All Roads Lead to Rome

— The Common Bottleneck of Major Open Questions in Contemporary Theoretical Physics

The last century has witnessed the quantum field theory framework in successfully quantizing systems with electromagnetic, weak and strong interactions. However, the same idea fails when encountering gravity. Plenty of efforts have been poured into solving the non-renormalizability of general relativity, yet quantum gravity still remains to be the most famous mystery in high energy physics. Another well-renowned problem is related to the strongly correlated systems that have attracted the attention of tons of condensed matter physicists. The strong interaction between electrons in the material poses great trouble in traditional perturbative techniques. Interestingly, in high energy physics, similar issues also cause obstacles in studying low energy confinement in QCD. And this problem subsequently prevents us from understanding systems at even lower energy scales by reconstructing emergent degrees of freedom from high energy theories.

The above problems, namely quantum gravity, strongly correlated electrons, confinement, and scale or complexity problems, are probably the most widely concerned open problems in contemporary theoretical physics. The purpose of this short essay is to clarify that all the above problems, despite drastically different appearances, originate from the same flaw of quantum field theory — the analytical structure of quantum theories with non-linear interactions.

Non-linear Interactions in Quantum Theories

Most physical theories exert their power by differential equations. In classical theories, the very differential equations are the Euler-Lagrange equations. For quantum theories, we in general have the Schrödinger equation. One of the benefits of differential equations is that one can always use numerical methods to simulate the physical processes when analytical solutions are unapproachable. However, this property ends when interactions kick in. Notice that the interactions here do not include the fixed potentials, as the backreaction on the potential backgrounds is not taken into account in this case.

Typically, one can solve the Schrödinger equation and obtain the so-called evolution operator . This is possible in free quantum theories by either analytical or numerical methods. However, the interactions will inevitably modify the structure of the Hilbert space. Specifically, the energy eigenbasis becomes ill-defined in this case due to the complicated entanglement induced by the interaction. Therefore, one can only make use of perturbative scattering theories and obtain the so-called scattering S-matrix . In scattering theories, interactions are restricted in a small region near . Thus the quantum states at and are away from the interaction, and we can resume the usage of the Hilbert space structure of free quantum theories.

All the above should look very familiar to all quantum field theory experts, and this technique has been proved useful in either elementary particles or some of the condensed matter materials. Despite that, weakness remains. Ostensibly, the perturbation series do not always converge. Essentially, a lack of understanding of the dynamics in the vicinity of an interaction as well as an analytical structure of the scattering process is the very culprit. The facts that quantum field theories have been widely applied to various aspects in physics, and that such an issue has remained unsolved for decades, have led to diverse open questions that look superficially disparate. In other words, many research efforts in different subdivisions in physics are actually towards the same direction, substantially speaking.

Connections with Open Questions

The first and most direct consequence of an ill-defined quantum field theory is the non-renormalizability of perturbative quantum gravity. In fact, non-renormalizability itself does not fail any theories. For example, the 4-fermion theory developed for the -decay is non-renormalizable. However, this simply means that there are new intermediate degrees of freedom (W bosons) generated at high energy. Therefore, the non-renormalizability of perturbative quantum gravity may also indicate new unknown degrees of freedom, and they can only be probed at either impractically high energy or in the vicinity of an interaction.

Another two open questions, i.e. strongly correlated electrons and confinement in QCD, are both the consequences of the strong interaction which fails the perturbative series. In strongly correlated systems, particles are constantly interacting with each other without the concept of "regions away from the interaction"; in QCD, the strong interaction produces large enough energy to create pairs of new particles even if the objects interacting are merely slightly separated. In both cases, the strong interaction makes it difficult to define a boundary where the free Hilbert space structure still approximately holds. And the understanding of these systems is only possible after we figure out what happens when interactions can not be omitted.

As a matter of fact, the confinement problem in QCD is closely connected to the mass gap problem. If an analytical theory of the dynamic process of confinement is available, then the mass gap should be a direct consequence. Since the confinement and mass gap problem remain unsolved, the attempts to construct low energy degrees of freedom at nuclear level are highly restricted — only phenomenological studies are possible at current stage.

Similar issues expand beyond QCD itself. In fact, all systems that depend on the perturbative methods in quantum field theory suffer. Without a proper analytical method, one can only obtain information at boundaries, i.e. regions where interactions are negligible. This prevents us from using numerical methods to simulate the dynamical process of the system. And thus, we are unable to theoretically search for emergent degrees of freedom at lower energy scales (or higher complexities). This is irrelevant to how much computational power needed — the methods do not exist yet.

Conclusions and Remarks

In this essay, we try to argue that many major open questions in physics, e.g. the non-renormalizability of perturbative quantum gravity, the properties of strongly correlated systems, the confinement and mass gap problem of QCD, the analytical methods for scale and complexity problems, have the same origin: the lack of understanding towards the quantum dynamics in the vicinity of an interaction. This has become a common bottleneck in diverse subdivisions in physics nowadays. In other words, we should not expect major breakthroughs without any progress towards this essential issue. On the other way round, however, one can also view the researches on the above related open questions as constituents of a collective research program whose ultimate goal is to solve the vicinity problem. This means that despite divergent appearances, efforts towards superficially disconnected open questions end up all together.

— August 28, 2021 at Fuzhou