No common complex domain/space

This question of mine was called overly vague and speculative by
Igor Khavkine <ikhavki@xxxxxx> Canada
Obviously, he understood some consequences although I wrote 'common
complex domain/space?'.

Most frequently, time domain is considered the original domain from
which Fourier transform leads into a complex frequency domain.
Alternatively it is also possible to perform FT of an original function
of frequency into a complex time domain. Original functions of reality
are always real and unilateral. Each of the two complex counterparts
exhibits Hermitean symmetry. In order to benefit from complex calculus,
one may choose either complex frequency domain or complex time domain.
While the original quantity is physically correct, the conjugate one in
complex domain must be unphysical in that, it is apparently symmetrical
and complex. This avoidable dilemma due to complexity has often been
overlooked. Of course, the uncertainty relation between the two
conjugate variables of a Fourier transform pair attracts more attention
since there is no remedy for it.

My related question refers to operators on a Hilbert space. They may be
complex in general. Recall that rewriting f(x) like a complex function
F(ix) is just a shorthand for the decision to replace e.g. 2 cos(x)dx
just by exp(ix)dx and omit exp(-ix)dx. Getting rid of the imaginary part
is one of two essential steps back to reality. 80 years ago, complex
functions of negative and positive frequency were considered unphysical
and therefore avoided which might have led to the assumption of a real
Hamiltonian, too. Who did deal with the question whether a particular
observable quantity belongs to either a real or a complex Hermitean
operator in Hilbert space? I merely found that Weyl worried about
possibly illusory symmetries.

Eckard Blumschein

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