Chiral symmetries associated with angular momentum

Abstract

In beginning quantum mechanics courses, symmetries of a physical system are usually introduced as operators which commute with the Hamiltonian. In this article we will consider chiral symmetries which anticommute with the Hamiltonian. Typically, introductory courses at the (under)graduate level do not discuss these simple, useful and beautiful symmetries at all. The first time a student typically encounters them is when the Dirac equation is discussed in a course on relativistic quantum mechanics, or when particle-hole symmetry is studied in the context of superconductivity. In this article, we will show how chiral symmetries can be simply elucidated using the theory of angular momentum, which is taught in virtually all introductory quantum mechanics courses.

Chiral symmetry in rotating systems

“…Therefore, the chiral symmetry is absent in this Hamiltonian. One can draw a similar conclusion if the prescription of Bhattacharya and Kleinert [20] is followed.…”

“…It is worthwhile to mention here that recently, Bhattacharya and Kleinert have established the correlation between chiral symmetry and angular momentum [20]. According to their prescription, the chiral symmetry prevails in those systems whose Hamiltonian anti-commute with angular momentum operator.…”

“…The chiral symmetry in this reformulated Hamiltonian arises mainly due to vacuum (whichever of the two ground states we care to work with) does not share the symmetry of the Hamiltonian. Again, we would like to emphasize that a similar judgment follows by using the guidelines of Bhattacharya and Kleinert [20].…”

“…We would like to emphasize that Hamiltonians (19) and (21) represent exactly the same physical system; all we have done is to change the notation (20). The first Hamiltonian (19) is even in J 3 and hence the chiral symmetry is removed.…”

Topological insulators and the Kane–Mele invariant: Obstruction and localization theory

“…Examples are rotations about a given axis by angle π and certain Hamiltonians built from the corresponding angular momentum operators, since on an eigenspace of J we have R(π) J R(π) −1 = (−1) 2l J, where R(α) denotes the rotation about the given axis by an angle α ∈ [0, 2π), and J is the angular momentum operator with l ∈ 1 2 N 0 the eigenvalue of J on the respective subspace. See [4] for an account of chiral symmetries in quantum mechanics together with some explicit examples. ⊳…”