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’t Hooft on Cellular Automata and String Theory | Not Even Wrong
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In mid s, I would be intrigued by the special role of permutations, too. That's why I was also attracted by an essay about the event-symmetric spacetime by a writer I didn't know too well at that time, Phil Gibbs. This URL in my directory has been there for 18 years and I couldn't even erase it for 15 years. As someone who had a web page since , you may call me a web fossil. This "increased affinity" to permutation matrices may have driven me closer to Matrix Theory and Matrix String Theory.
I can't resist to say that even though Matrix Theory and Utrecht share the discussion of the relationships between the unitary and permutation operators, they are actually sending physics along very different paths. City 1 However, the evolution operator is a different thing. And because it is such a bizarre hypothesis involving the infinity, you could fool yourself into thinking that it's plausible.
But wake up. You should get back to your senses.
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I wanted to say that it is intrinsically a "discrete" set but I decided to avoid this claim because what I normally mean by this adjective is that the set of possible elements is countable. The weird letter is the Aleph, the first letter of the Hebrew alphabet, and is used to represent the particularly infinite "cardinal" number of elements of the set of integers.
There can't be any solution. It is totally self-evident. You may also speculate that the time may be discrete as well, and if it is discrete, you will avoid the need for continuous functions. But be sure that such theories with a fundamentally discrete time will avoid any agreement with the basic observations, too.
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Such theories may resemble castling in chess — or other moves — but the real world significantly and conceptually differs from chess and everyone who has made at least basic observations of the real world has already noticed the profound differences. If you "hope" in something else, you are just deluding yourself into thinking that it is fine to deny even the very basic experimental data. The time coordinate is manifestly continuous and even if it were not continuous in some sense, there must be an explanation why it looks "almost fully continuous". There isn't any such explanation here. The position is indefensible.
If you really want to deny the basic properties of all observations, such as the continuous evolution of things the state vectors in time, and defend the existence of a loophole, you should have at least a feeble clue what such a loophole could look like and why such a complete coup in physics could still explain the observable data. You are starting physics from scratch and you should realize that until you have a "complete new theory", every single fact and explanation that children and adults are learning at schools and outside schools is against you.
City 2. Doesn't it look exactly like City 1? Too bad that the Copenhagen and Utrecht interpretations are so different, especially in their ability to agree with basic features of the observations of Nature. The position is equally indefensible if we use 't Hooft's other definition of his ontological basis, one saying that the relative phases are unmeasurable. This may be true in approximate effective classical descriptions of a physical system; and it may be true for wave functions combining vectors from different superselection sectors which you shouldn't combine, anyway.
However, small enough physical objects such as elementary particles and their small bound states are demonstrably elements of a single superselection sector and no classical approximation is good enough.
All superpositions may really be prepared by an experimental procedure I may describe and all relative phases are completely physical. All the phases, except for the overall phase if you change all phases in the same way , matter. They have observable consequences. It's easy to demonstrate it for every real-world "small" or "elementary" physical object and an undergraduate student of quantum mechanics should be able to figure these things herself.
But let me pick a trivial example, the spin of an electron. When the other observables of the electron i. This leaves us with three real parameters. And they're identified within "one-dimensional" equivalence classes by the redefinition of the overall phase. So the equivalence classes are still labeled by two real parameters : e. But both parameters are equally physical and equally important. I included the factor for the reader to see that all these things are totally observable quantities, the same angular momentum that can make a gyroscope or the Earth spin a little bit faster, too.
But both latitude and longitude are real and equally important to describe a point on a sphere which, in this case, encodes the direction of the spin!pt.exaqucujem.tk
The Cellular Automaton Interpretation of Quantum Mechanics
It's that simple. You can't dismiss the second, relative phase variable as being less physical than the first one. Doing so would be like telling your pilot that the latitude of his destination is enough.
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It's not enough! The longitude matters, too. The spin of the electron is an extremely simple system you should fully master and, much like the double slit experiment, it is totally enough to see that the complex amplitudes in quantum mechanics have to have the usual "probability amplitude" interpretation clarified by the Copenhagen folks! Whenever you try to "reinterpret" the spin in an "ontological way", you inevitably make the choice of the original up-down axis visible and you break the rotational symmetry. The rotational symmetry of a theory with a spinor describing a single particle may only hold if the components of the spinor are probability amplitudes and if any basis is treated on par with any basis.
After all, as I reminded you, the choice of an orthonormal basis on the space of 2-complex-component spinors is exactly the same thing as the choice of the axis with respect to which we may talk about "up" and "down" spin states. It should be totally clear to you why the relative phase is as important for the spin as the real-valued parameters connected with absolute values. If you can't get this point, you should ask your instructor to give you a failing grade in your undergraduate quantum mechanics course.
And if Prof 't Hooft wants to propose that there is a preferred ontological basis of the two-dimensional Hilbert space of the electron's spin, he should fail in the undergraduate quantum mechanics course, too.
The Nobel prizes may be given to people without the bachelor degrees, however, so the failing grade is just fine. In the case of the spin, the meaning of the relative phase was particularly clear — it informs us about the longitude of the end point of the axis with respect to which the electron is spinning up. But in all other examples in quantum mechanical theories including quantum field theory and string theory, of course , one may show that relative phases always influence predictions of observable phenomena.
In the double slit experiment which is also enough to understand all of quantum mechanics if you think about it carefully enough, as Feynman pointed out, the relative phase between the amplitudes for the particle to be near the "left slit" and for the particle to be near the "right slit" informs you about the position of the interference maxima and minima. The relative phase just moves the "strips" in the interference pattern in the transverse direction to these strips.