Can we fully describe the physical world as a collection of fields? I'm new to the Physics SE, coming from a pure math background. I think my question is full of incomprehension and lack of basic knowledge, but here goes.
I'm trying to wrap my head around the modern description of our physical world. My basic understanding is that there are $n$ known elementary particles (that might be 17/24/25/31 depending on what counts, but the details are not so important to me). Particles can be understood as 'ripples' or 'excitations' of their underlying fields. I assume the nature of this field might change depending on the particle (scalar/vector/tensor?), but what does excitation mean? Above/below some kind of threshold that brings the field to a 'non-ground' state?
Now assuming that each elementary field is given by $F_i : \mathbb{R}^4 \to \mathbb{R}^{k_i}$ for each $i \in \{1, \ldots, n\}$, does this mean that the physical world we live in can be fully described as the collection $\{F_i\}$ which takes in a point in spacetime $(x,y,z,t)$ and returns the 'excitation' level of each elementary field at that point?
Edit: following G.Smith's comment, quantum fields map a point in spacetime to an operator on a Hilbert space. Is there a finite number of such quantum fields for each elementary particle, and so can we describe fully the physical universe as a finite collection of such maps in principle?
 A: The Standard Model of particle physics describes the world as 17 interacting quantum fields: 1 scalar field with spin 0, 12 spinor fields with spin 1/2, and 4 vector fields with spin 1. (There are various ways to count the fields, but this is one common way.) It includes three of the four known forces. It omits gravity, which General Relativity explains non-quantum-mechanically as due to the curvature of the 4D pseudo-Riemannian manifold that is spacetime.
Each quantum field describes one kind of elementary particle. (And also its antiparticle, but for some fields the particle and antiparticle are the same.) The particle associated with the one scalar field is the Higgs boson. The particles associated with the 12 spinor fields are the 6 quarks, 3 charged leptons, and 3 neutrinos. The particles associated with the 4 vector fields are the photon, gluon, and $W$ and $Z$ weak bosons. We can do experiments on all of these particles with current technology. There may be other fields (and their particles) that we cannot yet observe.
In some quantum gravity theories, gravity is described by a quantized tensor field with spin 2. The associated particle is called a graviton. This is speculative as we cannot detect individual gravitons with current technology.
A: It would probably depend on what you consider "dimensions" to be. If it is, spatial and temporal dimensions, 4 is the most popular choice, although string theorists may disagree. Quantum Mechanics operates in Hilbert space, potentially requiring infinite "dimensions" (orthogonal axis).
I'm guessing your question is about "how many different parameters would I need to fully describe the universe?". If that is the case, the Standard Model says " a lot of them".
