplied by the universal Gaussian in Eq. A.7,

2 2 2 푒−(푥−푘푦푙퐵) ⁄2푙퐵. Clearly a large magnetic field 퐵⃗ = (0,0, 퐵) forces the fermions into “tight”

2 distributions with width 푙퐵 = √ℏ⁄푞퐵 in the 푥-direction end centered at 푥푐 = 푘푦푙퐵. If we had 푦퐵 푥퐵 chosen a “symmetric” gauge where 퐴 = (− , , 0), the (푥, 푦) symmetry of the problem 2 2 would be more apparent, eigenfunction by eigenfunction. However, the gauge choice here will be sufficient and has particularly simple analytics. Using the energy eigenfunction Eq. A.7 one

18 can construct functions with more interesting 푦 dependences. The large degeneracy of the states Eq. A.7 makes this possible as we will now discuss.

Since the dynamics in the 푧-direction is unaffected by translations in the 푧-direction, we can restrict our analysis to the case 푘푧 = 0. The energy Eq. A.6 is then given by the 1- dimensional harmonic oscillator with a frequency 휔퐵,

1 퐸 = ℏ휔 (푛 + ) A.8 푛 퐵 2

These are the famed “Landau” levels. These levels clearly have a huge degeneracy: 퐸푛 does not depend on 푘푦. To count states let’s consider a rectangle in the (푥, 푦) plane of dimension 퐿푥 and

푖푘푦푦 퐿푦. The functions 휓푛,푘 are simple plane waves in the 푦-direction, 푒 . In particular, the wavefunctions in the (퐿푥, 퐿푦) box should satisfy the boundary condition,

휓(푥, 푦 + 퐿푦, 푧) = 휓(푥, 푦, 푧) A.9a which implies

푒푖푘푦퐿푦 = 1 A.9b

So the wave number 푘푦 is quantized in units of 2휋⁄퐿푦. But the wavefunctions are not plane

2 waves in the 푥-direction: they are Gaussians of width 푙퐵 = √ℏ⁄푞퐵 localized at 푥푐 = 푘푦푙퐵. But 푥 ranges from 0 to 퐿푥 in the box used for counting, so the largest possible value of 푘푦 for 푥 inside 2 the box is 퐿푥⁄푙퐵. So, the number of states in each Landau level in the box is,

푑푦 푑푘 퐿 퐿 ⁄푙2 푞퐵퐿 퐿 푁 = ∬ 푦 = 푦 ∫ 푥 퐵 푑푘 = 푥 푦 A.9c 2휋 2휋 0 푦 2휋

푞퐵 So, the number of Landau levels per transverse area is . We will use this counting argument to 2휋 make the 3+1 dimensional chiral anomaly out of 1+1 dimensional anomalies in the text.

Let’s look briefly at some of the magnitudes here which are relevant experimentally. For a −8 1.0 Tesla B field, the magnetic length is very small, 푙퐵 ≈ 2.5 × 10 m. From Eq. A.9c, the 2 10 number of degenerate states per Landau level in a region 퐿푥퐿푦 = 1.0. cm. is roughly 10 for a magnetic field 퐵 = 0.10 T. This huge degeneracy is responsible for much of the physics of the Hall effect in condensed matter physics.

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References 1. J. Steinberger, PR 76, 1180 (1949). 2. J. Schwinger, PR 128(5), 2425 (1962). 3. A. Casher, J.B. Kogut and L. Susskind, PRL 31, 792 (1973); PR D10, 732 (1974). 4. J.B. Kogut and L. Susskind, PR D 10, 3468 (1974). 5. J.B. Kogut and L. Susskind, PR D 11, 3594 (1975). 6. H.B. Nielsen and M. Ninomiya, PL B130, 389 (1983). 7. L. Alverez-Gaume and M.A. Vazquez-Mozo, An Invitation to Quantum Field Theory, Springer-Verlag, Berlin, 2012. 8. G. ‘tHooft, PR D14, 3432 (1976). 9. R. Shankar, Principles of , Springer Science+Business Media, New York, 1980.

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