The author | Hu Exhaustion
Edited | Trader Joe's
Symmetry occupies a central place in modern physics, and the most powerful tool for describing symmetry is group theory.
In this article we will briefly introduce a special kind of group, the Lie group.
Physically, we often encounter some symmetries that can change continuously, and in order to describe the symmetry of this continuous change, we need to use the Lie group.
Lorentz symmetry, for example, is such a symmetry that, with the help of the concept of lie groups (and their representation theory), we can quantitatively describe the Lorentz transform and even derive the concept of spin from it.
On the other hand, modern particle physics has a very important idea that theory tells us what we can see in the experiment, which of course does not mean that the theory can be made up without being responsible for the results of the experiment, it should be said that only through theory can the experimental data be given meaning.
From this point of view, the concept of "particles" that we are talking about today actually refers to such a thing as "the basis of the irreducible representation of space of the Lie group".
Therefore, even without quantitative calculations, it is necessary to understand modern particle physics from a conceptual point of view.
Furthermore, the most accurate physical theory of mankind at present, the Standard Model, is essentially a gauge theory, and the core element of this gauge theory is a Lie group.
In short, the mathematics that physicists can not use must not be used, and the fact that Lie group Lie algebra appears so widely in physical theory shows that modern particle physics really cannot do without it.
The purpose of this article is to briefly introduce lie group Lie algebra:
In the first section we review the basic definition of a group
The second section gives the definition of Li Qun
The third section introduces lie algebra and its relationship to lie groups
First, groups and symmetry
Symmetry is an extremely common concept, but how to accurately describe this concept mathematically is not a simple problem.
To describe this concept precisely, we first resort to intuition.
Considering a square, we would say that it is symmetrical along a diagonal or midline (a line connecting the midpoints of two opposite sides) because both sides of the line "look the same".
If at this time we flip it along the axis of symmetry, then since the left and right sides look the same, we cannot see any difference between the two squares before and after the flip operation, and since this is the case, we can say that the square is unchanged under this operation.
If we continue to flip along the previous axis of symmetry, it is clear that the square is still unchanged, and for the same operation (flipping along the previous axis of symmetry) no matter how many times we flip, the square is unchanged.
Now we can define the symmetry:
If an operation leaves the object being manipulated unchanged, then we call it a symmetric operation.
Sometimes it is said that the object has the symmetry of the corresponding operation.
Having figured out what symmetry means, we now need to find a theory to describe it, and that theory is group theory.
The definition of the group is given first
Set a set, between any two elements in the set, there is such a binary operation
1. Closedness
That is, the result of the operation of any two elements is still in this set.
2. Binding law
3. There is a unique unit element
4. There is a unique inverse element in any element in the collection
Then we say that this set and binary operations form a group.
The symmetric transformation for a square has a total of 8 elements, which are:
Identity transformation, clockwise rotation, clockwise rotation, clockwise rotation, folding along the four axes of symmetry.
It is easy to verify that they meet the definition of a group, which we customarily call this group.
Another example is that all integers and addition operations form a group called an additive group.
There are only finite group elements, and additive groups have infinite numbers.
Why do groups correspond to symmetry?
We can imagine that since symmetric operations do not change objects, then no matter what kind of symmetric operations are successively added to the object, they are still symmetrical operations, and there is always an identity operation - doing nothing.
It is also very natural for the requirement of inverse operation (in fact, for a set that does not satisfy only the group definition 4, we call it the (unitary) semigroup, which is very important in modern physics. ）。
By comparing this, we find that symmetric operations usually naturally have a group structure, so we can study it with group theory.
2. Li Qun
Now that we know that groups can correspond to symmetrical transformations of squares, what about a symmetry group of a circle?
Intuitively, we would think that a circle is more symmetrical than a square, because with the center of the circle as the axis of rotation, rotating at any angle, the circle remains unchanged, so we can say that there are countless symmetrical operations. Then the corresponding group should also contain an infinite number of group elements.
It is not a strange situation that a group with infinite groups of elements is familiar with an infinite number of group elements.
But for the symmetry operation of circles, there seems to be anything different from the infinite group of addition groups, what is it?
