Probability distribution of the vacuum energy density

As the vacuum state of a quantum field is not an eigenstate of the Hamiltonian density, the vacuum energy density can be represented as a random variable. We present an analytical calculation of the probability distribution of the vacuum energy density for real and complex massless scalar fields in Minkowski space. The obtained probability distributions are broad and the vacuum expectation value of the Hamiltonian density is not fully representative of the vacuum energy density.

Figure 1

The probability density functions of the vacuum energy density for real (solid) and complex (dashed) massless scalar fields. Variable $x$ is equal to the ratio $ϵ/\overline{ϵ}$, where $ϵ$ is the energy density and $\overline{ϵ}$ is the expectation value which is different for the real and complex cases. Mean values for both curves are $\overline{x}=1$ and the standard deviation for the real case is $\sigma =\sqrt{2/3}=0.816$ and for the complex case is $\sigma =\sqrt{1/3}=0.577$. For the real field maximal probability is at $x=0.419$ and for the complex field at $x=0.664$.

Figure 2

The probability density functions of the normally ordered vacuum energy density for real (solid) and complex (dashed) massless scalar fields. Variable $x$ is equal to the ratio $ϵ/\kappa$ for the real and $ϵ/2\kappa$ for the complex field. Parameter $ϵ$ is the energy density and $\kappa ={\Lambda }^4/(16{\pi }^2)$ in three-dimensional cutoff regularization, where the momentum cutoff is $\Lambda$.