On psychological theories of personal identity, personal identity is constituted by diachronic psychological relations, such as memory or concern. As it stands, the theory is silent on what constitutes transworld identity: what makes person x in world w1 be the same as person y in world w2. But let us think about what could be send in the vein of psychological theories about transworld identity.
Perhaps we could say that x in w1 is the same as y in w2 provided that x and y have the same earliest psychological states. But now sameness of psychological states is either type-sameness or token-sameness. If it's type-sameness, then we get the absurd conclusion that had your earliest psychological states been just like mine, you would have been me. Moreover, it is surely possible to have a world that contains two people who have the same earliest types of psychological states. But those two people aren't identity.
On the other hand, if we are talking of token-sameness, then we seem to get circularity, since the token-sameness of mental states presupposes the identity of the bearers. But there is a way out of that difficulty for naturalists. The naturalist can say that the mental states are constituted by some underlying physical states of a brain or organism. And she can then say that token-sameness of mental states is defined in terms of the token-sameness of the underlying physical states. This leads to the not implausible conclusion that you couldn't have started your existence with a different brain or organism.
But I think any stories in terms of initial psychological states face the serious difficulty that it is surely possible for me to have been raised completely differently and to have had different initial psychological states. This is obvious if the initial psychological states that count are states of me after the development of the sorts of cognitive functions that many (but not me) take to be definitive of personhood: for such functions develop after about one year of age, and surely I could have had a different life at that point.
In fact this line of thought suggests that no psychological-type relation is necessary for transworld identity. But if no psychological-type relation is necessary for transworld identity, why think it's necessary for intraworld identity?