Think about the wavefunction of a single electron. It specifies an amplitude for the electron to be found at any point in space. And there are continuum many such points. Now, the sense in which MWI states that there are many different ‘copies’ of you all simultaneously existing corresponds to the sense in which ‘copies’ of the electron are found (with various different amplitudes) at every point. A ‘copy’ of the electron = one of its possible locations.
So since there are continuum-many copies of an electron, I think it’s fairly safe to assume there are at least continuum-many copies of a ‘human’.
Of course, there may be a sense in which, although every possible configuration of the elementary particles in your body is assigned some amplitude, almost all of the amplitude gets ‘concentrated’ into a small number of ‘rivers’ of relatively much higher probability. For instance, the ‘river of probability’ for Schrödinger’s cat will split into a ‘live cat’ branch and a ‘dead cat’ branch. Each branch is smeared over infinitely many configurations, and there are infinitely many configurations not belonging to either branch, which also get some non-zero amplitude, but nonetheless the two ‘main branches’ cover almost all of the probability mass, and thus they stand out as real patterns—as real as ‘planets’ or ‘stones’. It is in this sense, I think, that you mean to say there may only be “finitely many” copies of a person.
But this kind of finiteness doesn’t suffice to defuse the reasoning which led you to say “if there are really infinitely many copies of you then your decision makes no difference”. (The resolution, of course, is to abandon that reasoning.)
Think about the wavefunction of a single electron. It specifies an amplitude for the electron to be found at any point in space. And there are continuum many such points.
It does not specify an probability for each point, but a density, which is only turned into an probability by integrating. The probability of being at a particular point is zero. More strongly, the system has countably many qubits.
So we agree: far from there being ‘only finitely many copies’, if there’s any sense at all to be made of the ‘number of copies’ then it is infinite and any (or ‘nearly any’) possible configuration of the matter making up your body gets some non-zero density or probability depending on whether by configuration we mean a ‘point’ in phase space (if that’s the right term...) or a little ‘cube’.
Your original post is correct. I think “continuum-many” is misleading, but I can’t object much given the context. I don’t remember what I was thinking.
ETA: I would think of an eigenbasis and say that there are countably many electrons, none of which is localized.
I am not certain what you mean by ‘continuum-many’; it sounds as though it could be either ‘infinite’ or ‘the large number you get from a lot of combinatorics’. However, I must point out that quantum theory has the interesting property of being quantized. (Sounds almost like a lolcat slogan. “Kwantum fyziks… iz kwantised.”) A particle in a bound state does not have infinitely many degrees of freedom, and since our local spacetime is apparently closed (if only just) every particle is in a bound state, it’s just not obvious.
I am not certain what you mean by ‘continuum-many’
It refers to cardinality. You know Cantor showed that, while natural numbers and rationals can be put into one-to-one correspondence, there is no way to put the reals into one-to-one correspondence with the naturals, because there are ‘too many’ real numbers? Well, “continuum-many” means “the same cardinality as the real numbers”.
Still, Douglas Knight makes a fair point—it is somewhat misleading to talk about continuum-many copies if each one has zero probability. In truth, I guess the concept of a ‘number of copies’ is too simple to capture what’s going on.
As for particles being in bound states and having finitely many degrees of freedom: I’d be surprised if it altered the ‘bigger picture’ whereby all possible rearrangements of the matter in your body (or in the solar system as a whole, say) get some (possibly minuscule) amplitude assigned to them. (Of course, ideally it would be someone who actually knows some physics saying this rather than me.)
“continuum-many” means “the same cardinality as the real numbers”.
Ok, fair enough. In that case I must merely disagree that there exist this many possible arrangements of matter; it seems to me that the arrangements are actually countably infinite.
As for particles being in bound states and having finitely many degrees of freedom: I’d be surprised if it altered the ‘bigger picture’ whereby all possible rearrangements of the matter in your body (or in the solar system as a whole, say) get some (possibly minuscule) amplitude assigned to them.
That’s true, but the question is whether that number has the cardinality of the reals or the integers. I think it’s the integers, due to the quantisation phenomenon in bound states; everything is in a bound state at some level. After my last post it occurred to me that the quantised states might be so close together that they’d be effectively indistinguishable; however, there would still be a finite number of distinguishable states. Two states are not meaningfully different if a quantum number changes by less than the corresponding uncertainty, so in effect the wave-function is quantised even in a continuously-varying number. Once you quantise it’s all just combinatorics and integers.
::Shrug:: There’s something important about the Planck distance, but I don’t know enough physics to be able to say much more. Like Hawking radiation, It’s something that only crops up when you start trying to do ‘quantum gravity’.
It’s tempting to imagine that the universe is something like the “Game Of Life” but with Planck sized cells, but what little I know about string theory makes this idea seem extremely naive. (And anyway, space could be both discrete and infinite.)
IANAPhysicist, but I’m fairly sure that space and time are entirely continuous in standard QM or QFT, though they are discrete in loop quantum gravity and possibly other theories of QG.
