Covariant (Dirac method) Bosonic String Quantization | String Theory @DietterichLabs

Uploaded September 2021 | Updated September 2026, 2 hours ago

In this video, I cover the covariant Dirac method for first quantizing the free bosonic string, sometimes called the "old covariant" method.

My Quantum Field Theory Lecture Series:
youtube.com/playlist?list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou

Making this video was a bigger job than any I had made yet, so if you know anyone who would like it, please don't hesitate to share it with them.

There is a typo in this video at 49:31. I meant to write and say "and h_γδ describes a massive analogue of a graviton (the traceless part of h_γδ) and a massive scalar (the trace of h_γδ)." The massless case shows up in the closed string.

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$Did I Get This Right? Tell Me In The Comments (see description for corrections/additional details) Attempting To Break Down The Quantum Many Body Problem | Schrodinger Equation | Dirac Equation In this video, I try to break apart the multi-electron atom many body problem in quantum mechanics. I attempt to do this using kind of a weird ansatz that I constructed specifically for this application. Let me know what you think of my attempt in the comments! There is a typo in the spherical harmonics normalization constant. In the second fraction under the square root, the 1s are supposed to be ls. Also, its worth noting that if the general solution is right, you could generalize to an arbitrary quantum Coulombic many-body problem, by just including coulomb potential terms with the appropriate sign and charge magnitude, and then propagating that accordingly throughout the solution. One could also include no stationary charges. This is just a change of parameterization. Pairings involving differing masses requires an obvious adjustment of the coordinates to resolve the kinetic terms in the same way. 3/19/2023: I realized recently that I know longer remembered how I justified the formulas I gave in this video for the hatted quantum numbers. In the video, I gave no detailed explanation. I simply said something about normalizability. I therefore decided to sit down and work it out again. I found that the formulas I gave in the video are wrong, and the fully detailed justification (which I meticulously recorded this time) for the justification for replacement formulas leverages much more than normalizability. The following link takes you to a page where I have a new educated guess for the correct formulas, and their justification: https://twitter.com/DietterichLabs/status/1638026174969389056 3/20/2023: It is worth noting that I only chose the hydrogen-like solutions for phi as a starting point. I figured that I might find solutions with the right properties within (a generalized set) of them. Had this not worked, I would have looked at other solutions. I did not assume that a solution form from a bound state problem would necessarily be the correct idea for a repulsive potential. 8/7/2023: In the general multi-electron atom solutions that I gave to the Schrodinger and Dirac equations, I included a number of summation and product symbols, which are supposed to generate all of the correct factors and terms. Im not entirely sure that they do so perfectly, but the intended design should be clear. In the Schrodinger case, there are the usual H-atom factors for each nucleus-electron interaction, as well as a difference-coordinate H-atom factor and a corresponding greens function term for each electron-electron interaction. In the Dirac equation, the greens function terms are unnecessary. 1/26/2024: It turns out that these new eigenvalues are also wrong. Here is a link to an updated video: https://youtu.be/-xLhrYA6TNU Here is the link to my video on solving the Dirac equation for the hydrogen atom: https://www.youtube.com/watch?v=jWYtP-rAoYA Here is my video on solving the Schrodinger equation for the hydrogen atom: https://www.youtube.com/watch?v=MaXFT-c8u1E In both of these videos, I actually address the general one-electron atom. I dont bother setting Z=1. This, of course, includes the hydrogen atom. For those who are curious, I also have a video on how to solve the Klein-Gordon equation for the one-electron atom: https://www.youtube.com/watch?v=9_3uQdF-tEs Superfluid Helium Resonance Experiment video: https://youtu.be/unUNQNmuvUQ Quantum Field Theory Lecture Series: https://www.youtube.com/playlist?list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou$ $Spherical Quantum Harmonic Oscillator | Schrodinger Equation | Quantum Mechanics In this video, I show you how to solve the Schrodinger equation for the 3D spherical quantum harmonic oscillator. One Clarification And 1 Typo: The Clarification: It is apparent from the math, but the quantum number n here must be a positive even integer. The Typo: The typo is in the normalization constant of the radial wavefunction. There is supposed to be a double factorial after the parentheses in the denominator. My Quantum Mechanics Lecture Series: https://www.youtube.com/playlist?list=PLSpklniGdSfSsAFCzO-YWGlJ4TNv8sMdy There is a typo in the spherical harmonics normalization constant. In the second fraction under the square root, the 1s are supposed to be ls. Superfluid Helium Resonance Experiment video: https://youtu.be/unUNQNmuvUQ Quantum Field Theory Lecture Series: https://www.youtube.com/playlist?list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou$

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Covariant (Dirac method) Bosonic String Quantization | String Theory