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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, naly Phase Space.
Every friend who has studied high school mathematics should have constructed a two-dinsional 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-dinsional plane, two numbers can uniquely determine a point. To describe a point in three-dinsional 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 dinsions.
Let us extend our thinking: how should we describe a point in four-dinsional 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 dinsions, and the sa would apply to n-dinsional space. You don't need to strain your brain trying to visualize how a space in four or even eleven dinsions 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-dinsional space can be uniquely described by n variables, and conversely, n variables can be encapsulated by a point in an n-dinsional space.
Now let us return to the physical world, how do we describe an ordinary Particle? At each mont t, it should have a certain position coordinate (q1,q2,q3) and also have a definite montum p. Montum, which is velocity multiplied by mass, is a vector and has components in each dinsional direction. Hence, to describe montum 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 mont t, we need a total of six variables. As we've seen before, these six variables can be summarized by a point in six-dinsional space. Hence, with a point in six-dinsional space, we can describe the classical behavior of one ordinary physical Particle. The high-dinsional space we construct with intent is the System's Phase Space.
Imagine a system composed of two Particles, at each mont t, this system must be described by twelve variables. However, likewise, we can use a point in twelve-dinsional space to replace it. For so macroscopic objects, such as a cat, it contains far too many Particles. Let's assu there are n particles, but this isn't a fundantal problem. We can still describe it with a point in a 6n-dinsional Phase Space. In this way, any activity of a cat over any period of ti can be equated to the movent of a point in 6n-dinsional 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-dinsional 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"
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