THE WAVE FUNCTION OF THE UNIVERSE

Until very recently, fundamental questions such as the creation of the Universe or the ultimate nature of space and time belonged to the realm of metaphysics or philosophy. However, the advancement of fundamental physics has not only included these questions within the scope of science but has also been able to propose plausible physical models to explain them. Quantum cosmology attempts to explain the beginning of the Universe by applying the principles of quantum mechanics (QM) to the entire Universe. In this article, we will examine some of the most fascinating consequences of this branch of fundamental physics, including concrete proposals regarding the creation of the Universe and the ultimate nature of space-time.

Quantum Cosmology

The principles of quantum mechanics include one of the strangest physical effects known: quantum superposition or entanglement. This principle implies that any non-isolated quantum system will become entangled with its surroundings. The Universe is not an isolated system, so it is expected that the initial quantum states that gave rise to the Universe we know were affected by quantum entanglement. This fact, along with the fact that the Universe is quantum at a fundamental level, leads us to the concept of the Universe's wave function. But how can we define something like this?

As we know, the Universe on large scales is described by general relativity (GR), but on small scales it is described by quantum mechanics. So, to approximate a wave function for the entire Universe, we must attempt to quantize the equations of general relativity. This can be done in three steps:

1) In MC two fundamental operators are used: the position operator q and the momentum operator p, so we must find the equivalent aqyp in the GR equations. To do this we must separate space and time in GR: assuming the x-axis is space and the y-axis is time, we split the x-axis into horizontal (space-like) "slices". Each "slice" (which represents all space at a given time) corresponds to a metric h(x), this will be the GR variable that plays the role of position q. The momentum p describes how the position changes over time so we must find the equivalent to the change in position h(x) over time in GR, this is given precisely by the extrinsic curvature pi(x) that defines how the metric (the equivalent aq) changes over time. In this way we already have the equivalent aqyp in GR: h(x) and pi(x).

2) One of the main characteristics of GR is that all points in space-time are identical; there are no privileged reference systems. Each point in space-time is defined by four coordinates: three spatial and one temporal. The temporal coordinate must meet the following requirement (this is equivalent to the classical requirement p2+m2=0, which represents invariance under time reparameterizations):

Where raizh=root of the determinant of the 3-dimensional metric, R=scalar of curvature of the 3 spatial dimensions and Hm=Hamiltonian of the non-gravitational fields.

Gabcd is metric-dependent and represents a particular metric within the space of all possible metrics.

3rd) As is known, in MC the operators p and q do not commute: [p,q]= ih, Therefore, we must impose that h(x) and pi(x) satisfy the standard commutation relations of the MC, that is:

With just these three steps we already have quantized (in semiclassical approximation) the RG:

This equation is called the Wheeler-DeWitt equation and is one of the fundamental equations of (semiclassical) quantum gravity. It consists of two distinct parts: the first part in parentheses (up to Hm) is the Hamiltonian corresponding to the gravitational fields, while the second part, Hm, represents the Hamiltonian of the matter fields. The equation can thus be expressed as:

Where Htot is the sum of the gravitational fields and the matter fields.

The wave function of the Universe

The wave function appearing in the Wheeler-DeWitt equation is the wave function of the Universe. It has several characteristics that can describe fundamental qualities of the Universe we inhabit:

1) The wave function of the Universe depends only on the geometry of the three spatial dimensions and does not depend on time. Furthermore, it does not depend on a specific geometry but on all possible three-dimensional geometries ; this space of all possible geometries is called superspace.

2nd) It implies the existence of a minimum length: the Planck length below which the wave function has no meaning.

In practice, the Wheeler-DeWitt equation is very difficult to solve and can only be done approximately by considering finite degrees of freedom and establishing certain boundary conditions.

The emergence of time and the classical Universe

In the Wheeler-DeWitt equation, in the part corresponding to gravitational fields, the natural scale that appears is the Planck length, which is much smaller than any scale that appears in matter fields. This allows us to express this part of the equation in powers of h and thus calculate the equation in different orders of approximation:

- In the first order of approximation, we can consider only the gravitational fields, since their overall contribution to the equation is much greater than that of the non-gravitational fields. An analogy would be that of the hydrogen atom: the nucleus would be the gravitational part, while the electron, much lighter, represents the non-gravitational part (1). In this first order of approximation, we have:

In this equation, So is the Hamilton-Jacobi equation of gravity and represents the field equations of general relativity. We can assign to the previous equation a series of classical trajectories orthogonal to So, each trajectory representing a complete spacetime, In this way the previous equation represents a superspace with all possible space-times that satisfy the Wheeler-DeWitt equation.

