Introduction to Group Representations

1. From Abstract Symmetry to Concrete Action

In quantum mechanics, the state of a system is a vector in a Hilbert space. A symmetry operation, like a rotation or a reflection, must transform this state vector into another valid state vector. Furthermore, if we apply one symmetry operation followed by another, the result should correspond to applying the single composite symmetry operation. This mapping from abstract group elements to concrete linear operators acting on a vector space is the essence of a representation.

Before diving into the formal mathematics, let's consider the physical requirements for a symmetry transformation in quantum mechanics. A symmetry should not change the observable outcomes of experiments. The most fundamental observable is the transition probability between two states. If a transformation $T$ takes states ${\psi }_1$ and ${\psi }_2$ to $T{\psi }_1$ and $T{\psi }_2$, then a core requirement of symmetry is that the probability of transitioning between the original states is the same as for the transformed states:

A celebrated result by Eugene Wigner addresses the profound consequences of this seemingly simple requirement.

Wigner's theorem is a cornerstone of quantum theory. It dramatically narrows down the possibilities for symmetry operators. Instead of dealing with any arbitrary (and potentially non-linear) function on the Hilbert space, we are guaranteed that symmetries must be realized by linear unitary operators or anti-linear anti-unitary operators. For most of our purposes, we will focus on the unitary case. This provides the physical justification for why the group of unitary operators on a Hilbert space, denoted $U(H)$, is what the video's author calls the "master group of quantum mechanics."

2. Defining a Group Representation

With the physical motivation in place, we can now formalize the definition. A representation of a group $G$ on a vector space $V$ is a mapping that assigns an invertible linear operator to each group element, in a way that respects the group's multiplication structure.

Definition: A linear representation of a group $G$ on a vector space $V$ is a homomorphism $\rho$ from $G$ into the general linear group $GL(V)$ of all invertible linear operators on $V$.

This means for any $g_1,g_2\in G$, the map must satisfy:

1. $\rho (g_1{g}_2)=\rho (g_1)\rho (g_2)$ (The composition of operators mirrors the group product)
2. $\rho (e)=I$ (The group identity maps to the identity operator)

If we choose a basis for the $n$-dimensional vector space $V$, each operator $\rho (g)$ can be written as an $n\times n$ invertible matrix, which we'll often call $D(g)$. In this case, the representation is a map $\rho :G\to GL(n,\mathbb{C})$, the group of invertible $n\times n$ complex matrices.

In quantum mechanics, since symmetries are represented by unitary operators acting on a Hilbert space $H$, we are specifically interested in unitary representations, which are homomorphisms into the group of unitary operators $U(H)$.

The point about the Lorentz group is worth emphasizing. The fact that it cannot be faithfully represented by finite-dimensional unitary matrices is a deep mathematical constraint with enormous physical implications. It forces the marriage of special relativity and quantum mechanics into the framework of infinite-dimensional Hilbert spaces, which is the domain of Quantum Field Theory. This is a prime example of how representation theory isn't just a classification tool; it dictates the very structure of our physical theories.

To solidify these definitions with more examples, please review the following text.

3. Equivalent Representations: Same Symmetry, Different Coordinates

The matrix form of a representation is not unique; it depends on the basis you choose for the vector space $V$. If you change the basis, the matrices representing the group elements will also change. However, the underlying physical transformation remains the same. A rotation of a molecule is still the same rotation, regardless of how you orient your coordinate system.

This leads to the concept of equivalent representations. Two representations are considered equivalent if they are related by a simple change of basis.

Definition: Two representations ${\rho }_1$ and ${\rho }_2$ of $G$ on vector spaces $U$ and $V$ are equivalent if there exists an invertible linear map (an isomorphism) $S:U\to V$ such that for all $g\in G$:

In terms of matrices $D_1(g)$ and $D_2(g)$, this means they are related by a single, fixed similarity transformation for all group elements.

The physical meaning is that $D_1$ and $D_2$ are simply different descriptions of the same set of symmetry operations, expressed in different coordinate systems.

The following reading formalizes this idea and provides a clear example.

The example of $D_4$ is particularly illuminating. The matrices for reflection change depending on whether the axes are aligned with the square's edges or its diagonals, but both sets of matrices form a valid representation, and these two representations are equivalent. This is crucial because when we later classify representations, we only care about listing the inequivalent ones—the truly distinct ways a group can act on a system.

Conclusion

In this lesson, we have laid the formal groundwork for representation theory, the primary tool for applying group theory in quantum physics. We've bridged the gap from abstract groups to the concrete world of matrices acting on vector spaces.

The key takeaways are:

• A group representation is a homomorphism $\rho :G\to GL(V)$, which maps abstract group elements to invertible linear operators that act on a vector space $V$ and obey the same multiplication rules.
• Wigner's theorem provides the physical justification for focusing on unitary representations ($\rho :G\to U(H)$) in quantum mechanics, as they preserve transition probabilities.
• Finding a valid representation is a non-trivial task with physical consequences, as demonstrated by the non-existence of a finite-dimensional unitary representation for the Lorentz group.
• Two representations are equivalent if they are related by a similarity transformation, ${\rho }_2(g)=S{\rho }_1(g)S^{-1}$. This corresponds to describing the same physical symmetry operation in a different basis.

Now that we understand what a representation is, a natural question arises: can a representation be broken down into smaller, simpler pieces? Just as we can decompose a vector into its basis components, we might be able to decompose a representation into more fundamental building blocks. These building blocks are called irreducible representations, and they will be the focus of our next lesson.