[Translator's note: Reading this article requires a certain mathematical foundation. You can refer to: notation of infinite sums, some properties of infinite series]
Once upon a time, there was a blinding mathematical conclusion that was popular all the time. It said: Add up all natural numbers (1+2+3+4...), and the result is -1/12. The following video is about this. It says it proves the result, and it also says that this result is very commonly used in physics. Everyone felt that their primary school arithmetic was in vain, and even the New York Times (need to overturn the wall - translator's note) reported it. So, what exactly is that video talking about?
First of all, I can give you a reassurance: the sum of all natural numbers does not equal -1/12. Pick up the calculator, calculate these parts and you will understand:
and so on. The larger n is obtained, that is, the more you add, the larger Sn is. As long as n is big enough, you can make Sn as big as it is. For example, for n=1000, you can get
. For n=10000, you can get
. Therefore, mathematicians believe that 1+2+3+4+... diverges to infinity. Or simply put it is equal to infinity.
So where did -1/12 come out? This error result was actually created by the famous Indian mathematician Ramanujin (pictured above) in 1913. Ramanujan knew very well what he was doing, and he had a reason to write this result. He was studying the Euler ζ function at that time. Regarding this function, let’s first look at the sum of
below. It can be seen that it adds the sum of each natural number squared and then takes the reciprocal number:
The sum of this sum does not diverge. If you write its part and sequence as above: the result after
can be arbitrarily close to (not exceed) π^2/6=1.644934…. Mathematically, its sum converges to π^2/6, or its sum equals π^2/6.
What if the denominator does not take the power of 2, but takes x? The corresponding sum is listed as follows:
As long as x is a number greater than 1, the above formula can converge to a finite value. For each x1, S(x) has a definite value. Such an equation is called a function, while S(x) is specifically named Euler's ζ function after the surname Euler, an outstanding mathematician of the 17th century.
is still fine until now. But what if you substitute a number less than 1 into it? For example, if you substitute x=-1, there will be
, which will return to our initial sum, and we know that it is divergent. The same result is the same for other xs less than or equal to 1: the sum formula diverges.
However, we can still make some achievements. Using point-level mathematics (called complex analysis, see tips) we can expand the domain of the Euler ζ function to x less than or equal to 1 and let it give a finite value. In other words, there is a way to define a new function ζ(x), with a defined finite value for x1
and x≤1 function ζ(x). This method of expansion is called analytical extension, and the new function obtained from this is called Riemann ζ function (named after the surname of Riemann, a mathematician in the 18th century). (It uses a clever method to give a finite value when x≤1, that is, subtract the primary divergence sum and subtract the two infinities to obtain a finite quantity.)
Tips
As mentioned earlier, the Euler ζ function S(x) is defined as a real number whose domain is greater than 1. Real numbers are part of the large number field of complex numbers. Real numbers correspond to points on an infinitely long line, and complex numbers correspond to points on the entire plane containing the real number axis. This plane is called a complex plane. We can define functions where independent variables take real numbers, and we can also define functions where independent variables are complex numbers (called complex variable functions).
A strange property of complex variable functions is: if you know that the function is meaningful to a set of independent variables, then (through some technical means) you can know the value of the function at other points in the complex plane. The method of expanding the domain of function is called analytical extension. The Euler's ζ function is defined in the real domain greater than 1. Because real numbers are also complex numbers, we can regard it as a complex variable function, and then use the analytical extension method to obtain a new function that combines the Euler's ζ function on the entire complex plane but on the real domain greater than 1. This new function is the Riemann ζ function.
is OK. Now our ζ(x) and Euler ζ function S(x) are the same at x1.When substituting a value of x≤1, the ζ function can obtain a finite value. So, what value will you get if you substitute x=-1? You may have guessed:
If you mistakenly equate ζ(x) with S(x) when x=-1, you will come up with this (wrong) formula:
This is an explanation of the expression of the Ramanujian mystery.
Little trick
So, how do people in the video "prove" that the sum of all natural numbers is equal to -1/12? The answer is, they actually didn't prove it. Watching their videos is like watching how a magician stuffs a rabbit into a magic hat. The first step in their "proving" is to make you believe that the following infinite and
are equal to 1/2. The
video did not explain it here, as if this result was taken for granted. But let's take a closer look at whether this is the case. Suppose 1-1+1-1+1-1... has a finite value Z, add it once to get
, but this is the initial sum (this conclusion is problematic. If the spliced formula is still 1-1+1-1..., then it can be determined that the previous sequence ends with +1-1 - the translator's note), and thus get
Therefore, if Z=1/2, you will come to the ridiculous conclusion that 1/2=1. Therefore, it is wrong to think that infinity and 1-1+1-1-… are equal to 1/2. In fact, you can use a divergent infinite and to make up various results. This is just a trick. What appears in
physics?
So, why does this weird error result appear in the physics textbook like in the video? This is what makes this question interesting. Imagine you have two conductive metal plates and place them parallel to the vacuum. According to classical physics, there should be no net interaction force between these two plates. (Not considering gravity - translator's note.)
But classical physics cannot handle strange phenomena that occur at tiny scales. At this time, you need quantum physics. Quantum physics will tell us all kinds of strange things, one of which is that vacuum is not empty, but also full of vitality: every moment, the so-called virtual particles roll and appear and disappear. These activities bring what is called zero energy: this is the lowest energy that matter has at absolute zero degrees.
When you use quantum physics mathematics to calculate the energy density between these two metal plates, you will get the following infinite sum:
This formula is the formula obtained when substituting x=-3 into the Euler ζ function:
How sad! This sum diverges (it diverges faster than S(-1), which means that the energy density is infinite and obviously meaningless. What if I silly think that this infinite sum equals the value of the Riemann ζ function (rather than the Euler ζ function) at x=-3? This will give you a limited energy density, indicating that there is an attraction between the two metal plates. This is also very strange, because according to classical physics, this force does not exist.
But the banner has been raised! Physicists did an experiment and really discovered this force, and its corresponding energy density was exactly equal to ζ (-3)!
This amazing result is called the Casimir effect, named after the surname of Dutch physicist Hendrik Casimir. (The illustration is as follows: Casimir plates are Casimir plates, and Vacuum fluctuations are vacuum fluctuations.)
Let's take a little time to take a look. Quantum physics thinks that the energy density is
is meaningless, but experiments have proved that if you (incorrectly) treat this sum as the value of ζ(x) when x=-3, you can get the correct result. It seems that we naturally used the method mentioned above, which extends the Euler's ζ function to the range where x is less than or equal to 1, subtracts an infinity by cleverly, and then obtains a finite value. Submit, mortal!
As for the reason why ζ(-1) and S(-1) rather than ζ(-3) and S(-3) appear in the videos and textbooks, it is because the energy density of the one-dimensional Casimir effect is calculated using ζ(-1). (Three-dimensional is ζ(-3).) (As the textbook, it focuses on basic theory rather than practice. If it can be simple, it is simple - the translator's note.)
So why do the guys in the video post such a video? Of course they know that parsing extensions can allow functions to get definitions outside the original definition domain, but these are a bit more technical for their videos.They knew that the final result obtained by analyzing the method of sequel was correct, so they did not say that these were difficult, but used some tricks. In this way, their videos got a million views, which also made everyone start talking about ζ functions and mathematics. I congratulate them on this point. The ζ function is very wonderful, and what we are talking about above is just the beginning of a series of magical mathematical properties. (There is a relatively professional popular science book related to Riemann functions in China: "The Riemann Conjecture Talk" - Translator's Note.) When disseminating mathematics and physics to the masses, we always have to make some choices about how to explain, and we also have to weigh the boundary between precision and easy-to-understandness based on our conscience.
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