If a world splits, the energy of each split-off world would have to be less than the original world.
No, you are misunderstanding the argument. I am a MWI opponent but I know you are getting this wrong. If we switch to orthodox QM for a moment, and ask what the energy of a generic superposition is, the closest thing to an answer is to talk about the expectation value of the energy observable for that wavefunction. This is a weighted average of the energy eigenvalues appearing in the superposition. For example, for the superposition 1/sqrt(2) |E=E1> + 1/sqrt(2) |E=E2>, the expectation value is E1/2 + E2/2. What Q22 in the Everett FAQ is saying is that the expectation value won’t apriori increase, even if new worlds are being created within the wavefunction, because the expectation value is the weighted average of the energies of the individual worlds; and in fact the expectation value will not change at all (something you can prove in a variety of ways).
To use an extreme example, that’s supposed to be why you can’t see anyone in a world where the Nazis won WWII: That part of the wavefunction is so decoherent from yours that any interference is just random noise and there is therefore no meaningful interference.
Well, this is another issue where, if I was talking to a skilled MWI advocate, I might be able to ask some probing questions, because there is a potential inconsistency in the application of these concepts. Usually when we talk about interference between branches of the wavefunction, it means that there are two regions in (say) configuration space, each of which has some amplitude, and there is some flow of probability amplitude from one region into the other. But this flow does not exist at the level of configurations, it only occurs at the level of configuration amplitudes. So if “my world”, “this world”, where the Nazis lost, is one configuration, and the world where the Nazis won is another configuration, there is no way for our configuration to suddenly resemble the other configuration on account of such a flow—that is a confusion of levels.
For me to observe interference phenomena, I have to be outside the superposition. But I wasn’t even born when WWII was decided, so I am intrinsically stuck in one branch. Maybe this is a quibble; we could talk about something that happened after my birth, like the 2000 US election. I live in a world where Bush won; but in principle could I see interference from a world where Gore won? I still don’t think it makes sense; the fact that I remember Bush winning means that I’m in that branch; I would have to lose the memory for the probability flow here to come into contact with the probability flow in a branch where Gore won. More importantly, the whole world configuration would have to morph until it came to resemble a world where Gore won, for some portion of the probability flow “here” to combine with the probability flow there.
I’ll try to explain what I’m talking about. The wavefunction consists of a complex-valued function defined throughout configuration space. Configuration space consists of static total configurations of the universe. Change exists only at the level of the complex numbers; where they are large, you have a “peak” in the wavefunction, and these peaks move around in configuration space, split and join, and so on. So really, it ought to be a mistake to think of configurations per se as the worlds; instead, you should perhaps be thinking about the “peaks”, the local wavepackets in configuration space, as worlds. Except, a peak can have a spread in configuration space. A single peak can be more like a “ridge” stretching between configurations which are classically inconsistent. This already poses problems of interpretation, as does the lack of clear boundaries to a peak… Are we going to say that a world consists of any portion of the wavefunction centered on a peak—a local maximum—and bounded by regions where the gradient is flat??
But here I can only throw up my hands and express my chronic exasperation with the fuzzy thinking behind many worlds. It is impossible to intelligently critique an idea when the exponent of the idea hasn’t finished specifying it and doesn’t even realize that they need to do more work. And then you have laypeople who take up the unfinished idea and advocate it, who are even more oblivious to the problems, and certainly incapable of answering them.
Paul, if I could convey to you one perspective on MWI, it would be as follows: Most people who talk about MWI do not have an exact definition of what a world is. Instead, it’s really an ideology, or a way of speaking: QM has superpositions in it, and the slogan is that everything in the superposition is real. But if this is to be an actual theory of the world, and not just an idea for a theory, you have to be more concrete. You have to say exactly what parts of a wavefunction are the worlds. And when you do this, you face new problems, e.g. to do with relativity and probability. The exact nature of the problems depends on how the MWI idea is concretized. But if you give me any concrete, detailed version of MWI, I can tell you what’s wrong with it.
First, let me say beautifully clear explanation of what MWI is and especially what questions it needs to answer.
Except, a peak can have a spread in configuration space. A single peak can be more like a “ridge” stretching between configurations which are classically inconsistent. This already poses problems of interpretation, as does the lack of clear boundaries to a peak… Are we going to say that a world consists of any portion of the wavefunction centered on a peak—a local maximum—and bounded by regions where the gradient is flat??
I don’t think this is any more unreasonable than talking about firing two separate localized wave-packets at each other and watching them interfere, even if we don’t have a specific fixed idea of what in full generality counts as a “wave-packet”. Typically, of course, for linear wave equations we’d use Gaussians as models, but I don’t think that’s more than a mathematically convenient exemplar.
