Can a mathematical statement/equation describe the following?

 

If a thing displays at least two verified characteristics which are impossible to reconcile mathematically, then it does indeed have each of those characteristics -- simultaneously.

 

The reason I ask concerns the first time I heard about light's wave-particle duality in my studies. I had immediately resisted the idea of a contradiction, that light just had to be either a wave or a particle. For there was no dilemma, because a photon could travel as a particle...yet in a wave-like motion.

 

See the images for an example of this. (It's difficult to imagine I'd be the first person to offer this solution. But even so, I'm really interested in a general mathematical statement for dualities to coexist/reconcile)

 

Above: photon traveling like a wave, but clearly a particle. Below: it leaves a trail behind (to help see the wave )

Put a double slit in the way, it doesn't travel through both slits...

 

We've discussed this before, IIRC there's a very nice argument as to why this cannot be which is better than my thought experiment above...

Ooo, I'd like to read that discussion/argument.

The problem with this idea is that a photon is not going to follow a sinusoidal path in the absence of electromagentic or gravitational forces strong enough to cause it to change direction. Even though you could make a device to create a field like that, this is obviously not the way that light propagates in free space. In the absence of an external force, a photon (like any other particle) will move in a straight line.

 

It may be easier if you think of "waves" and "particles" as both being just approximations or models for the underlying actuality, and that the photon itself is actually neither.