Quantum Field Theory I

1.3. RELATIVITY AND QUANTUM THEORY 43
The Hilbert space spanned over the eigenvectors |p,σ〉 of the translation
generators P µ is not the only possible choice of a playground for the relativistic
quantum theory. Another quite natural Hilbert space is provided by the functions
ϕ(x). A very simple representation of the Poincaré group is obtained by
the mapping
ϕ(x) → ϕ(Λx+a)
This means that with any Poincaré transformation x → Λx+a one simply pulls
back the functions in accord with the transformation. It is almost obvious that
this, indeed, is a representation (if not, check it in a formal way). Actually,
this representation is equivalent to the scalar representation discussed above, as
we shall see shortly. (Let us remark that more complicated representations can
be defined on n-component functions, where the components are mixed by the
transformation in a specific way.)
It is straightforward to work out the generators in this representation (and
we shall need them later on). The changes of the space-time position x µ and
the function ϕ(x) by the infinitesimal Poincaré transformation are δx µ and
δx µ ∂ µ ϕ(x) respectively, from where one can directly read out the Poincaré
generators in this representation
J i ϕ(x) = (J i x) µ ∂ µ ϕ(x)
K i ϕ(x) = (K i x) µ ∂ µ ϕ(x)
P µ ϕ(x) = i∂ µ ϕ(x)
Using the explicit knowledge of the generators J i and K i one obtains
J i ϕ(x) = i 2 ε (
ijk δ
µ
j η kν −δ µ k η jν)
x ν ∂ µ ϕ(x) = −iε ijk x j ∂ k ϕ(x)
K i ϕ(x) = i(δ µ i η 0ν −δ µ 0 η iν)x ν ∂ µ ϕ(x) = ix 0 ∂ i ϕ(x)−ix i ∂ 0 ϕ(x)
or even more briefly ⃗ J ϕ(x) = −i⃗x×∇ϕ(x) and ⃗ Kϕ(x) = it∇ϕ(x)−i⃗x ˙ϕ(x).
Atthispointthereadermaybetempted tointerpretϕ(x) asawave-function
of the ordinary quantum mechanics. There is, however, an important difference
between what have now and the standard quantum mechanics. In the usual formulation
ofthe quantum mechanicsin termsofwave-functions, the Hamiltonian
is specified as a differential operator (with space rather than space-time derivatives)
acting on the wave-function ϕ(x). Our representation of the Poincaré
algebra did not provide any such Hamiltonian, it just states that the Hamiltonian
is the generator of the time translations.
However, if one is really keen to interpret ϕ(x) as the wave-function, one
is allowed to do so 44 . Then one may try to specify the Hamiltonian for this
irreducible representation by demanding p 2 = m 2 for any eigenfunction e −ipx .
44 In that case, it is more conveniet to define the transformation as push-forward, i.e. by
the mapping ϕ(x) → ϕ ( Λ −1 (x−a) ) . With this definition one obtains P µ = −i∂ µ with the
exponentials e −ipx being the eigenstates of these generators. These eigenstates could/should
play the role of the |p〉 states. And, indeed,
e −ipx → e −ipΛ−1 (x−a) = e −iΛ−1 Λp.Λ −1 (x−a) = e i(Λp)(x−a) = e −ia(Λp) e ix(Λp)
which is exactly the transformation of the |p〉 state.