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This article comes from the official account of Wechat: ID:fanpu2019, author: an Yusen

Many people should have heard of wormholes, whether from the reverie of science fiction that they can travel through time and space, or from the academic news at the forefront of theoretical physics that they do not understand but think it is very powerful, but what on earth is a wormhole? How does it become a structure that connects time and space, is it just a physicist's toy? In fact, wormholes have hidden implications that we have not yet discovered in the study of quantum gravity in recent years.

Wormhole (wormhole) is a magical space-time structure, at the same time, physical research has increasingly proved that wormhole is the key to connect quantum theory and gravity theory. This paper intends to introduce the basic concept of wormhole and its role in theoretical physics from two aspects of Lorentz (including time and space) wormhole and Euclid wormhole.

Lorentz wormhole first, we introduce Lorentz wormhole. Lorentz wormhole is a possible wormhole structure in space-time, and it is a real physical object.

The earliest research on "Cave" inspired Carl Sagan's "contact" (Contact), which has also been successfully filmed and televisionized, and the film "Contact", directed by Robert Zerigis, has been well received. In the original manuscript of Zi Shuo, the author tries to realize the tunnel of time and space. But his friend Kip Thorne expressed concern. As an expert in the study of the theory of relativity, he knows very well that it is very difficult for a hole to be a tunnel of time and space. But this aroused the interest of Kip Thorne, and the initial series of studies on holes were carried out later.

Wormhole and energy conditional time travel is an eternal interest of science fiction enthusiasts, and "traversable" holes seem to be a good way to achieve it. Therefore, one of the most important things in the study of the hole is to study its passability. In general, in the study of semantic relativity, we know the distribution of matter, and then study the shape of space-time that this distribution will give; however, in the study of wormholes, physicists realize a specific shape of space-time-so Morris and Thorne consider the opposite, first giving restrictions on the structure of space-time, and then solving the distribution of matter through the Einstein field.

The initial calculation was carried out in a spherically symmetric coordinate system, and they found that if the space-time structure of a "specific" hole is to be filled, then the distribution of matter required must violate the energy condition, and generally speaking, it is necessary to introduce strange negative energy matter [1]. This can be easily seen by geodesic confluence. In the theory of semantic relativity, in order to explore some properties of space-time, some conclusions can be drawn through the change of geodesic convergence without understanding Einstein's equation. For example, if you need a hole structure to connect two different space-time regions and cross it, then the light passing through it needs to converge to the throat of the hole (that is, the narrowest part of the wormhole structure) and then emit from the throat. In general relativity, whether light converges or diverges can be given by the expansion of light-like geodesic sinks. The equation describing it is often called the Ray-Chaudhuri equation, which is as follows:

We can choose the line convergence where both rotation and shear are 0, σ = ω = 0, so that according to the characteristics of the line convergence passing through the hole, we can know that there must be a position of d θ / d λ = 0 in the throat of the hole, which darkens the following process.

According to the theory of relativity,

This destroys the condition of light-like energy, so the existence of a hole must require the introduction of strange matter with negative energy in its throat.

The introduction of this strange material makes the structure of the hole very difficult. This kind of material which violates the condition of light-like energy is only allowed in the theory of quantity, and it is usually "differential". At the same time, if it is satisfied that the hole can pass through, we also need to consider the tidal effect caused by the hole acting on the body. Under the condition of "the tide that the body can bear", the theory predicts that the hole will be "often huge". It is more difficult for such a huge space to be propped up by strange matter. But perhaps, as science fiction calls the "three-body" fantasy, a well-developed vision can do anything free of technical barriers under the conditions permitted by the laws of physics-it is still possible to build a sinkhole.

Wormhole and time machine since the wormhole can be seen as a shortcut between two distant points in the universe, maybe the wormhole can be transformed into a time machine. [2] in the discussion of the time machine, we ignore some details and only regard the sinkhole as a machine connecting two points in space-time (tmem0) and (tMagol L). The entrance of the sinkhole corresponds to (tMagol 0) and the exit corresponds to (tMagol L). If we let the exit move at the speed of the entrance relative to the entrance, then according to the clock slow effect of special relativity (such as the double paradox), there will be a time difference T between the exit and the mouth; then we shorten the space distance L to 0, let the exit and the mouth return to the point, then from the mouth to the exit, time will make a T jump, which completes the operation of traversing to the past or the future. This is the simplest version of building a time machine through holes.

Time machines may be more likely to stimulate their interest because they are always full of all kinds of regrets. When there are all kinds of regrets in their twilight years, the time machine may give them a chance to start all over again to make up for these regrets. Therefore, many beautiful love stories can be spread out in this context.

