Voting (specifically up/down-votes, not votes for closure or other review voting) has always been anonymous to avoid negative (or positive) consequences that result when people target specific users. This type of targeted voting is called serial voting (or voter fraud which stands opposite to regular, expected voting rationale) and Stack Exchange has attempted to automate the reversal of such habits using a script that runs every day.
With the above in mind, there is no way of identifying who downvoted post. At best you can guess who did this by tracking people's reputation in real time and seeing a -2 difference if there's a downvote on an answer of yours (question downvotes a free). But this is speculative because reputation changes due to many site interactions (editing, up/down votes cast and downvotes received) and there could be new users you don't know about casting downvotes...
Also, let's assume you know who downvoted your post. What is your next step? Sure, you want them to tell you why they did that. They don't have to if they don't want to, leaving you now with the same negative feeling... which often results in retaliation (a completely human instinct) through downvoting that person in a commensurate fashion. Such behaviour doesn't help anyone but you; the community (or greater good) either doesn't notice it, or when they do and don't agree with it, would easily reverse it.
Of note here is that there's also a distinction between voting behaviours on Meta vs Main: The former has a focus on the Main site and often discusses ideas. Upvotes are therefore commonly seen as support for or in favour of the suggestion (question and/or answer). Since the voting is somewhat different from the Main site, it also provides a reason why Meta doesn't have its own reputation. Of course one can equate the two voting systems (a -1 on Meta might feel like people don't like you, personally, just like a -1 on Main feels like people don't like you rather than your question/answer).
The voting system and its associated behaviour is something that is perceived very differently by different people. So it will probably be a point of discussion for the foreseeable future. And since it's perceived so differently, it's probably equally difficult to persuade people to think of it in a less threatening or negative manner if they've been "done in" by other's negative voting. We've lost a number of community members as a result. But from a bigger perspective, the principle remains solid and mostly objective - people get to express their opinions through voting and the general consensus should prevail in the long run.
You can view a graph of your site-specific reputation changes by viewing the reputation tab of your network profile:
The interactive chart allows you to zoom in and see the reputation changes on a day-to-day basis.