Euclideanization without Complexification of the Spacetime

Abstract: Minkowski spacetime can be mapped by a series of projections in a higher-dimensional spacetime to a Euclidean space, constituting a process of Euclideanization shown here in detail for two dimensions. The result allows regularizations and computations of integrals that appear in quantum field theory (QFT) without performing the standard Wick rotation of time to imaginary values. However, there is no physical spacetime transformation that produces a Wick rotation. In avoiding this complexification process, the new Euclidenization procedure has important advantages in the transformations of the action principles, including fermionic fields and theories at constant chemical potential. In all cases, complex-valued amplitudes of the form $\exp(iS/\hbar)$ are mapped to real statistical weights $\exp(-S_{\rm E}/\hbar)$ with a Euclidean action $S_{\rm E}$. The procedure is also amenable to fields on curved background spacetimes as well as gravitational interactions.

An Overview of Euclideanization Without Complexification of Spacetime

The paper "Euclideanization without Complexification of the Spacetime" introduces a geometrical approach to transform Minkowski spacetime into Euclidean space without employing the traditional complexification of time. The authors propose an alternative method that avoids complex numbers and instead utilizes a series of projections in higher-dimensional auxiliary spacetimes. This method presents notable advantages, especially in contexts where traditional Wick rotations face limitations, such as field theories on non-static curved spacetimes and theories involving gravitational interactions.

Key Concepts and Methodology

Fundamentally, the authors challenge the conventional necessity of implementing a Wick rotation in quantum field theory (QFT) computations, which traditionally transforms time to imaginary values to reconcile Minkowskian and Euclidean descriptions. This standard practice, although effective, lacks a direct physical spacetime interpretation and introduces several technical disadvantages, particularly in complex extensions.

The paper develops a geometry-based procedure by embedding the original 2-dimensional Minkowski plane in an auxiliary 3-dimensional Minkowski space, utilizing properties such as hyperbolae and cones. In this auxiliary space, a central cone is defined, intersected by half-cones over spacelike and timelike events. A sequence of projections then maps points in the original spacetime plane onto these cones and finally back to a plane that inherits Euclidean geometry, without introducing complex numbers for time.

Numerical Results and Implications

One significant result of this transformation is converting the complex exponentiated actions in path integrals into real statistical weights, thus maintaining a real-valued Euclidean action $S_E$ from the complex Minkowskian action $S$. Through detailed steps demonstrated within the paper, the method ensures that regularized integrals appearing in fields such as QFT can be computed without invoking the imaginary time prescription, all while producing results consistent with those attained via Wick rotation.

The applicability to fermionic theories, including those at constant chemical potentials such as lattice QCD, is also highlighted. Although non-unique aspects remain due to ambiguities in how spinors transform along with the metric under this geometrical procedure, proposed avenues for their rectification are discussed. Through a specific spinor transformation, fermionic theories can be formulated with real Euclidean actions, elegantly addressing and potentially mitigating computational problems like the sign problem.

Theoretical and Practical Implications

The implications of this research underscore both theoretical and practical advancements in treating spacetime transformations within quantum field theories and gravity. By retaining real attributes in processes traditionally reliant on concessional complex values, the findings offer new pathways for future explorations, including potential expansions to non-static curved spacetimes and dynamically interacting gravity.

Moreover, the pathway toward an intuitive geometrical understanding of Euclideanization widens the scope for applying the proposed techniques in further theoretical investigations. It could lead to new insights, especially in scenarios where traditional Wick rotations are less effective without solid theoretical backing.

Conclusion and Future Directions

This novel geometrical approach essentially bridges a conceptual gap in reconciling the computational techniques of Minkowskian and Euclidean space within QFT ambit, without necessitating the complexification of spacetime. Broadening the framework to incorporate higher-dimensional considerations offers a compelling foundation for continued exploration, particularly in the field of gravitational interactions and non-static cosmological models. Future research should expand on the outlined methodology's application to complex fields and extend to higher-dimensional spacetimes. The authors present a promising step towards a more nuanced understanding and utilization of Euclidean methodologies in theoretical physics, reinforcing the potential for innovative breakthroughs in mathematical physics and adjacent disciplines.