A Popular Science Explanation About Hilbert Space

This is Daozhang's Science Popularization Channel!

In the main text, the golden finger our protagonist Wang Qi is using for the second time is the Hilbert space from the great mathematician David Hilbert of Earth.

Since I don't want to pad the word count in the main text, I, this humble Daoist, am posting this popular science explanation of the mathematical method here! Interested book friends are welcome to take a look~

A phase space isn't a physically existing space, but an abstract tool for calculation, the phase space.

Every friend who has studied middle school math should have established a two-dimensional Cartesian plane: draw an x-axis and a y-axis perpendicular to it, and add arrows and scales (this is what's usually called a plane rectangular coordinate system). In such a planar system, every point can be represented by a coordinate (x,y) containing two variables, such as (1,2) or (4.3, 5.4). These two numbers represent the projection of the point onto the x-axis and y-axis, respectively. Of course, it's not necessary to use a rectangular coordinate system; polar coordinates or other coordinate systems can also be used to describe a point. But regardless, for a 2D plane, two numbers are sufficient to uniquely specify a point. To describe a point in three-dimensional space, our coordinates would need three numbers, like (1,2,3). These three numbers represent the projection of the point onto three mutually perpendicular dimensions, respectively.

Let's expand our thinking: how would we describe a point in a four-dimensional space? Clearly, we would need coordinates with four variables, such as (1,2,3,4). If we use a rectangular coordinate system, these four numbers would represent the projection of the point onto four mutually perpendicular dimensions. The situation is the same when generalizing to n dimensions. You don't need to strain your mind trying to visualize how a 4D or 11D space is mutually perpendicular in four or even eleven directions. In fact, this is just a hypothetical system we construct mathematically.

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

Now let's return 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 mass times velocity, a vector with components in each dimensional direction. Therefore, describing momentum p also requires three numbers: p1, p2, and p3, representing its velocity in the three directions. In summary, to fully describe the state of a physical point mass at time t, we need a total of six variables. As we saw earlier, these six variables can be summarized by a point in a 6-dimensional space. Thus, with a point in a 6-dimensional space, we can describe the classical behavior of one ordinary physical particle. This high-dimensional space we deliberately constructed is the system's phase space.

If a system consists of two particles, then at each moment t, the system must be described by twelve variables. However, similarly, we can use a point in a 12-dimensional space to represent it. For some macroscopic objects, like a cat, the number of particles is immense. Let's assume there are n particles. This isn't a fundamental issue; we can still describe it with a point in a 6n-dimensional phase space. In this way, the activity of a cat over any period can actually be equivalent to the motion of a point in a 6n-dimensional space (assuming the number of particles making up the cat remains constant). We do this not because we have too much free time, but because mathematically, describing the motion of a point, even a point in a 6n-dimensional space, is more convenient than describing a cat in ordinary space. In classical physics, for such a point representing the entire system in phase space, we can use the so-called Hamiltonian equation to describe it and derive many useful conclusions.

--Partially selected from Cao TianyuanHistory of Quantum Physics