only 4 dimensins could be considered almost retrograde, but I think you

have a much better handle on 4 than most of us already. The Dirac spinor has 4 components

(relativistic QM)

and a 4 by 4 tensor is used in electrodynamics. So 4 is indeed comon in anything to do with SRT.

6 or 7 may be much more comon for phase space of just one particles, or 6N for a gas of particles.

Of course mathematicians have their field day on more than 2 dimensions, generalized Pythagoras

with vectors, arbitrary vector spaces, then more generalization into function spaces like Hilbert,

seemingly with infinite dimensions. Then functional analisis starts to work in the infinite dimensional realm.

The math of E. Cartan, the p forms, take vector analisis into the realm of the outer product and higher dimensions

than 3.

Anyway, 4 is a good starting place.

There's nothing special about four components. It's not the number of components that's relevant, it's the symmetries. From the symmetries it's then possible to deduce what are the particular properties of four-component objects.

Yes, there are several mathematical models that incorporate a fourth dimension to describe physical phenomena beyond space and time. Here are a few examples:

Kaluza-Klein Theory: In this theory, proposed by Theodor Kaluza and Oskar Klein in the 1920s, an extra spatial dimension is introduced to unify electromagnetism and gravity. The idea is that this extra dimension, in addition to the three familiar spatial dimensions, allows for the unification of these fundamental forces.

String Theory: String theory is a theoretical framework that postulates the existence of additional spatial dimensions beyond the three we experience. In most versions of string theory, the number of spatial dimensions is typically ten or eleven. These extra dimensions are typically compactified or curled up at incredibly small scales, making them effectively unobservable at our macroscopic level.

Braneworld Models: Braneworld models, inspired by string theory, propose that our observable universe is a four-dimensional "brane" embedded within a higher-dimensional space (the bulk). These models suggest that gravity is confined to the brane, while other fundamental forces can propagate in the full higher-dimensional space.

Grand Unified Theories (GUTs): Some Grand Unified Theories, which attempt to unify the fundamental forces of nature, introduce additional dimensions beyond the three spatial dimensions and time. These extra dimensions often manifest at very high energies, such as the early universe, and play a role in the unification of forces.

It's important to note that these models are highly speculative and still the subject of active research. They are attempts to understand the fundamental nature of the universe, and their mathematical frameworks go beyond the traditional understanding of space and time.

“Apart from space-time, are there any other mathematical models of physical phenomena using a fourth dimension?”

- to answer to this question it is necessary before to understand “what is “dimension”?”, which is in physics applied scientifically only as “a space dimension”, and so to answer to the question it is necessary to understand “what is “space”?”.

The last question is fundamental one, and so cannot be answered really, besides that the fundamental phenomenon/notion “space” is, of course, used in physics – and in everyday all humans’ practice, however that happens purely instinctively, since in mainstream philosophy and sciences the fundamental phenomena/notions “Space” and “Time” [and “Matter”, “Consciousness”, “Energy”, “Information” as well, while all they can be defined only together] are fundamentally transcendent/uncertain/irrational.

As that these phenomena/notions are quite equally are fundamentally transcendent/uncertain/irrational for, say, animals, but they instinctively know that they live in 3D space, where some “distances”, velocities, etc., are;

- and what is in the mainstream physics differs from animals’ knowledge only in that in mainstream physics instead of the quite correct instinctive knowledge that Matter’s space and time compose some real mathematically Euclidian 4D spacetime, in the mainstream physics it is postulated that real Matte’s spacetime is imaginary mathematically, and so rather strange, 4D Minkowski and pseudo-Riemannian spaces.

Or, say, yeah, in mainstream physics quite legitimately a lot of other strange things exist as

“….String Theory: String theory is a theoretical framework that postulates the existence of additional spatial dimensions … Braneworld Models: Braneworld models, inspired by string theory, propose that our observable universe is a four-dimensional "brane" embedded within a higher-dimensional space (the bulk)… Grand Unified Theories (GUTs): Some Grand Unified Theories, which …, introduce additional dimensions beyond the three spatial dimensions and time. These extra dimensions often manifest at very high energies, such as the early universe, and play a role in the unification of forces….”, etc.,

- while yeah

“…It's important to note that these models are highly speculative and still the subject of active research. They are attempts to understand the fundamental nature of the universe, and their mathematical frameworks go beyond the traditional understanding of space and time.….”

- what looks again rather strange – really all that has to physics no any scientific relation.

The fundamental phenomena/notions above can be, and are, scientifically defined only in framework the philosophical 2007 Shevchenko-Tokarevsky’s “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

- and more concretely at application to the studied by physics “Matter” in the Shevchenko-Tokarevsky’s informational physical model , which is based on the conception and is in complete accordance with all existent reliable experimental data, see at least first dozen of pages in

https://www.researchgate.net/publication/355361749_The_informational_physical_model_and_fundamental_problems_in_physics

,

- where [in the conception] it is rigorously scientifically proven that there exist nothing else than some informational patterns/systems of the patterns that are elements of the absolutely fundamental and absolutely infinite “Information” Set; including “Matter” is nothing else than some informational system/the Set’s element.

The utmost general scientific definition of “Information” is

“Information is something that is constructed in accordance with the set/system of absolutely fundamental Rules, Possibilities, Quantities, etc. – the set/system “Logos” in the conception” .

- while “Space”, “Time”, and “Energy/Momentum” are the Logos set elements: and so any/every informational pattern/system, including an informational system “Matter”,

- can be created only by spending some portion of Energy/Momentum,

- can be created and exist only in absolutely “automatically ” appeared at creation its specific concrete space, and

- change in the absolutely fundamentally always existent fundamentally unique, universal for all the Set’s elements, “time dimension”.

In contrast to unique time dimension, concrete space can be composed from arbitrary number of arbitrary dimensions, the number and specificity of the dimensions is completely determined by degreases of freedom at changing states of basic an informational pattern/system elements,

- in Matter that are the primary elementary logical structures – (at least) [4+4+1]4D binary reversible fundamental logical elements [FLE], which have (4+4+1) degreases of freedom at their binary flips,

- and which compose the (at least) [4+4+1]4D dense lattice, which is placed in the corresponding Matter’s fundamentally absolute, fundamentally flat, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct),

The post is rather long already, so more see the links above.

Cheers