Cramer's TI (was Re:Quantum Theory interpretation)
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Time Doesn't Exist
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· Score: 1
Cramer's transactional interpretation always rather reminds me of the data assimilation process in modern weather forecasting systems.
The measurements from a single time point are not enough to fix the state of the weather model, so the model state is estimated from all the data gathered over a 24-48 hour period. One approach (4D-var) is to run the model forward, note the differences between the predictions and the observed data at each stage, then propagate these difference fields backwards in time to produce an improved estimate of the initial conditions. After several trips forward and back, the model should coverge to its best estimate of the historical path.
Cramer's model is rather like that: the measurement system is a vital boundary condition which changes our best estimate of the entire history, so Cramer assimilates it by propagating the mismatch backwards and forwards to convergence. This is a great way to picture the Copenhagen interpretation, underlining its insistence that an experimental quantum mechanical system is not completely specified until the macroscopic measurement apparatus is specified.
But I do prefer to think of TI as how an imaginary computer might cope with assimilating the asynchronous information into a model, rather than having to invest too much ontological reality into his 'propose' and 'accept' waves.
For myself, I believe the most fundamental thing about QM is that it forces us to abandon the idea of a space-time of completely independent points -- the universe just doesn't contain that much information -- and instead there is always a sharing of information between a point and its neighbours. So I'm not too bothered that there appear to be, at the least, a non-local superselection rule connecting a quantum object and its measuring device. Treating them as probabilistically independent is an approximation -- useful, but wrong.
If we now alter our approximation to reflect the interdependence, we should not be surprised that we must also alter our best estimate of what the test object is doing/has done. But this is a sudden change in our knowledge (epistemology), not a sudden change in reality (ontology). Doing so, we will infer a new, consistent trajectory for the combined object+apparatus system, which is just like the final version of history left in Cramer's TI, once all the to-ing and fro-ing is done: a smooth, deterministic evolution, without any sudden jumps or discontinuities, and no identifiable quantum measurement 'event'.
((Footnote: to hedge my bets, I should also say that I'm quite impressed by Zurek's decoherence calculations, which I suppose support MWI; though I'm not sure whether they help much with EPR))
Localisation (was Re:Quantum Theory interpretation
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Time Doesn't Exist
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· Score: 2
Bzzt! If Schroedinger's equation is a pure diffusion equation -- meaning that given a start point for where a particle is, it's eventual location spreads over time. By your argument, all matter would have diffused to everywhere in the Universe by now, and there'd be no way to distinguish between your monitor, your keyboard, and your navel.
Actually, it wouldn't diffuse:-)
But it is very easy to think that it would, from the typical undergraduate wave mechanics course -- I know I did!
The very first exercise everyone does is to show how a gaussian wavepacket inexorably spreads out; and then after that the only localised states you ever encounter are bound states, with the quantum object penned in by an ever growing potential -- which seems quite inappropriate for a macroscopic object, because the potential would have to be enormous.
However, in fact quantum mechanics doesn't predict that everyday objects should diffuse off to infinity; and the reason why is a purely wave thing -- it works just as well for sound and vibration. Finding this out probably gave me a bigger shock than any other misconception I suddenly realised in my undergraduate physics course, and I think it's something which is criminally under-emphasised in the teaching of wave mechanics: you don't need quantum measurement to explain why apparently free macroscopic objects are localised.
The revelation was a phenomenon discovered by Philip Anderson in 1958 in a paper he called "Absence of diffusion in certain random lattices". (They later gave him a Nobel prize for it.) Anderson's result was that the eigenstates of quantum objects actually are spatially localised, even for very small absolute differences in the potential function (much less than would be needed for a bound state), if the potential is not smooth but disordered.
In fact it is now known that in 1D and 2D any magnitude of disorder, however small (so long as it persists throughout the surface), will produce exponential localisation. And this is not some strange quantum phenomenon; it happens for any waves -- the same mechanism for example can cause vibration energy to get trapped in a small part of a large engineering structure. In 3D a rule of thumb is that the energy will be unable to diffuse away if the effective scattering length is less than one wavelength (the Ioffe-Regel criterion).
