Selecting the complex structure for quantization in the quantum case

Determine a principled prescription for choosing the complex structure that underlies the quantization of field theories, such as the Klein–Gordon theory on curved spacetimes, in the genuinely quantum case beyond static or stationary scenarios, providing criteria that ensure mathematical consistency and physical viability.

Background

The complex structure determines the Kähler geometry used for quantization and fixes the Gaussian measure defining the quantum state space. In the classical setting, the authors adopt a dynamical prescription derived from the Hamiltonian vector field.

For the quantum theory, the authors note that while various choices exist (and coincide in special cases such as static or stationary spacetimes), a general, widely accepted criterion for selecting the complex structure remains unresolved.

References

As a matter of fact, how to choose this prescription in the quantum case is still an open problem [MO21].

— Geometric Flavours of Quantum Field Theory on a Cauchy Hypersurface: Gaussian Analysis for the Hamiltonian Formalism and Applications to Cosmology  (2502.12760 - Martínez-Crespo, 18 Feb 2025) in Chapter 7, Section 7.1

However, we still can say nothing about the form or properties of the modes even on the horizon. Indeed, even though we can find some solutions with necessary properties in the vicinity of the horizon (like solution Umode2), it does not mean that these solutions are indeed represent actual modes possessing the necessary orthogonality conditions (the form of the latter is still unknown for Eq.~KSqescf).

Umode2:

ϕ^l(T,X,ω)∣T+X→0=1rs4πω e−iω(T−X)(1+T2−X24ers2−il(l+1)+14ωers2(T+X))\hat\phi_{l}(T,X,\omega)|_{T+X\to 0}=\frac{1}{r_{s}\sqrt{4\pi\omega}}\,e^{-i\omega(T-X)}\left(1+\frac{T^{2}-X^{2}}{4er_{s}^{2}}-i\frac{l(l+1)+1}{4\omega er_{s}^{2}}(T+X)\right)

KSqescf:

∂2ϕl∂T2−∂2ϕl∂X2+2r(∂r∂T∂ϕl∂T−∂r∂X∂ϕl∂X)+l(l+1)rse−rrsr3ϕl=0,\frac{\partial^{2}\phi_{l}}{\partial T^{2}}-\frac{\partial^{2}\phi_{l}}{\partial X^{2}} +\frac{2}{r}\left(\frac{\partial r}{\partial T}\frac{\partial\phi_{l}}{\partial T}- \frac{\partial r}{\partial X}\frac{\partial\phi_{l}}{\partial X}\right)+\frac{l(l+1)r_{s}e^{-\frac{r}{r_{s}}}}{r^{3}}\phi_{l}=0,

— Remarks on quantum scalar field theory for a uniformly accelerated reference frame and a Schwarzschild black hole  (2608.23108 - Smolyakov, 24 Aug 2026) in Section 5.3, “Is there a consistent quantum scalar field theory in Kruskal-Szekeres spacetime?”; Conclusion

It means that even if there exists a consistent quantum field theory in Kruskal-Szekeres spacetime for a free scalar field satisfying the necessary canonical commutation relations, at the moment it is not clear how to define particles in such a theory or to build the corresponding Fock space. This problem still deserves a detailed analysis.

— Remarks on quantum scalar field theory for a uniformly accelerated reference frame and a Schwarzschild black hole  (2608.23108 - Smolyakov, 24 Aug 2026) in Section 5.3, “Is there a consistent quantum scalar field theory in Kruskal-Szekeres spacetime?”; Conclusion