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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), author: Dong Weiyuan

As we all know, quantum mechanics is a compulsory course for physics majors. In many classical textbooks, almost all the quantum theories learned are derived from wave dynamics and matrix mechanics, which were born in the 1920s, and their explanations are based on Copenhagen interpretation. Of course, many people have heard of other interpretations, such as multi-world interpretation or Bohm mechanics, which will be introduced in this article.

As a completely different theory of quantum mechanics from ordinary textbooks, Bohm mechanics has been developed for many years, and even in recent years, relevant researchers have done a good job in experimental testing and theoretical prediction. it shows the "power" of this theory. However, there is little introduction to this theory and popular science in China. This paper will start with the origin of Bohm mechanics, introduce its key concept of "quantum potential", and hope that readers can quickly understand Bohm mechanics.

The problem of quantum measurement has always been one of the important problems that all quantum theories must face. In the orthodox textbook of quantum mechanics, the observable physical quantities such as the position and momentum of particles are determined only at the moment of being measured, and usually evolve together with a number of possible values.

As long as we do some calculations around the quantum state, we can predict the results of the measurement. But the quantum state itself is not an observable object, just as a person's behavior is dominated by his own character, but we cannot directly see his character through dissection.

Just as character only probabilistically dominates behavior, quantum states are probabilistic descriptions of the possible behavior of particles. From this point of view, the quantum state is indeed very similar to the character of particles, but the inappropriateness of this metaphor is also very obvious. Foodie, which loves both Chinese food and western food very much, will not give up Chinese food completely just because of a meal of western food. However, a particle that originally contains both spin-up and spin-down states will completely collapse into a single up or down state after a single measurement.

For this quantum state evolution process that produces jumps in the process of measurement, we can either retain all causal connections in general with the help of multi-world theory, or simply embrace randomness and put probability at a level that is even more basic than shoulder-to-shoulder causality. Apart from that, do we have any other angles of understanding?

Bohm mechanics (Bohmian mechanics) provides a completely different perspective. With the help of a mysterious concept of "quantum potential (Quantum potential)", Bohm mechanics can restore the process of quantum measurement to intuitive classical images. Each particle has a definite position and velocity at all times, and evolves under the restriction of Newton's law, and its state has nothing to do with whether it is measured or not, let alone jump because of the measurement. This theory firmly maintains the core and basic position of deterministic causality in the laws of physics, and does not require particles to have the art of separation like Sun WuKong to support probability. As for the probabilistic presentation of the experimental phenomena, it is only a statistical effect similar to thermodynamics in this theory.

The path of particles passing through a double slit in Bohm theory: the origin of Wikipedia Bohm mechanics, also known as de Broglie-Bohm theory (De Broglie-Bohm theory), was originally proposed as an interpretation of quantum theory. Since the Schrodinger equation was published in 1926, physicists have been trying to understand what the wave function in the equation represents. Born's probability wave explanation, though strongly opposed by the giants Einstein and Schrodinger, was quickly adopted by the Copenhagen School headed by Bohr.

At the same time, de Broglie, the creator of the concept of matter wave, is also constructing an interpretation of wave-particle duality. He believes that the vibrating field in space-time should be an objective entity, while particles are singularities or regions formed by very special modes of vibration. These singularities can walk forward perpendicular to the wave array, which makes us think we see particles. In fact, particles are not objective entities in nature, but just singular geometric points in the wave field.

As for the probability characteristics of particles, it can be explained by radiation intensity. Roughly speaking, the square of the amplitude of that solid wave is the radiation intensity, and singularities are more likely to occur in places with high radiation intensity. The probabilistic presentation seen when measuring particles is only a statistical effect, not what born calls the fluctuating content itself.

De Broglie calls his interpretation "double solution" because it is based on two different solutions of the same wave equation, one is an ordinary continuous solution, which carries volatility, and the other is a singular solution, which is regarded as a particle, showing particle nature.

