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On the Topological Nature of Noether's Theorem (2)

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Posted 2025-11-27 01:37:52

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Topological Vortex Theory (TVT) provides an ideal mathematical model for this. Within this theory, the vacuum is not empty but a dynamic network composed of countless interconnected topological vortex-antivortex pairs, constantly being created and annihilated [1, 2, 10]. The symmetries described by Noether's theorem are precisely the macroscopic, continuous-limit statistical average manifestation of this topological structure.

2. Topological Vortex-Antivortex Pairs and the Topological Realization of PCT Symmetry

A topological vortex is a non-trivial, stable defect in the spacetime metric or field configuration, analogous to quantum vortices in superfluid or cosmic strings [1, 10]. Its stability is guaranteed by topological invariants (e.g., winding number, Chern number) and cannot be eliminated through continuous deformation [2, 3].

We postulate that, at the most fundamental level, fluctuations in spacetime spontaneously give rise to topological vortices and their twin anti-topological vortices. At the moment of their creation, this vortex-antivortex pair constitutes a perfectly symmetric system:

a) Parity Conservation (P) Symmetry: The vortex and antivortex possess opposite chirality. For instance, one might be left-handed, the other necessarily right-handed. When the entire system undergoes spatial inversion, the left-handed vortex becomes right-handed and vice versa, yet the overall topological configuration of the system remains unchanged because both coexist.

b) Charge Conjugation (C) Symmetry: In the topological context, "charge" can be broadly interpreted as a topological charge (e.g., winding number). The vortex carries a positive topological charge, while its antivortex carries an equal negative charge. Under a C transformation, they interchange; positive and negative topological charges are swapped, but the total topological charge of the system remains zero, preserving symmetry [9].

c) Time Reversal (T) Symmetry: The formation process of a vortex, under time reversal, corresponds to the annihilation process of an antivortex. However, at the instant of creation, we consider the "formation" event itself; the dynamical paths of the vortex and antivortex are mapped onto each other under time reversal.

According to the fundamental CPT theorem of quantum field theory, any local Lorentz-invariant quantum field theory must be invariant under the combined PCT transformation [8]. Our model suggests that the physical carrier of this ultimate symmetry is precisely the topological configuration of the vortex-antivortex pair at the point of creation. At this moment, they form an inseparable, topologically entangled state satisfying perfect PCT symmetry. The conservation laws corresponding to Noether's theorem are the macroscopic manifestation of this topological entanglement symmetry in the continuous limit.

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