Some time ago, I received an email from an enthusiastic reader. The letter mentioned that if 1-1+1-1…=1/2 is determined as fact, one will draw an incomprehensible conclusion such as 1+2+3+…=-1/12. The problem of summing natural numbers mentioned by this reader happens to play a very important role in quantum theory and string theory. From the energy of vacuum to the dimensional number of space and time, there is a subtle connection with the sum of natural numbers. In this little mathematical magic, there is even a secret of discontinuity in time and space.
Written by | Dong Weiyuan
Mathematics teacher once told us that only the converging -level number can solve the sum of infinite terms. However, in some popular science books, you will encounter a magical sum of:
How can the sum of all natural numbers be negative numbers, and it is a fraction? Is this a distortion of human nature or a degradation of morality?
Treat the axis of symmetry as series and
If you want to understand this strange conclusion, let's first look at a simple example: 1, -1, 1, -1, ...Can this sequence find the sum of infinite terms? Italian mathematician Grandi (Dom Guido Grandi, 1671-1742) began to seriously think about this problem as early as 1703. It can be said that this is the starting point for the research on summing divergent series. This sequence was later named "Grandi Series".
Italian mathematician Grandi丨Image source: Wikipedia
Maybe some friends may guess that since the number of 1 and -1 in this sequence is equally large, the sum should be equal to 0. Unfortunately, such a guess is wrong. The infinite set is like a amoeba with strong regeneration ability, with as many parts as the whole. After we take any 1 or -1 from the sequence, the remaining 1 and -1 numbers are still the same. If the remaining 1 and -1 are added to zero, then isn't the total sum determined only by the number of 1 or -1 taken out first - that is, any integer. This is obviously too unreliable. It seems that we cannot rely on comparing the number of 1 and -1 to sum it.
There is another way, which is to use convergent series to find clues. We know that at |q|1,
Now we roughly let q=-1, so
This result seems to be acceptable, but q=-1 is an "illegal" condition after all, and we need a more reasonable way to calm our inner anxiety. If the first n term sum of this series is recorded as A(n), we will now start to find A(∞).
Ha! According to this equation, we once again get the result of A(∞)=½. This time, there seems to be no obvious illegality, and the police are not afraid of coming. But, I still feel something is wrong.
A(1)=1
A(2)=1-1=0
…
You can see that A(n) jumps back and forth between 1 and 0. According to the definition of limit,
This limit does not exist. When we write down the symbol A(∞), there is no clear definition of what it refers to. In fact, this is also the basic problem of summing divergent series: how to define the sum of divergent series. There are more than one definition related to
. Generally speaking, there are mainly two types of Cesaros sum and Abel sum. In addition, , Ramanujan, , Riemann and others have also developed many more general theories, with many contributions derived from Euler.Although those mathematical languages are strict, the side effects of hypnosis and persuasion are not small, so this article does not intend to focus on the basic definitions from the set theory, but only uses a very "physical" perspective to define them: A(∞) represents the average value of all A(n).
uses the sum method defined by the "average value", so that many divergent series can be summed. For example, the series
can also be directly glared out with the result with your eyes in the same way. We use B(n) to represent the sum of the first n terms, that is,
then
B(0)=0
B(1)=1
B(2)=1-2=-1
B(3)=1-2+3=2
…
After drawing the dots and strokes of these B(n) on the figure, there is no need to start writing to calculate. You can directly see with your eyes that the average value of all B(n) is 1/4.
If you only look at the picture, we can also use the conclusion of the previous A(∞)=½ to calculate
slightly adjust the calculation order on the right side of the equation, first subtract the nth term in the previous brackets from the nth term in the subsequent brackets, and then do the sum.
i.e.
A(∞)-BB
B(∞)=B(∞)
htt ml3 So
B(∞)=½A(∞)=¼
Htm3Of course, the method of drawing points and glaring directly at the result with your eyes is sometimes also required. Take the sum of the entire natural numbers as an example, we also let C(n) represent the first n term and
troubles appear! Obviously, the points corresponding to C(n) are distributed on an upward parabola, and there is no way to directly see the average value, and it seems that there is no finite average value at all! Don't worry, we can continue to deform.
In this way, we will divide the points corresponding to each C(n) into two "half points" corresponding to the green term and purple term in the above formula, and we can actually make up two symmetrical curves.
After we finish drawing all the infinite "half points". You can point to the axis of symmetry between the two curves and announce:
Because the average value of all C(n) is equal to the average value of all "half points", and the distribution of "half points" on the two curves is completely symmetrical, only one irrelevant 0 at the beginning of the green curve.
In addition to guessing the value from the picture, we can also use the result just now, and calculate the result (∞)=¼.
