The standard says something like this:

*The process performance index is Pp = (U-L)/Delta*

[...]

well we would need an actual quote to assess what the standard says and not just your paraphrase of it...in any event the formula you provided is not the accepted formula for Pp in the vast majority of the literature and other governing body / major industry company manuals or leading statistical software packages. in these documents, delta is replaced by SD, where SD is the total standard deviation of the individual values (and not the average within subgroup standard deviation. I don't know if you remembered incorrectly or if the standard stated it that way in error or as a proposed alternative to using a standard deviation.

I am not sure of this statement:

If a process is under statistical control, a capability index can be assigned. The formula is the same as the Pp.

Although the quoted has issues, I am not sure why this would cause confusion. the statemetn is 'correct' in that if the process is in statistical control, one can assign (calculate) the capability index.

Can you explain more why? Also what do you use then?

Well, it's just insensical to describe variation as a single number as if it were accurate and true. These indices tell you very little about the 'capability' of yoru process. it is really just a lazy manager's need for a basic yes/no answer. The index isn't actionable. once it's bad you go back to plotting your data and understanding the original structure of the data collection to figure ou twhat to do next. If it's good but the structure was bad you are fooled into complacency as the index can't tell you the structure. but a time series plot of the data can. so why not just use that?

other things that just make this worse:

The common indices assume a Normal distribution which often doesn't exist. most people don't understand and/or comply with the requisite analysis structures: statistical control first, samples are representative, small samples are horribly in accurate and imprecise. oen need look no further than thsi forum for evidence of the abuse and mis-use of these indices.

What do I use? a multi-vari chart of the data. multi-vari shows me within positional (if applicable), piece to piece, time time in a lot, lot to lot, operator to operator, raw material lot to lot, equipment to equipment, time to time (days weeks, months, seasons). Plot the individual data poitns against the spec limits. I dont' have to do any math. just collect the data, plot the data, look at the data and THINK about the data.