# Lecture 3 | String Theory and M-Theory | Summary and Q&A

## Summary

In this video, the professor discusses the harmonic oscillator and the spin of massless particles in relation to string theory. He explains the energy and Lagrangian of a harmonic oscillator and how it can be described by creation and annihilation operators. The professor then introduces the spin of massless particles and how they differ from massive particles. He discusses the number of states and polarization of massless particles, particularly photons. Finally, he highlights the issue of negative mass squared in string theory and the need for further exploration.

## Questions & Answers

### Q: What is the energy of the ground state of a vibrating string?

The energy of the ground state of a vibrating string can be represented by the mass squared of the unexcited oscillators. We denote this as M naught squared, and its exact value is not known yet. It is the starting point for the energy spectrum of a string.

### Q: How does the energy of a string increase when it is excited?

The energy of a string increases by integer units when it is excited by specific creation operators. The lowest energy excitation is given by the raising operator a_1, which increases the energy of the ground state by one unit. This corresponds to the frequency of that particular oscillator being one unit.

### Q: Can there be other states with the same energy as the first excitation?

Yes, there can be other states with the same energy as the first excitation. Any linear combination of the first excitation state and the ground state will have the same energy. These linear combinations form a vectorial character and are similar to the polarization states of photons.

### Q: How do the a and b oscillators transform under rotations?

The a and b oscillators, which correspond to the x and y components of a vector, transform under rotations like the components of a two-dimensional vector. They have a vectorial character and can be thought of as similar to the polarization states of photons.

### Q: How many polarization states are there for massless particles in string theory?

There are exactly two polarization states for massless particles in string theory. These polarization states correspond to the oscillations of the string in the x and y directions. They behave similarly to the polarization states of photons.

### Q: What is the issue with negative mass squared in string theory?

Having a particle with negative mass squared is problematic because mass squared is generally positive. This inconsistency raises questions about the validity of string theory and the need for further exploration and understanding.

## Takeaways

The study of string theory involves understanding the energy spectrum and polarization of massless particles in relation to the oscillations of strings. Linear combinations of ground states and excitations form polarization states similar to photons. However, the issue of negative mass squared in string theory highlights the need for deeper understanding and further investigation into its validity.