For non-linear models, (e.g. KdV) we have soliton solutions that have rather different properties, such as being self-focusing, rather than spreading out. I guess I don’t see why it matters whether you have an exact definition for “world” or not—so long as you can plausibly exhibit them.
The question in my mind is whether evolution on configuration space preserves wave-packet localization, or under what conditions they could develop. I find it hard to even formalize this, but given that we have a linear wave-equation, I would tend to doubt they do.
e.g. to do with relativity
Of course relativity will be an issue. QM is not Einsteinian relativistic, only Galilean (relabeling phases properly gives a Galilean boost), and that’s baked into the standard operators and evolution.
I do admit to over-generalizing in saying that when a world splits, the split-off worlds each HAVE to have lower energy than the “original world”. If we measure the energy associated with the wavefunction for individual worlds, on average, of course, this would have to be the case, due to the proliferation of worlds: However, I do understand, and should have stated, that all that matters is that the total energy for the system remains constant over time, and that probabilities matter.
Regarding the second issue, defining what a world is, I actually do understand your point: I feel that you think I understand less on this than is actually the case. Nevertheless, I would say that getting rid of a need for collapse does mean a lot and removes a lot of issues: more than are added with the “What constitutes a world” issue. However, we probably do need a “more-skilled MWI advocate” to deal with that.
I do admit to over-generalizing in saying that when a world splits, the split-off worlds each HAVE to have lower energy than the “original world”. If we measure the energy associated with the wavefunction for individual worlds, on average, of course, this would have to be the case, due to the proliferation of worlds
Let me see if I am understanding you. You’re now saying that the average energy-per-world goes down, “due to the proliferation of worlds”? Because that still isn’t right.
The simplest proof that the average energy is conserved is that energy eigenstates are stationary states: subjected to Hamiltonian evolution, they don’t change except for a phase factor. So if your evolving wavefunction is Psi(t), expressed in a basis of energy eigenstates it becomes sum_k c_k exp(-i . E_k . t) |E_k>. I.e. the time dependence is only in the coefficients of the energy eigenstates, and there’s no variation in their norm (since the time dependence is only in the phase factor), so the probability weightings of the energy eigenstates also don’t change. Therefore, the expectation value of the energy is a constant.
There ought to be a “local” proof of energy conservation as well (at least, if we were working with a field theory), and it might be possible to insightfully connect that with decoherence in some way—that is, in a way which made clear that decoherence, the process which is supposed to be giving rise to world-splits, also conserves energy however you look at it—but that would require a bit more thought on my part.
ETA: Dammit, how do you do subscripts in markdown? :-)
No, I think you are misunderstanding me here. I wasn’t claiming that proliferation of worlds CAUSES average energy per-world to go down. It wouldn’t make much sense to do that, because it is far from certain that the concept of a world is absolutely defined (a point you seem to have been arguing). I was saying that the total energy of the wavefunction remains constant (which isn’t really unreasonable, because it is merely a wave developing over time—we should expect that.) and I was saying that a CONSEQUENCE of this is that we should expect, on average, the energy associated with each world to decrease as we have a constant amount of energy in the wavefunction and the number of worlds is increasing. If you have some way of defining worlds, and you n worlds, and then later have one billion x n worlds, and you have some way of allocating energy to a world, then this would have to happen to maintain conservation of energy. Also, I’m not claiming that the issue is best dealt with in terms of “energy per world” either.
Now you are saying what I first thought you might have meant. :-) Namely, you are talking about the energy of the wavefunction as if it were itself a field. In a way, this brings out some of the difficulties with MWI and the common assertion that MWI results from taking the Schrodinger equation literally.
It’s a little technical, but possibly the essence of what I’m talking about is to be found by thinking about Noether’s theorem. This is the theorem which says that symmetries lead to conserved quantities such as energy. But the theorem is really built for classical physics. Ward identities are the quantum counterpart, but they work quite differently, because (normally) the wavefunction is not treated as if it is a field, it is treated as a quasiprobability distribution on the physical configuration space. In effect, you are talking about the energy of the wavefunction as if the classical approach, Noether’s theorem, was the appropriate way to do so.
There are definitely deep issues here because quantum field theory is arguably built on the formal possibility of treating a wavefunction as a field. The Dirac equation was meant to be the wavefunction of a single particle, but to deal with the negative-energy states it was instead treated as a field which itself had to be quantized (this is called “second quantization”). Thus was born quantum field theory and the notion of particles as field quanta.
MWI seems to be saying, let’s treat configuration space as a real physical space, and regard the second-quantized Schrodinger equation as defining a field in that space. If you could apply Noether’s theorem to that field in the normal way (ignoring the peculiarity that configuration space is infinite-dimensional), and somehow derive the Ward identities from that, that would be a successful derivation of orthodox quantum field theory from the MWI postulate. But skeptical as I am, I think this might instead be a way to illuminate from yet another angle why MWI is so problematic or even unviable. Right away, for example, MWI’s problem with relativity will come up.