Standard QFT doesn’t have discrete space and QCD may make sense with continuum of space-time, but models with a Landau pole, like QED and the standard model, don’t make sense at small length scales. The length at which the Landau pole appears in QED is smaller than the Planck length, so no one cares about it, since they expect bad things to happen already at the Planck scale.
Think about the wavefunction of a single electron. It specifies an amplitude for the electron to be found at any point in space. And there are continuum many such points. Now, the sense in which MWI states that there are many different ‘copies’ of you all simultaneously existing corresponds to the sense in which ‘copies’ of the electron are found (with various different amplitudes) at every point. A ‘copy’ of the electron = one of its possible locations.
So since there are continuum-many copies of an electron, I think it’s fairly safe to assume there are at least continuum-many copies of a ‘human’.
Of course, there may be a sense in which, although every possible configuration of the elementary particles in your body is assigned some amplitude, almost all of the amplitude gets ‘concentrated’ into a small number of ‘rivers’ of relatively much higher probability. For instance, the ‘river of probability’ for Schrödinger’s cat will split into a ‘live cat’ branch and a ‘dead cat’ branch. Each branch is smeared over infinitely many configurations, and there are infinitely many configurations not belonging to either branch, which also get some non-zero amplitude, but nonetheless the two ‘main branches’ cover almost all of the probability mass, and thus they stand out as real patterns—as real as ‘planets’ or ‘stones’. It is in this sense, I think, that you mean to say there may only be “finitely many” copies of a person.
But this kind of finiteness doesn’t suffice to defuse the reasoning which led you to say “if there are really infinitely many copies of you then your decision makes no difference”. (The resolution, of course, is to abandon that reasoning.)
It does not specify an probability for each point, but a density, which is only turned into an probability by integrating. The probability of being at a particular point is zero. More strongly, the system has countably many qubits.
Sure.
So we agree: far from there being ‘only finitely many copies’, if there’s any sense at all to be made of the ‘number of copies’ then it is infinite and any (or ‘nearly any’) possible configuration of the matter making up your body gets some non-zero density or probability depending on whether by configuration we mean a ‘point’ in phase space (if that’s the right term...) or a little ‘cube’.
Your original post is correct. I think “continuum-many” is misleading, but I can’t object much given the context. I don’t remember what I was thinking.
ETA: I would think of an eigenbasis and say that there are countably many electrons, none of which is localized.
I am not certain what you mean by ‘continuum-many’; it sounds as though it could be either ‘infinite’ or ‘the large number you get from a lot of combinatorics’. However, I must point out that quantum theory has the interesting property of being quantized. (Sounds almost like a lolcat slogan. “Kwantum fyziks… iz kwantised.”) A particle in a bound state does not have infinitely many degrees of freedom, and since our local spacetime is apparently closed (if only just) every particle is in a bound state, it’s just not obvious.
It refers to cardinality. You know Cantor showed that, while natural numbers and rationals can be put into one-to-one correspondence, there is no way to put the reals into one-to-one correspondence with the naturals, because there are ‘too many’ real numbers? Well, “continuum-many” means “the same cardinality as the real numbers”.
Still, Douglas Knight makes a fair point—it is somewhat misleading to talk about continuum-many copies if each one has zero probability. In truth, I guess the concept of a ‘number of copies’ is too simple to capture what’s going on.
As for particles being in bound states and having finitely many degrees of freedom: I’d be surprised if it altered the ‘bigger picture’ whereby all possible rearrangements of the matter in your body (or in the solar system as a whole, say) get some (possibly minuscule) amplitude assigned to them. (Of course, ideally it would be someone who actually knows some physics saying this rather than me.)
Ok, fair enough. In that case I must merely disagree that there exist this many possible arrangements of matter; it seems to me that the arrangements are actually countably infinite.
That’s true, but the question is whether that number has the cardinality of the reals or the integers. I think it’s the integers, due to the quantisation phenomenon in bound states; everything is in a bound state at some level. After my last post it occurred to me that the quantised states might be so close together that they’d be effectively indistinguishable; however, there would still be a finite number of distinguishable states. Two states are not meaningfully different if a quantum number changes by less than the corresponding uncertainty, so in effect the wave-function is quantised even in a continuously-varying number. Once you quantise it’s all just combinatorics and integers.
I don’t think you’re right… isn’t it broken down into plank lengths or something?
::Shrug:: There’s something important about the Planck distance, but I don’t know enough physics to be able to say much more. Like Hawking radiation, It’s something that only crops up when you start trying to do ‘quantum gravity’.
It’s tempting to imagine that the universe is something like the “Game Of Life” but with Planck sized cells, but what little I know about string theory makes this idea seem extremely naive. (And anyway, space could be both discrete and infinite.)
IANAPhysicist, but I’m fairly sure that space and time are entirely continuous in standard QM or QFT, though they are discrete in loop quantum gravity and possibly other theories of QG.
Standard QFT doesn’t have discrete space and QCD may make sense with continuum of space-time, but models with a Landau pole, like QED and the standard model, don’t make sense at small length scales. The length at which the Landau pole appears in QED is smaller than the Planck length, so no one cares about it, since they expect bad things to happen already at the Planck scale.