Timeline for Anticommutator Relation of Quantized Fermionic Field and Fermi–Dirac statistics: How are these related?
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9 events
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| Jul 11, 2023 at 16:13 | comment | added | user267839 | to us which bracket relation we should take for associated creation/annihilation op's? In other words why we would obtain something absurd if we would try to assign naively just any random (only formally mathematically meaningful) bracket relation coming into mind , ie under "ignoring" the given symmetries relative to particle exchange our wave function have. Where in the construction something breakes badly down if we would try to do our quantization it that way? (my motivation is if it's possible to refute this suggested "approach" by certain reductio ad adsurdum argument) | |
| Jul 11, 2023 at 16:05 | comment | added | user267839 | So far I know the "why particles with half-spin have to have asymmetrical wave function relative to particle exchange" part is exactly the of spin-statistic theorem I'm familar with. So that's not the issue of my concern. The point which I still not understood is, say we start with some many-particle wave function and want to perform second quantization on it. As you said it posses as a honest function certain symmetries relative to particle exchange, which we a priori know, since this function is given. Now the funny question is why these symmetries on level of this wave function "dictate" | |
| Jul 11, 2023 at 2:33 | comment | added | freude | You may ask why particles with half-spin have to have asymmetrical wave function relative to particle exchange, but this is another question. It has something to do with the phase of the wave function and rotations I believe. What we have established here is the commutator rules are governed by the symmetry of the wave function relative to particle exchange and also this symmetry defines what statistics should be. | |
| Jul 11, 2023 at 2:31 | comment | added | freude | In other words, to build a many-body theory of particles you need to know what those particles are in terms of spin and mass and how they interact. | |
| Jul 11, 2023 at 2:29 | comment | added | freude | If many many-particle wave function of indistinguishable particles is given (you say you start with it), it possesses certain symmetry relative to particle exchange in it. This will affect the commutation rules for the second-quantization operators. | |
| Jul 10, 2023 at 15:53 | comment | added | user267839 | then we would be able to deduce that the wave function we initially started with should have been asymmetric wrt particle exchange? Do I understand your statement correctly? | |
| Jul 10, 2023 at 15:53 | comment | added | user267839 | I'm not completely sure if I understand you first statement correctly. Say we start with a many-particle wave function and want to turn it into an operator via second quantization. At this stage we a priori "don't know yet" from naive point of view, if we should impose commutator or anticommutator rule. Should your statement be read as that if we would be "somehow" able to figure out, that the only possible relation we can impose for the quantized field must be anticommutative, | |
| Jul 10, 2023 at 12:36 | history | edited | freude | CC BY-SA 4.0 |
added 117 characters in body
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| Jul 10, 2023 at 12:29 | history | answered | freude | CC BY-SA 4.0 |