Is time really flowing? Einstein: The difference between the present and the future is just an illusion.

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The laws of physics imply that the passage of time is only an illusion. In order to avoid this conclusion, we may have to rethink the reality composed of infinitely accurate numbers. Although there seems to be a knife edge between the fixed past and the open future, we have been crossing it. But strangely, this knife edge-that is, "now"-cannot be found in the existing laws of physics.

For example, in Einstein's theory of relativity, time and three-dimensional space are intertwined to form a curved four-dimensional space-time continuum, that is, the "block universe", which contains all the past, present and future. Einstein's equation describes that everything in the universe has been decided from the beginning; The initial conditions of the universe determine what happens next, and surprises don't happen-they just seem to happen. In 1955, a few weeks before his death, Einstein wrote: "For those of us who believe in physics, the difference between the past, the present and the future is just a stubborn illusion."

Nicholas Gissing is in his home office overlooking the garden.

Einstein believes that reality is eternal and predetermined, and this view is still very popular today. Marina Cortes, a cosmologist at the University of Lisbon in Portugal, said: "Most physicists believe in the theory of massive universe because it is predicted by general relativity."

However, she also pointed out that "if someone is asked to think more deeply about what a massive universe means, they will begin to question and shake its meaning."

Quantum mechanics is a branch of physics that describes the probabilistic behavior of particles. In the eyes of physicists who think about time carefully, quantum mechanics has brought a lot of trouble. On the quantum scale, irreversible changes distinguish the past from the future: a particle can keep multiple quantum States at the same time, and before you measure it, the particle will fall into one of them. Incredibly, although particle behavior usually follows statistical patterns, individual measurement results are random and unpredictable. The obvious inconsistency between the nature of time in quantum mechanics and the operation mode of time in relativity has caused uncertainty and confusion.

In 20 19, the Swiss physicist Nicholas Gissing published four papers, trying to dispel the fog that lingers in physics. In Gissing's view, this problem has always been a mathematical problem. In his view, both time in the general sense and what we call "the present" can be easily expressed by "intuitive mathematics". This is a mathematical language with a history of one hundred years, which opposes the existence of infinite numbers. Gissing believes that when intuitive mathematics is used to describe the evolution of physical systems, it can clearly show that "time is really passing and new information is being created". In addition, under this formal system, the strict determinism implied by Einstein's equation gives way to the unpredictability of quantum form. If the accuracy of numbers is limited, then nature itself is inaccurate and therefore unpredictable.

Few people try to revise the laws of physics with new mathematical language. Physicists are still digesting Gissing's work, but many of them who participated in the debate think that they may be able to bridge the conceptual difference between the determinism of general relativity and the randomness inherent in quantum scale.

"I think this is very interesting," said Nicole halpern, a quantum information scientist at Harvard University, in response to a recent article by Nicholas Gissing in Nature-Physics. "I want to give intuitive mathematics a chance."

Marina Cortes said that Gissing's method was "very interesting" and its meaning was "shocking and provocative". She said: "This is indeed a very interesting formal system, which solves the problem of limited accuracy in nature."

Gissing said that everything we have experienced shows that the future is open and the present is very real, so it is very important to formulate the laws of physics. "I am a down-to-earth physicist," he said. "Time is running out, as we all know."

Information and time

The 67-year-old Gisin was first and foremost an experimenter. He runs a laboratory at the University of Geneva and has done some pioneering research in quantum communication and quantum cryptography. But he is also a rare cross-border physicist, famous for his important theoretical insights, especially those involving quantum opportunities and nonlocality.

On Sunday morning, Ji Xin didn't go to church, but used to sit quietly in a chair at home, holding a cup of oolong tea and thinking about profound conceptual puzzles. One Sunday about two and a half years ago, he realized that in Einstein's theory and other "classical" physical theories, the deterministic picture of time implicitly assumed the existence of infinite information.

