Narrow Results

We consider the Klein–Gordon equation on a static spacetime and minimally coupled to a static electromagnetic potential. We show that it is essentially self-adjoint on $C_{\mathrm{\text{c}}}^{\infty }$. We discuss various distinguished inverses and bisolutions of the Klein–Gordon operator, focusing on the so-called Feynman propagator. We show that the Feynman propagator can be considered the boundary value of the resolvent of the Klein–Gordon operator, in the spirit of the limiting absorption principle known from the theory of Schrödinger operators. We also show that the Feynman propagator is the limit of the inverse of the Wick rotated Klein–Gordon operator.

For the interacting Feynman propagator ${\mathrm{\Delta }}_{F,\mathrm{\text{int}}}(x,y)$ of scalar electrodynamics, we show that the sign property, $\mathrm{\text{Re}}{i\mathrm{\Delta }}_{F,\mathrm{\text{int}}}\ge 0$, may hinge on the reversibility of time evolution. In contrast, $\mathrm{\text{Im}}{i\mathrm{\Delta }}_{F,\mathrm{\text{int}}}$ is indeterminate. When we switch to reduced dynamics under the weak coupling approximation, the positive semidefinite sign of $\mathrm{\text{Re}}{i\mathrm{\Delta }}_{F,\mathrm{\text{int}}}$ is generally lost, unless we impose severe restrictions on the Kraus operators that govern time evolution. With another approximation, the rotating wave approximation, we may recover the sign by restricting the test functions to exponentials under certain conditions.

By using the magnetic entangled state 〈λ| representation and the squeezing transform we derive Feynman propagator for a charged particle in a quantum dot in the presence of a uniform magnetic field. It is the neat expression of the squeezing operator in 〈λ| representation that provides the concise way and the brevity. The partition function of this sytem is thus obtained.

At mesoscopic scales close to, but somewhat larger than, Planck length, one could describe quantum spacetime and matter in terms of a quantum-corrected geometry. The key feature of such a description is the introduction of a zero-point length into the spacetime. When we proceed from quantum geometry to quantum matter, the zero-point length will introduce corrections in the propagator of matter field in a specific manner. On the other hand, one cannot ignore the self-gravity of matter fields at the mesoscopic scales and this will also modify the form of the propagator. Consistency demands that these two modifications coming from two different directions are the same. I show that this nontrivial demand is actually satisfied. Surprisingly, the principle of equivalence, operating at Planck scales, ensures this consistency in a subtle manner.