Now for a macroscopic object, the de Broglie wavelength is very very small. But I think the effective scattering length is even shorter. The key is the disorder of the object itself. Even if each of its constituent particles are experiencing quite small local potentials with a slow spatial variation, these all have to be added together to give the potential seen by the centre of mass. Even quite a gently changing potential has spectral power implicit in the higher modes of its fourier transform; but adding together all the different contributions effectively amplifies and phase randomises these frequencies. So the potential the centre of mass sees will have real high spatial frequency oscillations -- it won't be smooth at all!
That is why the effective scattering length for macroscopic objects in a typical environment is in fact even shorter than the de Broglie wavelength; and thus why the "good quantum states" for my monitor, my keyboard and my navel are all very strongly Anderson localised.
Ooops... the original mathematical demonstration of supersymmetry was Dimopolous, Fayet, Gol'fand and Lichtman, 1971; and the first mainstream supersymmetric particle model was Wess and Zumino, 1973.
Unfortunately, renormalization consists of a somewhat more sophisticated version of the following equations:
A/0 = B/0 {yes, that's dividing both sides by zero}
Therefore A = B.
Or rather, a + A/0 = b + B/0, so a=b (Feynman, Schwinger, Tomonaga, 1947).
But you could ask whether this is really so much worse than A.0 = k B.0 (Newton, Leibniz c.1680).
In both cases, once you have found the right way to show that the cancellations work for all finite values, the limit starts to look plausible (and you can start isolating just what situations would break it). 't Hooft gave the fundamental proof that all gauge theories genuinely are renormalisable (including the electroweak theory and QCD).
The important thing about renormalisation is that the problem isn't with the interactions, it's that the set of basis functions that you're using to expand space are getting more and more nearly orthogonal to reality. That means you end up with something rather like a very very ill-conditioned matrix to invert. (If you like, this is the O^(-1) (B-A)). You don't have to expand in such a bad basis, though. 't Hooft was able to show that if certain symmetry properties hold then all the nasties cancel, whatever the basis expansion.
So it probably isn't still true that all physicists are "exceedingly unhappy" about renormalisation; but a better sequence of basis functions for thinking about small-scale reality would certainly be nice.
't Hooft also found a quite unexpected mathematical gauge symmetry between bosons and fermions, which probably deserves a Nobel prize by itself. We still don't know whether the laws of nature have this supersymmetry or not, but the idea has fascinated theoretical physicists for 25 years, and is built in to superstrings in their very foundations.
Slashdot Mirror
http://www.theregister.co.uk/991018 -000001.html
And Ret Hat is going to be poor soon enough :-)
The measurements from a single time point are not enough to fix the state of the weather model, so the model state is estimated from all the data gathered over a 24-48 hour period. One approach (4D-var) is to run the model forward, note the differences between the predictions and the observed data at each stage, then propagate these difference fields backwards in time to produce an improved estimate of the initial conditions. After several trips forward and back, the model should coverge to its best estimate of the historical path.
Cramer's model is rather like that: the measurement system is a vital boundary condition which changes our best estimate of the entire history, so Cramer assimilates it by propagating the mismatch backwards and forwards to convergence. This is a great way to picture the Copenhagen interpretation, underlining its insistence that an experimental quantum mechanical system is not completely specified until the macroscopic measurement apparatus is specified.
But I do prefer to think of TI as how an imaginary computer might cope with assimilating the asynchronous information into a model, rather than having to invest too much ontological reality into his 'propose' and 'accept' waves.
For myself, I believe the most fundamental thing about QM is that it forces us to abandon the idea of a space-time of completely independent points -- the universe just doesn't contain that much information -- and instead there is always a sharing of information between a point and its neighbours. So I'm not too bothered that there appear to be, at the least, a non-local superselection rule connecting a quantum object and its measuring device. Treating them as probabilistically independent is an approximation -- useful, but wrong.
If we now alter our approximation to reflect the interdependence, we should not be surprised that we must also alter our best estimate of what the test object is doing/has done. But this is a sudden change in our knowledge (epistemology), not a sudden change in reality (ontology). Doing so, we will infer a new, consistent trajectory for the combined object+apparatus system, which is just like the final version of history left in Cramer's TI, once all the to-ing and fro-ing is done: a smooth, deterministic evolution, without any sudden jumps or discontinuities, and no identifiable quantum measurement 'event'.