Anyway, that’s all rather esoteric, but the bottom line is that you don’t use this “Noetherian configuration-space energy” in quantum mechanics, you use a concept of energy which says that energy is a property of the individual configurations. And this is why there’s no issue of “allocating energy to a world” from a trans-world store of energy embodied in the wavefunction.
No, you are misunderstanding the argument. I am a MWI opponent but I know you are getting this wrong. If we switch to orthodox QM for a moment, and ask what the energy of a generic superposition is, the closest thing to an answer is to talk about the expectation value of the energy observable for that wavefunction. This is a weighted average of the energy eigenvalues appearing in the superposition. For example, for the superposition 1/sqrt(2) |E=E1> + 1/sqrt(2) |E=E2>, the expectation value is E1/2 + E2/2. What Q22 in the Everett FAQ is saying is that the expectation value won’t apriori increase, even if new worlds are being created within the wavefunction, because the expectation value is the weighted average of the energies of the individual worlds; and in fact the expectation value will not change at all (something you can prove in a variety of ways).
Well, this is another issue where, if I was talking to a skilled MWI advocate, I might be able to ask some probing questions, because there is a potential inconsistency in the application of these concepts. Usually when we talk about interference between branches of the wavefunction, it means that there are two regions in (say) configuration space, each of which has some amplitude, and there is some flow of probability amplitude from one region into the other. But this flow does not exist at the level of configurations, it only occurs at the level of configuration amplitudes. So if “my world”, “this world”, where the Nazis lost, is one configuration, and the world where the Nazis won is another configuration, there is no way for our configuration to suddenly resemble the other configuration on account of such a flow—that is a confusion of levels.
For me to observe interference phenomena, I have to be outside the superposition. But I wasn’t even born when WWII was decided, so I am intrinsically stuck in one branch. Maybe this is a quibble; we could talk about something that happened after my birth, like the 2000 US election. I live in a world where Bush won; but in principle could I see interference from a world where Gore won? I still don’t think it makes sense; the fact that I remember Bush winning means that I’m in that branch; I would have to lose the memory for the probability flow here to come into contact with the probability flow in a branch where Gore won. More importantly, the whole world configuration would have to morph until it came to resemble a world where Gore won, for some portion of the probability flow “here” to combine with the probability flow there.
I’ll try to explain what I’m talking about. The wavefunction consists of a complex-valued function defined throughout configuration space. Configuration space consists of static total configurations of the universe. Change exists only at the level of the complex numbers; where they are large, you have a “peak” in the wavefunction, and these peaks move around in configuration space, split and join, and so on. So really, it ought to be a mistake to think of configurations per se as the worlds; instead, you should perhaps be thinking about the “peaks”, the local wavepackets in configuration space, as worlds. Except, a peak can have a spread in configuration space. A single peak can be more like a “ridge” stretching between configurations which are classically inconsistent. This already poses problems of interpretation, as does the lack of clear boundaries to a peak… Are we going to say that a world consists of any portion of the wavefunction centered on a peak—a local maximum—and bounded by regions where the gradient is flat??
But here I can only throw up my hands and express my chronic exasperation with the fuzzy thinking behind many worlds. It is impossible to intelligently critique an idea when the exponent of the idea hasn’t finished specifying it and doesn’t even realize that they need to do more work. And then you have laypeople who take up the unfinished idea and advocate it, who are even more oblivious to the problems, and certainly incapable of answering them.
Paul, if I could convey to you one perspective on MWI, it would be as follows: Most people who talk about MWI do not have an exact definition of what a world is. Instead, it’s really an ideology, or a way of speaking: QM has superpositions in it, and the slogan is that everything in the superposition is real. But if this is to be an actual theory of the world, and not just an idea for a theory, you have to be more concrete. You have to say exactly what parts of a wavefunction are the worlds. And when you do this, you face new problems, e.g. to do with relativity and probability. The exact nature of the problems depends on how the MWI idea is concretized. But if you give me any concrete, detailed version of MWI, I can tell you what’s wrong with it.
First, let me say beautifully clear explanation of what MWI is and especially what questions it needs to answer.
I don’t think this is any more unreasonable than talking about firing two separate localized wave-packets at each other and watching them interfere, even if we don’t have a specific fixed idea of what in full generality counts as a “wave-packet”. Typically, of course, for linear wave equations we’d use Gaussians as models, but I don’t think that’s more than a mathematically convenient exemplar. For non-linear models, (e.g. KdV) we have soliton solutions that have rather different properties, such as being self-focusing, rather than spreading out. I guess I don’t see why it matters whether you have an exact definition for “world” or not—so long as you can plausibly exhibit them. The question in my mind is whether evolution on configuration space preserves wave-packet localization, or under what conditions they could develop. I find it hard to even formalize this, but given that we have a linear wave-equation, I would tend to doubt they do.
