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QFT in Klein space

Bin Chen, Zezhou Hu, Xin-Cheng Mao

Abstract

In this paper, we investigate the quantum field theory in Klein space that has two time directions. To study the canonical quantization, we select the ``length of time" $q$ as the evolution direction of the system. In our novel construction, some additional modes beyond the plane wave modes are crucial in the canonical quantization and the later derivation of the LSZ reduction formula. We also derive the free two-point function by using Wick contraction in the canonical quantization formalism. Moreover, we introduce the path-integral formalism in which we can redefine the vacuum states and rederive the correlation functions. We show that all the results in the Klein space derived in our novel approach match those obtained via analytical continuation from the Minkowski spacetime.

QFT in Klein space

Abstract

In this paper, we investigate the quantum field theory in Klein space that has two time directions. To study the canonical quantization, we select the ``length of time" as the evolution direction of the system. In our novel construction, some additional modes beyond the plane wave modes are crucial in the canonical quantization and the later derivation of the LSZ reduction formula. We also derive the free two-point function by using Wick contraction in the canonical quantization formalism. Moreover, we introduce the path-integral formalism in which we can redefine the vacuum states and rederive the correlation functions. We show that all the results in the Klein space derived in our novel approach match those obtained via analytical continuation from the Minkowski spacetime.

Paper Structure

This paper contains 14 sections, 140 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basics of Klein space
  3. Canonical quantization for real scalar in Klein space
  4. Free real scalar field
  5. Mode expansion
  6. Hamiltonian and evolution operator
  7. Correlation functions
  8. The interacting real scalar field and LSZ reduction formula
  9. Path-integral formalism
  10. Conclusions and discussions
  11. Amplitudes and correlation functions from analytical continuation
  12. Self-consistency check of the propagator
  13. Perturbative expansion and Wick theorem
  14. Causal structure

Figures (4)

  • Figure 1: The Penrose diagram of Klein space.
  • Figure 2: The path integral approach to define vacuum states and correlation functions in the Euclidean signature. The complex plane here is described by the complexified $q_E$.
  • Figure 3: The path integral approach to define vacuum states and correlation functions in the Klein signature. The complex plane here is described by the complexified $q$ with $q= - iq_E$
  • Figure 4: The integral contour for the complexified $\omega$.