At the Solvay Conference, unprecedented in the history of physics in 1927, in addition to the wonderful debate between Einstein and Bohr, de Broglie also demonstrated his idea of "double solution". Since the title of his speech was "pilot wave", this theory was later called more navigational wave theory than double solution theory.

Of the participants in the Fifth Solvay Conference in 1927, 17 were Nobel laureates. Wikipedia's appearance at the Solvay conference was a disaster for navigation wave theory. The quick-witted Pauli pointed out on the spot that although this theory can explain the double-slit interference experiment, it can only describe the behavior of a single particle in a more general scene, and when considering the collision and scattering of two particles, the theory will collapse instantly, not to mention a complex multi-particle system.

De Broglie, who was so speechless as to be refuted, soon gave up his idea and succumbed to the Copenhagen interpretation, following the born rule as the standard. Other top celebrities who attended the meeting also witnessed this scene, and for a long time since then, that generation of researchers have stayed away from the navigation wave theory. It was not until 25 years later, in 1952, that the theory was rediscovered and repaired by David Bohm.

Different from de Broglie's idea, Bohm believes that particles are real objective entities, not singularities in the wave field, and waves are also objective entities. The so-called "wave-particle duality" means that each solid particle is held hostage by its own navigation waves like a puppet. The things that hit the wall are done by particles, showing the nature of particles in the exchange of energy, while the path of motion is constrained by navigation waves, showing volatility. Of course, if there are multiple particles, the navigation waves will interfere with each other, and the motion of each particle will also be affected by other navigation waves.

This picture is a rough sketch of the author's own graffiti. Please ignore the imprecise details of the tons in the picture. In Bohm's hands, this theory not only solves the problem of multi-particle system and perfectly answers Pauli's query at that time, but also can explain quantum phenomena such as quantum tunneling, chemical bond formation mechanism, AB effect, superfluid, superconductivity, Wheeler delayed selection experiment and so on.

Unfortunately, Bohm himself encountered a lot of bad luck in the golden period of academic research. During the Cold War, he was suspected of being a communist by the US government. In 1950, when McCarthyism was in its heyday, Bohm was arrested and imprisoned. Although he was released the following year, he lost the opportunity to continue his academic research in the United States and had no choice but to live in Brazil.

The 2.0 version of the navigation wave theory developed by Bohm is the result of his research during his time in Brazil. Bohm moved to Israel in 1955 and moved to England in 1957. Years of wandering did not stifle his enthusiasm for research. During his stay in Israel, he combed and reinterpreted the EPR paradox in detail, which paved the way for Bell's discovery of the famous Bell inequality later.

David Bohm (David Bohm, 1917-1992)? photo Source: Wikipedia, but Bohm himself has been trapped on the fringes of academia for non-academic reasons. Although Bohm, in his early 30s, was an assistant professor at Princeton University at the end of World War II, it was not until 1961 that the 44-year-old got a professorship at Birkbeck College, University of London. When Bohm and his student Aharonov discovered the amazing AB effect in 1959, he was a general researcher at the University of Bristow in England.

In this context, although the mainstream physics circles have paid some attention to Bohm mechanics, the real follow-up research is very few, far less than the degree that they deserve. In the 40 years from his theory in 1952 to his death in 1992, Bohm has been ploughing the land almost all alone, except for the support of a few physicists such as Bell.

The reason why many physicists ignore Bohm's theory is that "quantum potential" is too difficult to understand. Einstein once described it as "a fairy tale in physical theory". But is this really the case? Let's first take a look at what this "quantum potential" is, which has attracted a lot of complaints from people in physics.

The concept of quantum potential was first introduced by Bohm and did not exist in de Broglie's navigation wave theory. The process of deriving the quantum potential mathematically is very simple, as long as the wave function is written in the form of ψ = ReiS/ potential, and then substituted into the Schrodinger equation.

You get two relationships:

And

The second equation is very intuitive. ▽ S is momentum, ▽ S / m is the velocity of particles, and the whole equation is the conserved flow equation of R2. The first equation is more interesting. It is easy to see that in the macro world, when → 0, this equation naturally returns to the Hamilton-Jacobi equation of classical mechanics.

