I have more or less come to terms with the use of negative frequencies in Fourier transform in exponential form, but then I have a problem with some opinions stating that negative frequencies move backward in time.
My understanding is: if what you analyze is an oscillation, the equivalent of the spatial dimensions is either time points or frequencies, depending on the angle of the basis or reference system that you choose (sequential or cyclical, respectively); if you choose the cyclical basis, then you get a complex magnitude for the signal in each dimension (radius and phase); your basis vectors can be sinusoids (but then these do not capture phase; if you want phase, you need to use two sinusoids per dimension, separated by 90 degrees, i.e. sine and cosine, whose combination in a complex vector renders the phase) or complex exponentials (which do capture phase, but then, if you wish to capture real signals without phase, you need negative and positive exponentials, moving clockwise and anticlockwise respectively, so that their combination annuls the phase).
So, if this understanding is right (please correct me to the extent that it isn’t), the key for using negative frequencies is that they are necessary to capture all aspects of the object in question through the exponential form.
Now some people say that negative frequencies, traveling clockwise, “move back in time”. But then if they do, an exponential of a given frequency that is positive and moving anticlockwise forward in time will be visiting the same places and at the same times as a negative exponential moving clockwise and backward in time. Hence one of them would not be of much added value as a way of capturing the signal...
So could we conclude that it is wrong to say that negative frequencies travel backward in time?