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Notes on Gauge Theories Notes on Gauge Theories Leonardo Almeida Lessa Abril de 2019 Contents 1 Motivating Examples 2 1.1 Classical Electromagnetism . .2 1.2 Quantum Electrodynamics . .5 1.3 Dirac Monopole . .6 1.3.1 Magnetic Charges . .6 1.3.2 Electromagnetic Interaction in Quantum Mechanics .8 1.3.3 QM and the Dirac Monopole . .9 2 Fibre Bundles 11 2.1 Motivation: Tangent Bundle . 11 2.2 Fibre Bundles . 13 2.2.1 Triviality of Bundles . 16 2.3 Principal Bundles . 17 2.3.1 Associated Vector Bundle . 18 3 Connections in Fibre Bundles 20 3.1 Fundamental Vector Field . 20 3.2 Connection . 22 3.3 Parallel Transport . 22 3.4 Curvature . 24 3.5 Local Forms . 24 3.5.1 Gauge Potential . 24 3.5.2 Field Strength . 27 3.6 Covariant Derivative . 28 4 The Examples from Another View 30 4.1 QED and Yang-Mills Gauge Theories . 30 4.2 Dirac Monopole . 31 Here are some notes on gauge theories. We begin by discussing some examples from Physics and their common features. Then, we detour to the theory of fibre bundles and connections on principal bundles, which form the mathematical basis of gauge theory. Finally, we review our previous examples in this new framework. 1 Leonardo A. Lessa I personally find fascinating that many phenomena that appear uncorre- lated at first can be explained in an unified way. In the case of gauge theory, we see another intersection of Geometry and Topology with Physics. The text is heavily based on “Geometry Topology and Physics”, by Mikio Nakahara [1]. Other references are included at the end. 1 Motivating Examples The notion of gauge appears when we have more degrees of freedom in the description of the theory than there is in the physics of the problem. Sometimes, the underlying mathematical objects, confined to our abstraction of the world, are redundant and unphysical, even though derived observable results are unambiguous. We artificially distinguish quantities that are physically equivalent. Thus, the observables of our theory have to be symmetric upon gauge transforma- tions in these equivalent quantities. They possess gauge symmetry described by gauge groups, similarly to the familiar symmetries of spacetime, but qual- itatively different. A gauge transformation changes our description of the modeled system while a spacetime symmetry changes the modeled system itself, although not modifying the physical laws. Before presenting the mathematical details, we will now discuss some examples. Our goal now is to create a clear image of what we mean by a gauge theory, to then formalize it in more generality. 1.1 Classical Electromagnetism The first contact one may have with gauge transformations is with Elec- tromagnetism, whose dynamics are determined by Maxwell’s equations. In natural (Lorentz-Heaviside) units, by which we set c = 0 = µ0 = ~ = 1, Maxwell’s equations can be written as @B r × E = − ; r · B = 0; (1.1) @t @E r · E = ρ, r × B = + J: (1.2) @t Poincaré’s Lemma applied to the homogeneous equations (1.1) tells us there exists at least locally1 a scalar field V and a vector field A, also known as potentials, satisfying (1:1) Exercise 1.1 3 In R vector calculus, Poincaré’s @A Lemma amounts to E = − − rV; (1.3) @t r × v = 0 ) v = rf; B = r × A: (1.4) r · v = 0 ) v = r × u; for some locally defined real Although we are guaranteed of their existence, the potentials V and A function f and vector field u. are far from unique. For every arbitrary function Λ, we may transform the Adapt this to prove equations (1.3) and (1.4). 1We will see later that the Dirac monopole is an example where the vector field A cannot be defined in the whole space. This has to do with the topology of the problem and has interesting consequences. 2 1.1 Classical Electromagnetism Leonardo A. Lessa potentials by @Λ V 0 = V − ; (1.5) @t A0 = A + rΛ; (1.6) but still get the same electric E0 = E and magnetic fields B0 = B(1:2). Since Exercise 1.2 E and B control the dynamics of charged particles via Lorentz force, we Verify this arrive at the same physics after this transformation. Indeed, this is a case of gauge transformation. In the manifestly covariant formulation of Electromagnetism [2], the po- tentials V and A are part of a (four-)potential Aµ = (V; A); (1.7) as well as the four-current J µ = (ρ, J), which transform like a four-vector under Lorentz transformations. Similarly, the electric and magnetic fields can be grouped into a antisymmetric (0, 2)-tensor Fµν , called the Faraday