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In the main text, the second cheat our protagonist Wang Qi used was David Hilbert's Hilbert space, brought from Earth.
Since I don't want to pad the main text with more words, this humble Daoist has posted this introduction to the mathematical method here! Interested readers, feel free to come in and take a look~
Albert space did not truly exist; it was an abstract tool used for calculation—that is, phase space.
Everyone who studied middle-school mathematics should have constructed a two-dimensional Cartesian plane: draw an x-axis and a y-axis perpendicular to it, then add arrows and scales [what is commonly called a plane rectangular coordinate system]. In such a planar system, every point could be represented by coordinates containing two variables, (x, y), such as (1, 2) or (4.3, 5.4). These two numbers respectively represented the point's projections onto the x-axis and y-axis. Of course, one did not have to use a rectangular coordinate system; polar coordinates or other coordinate systems could also describe a point. But no matter what, for a two-dimensional plane, two numbers could uniquely specify one point. To describe a point in three-dimensional space, our coordinates needed three numbers, such as (1, 2, 3). These three numbers respectively represented the point's projections along three mutually perpendicular dimensions.
Let us stretch our thinking a little: if there were a point in four-dimensional space, how should we describe it? Clearly, we would need coordinates containing four variables, such as (1, 2, 3, 4). If we used a rectangular coordinate system, then these four numbers would represent the point's projections along four mutually perpendicular dimensions. Extending this to n dimensions, the situation was the same. There was no need for everyone to rack their brains trying to picture how four-dimensional or eleven-dimensional space could be perpendicular in four or even eleven directions. In truth, this was merely a hypothetical system constructed in mathematics.
What concerned us was this: a point in n-dimensional space could be uniquely described by n variables; conversely, n variables could also be encompassed by a point in n-dimensional space.
Now let us return to the physical world. How did we describe an ordinary particle? At every moment t, it should have a definite position coordinate (q1, q2, q3), as well as a definite momentum p. Momentum was velocity multiplied by mass, and it was a vector with components in every dimensional direction. Thus, describing momentum p also required three numbers: p1, p2, and p3, respectively indicating its velocity in the three directions. All in all, to fully describe the state of a physical point mass at time t, we needed six variables in total. As we had already seen, these six variables could be summarized by one point in six-dimensional space. Therefore, with one point in six-dimensional space, we could describe the classical behavior of one ordinary physical particle. This deliberately constructed high-dimensional space was the system's phase space.
If a system consisted of two particles, then at every moment t it had to be described by twelve variables. Yet likewise, we could replace it with one point in twelve-dimensional space. For certain macroscopic objects, such as a cat, the number of particles it contained was far too great—let us suppose there were n of them. But this was not an essential problem; we could still use a point mass in a 6n-dimensional phase space to describe it. In this way, a cat's activity over any period of time could in fact be equivalent to the motion of a point in 6n-dimensional space (assuming the number of particles composing the cat remained unchanged). We did this not because we had eaten our fill and had nothing better to do, but because mathematically, describing the motion of a point—even one in 6n-dimensional space—was more convenient than describing a cat in ordinary space. In classical physics, for such a point in phase space representing the entire system, we could use the so-called Hamiltonian equation to describe it and derive many useful conclusions.
—Partly excerpted from Cao Tianyuan's A History of Quantum Physics
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