Quantum Field Theory I

42 CHAPTER 1. INTRODUCTIONS
the scalar representation of the Poincaré group
Investigations of the Poincaré group representations are of vital importance
to any serious attempt to discuss relativistic quantum theory. It is, however,
not our task right now. For quite some time we will need only the simplest
representation, the so-called scalar one. All complications introduced by higher
representations are postponed to the summer term.
Let us consider a Hilbert space in which some representation of the Poincaré
group is defined. Perhaps the most convenient basis of such a space is the
one defined by the eigenvectors of the translation generators P µ , commuting
with each other. The eigenvectors are usually denoted as |p,σ〉, where p =
(p 0 ,p 1 ,p 2 ,p 3 ) stands for eigenvalues of P. In this notation one has
P µ |p,σ〉 = p µ |p,σ〉
and σ stands for any other quantum numbers. The representation of space-time
translations are obtained by exponentiation of generators. If U (Λ,a) is the
element of the representation, corresponding to the Poincaré transformation
x → Λx+a, then U (1,a) = e −iaP . And since |p,σ〉 is an eigenstate of P µ , one
obtains
U(1,a)|p,σ〉 = e −ipa |p,σ〉
And how does the representation of the Lorentz subgroup look like Since the
p µ is a fourvector, one may be tempted to try U (Λ,0)|p,σ〉 = |Λp,σ〉. This
really works, but only in the simplest, the so-called scalar representation, in
which no σ is involved. It is straightforward to check that in such a case the
relation
U (Λ,a)|p〉 = e −i(Λp)a |Λp〉
defines indeed a representation of the Poincaré group.
Let us remark that once the spin is involved, it enters the parameter σ and
the transformation is a bit more complicated. It goes like this: U (Λ,a)|p,σ〉 =
∑
σ ′ e −i(Λp)a C σσ ′ |Λp,σ ′ 〉 and the coefficients C σσ′ do define the particular representation
of the Lorentz group. These complications, however, do not concern
us now. For our present purposes the simplest scalar representation will be
sufficient.
Now it looks like if we had reached our goal — we have a Hilbert space
with a representation of the Poincaré group acting on it. A short inspection
reveals, however, that this is just the rather trivial case of the free particle. To
see this, it is sufficient to realize that the operator P 2 = P µ P µ commutes with
all the generators of the Poincaré group, i.e. it is a Casimir operator of this
group. If we denote the eigenvalue of this operator by the symbol m 2 then the
irreducible representations of the Poincaré group can be classified by the value
of the m 2 . The relation between the energy and the 3-momentum of the state
|p〉 is E 2 − ⃗p 2 = m 2 , i.e. for each value of m 2 we really do have the Hilbert
space of states of a free relativistic particle. (The reader is encouraged to clarify
him/herselfhow should the Hilbert space ofthe states of free relativistic particle
look like. He/she should come to conclusion, that it has to be equal to what we
have encountered just now.)
Nevertheless, this rather trivial representation is a very important one — it
becomes the starting point of what we call the particle-focused approach to the
quantum field theory. We shall comment on this briefly in the next paragraph.