The theory of thermodynamics has significantly advanced our understanding of various devices, particularly engines, for over two centuries. However, its mathematical framework has often been criticized for lacking precision. Currently, researchers are reinforcing the foundation of this seminal theory, incorporating modern mathematical approaches to describe quantum fields more effectively.

Thermodynamics remains one of the most pertinent branches of physics in daily life. Its evolution was propelled by engineers aiming to enhance the performance of heat engines—basic devices exemplified by ubiquitous technologies such as car engines and refrigerators.

Despite its success, thermodynamics has historically struggled with mathematical rigor. Brian Roberts from the London School of Economics has embarked on a mission to reconstruct thermodynamic principles using innovative mathematical concepts rooted in geometry and quantum field theory—an approach that deviates significantly from traditional teachings.

The cornerstone of Roberts’ new methodology is “gauge theory,” which addresses characteristics of entities that are typically unobservable or unmanageable.

A straightforward example featuring a marble rolling on a surface serves to clarify this theory. Although each marble appears identical, their centers conceal different colors.

Gauge theory delineates a mathematical terrain defined by the numbers governing the marbles’ movements—referred to as “observable” space—and an alternative domain, known as “bundle” space, which contains hidden information about the internal colors of each marble.

These two mathematical realms are profoundly interconnected; the observable space serves as a projection of the unobservable bundle space. According to Roberts, this concept parallels the phenomenon of illuminating an object: even if the object itself remains concealed, its properties can still be inferred by observing its projection—its shadow.

Roberts proposes that this framework may be instrumental in examining thermodynamics since the discipline also encompasses both visible and hidden quantities.

“Thermodynamics has two levels,” he explains. “Some levels are more accessible, allowing for work extraction—akin to movable pistons in an engine. Others are less overt, like the heat generated or lost within a system, which cannot be manipulated directly.” He identifies this as a concealed contribution to energy.

This distinction is not emphasized in conventional thermodynamics where “work” and “heat” are treated equivalently, aiding in the total energy calculations of an object. However, for Roberts, recognizing the concealed aspects of thermodynamic energy permits aligning thermodynamics with gauge theory structures inside bundle space.

This innovative perspective offers a pathway to harness foundational principles established in other physics domains relating to gauge theory, ultimately leading to a comprehensive grasp of thermodynamics.

For instance, crucial thermodynamic concepts such as temperature and entropy can be articulated through specific projections from bundle space to observable space. Roberts contends that this geometric definition of entropy is more intuitive, facilitating its application across diverse systems, from engines to black holes.

Furthermore, gauge structures have connections to experiments in the quantum theory of electromagnetic fields. Roberts anticipates that a similar interplay may emerge within thermodynamics. Preliminary investigations involving specific molecular junctions hint at a thermodynamic version of the Aharonov-Bohm effect—an iconic experiment in which charged particles seem to encounter a hidden magnetic field.

Roberts presented this work at the Basics of Physics conference held on June 16 in Irvine, California.

Lucas Cerelli, a professor at the Federal University of Goiás in Brazil, expressed admiration for Roberts’ concept, emphasizing its compatibility with evolving efforts to decode thermodynamics in the quantum framework as a gauge theory.

With the application of gauge theory to quantum phenomena, thermodynamics presents even greater complexities. Cerelli cautions, “I have reservations about quantum thermodynamics, given the multitude of definitions regarding heat and work. Establishing a rigorous mathematical framework could yield a consistent and unique understanding.”

Cerelli and his team have been pursuing this by leveraging gauge theory, reportedly achieving success in reconciling some outcomes of standard quantum thermodynamics.

A significant challenge ahead for both quantum and classical thermodynamics will be integrating these with Albert Einstein’s special theory of relativity. Fortunately, Cerelli asserts, gauge mathematics may offer advantages over traditional methodologies.