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The Chiral Anomaly and the Dirac Sea

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The Chiral Anomaly and the Dirac Sea Supplemental Lecture 20 The Chiral Anomaly and the Dirac Sea Introduction to the Foundations of Quantum Field Theory 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 quantum electrodynamics. 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 fermion 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: Special Relativity, 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 Dirac Equation, Chirality, Massless Fermions, The Dirac Sea, Chiral Symmetry, Axial Anomaly, Quantum Electrodynamics in 1+1 and 3+1 Dimensions, Landau Levels. --------------------------------------------------------------------------------------------------------------------- 1 Contents 1. The Dirac Equation yet again. ................................................................................................ 2 2. The Vector and Axial Vector Currents. .................................................................................. 5 3. The Chiral Anomaly. .............................................................................................................. 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 gamma matrices, 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 0 1 푖 0 휎푖 훾 = ( ), 훾 = ( ) 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. 2 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 Lorentz group 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 Action, 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 푖 0 1 0 휎푖 −휎푖 0 훾 훾 = ( ) ( ) = ( ) 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 4 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 conservation law 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 mass 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 electromagnetic field, 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.
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