So we can feel that the strange quantum properties of the microscopic world should all be contained in that purple term. Yes, this term is the "quantum potential" defined by Bohm.

We can use this quantum potential in the same way as ordinary potential energy, such as writing the equation of motion of particles as

A familiar Newton's law properly describes the trajectory of particles! Among them,-▽ UQ is a force that produces various quantum effects in the microscopic world.

Why is it so difficult for top physicists to understand that such a model that saves effort and works almost seamlessly with macroscopic classical mechanics?

If we take a closer look at the expression of the quantum potential UQ, we will find that although there is an amplitude R representing the vibration energy of the navigation wave, the UQ will not change no matter how many times R is magnified or reduced. Readers familiar with geometry may also see that UQ seems to correspond to a certain curvature of R.

In other words, the magnitude of the quantum potential has nothing to do with the vibrational energy carried by the navigation wave, and the navigation wave does not use its own energy when acting on the particles. It can be further clarified that in Bohm's theory, the action between the navigation wave and the particle is one-way, the navigation wave is the active side, the particle is the passive side, and there is no reverse effect. The navigation wave shows the curvature of its amplitude to the particle, thus "telling" the particle how to move.

This one-way function, which is not based on energy exchange, has gone beyond the definition of "interaction" in physics. Bohm believes that this is a mechanism based on information transmission, so it is sometimes simply called "information potential (information potential)".

To take a simple example: if I see a fascinating article in my life, I will jump excitedly, and the article will have an effect on me and change my exercise state, while I have no reverse effect on this article. Although these experiences can help us understand what Bohm called the new mechanism of action, if we are to be seriously integrated into the theoretical system of physics, we do need to muster up a lot of courage, overcome a lot of anxiety and, more importantly, we still need to face many of the resulting problems.

For example, if a particle determines its state of motion by receiving information from navigation waves, does it mean that the particle itself must have a sufficiently delicate and complex internal structure? At the very least, it is necessary to support the ability to receive, identify and process information. If so, how can particles be the most basic units that make up the physical world?

Bohm himself thought carefully and deeply about many related issues. on his deathbed, he and his colleague Basil Healy co-wrote The Undivided Universe, an important book on Bohm's mechanics, which was officially published after Bohm's death. A considerable part of the book talks about this kind of special mechanism.

In my opinion, although the questioning of this new mechanism seems conservative and narrow, it also has some reasonableness. But other doubts caused by lack of understanding are somewhat unfair or even untrue.

Prejudices and misunderstandings even among physics professionals, the reference to Bohm mechanics is often criticized as follows:

The non-locality of the quantum potential will lead to superluminal speed, which violates the theory of relativity.

The theory has no covariance and cannot be reconciled with relativistic space-time.

Can not provide particle generation / annihilation mechanism, can not be reconciled with field theory.

It is essentially equivalent to the textbook version of quantum theory, and can not provide independent experimental prediction.

In fact, all these arguments are not reasonable.

If we only talk about non-locality in general, this is not a unique attribute of Bohm's theory at all, but an inevitable feature of any kind of quantum theory interpretation. In fact, studies from von Neumann to Bell have long proved that any localized theory is doomed to fail to conform to the experimental facts.

Admittedly, the non-locality in Bohm's theory is essentially different from that in the Copenhagen interpretation, which avoids the superluminal transmission of information through vagueness, while Bohm's theory can not avoid this taboo. In a system of two identical particles, the quantum potential becomes

Where R is the amplitude of the wave function ψ (R1, R2, t) = ReiS/ moment in the phase space, R1 and R2 are the positions of two particles, and ▽ 1 and ▽ 1 correspond to the spatial derivatives of the positions of the two particles at t time, respectively.

We can read that the quantum potential of a multi-particle system is, by definition, a non-local global quantity. If each particle obtains information from this global quantum potential to determine its own mode of motion, it must first admit that information can be transmitted in space without the speed of light.

