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I have wanted to write the Nott theorem for a long time, but I haven't started writing it for a long time. I think it's time to give myself an account. The Nott theorem put forward by Eminot is one of the most wonderful mathematical ideas so far. But it is not well known to the general public.

Einstein and Hilbert tried to get the equation of general relativity, but struggled for a long time. Because, before general relativity, there is such a special paradox, the paradox is like this:

If energy distorts space-time and space-time contains energy, then space-time will distort space-time.

Eminot solved the problem, but how did she solve it?

What happens if everything in the universe is translated to the right of it now? We know it's not going to happen. But you still ask, "what's changed?" But more importantly, what remains the same?

Such thought experiments seem to be an infinite "abuse" of time. But after you understand Nott's theorem, you don't think so. The usual understanding of Nott's theorem is:

Symmetry means conservation.

But what is symmetry? What is the law of conservation? How are they connected?

"Symmetry" in everyday language refers to a sense of harmony and beauty of proportion. In mathematics, "symmetry" has a more precise definition and is usually used to refer to objects that remain unchanged under certain transformations, including translation, reflection, rotation, or scaling.

Mathematically, symmetry is established when an object is arbitrarily transformed without a change in direction. Rotate a ball 90 degrees around the center, and the result is the same as at the beginning. The sphere is rotationally symmetrical. Take a straight line (infinite length), move 4 units to the right, the straight line does not change. This is regarded as "translational symmetry". This seems to be an insignificant nature, but this is not the case.

Eminot's idea of symmetry goes like this:

Suppose the object is a system that constantly converts according to our wishes. This "system" is a part of the universe, or the universe itself. We stretch all the distances or rotate all the angles with the lambda value. Nott's question is, will the system remain the same anyway?

Note: Nott's theorem limits "symmetry" to continuous symmetry. Continuous symmetries are described by continuous changes in functions, and they are discrete. The sphere is continuous symmetrical, while the triangle is not. For details, please read: lie Algebra-- one of the most important tools in physics, the simplest explanation.

First of all, Nott is particularly interested in energy. The definition here makes some adjustments: we say that if the total energy of the object in the system does not change under any transformation, then the system is symmetrical. For example, if I separate a mass and compare it with a shifted mass, the energy will remain the same.

So this system is considered to be "translational symmetric" because the "arbitrary transformation" we make to it is to translate the lambda unit to the right. On the other hand, a small change to the system can invalidate this symmetry. Suppose there is a planet in this system. The closer the mass to the planet, the smaller its gravitational potential energy, and the farther away from it, the greater its gravitational potential energy. Therefore, the system is not considered to be translational symmetric.

This is the symmetrical part. What does the law of conservation mean? Simply put, conserved quantities are those that can neither be destroyed nor created, but are transformed from one form to another. Some conserved quantities are energy, momentum and electric charge.

Conserved quantity refers to something that remains the same in quantity for a period of time and can neither be created nor destroyed. The laws that describe these quantities are called conservation laws.

But why? Why are these quantities conserved? Why can't we create energy? The Nott theorem answers all these questions. It explains very accurately where conservation comes from.

Specifically, it assumes that translational symmetry means conservation of momentum. In fact, if each atom in the universe moves one meter to the right, we cannot tell the difference between them. But here's the problem. When is momentum not conserved?

Let's assume that an apple falls. If the apple moves 2 units to the ground, its gravitational potential will be smaller and its speed will be higher. Therefore, p=mv is not the same in both cases. This system is considered to be translational asymmetry.

But if we consider the entire universe. Move everything a few units to the left or right, and you won't notice that the position of anything has changed. This must mean that energy is conserved, so momentum is also conserved. The point is that translational symmetry means conservation of momentum.

Translational symmetric → momentum conservation

But what about rotational symmetry? The same thing. Think of the earth and Apple as one system. But the apple did not fall, but revolved around the earth. We say that the conserved quantity here is angular momentum. If the object moves along a spherical orbit, as shown in the following picture, the total energy of the object at any given position remains the same. So it has rotational symmetry on this axis.

Conservation of angular momentum of rotationally symmetric →

We have discussed the transformation of systems in space. But what does it mean to translate the system in time? In other words, suppose there is a system that compares t at a particular time with the system at t + lambda time. If the energy of the system is constant, then it is considered to be time-translational symmetry. What does Nott say about conservation? Energy.

Energy conservation of time symmetric →

The following content, you do not need to bother to fully understand, I just show it. This is one of the most profound proofs of a series of ingenious mathematical operations.

It can be regarded as a function of ε, and the derivative at ε'= 0 is calculated. Using Leibniz's rule, we get:

Notice what the Euler-Lagrange equation implies:

Put this into the previous equation and get:

Once again, by applying the Euler-Lagrange equation, we get:

By substituting the previous equation, we get:

From which we can see

It's a motion constant, it's a conserved quantity. Due to

Get

So the conservation quantization is simplified to

This is Nott's derivation of independent variables.

The above mathematical route includes the derivation of an independent variable. There are also translation and rotation invariants similar to those discussed earlier. For quantum mechanical systems, there is a version of field theory, which is the basis of modern particle physics. But it's more than that. From Maxwell equations to general relativity, you will find it everywhere.

As she hypothesized, not all energy distorts space-time. Only the energy in the stress-energy tensor is important. The stress-energy tensor is a mathematical object that contains all the information about the energy that distorts space-time.

Tensor has four kinds of information. Two of them are "charges", which are conserved things:

1. Energy of the field

2. Momentum of the field

By the way, this tensor is named after these two. They are exactly similar to the energy and momentum of particles, but for a continuous system: field → spacetime itself.

So far, there are four components: one energy and three momentum (because momentum is a vector, it can be decomposed in any x, y, and z direction). The other two types of information are "flux":

1. Energy flux

2. Momentum flux

But what is flux? For example, our electromagnetic waves carry energy, but we also have power (which may not be easy to understand), because photons are massless, and if m = 0, then in p = mv, we get p = (0) v. The answer is a little more complicated than that. The important thing is that waves have momentum. Flux is an exact expression of the energy flowing in each direction and the amount of momentum vector flowing in each direction.

This is all the information about the energy-momentum tensor.

It doesn't explain everything. It just sums up the gravitational influence of matter. Gravity couples energy and momentum as well as the flux of energy and momentum. It doesn't care whether it comes from matter or electromagnetic field, or what matter it comes from. The important thing is that it is symmetrical.

This tensor is the result of a general spatio-temporal transformation caused by the symmetry of space-time.

However, in quantum field theory, the Nott theorem can be violated because of the quantum effect. The symmetry at the classical level broken by quantum correction is called "anomaly". Abnormal symmetry is one of the many predictions of the standard model. The following article will explore the violation of symmetry and their relationship with quantum mechanics.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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