Wheeler-De Witt equation and the Canonical Construction of the Glauber-Sudarshan States in Quantum Gravity

In the vast landscape of modern physics, two giants stand apart: quantum mechanics, which governs the behavior of the smallest particles, and general relativity, which describes the curvature of space and time caused by massive objects. For decades, scientists have struggled to merge these two frameworks into a single theory of quantum gravity. A central difficulty arises because, in the quantum realm of gravity, the usual rules for how things change over time break down. In standard physics, we track the evolution of a system by watching it move forward in time, driven by a quantity called energy. However, in a universe governed by the equations of quantum gravity, the total energy of the system is often constrained to be zero. This creates a profound puzzle: if the energy is zero, the mathematical operators that usually drive time forward become ineffective, leaving the universe seemingly frozen in a state where nothing happens. This is known as the "problem of time," and it has long blocked physicists from describing how a quantum universe could evolve from a static beginning into the dynamic, expanding cosmos we observe today.

A team of researchers at McGill University has proposed a new way to navigate this frozen landscape. They are not trying to force the old rules of time to work in a place where they do not fit. Instead, they have constructed a specific type of quantum state, known as a Glauber-Sudarshan state, which allows the universe to evolve without relying on the traditional, broken concept of a global time clock. Their work suggests that by carefully accounting for the subtle, invisible fluctuations that occur in the fabric of space-time, they can describe a transition from a quiet, supersymmetric background into a transient, expanding phase that resembles our own universe. This approach does not just offer a mathematical trick; it provides a mechanism to calculate the value of the cosmological constant—the energy density of empty space that drives the expansion of the universe—showing how a positive value can emerge from a system that appears static at first glance.

The researchers began by confronting the limitations of the standard tools used in quantum field theory. In ordinary physics, scientists calculate how particles interact by looking at how they change over time, using a mathematical tool called a Hamiltonian to track their progress. In quantum gravity, however, this Hamiltonian vanishes, rendering the standard method of tracking time useless. To get around this, the team introduced a new way of thinking about the universe's state. Instead of trying to evolve the universe step-by-step using a clock that doesn't exist, they defined the state of the universe by "displacing" it from a stable, quiet minimum. Imagine a ball sitting at the very bottom of a valley; in the standard view, if the ball has no energy, it stays there forever. The researchers proposed that by applying a specific, carefully tuned disturbance to this ball, they could shift it into a new configuration. This disturbance is not a random push but a structured operation that respects the deep constraints of the theory, effectively creating a new state that is no longer frozen.

A critical part of their discovery involves the role of "ghosts." In physics, the term ghost does not refer to a spirit, but to mathematical entities introduced to handle the redundancies in the equations of gravity. In most theories, these ghosts can be ignored or removed once the calculations are done. However, the authors found that in their construction, these ghosts are essential and cannot be discarded. They act as a necessary scaffolding that holds the structure of the theory together. In fact, the researchers showed that these ghosts play a dual role: they are required to define the initial state of the universe, but they also disappear from the final description of the expanding phase. This disappearance is not a loss of information but a sign that the system has reorganized itself. The ghosts help convert the initial, static supersymmetric background into a dynamic, non-supersymmetric state that can expand.

The paper demonstrates that this transition is not just a theoretical possibility but can be quantified with precision. By using a sophisticated mathematical technique called Borel-Écalle resummation, the team was able to sum up an infinite series of complex interactions that would otherwise be impossible to calculate. This method allowed them to move beyond simple approximations and find a closed-form expression for the behavior of the universe. The result was striking: the back-reaction of the quantum fluctuations—the way the universe responds to its own internal disturbances—naturally generates a positive cosmological constant. This is the energy density that causes space to expand. The researchers found that this value emerges only when all the non-perturbative effects, the subtle quantum corrections that are usually ignored, are included in the calculation. Without this complete summation, the cosmological constant would appear to be zero or negative, failing to match our observations.

The study also clarifies the relationship between these new states and the traditional descriptions of string theory. In string theory, particles are often described as vibrations of tiny strings, and their interactions are defined by "vertex operators." The researchers showed that their Glauber-Sudarshan states share a structural similarity with these vertex operators but are fundamentally different in their origin and behavior. While standard vertex operators describe small perturbations around a fixed background, the new states are designed to describe a background that is itself changing and evolving. They are constructed from the underlying degrees of freedom of M-theory, a more fundamental framework that unifies different versions of string theory. This construction allows the team to describe a universe that starts in a supersymmetric state and evolves into a transient de Sitter phase, a period of accelerated expansion that is consistent with the current state of our universe.

One of the most significant findings is the explanation for why the cosmological constant is small but positive. The researchers argue that this value is an emergent phenomenon, meaning it arises from the collective behavior of the system rather than being a fixed parameter written into the laws of physics. At the level of simple, step-by-step calculations, the cosmological constant is invisible. It only becomes apparent when the full, complex web of quantum interactions is taken into account. The team's calculations suggest that the value of this constant is determined by the specific way the quantum states are constructed and how the ghosts and other mathematical components interact. This provides a potential pathway to understanding why the universe is expanding at the rate it is, without needing to fine-tune the initial conditions of the cosmos.

The work also addresses the "problem of time" by showing that time can be defined relationally. Instead of an external clock ticking away in the background, time emerges from the relationship between different parts of the system. The researchers showed that by choosing a specific internal variable to act as a clock, the evolution of the rest of the universe can be described consistently. This relational time is not an illusion but a physical reality that arises from the constraints of the theory. The study confirms that the Wheeler-De Witt equation, which governs the quantum state of the universe, can be satisfied at two different levels: the initial supersymmetric state and the final expanding state. The transition between these two levels is driven by the back-reaction of the quantum fluctuations, which are carefully accounted for in their model.

In conclusion, this research offers a fresh perspective on one of the most difficult problems in theoretical physics. By constructing a specific quantum state that respects the fundamental constraints of gravity, the authors have shown how a universe can evolve from a static beginning into an expanding one. They have demonstrated that the cosmological constant is not a random number but a consequence of the deep quantum structure of space-time. The use of ghosts, the resummation of infinite series, and the definition of relational time are not just mathematical curiosities but essential ingredients in a coherent picture of quantum gravity. While the model is currently a simplified version of the full theory, it provides a concrete framework for understanding how the universe could have emerged from a state where time seemed to stand still. The findings suggest that the dynamic nature of our universe is a natural outcome of the quantum laws that govern it, waiting to be fully understood as the theory is extended to more complex scenarios.