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Special Relativity and Classical Field Theory Special Relativity and Classical Field Theory Notes on Selected Topics for the Course \Klassische Feldtheorie" Matthias Blau Version of May 5, 2021 Contents 1 Introduction 4 1.1 Overview . 4 1.2 Notation and Conventions . 5 2 Minkowski Space(-Time) and Lorentz Tensor Algebra 7 2.1 Einstein Principle of Relativity as an Invariance Principle . 7 2.2 Warm-Up: Euclidean Geometry, Euclidean Group and the Laplace Operator . 8 2.3 From Invariance of to Minkowski Geometry and Lorentz Transformations . 14 2.4 Example: Lorentz Transformations in (1+1) Dimensions (Review) . 16 2.5 Minkowski Space, Light Cones, Wordlines, Proper Time (Review) . 20 2.6 Lorentz Vectors and Minkowski Geometry . 22 2.7 Lorentz Scalars and Lorentz Covectors . 24 2.8 Higher Rank Lorentz Tensors . 27 2.9 Lorentz Tensor Algebra . 28 2.10 Lorentz Tensor Fields and the Lorentz-invariance of Tensorial Equations . 32 2.11 Lorentz-invariant Integration . 33 2.12 Lorentz-invariant Differential Operators . 34 3 Lorentz-Covariant Formulation of Relativistic Mechanics 37 3.1 Covariant Formulation of Relativistic Kinematics and Dynamics . 37 3.2 Energy-Momentum 4-Vector . 39 3.3 Minkowski Force? (how not to introduce forces and interactions) . 41 3.4 Lorentz-invariant Action Principle for a Free Relativistic Particle . 42 3.5 Noether Theorem and Conservation Laws (Review) . 47 3.6 Noether Theorem for the Relativistic Particle . 50 4 Lorentz-Covariant Formulation of Maxwell Theory 54 4.1 Maxwell Equations (Review) . 54 4.2 Lorentz Invariance of the Maxwell Equations: Preliminary Remarks . 55 4.3 Electric 4-Current and Lorentz Invariance of the Continuity Equation . 56 4.4 Inhomogeneous Maxwell Equations I: 4-Potential . 57 4.5 Inhomogeneous Maxwell Equations II: Maxwell Field Strength Tensor . 58 4.6 Homogeneous Maxwell Equations I: Bianchi Identities . 61 1 4.7 Homogeneous Maxwell Equations II: Dual Field Strength Tensor . 62 4.8 Maxwell Theory and Lorentz Transformations I: Lorentz Scalars . 65 4.9 Maxwell Theory and Lorentz Transformations II: Transformation of E;~ B~ . 67 4.10 Example: The Field of a Moving Charge (Outline) . 68 4.11 Covariant Formulation of the Lorentz Force Equation . 70 4.12 Action Principle for a Charged Particle coupled to the Maxwell Field . 72 5 Classical Lagrangian Field Theory 76 5.1 Introduction . 76 5.2 Variational Calculus and Action Principle for Fields . 76 5.3 Poincar´e-invariant Actions for Real Scalar Fields . 80 5.4 Actions and Variations for Complex Scalar Fields . 85 5.5 Action for Maxwell Theory . 87 6 Symmetries and Lagrangian Field Theories 91 6.1 Noether's 1st Theorem: Global Symmetries and Conserved Currents . 91 6.2 Gauge Invariance and Minimal Coupling . 94 6.3 Spacetime Symmetries and Variations I: Translations . 98 6.4 Spacetime Translation Invariance and the Energy-Momentum Tensor . 101 6.5 Energy-Momentum Tensor for a Scalar Field . 103 6.6 Energy-Momentum Tensor for Maxwell Theory . 105 7 Symmetries and Gauge Theories: Selected Advanced Topics 113 7.1 Higher Dimensional and Higher Rank Generalisations of Maxwell Theory . 113 7.2 Abelian Chern-Simons Gauge Theory . 115 7.3 Spacetime Symmetries and Variations II: Lorentz Transformations . 117 7.4 Some Properties of the Gauge Covariant Derivative . 120 7.5 Spontaneously Broken Symmetries (Goldstone and Higgs): Toy Models . 121 8 General Structure of Theories with Local Symmetries: Noether's 2nd Theorem 126 8.1 Maxwell Theory Revisited . 126 8.2 Noether Charges for Local Symmetries are Identically Zero . 130 8.3 Noether's 2nd Theorem . 130 8.4 Local Symmetries lead to Identically Conserved Noether Currents . 133 2 8.5 Converse of Noether's 2nd Theorem . 135 8.6 Epilogue and Outlook . 136 3 1 Introduction 1.1 Overview These are notes on selected topics covered in the 3rd year (6th semester) course \Klassische Feldtheorie". Prerequisites for this course are: • Basic Calculus and Linear Algebra • Basics of Special Relativity • Maxwell Theory (Electrodynamics) • Lagrangian Mechanics and Action Principle In general the new subjects covered in this course are (usually a strict subset of) those indicated in the table of contents: 1. At the beginning of the course I give a lightning review of the physical foundations of special relativity (definition of inertial systems, Galilean relativity principle, propagation of light, Maxwell, Michelson-Morley, Lorentz, Einstein etc.). However, since this is 1st year undergraduate material, I do not cover it in these notes, and I assume familiarity with these topics. 