Summary:
The world surrounding us has an arrow of time, yet all fundamental equations of motion are time-symmetric. To solve this apparent contradiction is a long standing important problem in physics. In this lecture we will summarize a brief history of this problem, especially in terms of the complex spectral representation of the fundamental equations, which is developed by the Austin-Brussels group led by Ilya Prigogne. This formulation shows that the irreversibility is an exact property of the time-symmetric fundamental equation of motion due to the resonance singularity (which is related to the well-known small denominator problem in chaotic dynamics). That is, contrary to the traditional explanation, the irreversibility is not a result of coarse graining operation to the system because of our limited manipulation of microscopic system with a huge degrees of freedom.