Please note that, while an appealing idea, timeless physics is not a physical theory but only a hope for one. As far as I know, it has no solid mathematical foundations separate from the mainstream and makes no new testable predictions.
With this caveat, why do you want to ignore the complex numbers, given that they are just a set of real commuting 2x2 matrices? Or do you want to get rid of all matrices (i.e. of all linear algebra) in physics? If so, what would be your motivation?
Please note that, while an appealing idea, timeless physics is not a physical theory but only a hope for one.
Really? Isn’t it just the idea that Schroedinger’s time-independant equation is the correct one? Is there not a time-independent version of the correct one (as opposed to the non-relativistic approximation I’ve seen)?
With this caveat, why do you want to ignore the complex numbers, given that they are just a set of real commuting 2x2 matrices?
If you can remove complex numbers in this manner, you could use R, or R^2 in the same manner as you’d use C, but you could also use R^3, R^4, R^5, etc. It must be more likely that any of those is correct than that R^2 is correct in particular.
Really? Isn’t it just the idea that Schroedinger’s time-independant equation is the correct one? Is there not a time-independent version of the correct one (as opposed to the non-relativistic approximation I’ve seen)?
The best of what people have come up with so far is the Wheeler-de Witt equation. Unfortunately it is not good enough.
If you can remove complex numbers in this manner, you could use R, or R^2 in the same manner as you’d use C, but you could also use R^3, R^4, R^5, etc. It must be more likely that any of those is correct than that R^2 is correct in particular.
The best of what people have come up with so far is the Wheeler-de Witt equation. Unfortunately it is not good enough.
From what I can find, that looks like some attempt at quantum gravity. We can’t do that with MWI either, as far as I know. Am I mistaken about this?
Commuting 2x2 real matrices are the smallest real representation of C.
I guess I’ll take your word for it.
Do you really need C for quantum physics though? You can’t multiply two amplitudes together. The only thing I’ve seen is rotating by 90 degrees in Schrodinger’s time-dependent equation. If I accept timeless physics, it doesn’t even do that.
You do this whenever you calculate the amplitude contributed by a single history within a sum over histories. The amplitude for an event is exp(i.action) and action is additive, so the amplitude for two events forming a single history is the product of their individual amplitudes, exp(i.action1+i.action2). In this respect it’s just like ordinary probability theory, where you multiply probabilities for conjunction of events and add them for disjunction.
I don’t understand the motivation or the assumptions for what you are doing. Quantum cosmology is such a guessing game that even unusual formal generalizations might lead somewhere, and I would like to offer you useful feedback, but I’m wondering if there’s some basic misconception about QM motivating you.
the amplitude for two events forming a single history is the product of their individual amplitudes
I’m not sure I understand. What is an “event”?
I’ve noticed that the amplitude of a system is equal to the product of the amplitudes of the component particles, but that’s just mathematical shorthand. Individual particles don’t have their own amplitude. Only the universe does.
I don’t understand the motivation or the assumptions for what you are doing.
I’m trying to make it simpler. It’s not much, but each bit you can shave off of the equations doubles the probability.
An event is a “thing that happens”. Relativity made discussion of “events” routine in physics, because one wants to talk about something—the tick of a clock, the emission or absorption of a photon—that is localized in space and time. “Event” is a completely standard term of art in relativity—thus “event horizon”. Of course, it is also an elementary everyday word and concept, independent of its use in physics.
In standard probability calculus, P((A and B) or (C and D)) = P(A) x P(B) + P(C) x P(D). You’re summing the probabilities of two possibilities: the possibility that A and B occur together, and the possibility that C and D occur together. Feynman’s reformulation of quantum mechanics as a “sum over histories” has this schematic form as well, except that it is the complex-valued “probability amplitudes” that we multiply and add. The basic events are “a particle moves from one place to another place” and “a particle is emitted or absorbed”, and the amplitude or these events is “e” to the power of “i” times the “action” for this event, action being a concept from classical physics which carries over to quantum theory, and which in fact assumes a fundamental role there.
A standard example of a timeless-looking construction from quantum cosmology is the Hartle-Hawking wave-function of the universe, derived from a “no-boundary condition”. This wavefunction assigns an amplitude of three-dimensional configurations of the universe; which sounds like Barbour’s “Platonia”, But how are these amplitudes calculated? By summing over space-time histories which evolve to the three-dimensional configuration of interest. The amplitude for a configuration X is the sum of the amplitudes of every space-time history which starts from “nothing” (that’s why this is the “no-boundary condition”) and which evolves to X.
In a timeless framework, you could possibly conceive of an event as the difference between one configuration and its neighbor in configuration space, and the amplitude for the “event” as the weighting for the contribution made by the first configuration to the amplitude of the second configuration, via timeless amplitude “flow”. That is, if you have one configuration of particles, and then a neighboring configuration which is the same except that there is now an extra photon on top of one of the electrons, then the “event” corresponding to this configurational difference would be “emission of a photon by that electron”, and the usual Feynman amplitude for this event would define the proportional contribution to the amplitude flow entering timelessly into the second configuration’s point in configuration space.
It’s a standard fact about the Schrodinger and Feynman formulations of quantum mechanics that they are equivalent—the evolution of the Schrodinger wavefunction is equivalent to the cumulative flow of amplitude produced by the converging and diverging Feynman histories—and this should carry over to the timeless case of quantum cosmology… in principle. But in practice, the Feynman formulation seems more relativistic so perhaps it’s more fundamental. In any case, you do sometimes multiply amplitudes when you do quantum mechanics as Feynman did it.
