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math-physicist

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I am reading Peskin-Schroeder's QFT text, and there on pg. 98 Equation (4.56) they derive the expression for the vacuum energy density (relative to the zero of energy set by ##H_0|0\rangle = 0##):

$$ \frac{E_0}{\rm{Volume}} = \frac{i\,\sum\text{(all disconnected pieces)}}{(2\pi)^4\,\delta^4(0)}\ . $$

My questions are:

$$ \frac{E_0}{\rm{Volume}} = \frac{i\,\sum\text{(all disconnected pieces)}}{(2\pi)^4\,\delta^4(0)}\ . $$

My questions are:

- What do they mean by "relative to the zero of energy set by ##H_0|0\rangle = 0##" ?
- The right side of the above expression has a delta function evaluated at 0. Doesn't that imply that the right side is essentially negligible/zero? Then how does one interpret this equation? The vacuum has zero energy?
- How does one derive Equation (4.56) starting from Equation (4.55) which states:

$$ e^{\sum V_i}\propto e^{-2iE_0T}\ ? $$

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