Lorentz covariance + Open string

[wam_mi]

Hi there,

Can I ask

(i) Is Lorentz covariance the same thing as Lorentz invariance? They seem to appear everywhere whenever we talk about space-time coordinates... what is the difference?

(ii) Is an open string the same as a bosonic string? Do bosons only appear as open strings or as closed strings? Sorry I'm confused!

Thanks!

 

[Ben Niehoff]

1. Invariant objects are just that: invariant. In order to be invariant, they must have no free indices; that is, they must be scalars.

Covariant objects, on the other hand, have free indices. They are not invariant under Lorentz transformations, but rather, they transform in a specific way

$$A_{\mu'} = {L^{\mu}}_{\mu'} A_{\mu}$$

Invariant objects can be built out of covariant objects by contracting all available indices:

$$\epsilon^{abcd} F_{ae} A^e S_{bcf} T^f_d$$

This one is invariant...not sure if it's ever useful, though. :)

Note that some objects that have indices are not covariant; for example, the Christoffel symbols [itex]\Gamma^{\mu}_{\nu\sigma}[/tex]. Also, objects built out of non-covariant objects by contracting all indices are not generally Lorentz scalars...you have to check using the transformation rule of whatever objects you're contracting.

2. No. Bosonic strings may be either open or closed. Superstrings may also be open or closed. Open just means the string has endpoints, as opposed to closed strings which form loops.

 

[Entropia]

Hi there!

To answer your questions:

(i) Yes, Lorentz covariance and Lorentz invariance are essentially the same thing. They both refer to the idea that the laws of physics should appear the same in all inertial reference frames, regardless of their relative motion. This is a fundamental principle in special relativity and is closely related to the concept of space-time coordinates.

(ii) An open string is a type of string in string theory that has two endpoints, while a closed string is a string that is connected at both ends to form a loop. So, an open string and a bosonic string are not the same thing. A bosonic string can be either open or closed, while a fermionic string can only be closed. Bosons and fermions are both types of particles, and they can appear in both open and closed strings. However, in string theory, the most commonly studied strings are bosonic, which is why you may see them referred to as "open strings."

I hope this helps clarify things for you! Let me know if you have any other questions.