Scalar and fermion representations of the Lorentz group in Minkowski plane, QFT correlators, pair creation in electric field and the Unruh effect

Abstract

Boost modes [Formula: see text] are eigenfunctions of the Lorentz transformations generator in two-dimensional (2D) Minkowski space (MS). We demonstrate and discuss deep interrelation between the boost modes and the field correlators, also known as Wightman functions. In the case of a massive scalar field, the boost modes, as functions of the spectral parameter [Formula: see text], contain the Dirac delta-function singularity [Formula: see text] at the light cone. The zero boost mode coincides up to a constant factor with the Wightman function. The light cone singularity of boost modes for a fermion field is stronger. For this case, they contain the Gelfand [Formula: see text]-function of complex argument [Formula: see text], while the Wightman function components coincide with analytical continuation of the boost modes set towards the spectral values [Formula: see text]. We argue that due to the discovered properties of the boost modes the so-called Unruh modes, which are at the core of the Unruh effect derivation, do not constitute a complete set in MS and thus cannot be used for quantization of neither scalar, nor fermion field. Finally, we discuss boost modes for the case of the constant electric background and rederive the well-known result for spontaneous pair creation rate. Solution of this problem in the boost modes representation reveals distinctions between the Unruh problem and the effect of pair creation by an electric field in vacuum.