The Vacuum as a Lagrangian subspace

We unify and generalize the notions of vacuum and amplitude in linear quantum field theory in curved spacetime. Crucially, the generalized notion admits a localization in spacetime regions and on hypersurfaces. The underlying concept is that of a Lagrangian subspace of the space of complexified germs of solutions of the equations of motion on hypersurfaces. Traditional vacua and traditional amplitudes correspond to the special cases of definite and real Lagrangian subspaces, respectively. Further, we introduce both infinitesimal and asymptotic methods for vacuum selection that involve a localized version of Wick rotation. We provide examples from Klein-Gordon theory in settings involving different types of regions and hypersurfaces to showcase generalized vacua and the application of the proposed vacuum selection methods. A recurrent theme is the occurrence of mixed vacua, where propagating solutions yield definite Lagrangian subspaces and evanescent solutions yield real Lagrangian subspaces. The examples cover Minkowski space, Rindler space, Euclidean space, and de Sitter space. A simple formula allows for the calculation of expectation values for observables in the generalized vacua.

Figure 1

The region $M$ determined by the time interval $[t_1,t_2]$ in Minkowski space. A source $\mu$ is located within the region. The solution $\eta$ of the corresponding inhomogeneous equations of motion is a negative energy solution before $t_1$ and a positive energy solution after $t_2$.

Figure 2

The region $M$ is bounded by the hypercylinder of radius $R$. The interior solutions of Klein-Gordon theory are given in terms of spherical Bessel functions $j_{\ell }$, the vacuum on the exterior in terms of propagating $h_{\ell }$. For the massive case in the interior there are additionally the functions $a_{\ell }$ and in the exterior the evanescent $k_{\ell }$.

Figure 3

(a) Solutions in a time-interval region $M$ yield a real Lagrangian subspace $L_M\subseteq L_{\partial M}$. (b) The future vacuum state ${\psi }_0$ on a spacelike hypersurface $\mathrm{\Sigma }$ is encoded through a definite Lagrangian subspace ${\tilde{L}}_X\subseteq L_{\partial X}^{\mathbb{C}}=L_{\mathrm{\Sigma }}^{\mathbb{C}}$. (c) The corresponding past vacuum.

Figure 4

Generic setting with a compact region $M$ and noncompact exterior $X$. In $M$ we have an amplitude map while in $X$ we have a vacuum.

Figure 5

Rindler space corresponds to the right wedge of Minkowski space; it is bounded by the half-lines $\eta =\pm \infty$. In this spacetime we consider the region $X_s=[0,s]\times {\mathbb{R}}^{+}$.

Figure 6

Spacetime region bounded by two hyperbola of constant Rindler spatial coordinates at ${\rho }_1$ and ${\rho }_2$ with ${\rho }_2>{\rho }_1$. The boundary of Rindler corresponds to $\rho =0$.

Figure 7

Annulus region bounded by two circles of radii $R$ and $\widehat{R}$ in Euclidean space.