The simplest natural number we are familiar with (from the network)
At the beginning of the establishment of quantum mechanics, Schrödinger would be imaginary
Equations were introduced to describe the peculiar behavior of microscopic particles. However, whether the plural is a mathematical technique or an objective reality has never been answered. Is it feasible if we don't use complex numbers, but only real numbers to describe the quantum world? Are complex numbers necessary in quantum mechanics?
Mathematics accompanies almost every one of us in cognition. When we were very young, parents would probably teach us to count with apples and fingers; later, the range of numbers continued to expand, and the understanding of numbers established through a few apples and fingers could no longer cover all the scenarios that people encountered. Thousands of years ago, out of the needs of production and life, we needed to represent not only "surplus", but also "deficit", so human beings crossed the concept of positive numbers and zeros, and negative numbers were produced; also thousands of years ago, when we needed to describe the unit "1" into several parts, the fraction was generated. Fractions can be reduced to finite decimals or infinite looping decimals. There are also numbers that are "unreasonable", and they become decimals that neither terminate nor repeat, but are infinitely non-cyclical, and we call them irrational numbers. The more familiar irrational numbers are Pi and the length of a square diagonal with a side length of 1
。 It can be seen that even the most "unreasonable" irrational numbers can be found in everyday life scenes such as circles and squares. As long as you acknowledge the existence of circles and squares, you have to admit the existence of irrational numbers.
Image from Wikipedia
The application of numbers is too important in life, and all civilizations in the world, even if they are thousands of miles apart, have invariably produced brilliant cultures and long histories in digital.
But in any case, no matter how abstract these types of numbers mentioned above, they can always find corresponding meanings in reality. Until, you come across — plural.
The complex number consists of a real part and an imaginary part, of which the imaginary part is confusing
Although you know that it represents the square root of -1, most people probably can't say what it means and what scenario it corresponds to in the real world.
This question has puzzled even great mathematicians.
In the 16th century, the Italian mathematician Girolamo Cardano, in one of his books called The Great Arts, introduced the method of taking the square root of negative numbers in order to discuss the problem of "dividing 10 into two parts so that their product is 40", and he wrote the two numbers as (
) and (
), which solves the problem. At the time, he had only thought it was a mathematical technique for convenient calculations, and had not yet realized that he had touched the gate of the Palace of Plurals. Later, Descartes named the expression of a negative number taking the square root as an imaginary number. This imaginary number, like a seemingly innocent ghost, mathematicians at that time had difficulty understanding its secrets, after more than two hundred years of many mathematicians to follow, the theory of complex numbers was established. Its importance is amazing, and it is no wonder that the French mathematician Adama said: "In the field of real numbers, the shortest path connecting two truths is through the field of complex numbers." ”
The theory of complex numbers is gaining prominence in mathematics and is an important tool in physics and other engineering techniques. We calculate current and process signals without the tool of complex numbers. However, it is only a tool. What does that mean? That is, with it we can deal with the problem more conveniently, without it, we can also get it done, but it is more troublesome. After all, the current we calculate in the end should always be a real number.
But one day, this certainty began to waver here in quantum mechanics. Like mathematicians in the 16th century, confusion about imaginary numbers came to the minds of physicists.
In 1926, when the physicist Schrödinger was building the wave equation, he originally wrote down the differential equation of mechanical particles with reference to the model of wave optics, but this equation had no physical significance, but when he put the square root of minus 1
When put into an equation, the wave function in complex form instantly becomes meaningful, helping us to accurately describe the quantum behavior of particles. And the invisible and intangible abstract concept of the wave function, whether it is Schrödinger himself or other physicists, no one can tell what its essence is.
But for us, physics is the study of the real world, and all the physical quantities you can imagine in the real world should be measurable, and how can there be imaginary numbers in the measurable quantities? We know that the modulus of the wave function describes the probability of particle occurrence, so although the wave function is written as a complex number, the probability itself is still a real number. Well, imaginary numbers
Is it really necessary to describe the real world? Schrödinger wasn't sure either. In his letter to Lorenz, he seems inclined to imaginary numbers
It is just a mathematical processing method, and the measured physical quantities in reality should be in the form of real numbers. At that time, he said in his letter that the wave function introduced complex numbers, which he was not very sure of, and that the quantum wave function should be a real function. Schrödinger has been trying to erase the complex number from his wave equation, but without success.
Are complex numbers necessary in quantum mechanics (image from Quanta Magazine)
So, what is the relationship between the wave function of the complex number and the real quantum world? What role does the plural play in it?
To answer this question, we can look back at the birth of irrational numbers. More than two thousand years ago, irrational numbers were born to describe the diagonal length of a square with a side length of 1, as long as you admit the existence of a square, you must admit that irrational numbers exist objectively, otherwise, what is the diagonal length used to represent? It can be seen that irrational numbers are not "unreasonable". So what about the imaginary (complex) number, is it really imaginary, or does it have objective reality?
