One thing that’s interesting about the physical criterion for continuity of identity (the claim that it’s the continuation of the body that counts, not the continuation of any mental content–cf. Olson, Wiggins) is that most of those supporting this position would hold that a person is the same person even if he or she loses a leg, an arm, or really, almost any part of the body except the functioning brain. (I’ve had students claim that loss of limbs would involve loss of continuity of identity, which I think is an interesting idea which requires a greater exploration of what they mean in that case by “continuity of identity.”)
This is discussed by Parfit in ch.10 of Reasons and Persons, and it’s a fair assessment of the general consensus. So what counts isn’t the body, as such, but rather the part of the body that produces psychological content. Further, if the brain is rearranged enough, even some physicalists would accept that the person (if not the human animal) has ceased to be.
What all this points to, I think, is that for many purposes the physicalists have a psychological criterion or critera: that is, they’re just focusing on the physical area where psychological effects are produced. This is a reasonable strategy, especially if you want to hold an eliminativist position in regard to psychological terms. Still, the claim of the physicalists who make this move is that the body part that produces psychological effects is what must continue to exist for continuity of identity (and probably it has to continue without certain changes, like those that would cause the death of the organism…there’s an interesting set of end-of-life problems here, and some questions about total loss of brain function, or loss of all higher function/all function other than brain-stem function, etc.)
Some physicalists would probably accept that a person remains the same person even in the event of massive brain injury if the body survives…this is what Marya Schechtman was getting at in “Personhood and the Practical,” for example. Schechtman, though, accepts that sometimes the physical criterion is the relevant one, and sometimes it’s not. I think this is a fairer approach than picking one criterion and assuming it’s relevant in all cases or for all questions and concerns.