The basic events are “a particle moves from one place to another place” and “a particle is emitted or absorbed”, and the amplitude or these events is “e” to the power of “i” times the “action” for this event
Is that like the idea that a particle being in a certain position has an amplitude? It doesn’t. The universe does. It’s just that if you pretended that a decohered particle was its own universe, you’d get the same results from much simpler calculations.
This does explain why physicists tend to write amplitude as a complex number. I’ve wondered that for a while.
Please note that, while an appealing idea, timeless physics is not a physical theory but only a hope for one. As far as I know, it has no solid mathematical foundations separate from the mainstream and makes no new testable predictions.
With this caveat, why do you want to ignore the complex numbers, given that they are just a set of real commuting 2x2 matrices? Or do you want to get rid of all matrices (i.e. of all linear algebra) in physics? If so, what would be your motivation?
Really? Isn’t it just the idea that Schroedinger’s time-independant equation is the correct one? Is there not a time-independent version of the correct one (as opposed to the non-relativistic approximation I’ve seen)?
If you can remove complex numbers in this manner, you could use R, or R^2 in the same manner as you’d use C, but you could also use R^3, R^4, R^5, etc. It must be more likely that any of those is correct than that R^2 is correct in particular.
The best of what people have come up with so far is the Wheeler-de Witt equation. Unfortunately it is not good enough.
Commuting 2x2 real matrices are the smallest real representation of C.
From what I can find, that looks like some attempt at quantum gravity. We can’t do that with MWI either, as far as I know. Am I mistaken about this?
I guess I’ll take your word for it.
Do you really need C for quantum physics though? You can’t multiply two amplitudes together. The only thing I’ve seen is rotating by 90 degrees in Schrodinger’s time-dependent equation. If I accept timeless physics, it doesn’t even do that.
You do this whenever you calculate the amplitude contributed by a single history within a sum over histories. The amplitude for an event is exp(i.action) and action is additive, so the amplitude for two events forming a single history is the product of their individual amplitudes, exp(i.action1+i.action2). In this respect it’s just like ordinary probability theory, where you multiply probabilities for conjunction of events and add them for disjunction.
I don’t understand the motivation or the assumptions for what you are doing. Quantum cosmology is such a guessing game that even unusual formal generalizations might lead somewhere, and I would like to offer you useful feedback, but I’m wondering if there’s some basic misconception about QM motivating you.
I’m not sure I understand. What is an “event”?
I’ve noticed that the amplitude of a system is equal to the product of the amplitudes of the component particles, but that’s just mathematical shorthand. Individual particles don’t have their own amplitude. Only the universe does.
I’m trying to make it simpler. It’s not much, but each bit you can shave off of the equations doubles the probability.
An event is a “thing that happens”. Relativity made discussion of “events” routine in physics, because one wants to talk about something—the tick of a clock, the emission or absorption of a photon—that is localized in space and time. “Event” is a completely standard term of art in relativity—thus “event horizon”. Of course, it is also an elementary everyday word and concept, independent of its use in physics.
In standard probability calculus, P((A and B) or (C and D)) = P(A) x P(B) + P(C) x P(D). You’re summing the probabilities of two possibilities: the possibility that A and B occur together, and the possibility that C and D occur together. Feynman’s reformulation of quantum mechanics as a “sum over histories” has this schematic form as well, except that it is the complex-valued “probability amplitudes” that we multiply and add. The basic events are “a particle moves from one place to another place” and “a particle is emitted or absorbed”, and the amplitude or these events is “e” to the power of “i” times the “action” for this event, action being a concept from classical physics which carries over to quantum theory, and which in fact assumes a fundamental role there.
A standard example of a timeless-looking construction from quantum cosmology is the Hartle-Hawking wave-function of the universe, derived from a “no-boundary condition”. This wavefunction assigns an amplitude of three-dimensional configurations of the universe; which sounds like Barbour’s “Platonia”, But how are these amplitudes calculated? By summing over space-time histories which evolve to the three-dimensional configuration of interest. The amplitude for a configuration X is the sum of the amplitudes of every space-time history which starts from “nothing” (that’s why this is the “no-boundary condition”) and which evolves to X.
In a timeless framework, you could possibly conceive of an event as the difference between one configuration and its neighbor in configuration space, and the amplitude for the “event” as the weighting for the contribution made by the first configuration to the amplitude of the second configuration, via timeless amplitude “flow”. That is, if you have one configuration of particles, and then a neighboring configuration which is the same except that there is now an extra photon on top of one of the electrons, then the “event” corresponding to this configurational difference would be “emission of a photon by that electron”, and the usual Feynman amplitude for this event would define the proportional contribution to the amplitude flow entering timelessly into the second configuration’s point in configuration space.
It’s a standard fact about the Schrodinger and Feynman formulations of quantum mechanics that they are equivalent—the evolution of the Schrodinger wavefunction is equivalent to the cumulative flow of amplitude produced by the converging and diverging Feynman histories—and this should carry over to the timeless case of quantum cosmology… in principle. But in practice, the Feynman formulation seems more relativistic so perhaps it’s more fundamental. In any case, you do sometimes multiply amplitudes when you do quantum mechanics as Feynman did it.
Is that like the idea that a particle being in a certain position has an amplitude? It doesn’t. The universe does. It’s just that if you pretended that a decohered particle was its own universe, you’d get the same results from much simpler calculations.
This does explain why physicists tend to write amplitude as a complex number. I’ve wondered that for a while.