In her recent __ Physics Today__ article April, p. 38), contributor Hanna Price writes;

"What would the universe be like if it had four spatial dimensions instead of three? Experimentalists are starting to explore the physics of higher dimensions with the help of recently developed tricks that synthetically mimic an extra fourth dimension in platforms such as ultracold atoms, photonics, acoustics, and even classical electric circuits. Although any such trick necessarily has limitations, as the fourth spatial dimension is always artificial, those approaches have proven that they can simulate some four-dimensional effects in controlled experimental systems."

In nonrelativistic physics, in which space (x,y,z) and time (t) are distinct, a spatial dimension is simply a direction along which objects can move both forward and backward, e.g. such as depicted in this 3 D reference frame:

*x*,

*y*,

*z*)—that must be specified to define where an object is at a particular moment in time, e.g.

*What would happen with an increase in the number of spatial dimensions to four or more?*Mathematicians can do this using Algebraic homology which is a branch of topology that is used to analyze higher-dimensional structures. This is accomplished by first converting them into flat, two-dimensional configurations, then assigning algebraic symbols to each 'dimension' (chain). Let's consider a relatively simple example: the basic 2D torus pattern shown below:

*Greek letters*can be assigned to the box sides. This is for ease of identification of the particular equivalence classes. For example, arrows assigned to segments AB and BA on both sides of the shape shown above are made to point in the same direction, say top to bottom. The same direction implies two sides have to blend together when connected. A similar consideration applies to the bottom ends (AD + DA) when joined. So that the arrow from A to D on top would match an arrow direction from A to D on the bottom.

Thus, for the ‘top’ side of the torus:

A ---->-----D ----->------ A

and, for the ‘bottom’ side:

A ---->-----D -----> ------ A

One could go one step further, as I indicated, and assign Greek letters to the different segments. For example:

a : A ---->-----D -----> ------ A

b : A ---->-----D -----> ------ A

We now have a one-dimensional homology space (H1) denoted by:

H1 = ( a + b )

The same applies to the complementary homology space (H1') that runs vertically so as to join the left and right sides, which we might denote by:

H1' = (d + g)

*x*,

*y*,

*z*,

*w*). Often that extension leads to no new phenomena. But in certain fields, new effects are predicted to emerge, such as so-called topological insulators. In mathematics, topology is basically a framework to classify different surfaces. For example, this toroidal surface ("donut") I created using Mathcad 14, e.g.

*one hole*, so in topology we assign it an index or

*genus*of 1. On the other hand, this sphere which I also created with Mathcad,

*edge currents*", i.e. currents circulating around the edge of a material despite the bulk remaining insulating, or what we call a "topological insulator." By analogy to my earlier squishing a sphere of putty or hard clay, these indices are hard to change, and so are topological properties.

*out-of-plane*magnetic (

**B-**)

**After a current (I) was passed through the device and measured the Hall voltage ( V**

**) across it could be obtained to find the Hall conductance . i.e.**

_{H }_{xy }

**=**I

**/**V

_{H}**=**e

^{2 }n

**/**h

*filling factor*" e.g.

**B**, n

_{ B}~

**B**, the Landau levels increase in energy and the number of states in each level grow. Thereby fewer and fewer electrons occupy the top level until it becomes empty. If the magnetic field keeps increasing, eventually, all electrons will be in the lowest Landau level ( n<< 1 ) which is called the magnetic quantum limit.

_{xy}) exhibited well-defined plateaus that were precisely quantized by integer multiples of e

^{2 }In fact, that quantization was so precise that it became part of the 2019 re-definition of the kilogram in SI units. (See the article by Wolfgang Ketterle and Alan Jamison,

*Physics Today*, May 2020, page 32.)

**B**-field out of the page. Note the circular center (clockwise) ring which denotes a closed, circular cyclotron orbit in the bulk material. This is incepted classically when a charged particle is subjected to an out of plane magnetic field. Now, the topological currents around the edge - called "

*skipping currents"*- are also induced and continue moving in the direction dictated by the B-field. Quantum mechanically the behavior translates to the characteristic bulk energy bands and conducting edge state of a topological insulator. In other words, a 2D quantum Hall system is an example of what would now be called a topological insulator, with the strong Hall conductance being one of its key experimental signatures.

*ibid*..). By 2019 the theorized 3D quantum Hall effect was indeed observed in bulk zirconium penta-telluride crystals. But the 3D quantum Hall effect is what’s often referred to as a "

*weak topological phenomenon"*because key properties, such as the first Chern numbers, remain essentially 2D concepts even though the system is 3D.

^{3}This 4D quantum Hall effect featured a different form of quantized Hall conductance from its 2D cousin and was instead related to a 4D topological invariant called the

*second*Chern number, which generates 3D conducting surface volumes, as shown below:

*op. cit*., p. 40), there have been various 4D quantum Hall models proposed to date. Some based on earlier Yang-Mills theory. But it is a mistake to push the analogy of the mathematical hyper-dimensionality to the physics 4D topology too far. In the end, theoretical (purely mathematical) excursions without experimental confirmation or employing "tricks" (e.g. synthetic dimensions) to create 4D analogs, is not really going to get us far. Even Price admits this esoteric area of physics is "still in its infancy" and that usually means we will have to wait and see if the experimental infrastructure and other resources can catch up to the 4 dimensional aspirations. Meanwhile it is still nice to see that at least some claims are empirically validated, if only in very limited domains. .

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