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I've been asked to work out a problem about vacuum energy <0|H|0> where H is the energy density of harmonic oscillator. When I integrate this expectation value over space of finite dimensions L, I get that the expectation value for the vacuum energy scales as L^3 .

Does anyone know why, as I increase the box, we get higher vacuum energy. Also, why does vacuum state have energy at all? I know particles and antiparticles can be created and destroyed but shouldn't the overall energy of the total vacuum at any time be zero?

Thanks for any help on understanding this explanation!

imanbk