# Bridges in Physics. How would you build them?

with mathematical formulas? not enough! also with cartoons.

 The Pont du Gard is a bridge Romans completed during the period of Augustus. It is located near Avignon, in southern France. Do you know how long it took Romans to build this bridge? Ancient Greeks talked about digging the Corinthian Canal across the neck of Peloponnese Peninsula.

• God built mountains and oceans. Humans can build bridges and canals.

• Isaac Newton built a bridge between elliptic and hyperbolic orbits for planets and comets by writing down a mathematical formula.

• James Clerk Maxwell combined electricity and magnetism by writing four equations.

• Einstein was a great bridge builder.

1. Massive Particles and Massless Particles.

2. Frequency and Energy.

3. Mass and Energy.

4. Einstein was a great bridge builder, but he could not build a bridge between him and Bohr. Click here for a story.

• Heisenberg built a bridge between Particles and Waves. He attempted to talk to Einstein. Click here for a story.

 Dirac and Feynman in Poland (1962). Poet and Cartoonist. photo from the Caltech photo lab.

• Needless to say, Heisenberg was interested in building a bridge between him and Einstein, but he failed.

In the 20th Century, humans discovered two big mountains. One is Relativity and the other is called Quantum Mechanics. It has been and still is a challenge for physicists to build a bridge between them. I have been talking about this problem for many years, but I am not the first person to worry about the problem. Quantum field theory is a case in point. This theory takes care of scattering problems based on the Lorentz-covariant scattering matrix.

Yet, Paul A. M. Dirac was never happy with the renormalization process in QFT. His main interest was space-time symmetries of localized waves. How would the hydrogen atom look to a moving observer? How can we extend the Heisenberg brackets to the Lorentz-covariant world? He addressed this issue in his papers starting from 1927.

1. Dirac (1927). The Quantum Theory of the Emission and Absorption of Radiation, Proc. Roy. Soc. (London) A [114], 243 - 265 (1927).

2. Dirac (1945). The Quantum Theory of the Emission and Absorption of Radiation, Proc. Roy. Soc. (London) A [A183], 284 - 295 (1945).

3. Dirac (1949). Forms of Relativistic Dynamics, Rev. Mod. Phys. [21] 392 - 399 (1949).

In these papers, Dirac attempted to solve the problem by writing down beautiful formulas. In so doing, Dirac ended up with building three mountains. Dirac never attempted to build the bridges from one mountain to another. Why could he not do it? The answer is very simple. Dirac never attempted to explain his ideas with pictures. Let us go back to QFT. Without Feynmam diagrams, it is a useless theory.

• Thus it is a great challenge to build bridges between Dirac's papers using pictures. We consider the following figure. You may expand the picture by clicking on it.

• In this figure, it is possible to combine Dirac's time-energy uncertainty relation of 1927 with the well-known Heisenberg relation. Dirac used a Gaussian function to combine them in his paper of 1945, but he did not draw a circle to tell what he was doing. With a circle, it is possible to combine his 1945 paper with the light-cone system he introduced in 1949. It is very easy to do synthetical (combining separate elements to form a coherent whole) process with pictures, but Dirac never used pictorial language. He was not like Feynman.

Feynman was a cartoonist. Feynman diagrams!

• In the above figure, the Lorentz boost transforms a circle into an ellipse. The question then is whether it is a science fiction or observable in the labs. The following figure will tell that this effect is amply demonstrated in high-energy labs during the latter half of the 20th century, as shown in the following figure.

• Next Question. If it is possible to synthesize quantum mechanics and special relativity, is it possible to derive both quantum mechanics and special relativity from a single set of formulas? This is an ambitious project Dirac started in his 1963 paper:

Until 1963, the concept of symplectic transformations (squeezes and rotations) was strange to physicists, even to Dirac. By now, it is well known that the single harmonic oscillator has the symmetry of the Lorentz group applicable to two space-like and one time-like dimensions. In 1963, Dirac considered two coupled oscillators. He discovered that they lead to the Lorentz group applicable to three space-like and two-time like directions with ten generators. Among them are

1. three for rotations in the three-dimensional space-like space,
2. three for boosts with respect to the first time variable,
3. three for boosts with respect to the second time variable,
4. one rotation between the two time variables.

