Transverse waves vibrate at right angles to their movement, so if light vibrated in a physical direction, it couldn’t move that way. Light moves in every direction and space is isotropic (the same in every direction), so light can’t vibrate in space. Once space lets light move in three dimensions, there are no free directions for a transverse wave to vibrate into. Light is then a transverse wave that moves in space but doesn’t vibrate within it, yet complex numbers can describe it, so why is that?

Consider Abbot’s story of Flatlanders who lived on a two-dimensional surface that was all they knew (Abbott, 1884), so they could see a circle but had to imagine a sphere. Now suppose a point entity moved in their world that set values in a transverse circle at right angles to the surface of their space (Figure 3.8a). They couldn’t see that rotation but could explain it as a vibration in an imaginary plane just as we explain light with complex numbers. As the point moves, it would have a polarization plane in their space just light has in ours (Figure 3.8b). If they called this moving point a photon, it would be a sine wave with a frequency and an imaginary amplitude just as a photon is for us (Figure 3.8c). We then describe light in our space just as Flatlanders might describe a wave that vibrates outside their space, so are we in effect three-dimensional Flatlanders?

If our space is a surface, light could move on it as waves do on a lake but in three dimensions not two. Complex numbers then explain electro-magnetism because light really does vibrate outside space, so:
“In quantum mechanics there really are complex numbers, and the wave function really is a complex-valued function of space-time.” (Lederman & Hill, 2004), p346.
Complex numbers describe a rotation that occurs outside our space (Note 1) (Figure 3.9). We call this rotation imaginary because our space doesn’t allow it, but Flatlanders would say the same of a vibration in a third dimension that we know is possible. There is then no logical reason why our space can’t be a surface in a higher dimensional reality, so light can vibrate into another dimension as the equations say.
In this model, our space is a surface inside a four-dimensional quantum network, so light can vibrate transversely. It can’t leave that surface, just as water waves can’t leave the surface of a pond, so we can’t see anything outside our space. We are three-dimensional Flatlanders who call vibrations transverse to our space, like light, imaginary, but what vibrates when the electro-magnetic field does?
Note 1. Complex number theory describes a rotation into an imaginary plane. In normal multiplication, multiplying a number by two doubles it, e.g. 5 x 2 = 10. Multiplying by 4 adds it four times, e.g. 5 x 4 = 20. In complex multiplication, i is a 90° rotation into an imaginary plane, so times 2i is a 180° rotation that turns a complex number into its negative, e.g. 5 x 2i = -5. Times 4i is a 360° rotation that has no effect, so 5 x 4i = 5.