2. The first aim of these notes is to arrive at a Lorentz covariant formulation of special relativity and the laws of classical phyics (primarily mechanics and electrodynamics or Maxwell theory) in terms of what are known as Lorentz tensors. After all, special relativity is (regardless of what you may have been taught) not funda- mentally a theory about people changing trains erratically, running into barns with poles, or doing strange things to their twins; rather, it is a theory of a fundamental symmetry principle of physics, namely that the laws of physics are invariant under Lorentz transfor- mations. They should therefore also be formulated in a way which makes this symmetry manifest. This is achieved by the use of objects which transform in a simple (multi-)linear way under Lorentz transformations, and such objects are called Lorentz tensors. 3. The second aim of these notes is to provide an introduction to classical Lagrangian field the- ory, in order to introduce some fundamental concepts involved in the modern formulation of theoretical physics, like the Noether theorem for field theories, the energy-momentum tensor, and the idea of minimal coupling. 4. Moreover, I usually end with some remarks and reflections on gravity and relativity, as an outlook on general relativity. This is described in detail in the first part of my (voluminous) Lecture Notes on General Relativity and will therefore also not be covered in these notes. 5. Sections 7 and 8 contain supplementary and more advanced material that will not be covered in the course. 4 1.2 Notation and Conventions Please do not be scared off by this section. Notation is mainly a book-keeping device, a language that one needs to get used to and that one learns by using it. • Good notation is one that is at the same time informative, unambiguous (in the situation at hand), and easy to use. • Bad notation is one in which objects that appear are undefined, ill-defined, or one that is uninformative or difficult to understand or remember and therefore difficult to use. How detailed or specific the notation should be will very much depend on the context (and the person using it) and should therefore permit a certain amount of flexibility: it should be sufficiently precise to be able to perform the task at hand in an efficient and accident-free manner, but it does not have to be more precise than that. Having said this, here are some notational conventions that I will (try to more or less consistently) adhere to in the following: • As is common in physics, instead of using some abstract coordinate-free notation (beloved by mathematicians) we will usually work in components that refer to a specific (orthonor- mal) basis or (Cartesian) coordinate system. I usually use lower-case Roman letters from the beginning of the alphabet (a; b; c; : : :) for spacetime indices, and Roman letters from the middle of the alphabet i; j; k; : : : for spatial indices. In particular, Cartesian coordinates for a point x of the Euclidean space R3 are denoted by ~x = (xi) = (x1; x2; x3) with i; j; : : : 2 f1; 2; 3g ; (1.1) and inertial spacetime coordinates of an event in Minkowski spacetime will be denoted by (xa) = (x0 = ct; xi) ≡ (x0; ~x) with a; b; : : : 2 f0; 1; 2; 3g : (1.2) • You see that, as is customary, we have already tacitly (and now explicitly) identified a point x in R3, given by the coordinates (xi) = (x1; x2; x3), with the position vector ~x (pointing from the origin to the point x). Once one has decided to denote the components of the position vector ~x by xi, it is reasonable to extend this notation to other vectors ~v 2 R3, i.e. to denote its components by ~v = (vi) = (v1; v2; v3), with \upper" indices. • We will often deal with (linear) transformations of coordinates or vectors. In this case, one needs a notation to distinguish the new from the old coordinates. Here there are several options, and which one is the most useful may depend on the circumstances (recall the discussion above), but may also be a question of personal taste. { In vectorial notation, one can try to distinguish the new coordinates from the old coordinates ~x, by writing something like ~x0 or or ~x¯, but this can quickly become 5 somewhat inconvenient (and is also not ideal on the blackboard, unless the backboard is really clean). Thus, in vectorial notation, it is often more convenient to use a new letter for the new coordinates, such as ~y or ~z etc. This is at least easy to read. { In components, with initial coordinates xi, one can also follow the above convention and simply denote the new coordinates by yi. However, in that case it is also occa- sionally convenient to just use \barred" or \primed" x-coordinates instead, such as x¯i (which is easy to read). For certain purposes, it is also useful to employ a different kind or range of indices for different coordinate systems, say xi; xj;::: for the original coordinates, and something likex ¯m; x¯n;::: or ym; yn;::: for the new coordinates. This has the advantage that writing something like vi makes it clear that these are the components of a vector ~v with respect to the original basis, while something like vm orv ¯m would then obviously refer to the coordinates of the same vector ~v with respect to the new basis.
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