From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 81

From the definition of the domain ofPand passing to the Fourier-Plancherel
transform, one finds again (it is not so easy to see it)

〈Pψ,φ〉=〈ψ,Pφ〉 for allψ,φ∈D(P) , (2.18)

so that, eigenvalues are real if exist. HoweverPdoes not admit eigenvectors. The
naive eigenvectors with eigenvaluep∈Rare functions proportional to the map
Rx→eipx/, which does not belong toL^2 (R,dx)norD(P). We will tackle all
these issues in the next section in a very general fashion.
We observe that the space ofSchwartz functions,S(R)^3 satisfies
S(R)⊂D(X)∩D(P)

and furthermore S(R)isdenseinL^2 (R,dx)andinvariant under X andP:
X(S(R))⊂S(R)andP(S(R))⊂S(R).

Remark 2.1.13. Though we shall not pursue this approach within these notes,
we stress thatXadmits a set of eigenvectors if we extend the domain ofXto the
spaceS′(R)ofSchwartz distributionsin a standard way: IfT∈S′(R),

(X(T),f):=(T,X(f)) for everyf∈S(R).

With this extension, the eigenvectors inS′(R)ofXwith eigenvaluesx 0 ∈Rare
the distributionscδ(x−x 0 )(see the first part of this book) This class of eigenvectors
can be exploited to build a spectral decomposition ofXsimilar to that in (2.4).
Similarly, P admits eigenvectors inS′(R)with the same procedure. They are
just the above exponential functions. Again, this class of eigenvectors can be used
to construct a spectral decomposition of P like the one in (2.4). The idea of
this procedure can be traced back to Dirac [ 3 ]and, in fact, something like ten
years later it gave rise to the rigoroustheory of distributionsby L. Schwartz.
The modern formulation of this approach to construct spectral decompositions of
selfadjoint operators was developed by Gelfand in terms of the so calledrigged
Hilbert spaces[ 4 ].

Referring to a quantum particle moving inRn, whose Hilbert space isL^2 (Rn,dxn),
one can introduce observablesXkandPkrepresenting position and momentum
with respect to thek-th axis,k=1, 2 ,...,n. These operators, which are defined
analogously to the casen= 1, have domains smaller than the full Hilbert space.
We do not write the form of these domain (where the operators turn out to be
properly selfadjoint referring to the general definition we shall state in the next

(^3) S(Rn) is the vector space of theC∞complex valued functions onRnwhich, together with
their derivatives of all orders in every set of coordinate, decay faster than every negative integer
power of|x|for|x|→+∞.