After adjusting the order,
so there are many ways to get -1/12 result. For example, the magical Zeta function
This function with plural s as variables has been repeatedly studied due to the famous Riemann conjecture and its close connection with number theory .Mathematicians can write many variations of this function. One of the forms that analyze and extend to all complex planes is
Use this form to calculate
Since we have gone through so many different ways, we all have the same path to the result of -1/12. Can we write 1+2+3+…=-1/12 into middle school textbooks in a grand manner? I believe many people will be like me and are still worried about this. Because in all the above deductions, there is a rather hidden problem, that is, the meaning of the equal sign. It seems natural to write
or
directly into
, but in fact, in the two formulas, the previous "=" represents "defined as", not the same magnitude. Therefore, the clearer way to write it should be
and
so that you can see that the value of -1/12 is not natural and natural like 1+1=2. Instead, it is necessary to assume in advance that "the sum of all natural numbers is a certain number", and then carefully select a value with the best logical consistency and specify it as the sum of all natural numbers . But when logical self-consistentness and intuition are clearly conflicting, we are all surprised that this is no longer new on the road to mathematical development.
extends to infinite scissors
In the previous discussion, we directly ignored the concept of mathematical limits and roughly used the average value as the sum of divergent series. Now let’s pick up the limit concept again and see from another perspective how -1/12 came out.
versus C(n) This divergence series, we can introduce a certain scissor function f(x) to suppress those terms that tend to infinity, so that the divergence trend stops near a specific position N and finally converges to a certain limit S(N). In this way, we use the standard limit concept to construct an S(N). When N is finite, S(N) is a finite value, and when N tends to infinity, S(N) corresponds to the sum of the entire natural numbers.
can act as scissors. For example, we take
At this time,
Through numerical calculation, we found that S(N) is moving towards infinite as N increases. This is in line with our previous intuition, but what about -1/12? Don't worry, let's expand S(N) with 1/N. We found that the numerical results of S(N) in large N can be fitted well by the expansion below
ha! I actually saw this -1/12 again, which is a constant term in the S(N) expansion formula. That is to say, the component that is independent of the change in N in S(N) is -1/12. When N is large enough, those terms containing 1/N can be ignored, and S(N) can be regarded as a parabola with the lowest point at -1/12.
We then take the scissors function f(x)=e^(-x) Try it. At this time, the sum of
can be strictly calculated. Let’s first find the derivative of β on both sides of the following equation
You can get
Also do 1/N expansion under large N conditions, and you will get
5
Take β=1 and get
also appears the constant term -1/12, and it is also a parabola where the root droops to -1/12
f(x) is directly taken as the jump function, that is, suddenly truncate at n=N , then
will not have the constant term -1/12.
It seems that in addition to the sudden truncation of the jump function, other smooth truncation methods can get the interesting result of
. This seems to tell us that even though the sum of the entire natural number is destined to be unable to escape the fate of going to infinity, it has always been in love with -1/12 for some mysterious reason. Or it can be said that
divergence terms are just mediocre background colors, and -1/12 is the character kernel engraved on the background colors.
Vacuum energy
From a practical perspective, we sometimes need to process the sum of natural numbers like using convergent series, so we have to find a certain "rein" to control it. For example, when studying vacuum energy, physicists encountered the sum of all natural numbers, and they really hoped that this sum is a certain number.
In the theoretical model of quantum field theory, the vacuum is like a three-dimensional spring net, composed of countless small springs interwoven horizontally and vertically. The so-called particles are that some of the small springs vibrate violent enough that from a distance, I think something foreign object appears in the spring net, but as long as I get close, I will see that there is nothing else but the vibration itself. In other words, particles are essentially the vibration of vacuum. Therefore, when energy changes, the number of particles does not have to be bound by any conservation law and can be increased or decreased out of thin air. However, whether particles can be generated or disappeared is related to the vibration frequency of of the small spring. When the vibration frequency is ω, there is a relationship between the number of particles n and the energy E of the field:
It can be seen from the relationship that every time the vacuum accumulates a portion of energy of ћω, a particle will be generated; conversely, every time one portion is reduced, a particle will be erased. Or simply put, each particle is actually an energy pack of ћω the size of energy. Interestingly, when n=0, it corresponds to the situation where there are no particles in the vacuum, and the energy is ½ћω. In other words, when the energy of the vacuum cannot be lower, the energy is still not 0. This is vacuum zero point energy . Let’s calculate the vacuum energy in a finite space and see what it really has to do with the sum of the entire natural number.
3 adds up all the zero energy of the frequency, and the total energy in the vacuum is
Look, the sum of natural numbers
That's it, now you should be able to understand how much physicists hope that
is a definite value. What's more interesting is that if we just think that the sum of natural numbers is -1/12, we can even design a physical experiment to verify this conclusion.
3 As shown in the figure below, three parallel metal plates are placed so that the distance between A and B is a and B is b.