Of course relativity will be an issue. QM is not Einsteinian relativistic, only Galilean (relabeling phases properly gives a Galilean boost), and that’s baked into the standard operators and evolution.
I do admit to over-generalizing in saying that when a world splits, the split-off worlds each HAVE to have lower energy than the “original world”. If we measure the energy associated with the wavefunction for individual worlds, on average, of course, this would have to be the case, due to the proliferation of worlds: However, I do understand, and should have stated, that all that matters is that the total energy for the system remains constant over time, and that probabilities matter.
Regarding the second issue, defining what a world is, I actually do understand your point: I feel that you think I understand less on this than is actually the case. Nevertheless, I would say that getting rid of a need for collapse does mean a lot and removes a lot of issues: more than are added with the “What constitutes a world” issue. However, we probably do need a “more-skilled MWI advocate” to deal with that.
Let me see if I am understanding you. You’re now saying that the average energy-per-world goes down, “due to the proliferation of worlds”? Because that still isn’t right.
The simplest proof that the average energy is conserved is that energy eigenstates are stationary states: subjected to Hamiltonian evolution, they don’t change except for a phase factor. So if your evolving wavefunction is Psi(t), expressed in a basis of energy eigenstates it becomes sum_k c_k exp(-i . E_k . t) |E_k>. I.e. the time dependence is only in the coefficients of the energy eigenstates, and there’s no variation in their norm (since the time dependence is only in the phase factor), so the probability weightings of the energy eigenstates also don’t change. Therefore, the expectation value of the energy is a constant.
There ought to be a “local” proof of energy conservation as well (at least, if we were working with a field theory), and it might be possible to insightfully connect that with decoherence in some way—that is, in a way which made clear that decoherence, the process which is supposed to be giving rise to world-splits, also conserves energy however you look at it—but that would require a bit more thought on my part.
ETA: Dammit, how do you do subscripts in markdown? :-)
ETA 2: Found the answer.
No, I think you are misunderstanding me here. I wasn’t claiming that proliferation of worlds CAUSES average energy per-world to go down. It wouldn’t make much sense to do that, because it is far from certain that the concept of a world is absolutely defined (a point you seem to have been arguing). I was saying that the total energy of the wavefunction remains constant (which isn’t really unreasonable, because it is merely a wave developing over time—we should expect that.) and I was saying that a CONSEQUENCE of this is that we should expect, on average, the energy associated with each world to decrease as we have a constant amount of energy in the wavefunction and the number of worlds is increasing. If you have some way of defining worlds, and you n worlds, and then later have one billion x n worlds, and you have some way of allocating energy to a world, then this would have to happen to maintain conservation of energy. Also, I’m not claiming that the issue is best dealt with in terms of “energy per world” either.
Now you are saying what I first thought you might have meant. :-) Namely, you are talking about the energy of the wavefunction as if it were itself a field. In a way, this brings out some of the difficulties with MWI and the common assertion that MWI results from taking the Schrodinger equation literally.
It’s a little technical, but possibly the essence of what I’m talking about is to be found by thinking about Noether’s theorem. This is the theorem which says that symmetries lead to conserved quantities such as energy. But the theorem is really built for classical physics. Ward identities are the quantum counterpart, but they work quite differently, because (normally) the wavefunction is not treated as if it is a field, it is treated as a quasiprobability distribution on the physical configuration space. In effect, you are talking about the energy of the wavefunction as if the classical approach, Noether’s theorem, was the appropriate way to do so.
There are definitely deep issues here because quantum field theory is arguably built on the formal possibility of treating a wavefunction as a field. The Dirac equation was meant to be the wavefunction of a single particle, but to deal with the negative-energy states it was instead treated as a field which itself had to be quantized (this is called “second quantization”). Thus was born quantum field theory and the notion of particles as field quanta.
MWI seems to be saying, let’s treat configuration space as a real physical space, and regard the second-quantized Schrodinger equation as defining a field in that space. If you could apply Noether’s theorem to that field in the normal way (ignoring the peculiarity that configuration space is infinite-dimensional), and somehow derive the Ward identities from that, that would be a successful derivation of orthodox quantum field theory from the MWI postulate. But skeptical as I am, I think this might instead be a way to illuminate from yet another angle why MWI is so problematic or even unviable. Right away, for example, MWI’s problem with relativity will come up.
Anyway, that’s all rather esoteric, but the bottom line is that you don’t use this “Noetherian configuration-space energy” in quantum mechanics, you use a concept of energy which says that energy is a property of the individual configurations. And this is why there’s no issue of “allocating energy to a world” from a trans-world store of energy embodied in the wavefunction.