The journal Physics and Engineering recently published a paper online, and it took only 5 days from reception to publication, and the authors believe that this is the fastest record for the publication of papers. The content of the paper is how to understand some cutting-edge problems in quantum mechanics and thermodynamics based on the simplest knowledge of classical differential geometry. The main part of this article is to introduce two typical results that the author considers to be successful in teaching after more than 20 years of learning and using differential geometry in scientific research.
Written by | Quanhui Liu (Ph.D. in Theoretical Physics, Professor, School of Physics and Microelectronics Science, Hunan University)
<h1 class="pgc-h-arrow-right" data-track="5" >1 Paper published the fastest world record: five days</h1>
Recently, there was a small paper, from the time of submission to publication, five days, setting a record for the fastest speed of articles published in Chinese (and perhaps internationally) academic journals. This is the humble work "Those who understand geometry, there is no disadvantage in physics" in "Physics and Engineering" (an academic journal of the Teaching Steering Committee of The Physics Curriculum of Colleges and Universities of the Ministry of Education, edited and published by Tsinghua University). Received on March 13, 2021 and launched on March 18, 2021. This article is neither a submission nor an appointment, but the editorial department of Physics and Engineering and the author discussed writing an article for their WeChat public account, and after the article was drafted, the editorial department thought it was a "good article" and then "upgraded" to the publication. It took only two days to concoct this article, but the accumulation of material lasted for more than 20 years, and the content was conceived as an abdominal manuscript for a while. During the same time period, an article submitted to University Physics, "Thermodynamics from a Geometric Perspective", was also well received, and the editorial board will arrange for the first article of a certain issue to be launched.
Figure 1 The fastest historical record of the speed of articles published by Chinese academic journals: five days.
There are a few points in my life that are not so bleak, and they all have something to do with geometry. In addition to papers, funding, and other mundane matters directly related to making a living, there are also the following:
1. Directed the research conducted by several undergraduates, and the best three papers were related to geometry. Undergraduates can conduct cutting-edge research related to geometry, and it is not difficult to explain geometry. At the same time, it also shows that geometry is one of the most important stepping stones for the study of physics.
2. In 1989, he received a master's degree, the thesis was the phase structure of the Z2 gauge field, which is a kind of geometry.
3. Ph.D. in 1999, the thesis was on biofilm geometry and general relativity, requiring classical differential geometry.
4. In 2000, the application for the ICTP's Regular Associate (equivalent to a visiting scholar or postdoc) was the second application, just a few words more than the first application in 1999: Dowling et al. in a 1999 PRL and Aharonov and other authors in a PRL in 2000, both affirmed the author's CPL article on Berry geometric phase.
5. Strangers know each other. In 2016, I came across Dowling in Shanghai, and when I talked about the past, he was happy like a child, see the photo (Figure 2). Sadly, Dowling passed away in 2020, and Nature Photonics posted a special commemorative article. In 2018, I had the great honor of hosting Berry at Lake University and presiding over his lecture (Yuelu Pulpit), see photo (Figure 3).
Figure 2 I pose with Dowling
Figure 3 Group photo of Berry and teachers and students of Hunan University Institute of Materials and Electricity
<h1 class="pgc-h-arrow-right" data-track="24" >2 Physics images, often referring to geometric images</h1>
The great physicist Dyson thinks he learned more from Fermi's 20 minutes than he learned from Oppenheimer in 20 years. Fermi's casual words are understood as Fermi's secrets, which are not only widely disseminated in seconds, but also enshrined as the Fermi guidelines for teaching and research in physics. The original words are: "One way, and this is the way I prefer, is to have a clear physical picture of the process that you are calculating. The other way is to have a precise and self-consistent mathematical formalism." Physical differs from physics in that the adjective physics refers to full and living physics, while physics is a description of state. Mr. Peng Huanwu emphasized the difference between physicic and physics.
Figure 4 Cover of a "College Physics" textbook edited by Thorne and Blandford
The role of geometry in physics education is highly emphasized, and the "University Physics" teaching group led by Thorne is the only one. If it weren't for Thorne winning the 2017 Nobel Prize in Physics for the successful detection of gravitational waves, probably not much attention would have been paid to a college physics textbook he published in 2017 with a very odd name— Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics), see photo (Figure 4). In fact, the content of this book has been tempered for 37 years, and it has been taught at Caltech and Stanford University for 37 years, and the characteristics of the book are to reshape classical physics with geometry. The preface reads: "Geometry is the deep thread in this book, and the very important latitude and longitude. We will see how the fundamental principles of classical physics can be determined or strongly limited by refined geometric thinking. Geometry not only highlights the characteristics of classical principles, but also helps to relate them to the corresponding quantum principles. Further, the geometric method avoids lengthy analytical calculations. Although the associated lengthy, routine calculations are sometimes difficult to avoid, in this case we sometimes resort to modern symbolic arithmetic software Maple, Mathematica, and Matlab to save space. In other words, blow away the dust off the computational difficulty of physics and discover that there is geometry everywhere in physics.
Geometry in college must be understood with the help of calculus. Therefore, physical images in universities should be differential geometric images. The so-called physics is profound, and it is likely that it is nothing more than simple geometry. The next two sections of this article, through two examples, hope to illustrate the following truth: the important and difficult physics problems in physics courses require only simple differential geometry to turn decay into magic, becoming exquisite and interesting.