The answer is that when we talk about the transformation of a circle, we can talk about a circle turning at an infinitesimal angle. The concept of " infinitesimal " does not represent the concept of " infinitesimal " for the algebraic structure of groups , because infinitesimals involve limits , and the concept of limits depends on topology rather than groups.
So we need to use both group structure and topology to accurately describe this transformation.
In practical applications, we usually need not only topology, but also differential structures built on topology, and the combination of the two leads to the concept of Lie group. definition
A Li group is a set, satisfied
1. It's a group
2. It is a differential manifold
3. The group structure is compatible with the differential structure.
In general, we will correspond each group element to a point on the manifold and place the unit element at the origin. The compatibility condition tells us about group operations
1）
Can be written as a binary map on a manifold
And the mapping is smooth.
2), existence satisfied
。
The Lie group has both a group structure and a differential structure, which allows us to study it simultaneously using the method of group and manifold.
Let's look at a few examples
1. Matrix group
A one-dimensional positive group is a satisfying group. We can write it straight out,
So the manifold it corresponds to is the circle.
2. Matrix group
Two-dimensional monophyllum groups are satisfied groups. If we require that its matrix representation is also satisfied, then we call this subgroup a matrix group.
Its group element satisfies
therefore
So there are only 3 independent real parameters that can be written
The constraint tells us that this is a three-dimensional sphere.
Third, Lie algebra
Now let's see what can be achieved by means of both groups and manifolds.
For differential manifolds, we know that there can be tangent vectors on top of them.
Now consider a tangent vector at the origin, since each element of the Lie group is also a group element, so any group element can act on the origin.
Such a group action is a smooth map, so we can ask what effect does this map have on tangent vectors?
From the knowledge of differential manifolds we know that any one has a smooth mapping of manifolds
both can generate a predicate map of tangent vectors,
Thus the group element is accompanied by such a prevarication map, which reflects the tangent vector at the origin to a tangent vector at the point (note that this is because we set the origin as a unit element, so the effect on the origin turns out to be a point. ）。
In this way, we act on all the group elements on the tangent vector, thus making a tangent vector field.
This tangent vector field has a very special property, that is, the group action remains constant to it, so we call it (left) invariant vector field. A simple proof is given below
Let this vector field be the vector at any point, and from the above construction we know that this is a prepolation mapping from the origin
Because of the group arithmetic, we know, therefore
So the group action does not change the tangent vector field we construct, the proof is complete.
For any tangent vector at the origin, we can construct a vector field in this way. For vector fields , their Lie derivative can be found
We note the value of the derived Lie derivative at the origin as a binary operation in the tangent space at a new origin
It can be proved that this new operation also satisfies the allocation rate
We call this new algebraic relation the Lie algebra of the Lie group.
Mathematically, Lie algebra does not depend on Lie groups, and we can learn to use Lie algebras independently of Lie groups. However, in physics, lie groups and lie algebras are generally combined, so let's briefly talk about the relationship between lie groups and lie algebras.
From the process of constructing the Lie algebra, we can see that the Lie algebra can be transformed locally into a Lie group element, because the Lie algebra can be regarded as a tangent vector on the Unit element of the Lie group, so this tangent vector can generate a local flow satisfaction
Solving this flow, we find that the result is an exponential map
Thus knowing the elements on this curve near the origin, taking different tangent vectors, we get the flow in different directions, so that all the elements near the origin can be solved.
However, this method generally cannot obtain all the elements of the Lie group, and only the tightly connected Lie group exponential map is fully emitted.
Sometimes we also call it a generator, and when we call it a generator, we emphasize its Lie algebraic properties rather than the tangent vectors on the manifold.
Lie algebra transforms a nonlinear object such as dealing with a lie group into its own linear space, which greatly facilitates many problems, but lie algebra can only reflect the local properties of the lie group, and cannot do anything about the overall nature, in fact, different lie groups can have the same lie algebra, so the study of lie groups can not only rely on lie algebra.
The reproduced content represents the views of the author only
Does not represent the position of the Institute of Physics, Chinese Academy of Sciences
Source: yubr
Edit: Herding fish