Take the weather as an example, because it is a chaotic system, or it is highly sensitive to small differences, so we can't accurately predict the weather in a week. But because this is a classic system, the textbook tells us that in principle, we can predict the weather in a week, as long as we can accurately measure every cloud, every gust of wind and the flapping of butterfly wings. The actual physical laws of the weather unfold like a clock, but due to our own shortcomings, we can't measure the weather with enough accurate decimal places, so we can't make accurate forecasts.

Now, extend this idea to the whole universe. In a predetermined world, time seems to only unfold, and what will happen at any time must actually be set from scratch. The initial state of each particle is coded with countless precise numbers. Otherwise, in the distant future, this predictable universe itself will collapse.

However, information is tangible. Modern research shows that information needs energy and takes up space. Any volume of space has limited information capacity. Gissing realized that the initial conditions of the universe needed to cram too much information into a limited space. He said: "A real number with infinite numbers is meaningless in physics." Block universe implicitly assumes the existence of infinite information, which will inevitably fall apart.

Gissing found a new way to describe time in physics. This method does not assume infinite accuracy of initial conditions.

The logic of time

Modern academic circles have accepted the view that there is a real number continuum, that is, most real numbers have infinite numbers after the decimal point, but this view does not reflect the fierce debate on this issue in the first few decades of the 20th century. David Hilbert, a great German mathematician, believes that real numbers exist and can be manipulated as complete entities. Those who oppose this view are mathematical "intuitionists" led by Ruiz Brouwer, a famous Dutch topographer. They think that mathematics is a structure. Brouwer insists that numbers must be constructable, their numbers can be calculated, or they can be selected or randomly determined. He pointed out that numbers are limited and a process: when more numbers appear in the form of what he called selection sequences, they will become more accurate; The selection sequence is a function used to produce more and more accurate values.

Intuitionism bases mathematics on constructibility, which has a far-reaching impact on mathematical practice and deciding which statements are correct. The most fundamental difference between intuitive logic and standard mathematics is that it does not recognize law of excluded middle, who has been boasted since Aristotle's time. Law of excluded middle means that a proposition is either true or its negative proposition is true. This is a clear set of choices and provides a powerful reasoning model. However, in Brouwer's framework, statements about numbers may not be "true" or "false" in a given time, because the exact values of numbers are not displayed.

When it comes to numbers like 4, 1/2 or π, there is no difference in standard mathematics. Although π is an irrational number without a finite fractional part, an algorithm can be used to generate its fractional expansion so that π is determined as 1/2. But what if it is another number x similar to 1/2?

Suppose the value of x is 0.4999, and the following numbers are expanded in the selection sequence. Maybe the sequence of 9 exists forever, in which case X converges to 1/2.

However, if a number other than 9 appears at a certain point in the following sequence, for example, the value of X becomes 4.999999999997, then no matter what happens, X is less than 1/2. Before that, when we only knew 0.4999, "we didn't know whether numbers other than 9 would appear," explained Karl Posey, a mathematical philosopher and an authoritative expert in the field of intuitive mathematics at Hebrew University in Jerusalem. "When we consider this value of X, we can neither say that X is less than 1/2 nor that x = 1/2." The proposition "x = 1/2" does not hold, nor does its negative proposition. Law of excluded middle is not established.

In addition, the continuum cannot be clearly divided into two parts, one part is that all numbers are less than 1/2, and the other part is greater than or equal to 1/2. Posey said: "If you try to cut the continuum in half, the number X will stick to the knife, and it will not be divided into left or right." "This continuum is sticky and constantly being cut."

Hilbert compared taking law of excluded middle from mathematics to "forbidding boxers to use their fists", because this principle is the basis of mathematical reasoning. Although Brouwer's intuitive framework attracted people like Kurt G?del and Herman Weil, standard mathematics was dominated by real numbers because of its ease of use.

The unfolding of time

20 19 May, Gissing first came into contact with intuitive mathematics at a meeting attended by Karl Posey. When they started talking, Gissing soon discovered that there was a connection between the decimal places described in this mathematical framework and the physical concept of time in the universe. When the uncertain future becomes a concrete reality, materialized numbers seem to naturally correspond to the time series that defines the present. Law of excluded middle's absence is similar to the indeterministic proposition about the future.

