Welcome to the Daoist's popular science channel!

In the main text, our protagonist Wang Qi's second use of the "Golden Finger" was the Hilbert Space originated from David Hilbert, the great mathematician from Earth.

As I don't want to pad the word count in the main text, I, the humble Daoist, am posting this popular science knowledge here! Interested readers might want to take a look~

Albert Space doesn't actually exist; rather, it's an abstract tool used for calculus, namely Phase Space.

Every friend who has studied high school mathematics should have constructed a two-dimensional Cartesian Plane: draw an x-axis and a perpendicular y-axis, add arrows and graduations (the usual so-called cartesian coordinate system). Within such a plane system, every point can be represented by coordinates that contain two variables (x,y), such as (1,2) or (4.3,5.4). These two numbers represent the projection of the point on the x-axis and y-axis, respectively. Of course, it's not necessary to use a Cartesian coordinate system; one can also use Polar Coordinates or other coordinate systems to describe a point. But in any case, for a 2-dimensional plane, two numbers can uniquely determine a point. To describe a point in three-dimensional space, our coordinates will need to contain three numbers, for instance, (1,2,3). These three numbers represent the projections of the point in three mutually perpendicular dimensions.

Let us extend our thinking: how should we describe a point in four-dimensional space? Clearly, we would use coordinates with four variables, such as (1,2,3,4). If we are using a Cartesian coordinate system, then these four numbers would represent the projections of the point in four mutually perpendicular dimensions, and the same would apply to n-dimensional space. You don't need to strain your brain trying to visualize how a space in four or even eleven dimensions can be mutually perpendicular in four or eleven directions; in fact, this is just a hypothetical system we construct in mathematics.

What we are concerned with is: a point in n-dimensional space can be uniquely described by n variables, and conversely, n variables can be encapsulated by a point in an n-dimensional space.

Now let us return to the physical world, how do we describe an ordinary Particle? At each moment t, it should have a certain position coordinate (q1,q2,q3) and also have a definite momentum p. Momentum, which is velocity multiplied by mass, is a vector and has components in each dimensional direction. Hence, to describe momentum p, one would need three numbers: p1, p2, and p3, representing the velocity in three directions. In summary, to completely describe the state of a physical Particle at moment t, we need a total of six variables. As we've seen before, these six variables can be summarized by a point in six-dimensional space. Hence, with a point in six-dimensional space, we can describe the classical behavior of one ordinary physical Particle. The high-dimensional space we construct with intent is the System's Phase Space.

Imagine a system composed of two Particles, at each moment t, this system must be described by twelve variables. However, likewise, we can use a point in twelve-dimensional space to replace it. For some macroscopic objects, such as a cat, it contains far too many Particles. Let's assume there are n particles, but this isn't a fundamental problem. We can still describe it with a point in a 6n-dimensional Phase Space. In this way, any activity of a cat over any period of time can be equated to the movement of a point in 6n-dimensional space (assuming the number of particles composing the cat remains unchanged). We do so not because we are idly full and bored, but because in mathematics, describing the motion of a point, even one in 6n-dimensional space, is more convenient than describing a cat in ordinary space. In classical physics, for such a point in the Phase Space that represents the entire System, we can use the so-called Hamiltonian Equations to describe it and obtain many useful conclusions.

— Excerpt from Cao Tianyuan's "Quantum Physics Story"