44.2.2 Rotations
We can use the same method as for translations to find the quadratic combi-
nations of coordinates onMcorresponding to the Lie algebra of the rotation
group, which after quantization will provide the angular momentum operators.
The action on Klein-Gordon solutions will be given by the operators
L=X×P=x×−i∇ position space
=−i∇p×p momentum spaceThe corresponding quadratic operators will be
̂L=−∫
R^3:̂π(x)(x×∇)φ̂(x):d^3 x=
∫
R^3a†(p)(p×i∇p)a(p)d^3 p (44.8)which, again, could be found using the moment map method, although we
will not work that out here. One can check using the canonical commutation
relations that the components of this operator satisfy theso(3) commutation
relations
[−iL̂j,−iL̂k] =jkl(−iL̂l) (44.9)
and that, together with the momentum operatorsP̂, they give a Lie algebra
representation of the Euclidean groupE(3) on the multi-particle state space.
Note that the operatorsL̂commute with the HamiltonianĤ, and so will act
on the energy eigenstates of the state space, providing unitary representations
of the groupSO(3) on these energy eigenspaces. Energy eigenstates will be
characterized by the irreducible representation ofSO(3) they are in, so as spin
s= 0,s= 1,...states.
44.2.3 Boosts
From the point of view that a symmetry of a physical theory corresponds to
a group action on the theory that commutes with time translation, Lorentz
boosts are not symmetries because they do not commute with time translations
(see the commutators in equation 42.2). From the Lagrangian point of view
though, boosts are symmetries because the Lagrangian is invariant under them.
From our Hamiltonian point of view, they act on phase space, preserving the
symplectic structure. They thus have a moment map, and quantization will
give a quadratic expression in the field operators which, when exponentiated,
will give a unitary action on the multi-particle state space.
Note that boosts preserve not only the symplectic structure, but also the
relativistic complex structureJr, since they preserve the decomposition of mo-
mentum space coordinates into separate coordinates on the positive and negative
energy hyperboloids. As a result, when expressed in terms of creation and anni-
hilation operators, the quadratic boost operatorsK̂will have the same form as