![[Assets/Quantum Theory Groups and Representations - Peter Woit.jpg#outline|150]] # Notes on *Quantum Theory, Groups and Representations* by Peter Woit --- ### Chapter 1: Introduction and Overview #### Section 1.2: Basic principles of quantum mechanics Page 5… The observables (and thus operators) with important physical significance will turn out to correspond to some group action on the physical system. Expanded on in section 1.4. Linear Momentum → Action of translation in space. $\mathbb{R}^3$ Energy → Action of translation in time. $\mathbb{R}$ Angular Momentum → Action of rotation in space. $SO(3)$ Charge → Action of rotation in complex plane. $U(1)$ #### Section 1.3: Unitary group representations Page 8… If we have a space M, we can look at the set of functions on that space, F(M), to better understand the space. This is because F(M) is automatically a vector space, so it linearizes the problem no matter the geometry of M. #### Section 1.4: Representations and quantum mechanics > The fundamental relationship between quantum mechanics and representation theory is that whenever we have a physical quantum system with a group G acting on it, the space of states H will carry a unitary representation of G. This makes intuitive sense, because a symmetry transformation (that forms a symmetry group) will not change the “outcomes of measurements,” or the inner product structure of the state space. In the physical, a symmetry transformation takes you from one state of the system to another without changing the outcome of any possible measurement you could make on the system. > [!tip] Clarification > When I say symmetry here, I don’t mean one that preserves the hamiltonian (and thus equations of motion), I just mean one that preserves measurement outcomes (inner products). In the abstract, each state is a vector, and thus a transformation between those states/vectors is a linear map. Since the outcome of measurements is preserved, that map preserves inner products. Therefore the map is unitary. A representation is a homomorphism (group structure preserving map) between group elements (ie a symmetry transformation) and linear maps. So a representation is how we formalize the relationship between the physical transformations and the associated linear maps. Therefore that state space carries a unitary representation of the symmetry group. > For a representation $\pi$ and group elements $g$ that are “close to the identity,” exponentiation can be used to write $\pi(g) \in GL(n,\mathbb{C})$ as $\pi(g)=e^{A}$ where $A$ is also a matrix close to the zero matrix. What he means by close to the identity is that it belongs to the tangent space of the group manifold at the identity. See [Wikipedia: Exponential map](https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)). We will see that when $\pi(G) \in U(n)) \subset GL(n, \mathbb{C})$, or when the representation is unitary, that $A$ will be skew-adjoint. This allows us to define a self-adjoint operator from it with $B = iA$. Self-adjoint operators are an observable of the system. > Lie group actions thus provide us with a class of quantum mechanical observables, with the self-adjointness property of these operators corresponding to the unitarity of the representation on state space. This allows us to take an abstract group element, like a rotation by $\theta$ around a particular axis, and see what the corresponding transformation is between vectors in our state space. Physical rotation → map between states. Furthermore, we get an associated observable from that group’s acting on our system. In other words, we get a group action on the state space, because an action takes a group element and a vector, and gives us the output vector. ### Chapter 2: The Group $U(1)$ and its Representations U(1) is the circle group, and can be thought of as the rotations in the complex plane, so it is parameterized by a single angle $\theta$. We can identify the group elements with points on the imaginary unit circle, $e^{i \theta}$. The group action would then be multiplication, $e^{\theta_{1}}e^{\theta_{2}} = e^{\theta_{1} + \theta_{2}}$. We could just as well identify the group elements with angles $\theta \in [0, 2\pi]$. The group action would then be addition mod $2\pi$, $\theta_1\theta_2 = \theta_1 + \theta_2$ (mod $2\pi$). This is a 1D lie group. This group is isomorphic to the 2D rotation group SO(2) for obvious reasons. We will see that if a state space is a unitary representation of U(1), it can be decomposed into a direct-sum of 1D subspaces that each represent U(1) (subrepresentations), where each 1D subspace can be characterized by an integer q. That integer will be the eigenvalue of a self-adjoint operator Q, which we call the charge operator. Examples of U(1) groups acting on physical systems include: - U(1) acts on complex valued wavefunctions by point-wise transformation. This can be used to understand how particles interact with electromagnetic fields. The physical interpretation of the eigenvalue of the Q operator will be the electric charge of the state. - If one chooses a particular direction in three-dimensional space, then the group of rotations about that axis can be identified with the group U(1). The eigenvalues of Q will have a physical interpretation as the quantum version of angular momentum in the chosen direction. #### Section 2.1: Some representation theory Definition: A representation is called irreducible if it has no subrepresentations, or proper subspaces that are themselves representations. **Theorem 2.1**: Any unitary representation (that is, it preserves the inner product) can be written as a direct sum of irreducible subrepresentations. Notice Thm. 2.1 applies to all U(n), not just U(1). **Theorem (Schur’s Lemma)**: Schur’s lemma says that for an irreducible representation, if a matrix M commutes with all the representation matrices π(g), then M must be a scalar multiple of the unit matrix, $M = \lambda I$. **Theorem 2.2**: If G is commutative, all of its irreducible representations are one dimensional. #### Section 2.2: The group U(1) and its representations Since U(1) is commutative, all its irreducible representations will be one dimensional by **Thm. 2.2**, π : U(1) → GL(1, C). Note that U(n) in general is not commutative. π is a map from one manifold (the group U(1)) to another manifold (the matrix group GL(1, C)). So it is differentiable. A differentiable map π that is a representation of U(1) must satisfy homomorphism and periodicity properties which can be used to show: **Theorem 2.3**: All irreducible representations of the group U(1) are unitary (hence the U), and given by $ \pi_{k} : e^{i\theta} \in U(1) \rightarrow \pi_{k}(\theta) = e^{ik\theta} \in U(1) \subset GL(1, \mathbb{C}) \cong \mathbb{C^*} $ for some $k \in \mathbb{Z}$. Homomorphism gives exponential form, periodicity gives integrality of k, and the i in the exponential gives unitarity. Remember that $GL(1, \mathbb{C})$ is the invertible 1x1 complex matrices, so pretty much invertible complex numbers. > [!info] Notation ambiguity > Because elements of U(1) can be identified with both complex exponentials or angles, there is ambiguity in the input notation. But the output is always the same, since it takes us to GL(1, C). > > $\pi_{k}:U(1)\to GL(1,\mathbb{C)}$ > > $\pi_k(e^{i\theta}) = e^{ik\theta}$ > $\pi_k(\theta) = e^{ik\theta}$ > > Keep in mind our representation would be $(\pi_{k},\mathbb{C})$. Where $\pi_{k}$ is the map and $\mathbb{C}$ is the vector space. #### Section 2.3: The charge operator > [!caution] QUESTIONS > I see how Thm. 2.2 implies all representations are scalar matrices, but why does that imply 1D scalar matrices? > > I don’t fully understand the motivation behind the tangent space stuff, but maybe that will be covered fully in chapter 5. $\pi(e^{i\theta}) = e^{iQ\theta} \in U(n) \subset GL(n ,\mathbb{C})$. This allows us to take an abstract group element, like a rotation by $\theta$ around a particular axis, and see what the corresponding transformation is between vectors in our state space. Physical rotation → map between states. Furthermore, we get an associated observable from that group’s acting on our system. In other words, we get a group action on the state space, because an action takes a group element and a vector, and gives us the output vector. #### Section 2.4: Conservation of charge and U(1) symmetry j ---