Quantum field theory in flat spacetime with multiple time directions

This work serves as the generalization of the work [1], where we investigated the quantum field theory in Klein space which has two time directions. We extend studies to the general spacetime <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi mathvariant="double-struck">R</a:mi><a:mrow><a:mi>n</a:mi><a:mo>,</a:mo><a:mi>d</a:mi><a:mo>−</a:mo><a:mi>n</a:mi></a:mrow></a:msup><a:mo stretchy="false">(</a:mo><a:mi>n</a:mi><a:mo>,</a:mo><a:mi>d</a:mi><a:mo>−</a:mo><a:mi>n</a:mi><a:mo>≥</a:mo><a:mn>2</a:mn><a:mo stretchy="false">)</a:mo></a:math> in a similar manner; the “length of time” <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>q</f:mi></f:math> is regarded as the evolution direction of the system and additional modes beyond the plane wave modes are introduced in the canonical quantization. We show that this novel formulation is consistent with the analytical continuation of the results in Minkowski spacetime.