<h1 class="pgc-h-arrow-right" data-track="31" >3 The third law of thermodynamics and the tangent between two functions</h1>
Figure 5 Geometric understanding of a function and its Taylor unfolding (image taken from the network)
Below is a schematic diagram of a law discovered by Thomson Bertello in thermodynamic Thomson-Bertello during cryogenic experiments (Figure 6). What does this diagram tell us? The textbooks contain tortuous and difficult analyses that lead to the Nernst formulation of the third law of thermodynamics. Once the concept of tangent contact is established, it is immediately found that the two lines have a first-level tangent touch at zero temperature, that is, the two functions have the following relationship
Thus, the Thomson-Bertrow principle reveals: 1, when T → 0, ΔS →0, the third law of thermodynamics; 2, it is precisely because of the first-order tangent contact that this T→0 can actually be as high as room temperature. This room temperature can be quantitatively solved by analyzing advanced tangent contacts.
This example shows that geometry must be combined with calculus, and that complete elementary geometry is not sufficient.
Figure 6 Two functions have first-order tangent contacts (Image from the famous book Herbert B. Callen, Thermodynamics and an Introduction to Thermostatistics 2nd Edition)
<h1 class="pgc-h-arrow-right" data-track="37" >4 the average curvature implied in the momentum operator</h1>
The three generalized momentum operators under the spherical coordinates are,
These three operators are found in general elementary quantum mechanics textbooks, and the main part is the usual differential operator. The extra part of the first two operators, which is usually understood as a function that is added in order for the differential part to become a secret, has no deep meaning, and may be far from it. These two functions are actually the average curvature of two different planes. Mean curvature is an entry-level concept in classical differential geometry.
First, a little introduction to what is the average curvature. The way to look at curved shapes of surfaces is to transform surfaces into lines. Cut the surface apart, and the surface on the cross-section is a curve one by one, referred to as a cross-section. However, when intercepting, it should be cut along the normal. A two-dimensional surface, you can find two orthogonal intercepts (normal intercepts), and use the curvature of these two intercepts (principal curvature) to mark the degree of curvature of the surface, referring to Figure 7.
Figure 8 Spherical coordinates. r is constant, a spherical surface with a radius of r with an average curvature of 1/r. When θ is constant, it is a cone with an average curvature of cot θ/2. φ is constant, it is a plane with an average curvature of zero. (The picture is taken from the network and edited)
Fig. 9 When the cone at the P point is split into two lines, one is a busbar with zero curvature; when the other knife is cut down, the knife face is orthogonal to the busbar, and the radius of curvature of the normal cut-off line is r tan θ and the curvature is cotθ/r. Thus, the average curvature of the cone at r=1 is cot θ/2.
One might argue that knowing an expression solves almost all problems for understanding three generalized momentum operators does not require knowing whether these quantities are average curvature. For these three operators, it seems to be true. In fact, this is not the case, because physical problems often have to be examined from a larger scope, and then look at the problem in reverse, in order to obtain a larger field of view and the correct perspective. There are many questions related to this curvature. First, in a straight space, moving a quantum state needs to be done by moving the operator constructed by the momentum operator, but in the straight space, the momentum does not contain superfluous functions, so the quantum state ring returns to the original place for a week, and the quantum state does not change. On curved surfaces, it is precisely because of the extra part of momentum that the quantum state will have an additional phase factor. Second, here it is about the "concatenation of the century": the observational significance of the radial momentum operator Pr. To solve this problem, it is necessary to resort to geometry, see the document Int. J. Geom. Meth. Mod. Phys., 2015,12: 1550028.
This example gives us at least three lessons: first, the average curvature occurs directly in the most common operators, then differential geometry is at the beginning of quantum mechanics, and thus in the whole of the course; second, the operator generally contains a geometric part, that is, a momentum in the general sense, which should be geometric momentum, see the literature, Euro. Phys. J.C., 2019, 79: 712 and the references therein; third, it is generally believed that the geometric basis of modern physics is Riemannian geometry, while the average curvature is external curvature, insignificant, this view is one-sided.
<h1 class="pgc-h-arrow-right" data-track="48" >5 assertions and conclusions</h1>
The article is nearing the end, but the interest is not gone, and some assertions are recorded here.
If you teach a course for 50 or even 80 years, and you are very familiar with it, you will never be able to reach the point where Thorne reshapes the classical physical concept system with geometry, nor can you find that ordinary momentum arithmetic actually contains simple geometry but profound physics, and so on. Any university teacher must have cutting-edge scientific research experience that is closely related to the teaching content of this course. A purely teaching-type university teacher can be a serious teacher or a good teacher, but if he cannot be fascinated in the teaching content, cannot grasp major topics, and cannot understand the latest progress, he cannot be a scholar-type teacher.
On the Hexi stage of Yuelu Academy, there is a pair of lianyun "He'an Li Mian for learning, through the heavens and the earth people are called talented.". However, "Amway Mian" is too much into the world for learning the three ways, and it is necessary to have a bit of a spirit of birth to do learning. The author is in charge of the "Yuelu Forum" at Hunan University, insisting on inviting some scientists with both liberal arts and sciences to come and give lectures, firmly believing that college students are in the stage of growth, the scope of interest cannot be too narrow, and they should "turn to more teachers and be Ru teachers." At the same time, there has to be a little tough curriculum in college. Nowadays, the number of courses in universities is numerous but easy to lose. Even simple differential geometry is not in the curriculum system of the university physics department, is this a problem of the university curriculum?
The problem may have to be looked at from a different perspective. Differential geometry and its development history are the most brilliant parts of human civilization, and this part should be one of the books that college students must read like Tang poems and Song poems, classic novels, etc. This is especially true for science and engineering students.
Greek stories, ancient and modern sayings, many examples, and so on, are crammed into the narrow passage of judgment, and the following converges on the subject of this article. Repeat two sentences from the text as the conclusion of the article: not only do physical images often refer to geometric images, but the profundity of physics is likely to be simple geometry.