Quantum Mechanics for Mathematicians

A quantum system corresponding to indistinguishable particles interacting
with each other with an interaction energyv(x−y) (wherex,yare the positions
of the particles) is given by adding a term

1

2

∫∞

−∞

∫∞

−∞

Ψ̂†(x)Ψ̂†(y)v(y−x)Ψ(̂y)Ψ(̂x)dxdy

to the free particle Hamiltonian. Just as the free-particle Hamiltonian has an
expression as a momentum space integral involving products of annihilation and
creation operators, can you write this interaction term as a momentum space
integral involving products of annihilation and creation operators (in terms of
the Fourier transform ofv(x−y))?

Problem 3:
For non-relativistic quantum field theory of a free particle in three dimen-
sions, show that the momentum operatorsP̂(equation 38.11) and angular mo-
mentum operatorsL̂(equation 38.12) satisfy the commutation relations for the
Lie algebra ofE(3) (equations 38.13).

Problem 4:
Show that the total angular momentum operators for the non-relativistic
theory of spin^12 fermions discussed in section 38.3.3 satisfy the commutation
relations for the Lie algebra ofSU(2).

B.18 Chapters 40 to 42

Problem 1:
IfP 0 ,Pj,Lj,Kj are the operators in any Lie algebra representation of the
Poincar ́e group corresponding to the basis elementst 0 ,tj,lj,kjof the Lie algebra
of the group, show that the operator

−P 02 +P 12 +P 22 +P 32

commutes withP 0 ,Pj,Lj,Kj, and thus is a Casimir operator for the Poincar ́e
Lie algebra.

Problem 2:
Show that the Lie algebraso(4,C) issl(2,C)⊕sl(2,C). Within this Lie
algebra, identify the sub-Lie algebras of the groupsSpin(4),Spin(3,1) and
Spin(2,2).

Problem 3:
Find an explicit realization of the Clifford algebra Cliff(4,0) in terms of 4 by
4 matrices (γmatrices for this case) and use this to realize the groupSpin(4) as
a group of 4 by 4 matrices (hint: recall that the Lie algebra of the spin group is
given by products of two generators). Use these matrices to explicitly construct
the representations ofSpin(4) on two kinds of half-spinors, on complexified
vectors (C^4 ), and the adjoint representation on the Lie algebra.