The Challenge of Infinities in Physics

The discussion surrounding scale leads to another perplexing issue: the appearance of 'infinities' in physical theories. To illustrate this, let's consider a basic physical system: a closed, empty box within a field. The mathematics suggests that the energies within this space can become infinite.

In physics, a field is a function that assigns values at specific points in space and time. For instance, I can visualize the temperature in my room at various locations. Quantum field theory examines how these fields can be represented as particles and vice versa. For our example, think of the area as an abstract mathematical construct utilized to develop physical predictions.

Inside this empty box, the scalar field oscillates like a spring, with frequencies that increase as integer multiples of a base frequency. There’s nothing inherently special about any specific frequency in this scenario. When we sum these frequencies to determine total energy, we encounter an infinite result. However, we know that infinite quantities are not measurable, leading us to question where the error lies.

In reality, the concept of a perfectly sealed box is flawed. Any box we construct is influenced by the materials from which it is made. Consequently, the ‘energy’ derived from this scenario is not physical or measurable. If it were, sealing a box would lead to catastrophic consequences, known as the ultraviolet catastrophe.

The underlying issue is that the infinite energy calculated must be seen as mathematical abstraction, serving primarily to facilitate the computation of finite, physical quantities in subsequent calculations.

Why the Mathematical Model Fails

If we assume the box is made of solid material, it would contain all possible frequency modes, resulting in infinite energy. However, we have neglected crucial aspects of physics in this model. A truly sealed box cannot exist, as it must be composed of atoms. Therefore, the high-frequency modes from the vacuum will inevitably escape through the walls.

This assumption is critical — it is reasonable to propose that physical models inherently experience some degree of leakage concerning higher frequency waves. By adopting this perspective, we might reach a more plausible conclusion if we impose a cutoff on the frequencies allowed to persist.

Interestingly, if we limit the frequencies within the box to a maximum cutoff, Lambda, we can express the energy within the box as follows:

In this context, the box's energy depends on both the cutoff level (in the first term) and its size (in the second term). Remarkably, the force derived from this energy—obtained by differentiating the energy concerning the box's length—remains independent of the frequency cutoff. Thus, we discover a quantum mechanical force from a seemingly dubious calculation, leading us to establish an arbitrary number, Lambda, to restrict frequencies.

This force, known as the Casimir force, is indeed observable and was experimentally confirmed by Lamoreaux in 1997. In his landmark paper, Casimir demonstrated that the method used to exclude higher frequency modes does not significantly affect the resulting force under most conditions, provided some form of cutoff is implemented.

The Casimir force exemplifies the separation between small-scale and large-scale physics. The micro-interactions of high-energy frequency modes with the walls are initially excluded from the long-distance macro-physics, ultimately leading us to conclude that they were never significant in the first place. This framework supports the central theme of this article: large-scale physics operates independently of small-scale phenomena.

The Concept of Renormalization

The decoupling of long and short wavelength physics is a remarkable conclusion. Initially, we sought to compute an observable quantity, necessitating the management of infinity through a cutoff. We then derived a physical value by differentiation to obtain a force. This method, which allows for physical predictions about macroscopic phenomena through an intermediate step of cutting off short-distance physics, is known as renormalization.

In summary, I hope to have conveyed how the Casimir effect serves as a minimal example illustrating the independence of macroscopic physics from microscopic phenomena. Renormalization is a foundational aspect of modern physics, with significant contributions from Feynman, Schwinger, and Tomonaga in addressing infinities in quantum electrodynamics.

The second video, "Martin Hairer Colloquium Talk: Taming Infinities," provides further insights into the challenges and methodologies in modern physics regarding infinities.