Is it correct to use Pearson correlation for variables measured on a "Likert scale"?
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Technically, no. The distribution of Pearson's r under the null hypothesis is calculated assuming that the variables are normally distributed. That won't be a good approximation to a stardard 5-value Likert scale.
Unfortunately, I don't really have a clearly correct alternative to offer. Statistics on ordinal sets is a bit outside the mainstream, and as a physical scientist, I deal mostly in continuous quantities, so I have had little reason to delve into it. But I can say two things...
First, of the trio Pearson, Spearman, and Kendall tests of association, Kendall would be closest to being right. Spearman is non-parametric, but since it depends on ranks and since you have only a few possible values, there will be many ties, and the way different Spearman implementations deal with ties is rather arbitrary. (Ties are infinitely unlikely for a continuous variable, so the traditional derivation ignores them.) Kendall is also non-parametric, and has a better-developed theory of ties.
Second, if your N is large and your P very strong, from a practical perspective these quibbles don't matter. The association is real. Only when N is small or P is on the edge of significance are they going to move you across the significance threshold.
