English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In a standard quantum field theory the norm $\langle \Omega\vert\Omega\rangle$ of the vacuum state is taken to be finite. In this paper weprovide a procedure, based on constructing an equivalent wave mechanics, fordetermining whether or not $\langle \Omega\vert \Omega\rangle$ actually isfinite. We provide an example based on a second-order plus fourth-order scalarfield theory, a prototype for quantum gravity, in which it is not. In thisexample the Minkowski path integral with a real measure diverges though theEuclidean path integral does not. Thus in this example contributions from theWick rotation contour cannot be ignored. Since $\langle \Omega\vert\Omega\rangle$ is not finite, use of the standard Feynman rules is not valid.And while these rules not only lead to states with negative norm, they in factlead to states with infinite negative norm. However, if the fields in thattheory are continued into the complex plane, we show that then there is adomain in the complex plane known as a Stokes wedge in which one can define anappropriate time-independent, positive and finite inner product, viz. the$\langle L\vert R\rangle$ overlap of left-eigenstates and right-eigenstates ofthe Hamiltonian; with the vacuum state then being normalizable, and with therebeing no states with negative or infinite $\langle L\vert R\rangle$ norm. Inthis Stokes wedge it is the Euclidean path integral that diverges while theMinkowski path integral does not. The concerns that we raise in this paper onlyapply to bosons since the matrices associated with their creation andannihilation operators are infinite dimensional. Since the ones associated withfermions are finite dimensional, the fermion theory vacuum is automaticallynormalizable. Our results are relevant to the construction of a consistent,unitary and renormalizable quantum theory of gravity.

Determining the normalization of the quantum field theory vacuum, with implications for quantum gravity