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Groups

In this section, we will review the concept of groups, which is a fundamental concept in mathematics and physics. They are the core of quantum field theory, and it is essential to understand them in order to understand the mathematics of quantum field theory.

Table of Contents​

What is a Group?​

Consider an equilateral triangle in the plane. There are a few ways to transform the triangle, such as rotating it or reflecting it. Group theory is the study of such transformations.

In particular, we will study the set of all transformations that can be applied to the triangle, and how these transformations can be combined. Groups fundamentally represent different ways to transform an object. As such, we naturally expect certain properties to hold;

  1. There is always one transformation that does nothing to the object. For the rotation group for the triangle, for example, it is akin to rotating the triangle by degrees. This is called the identity transformation.
  2. The transformations can be combined to form new transformations. For example, if we rotate the triangle by degrees and then by another degrees, we can combine these two transformations to form a new transformation that rotates the triangle by degrees. This is called the closure property.
  3. The way we combine the transformations is associative. For example, if we rotate the triangle by degrees and then reflect it, we can combine these two transformations in any way to get the same result. This is called the associative property.
  4. Each transformation has an inverse transformation that undoes it. For example, if we rotate the triangle by degrees, we can undo this transformation by rotating it by degrees. This is called the inverse property.

Given that we expect these properties to hold, we can define a group as follows:

Group: A group is a set together with a binary operation that combines two elements to form a third element, such that the following properties hold:

  1. Closure: For all , .
  2. Associativity: For all , .
  3. Identity: There exists an element such that for all , .
  4. Inverse: For each , there exists an element such that .

In this definition, is the set of elements, and is the binary operation that combines the elements. and are elements of the group, which in our example are the rotations of the triangle. is the identity element, which in our example is the rotation by degrees.

Example: Number Line​

Consider the set of all integers . When two numbers are added, this can be visualized as moving the number line twice. For example, if we are at and we add , we shift the number line to the left by ;

When we add multiple numbers, this is just a matter of moving the number line multiple times, which can be combined into a single movement. For example, if we add and then , we can combine these two movements into a single movement of .

When we visualize addition like this, this gives us a clue that addition is an operation that satisfies the properties of a group. While it can be a good exercise to algebraically verify that addition satisfies the properties of a group, it is more insightful to notice the link between addition and its geometric representation. In particular, we can see that the number line is a one-dimensional space, and the operation of addition is a way to move along this space. This allows us to extend this group to higher dimensions, eventually leading to the concept of a vector space.

In addition to the properties of a group, addition is also commutative. In other words, the order in which we add the numbers does not matter. Groups that are commutative are called abelian groups.

Example: Rotation of a Square​

Consider the set of all rotations of a square. We can rotate the square by , , , and degrees.

We can see that the set of all rotations of a square satisfies the properties of a group.

  1. Closure: If we rotate the square by degrees and then by degrees, we can combine these two transformations to form a new transformation that rotates the square by degrees.
  2. Associativity: The way we combine the transformations is associative. For example, if we rotate the square by degrees and then reflect it, we can combine these two transformations in any way to get the same result.
  3. Identity: The identity transformation is the rotation by degrees.
  4. Inverse: The inverse transformation of a rotation by degrees is a rotation by degrees.

Additionally, the set of all rotations of a square is also abelian, since the order in which we apply the transformations does not matter.

Continuous Transformations​

When we talk about groups, we make a distinction between discrete and continuous transformations. Discrete transformations are transformations that can be applied in a finite number of steps, while continuous transformations are transformations that can be applied in an infinite number of steps.

For example, the set of all rotations of a square is a discrete group, since we can only rotate the square by , , , and degrees. However, if we allow rotations of any angle, then the set of all rotations of a is a continuous group. We call these Lie groups.

Previously, we mentioned in quantum mechanics that spinors transform under matrices, whereas physical vectors transform under Euler rotations, which are matrices. Together, they form a continuous group, and are hence called Lie groups.

Each Lie group has a corresponding Lie algebra. Generally, for a Lie group , the Lie algebra is the set of generators of the group. More specifically, if is a generator of the group, then the Lie group can be expressed as .

