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jackhttp://www.scottaaronson.com/democritus/lec19.html
ETA: http://scottaaronson.com/blog/?p=368
What I thought
If I live in world A and go back in time and alter things, then I leave a new world A'. What if the new me in world A' ALSO goes back in time to alter things, leaving world A''? If me' changes them back again, you have some kind of grandfather paradox.
I had always had an idea that the obvious resolution is that it will "settle down" to some steady state A'''''''''''''''' which leads to the same A''''''''''''''''. I envisaged this as moving through the possible states until you find a stable world to stop in even though "moving through" would not be relative to the normal timeline.
A proper formulation
However, I was enchanted to read that very nearly this was seriously investigated by someone. Specifically, if you ask "what happens in quantum mechanics, if there is a loop in space where the future leads back round to the past"? Well, then it's obvious: you solve the differential equation of how the world evolves over time, with the constraint the state at the start and end of the loop have to match up. QM (and any theory where you solve differential equations) is full of that sort of thing anyway. A priori "satisfying the differential equation" means a ball doesn't suddenly stop in mid-air abandoning all its momentum -- you just can't imagine the world different -- and just as much, a world where trajectories of objects/probabilities are consistent over the loop.
You don't even need quantum probabilities, classical ones do just as well. Imagine that all the different histories we consider are assigned probabilities p1, p2, etc. And world 1 leads to world 513, world 2 leads to world 31, etc. (Or, if you prefer, world 1 leads with chance x to world 513 and chance y to world 774, etc) Then we have a continuous markov chain where p1 is defined by the sum of the probabilities of all the worlds that lead to world 1. Ie. each the probabilities "flow" from world to world, according to which worlds lead to which other world.
What happens? Well, clearly we end up in a steady state. If only one world is non-paradoxical -- ie. leads to itself 100% -- and other setups lead there, we end up there. Or a superposition of independent steady-states. In this model the universe is all nice and correcting.
How this deals with paradoxes and popular films
What happens if world A is "kill grandfather" and world B is "never born"? Then the steady state is 50% A and 50% B. That's as good an answer as you're going to get.
What happens with events that cause themselves, like skynet or transparent aluminium in startrek? Imagine two steady states "world where skynet is never created" and "world where skynet is created based on time-travelling technology from terminator, then goes on to invent terminator and time-travel". Well, no paradox, you're in one or the other or a superposition of both.
However, that's not quite true. If you imagine you continuous markov chain as little glass bulbs full of water (and you should) then both worlds are little bulbs where the water flows right back in, supporting yourself. But there are leaks. In a world without time-travel, there's a chance you'll invent skynet anyway -- that's a leak from the first to the second. And in a world with skynet, there's a chance John Connor will send Terminator II back in time to blow up proto-skyne. That's a leak from the second to the first. So in fact, it depends which of those probabilities is greater -- we'll end up in one steady world or the other.
Terminator I film represented the closed-loop skynet world. Terminator II film represented the "blow up skynet" escape from the loop. Terminator III represented "invent skynet anyway" (er, I may misremember). Termator IV represented a similar but different alternative history, where there's a non-time-travelling skynet.
Incidentally I say "loop" because it's not clear skynet *precisely* causes itself -- it may cause a *better* skynet, if it knows what it's doing, until it reaches some sort of transcendent steady state, where it sends back in time a few seconds an ultimate skynet seed, which grows in seconds into an ultimate skynet.
I don't know why people knock Terminator's time-travel physics -- to me it seems a wonderful example of how things can work.
You might even be able to represent Marty McFly fading out of existence as a probability shift from one world to another, but I wouldn't bet on it.
The maths
Of course, this is an utterly un-technical approach. I'm just imagining how things could work. The original paper actually lead to mathematics about quantum mechanics (and specifically quantum computing) I'm not equipped to understand. Nor do I pretend to know what a superposition of "kill grandfather" and "never born" is.
ETA: http://scottaaronson.com/blog/?p=368
What I thought
If I live in world A and go back in time and alter things, then I leave a new world A'. What if the new me in world A' ALSO goes back in time to alter things, leaving world A''? If me' changes them back again, you have some kind of grandfather paradox.
I had always had an idea that the obvious resolution is that it will "settle down" to some steady state A'''''''''''''''' which leads to the same A''''''''''''''''. I envisaged this as moving through the possible states until you find a stable world to stop in even though "moving through" would not be relative to the normal timeline.
A proper formulation
However, I was enchanted to read that very nearly this was seriously investigated by someone. Specifically, if you ask "what happens in quantum mechanics, if there is a loop in space where the future leads back round to the past"? Well, then it's obvious: you solve the differential equation of how the world evolves over time, with the constraint the state at the start and end of the loop have to match up. QM (and any theory where you solve differential equations) is full of that sort of thing anyway. A priori "satisfying the differential equation" means a ball doesn't suddenly stop in mid-air abandoning all its momentum -- you just can't imagine the world different -- and just as much, a world where trajectories of objects/probabilities are consistent over the loop.
You don't even need quantum probabilities, classical ones do just as well. Imagine that all the different histories we consider are assigned probabilities p1, p2, etc. And world 1 leads to world 513, world 2 leads to world 31, etc. (Or, if you prefer, world 1 leads with chance x to world 513 and chance y to world 774, etc) Then we have a continuous markov chain where p1 is defined by the sum of the probabilities of all the worlds that lead to world 1. Ie. each the probabilities "flow" from world to world, according to which worlds lead to which other world.
What happens? Well, clearly we end up in a steady state. If only one world is non-paradoxical -- ie. leads to itself 100% -- and other setups lead there, we end up there. Or a superposition of independent steady-states. In this model the universe is all nice and correcting.
How this deals with paradoxes and popular films
What happens if world A is "kill grandfather" and world B is "never born"? Then the steady state is 50% A and 50% B. That's as good an answer as you're going to get.
What happens with events that cause themselves, like skynet or transparent aluminium in startrek? Imagine two steady states "world where skynet is never created" and "world where skynet is created based on time-travelling technology from terminator, then goes on to invent terminator and time-travel". Well, no paradox, you're in one or the other or a superposition of both.
However, that's not quite true. If you imagine you continuous markov chain as little glass bulbs full of water (and you should) then both worlds are little bulbs where the water flows right back in, supporting yourself. But there are leaks. In a world without time-travel, there's a chance you'll invent skynet anyway -- that's a leak from the first to the second. And in a world with skynet, there's a chance John Connor will send Terminator II back in time to blow up proto-skyne. That's a leak from the second to the first. So in fact, it depends which of those probabilities is greater -- we'll end up in one steady world or the other.
Terminator I film represented the closed-loop skynet world. Terminator II film represented the "blow up skynet" escape from the loop. Terminator III represented "invent skynet anyway" (er, I may misremember). Termator IV represented a similar but different alternative history, where there's a non-time-travelling skynet.
Incidentally I say "loop" because it's not clear skynet *precisely* causes itself -- it may cause a *better* skynet, if it knows what it's doing, until it reaches some sort of transcendent steady state, where it sends back in time a few seconds an ultimate skynet seed, which grows in seconds into an ultimate skynet.
I don't know why people knock Terminator's time-travel physics -- to me it seems a wonderful example of how things can work.
You might even be able to represent Marty McFly fading out of existence as a probability shift from one world to another, but I wouldn't bet on it.
The maths
Of course, this is an utterly un-technical approach. I'm just imagining how things could work. The original paper actually lead to mathematics about quantum mechanics (and specifically quantum computing) I'm not equipped to understand. Nor do I pretend to know what a superposition of "kill grandfather" and "never born" is.