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Why photons (those beautiful quanta of energy) have mass

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I have always been interested in physics especially the quantum kind. I was recently reading Hidden in Plain Sight: The simple link between relativity and quantum mechanics and a simple string stuck out at me, "mass less photons"(I will insert exact location here). Unfortunately those words where a sticking point in the book for me. Here is my proof that photons have mass(higgs bosom).

1. It must have very little mass
2. Particle wave duality
3. At the macro level light waves are commonly compared to sound waves
4. At the micro level light behaves like a particle
5. Solar sails
6. (Newtonian physics) Mass x Energy = Force
7. Solar sail rely on light (photons)
8. if mass = 0 then (0 x c) = force which is 0
9. solar sails require huge surface area
10. Photons and normal particles are both affected by gravity

Sorry Andrew Thomas... Photons have mass

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