Master Dissertation

5.2 Example - In the Hilbert Space Setting of
Quantum Mechanics
Before going on we illustrate the significance of the table by applying it to
the more transparent theories of Hilbert spaces and quantum mechanics.
In quantum mechanics we substitute distributions by functions and replace
x by t.
Now, assuming T1(t) is given we want to construct T2(t). We follow the
rules given from the box on the previous page and construct the advanced
and retarded functions.
and
A ′ 2(t1, t2) = ˜ T1(t1)T1(t2) = −T1(t1)T1(t2) (5.13)
R ′ 2(t1, t2) = T1(t2) ˜ T1(t1) = −T1(t2)T1(t1), (5.14)
where the last inequalities follow from (5.1). Then
and
A2(t1, t2) = A ′ 2(t1, t2) + T2(t1, t2) (5.15)
R2(t1, t2) = R ′ 2(t1, t2) + T2(t2, t1)
From Theorem 5.1 we know that if ti > tj, then
T2(ti, tj) = T1(t1)T1(tj).
Hence A2 vanishes for t1 > t2 and R2 vanishes for t1 < t2.
Now
D2 = R2 − A2 = R ′ 2 − A ′ 2
is known from equations (5.13) and (5.14). The T2 cancel out since they
are symmetric.
Finally, R2 and A2 can be determined by their support properties. Let
Θ(t) be the Heaviside function
Then
Θ(x) =
1, t ≥ 0,
0, t < 0.
A2(t1, t2) = Θ(t2 − t1)D2(t1, t2)
= Θ(t2 − t1)(T1(t1)T1(t2) − T1(t2)T1(t1)).
Hence A2 is uniquely determined up to its value at t1 = t2.
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