Enter the Immortal Cultivation

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In the main text, our protagonist Wang Qi used the golden finger for the second time, which was the Hilbert space of David Hilbert, a great mathematician from Earth.

Since I don't want to waste words in the main text, I will post the popular science of this mathematical method here! Interested readers may wish to come and have a look~

Albert space does not really exist, but is an abstract tool used for calculation, namely phase space.

Every friend who has studied middle school mathematics should have established a two-dimensional Cartesian plane: draw an x-axis and a y-axis perpendicular to it, and add arrows and scales [that is, the so-called plane rectangular coordinate system]. In such a plane system, each point can be represented by a coordinate (x, y) containing two variables, such as (1, 2), or (4.3, 5.4), and these two numbers represent the projection of the point on the x-axis and y-axis respectively. Of course, it is not necessary to use the rectangular coordinate system, and polar coordinates or other coordinate systems can also be used to describe a point, but in any case, for a 2-dimensional plane, two numbers can uniquely indicate a point. If we want to describe a point in three-dimensional space, then our coordinates must have three numbers, such as (1,2,3). These three numbers represent the projection of the point in three mutually perpendicular dimensional directions.

Let's expand our thinking: if there is a point in four-dimensional space, how should we describe it? Obviously, we have to use coordinates containing four variables, such as (1,2,3,4). If we use a rectangular coordinate system, then these four numbers represent the projection of the point in four mutually perpendicular dimensional directions. The same is true for n-dimensional space. You don't have to bother trying to imagine how 4-dimensional or 11-dimensional space is perpendicular to each other in 4 or even 11 directions. In fact, this is just an imaginary system we constructed mathematically.

What we care about is: a point in n-dimensional space can be uniquely described by n variables, and conversely, n variables can also be covered by a point in n-dimensional space.

Now let's go back to the physical world. How do we describe an ordinary particle? At each moment t, it should have a definite position coordinate (q1, q2, q3) and a definite momentum p. Momentum is velocity multiplied by mass, and is a vector with components in each dimensional direction. Therefore, to describe momentum p, we need to use three numbers: p1, p2 and p3, which represent its velocity in three directions respectively. In short, to fully describe the state of a physical particle at time t, we need a total of 6 variables. As we have seen before, these 6 variables can be summarized by a point in 6-dimensional space, so using a point in 6-dimensional space, we can describe the classical behavior of an ordinary physical particle. This high-dimensional space that we deliberately constructed is the phase space of the system.

If a system consists of two particles, then at each moment t, the system must be described by 12 variables. But similarly, we can replace it with a point in 12-dimensional space. For some macroscopic objects, such as a cat, it contains too many particles, assuming there are n, but this is not an essential problem. We can still describe it with a particle in a 6n-dimensional phase space. In this way, the activities of a cat in any period of time can actually be equivalent to the movement of a point in 6n space (assuming that the number of particles that make up the cat remains unchanged). We do this not because we have too much time to spare, but because mathematically, it is easier to describe the movement of a point, even a point in 6n-dimensional space, than to describe 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 equation to describe it and draw many useful conclusions.

——Partially selected from Cao Tianyuan's "History of Quantum Physics"