Show that indeed Lμν^e=Lνμ^e and that q^μLμν^e=q^νLμν^e=0 ....

Name: Problem 9, Chapter 8: Quarks and leptons: introductory course in modern particle physics
Uploaded: 2021-07-28T05:05:34-08:00
Duration: 7 min 28 s
Channel: Aidan Mcnabb
Description: Show that indeed Lμν^e=Lνμ^e and that q^μLμν^e=q^νLμν^e=0 .

Key Concepts

Tensor Symmetry

In field theory, a tensor is said to be symmetric if its components do not change when the indices are interchanged. For the electron tensor, L???????, proving its symmetry involves showing that the mathematical operations and physical interactions from which it is derived treat both Lorentz indices on an equal footing. This symmetry is crucial for ensuring that the physical observables computed from the tensor behave consistently under Lorentz transformations.

Current Conservation

Current conservation is a fundamental principle in electrodynamics and quantum field theory, which states that the divergence of the electromagnetic current must vanish. In this context, showing that q?L??????? = 0 (and similarly for the other index) is a demonstration of this conservation law. It implies that there is no net accumulation of charge, ensuring that the electron current used to construct the tensor is conserved in all interactions.

Gauge Invariance

Gauge invariance is a key principle in gauge theories like quantum electrodynamics, where the physical outcomes remain unchanged under local transformations of the phase of the electron field. This invariance leads to strong constraints on the form of interaction terms and observables, such as ensuring that the contraction of the momentum transfer vector with the electron tensor is zero. This requirement helps maintain the consistency and renormalizability of the theory.

Transcript

00:01 Hello and welcome to problem 27 of chapter 3, section 4.

00:04 In this problem, we're trying to find the run skin of e to the lambda t times cosine of and e to the lambda t times sine of muti.

00:13 If in your problem there is a 2 here instead of a lambda, note that it's just as hypo.

00:18 All right, so let's start off by talking about what is the run skin.

00:22 Well, the run skin is the 2x2 determinant of the, let's recall this first part y1, in this second part, y2.

00:31 What is just the determinants of y1, y2, as well as their derivatives, y1 prime, and y2 prime.

00:41 Great.

00:41 So let's calculate what y1 prime and y2 prime are.

00:46 We're going to have to use the product rule for this.

00:51 So let's start off with y1 prime.

00:56 Let's start off with, well, we know the product rule is the derivative of the first one, the second plus the derivative of the second times the first.

01:05 So it doesn't really matter which derivative you take first.

01:08 I'm going to start off with lambda e to the lambda t times cosine of muti.

01:16 And we'll subtract that from mu, sine of muti times e to the lambda t.

01:28 Great.

01:29 And then we're going to do the same thing with by 2 prime.

01:33 So we'll get lambda e to the lambda t times sine of mu t.

01:47 We'll add this to e to the lambda t times cosine it.

01:55 Well, times mu, cosine of mu t.

02:05 Great.

02:06 So now we can calculate the 2x2 determinant.

02:09 And if we're doing that, the ronskin is just y2 prime times y1 minus y1 prime times y2.

02:24 And this is just how you calculate the 2 .2 determinant.