However, the emergence of time machines will cause a lot of causal problems, so most of the time, time machines are only regarded as playful, serious scientific research topics. Maybe "I hate time machines". What physicists need to do is to find the corresponding physical principles to prove that time machines cannot be made.

Wormhole and quantum entanglement in 1997, Maldacena with his AdS / CFT primitive theory exploded a thunder in the field of theoretical physics. since then, more and more scholars began to study the holographic properties of citation. [3] later, based on Maldacena's conclusion [4], Raamsdonk first found through a simple argument that there is an essential relationship between "hole" and quantity entanglement, that is, ER=EPR conjecture. [5] (the name ER=EPR was formally proposed by Susskind and Maldacena in 2013, which is to solve the wall problem of the hole [6].) ER refers to the Einstein-Rosen Bridge, which is the area connecting two holes and can be seen as the forerunner of the study of holes. But it is impenetrable, and any move across the Einstein-Rosen Bridge will inevitably lead to a singularity. EPR refers to quantitative entanglement.

Einstein-Rosen Bridge. Source: arXiv: 2110.14958 Let's briefly introduce this point of view. In 2110.14958, the study of Maldacena made a discovery, the thermal field barystate TFD in quantum field theory.

Corresponding to the corresponding AdS Shixiang cave, its Penrose diagram is consistent with the Penrose diagram along the most advanced analytical path of the Shixiang cave. Of course, if you stare at a space intercept in the Penrose diagram, it can be understood as two holes connected by a hole structure in the middle.

The correspondence between the thermal field barystate and the Shifei hole source: arXiv: 1005.3035 We found that the thermal field barystate is an entangled state, and adjusting the temperature (that is, the β of the thermal field) corresponds to regulating the entanglement on the left and right sides. When the temperature is very low, the entangled state becomes a direct product state without entanglement; when the temperature is very low, it becomes the most entangled state. It is found that the throat in the middle of the wormhole structure gradually narrows and disconnects with the change of temperature from high to low. Therefore, we find that from the perspective of boundary theory, the operation of reducing the entanglement corresponds to the connection between the two holes. Therefore, there is a deep relationship between the entanglement and the hole, even if it is said that they are essentially the same thing.

The shape of the hole narrows with the decrease of temperature. Source: arXiv: 1005.3035ER=EPR conjectures that the source of time and space may be entangled by "quantity". Usually the measure of entanglement is entanglement entropy, but the growth time of ER bridge will exceed the thermal equilibrium time (entanglement entropy tends to be fixed after thermal equilibrium), so the concept of entropy seems difficult to describe the volume change of ER bridge. On the basis of this, physicists propose that some physical quantities which may have different properties from entropy are associated with "hole volume production", that is, computational complexity. Its physical meaning is to specify a series of operating gates, the number of operating gates required from the initial state to the final state.

At the same time, it is interesting to note that although the Einstein-Rosen bridge mentioned before is impenetrable, we can construct a corresponding model to implement the "traversable" hole, that is, introducing an operation called double trace deformation at the boundary, leading to the following operator perturbation

. This operation is equivalent to introducing a negative energy flow to the background space-time, and its energy will become normal near the event horizon because of the blue shift, so it will cause a very negative reaction to the background, affecting the position of the event horizon. Causes the event horizon of the hole to contract inward. So the light from the original singularity will run out of the event horizon and re-reach the other boundary. That is to say, the traversability of the hole is realized.

According to ER=EPR, this process is equivalent to the introduction of quantum teleportation, while double trace deformation is similar to the classical channel. In the teleportation of quantity, it seems that the measure is reconstructed by being entangled in another place; under the image of the introduction, it has a new understanding, that is, it is through the "hole crossing" connecting the two places [7].

Physical images of traversable traverses? source: arXiv: 1704.05333

The above Euclid wormhole introduces the necessary conditions and corresponding physics for the "hole" in space-time as a possible physical object. However, in recent years, in the study of quantitative introduction, a new kind of hole structure has aroused more interest, that is, Euclid wormhole.

Before introducing what is an Euclid wormhole, we first introduce the European operations that are often carried out in theoretical physics research. By analyzing the similarity between the path integral in quantum field theory and the partition function in statistical physics, we find that if we perform the following operation of wick rotation on time (for wick rotation, see "temperature and mysterious virtual time | wonderful gate"), that is, the imaginary number of time coordinates, we can equate the problem of quantum field theory with the problem of statistical physics, and then we can get Euclidean path integral. In the Euclidean path integral, there is no time direction, which can be regarded as physics on a certain time surface. (of course, we can also combine the Euclidean path integral with the Lorentz path integral. )

Euclidean path integral is a very effective tool to study many theoretical physics problems. Later, we will introduce that when using Euclidean path integration to calculate the fine entropy of Hawking radiation from a black hole, there will be wormhole structures that have not been found before. This kind of hole structure can help us to understand many difficult problems, such as the loss of information.

