Lorentz covariance + Open string

1. Nov 19, 2009

wam_mi

Hi there,

(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!

2. Nov 19, 2009

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.