The issue with negative frequency is not hard to understand. First off if I were to build a band-pass filter if this were to select for negative frequencies it would mean I need an inductor with a negative inductance. This frankly does not make sense. How one construct a device to measure or that would admit a negative frequency? Of course one can construct any type of mathematics with a negative frequency, but that is not the same as physics.
It is the case of quantum mechanics a negative frequency does correspond to negative energy. The energy of a photon E = ħω has sign dependency with the angular frequency ω = 2πν. Suppose we have a particle with mass 0 > m. Suppose this particle decays into photons, then the photons clearly have negative frequency
mc^2 --- > 2ħω,
and of course if we have negative frequency photons we can have the converse process. As a result we may generate a negative mass particle.
Negative mass particles are quirky. For one thing Newton's law of gravity F = ma = GMm/r^2 indicates that if we have M positive and M negative with
Ma = -GMm/r^2
ma = GMm/r^2
that the two masses are accelerated in the same direction. In fact if the two mass have the same magnitude the two race off indefinitely to "infinity."
This is related to the issue of tachyons. What is the problem with tachyons? The relativistic momentum-energy interval is
m^2 = E^2 - p^2
where I have set c = 1. The magnitude of the momentum is greater than the energy and so m^2 is less than zero. The energy E = sqrt{p^2 m^2}defines the velocity
v = ∂E/∂p = p/E
which since p > E is greater than unity. This has the meaning that the particle moves faster than light. The interval defines the Klein-Gordon equation ∂_t^2 - ∂_i^2 = □ so that □ψ = m^2ψ. The term m^2ψ comes from the potential V(ψ) = (1/2)m^2ψ^2, and the gradient of the potential ∂V(ψ)/∂ψ = m^2ψ is less than zero. Hence the vacuum state is not stable.
The relationship between the negative mass particle and the tachyon is seen if we consider the extreme boosted situation with p^2 = p_x^2 p_y^2 p_z^2 and p_z much larger than the other components. We may then write the energy as
E =sqrt{p_x^2 p_y^2 p_z^2 m^2 }
~= p_z sqrt(1 (p_x^2 p_y^2)/p_z^2 m^2/p_z^2}
and binomial theorem gives
E =~ p_z (p_x^2 p_y^2)/2p_z m^2/2p_z)
And p_z plays the role of a Lorentz factor. This is a very classical appearing equation and the last term is interpreted as a mass μ = m^2/p_z that is negative. So the negative mass physics is related to the tachyon physics in this extreme boosted situation.
So there are a number of strikes against this idea of negative mass and negative frequency. Ashoke Sen has found that tachyons can form condensates in D-brane physics. These are special cases, such as the M2-brane in a black hole interior, where the tachyons are not free to propagate.
Cheers LC