Ordinary people often ask, who is the greater mathematician or physicist? The answer is that there are many different opinions. People usually use the Nobel Prize to measure the achievements of a scientist. Unfortunately, the Nobel Prize does not establish a mathematics prize. However, if your math is as good as Penrose, you can win the Nobel Prize twice, once for others and the other for yourself.

Players tiling

aperiodic tiling means no translational symmetry tiles. Penrose discovered very early that a non-periodical tile can be achieved by combining floor tiles with four shapes: pentagon, pentagram, rhombus and boat. In 1974, Penrose discovered that two types of diamond-shaped floor tiles can be used to achieve this non-periodic tiling equivalently. Now this type of floor tile is named Penrose floor tile. In 2013, the entrance to the mathematics department of Oxford University was paved with Penrose floor tiles, as a noble tribute to Penrose’s mathematical attainments.

In 1982, Daniel Schechtman (Daniel Schechtman) discovered that there is a pentagonal structure in the holmium-magnesium-zinc alloy. Like the Penrose floor tiles, it has no translation invariance, but rotation 2π/5 invariance. There was a major controversy in the academic world at that time. Was this a special crystal structure, or was there an experimental error? It was later discovered that if the atoms are arranged like Penrose tiles, the theoretically calculated X-ray diffraction pattern is exactly the same as the experiment. As a result, quasicrystals were discovered, and Shechtman won the 2011 Nobel Prize in Chemistry.

Penrose also has a lot of games that look like tiling. For example, the ability to imagine things that are physically impossible is a wonderful feature of human consciousness. Using this kind of imagination, people can resonate in consciousness and inspire conceptually. In 1961, Penrose and his father (Lionel Penrose) conceived a kind of endless staircase, which is the famous Penrose triangle. The versatile Penrose is the Leonardo of our time, and Penrose has contributed too much. There are too many innovations named after Penrose, Penrose theorem, Penrose conjecture, Penrose generalized inverse matrix, Penrose graph, Penrose equation, Penrose flag, Penrose twist…the list goes on. Let’s start with the singularity theorem below!

Singularity theorem

Penrose was a mathematician whose original specialty was algebraic geometry. In the 1950s, he was influenced by Hermann Bondi and Dennis Sciama of Cambridge University, which aroused his interest in general relativity. It is his pure mathematics background that his research method is very different from others.

Until the 1960s, the research method of general relativity was only a local technique: Einstein’s field equations described how the energy and momentum tensor of matter determines the geometry at a certain event, and geometry tells how the matter moves. In other words, this is a kind of local physics, the solution of Einstein’s field equation is obtained through local integration in time. The non-gravitational laws of physics can be derived using the principle of equivalence in the local Lorentz frame of reference at each event in time and space. However, since 1963, Penrose and his collaborators’ research on black holes and singularities has revealed the overall nature and overall laws that are concise and beautiful, comparable to Einstein’s equivalent principle. Hawking pointed out that “Penrose first discovered that there is no need to solve equations to discover universal laws.”

When calculating in an asymptotically flat spacetime, it is necessary to consider the asymptotic form of the physical field at infinity. If you want to know how much energy gravitational waves and electromagnetic waves will take away when a supernova erupts, it is not enough to only consider the asymptotic form of the space infinity, you must also consider the asymptotic form of the metric at “future infinity”. In 1964, Penrose used the “conformal transformation” technique to change the “infinity” to a finite radius, and the asymptotic calculation became a finite calculation. Penrose divided the “infinity” in asymptotically flat space-time into five categories:

What is the end of gravitational collapse? The end point of spherically symmetrical collapse is the singularity, so can asymmetrical collapse avoid the singularity? Einstein believed that asymmetry could avoid singularities, so he did not believe in the existence of black holes. However, Penrose, Hawking and RP Geroch proved the singularity theorem. Regardless of symmetry or not, the singularity is a very common phenomenon, and there must be black holes in the universe. In 1965, Penrose gave the concept of a capture surface. The capture surface is no longer limited to a 2-dimensional sphere, but an arbitrary closed surface that captures light. In 1968, Jerome refined the definition of the singularity, and the space-time manifold cannot extend beyond the singularity. In 1969, Hawking and Penrose published their singularity theorem: within the framework of general relativity, if space-time still satisfies four natural physical conditions, then space-time must contain singularities.

Battle flag fluttering

Hawking, Penrose introduced the modern concept of spin and the overall amount of technology in general relativity. The application of spinor in the theory of relativity firstly arises from the analysis of space rotation, and then the analysis of space-time rotation. Consider a cube and rotate it around one axis by π/2, and then around another axis by π/2. As a result, the cube has shifted from the “initial” orientation to the “end” orientation. What is the law of rotation combination? If you use a vector to describe the spinner, it is obviously wrong. Because of using the vector synthesis rule, the result is: the synthesized vector is on the plane composed of two vectors; the size of the synthesized vector is π/. However, the single rotation of the cube from the initial point to the final point is: the axis of rotation points from the center of the cube to an apex angle; the angle of rotation is 2π/3.

This is the mathematical object called the spin matrix today, which is used to describe the change of the spinor.

