Quantum Mechanics for Mathematicians

B.20 Chapters 45 and 46

Problem 1:
Show that the Yang-Mills equations (46.9 and 46.11) are Hamilton’s equa-
tions for the Yang-Mills Hamiltonian 46.10.

Problem 2:
Show that the matrixP⊥with entries

(P⊥)jk=δjk−

pjpk

|p|^2

acts on momentum vectors inR^3 by orthogonal projection on the plane per-
pendicular top. Use this to explain why one expects to get the commutation
relations of equation 46.18 in Coulomb gauge (the condition∇·A= 0 in momen-
tum space says that the vector potential is perpendicular to the momentum).

Problem 3:
Show that the groupSO(2) acts on the space of solutions of Maxwell’s
equations by
E(x)→cosθE(x) + sinθB(x)
B(x)→−sinθE(x) + cosθB(x)

Forθ=π 2 this symmetry interchangesEandBfields and is known as electric-
magnetic duality. A much harder problem is to see what the corresponding
operator acting on states is (it turns out to be the helicity operator).

B.21 Chapter 47

Problem 1:
Show that complex-valued solutions of the Dirac equation 47.2 correspond
in the non-relativistic limit (energies small compared to the mass) to solutions
of the Pauli-Schr ̈odinger equation 34.3.Hint: write solutions in the form

ψ(t,x) =e−imtφ(t,x)

Problem 2:
For two copies of the Majorana fermion theory, with the same massmand
field operatorsΨ̂ 1 ,Ψ̂ 2 , show that the theory has anSO(2) symmetry, and find
the Lie algebra representation operatorQ̂for this symmetry. Compute the
commutators
[Q,̂Ψ̂ 1 ], [Q,̂Ψ̂ 2 ]

Problem 3: