The vertex algebra of one free boson: Part 5: Light-cone coordinates.
Channel: Edwin Steiner
Date: 08/04/2022

This video is the fifth part of a mini-series about the
appearance of the vertex algebra of one free boson from
the physical point of view.

In this part, we define light-cone coordinates and use
them to get more insight about the left- and right-moving
modes and about the zero mode.

The video is aimed at people who have some knowledge
of calculus and linear algebra but who don't necessarily
have much background in physics.

0:00 Introduction
0:56 Comparison with Zwiebach's String Theory book
4:41 Another path to the left/right-movers
6:04 Factoring a differential operator
10:18 Light-cone coordinates
12:19 The field expressed in the new coordinates
14:25 Derivatives in u and v
18:09 EOM in light-cone coordinates
19:19 A family of solutions
21:56 Generality of the solutions
26:59 Graphical summary of the proof
28:51 The solution in x and t
30:41 Ambiguity of the decomposition
31:11 The periodicity constraint
35:57 Implications of periodicity
37:05 Periodicity of derivatives
39:22 Mode expansions of the derivatives
42:15 Reality conditions
44:32 Integrating the left-/right-movers
48:41 Constraint on the zero mode
50:53 Completing the expansion
52:49 Expansion in x and t
54:10 Comparing results
55:55 Final clean-up
56:23 Summary
57:59 Fixing the field for all times
59:24 Outro

[r2]

Part 1: https://youtu.be/uukBDltkDz0
Part 2: https://youtu.be/-rzYFMwdNww
Part 3: https://youtu.be/_kV8PFCr0xw
Part 4: https://youtu.be/Qil91hXxXxw

Barton Zwiebach's book: https://www.amazon.com/Course-String-Theory-Students-International-dp-0521176336/dp/0521176336/ref=mt_other?_encoding=UTF8&me=&qid=

My series about quantum mechanics:
Episode 0:   https://youtu.be/aS-o2ysf_O4
Episode 1:   https://youtu.be/85TUPcL7aWQ
Episode 2:   https://youtu.be/7C0ggtMLRFs
Episode 3.1: https://youtu.be/S6Z4FILJVzM

My channel: https://www.youtube.com/c/EdwinSteiner
Physics videos playlist: https://www.youtube.com/playlist?list=PLaMCxumvPhAFb6JEmIDQx3w7XaN0p_YJ2

Support me on SubcribeStar: https://www.subscribestar.com/edwinsteiner