In Quantum Field Theory, elementary particles are represented like localized oscillations (localized transverse spherical standing waves) of their underlying fields, or superpositions of their normal modes of oscillation. Is there a way for relating this concept of ‘particles’ also to polytopes or tessellations, similarly to the way quarks are presented in Lattice Quantum Chromodynamics as points on a lattice and gluon ‘particles’ as lines on a lattice? And by similarly I mean something mathematically abstract similarly of course. Can for example the normal modes and quantum numbers be used to form some kind of polytopes or geometric functions? Is the amplituhedron something similar, describing scattering of free particles or it is not? Is there a mathematically viewable representation of a particle like a combination between the two pictures on the top of this page? Because If I understand the Copenhagen Interpretation; and Quantum Field Theory and the Standard Model are some of the most mainstream theories for elementary particles? Aren't they in conflict with each other? Lets say after a particle has been emitted wouldn't it have already been established as a particle and it's wave function established for a very specific trajectory if it doesn't experience a lot of interference from the envirronment? Why does the particle become a probability distribution before coming in front of a change of possibility for it being at two or more places every time it is in front of a possibility for a change of its state of being a particle at any particular single place or an abstract probability density function? Quantum Field Theory seems very intuitive and elegant, but I definitely cannot quite understand the Copenhagen Interpretation or more precisely the part of it that seems to consider elementary particles as solid? If there isn't such a mathematically viewable representation of a particle what would be good sources on the Copenhagen Interpretation? And is it really one of the most mainstream modern theories?
The theory of Causal Dynamical Triangulations (CDT) is a useful concept in, particularly, quantum gravity, which relies on modelling the evolution of spacetime (especially applied to the early Universe) by triangulating the smallest scale phenomena (around Planck scale). This does not mean, however, that every particle gets mapped onto a simplex ('triangle' unit), for at this scale, it is exactly the Heisenberg uncertainty principle that keeps us from assigning a definite point to a particle in space-time. All we can talk about is how to distribute these fluctuations.
For this, you have to properly understand how basic quantum mechanics works. The Copenhagen interpretation is just what it is: an interpretation. There are multiple views; this one assumes that it is the observation that compels the particle to take a certain position (D. Griffiths does a nice job explaining this in the first chapter of Introduction to Quantum Mechanics). In this way, the particle is always described by a probability distribution, before its wavefunction collapse. Elegantly, this distribution is just the distribution of the wave of the particle. On a large scale - sure, its trajectory is well-established, which is in agreement with the correspondence principle. The only difference is the accuracy by which we can measure actual values, which naturally means that we will encounter more uncertainty between complementary properties on smaller scales. It is just so. More so, it is actually already embedded in Quantum Field Theory, in the form of virtual pair production.
Lastly, the Standart Model is the standart quantum field theory. It also includes higher order terms in Dyson series, which describe the virtual pairs, just as every QFT does. So the only representation possible here is a probabilistic (-> deterministic) one. If we are including gravity (which is outside the scope of Standart Model), then the spacetime, as any manifold, triangulates easily and, in this scope, allows for the simplices to evolve by deterministic rules.