Technical caveat: I should have said it’s actually the Hamiltonian, not the Lagrangian that directly tells you the energy of a configuration. (Its easy to convert between Hamiltonians and Lagrangians though, and it turns out Lagrangians are handier for QFT.)
Stephen
Karma: 41
“As I understand it, an electron isn’t an excitation of a quantum electron field, like a wave in the aether; the electron is a blob of amplitude-factor in a subspace of a configuration space whose points correspond to multiple point positions in quantum fields, etc.”
It is hard to tell from the brief description, but it seems to me that you are talking about localized electrons and Wikipedia is talking about delocalized electrons. To describe particles in quantum field theory you have some field in spacetime. In the simplest case of a scalar field it is described by some function f(x,y,z,t). Note that f(x,y,z,t) is not a quantum wavefunction, it is just a classical field. Quantum mechanically, there is an amplitude corresponding to each possible configuration of this field. (Thus the wavefunction is technically a “functional”). Different configurations have different energies. The Langrangian tells you what energy corresponds to what configuration. The Lagrangian for a single field not interacting with anything looks sort of like the Lagrangian for material that can vibrate. (This is just an analogy, it has nothing to do with the aether.)
By a change of basis, we can write the Lagrangian in terms of normal modes, which each behave like harmonic oscillators, and which are decoupled from each other. As a one dimensional example, the normal modes for a violin string are the sine waves whose wavelengths are the length of the string, half the length of the string, 1⁄3 the length of the string, etc. These modes thus correspond to sinusoidal variation of the field. (This has nothing to do with string theory. The violin string is just a handy example of a vibrating system.) We know how to “quantize” the Harmonic oscillator. It turns out that the allowed energies are (n+1/2)h*omega, where n=1,2,3,..., and omega is the resonant frequency of the mode and h is planck’s constant. If the mode of frequency omega is excited to n=1 and the mode of frequency omega’ is excited to n=5 that corresponds to a six electron state with one electron of frequency omega and five electrons of frequency omega’. (Similarly for photons, or any other particles. For photons these frequencies correspond to colors.)
We can have superpositions of different such states. For example we could have quantum amplitude 1/sqrt(2) for mode omega to have n=1 and quantum amplitude 1/sqrt(2) for mode omega to have n=2. If we just have quantum amplitude 1 for a given mode omega to be in the n=1 state, and amplitude zero for all other configurations of the field, then this is a one electron state, where the electron is completely delocalized. What state corresponds to an electron in a particular region? A localized electron does not correspond to the field being nonzero in only a small region (e.g. the violin string has a localized bump in it like this ---^---). That would be a multi-electron state, because it decomposes into a classical superposition of many different sine waves, so we would have n>0 in multiple modes. Instead we can build a localized state of an electron by making a quantum superposition over different modes being occupied. It is important not to get the wavefunction confused with the field f(x,y,z,t). (If you have heard about the Dirac and Klein-Gordon equations, the solutions are analogous to f(x,y,z,t), not analogous to Schrodinger wavefunctions. Historically, there was some confusion on this point.)
Everything I have described so far is the quantum field theory of non-interacting particles. Although I may not have explained that well, it is actually not too complicated. However, if the particles interact, then the normal modes are coupled. Nobody knows how to treat this directly, so you need to use perturbation theory. This is where the complicated stuff about Feynman diagrams and so forth comes in.
I hope this is helpful.
Nick,
Thanks for your comment. If I understand correctly, by c) you are suggesting that consciousness is something like temperature or pressure, a property of physical systems, but one which you don’t need to know about if you are doing a completely detailed simulation. I was lumping this in with epiphenomenalism, since in that case, consciousness does not affect physical systems, it is a descriptor of them. However, I guess the key point is that one can subscribe to epiphenomenalism in this sense without concluding that zombies are logically possible. Because we understand temperature, it is obvious to us that imagining our world exactly as it is except without temperature is nonsensical. To make an even starker example, it would be like saying there are two identical universes that contain five things, but in one of the universes they don’t have the property of fiveness. Maybe if we understood in what way consciousness is a descriptor of physical systems, we would see that our world exactly as it is except without consciousness is a non-sequitur in the same way.
You might argue that the Born rule is an extra postulate dictating how experience binds to the physical universe, particularly if you believe in a no-collapse version of quantum mechanics, such as many-worlds.
While I don’t necessarily endorse epiphenomenalism, I think there may exist an argument in favor of it that has not yet been discussed in this thread. Namely, if we don’t understand consciousness and consciousness affects behavior then we should not be able to predict behavior. So it seems like we’re forced to choose between:
a) consciousness has no effect on behavior (epiphenomenalism)
or
b) a completely detailed simulation of a person based on currently known physics would fail to behave the same as the actual person
Both seem at least somewhat surprising. (b would seem impossible rather than merely surprising to a person who thinks physics is completely known and makes deterministic predictions. In the nineteenth century, most people believed the latter, and some the former. Perhaps this explains how epiphenomenalism originally arose as a popular belief.)
Taking a cue from some earlier writing by Eli, I suppose one way to give ethical systems a functional test is to imagine having access to a genie. An altruist might ask the genie to maximize the amount of happiness in the universe or something like that, in which case the genie might create a huge number of wireheads. This seems to me like a bad outcome, and would likely be seen as a bad outcome by the altruist who made the request of the genie. A selfish person might say to the genie “create the scenario I most want/approve of.” Then it would be impossible for the genie to carry out some horrible scenario the selfish person doesn’t want. For this reason selfishness wins some points in my book. If the selfish person wants the desires of others to be met (as many people do), I, as an innocent bystander, might end up with a scenario that I approve of too. (I think the only way to improve upon this is if the person addressing the genie has the desire to want things which they would want if they had an unlimited amount of time and intelligence to think about it. I believe Eli calls this “external reference semantics.”)
Eli,
I agree that G’s reasoning is an example of scope insensitivity. I suspect you meant this as a criticism. It seems undeniable that scope insensitivity leads to some irrational attitudes (e.g. when a person who would be horrified at killing one human shrugs at wiping out humanity). However, it doesn’t seem obvious that scope insensitivity is pure fallacy. Mike Vassar’s suggestion that “we should consider any number of identical lives to have the same utility as one life” seems plausible. An extreme example is, what if the universe were periodic in the time direction so that every event gets repeated infinitely. Would this mean that every decision has infinite utility consequence? It seems to me that, on the contrary, this would make no difference to the ethical weight of decisions. Perhaps somehow the utility binds to the information content of a set of events. Presumably, the total variation in experiences a puppy can have while being killed would be exhausted long before reaching 3^^^^^3.
Psy-Kosh: I think that is a great question. Here is my take on it:
The wavefunction for six particles will be a function of six variables, x1,y1,z1,x2,y2,z2. You could of course think of these as just six variables without thinking in terms of two particles with three coordinates apiece. However, from this point of view, the system would have certain strange properties that appear coincidental. For example, suppose the two particles are bosons. Then, if we exchange them, nothing happens to the wavefunction. This seems fairly natural. However, from the 6D point of view we have the strange property that if we swap three particular pairs of variables (x1 swapped with x2, y1 swapped with y2, and z1 swapped with z2) the wavefunction is unchanged, whereas in general if we pair the variables in any other way and swap them the wavefunction is changed. Similarly, the potential term in the Hamiltonian will often depend on the distance between the two particles (such as if they repel coulombically). This again seems natural. However, from the 6D point of view this is a mysterious property that the potential depends only on (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2, where we have subtracted variables in pairs in some particular way, rather than in any of the many other ways we could pair them.