Example: Rotations in ​

Consider the Lie group of all rotations in the plane:

This is a subset of the group, which is the group of all rotations in three-dimensional space. Recall that the rotations preserve the length of the vector, and hence are orthogonal (this can be proven using the definition of the inner product). The generator of this group is given by:

To exponentiate a matrix, we use the Taylor expansion of the exponential function:

If one works out the Taylor expansion of the above matrix, one can see that it is indeed equal to the rotation matrix. It's easy to verify that a given matrix is the generator of a Lie group by exponentiating it and checking that the resulting matrix is indeed a rotation matrix. In order to think of one, however, more effort is required. We can use a trick to find the generator of a Lie group;

  1. We know that we need to find such that , where is the rotation matrix.

  2. We can take the derivative of both sides with respect to :

  3. It's easy to prove that (as we expect), thus:

  4. Setting gives us:

As such, to find any generator matrix , we can take the derivative of the rotation matrix with respect to and set .

Example: Taylor's Formula​

We shall use Lie groups to derive an interesting result. Consider a function . Suppose we translate the function (to the left) by a small amount . This is equivalent to the new function .

We can see the translations of this function as a group of transformations. Let be the translation operator, which translates the function by . This set follows the group properties:

  1. Closure: .
  2. Associativity: .
  3. Identity: .
  4. Inverse: .

Now, because is the variable that we are translating by, we can find the generator of this group by taking the derivative with respect to and setting :

To find this out, we first apply both sides to a function :

This might seem a bit strange, but if introduce a new variable it becomes a bit clearer:

As such, , and we can see that the generator of the translation group is simply the derivative operator;

This means that, by the definition of the generator, we can write the translation operator as:

Interestingly, we are exponentiating the derivative operator, which is a bit strange. However, this indeed gives us the translation operator. This yields Taylor's formula:

To extend it to multiple dimensions, we simply replace with the gradient operator and with the vector :

One application of this is to find the translation operator in quantum mechanics. We want a translation operator such that . Treating the ket as a function of , we can see that this is the same as the translation operator we derived above. As such, we can write:

Group Homomorphisms​

Suppose we have two groups and . Is there a way to relate these two groups? If we can find a way to relate the two groups, we can use the properties of one group to study the other group. We introduce something known as a homomorphism:

This is a mapping between the two groups that preserves the group structure:

where and is the group operation (for the group on the left and for the group on the right). This means that if we take two elements and from group , and we apply the group operation to them, we can map the result to an element in group .

More specifically, we can look for a one-to-one mapping between the two groups, known as a bijection. This means that we can map each element of to a unique element of , and vice versa. These bijective homomorphisms are called isomorphisms.

We shall use this idea to derive Euler's formula without any calculus. Suppose we have . We know from the laws of exponents that . As we have done before, we can think of as a translation of the number line, and as a scalar multiplication of the number line (a stretch).

In other words, we can think of as a homomorphism between the group of translations and the group of stretches; . We formally write as:

where is the group of translations and is the group of stretches (positive real numbers under multiplication).

This means that we can take a translation and map it to a stretch . We can quickly prove that this is indeed a homomorphism:

Indeed, applying the group operation to the left and the group operation to the right gives us the same result. This means that we can use the properties of the group of translations to study the group of stretches.

Specifically, we will extend the additive group of translations from real numbers to that of complex numbers. We will introduce a new homomorphism , where is the group of translations in the complex plane and is the group of stretches in the complex plane. in other words, if we have a complex number , we can think of as a homomorphism between the group of translations in the complex plane and the group of stretches in the complex plane.

We know that with a translation along real numbers, , is a stretch. But what about the imaginary part ?

We first assert that , where is the complex conjugate of . Now consider :

This means that and are inverses of each other. But we also know that , so we can write:

When we multiply a complex number by its conjugate, we get the modulus squared of the complex number. This means that we can write:

Thus, is a complex number with modulus . It must lie on the unit circle. We can then write:

where is an angle that depends on . Recall that is a homomorphism, so it must satisfy . This means that we can write:

The left-hand side is just directly from Equation . Thus we have:

This implies that . This means that is a linear function of . We can write:

Suppose we set . Then we can write:

Thus, because was our definition of the exponential function, we can write:

which is Euler's formula.

As you can see, we derived this formula almost entirely from the properties of groups. Unfortunately, it did rely on some assumptions—that and that . A full proof comes from calculus.