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.
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.