The problem of copying wormholes and information loss is the most profound shield of quantum science and semantic relativity in the space-time of wormholes. Considering the collapse of pure matter into a hole and then radiation, we can see a positive evolution from pure state to mixed state, but it is not allowed by stoichiometry. The problem of hole information, as a hen who can lay eggs, has inspired physicists to keep creating.

Recently, inspired by holographic entanglement entropy, we have found a way to calculate the exact entropy of Hawthorne radiation in the introduction, which is called the island formula. (see "the riddle of the paradox of black hole information, has Hawking's last problem been solved?" The exact entropy obtained by this calculation is miraculously full of Page curves, and enters into the positivity of quantum science. We know that although the RT formula of holographic entanglement entropy is written as a semi-conjecture at first, it has been accurately proved by leading path integral later. Can the island formula obtained by "this" be proved by introducing path integrals? If so, which part of the path integral should it contribute to?

First of all, we introduce how to calculate the entanglement entropy in field theory, which can be calculated by a method called copy technology (replica trick). The system to be studied makes n copies, enters the calculation, and finally goes into the analytical extension method. The formula is as follows:

The first equals sign above is the definition of entanglement entropy, and the second equals sign is obtained by applying the Lobida rule, which is usually called copy technology (replica trick). Because the physical meaning of the path integral describes the probability amplitude from the initial state to the final state.

So the Euclidean path integral can define the wave function, and then define the density matrix. Under the expression of the Euclidean path integral, the calculation of the entanglement entropy can be transformed into the calculation of the partition function on the copy manifold, that is, the last step equation of the upper pole.

According to the above idea, if we integrate the density matrix of Hawthorne radiation into a graphic table through Euclidean path integration, it is necessary to consider all possible copy manifold configurations to accurately calculate its entropy (that is, partition function). Consider the pure state of radiation and the overall composition of the hole.

A "simple" intention: the boundary conditions formed by the radiation density matrix on the left (the solid line represents the boundary of the hole after tracing, and the dotted line represents the radiation), and the right side represents the leading configuration needed for calculation. The first graph is a connected configuration, and the first graph represents a connected copy hole configuration. Photo: arXiv: 1911.11977

When the connected configuration is not considered, the entropy which is consistent with Hawking's initial calculation can be obtained, which violates the positivity; if we consider the connected configuration (usually called the copy hole), we will get the entropy which is consistent with the expectation of positivity. Considering the fully connected configuration, we can get the results of the island formula in the late stage, but the contribution of the "real copy" hole will be more abundant. ) the meaning of this connected configuration is very similar to that of the hole, which connects different lead regions through a connected structure (except that the different regions of the structure are obtained by replica trick the system), but its physical meaning is different from that of the Lorentz hole, and its specific physical meaning remains to be understood and clarified.

Copy the characteristics of the holes, from the picture we can see that the holes on each boundary are connected to each other. Source: arXiv: 1911.12333

The calculation of copy hole is complex, in which only the simplest model can consider all possible configurations of Replica hole and sum them analytically to obtain the most accurate radiation exact entropy [8]. However, physicists have been able to prove the correctness of the previously obtained island formula by copying the "hole" formula. The appearance of the copy hole gives a new note to the study of the information problem of the hole, and many problems have been re-discussed, such as the correspondence problem of the introduction ensemble [9], the problem of global symmetry in the quantitative introduction, and the residual (remnant) after the radiation of the hole [10].

Perhaps the really interesting thing has just begun, and we look forward to more surprises from the future study of the hole.

reference

[1] M.S. Morris and K.S. Thorne, Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity, Am. J. Phys. 56 (1988) 395.

[2] M.S. Morris, K.S. Thorne and U. Yurtsever, Wormholes, Time Machines, and the Weak Energy Condition, Phys. Rev. Lett. 61 (1988) 1446.

[3] J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200].

J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112].

[5] M.V.Raamsdonk, Building up spacetime with quantum entanglement, Gen.Rel.Grav (2010) 2323-2329

[6] J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [1306.0533].

[7] P. Gao, D.L. Jafferis and A.C. Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [1608.05687].

[8] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [1911.12333].

[9] G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, 1911.11977.

[10] P.S Hsin, L.V.Illiesiu, Z.Yang,Violation of global symmetries from replica wormholes and the fate of black hole remanants. Class.Quant.Grav.38 (2021) 1951 194004.

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