How the Dirac delta formalises the moment of transformation in CRR
From sufficient distance—across years, from another's view, in historical retrospect—the rupture appears as a point. A crisis, a revelation, a death: discrete moments marking before and after.
From inside the transformation, there is always duration: confusion, processing, the slow dawn of reorganisation. What seems instantaneous contains worlds of felt time.
The idealised Dirac delta, written δ(t − t₀), represents the pure mathematical essence of rupture: the limit case of instantaneous transformation. However, no real process can happen in literally zero time.
To model this, we use the smooth approximation δε(t), where the small parameter ε controls how narrow the transition is. When ε is relatively large, the change unfolds gently; as ε becomes smaller, the curve becomes sharper, steeper, and more focused, until in the limit as ε approaches zero, the spike becomes infinitely thin and high, converging toward the true Dirac delta.
In this way, ε marks the timescale of transformation, reminding us that what seems "instantaneous" is always relative to the resolution from which we observe. The smooth delta captures both perspectives: the felt process and the formal ideal.