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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?

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  • $\begingroup$ Most particles are not scalar fields into $\Bbb R^{k}$. The solutions of the Dirac equation (for fermions) for instance are spinor fields. There's many answers already on the site that talk about how legitimate it is to think of particles in QFT as genuine excitations in some kind of field. This is the picture presented in popular literature but has to be applied carefully when doing actual QFT. Also in order to talk about quantum field theory you're going be dealing with Hilbert spaces and operators rather than classical fields, which is more what you're talking about in your question. $\endgroup$
    – Charlie
    Commented Mar 7, 2021 at 18:10
  • $\begingroup$ Quantum fields map a point in spacetime to an operator on a Hilbert space. $\endgroup$
    – G. Smith
    Commented Mar 7, 2021 at 18:16
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    $\begingroup$ How many dimensions are required to fully describe the physical world? Dimensions of what? Spacetime? $\endgroup$
    – G. Smith
    Commented Mar 7, 2021 at 18:18
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    $\begingroup$ @G.Smith Sorry, I forgot to edit the title of the question after revisiting what I was really asking about. What I'm really after is 1) what kind of field describes the reality of elementary particles, and you're saying that quantum fields actually map a point to an operator in Hilbert space. Do you have a reference I can look into to understand that? 2) Given this, can we describe the physical world as a finite collection of such maps? $\endgroup$
    – smalldog
    Commented Mar 7, 2021 at 20:17
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    $\begingroup$ Thank you @G.Smith. I see that my question contained too many questions within it, so I'll post separate and specific ones shortly. Would you like to put your comments together in an answer that I can accept, to bring this question to a close? $\endgroup$
    – smalldog
    Commented Mar 9, 2021 at 22:03

2 Answers 2

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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.

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    $\begingroup$ Really nice answer. $\endgroup$
    – Árpád Szendrei
    Commented Mar 10, 2021 at 17:10
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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".

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  • $\begingroup$ Thank you for this, but I unfortunately mis-titled my question. Please see my comment to G.Smith above. $\endgroup$
    – smalldog
    Commented Mar 7, 2021 at 20:21

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