About Casimir effect I don't know the exact math. treatment. But, about the Lamb shift, as far as I understand, there is no other explanation than virtual particles.
In the article I recommended, the affect that one single photon excites two atoms, also has no other explanation than the virtual particles.
Thank you Sofia. I also read only math.explanations with virtual particles. Surely they can not be observed directly. But you can not observe directly even the air.
You can observe directly the air. Here is a simple experiment in physics: take a candle, place it on the water in a vessel, and light it. Then take an empty jar of glass and place it over the candle with the mouth down so that the mouth of the jar touch the water.
When the candle dies out, the water level rises inside the jar.