The present complex number is similar to the quantum world as the original irrational number.
In general, each wave function corresponds to a distribution of physical states. In addition, we think that independently constructed systems have independent physical states, so naturally the total physical states composed of these independent systems can be expressed directly in their tensor product form, which is somewhat similar to how in mathematics we multiply two or more numbers to get a total result.
When two wave functions are prepared by two identical and independent systems, the study proves that their corresponding distributions do not overlap, that is, a physical state can only be encoded into a unique wave function, which means that the wave function is objectively real [1]. And if we can show that quantum mechanics (wave functions) must use complex numbers, then complex numbers are objectively real.
So, the question now boils down to: Does quantum mechanics really have to use complex numbers? In other words, if you don't use complex numbers, will the calculation result be different except for the troublesome process?
In the classical world, we know that plurals can generally be written
In theory, then, we can always use the two real numbers a and b to replace, but a complex number becomes two real numbers, which is more troublesome to deal with. For the quantum world, many scientists are also constantly trying to describe quantum mechanics in various ways that do not introduce complex numbers.
We know that quantum mechanics has a unique mathematical structure, in which different physical system states are described in different Hilbert spaces, and observable measurements such as position or momentum are used as linear operator representations of Hilbert spaces for systems. From the early days of quantum mechanics, scientists have argued that many of the features of quantum theory within the framework of complex numbers are represented by two alternative hypothetical theories, such as that the Hilbert space of complex numbers can be replaced by a Hilbert space of a real number or quaternion. This was made clear when Berkhof and von Neumann proposed the hypothesis of quantum logic in 1936 that the closed subspaces of the Hilbert space of the quantum state could construct an algebraic semantics similar to Boolean logic, based on which models of real and quaternions satisfies their assumptions as well as standard complex number theory.
On the other hand, in 1960, the Swiss physicist Ernst Steckelberg introduced special operators in order to realize standard complex quantum theory, and required observable quantities to be paired with the introduced operators, which is similar to the commutative law in mathematics. Because of this limitation on observable quantities, his special operator acts as imaginary numbers
The role, the rules, although troublesome, but the final result has no effect under the real number framework. Although at the time he had only demonstrated that the quantum theoretical predictions of all single-particle experiments could be deduced using only real numbers, his rule could be further extended to multiparty systems.
Other studies have shown that in the quantum world, without the use of complex numbers, by introducing universal qubits that can interact with anything in the system, doubling the dimensions of state and measurement space, just as in classical physics, using a and b numbers instead of a complex number, we can still perfectly predict the famous quantum physics experiment - the Bell experiment. (The Bell Experiment is an important test of the fundamental theory of quantum mechanics, which explores the fundamental properties of entanglement.) It sends entangled particles to Alice and Bob separately, like torturing a pair of twins separately, simultaneously, back-to-back, and based on their answers, to see if the twins are really entangled across time and space, or who secretly delivers the message. ）
In addition to Schrödinger and Steckelburg, Von Neumann, Freeman Dyson (yes, the one who wrote Birds and Frogs), Nicolas Gisin, and William Wootters also experimented with real quantum mechanics. These studies have led physicists to think that complex numbers in quantum mechanics are only a means of convenient computation for us, not a necessary existence, as if we could describe our world in terms of only real numbers.
Guessing is guessing, and the proof of physical laws always needs experimental data to support. In January 2021, a new proposal was proposed by a theoretical group of scientists from Spain, Austria and Switzerland. This scheme is unique in that it is an experimentally testable, quantitative, and similar criterion to Bell's inequality.
Image from APS Physics
The so-called entanglement exchange, that is, Alice, Bob, Clarie three people are not together, at this time, the two entanglement sources R and S, S send a pair of entangled particles to Alice and Bob; R sends another pair of entangled particles to Bob and Clarie, according to Bob's Bell measurements, Alice and Clarie's hands originally unrelated particles are finally entangled. In the early Bell tests, all the participants' particles came from a single source, and the information they carried extra was not a problem in the real description.
But in the newly designed Bell test, the two entanglement sources are independent of each other, and the three parties involved in each other make local measurements independently. When Bob does the full Bell measurement, Alice and Clarie perform their respective measurements, what are the statistical results of the tripartite association? The theoretical calculations of scientists show that if we take the so-called "real quantum theory" without imaginary numbers, and we agree that independent subsystems constitute the entire system in the form of tensor products, then the resulting predictions will be inconsistent with the predictions under the complex number model. Whether complex numbers are necessary to describe quantum mechanics in this way becomes a verifiable thing.
The theoretical work was initially submitted to the scientific preprint server arXiv in January and officially published in the journal Nature in December 2021.