tensor or electromagnetic field strength tensor. The components of Fµν in a reference frame where the electric and magnetic fields at some point are E = (Ex;Ey;Ez) and B = (Bx;By;Bz) are, in matrix notation, 2 3 0 Ex Ey Ez 6−Ex 0 −Bz By 7 Fµν = 6 7 : (1.8) 4−Ey Bz 0 −Bx5 −Ez −By Bx 0 We immediately see that the transformation laws of the electric and magnetic fields under change of reference frame are not as simple as with the four- potential Aµ, since they are mixed up in the coordinates of the Faraday tensor Fµν . Equations (1.3) and (1.4) can be written in terms of the four-potential (1.7) and the Faraday tensor (1.8) via Fµν = @µAν − @ν Aµ; (1.9) where we have used Einstein notation to sum repeated indices and lowered the potential with the Minkowski metric ηµν = diag(−1; +1; +1; +1) as such: ν Aµ = ηµν A = (−V; A): Furthermore, the gauge transformation of (1.6) and (1.5) simplifies to 0 Aµ = Aµ + @µΛ: (1.10) Since the electric and magnetic fields remain the same under a gauge transformation, the same happens with Fµν . Thus, the Faraday tensor is gauge invariant. Accordingly, we can write Maxwell’s equations in terms of Fµν and they are also gauge invariant: µν @µ(∗F ) = 0; (1.11) µν ν @µF = J ; (1.12) 3 1.1 Classical Electromagnetism Leonardo A. Lessa where ∗F is the dual field strength tensor, defined by ∗F µν = 1 µναβF (1:3). Exercise 1.3 2 αβ Interpret equation (1.11) know- We can solve equations for V and A and work with them, but the final ing that the dual field strength observable results have to be gauge independent. This is not a negative ∗F µν is F µν with E and B swapped. aspect of our theory, quite the contrary. The labor of dealing with non- gauge-invariant quantities like Aµ is compensated by the manifest Lorentz invariance of the theory. Unitarity and locality are also manifest when we keep this redundancy in the quantum realm (See [8]). Even fixing the gauge is a useful idea. It is often the case that simplifi- cation in the calculation occurs when we choose a particular gauge. In the case of Electromagnetism, we can set up a differential equation for Λ so as the potentials have some property we want, as long as it doesn’t contradict Maxwell’s equations. Figure 1: Illustration of the space of all potentials Aµ. A point in this gauge is connected to other points (dashed lines) by gauge transformations, and to fix a gauge is to choose independent representatives among those (bold line). For example, we may choose the Lorenz Gauge @ Aµ = 0(1:4) to arrive Exercise 1.4 µ What differential equation does at a wave equation for the potential, Λ have to satisfy? ν (1.12) µν J = @µF ; µ ν ν µ = @µ(@ A − @ A ); µ ν = @µ@ A : (1.13) Although the potential Aµ has four components, by solving (1.13), we get two independent solutions, corresponding to the possible polarizations of a electromagnetic wave. This is also a manifestation of gauge symmetry. 4 1.2 Quantum Electrodynamics Leonardo A. Lessa 1.2 Quantum Electrodynamics Having already seen how classical electromagnetism has gauge symmetry, it may not be clear what is the corresponding gauge group. In Quantum Elec- trodynamics (QED), however, the gauge group appears naturally. In fact, we won’t even quantize anything, (fortunately!). The only “new” concept that comes from the quantum world is spinors. If the reader is unfamiliar with this formalism, we recommend [6]. The main idea is to express our previous results in the Lagrangian for- malism. The potential Aµ(x) is our dynamical field, whose dynamics is controlled by the following Lagrangian 1 L = − F F µν − A J µ: (1.14) EM 4 µν µ More precisely, by the Euler-Lagrange equations associated to L (1:5) EM Exercise 1.5 @L @ @L Why is (1.11) automatically sat- isfied? − ν = 0: (1.15) @Aµ @x @(@ν Aµ) One may argue that LEM is not gauge invariant, because of the term µ R 4 AµJ , and it is true. However, the action SEM = LEMd x is gauge in- variant, because the gauge transformation of SEM gives off a surface term, which goes to zero as we integrate over the entire spacetime M = R4: Z Z µ 4 µ 4 − AµJ d x ! − (Aµ + @µΛ)J d x; M M Z Z µ 4 µ µ 4 = − AµJ d x − [@µ(ΛJ ) − Λ@µJ ]d x; M M Z Z = − AµJ µd4x − ΛJ µd3x; M @M Z = − AµJ µd4x; M where in the third line we used that J µ is conserved and in the fourth, that the integrand goes to zero at infinity. Since the Euler-Lagrange equations (1.15) come from the variation of the action, then we still have a gauge invariant theory.
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