As a geometric object, a vector can be drawn vividly as an arrow. Screw is also a kind of geometric object. How can the screw be represented visually? This seems too difficult, beyond the imagination of ordinary people, but Penrose did it. If O is somewhere on the earth, a light pulse is emitted to the center P of the Aristakes crater on the moon, and the laser is designed to produce not a light spot, but an arrow. Penrose referred to the zero vector OP as a flagpole, and the bright arrow as a flag. Therefore, the spin quantity is constituted by the trinity of the following geometric quantities: the direction-winding relationship between the zero flagpole, the flag, the flag and the surrounding things. By repeatedly emitting laser pulses, and “rotating” the flag between two shots, when the flag rotates 2π back to the original direction, the spin is reversed. When the flag wraps around the flagpole 4nπ, the spin amount returns to its original value.

When people study the influence of gravity on fermions, the spin analysis in curved space-time is an indispensable mathematical tool. The analysis of the bound state of fermions in the gravitational field requires the solution of the Dirac equation in Schwarzschild space-time, and also the Newman-Penrose equation that deals with the spinner of curved space-time. The Trinity spinner flag plays an important role in visually depicting the twisting of more complex geometric objects.

Father twist amount of

the basic equations of quantum theory is a complex form of differential equations, Penrose considered basic space-time structure of the complex should be. Those who have studied complex analysis must know that the entire complex plane including the point at infinity can be expressed by the Riemann sphere. In order to establish a complex space-time structure, Penrose developed the twist quantity concept, and he created the twist quantity theory.

An observer located somewhere in time and space, observing a star, he drew the star’s azimuth on the celestial sphere. If the second observer passes through the same point at the same time, if there is a relative speed between the two, then due to the aberration, the second observer will draw the star in a different direction. The different points on the Riemann sphere can be connected by the Mobius transformation. The light space through the space-time points naturally constitutes a Riemann sphere. The positive Lorentz group that connects the physics of different speed observers can be realized by the automorphism group of the Riemann sphere.

Penrose regarded light as a more basic object than the point of space-time, and extended this concept to the entire space-time. Therefore, twisting space (light space) is a more basic space than space-time, and space-time has become a subordinate concept. A light in time and space is a point in twisted space, and a point in time and space is represented by a collection of light. In other words, the space-time point becomes a Riemann sphere in the twist space.

Such twist space is real 5-dimensional, and complex space must be real even-dimensional. Penrose believes that the energy and helicity of the photon must also be considered, so a 3-dimensional complex projection space CP3 is used, which is the projected twist space (PT). Its 5-dimensional subspace is PN, and PN divides PT into two parts of left-handed helicity and right-handed helicity, PT- and PT+.

The space-time point is described by 4 real numbers, and the point in the projected twist space is given by 4 complex number ratios Zα. Using two complex spin quantities ω and π, Penrose expressed the twist quantity. Using Penrose flag, you can vividly draw geometrical figures of twist. If the twist function defined on PN is extended to PT+, it will have a positive frequency; if it is extended to PT-, it will have a negative frequency. This is exactly the concept needed to promote the quantized place. The positive frequency part propagates in the forward direction of time, and the negative frequency part propagates in the backward direction of time. The positive frequency part is the propagator composed of the positive energy part, which allows people to carry out quantum physics research in the torsion space. Along this path, physicists further developed the torsion diagram, which is a scheme similar to the space-time Feynman diagram describing the interaction.

The torque theory is conformal and invariant, so for a conformal flat curved space-time, the torque space can also be established. Penrose not only studied the torsion theory of curved space-time, but also further studied the torsion cosmology. The twist space is complex 4-dimensional. Penrose is extremely dissatisfied with the 10-dimensional superstring theory with additional dimensions. Penrose is a major critic of superstring theory. In 2003, during an encounter with Edward Witten, Penrose was worried that the two would have a heated argument. Unexpectedly, the string theory master told him that he was studying how to combine twist and string theory. theory.

Quantum measurement

Bohr, Heisenberg and Pauli in 1927, the Copenhagen meeting on the proposed standard interpretation of quantum mechanics. It includes uncertainty principle, wave-particle duality, probability interpretation, eigenvalue identification measurement value. In 1932, von Neumann added the final element to the Copenhagen interpretation: wave function collapse. However, physicists such as Einstein, Schrödinger, and De Broglie did not accept this interpretation. The collapse of the wave function from the superposition state to the eigenstate is instantaneous, and there is no physical mechanism.

Eugene Wigner is a supporter of Copenhagen interpretation. He designed an ideal experiment called Wigner’s Friends. He asked a friend to verify the results of a particle physics experiment, which had been recorded the day before. The friend reported the results to Wigner in time. So the question arises. This particle event is described by a wave function. When did the wave function collapse? If Wigner believes that the collapse occurred at the moment of reporting to him, then he has become a complete solipsist. Of course he would not be so ridiculous. Could it be his friend’s consciousness that caused the wave function to collapse? What makes us different from the rest of the physical world and obey different laws? Before the emergence of human beings, has the world been in an uncertain superposition? Collapse is a discrete process, and consciousness is a continuous process. Can wave function collapse really be related to continuous consciousness?