According to the conclusion just now, we know that the vacuum energy between A and B is
The vacuum energy between B and C is
Now we want to know which direction the metal plate B in the middle will be subjected to. The stress situation can be solved according to the partial conduction of energy to position. The results show that if
, metal plate B will be subjected to a rightward force; otherwise, it will be subjected to a leftward force.
In fact, the experimental device can be further simplified. We can take C on the rightmost point to the infinite distance, leaving only A and B, and then measure whether A and B are attracted or rejected. If they are mutually exclusive, it means
Otherwise, it means
This experimental idea was first proposed by the Dutch physicist Casimir (Hendrik Casimir, 1909-2000) in 1948. Of course, the purpose of the experiment was not to measure the sum of natural numbers, but to verify the existence of vacuum zero-point energy. In fact, when Casimir proposed this experiment, he had predicted that the two metal plates were attracted to each other, which was the corresponding situation of
, because his theoretical calculation process had already used the analysis of the extension of the Riemann Zeta function. In 1996, Lamoreaux from the University of Washington verified the existence of the Casimir effect with experiments, and the paper was published in the January 1997 "Physical Review Letter" (PRL) .
It needs to be clarified that the experimental verification of the Casimir effect can only indicate the existence of vacuum zero energy, but it cannot be really used to verify the sum of all natural numbers in the mathematical sense. In fact, in reality, metal plates can only block electromagnetic waves in a limited frequency range. When the frequency is greater than a certain value, metal plates cannot block such extremely high frequency waves. Therefore, when calculating the Casimir effect from a more precise perspective, this high-frequency truncation needs to be considered. However, the specific calculation will use hypnotic content such as the Euler-McLaurin formula and the Bernoulli number, and this article will no longer cover it.
Next we turn to string theory to see how the sum of all natural numbers is related to the number of dimensions.
Latitude of spacetime
mentioned earlier that only standing waves can exist on a spring fixed at both ends, and all vibration frequencies can only be integer multiples of the lowest frequency. For a string with completely free ends, the conclusion is also true. Fixed ends means that the endpoint speed is zero, while free ends means that the endpoint acceleration is zero. The difference between the two is nothing more than writing it as
or
. In other words, for strings with length L, there must be a discrete frequency spectrum like a sequence of natural numbers
In addition, the quantization method in string theory is exactly the same as the technical means used in quantum field theory, so there is also a
relationship. This means that the string with the lowest energy is not completely stationary, but has the energy of
, and in every dimension that can vibrate, there are these energy.
Assuming that the spatial dimension number is d, then the minimum total energy of a string excited into a photon is
is definitely here again
Theory of relativity tells us that the minimum energy of a photon should be zero, so the string theory compatible with the theory of relativity must meet the
Deduction here, we must use the ultimate move of
, and solve it with d=25, that is, the space dimension must be 25 dimensions, plus time, a total of 26 dimensions of space-time.
In superstring theory, due to the introduction of supersymmetry factors, the ground state energy of the string is increased to 3 times, and the photon energy constraint becomes
From this, d=9 is found, and a time dimension is added to form a total of 10 dimensions of time and space.
or above is the story summary of Bose string theory requiring 25+1 dimension space-time, and superstring theory requiring 9+1 dimension space-time. I hope readers can use these examples to establish some concrete understanding of the role of the sum of natural numbers in physics.
Discrete space-time
In order to maintain the convergence of the topic, many interesting details have been deliberately skipped in the previous discussion. For example, in the ground state vibration mode of the string, if there is the component of
, then there must be
. This means that in only one vibration mode of the string, infinite energy is contained. Similarly, the calculation of vacuum zero-point energy will inevitably contain components with infinite energy. This is obviously too inappropriate. Our theoretical model needs to have a boundary to prevent this "ultraviolet disaster" in extremely high frequency directions. The reason why
can generate infinitely large frequencies is because we allow infinitely small wavelengths. Then we will naturally realize that in the theoretical model that can eliminate "ultraviolet disaster", there must be a limited minimum scale in space. To put it bluntly, the space cannot be a continuous stage, but must be a discrete plum blossom pile. What is this minimum scale? A natural candidate, of course, is Planck's length.
If the wavelength of a particle is shorter than the Planck length, then the particle will turn itself into a black hole on the spot because it has too high energy, and the area covered by this black hole will exceed the Planck length. Therefore, Planck's length became the most natural basic pixel of space-time in existing theories.
Therefore, the previous
became
Obviously, the proportional part of the energy always cancels out each other. Only the second part has an inverse energy that produces an inverse force on the B board. It can be seen from this that the Casimir effect is an effect that appears on the second term after the two huge first terms just cancel each other out, so this force is extremely weak. Only when two metal plates with an area of magnitude of square meters are brought close to the distance between microns, can a measureable attraction be generated.
Now we have truly explained the relationship between the Casimir effect and the sum of natural numbers. If you encounter a civil science and technology in the future, you will try to use this experiment to prove that the sum of natural numbers is a negative number, and you can roll your eyes without hesitation.
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