What is the PIE reconstruction for word-initial alpha with rough breathing? My concern is the tendency (popularly) to mistake initial alpha as privative when it is not (e.g. hamartia): The easiest case, I suppose, is the rough breathing, in which case, am I right to imagine a reconstructed *ha? The question is relevant also, however, with any genuinely alpha-initial root. Thank you.
Hamartía "sin" is a hard example to use, because to my knowledge, hamartánō "to sin" has no known IE cognates. This means there's no way to check or confirm reconstructions, and it may not come from PIE at all. Instead, I'm going to use háls "salt".
/h/ in any Indo-European language (except Hittite!) has nothing to do with PIE laryngeals like
*h₁. The laryngeals disappeared very early and are only visible by the effects they had on surrounding sounds.
In Ancient Greek in particular,
/h/ usually came from an earlier
*s before a vowel. (Later, it disappeared when it wasn't at the start of a word; even later, it disappeared everywhere.) So the reconstructed PIE for "salt" was something like
*séh₂l-s, as seen in Latin sāl and English salt.
This is also where the prefix ha- "together" came from: the PIE ancestor is reconstructed as
*sṃ, as seen in Latin forms like simplex and semel. In Greek,
*ṃ turned into a, and the
*s turned into h, giving the ha- found in Hāidēs, haploûs, hápas, and so on.
P.S. Sometimes you won't see an
/h/ where you might expect one, as in adelphós, which has that same ha- prefix. This is due to "Grassmann's Law": if you have two or more aspirated consonants in a word, all but the last one lose their aspiration. This is why you see héksō alongside ékhō: in the future tense, the khs simplifies into k, which allows the original h at the beginning to show through.