Physics states that ‘symmetry dictates interaction’ [1][2]; Invariance, symmetry, and conservation are usually approximately the same concepts [3], and the objects of conservation are usually discrete. The basic conservation of energy corresponds to the energy quantum e = hν, the basic conservation of momentum to the momentum quantum P =h/λ, the conservation of charge to the integer charge e, the conservation of the spin number to ℏ/2, the conservation of the particle number to the lepton number, the baryon number [4], and so on.
1) Does Noether's theorem impose a limit on the continuity of energy and momentum [5]?
2) If we regard these discretisations as representing different energy forms, do the symmetries likewise convert when the energy forms convert?
3) Assuming that an abstract energy remains constant in all cases, should there likewise be any symmetries that remain constant all the time to support symmetry evolution?
4) Should these different discretisations have a common origin? If so, how are the relationships between them constructed? Or through what channels are they related?
5) Particle number conservation are all additive and empirical postulates [4], should there be theoretical support behind them?
6) Symmetries are classified into external and internal symmetries [6]; external symmetries are concerned with spacetime coordinate transformations and internal symmetries are concerned with gauge invariance. If they are united, how are inner space symmetries related to external space symmetries?
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References：
[1] Yang, C. N. (1996). Symmetry and physics. Proceedings of the American Philosophical Society, 140(3), 267-288.
[2] Gross, D. J. (1992). Gauge theory-past, present, and future? Chinese Journal of Physics, 30(7), 955-972.
[3] “Symmetry, Invariance and Conservation (1) - Who is the Primary?”；https://www.researchgate.net/post/NO20Symmetry_Invariance_and_Conservation_1-Who_is_the_Primary
[4] Krieger, P. (2006). Conservation Laws - PHY357_Lecture6. https://www.physics.utoronto.ca/~krieger/PHY357_Lecture6.pdf
[5] Kosmann-Schwarzbach, Y. (2011). The Noether Theorems. In Y. Kosmann-Schwarzbach & B. E. Schwarzbach (Eds.), The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century (pp. 55-64). Springer New York. https://doi.org/10.1007/978-0-387-87868-3_3
[6] Wess, J. (2000). From symmetry to supersymmetry. In The supersymmetric world: the beginnings of the theory (pp. 67-86). World Scientific. https://www.changhai.org/articles/translation/physics/sym_and_supersym3.php （中文版）
Dear Chian Fan , here are the short answers to the questions asked.
1. No, it does not.
2. The considered forms of discretization are not different forms of energies. It is the quantum of action h that takes on the various forms you are considering. For this (discretization and transformation of form) it turns out to be a necessary and sufficient condition. The only conservation law becomes the conservation of the number of action quanta in an isolated system.
3. Yes, if the words "abstract energy" are replaced by "the number of quanta of action".
4. As already stated in point 2, the quantum of action h is common to different forms of discretization. It can have different, one might say independent embodiments for translational motion in the form of momentum and energy of a particle, rotational motion in the form of angular momentum and spin, electric charge and magnetic flux in the form of quantum, discrete mass of the elementary particle and the curvature of the surrounding space created around it. For each of the 4 types of embodiment, the conservation of the number of action quanta works independently. I would venture to note that there are also other types of embodiment of action quanta within the space of the nucleus and their transformation into discrete elements of the space surrounding them. But it is too early to talk about this in the context of this discussion.
5. Conservation of the number of particles is not a strict conservation law, just like conservation of their mass. During nuclear transformations, these laws begin to be violated. But if we consider quanta of action h, then everything will be in order with the conservation of their number.
6. I cannot answer this question, since it goes beyond my ideas.
Sincerely yours, Dulin Mikhail.
Dear Chian, your questions explore deep connections between symmetry and conservation/discrete systems in physics. :}
> Noether"s theorem does not impose discreteness but links continuous symmetries to conserved quantities; quantization arises from boundary conditions and quantum mechanics.
>> Symmetries can transform with energy conversions, as seen in phase transitions and effective field theories.
>>> Some fundamental symmetries like for instance Poincare symmetry, persist across scales, while others evolve due to renormalization.
>>>> A common origin may lie in gauge/spacetime symmetries, potentially unified in higher-dimensional frameworks.
>>>>> Some empirical conservation laws such as baryon and lepton number, may have deeper theoretical explanations in grand unification or anomaly cancellations.
>>>>>> Unifying internal [i.e. gauge) and external (spacetime) symmetries could involve deeper geometric structures, as suggested in supersymmetry and some string theories.
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