Image from nature
Now that the rules of the game are in place, you only need to design some good experimental devices to complete this verification. It must meet many stringent conditions, such as: the need to achieve deterministic entanglement exchange (the need for deterministic CNOT gate), if it is to use photons as entangled particles, to be able to effectively measure the polarization of photons, Alice, Bob, Clarie to ensure the class space separation to prevent "mutual collusion", and so on.
In March 2021, a research team composed of Pan Jianwei, Lu Chaoyang, and Zhu Xiaobo of the University of Science and Technology of China conducted the first experimental test on the necessity of complex numbers in quantum mechanics based on a superconducting quantum system independently developed by the University of Science and Technology of China[3]. They employed an I-shaped Transmon qubit design to increase the spacing between qubits to reduce near-neighbor coupling between bits on the same superconducting chip. Through high-precision quantum manipulation techniques, two entangled pulse sequences are used to prepare two pairs of entangled states, distributing qubits to the participating three parties. Each party independently chooses the measurement operation to be performed on its qubit, where Bob makes the full Bell state measurement. Finally, calculating the "score" of a quantum game based on the joint statistical distribution of the measurement results, participants with only real numbers can score up to 7.66 points, while experimental results show that tripartite participants consisting of 4 superconducting qubits can score 8.09(1), demonstrating the necessity of complex numbers in the standard quantum mechanical form with experimental accuracy exceeding 43 standard deviations of the criterion. The advantage of this experiment is deterministic entanglement exchange and qubit measurements, closing potential loopholes in detection efficiency.
Experimental result graph: Different theories correspond to different numerical boundaries, and the experimental measurement results greatly exceed the real quantum mechanics model (picture from Chen Mingcheng, Wang Cang, Liu Fengming, etc. PRL 128, 040403 (2022)).
In October 2021, Fan Jingyun's research team at southern university of science and technology conducted a complex number test experiment on optical systems based on the same concept [4]. In the experiment, two independent sources on the same experimental bench produce entangled polarized photon pairs that are distributed to the game's three parties. Alice and Clarie use a local combination of waveplates to perform random measurements of their respective received photons. The prototype for this experiment is from the first entanglement exchange experiment performed by Pan Jianwei and colleagues in Innsbruck in 1998 using linear optics. The SUSTech research team modified the game protocol of complex and real numbers, so that Bob can use linear optics to make probabilistic Bell state measurements to complete the verification. In the end, the three participating parties came to the same conclusion based on the results of the joint measurements with an experimental accuracy of more than 4.5 standard deviations of the criterion, that is, quantum physics requires complex numbers.
Two independent research results were published simultaneously on January 24, 2022 in the internationally renowned academic journal Physical Review Letters, establishing that quantum mechanics requires complex numbers. However, in these two experimental studies, all the quantum state preparation and local measurements of the game tripartite did not comply with the strict space-like separation required by theoretical design, so that in the game of complex and real number games, theoretically, real participants can cheat, using potential loopholes to obtain the same score as the complex participants, resulting in experiments can not distinguish between real and complex numbers under the framework of quantum mechanics.
Based on this, Pan Jianwei, Lu Chaoyang, Zhang Qiang, and others of the University of Science and Technology of China have further carried out experimental tests based on the strict satisfaction of Einstein's localization under the photonic system [6]. In this experiment, the researchers used two independent sources in a quantum network of light to independently generate entangled photon pairs, which were distributed to three participants in the distance for high-speed random photon measurements. During the game, participants are not affected by the measurement choices and results of other participants, and independently perform their own local operations. The experimental results show that the participants under the real number description are incompatible with the data observed in the optical quantum network experiment, and the near-step support proves that complex numbers are indispensable for describing quantum physics.
Diagram of a non-localized experimental apparatus. The three parties to the experiment are in a class-space space and meet strict Einstein non-locality conditions. (From Wu Dian, Jiang Yangfan, Gu Xuemei, et al. arciv.2201.04177, PRL to appear).
Now, our experiment has been validated, imaginary
Not just a tool, but an essential presence. Under the natural assumption that independent systems constitute the total physical state in the form of tensor products, it is proved that the wave function of quantum mechanics is objectively real, and that complex numbers are necessary in quantum mechanics, which means that complex numbers have objective reality and are no longer just mathematical techniques. This is like what we said earlier, as long as you admit the existence of squares, admit that there is such a thing as the length of the diagonal of squares, you have to admit the objective reality of irrational numbers. Of course, as for whether to accept the tensor product hypothesis, just as you accept the existence of squares in the world, you can maintain your own opinion.
Returning to the perspective of basic physics, the author thinks that Mr. Yang Zhenning once mentioned it in a speech at Taiwan Central University
An important role after the development of quantum mechanics. He believes that
It should not be just a tool, but a basic idea. But why the underlying theory must be introduced
, but no one knows.
The second question raised by Mr. Yang, why this is so, will continue to attract physicists to continue to ask. It is possible that we are now as we were written back then
Like Girolamo Cardano, touched the gates of a new world, which was waiting for humanity to push open.
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