The Reality of the Virtual (1) --- Why Do We Need Fields?

Lately, the physics I have been in contact with is all field theory, laying the foundation for general relativity from various aspects. I feel that my mathematical foundation is quite good, but my physical “depth” is lacking. I can usually understand the mathematical description of physical theories, but I don’t quite understand the physical basis and origin of each step. It’s truly a case of “surplus of mathematics and deficiency of physics.” After immersing myself in the ocean of field theory for a while, I have gained a general impression of it. However, there is one fundamental question for which I only today obtained a relatively satisfactory answer: Why do we need to introduce fields?

In traditional Newtonian mechanics, there is no concept of a “field.” For example, in celestial mechanics, we only need to consider the mutual gravitational forces between celestial bodies to perfectly solve many problems; there is no need for a field at all. I suspect most readers first encountered the concept of a “field” when studying electricity in high school, where textbooks introduced us to many strange concepts like electric fields and field lines. In fact, this is exactly how it happened: historically, the “field” was born for the sake of electromagnetism—the field lines first introduced by Faraday possess a unique charm.

So why introduce fields? Classical mechanics has no fields, while electromagnetism does, and electromagnetism is relativistic. Therefore, one might guess that fields only appear with relativity. This is indeed the case. In the book Gravitation and Spacetime, the author clearly points out that fields and action at a distance are mathematically equivalent. Abandoning action at a distance and introducing fields is done so that we can perfectly maintain laws such as “momentum conservation” and “energy conservation” in relativistic situations.

The interaction between objects in classical mechanics implies action at a distance, which allows for the existence of momentum and energy conservation. If placed within relativistic effects, these conservation laws would no longer hold (if that were truly the case, it would be a regret for physics). To take an extreme example, consider a two-body system consisting only of the Sun and the Earth, both operating normally. At this point, assuming their total momentum and energy are conserved poses no problem. But suppose the Sun suddenly disappeared (or some super-civilization “kidnapped” the Sun). Note that according to relativity, no speed can exceed the speed of light. Therefore, for 8 minutes, we on Earth would never notice any abnormality—the Earth would move as usual, and the Sun would shine as usual.

However, the Earth’s motion is not perfectly circular. During these 8 minutes, its speed changes just as if the Sun were still present—in any case, the magnitude of the speed will change. Suppose the speed increases—that is, the kinetic energy increases. That would be quite strange; calculating from the moment the Sun disappeared, energy is not conserved at all! The Earth “out of thin air” increased some kinetic energy! Thus, we can only choose one of three hypotheses:

1. The law of conservation of energy does not hold.
2. Modify the definition of energy to make it continue to hold.
3. Assume there is another type of entity that stores some energy, so that total energy is conserved.

Action at a distance adopts the second hypothesis, while the field is the product of the third hypothesis. The two are mathematically equivalent, but the field does not require modifying definitions; it simply introduces a new entity—the field—to store energy and momentum, while also acting as the medium for transmitting interactions. Thus, originally there was no field in the world; for the sake of beauty, the field came to be. It is not the only choice, but it is certainly the most beautiful one.

The general view in physics is that a field is a substantial entity, but it is intangible; we can only know of its existence through its interaction with objects. More accurately, an object interacts with the field, changing the field, and then the field interacts with another object—the total effect is that one object interacts with another, with the field acting as the medium. This beautifully resolves both the puzzle of action at a distance and the puzzle of conservation laws! From a narrow perspective, except for the fact that we cannot directly observe it, a field is basically no different from an ordinary object—it has momentum, energy (and according to E=mc^2, it also has mass), and it can interact with matter. Therefore, we consider it to be real.