- In the next order of approximation, something transcendental happens. Now we must include the contribution of the matter fields, and we obtain:

where x depends on the matter fields and obeys the equation:

This equation is equivalent to the Schrodinger equation for matter fields propagating in our usual space-time!

By including matter fields the Wheeler-DeWitt equation appears to "collapse" and becomes the "usual" Schrodinger equation.

However, there is still a small problem to be solved: the previous equation, although already classical, still describes a superposition of classical geometries instead of a concrete space-time. For example, the equation also includes its complex conjugate, which makes it impossible to fully recover the usual Schrodinger equation. The problem is that the wave function of the Universe contains an enormous number of correlations (entanglement) between a large number of degrees of freedom, most of which are unobservable. The solution to this problem consists of using the so-called "density matrix," using only the relevant degrees of freedom. The main degree of freedom to consider is the scale factor. If we take only this factor into account, we find that unwanted interference is suppressed except for very small values of the scale factor. This means that the Universe becomes classical as it increases in size. This explains the emergence of our classical Universe from the quantum superposition of possible Universes!

The emergency of time

Next we face one of the most intriguing problems: How does time appear in the timeless Wheeler-DeWitt equation? The key lies in the relationship between matter fields and gravitational fields: the former obey the second law of thermodynamics and therefore tend to increase entropy (they tend to a homogeneous equilibrium) while the latter tend to decrease entropy (they tend to compress matter and form clumps). Experimentally we know that the classical starting conditions of our Universe consist of a homogeneous and isotropic Universe (FLRW metric), the key is then to find an initial condition of very low entropy of the Universe that evolves to a high entropy state . This condition can be satisfactorily met if we use the same requirement that we considered in the previous section: the condition that the wave function of the Universe depends only on the scale factor . (2) For small values of the scale factor the number of degrees of freedom is low and therefore the entropy is low. In this way a coherent and satisfactory resolution is achieved: at small scales the Universe is a superposition of possible geometries as dictated by the principles of quantum mechanics, as the scale factor increases, the available degrees of freedom increase and therefore decoherence implies the recovery of the classical Universe, the Schrodinger equation and its inherent time. In this way the arrow of time is linked to the expansion of the Universe . Some works based on the study of the CMB like this one point to the possibility that our Universe is closed, this would imply the possibility that the Universe would slow down its expansion and begin to contract which could imply something incredible: the arrow of time could be reversed! (3)

Interpretations of Quantum Mechanics

As a final section to this article, a brief reflection on what are known as "QM interpretations" is in order. As is well known, more than a century after its discovery, there is no consensus among physicists on the most appropriate way to interpret the strange characteristics of the quantum world. Probably the most widely used interpretation is the so-called "Copenhagen interpretation." However, based on our knowledge of quantum gravity and quantum cosmology, this interpretation does not seem to be the most consistent with what the theory dictates. This interpretation seems to point more in the direction of Richard Feynman's "sum of histories."

generally indicating the existence of a superposition of geometries and topologies. This points to the existence of a type of multiverse and, therefore, to an interpretation more aligned with that proposed by Hugh Everett in 1958.

In the next article, we will explore the consequences of this in more detail and how modern quantum cosmology describes a very different Universe from that presented by conventional classical cosmology.

Grades:

(1) This analogy is deeper than it seems, in fact, there is a way to obtain the Wheeler-DeWitt equation using this concept.

(2) This initial condition is also used in a similar way in the so-called "no boundary" solution that leads to the famous Hartle-Hawking state. This solution constitutes the first concrete proposal to explain the origin of the Universe. In this proposal, the Universe arises "out of nothing" (through a primordial quantum fluctuation) from a Euclidean Universe (a Universe with complex time without classical space-time). Moreover, very recently this solution has gained some support from recent works based on string theories: Baby Universes, Holography and the Swampland

(3) These conclusions are of course still quite speculative, however, they are based on general principles of quantum cosmology.

Sources: Quantum Cosmology and the emergence of a classical world