and momentum
α(p) =δ(p−p′)
eigenstates are not inH 1 =L^2 (R). In addition, as in the single-particle
case, there are domain issues to consider, since differentiating bypor
multiplying bypcan take something inL^2 (R) to something not inL^2 (R).- H 1 =S(R)
This choice, takingαto be in the well-behaved space of Schwartz functions,
avoids the domain issues ofL^2 (R) and the Hermitian inner product is
well-defined. It however shares the problem withL^2 (R) of not including
position or momentum eigenstates. In addition, the inner product no
longer provides an isomorphism ofH 1 with its dual. - H 1 =S′(R)
This choice, allowingαto be distributional solutions, will solve domain
issues, and includes position and momentum eigenstates. It however intro-
duces a serious problem: the Hermitian inner product on functions does
not extend to distributions. With this choiceH 1 is not an inner product
space and neither are its symmetric tensor products.
To get a rigorous mathematical formalism, for some purposes it is possible
to adopt the first choice,H 1 =L^2 (R). With this choice the symmetric tensor
product version of the Fock space can be given a Hilbert space structure, with
operatorsa†(α) anda(α) defined by equations 36.12 and 36.13 satisfying the
Heisenberg commutation relations of equation 36.14. We will however want to
consider operators quadratic in thea(α) anda†(α), and for these to be well
defined we may need to useH 1 =S(R).
If we ignore the problem with the inner product, and takeH 1 =S′(R), then
in particular we can takeαto be a delta-function, and when doing this will use
the notation
a(p) =a(δ(p′−p)), a†(p) =a†(δ(p′−p))
and write
a(α) =∫
α(p)a(p)dp, a†(α) =∫
α(p)a†(p)dp (36.15)The choice of the conjugations here reflects that fact thata†(α) is complex linear
inα,a(α) complex antilinear.
While the operatora(p) may be well-defined, the problem with the operator
a†(p) is clear: it takes in particular the vacuum state| 0 〉to the non-normalizable
state|p〉. We will, like most other authors, often write equations in terms of op-
eratorsa(p) anda†(p), acting as ifH 1 =S′(R). For a legitimate interpretation
though, such equations will always require an interpretation either
- using cutoffs which make the values ofpdiscrete and of finite number,pj
labeled by an indexjwith a finite number of values, as in section 36.2. In
this case thea(p),a†(p) are thea(pj),a†(pj) of that section.