In the paper published in Physical Review A on February 20 19, Gissing and his collaborator flavio Del Santo formulated another version of classical mechanics with intuitive mathematical language, made the same predictions as the standard equation, but described the events with indeterminism. This creates a new picture of the universe in which unpredictable things will happen and time will unfold.

Kind of like the weather. In retrospect, we can't predict the weather accurately, because we can't know the initial conditions of every atom on the earth with infinite accuracy. But in Gissing's non-deterministic version, these infinitely accurate numbers never existed. Intuitionistic mathematics captures this point: when the future unfolds in a selection sequence, the numbers that accurately specify the weather state and indicate its future development are selected in real time. Lei Natuo Lai Na, a quantum physicist at Swiss Federal Institute of Technology, said that Gissing's argument "points in one direction, that is, deterministic prediction is basically impossible".

In other words, the world is uncertain and the future is open. Gissing pointed out that time "will not unfold like a movie in a cinema." It is actually creatively developed, and with the passage of time, new figures are constantly produced. "

Fay Doak, a theorist of quantum gravity in Imperial College London, London, UK, said that she "very much agrees" with Gisin's point of view, because "he is on our side and thinks that physics is inconsistent with our experience, so he missed something." Doak agrees that mathematical language has shaped our understanding of time in physics. The standard Hilbert mathematics regards real numbers as a complete entity, which is "obviously static and has the characteristics of not being limited by time, which is undoubtedly a limitation for physicists, especially when we try to include some dynamic things, such as the experience of time passing".

For physicists like Doc who are interested in the connection between gravity and quantum mechanics, the most important revelation of this new view of time lies in how it will establish a connection between two world views that have long been considered incompatible. "For me, a revelation is that," Reiner said, "in some ways, classical mechanics is closer to quantum mechanics than we thought."

Quantum Uncertainty and Time

If physicists want to solve the mystery of time, they should not only wrestle with Einstein's space-time continuum, but also understand the quantum nature of the universe. The picture of time depicted by quantum theory is completely different from Einstein's theory. Reina said: "Our two major physical theories, quantum theory and general relativity, have made different statements." He and several other physicists pointed out that this inconsistency made it very difficult to establish the theory of quantum gravity and understand why the Big Bang happened. "Look at where there are contradictions, and then look at what we have. In the end, it all comes down to the concept of time, "Reina said.

Time in quantum mechanics is rigid, not curved, and intertwined with the spatial dimension in relativity. In addition, the measurement of quantum system "makes time in quantum mechanics irreversible, while in other respects, quantum theory is completely reversible," Lai Na said. "So time has played a role in this matter, but we don't know much about it yet."

In the understanding of many physicists, quantum physics tells us that the universe is uncertain. "You can find two uranium atoms, one decays after 500 years and the other decays after 1000 years, but they are exactly the same in any way," said Nima Akani-Hamid, a physicist at the Institute for Advanced Studies in Princeton, New Jersey, USA. "In any sense, the universe is uncertain."

Nevertheless, other popular explanations of quantum mechanics, including the "multi-world" explanation, are trying to keep the classical concept of deterministic time alive. These theories describe quantum events as an established reality. For example, the multi-world theory holds that every quantum measurement divides the world into many branches, so as to realize every possible result, and these results are preset.

Gissing thinks the opposite. He hopes to provide an indeterministic language for classical physics and quantum physics, instead of trying to turn quantum mechanics into a determinism. However, this method deviates from the standard quantum mechanics to a great extent.

In quantum mechanics, information can be destroyed, but it cannot be created or destroyed. However, if the number that defines the state of the universe increases with time, as Gissing proposed, then new information will appear continuously. Gissing said that he "absolutely" opposes the idea that information is preserved in nature, mainly because "new information is obviously produced in the process of measurement". He added: "I mean, we need to look at these ideas in another way."

This new way of thinking about information may solve the information paradox of black holes. This paradox raises a question: What will happen to the information swallowed by the black hole? General relativity holds that information will be destroyed, while quantum theory holds that information is preserved. If quantum mechanics expressed by intuitive mathematics allows information to be created through quantum measurement, then information is also allowed to be destroyed.

