Abstract
We calculate the vacuum (Casimir) energy for a scalar field with ϕ4 self-interaction in (1 + 1) dimensions non perturbatively, i.e., in all orders of the self-interaction. We consider massive and massless fields in a finite box with Dirichlet boundary conditions and on the whole axis as well. For strong coupling, the vacuum energy is negative indicating some instability.
Highlights
The notion of vacuum energy results from the ground state of a quantum system, particle, or field. This is the state with no excitations
The first detection of zero-point energy this way is known from early quantum mechanics as a difference in vapor pressure between certain neon isotopes
We demonstrate a simple example where we have such a parameter in empty space
Summary
The notion of vacuum energy results from the ground state of a quantum system, particle, or field. The first detection of zero-point energy this way is known from early quantum mechanics as a difference in vapor pressure between certain neon isotopes (see the contribution by Rechenberg in [2]) This feature slipped over to quantum field theory. Continuing, it turns out that the limit L → ∞ can be performed in a meaningful way delivering the vacuum energy of a field with self-interaction in flat, empty space. For decades this was a serious problem For example, it took 20 years to generalize the original calculation of Casimir [5], which was for flat, parallel plates, to a sphere [6], and nearly 30 more years until the first calculation of Universe 2021, 7, 55 vacuum energy in a three-dimensional background field [7].
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