 Mrs. Dirac was Wigner's younger sister. Bridge between Dirac and Wigner.
Indeed, the first two sets, with three rotation and three boost generators, lead to the Lorentz group applicable to three space-like and one time-variables. This group is very familiar to us.

What about the four remaining operators? Dirac could have asked Eugene Wigner about this issue. Wigner's answer could have been the group contraction. Indeed, there is another bridge to be built between these two famous brothers-in-law. If contracted, these four operators become the translation operators along the three space and one time.

Indeed, Dirac's O(3,2) group becomes the inhomogeneous Lorentz group (Lorentz group plus translations), which Dirac was so eager to obtain in his 1949 paper. Click here for a detailed story.

Many people are asking me why I talk about Dirac's 1963 paper so often, while it is totally unknown to the present generation of physicists. The reason is very simple. I had an audience with Dirac in 1962, after he completed his paper, and before its publication in January of 1963. Click here for a detailed story.

• Dirac was a great physicist and a great poet. If you wish to talk about him, you may submit you paper to a special issue. Click here.

• Let us stop here, and relax. Let us look at some of the bridges and canals around the world.

## Bridges around the World

• London's Tower Bridge is the most famous bridge in the world. It is a challenge to take a photo of this bridge while opening.

1. This bridge is often called the London Bridge. The London Bridge is somewhere else.

2. This is the Tower Bridge because it is near the Tower of London. Click here for photos of the London Tower and the Tower Bridge.

3. From the engineering point of view, the Jubilee Bridge is the world's first suspension bridge built in 1945. This bridge had several different names. Click here for its history.
Here is another photo of this bridge.

4. The Thames River runs through the city of London. There are many many bridges across this river. You are invited to my London page. I love London, and I go there often.

 I was there.

• Pont du Gard. I was there in February of 2017, and became convinced that physics is really an art of building bridges.

I met there a Chinese historian and talked about the Great Wall Emperor Chin started building in 300 BC.

These two projects had different purposes with different engineering bases. However, there was one thing in common. They needed enormous amounts of manual labors. It took humans 2,000 years to realize that the labor is an essential variable in economics. Karl Marx found out from his variable called "surplus value," meaning the price difference between the finished products and the raw materials.

• These days, human intelligence (inventiveness) affects our economics. I assume many people are worrying about quantitative approaches to this problem. Great opportunities for physicists!

 I was there.

• Dandong Bridge across the Yalu River is the most talked-about bridge in the mass media. This bridge connects North Korea to the Chinese mainland. The trade sanction against North Korea crucially depends on the traffic across this bridge.

1. This bridge was built by Japanese expansionists. Its first span was built in 1935, and the second span was completed in 1939.

2. The southern halves of these spans were destroyed by the U.S. Air Force during the Korean War (1950-3).

3. Chinese and North Koreans reconstructed one of the spans, and the other still remains half-destroyed and unusable, as shown in this this photo.

Many people go to the end of this broken bridge and take photos. I was there in April of 2014. This is a photo of myself standing there. In the background is the reconstructed span which Chinese and N. Koreans use for their trade purposes.

• I have photos of many other bridges in the world, including

 Curved bridge. Why curved?

1. Many bridges in Pittsburgh (USA).

2. Golden Gate Bridge in San Francisco (USA).

3. Verzano-Narrows Bridge in the New York Harbor (USA).

4. Charles Bridge in Prague (Czech Republic).

5. Across the Bosphorus Channel in Istanbul (Turkey).

6. Anachikov Bridge in Saint Petersburg (Russia).

7. Pont Vechia (old bridge) in Florence (Italy).

8. Realto Bridge in Venice (Italy).

• Coronado Bridge, San Diego (Clifford, USA).

1. This is a curved bridge as can be seen from an airplane.

2. We are accustomed to think the shortest distance between two points is a straight line, and all the bridges in the world are straight. Can you explain why this bridge in curved?

• China is a big country, and there are two long west-to-east rivers flowing into the Pacific ocean. These two rivers provided the major transportation system before the rail-road appeared in recent years. Thus, Chinese rulers were interested in connecting these two rivers by constructing a north-to-south canal. The project started during the time of Confucius, but it took about 1300 years for many different Chinese rulers to complete the construction during the Sui dynasty (600 AD).