Symmetry, conservation laws and compatible operators

Symmetry

A symmetry of a dynamical system is a transformation that leaves the action $S = \int \mathrm{d}^4 x \mathcal{L}$ invariant. A set of symmetry may form a group, the symmetry group. Most important symmetries in physics are from symmetry groups. Notable examples of the symmetry group include: the spacetime symmetry - Poincaree group and conformal group, the gauge groups, the lattice group and the supersymmetry.

A continuous group (Lie group) can be written in terms of the single parameter exponential map, $g = \exp(-i \theta_r X^r), \; r = 1, 2, ..., n$, where $\theta_r$ are scalars. $X^r$ are called the group generators. The group generators form a Lie algebra. $n$ is called the dimension of the Lie algebra. In quantum mechanics, the group transformation is represented by unitary or anti-unitary operators (Wigner theorem). Therefore, the group generators $X^r$ are Hermitian operators ("real operators"), hence are observables. The Lie algebra, especially the commutation relations between groups generators (commutators) $[X^r, X^s] = iC^{rst} X^t, r,s,t = 1,2,3..., n$ is dictated by the Lie group. If in a group, all the commutators vanish, the group is called an Abelian group, otherwise called Non-Abelian. Most important symmetry groups in physics are Non-Abelian.

It's worth mentioning the Hamiltonian $H$ is a generator of the Poincaree group.

Conservation laws

A quantity $A$ is called conserved if it does not change upon time, $\frac{\mathrm{d}}{\mathrm{d}t} A = 0$. In quantum mechanics, the time evolution is dictated by the Hamiltonian of the system. In Heisenberg picture, \[
\frac{\mathrm{d}}{\mathrm{d}t} A = i [ H, A ] + \frac{\partial}{\partial t} A.
\] Similarly, in classical mechanics, \[
\frac{\mathrm{d}}{\mathrm{d}t} A = -\{H, A\} + \frac{\partial}{\partial t} A
\] where $\{ A, B\}$ is the Poisson bracket. One may note the similarity between the Poisson bracket and the commutator. Indeed, in the canonical quantization, Poisson bracket is replaced by the commutator $\{ \cdot, \cdot \} \to \frac{1}{i}[ \cdot, \cdot ]$.

Note that a conserved quantity does not necessarily commute with the Hamiltonian, as it may have explicit time dependence init. One has to be especially cautious in a time-independent problem, as we normally only consider the operator at $t = 0$.

Let $U = e^{-i \theta A}$ be a symmetry. Then the principle of time evolution should not change for a transformed state: $U \left.|\psi(t)\right> = V(t) U\left.|\psi(0)\right>$, where $V(t) = \exp(-i H t)$ is the time-evolution operator, $V(t) \left.|\psi(0)\right>$. That implies $U V = V U$. So we conclude $[H, A] = 0$. There are two pitfalls: 1). a symmetry may be represented by anti-unitary operators. In that case, the commutator is non-vanishing. 2). the generator may be explicitly time dependent, especially the symmetry transformation involves time.

Noether's theorem

Symmetry and conservation laws are closely related by Noether's theorem. Noether theorem states, for each differential symmetry of a physical system, there exists a conservation law. More specifically, let $q_i$ be the generalized coordinates (for instance, the quantum fields). Under the infinitesimal symmetry transformation, $q_i \to q_i + \epsilon \Delta q_i$. The Lagrangian must takes the form, \[
\frac{\mathrm{d}}{\mathrm{d}t} L = \frac{\mathrm{d}}{\mathrm{d}t} G \] to keep the action $S = \int \mathrm{d} t L(q, \dot{q}, t)$ invariant. Then, there exists a conserved quantity (often called conserved charge), \[
Q = G - \frac{\partial L}{\partial \dot{q}_i} \Delta q_i; \\
\dot{Q} = 0
\] The conserved charges are the generators of the symmetry group.

Boost generator $\vec{K}$ is the conserved charge of boost transformation. But it does not commute with $H$. So clearly, it has explicit time dependence. In fact, in the Poincaree algebra, \[
K^i = M^{0i} = x^0 P^i - x^i P^0 = t P^i - x^i H.
 \] This is even clearer in non-relativistic quantum mechanics, as the boost at $t = 0$ is simply a translation in the momentum space \[
\vec{p} \to \vec{p} + m \vec{v} \\
\vec{x} \to \vec{x} + t \vec{v} \\
t \to t \] Therefore we must have $\vec{K} = t\vec{P} - \vec{X} M \simeq t \vec{P} - \vec{X} H$. The conserved charge (i.e. $\vec{K}$) is simply the mass center.

Compatible operators

According to linear algebra, two linear operators can be simultaneously diagonalized iff they commute. $A$ and $B$ are simultaneously diagonalizable, means there exists a matrix $S$, both $S^{-1} A S$ and $S^{-1} B S$ are diagonal. Note that matrix $S$ may not be unique. Similarly, a set of linear operators can be simultaneously diagonalized iff each pair commutes. In quantum mechanics, the the of compatible operators are very useful to identify quantum states. Indeed, if one can find a complete set of compatible operators $\{A_1, A_2, \cdots, A_n\}$, the mutal eigenstates can be identified by the eigenvalues, $\left.|\lambda_1, \lambda_2, \cdots \lambda_n \right>$.

In practice, one of the operator is often chosen to be the Hamiltonian. Therefore, Hamiltonian is block diagonal in the mutual eigenstates of a set of its compatible operators. If the compatible operators is complete, the eigenstate of the Hamiltonian is uniquely determined by their eigenvalues. For example, in the non-relativistic Hydrogen atom problem, a set of the compatible operator is $\{ H, J^2, J_z\}$. The eigenstates can be identified by $\left.|n, l, m \right>$, where the eigenvalues are $\{E_n, l(l+1), m\}$. The eigenvalues of a set of compatible operators for the Hamiltonian are often called "good quantum numbers". Note that there may be more than one set of compatible operators for the Hamiltonian.

Clearly, conserved observables are the natural candidates as compatible operators. The leading example is the momentum operator $\vec{P}$. A counterexample is the boost operator $\vec{K}$, as it does not commute with the Hamiltonian $[\vec{K}, H] = -i \vec{P}$. Another counterexample is $J_x$ or $J_y$ if we have already used $J_z$. For a symmetry group, the Casimir operators are also natural candidates. The leading example is the total angular momentum operator $\vec{J}^2$ from the rotation group. According to what we have analyzed above, the generators of a symmetry is a promising candidate for the compatible operator set for the Hamiltonian. In practice, a degeneracy is the sign of extra compatible operators. When a unexpected degeneracy is detected, it is often said an "unknown" symmetry protects the degeneracy.