Copenhagen’s wave function collapse is an instantaneous effect without any dynamic mechanism, which caused dissatisfaction among many physicists represented by Einstein. Penrose’s solution is mechanical. He associates the collapse with an appropriate level of complexity, which can be described by the number of particles or the total mass. His plan requires that a new term be added to Schrodinger’s equation, and the additional term only works when certain conditions are met. Penrose suggested that once the total mass of the particles involved in the wave function approaches Planck’s mass, the wave function will collapse spontaneously. Planck’s mass is about 10-5 grams, which is large for the atomic scale; but very small for the human scale.

Conformal cycle

Penrose is a conformal master, Penrose diagram, twisting amount is based on the conformal transformation. More importantly, he proposed the Conformal Cyclic Cosmology (CCC). The universe that began with the Big Bang ended up in an accelerating expansion of time and space, forming a generation; the end of each generation is the beginning of the next generation of the Big Bang. CCC depicts an infinite cosmic cycle. The Big Bang of our universe is a continuation of the distant future of the previous century. CCC believes that our universe will not expand forever, let alone a major tear. It will stop expanding and return to collapse one day in the future.

Both the big bang as the initial singularity and the black hole at the terminal singularity can be described by the conformal invariant Weyl curvature tensor, and the Weyl tensor becomes a geometric quantity describing the entropy of gravity. The conformal expansion of the Big Bang reduces the infinite density and temperature to a finite value, while the conformal contraction increases the infinitely low density and temperature to a finite value. As a result, the two transitioned on a smooth boundary, and the universe entered the present one from the previous generation.

The universe is circulating, and Penrose’s evidence is that Hawking’s point exists in the cosmic microwave background radiation. Penrose and his collaborators used Planck satellite observations and found about 30 Hawking points. Penrose and others pointed out that, according to the CCC, these dots carry the remains of the black hole after the evaporation of the universe when the universe collapsed in the previous generation. This proves that another generation has existed before our generation.

Penrose Hawking older than 11 years, he has served as Hawking defense of judges. Hawking learned the overall technique from Penrose and used it to prove the second law of black hole mechanics. During the black hole process, the surface area of all black holes involved will not decrease. The two of them also proved the singularity theorem together. Hawking once said: “It was Penrose’s first singularity theorem that guided me to study the structure of causality and inspired my classical research on singularities and black holes.”

However, Hawking and Penrose’s views on physics , Not exactly the same. In 1994, the two had a big debate at the Newton Institute of Mathematics. In a sense, this is the continuation of the debate between Bohr and Einstein. Although the content of the debate has become deeper and more complex, the argument cannot be separated from the philosophical point of view. Hawking said that Penrose was a Platonist and he himself was a positivist. Hawking is only concerned with whether the theoretical predictions are consistent with the measurement results, while Penrose is concerned about the rationality of the dead and alive Schrödinger cat. Penrose said that no matter what the “reality” is, people must explain how they perceive the world, and people must figure out why the cat they feel is either dead or alive, not a superposition cat that is both dead and alive. Penrose is also dissatisfied with Hawking’s use of “Vicker’s rotation” in general relativity, thinking that this is not the same as the time axis rotating from the real axis to the imaginary axis in quantum field theory. Collaboration and controversy in the history of science were originally commonplace. However, happy friends like Genzel and Gates are rare. They argued fiercely, but it is really rare to win the Nobel Prize on the same subject. Genzel, born in 1952, has long used the European Southern Observatory’s telescope to observe the orbits of stars near the center of the Milky Way. Genzel’s team first discovered that Newtonian mechanics could not explain these orbitals and must use general relativity. In other words, Schwarzschild precession occurred in the orbit, which inferred the existence of the black hole.

Compared with Genzel, Gates is a young female astronomer. Since 1995, Gates has been involved in black hole research. The use of new technologies is the key to her success. The Gates team developed the adaptive optics system (AO system) of the Keck Observatory Telescope. In 2005, the first laser-guided AO picture of the center of the Milky Way was taken, which greatly improved the clarity. The two teams often have tit-for-tat arguments over the observed phenomena. In the words of Gates, “Nothing can make people progress more than a competition!” Genzel and Gates finally shared the other half of the 2020 Physics Prize together.

Relativity and quantum theory are successful examples of pure science. They shaped the 20th century into a century that symbolized science, greatly promoted related technological revolutions, and still affected our lives. Most of the Nobel Prizes in Physics in the 21st century Related to these two theories. As versatile as Leonardo’s Penrose, his research is driven by curiosity and does not pay attention to practical applications. Penrose’s research hopes to understand the position of humans in the universe and to understand mysteries such as black holes. With the cooperation of many physicists and astronomers, mankind finally unveiled the mystery of black holes. Penrose’s originality is so strong that some people suspect that he is a deviant person. To this he replied: “For basic physics, I am quite conservative in most aspects. I am more willing to accept traditional wisdom than people who have made progress at the forefront.” It is this quality of accepting traditional wisdom and being curious about the universe that made Penrose, Genzel, and Gates finally succeed, didn’t they?