((Footnote: to hedge my bets, I should also say that I'm quite impressed by Zurek's decoherence calculations, which I suppose support MWI; though I'm not sure whether they help much with EPR))
Actually, it wouldn't diffuse :-)
But it is very easy to think that it would, from the typical undergraduate wave mechanics course -- I know I did!
The very first exercise everyone does is to show how a gaussian wavepacket inexorably spreads out; and then after that the only localised states you ever encounter are bound states, with the quantum object penned in by an ever growing potential -- which seems quite inappropriate for a macroscopic object, because the potential would have to be enormous.
However, in fact quantum mechanics doesn't predict that everyday objects should diffuse off to infinity; and the reason why is a purely wave thing -- it works just as well for sound and vibration. Finding this out probably gave me a bigger shock than any other misconception I suddenly realised in my undergraduate physics course, and I think it's something which is criminally under-emphasised in the teaching of wave mechanics: you don't need quantum measurement to explain why apparently free macroscopic objects are localised.
The revelation was a phenomenon discovered by Philip Anderson in 1958 in a paper he called "Absence of diffusion in certain random lattices". (They later gave him a Nobel prize for it.) Anderson's result was that the eigenstates of quantum objects actually are spatially localised, even for very small absolute differences in the potential function (much less than would be needed for a bound state), if the potential is not smooth but disordered.
In fact it is now known that in 1D and 2D any magnitude of disorder, however small (so long as it persists throughout the surface), will produce exponential localisation. And this is not some strange quantum phenomenon; it happens for any waves -- the same mechanism for example can cause vibration energy to get trapped in a small part of a large engineering structure. In 3D a rule of thumb is that the energy will be unable to diffuse away if the effective scattering length is less than one wavelength (the Ioffe-Regel criterion).
Now for a macroscopic object, the de Broglie wavelength is very very small. But I think the effective scattering length is even shorter. The key is the disorder of the object itself. Even if each of its constituent particles are experiencing quite small local potentials with a slow spatial variation, these all have to be added together to give the potential seen by the centre of mass. Even quite a gently changing potential has spectral power implicit in the higher modes of its fourier transform; but adding together all the different contributions effectively amplifies and phase randomises these frequencies. So the potential the centre of mass sees will have real high spatial frequency oscillations -- it won't be smooth at all!
That is why the effective scattering length for macroscopic objects in a typical environment is in fact even shorter than the de Broglie wavelength; and thus why the "good quantum states" for my monitor, my keyboard and my navel are all very strongly Anderson localised.
So, not 't Hooft... sorry.
A/0 = B/0 {yes, that's dividing both sides by zero}
Therefore A = B.
Or rather, a + A/0 = b + B/0, so a=b (Feynman, Schwinger, Tomonaga, 1947).
But you could ask whether this is really so much worse than A.0 = k B.0 (Newton, Leibniz c.1680).
In both cases, once you have found the right way to show that the cancellations work for all finite values, the limit starts to look plausible (and you can start isolating just what situations would break it). 't Hooft gave the fundamental proof that all gauge theories genuinely are renormalisable (including the electroweak theory and QCD).
The important thing about renormalisation is that the problem isn't with the interactions, it's that the set of basis functions that you're using to expand space are getting more and more nearly orthogonal to reality. That means you end up with something rather like a very very ill-conditioned matrix to invert. (If you like, this is the O^(-1) (B-A)). You don't have to expand in such a bad basis, though. 't Hooft was able to show that if certain symmetry properties hold then all the nasties cancel, whatever the basis expansion.
So it probably isn't still true that all physicists are "exceedingly unhappy" about renormalisation; but a better sequence of basis functions for thinking about small-scale reality would certainly be nice.
't Hooft also found a quite unexpected mathematical gauge symmetry between bosons and fermions, which probably deserves a Nobel prize by itself. We still don't know whether the laws of nature have this supersymmetry or not, but the idea has fascinated theoretical physicists for 25 years, and is built in to superstrings in their very foundations.
A thoroughly worthy winner.