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.<br /> <br /> 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.[/itex]