Entanglement of the vacuum between left, right, future, and past: Dirac spinor in Rindler and Kasner spaces

We study the relations of the positive frequency mode functions of Dirac field in 4-dimensional Minkowski spacetime covered with Rindler and Kasner coordinates, and describe the explicit form of the Minkowski vacuum state with the quantum states in Kasner and Rindler regions, and analytically continue the solutions. As a result, we obtain the correspondence of the positive frequency mode functions in Kasner region and Rindler region in a unified manner which derives vacuum entanglement.

Figure 1

Minkowski spacetime and the coordinates for the R region, the L region, the F region, and the P region.

Figure 2

Relations of the mode functions in each region. This figure shows the equivalence of ${\mathrm{\Psi }}_{\omega ,{\mathit{k}}_{\perp }}^{\text{I,n(F)}}$ (left-moving wave modes in the F region), ${\mathrm{\Psi }}_{\omega ,{\mathit{k}}_{\perp }}^{\text{I,n(R/L)}}$ (right Rindler modes), and ${\mathrm{\Psi }}_{\omega ,{\mathit{k}}_{\perp }}^{\text{I,n(P)}}$ (right-moving wave modes in the P region). Similarly, ${\mathrm{\Psi }}_{\omega ,-{\mathit{k}}_{\perp }}^{\text{II,n(F)}}$ (right-moving wave modes in the F region), ${\mathrm{\Psi }}_{\omega ,-{\mathit{k}}_{\perp }}^{\text{II,n(R/L)}}$ (left Rindler modes), ${\mathrm{\Psi }}_{\omega ,-{\mathit{k}}_{\perp }}^{\text{II,n(P)}}$ (left-moving wave modes in the P region) are equivalent.

Figure 3

Local Lorentz frame in each region which indicates the deviation of direction.