FROM COMPLEX ANALYSIS TO KNOTTED FIELD THEORY IN MODERN PHYSICS

Complex analysis acts as a key part of the undergraduate course, Methods of Mathematical Physics. One of its kernel concepts is the Cauchy integral defined in the complex plane <inline-formula id="M4"> <math id="mathml_M4" display="inline" overflow="scroll"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow></math></inline-formula>. The aim of this paper is to introduce a new point of knowledge to undergraduate students and junior postgraduate students, that is, a third axis (the real axis <inline-formula id="M1"> <math id="mathml_M1" display="inline" overflow="scroll"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow></math></inline-formula>, denoted as the time axis) can be introduced into the 2-dimensional complex plane <inline-formula id="M2"> <math id="mathml_M2" display="inline" overflow="scroll"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow></math></inline-formula> to form a (2+1)-dimensional space, <inline-formula id="M3"> <math id="mathml_M3" display="inline" overflow="scroll"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow></math></inline-formula>, such that the Cauchy integral in the complex plane is transformed to be a description for an entangle with knot winding number <i>Lk</i>= ±1. Winding numbers are known as a weak topological invariant in knot theory, able to be regarded as the simplest case of a higher-order tool, the Kontsevich integral. Furthermore, if “multiple correlators” are considered simultaneously in a Cauchy integral, it means obtaining the general form of the Kontsevich integral (where the so-called group factor is ignored). This new tool plays a crucial role in both knot theory in mathematics and quantum field theory in physics. In this paper we will detail the above mentioned relevant concepts and theoretical methods, in the hope of providing for the interested reader a shortcut from undergraduate fundamental courses of theoretical physics to the frontiers of scientific research in modern mathematics and physics.