I'm unfamiliar with the philosophies of personal identity. Which theories would postulate that a total interruption of consciousness/neural activity (e.g., a coma), but where the brain itself is completely undamaged, would be "death", in the sense of the person before the coma wouldn't be able to feel what happens after it? 

Reason is I need to make a decision about elective surgery under general anesthesia imminently. I'm concerned about the possibility that from my current perspective I will die as I'm put under even though from everyone else's perspective I'll wake up all the same, as I would be "rebooted" into a new "session" of consciousness and my current session won't be able to access/experience what happens in the new one the same way I can feel what happens to me 5 minutes from now. Of course this may happen every night during sleep. However, the risk is much greater under general anesthesia because of the much more complete loss of activity and information processing much like a coma, e.g. even during the deepest stage of sleep perhaps only 1 brain hemisphere sleeps at a time. Hence a coma being a much better proxy for this question: if sleep is okay anesthesia may not be, but if a coma's okay it definitely is too.

I realize LWers are broadly on board with cryonics and thus unconcerned with this, but I'd still like to know which specific theories are more in line with my intuitive concerns. 

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I've had a general anaesthetic. One curious thing about it is that when I woke up from it, it was like —bam— awake, with no sensation of time having passed since before. I seem to be the same person. If one accepts the physicalist hypothesis, and that what makes a mind be this mind is the pattern, then suspension of that pattern for a while has no implication that one person died and another woke up with their memories.

On the other hand, I have read that this experience of apparently having not existed at all for a while shook one person's faith in immaterial souls.

I’ve also been under them, and yes, there is a very strong sense of having been temporarily annihilated. While it didn’t affect my metaphysical beliefs, it’s easy to see how it could for some people.

I don't think there's a consensus on this, and I suspect it's because there's no truth to be had.  There's no territory that matches any map of these concepts.  

Personally, I model my identity as gradually changing, not constant.  Me and yesterday-me are very similar, but there are a lot of differences from 25-year-ago-me.  Legally, debts and assets follow, but psychologically that person is only partly-me.   Embedded in this model is that I'm using a distance model of brain-state, not a continuity measure of how gradual or smooth the changes have been.

Anesthesia is probably more change than a night's sleep, but less than a year's activity and sleeps.  

I don't remember it having any names, but this SEP article might help: https://plato.stanford.edu/entries/identity-time/

See e.g. "Four-dimensionlism," "Personal Identity."

But yeah I think most people around here have gotten used to the notion of identity as coming from a memory and perceptual relationship that might skip across time or space (every teleporter is a time machine), or even branch. Not really from cryonics so much as from computationalism about minds.

I find amnesia and impaired mental states more interesting cases than anaesthesia from a personal identity point of view, although I do recall one instance when they overlapped: I was in the recovery room, and wondering how long I had been out. I lifted my head up to look at the clock on the other side of the room, which read 1:06 pm. I then (slightly fuzzily) tried to remember when I went under and how many minutes ago that was, and then double checked the clock because clearly my mental arithmetic wasn't reliable, and it said 1:17 pm!

I had no memory of anything like 11 minutes having passed - it felt like maybe 10 seconds, and my head was still up so I don't think I passed out. I asked the nurse whether the clock was working, and she told me that yes it was correct, like last time I asked. Err what? Last time? I put my head back down and thought about that for a bit and went to sleep. When I woke (just a few minutes later) I could much more easily subtract the times, but still couldn't remember more than a few seconds between the times I looked at the clock, nor asking about the clock the first time.

From a personal identity point of view, what sort of consciousness existed during those missing 11 minutes? Is it part of "me" now? Is there a fact of the matter or is it simply a point of view?

I'm concerned about the possibility that from my current perspective I will die as I'm put under even though from everyone else's perspective I'll wake up all the same, as I would be "rebooted" into a new "session" of consciousness and my current session won't be able to access/experience what happens in the new one the same way I can feel what happens to me 5 minutes from now.

This is definitely a question of a personal perspective, and for some people this forward-looking idea of survival of identity is important: "from my current perspective I will die as I'm put under". For other people what matters is the backward-looking perspective: did the person who woke up feel like they are the same person who was put under? If yes, there is no issue, if not, well, you have to figure out whether to change your model of identity or live with the idea that you will die and someone else with your memories will be created, or maybe even forgo general anesthesia completely if neither of the options above works for you.