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S14d. Why use complex numbers in physics?
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Complex numbers are _the_ natural number system for all but
elementary physics; one needs them to make sense of many advanced
concepts. Avoiding complex numbers would make much
of what is done incomprehensible.
Already Fourier analysis is most natural with complex numbers,
though here it could be avoided by using trigonometric series
instead.
The time-independent Schroedinger equation defines the
Fourier components of real, measurable expectations. So it is
very natural that quantum mechanics is based on complex entities, too.
Dispersion relations in optics are natural only in a complex setting.
Spectra of nonhermitian operators, essential for dissipative systems
even in the classical case, are always complex.
Analytic continuation plays a significant role in some physical
theories. For example, lattice gauge theory works in a continuation
of quantum field theory to Euclidean space, and the results must be
continued back to Minkowski space to get physical meaning.
On the other hand, at first sight it seems that only real quantities
are measurable. However this only holds for the most direct measurements
where you read a number from a meter. Most measurements are of a
more indirect kind, and then this restriction no longer applies.
To measure a family of physical quantities x_l (l=1,...,n),
one measures some related real quantities r_1,...,r_m connected to
the x_l by a system of equations F(x,r)=0 (in the absence of
measurement errors). In fact, there will always be measurement errors,
hence one generally uses more equations than unknowns and solves
the least squares problem ||F(x,r)||^2=min (or a more complicated
related problem if a model of measurement errors is avaialble)
to get an estimate of x.
This recipe is universally used for all sorts of measurements and
works whether the x_l are real or complex.