In this way, it seems quite reasonable to be questioned. But don't forget that the information and causal correlation defined by relativity cannot exceed the speed of light. It is still assumed that information transmission is based on energy exchange. In essence, relativity limits only the speed of energy transfer. As mentioned earlier, the new mechanism introduced by Bohm's theory does not depend on energy exchange. If that new mechanism really exists, it is only a very natural result that it is not limited by the speed of light.

With regard to the harmony of the theory of relativity, it can be seen from the quantum potential of the multi-particle system that the statement of "the same moment in different spatial positions" is indeed against the idea of relativity. Bohm himself has long been aware of this, and it has always been one of his key tasks to incorporate the relativistic concept of time and space into Bohm mechanics. After continuous exploration, he has successfully achieved this goal, and the whole chapter is devoted to discussing the relevant content in the book The Undivided Universe.

In addition to Bohm's own work, after the 1990s, there have been some other efforts to relativize Bohm mechanics, all of which have achieved quite good results. German mathematician theoretical physicist Detlef D ü rr and his research team, Indian physicist Partha Ghose, Croatian physicist Hrvoje Nikoli Qing and other researchers have successfully extended Bohm mechanics to the relativistic space-time from different angles.

Among them, the Detlef D ü rr team and Hrvoje Nikoli Qing also introduced the generation / annihilation mechanism of particles in different ways, thus opening the relationship between Bohm mechanics and quantum field theory. British researchers George Horton and Chris Dewdney went further, successfully reconciling Bohm mechanics with the quantum field theory of flat space-time, and even developed the theory to the point where gravity could be accommodated in 2010 [9-11].

As for the equivalence between Bohm's theory and the textbook version of quantum theory, it is the misunderstanding of this theory by people who do not understand it. First of all, only from the perspective of interpretation, different interpretations are not necessarily equivalent; secondly, Bohm mechanics is different from the quantum theory in textbooks from mathematical form to physical connotation. So Bohm's theory is not only another interpretation of quantum phenomena, but also a completely different quantum theory. Of course, independent experimental predictions can be given.

The experimental test of Bohm mechanics has two main points that can be applied to the experimental test.

One is the probability of particle distribution. In the textbook version of quantum theory, the relationship between the probability of particle distribution ρ and the wave function must satisfy ρ = | ψ | 2, which is called born's rule and is one of the basic hypotheses of quantum theory. However, Bohm mechanics regards this as a statistical conclusion in a certain "equilibrium state". Just like the thermal equilibrium state of a thermodynamic system, it is entirely possible to have a "non-equilibrium state" of ρ ≠ under special conditions.

The other is the classical trajectory of particles, which is also the most intuitive and distinct feature of Bohm's theory. If it can be shown by experiments that particles do move only along a smooth trajectory, rather than diffusing in an area like a ghost, it will undoubtedly be strong evidence of the theory. Or to put it more bluntly, as long as in the double-slit interference experiment, it can be proved that the particle passed through only one of the slits without detection, rather than passing through two slits at the same time, as the textbook says, the balance of credibility can be tilted to Bohm theory.

However, the initial exploration was just the opposite. In 1992, four researchers, Englert,Scully,S ü ssmann and Walther, proposed a thought experiment, whose deduction cast a deep shadow on the credibility of Bohm's theory. Later, this experiment was called the ESSW experiment [12]. This thought experiment adds some special links on the basis of the double-slit interference experiment in order to "deceive" navigation waves and particles.

According to their theory, if the particle motion mechanism is like Bohm's theory, then in the ESSW experiment, the particle will show an incredible trajectory. Researchers call it a "surreal trajectory" (surreal trajectory). It can also be seen from this wording that although several studies have not directly denied Bohm's theory, they have obviously had deep doubts.

