Photo by patpitchaya / Shutterstock.

**Physics is often baffling, **but one principle seems rock-solid: the law of conservation of energy. The world contains this thing called “energy” whose amount never changes. It can change its form or go from one body to another, but its total amount remains constant. Everything from the arc of a well-kicked football to the purring of a car engine depends on this law. It makes energy a precious commodity, counted, hoarded, and fought over.

The quantum world is uncertain; attributes such as energy are ill-defined or fuzzy.

We physicists have learned that our bodies do not merely use energy, but consist of it. Einstein’s formula *E*=*mc*^{2} identifies mass as a form of energy, one that can be converted to other forms (by a nuclear bomb, say) or created from those forms (in a particle collider). The formula strengthens our intuition that energy is the basic stuff of which things are made. When one gets deeper into physics, one also learns that conservation laws are intimately tied to symmetries, as first appreciated by the German mathematician Emmy Noether nearly a century ago. Energy is conserved because the laws of nature are symmetric in time—they do not change from moment to moment.

But physics wouldn’t be physics if it did not continually question itself. Not long after Einstein derived his famous formula, he began to create a theory of gravitation, his general theory of relativity. Energy conservation became a bit dicey. Although individual observers can measure the energy density immediately around them and confirm that the total energy of localized systems remains constant, it is impossible to define an overall energy that is strictly conserved. It might sound strange to be able to define a local quantity of energy and not a global one. And it is.

**Our own expanding** **universe **is a good example of that strangeness. The energy density of matter decreases in inverse proportion to the volume of space. For instance, galaxies move apart, so that there are fewer of them in a given volume, in accordance with energy conservation. But the energy density of starlight and other forms of radiation decreases at a steeper rate. Their energy is lost. It does not go into some other form. This is allowed because an expanding universe is not symmetric in time; its growth differentiates past from future. So, general relativity makes it hard to sustain the view that energy is fundamental stuff from which everything else is made.

That is just the start. Consider the other theory that revolutionized the physics of the 20th century, quantum mechanics. The quantum world is uncertain; attributes such as energy are ill-defined or fuzzy. Worse, the theory has a very serious conceptual flaw, which must be taken into consideration when reviewing the ultimate fate of the conservation of energy.

Physics wouldn’t be physics if it did not continually question itself.

Namely, quantum mechanics involves two distinct and incompatible recipes to determine how a particle or system of particles evolves in time. The first applies when the system is not being observed, the second when it is observed. The theory is vague about which recipe to use. What exactly constitutes a measurement or observation? Need a conscious being be involved? Can a flea make a measurement? A virus? This issue is known as the measurement problem, which, as various critics have noted, should be referred to as the reality problem: The theory is unclear what exists “out there” independently of our perceptions.

As Tim Maudlin of New York University has discussed, approaches to dealing with the problem come in three types.^{1} One adds so-called hidden variables—ingredients beyond what ordinary quantum theory provides—to provide a fuller description of the state of a system. The best known example is the de Broglie–Bohm theory, which supposes that besides the wave function there are particles that have definite positions that the standard quantum formalism does not capture. The wave function simply guides them like a sheep dog.

A second kind of approach postulates a random process that collapses the system’s uncertainty and eliminates its fuzziness. A third solution involves a multiplicity of universes. What we call a measurement somehow corresponds to a splitting of our universe into many branches, one corresponding to each possible result. All these ideas dispense with the problematic recipe for measurement. None is without problems, but that is what there is.

**In 2019, Maudlin, **Elias Okon of the National Autonomous University of Mexico, and I set out to study the fate of conservation laws in these three approaches.^{2} Our analysis involved general considerations as well as various thought experiments.

Consider a standard experiment in which a quantum system—made of, for instance, a few photons—evolves into that characteristic quantum type of combination known as the “superposition” of two paths. These lead to situations that, at the classical level, correspond to different values of the energy. One path takes photons to a distant galaxy and back, thus making them lose energy due to cosmic expansion. The other path involves no change in their original energy. According to a central tenet of quantum theory, each photon takes both paths.

Dark energy is a sort of cumulative memory of all the violations of local energy conservation that have taken place in the universe’s history.

The standard story in quantum physics is that any non-conservation can be explained away by taking into account the energy supplied or absorbed by the measuring apparatus. We remove that option by using another quantum effect, entanglement, to let us make the measurement remotely.

The three interpretive approaches offer different accounts of what happens to the energy. In spontaneous-collapse theories, the system, after a sufficiently long time, undergoes a sudden collapse to one of the energy values, leading to energy non-conservation. In the de Broglie–Bohm approach, any notion of energy with any chance of being generally conserved must involve both the particles and the guiding wave function. The wave function is split and later reunited in the lab, and the interference that occurs at the reunion makes the photons behave in such ways that energy is not conserved. In the many-worlds setting, the average energy of all the branches into which the world splits might be conserved, but in each branch energy will not be conserved. From the point of view of each branch, what occurs is just the same as in the collapse theories.

In short, we concluded that no scheme offers a reasonable definition of the global energy of a system that is strictly conserved. None offered a notion of local energy conservation, either—which is bad, because general relativity requires local energy conservation to be internally consistent.

To reconcile quantum mechanics and general relativity will require a quantum theory of gravity. Physicists disagree vehemently on what such a theory will look like, but most agree on one thing: The notion of spacetime will disappear at the fundamental quantum-gravity level. In that case, conservation laws lose their relevance completely. How can you say a certain quantity does not change with time if there is no time at the fundamental level?

**At the practical level,** the deviations from strict conservation are expected to be minuscule and will not help with the concrete problems we humans face with energy. But for many theorists, any violation is blasphemous. Still, there may be a compensation.

Thibaut Josset and Alejandro Perez of the University of Marseille, James Bjorken of Stanford University, and I have shown that a modification to general relativity (originally considered by Einstein himself) permits small deviations from local energy conservation.^{3-5} And such a theory may offer a path to resolve one of the biggest mysteries in modern science: dark energy.

Dark energy is the mysterious component of the universe—some 70 percent of its total content—that is causing its expansion to accelerate. According to our analysis, dark energy is a sort of cumulative memory of all the violations of local energy conservation that have taken place in the universe’s history. In one of the specific models considered, the predicted value matches observations in a completely natural way.

Of course, the situation is far from settled. The exploration of these and related issues is still in its infancy. But when we look anew at a principle we used to take for granted, we expect to continue being surprised by its implications.

*
Daniel Sudarksy is a theoretical physicist at the National Autonomous University of México in Mexico City. He focuses on the interplay of Einstein’s general theory relativity and quantum physics, searching for clues about a deeper theory that my be unearthed by focusing on the friction points.
*

### References

- Maudlin, T. Three measurement problems.
*Topoi***14**, 7-15 (1995). - Maudlin, T., Okon, E., & Sudarsky, D. On the status of conservation laws in physics: Implications for semiclassical gravity. arXiv:1910.06473 (2019).
- Josset, T., Perez , A., & Sudarsky, D. Dark energy from violation of energy conservation.
*Physical Review Letters***118**, 021102 (2017). - Perez, A., Sudarsky, D., & Bjorken, J.D. A microscopic model for an emergent cosmological constant.
*International Journal of Modern Physics D***27**, 1846002 (2018). - Perez, A. & Sudarsky, D. Dark energy from quantum gravity discreteness.
*Physical Review Letters***122**, 221302 (2019).