Jonathan oppenheim, a theoretical physicist at University College London, believes that information is indeed lost in black holes. He didn't know whether Brouwer's intuitionism could be the key to this point as Gissing advocated, but he said there was reason to believe that the creation and destruction of information might be closely related to time. "As time goes by, information is destroyed; It will not be destroyed because you are moving in space. " There are great differences between the dimensions that make up Einstein's massive universe.

Besides supporting the concept of creative time, intuitive mathematics also provides a novel explanation for our conscious experience of time. Recall that in this framework, the continuum is sticky and cannot be split in two. Gissing relates this stickiness to our feeling that "the present" has a "thickness", which is a substantive moment, not a zero-width point that completely separates the past and the future. In standard physics based on standard mathematics, time is a continuous parameter, which can take any value on the number axis. However, Gissing said, "If the continuum is expressed by intuitive mathematics, then time cannot be divided into two." Very thick, "as thick as honey".

So far, this is just an analogy. Oppenheimer said that he "has a strong feeling about the concept of' now thick'. I don't know why we feel this way. "

The future of time

Gissing's idea caused a series of reactions from other theoretical physicists, who also had their own thought experiments and intuition about time.

Several experts agree that real numbers do not seem to hold water in physics, and physicists need a new formal system that does not depend on real numbers. Ahmed alm Li Hai, a theoretical physicist who studies black holes and quantum gravity at Princeton Institute for Advanced Studies, believes that quantum mechanics "excludes the existence of continuum". Quantum mathematics binds energy and other quantities together like a package, more like an integer than a continuum. Infinite numbers are truncated in black holes. "Black holes seem to have an infinite number of internal states, but these numbers are cut off," he said. "Because of quantum gravity, real numbers can't exist because you can't hide them in a black hole. Otherwise, they can hide endless information. "

Sandu Pops Ku, a physicist at the University of Bristol in England, often agrees with Gisin. He agreed with the latter's indeterministic worldview, but he also said that he did not believe that intuitive mathematics was necessary. Pops Ku opposes the idea that real numbers are regarded as information.

Akani-Hamid thinks that Gissing's application of intuitive mathematics is very interesting, and it may play a role in the obvious conflict between gravity and quantum mechanics such as black holes and big explosions. "These problems, such as the finiteness of numbers, the fundamental existence of things, whether there are infinite numbers, or the creation of numbers," he said, "may ultimately be related to how we should look at cosmology, especially when we don't know how to apply quantum mechanics." He also saw the need for a new mathematical language, which could "liberate" physicists from infinite precision and let them "talk about things that are always a little vague".

Gissing's idea has aroused many voices, but it still needs to be enriched. Next, he hopes to find a way to reconstruct relativity and quantum mechanics with limited and fuzzy intuitive mathematics, just as he did with classical mechanics, which may bring the two theories closer. He has some ideas about how to deal with quantum mechanics.

One of the ways in which infinity appears in quantum mechanics is the tail problem. When you try to locate a quantum system, such as an electron on the moon, "if you use standard mathematics, you must admit that the probability that an electron on the moon is detected on the earth is very small," Gissing said. The mathematical function representing the particle position will leave a "tail" and "exponential decline, but not zero".

But Gissing wanted to know, "What kind of reality should we attribute an ultra-small number to? Most experimentalists will say,' Let it go to zero and stop questioning'. But people who are more inclined to theory may say,' OK, but from a mathematical point of view, it makes sense'. "

"Now it depends on what kind of mathematics is used," Gissing continued. "There are some things in classical mathematics. In intuitive mathematics, there is nothing. " Electrons are on the moon, and the possibility of appearing on the earth is zero.

Since Gissing's paper was published, the future has become more uncertain. For him, every day is Sunday now, because the epidemic is hanging over the world. Because he can't go to the lab, he can only see his granddaughter on the screen. His next plan is to hold the teacup in his hand, face the garden landscape and continue to think about time.

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