The moment of twists and turns occurred in 2016, when the Canadian physicist Steinberg put the ESSW experiment into practice with the help of quantum weak measurement technology, and unexpectedly got the results that supported Bohm's theory [13]. Although this result can not claim the victory of Bohm theory, it also makes many researchers, including Steinberg himself, immediately increase their interest and confidence in Bohm theory. In a very short time since then, the enthusiasm of the relevant experiments has been significantly improved.

In 2019, the Detlef D ü rr research team found a new experimental method from the point of view of ρ ≠ | ψ | 2. Specifically, they suggest measuring the time it takes for electrons to move from point A to point B, or, more accurately, the probability distribution of this "flight time". According to the textbook version of quantum theory, no matter what changes take place in the surrounding environment during the movement of electrons, the distribution of flight time will always be strictly defined by the ρ = | ψ | 2 rule. According to Bohm's theory, sudden environmental changes will destroy the "quantum equilibrium", just like a sudden injection of hot water into cold water, the system needs a relaxation time to re-reach equilibrium, and before reaching the new equilibrium, ρ ≠ | ψ | 2 will occur.

With the improvement of experimental technology, a German research team is recently planning to put this experimental idea into action. This has attracted a lot of attention, and I hope this experiment can be started smoothly as soon as possible. Unfortunately, Detlef D ü rr himself passed away at the beginning of this year, unable to witness the actual results of his proposed experiment.

The theoretical predictions of Bohm mechanics are rich. If experiments can prove Bohm's theory to be correct, it will certainly be a major event in the field of basic theory. However, even before it has been confirmed by experiments, Bohm mechanics, which is only a theoretical model, has played an important role in many fields.

In the atomic / molecular scale, or in the quantum many-body problem, due to the large number of particles, the degree of freedom of the system also increases, and the amount of computation often explodes. If you can't see the physical process clearly, it is easy to approximate the process, and it is easy to miss the key factors hidden in the details.

The advantage of Bohm's theory is that it can not only smoothly connect the classical computational framework of the macro, but also fully provide the quantum effects and mechanisms at the micro level. moreover, the complex action or evolution process can be shown clearly and intuitively through the classical particle path. So in atomic / molecular physics, material science, chemical physics, optics. The extensive use of Bohm's theoretical tools can be seen in these fields.

Of course, the value of Bohm's theory is not only to simplify calculation methods and visualize physical processes, but also to help us understand the mysteries of the universe. In 2014, Cai Qingyu of the Wuhan Institute of Physics and Mathematics of the Chinese Academy of Sciences and his colleagues found that the universe can be produced spontaneously through quantum mechanisms [15], which explains the early surge of the universe. In their paper, by using the Bohm theory model, they found that in the early universe, the quantum potential naturally played the role of cosmological constant, and the quantum effect was the root cause of the explosion of the early universe.

In addition to delivering ammunition to various research fields, Bohm's theory itself is constantly absorbing nutrients from other research results. In 2019, the Norwegian researcher Gregory Duane, inspired by ER=EPR 's idea, deduced the quantum potential of Bohm mechanics from the space-time structure of general relativity, thus revealing that the non-locality of the quantum potential is originally given by the wormhole on the Planck scale. In this way, it seems that the non-locality of the quantum potential is not so difficult to understand.

Up to now, there are still many interesting and profound problems in Bohm mechanics waiting for people to explore. May more beautiful fruits continue to bear fruit on this fertile soil.

Thank you: thank the reviewers for their professional advice.

reference

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D ü rr, Detlef; Goldstein, Sheldon; Norsen, Travis; Struyve, Ward; Zangh trees, Nino (2014). "Can Bohmian mechanics be made relativistic?" Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 470 (2162): 20130699. ArXiv:1307.1714. Bibcode:2013RSPSA.47030699D. Doi:10.1098/rspa.2013.0699. PMC 3896068. PMID 24511259

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[14] arXiv:1901.08672v1 [quant-ph] 24 Jan 2019

[15] Phys. Rev. D 89, 083510 (2014)

[16] https://doi.org/10.1016/j.physleta.2018.12.015

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