Many Worlds and the Vacuum Energy Problem

1. Introduction

A key open question in fundamental physics is the cosmological constant problem. Experimental measurements show that the cosmological constant $\mathrm{\Lambda }$ is extremely small, in contrast with the theoretical prediction, which is some 120 orders of magnitude larger. This problem is often phrased instead in terms of the vacuum energy density ${\rho }_V$. Using experimental measurements of $\lambda$, ${\rho }_V$ is experimentally estimated to be around $10^{-9}$Jm${}^{-3}$, whereas naive estimates from field theory suggest a value of $10^{113}$Jm${}^{-3}$. The reason for such a large discrepancy (122 orders of magnitude) is due essentially to the fact that quantum fields consist of oscillators at every point in a region of spacetime, each of which makes a non-zero contribution to the total ground state when it is summed over. The cosmological constant problem and the vacuum energy problem are often mentioned interchangeably. However, one must remember that the vacuum energy density is not proportional to the cosmological constant. It is actually a sum of two terms, one of which is proportional to $\mathrm{\Lambda }$ and one of which is due to zero-point fluctuations of fields. The challenge to solving the vacuum energy problem is then to explain how those two terms could cancel exactly.¹

There have been many attempts to deal with this paradox, with no solution so far which has been universally accepted.² Hawking proposed that the cosmological constant is effectively variable, but that it is most probable for it to take a value close to zero.³ There have also been various suggestions that the Standard Model Higgs boson can be used to resolve the problem of the cosmological constant, where the Higgs particle is identified with the inflaton scalar field.^4,5,6,7,8 Another interesting possibility is that there is a symmetry argument which results in a vacuum state with vanishing vacuum energy.⁹ In supersymmetric models, the zero-point energies of bosonic fields give positive contributions to the vacuum energy which are exactly cancelled to all orders of perturbation theory by negative contributions from fermionic fields.¹⁰ There have also been proposals which rely on the existence of wormholes or similar constructions.^11,12 The main difficulty with solving the vacuum energy problem is that somewhere in the calculations one needs to naturally deal with the fact that the theoretical vacuum energy density must be suppressed by a factor of enormous magnitude. It is hard to see where such a large number could come from without being inserted by hand. The need to answer this question in a convincing way has motivated the development of string theory and supersymmetric models.¹³

In this note, we propose that if one assumes that the many worlds interpretation of quantum mechanics is physical and describes splittings of universes which actually occur, then it might be possible to resolve the vacuum energy problem. We should distinguish the approach which we are suggesting from other attempts to solve the cosmological constant problem using ensemble-based approaches to cosmology.¹⁴ In many worlds terms, the usual idea is to argue that there are many Big Bangs and many different universes with different initial values such that the vacuum energy evolves differently in each one. We then happen to be in the universe with the observed vacuum energy, so this approach essentially relies on the anthropic principle. The related studies of Martel et al. also consider the possibility that the cosmological constant takes different values in different subuniverses.¹ This is distinct from what we propose here, which is that the usual theoretical value for the vacuum energy density is the value for all the parallel universes in a superposition. Since there could be an enormous number of such universes and experimental measurements only take place in the universe which we inhabit, this could explain why the measured value is smaller than the predicted value by such an enormous order of magnitude (by far the largest disagreement between theory and experiment in the history of physics).

2. Superpositions in Many Worlds

The ontological status of parallel universes in the many worlds interpretation of quantum mechanisms is somewhat controversial.¹⁵ The official line is that the many worlds interpretation does not make any additional physical claims which are not contained in other interpretations of quantum mechanics, so that the alternative worlds are not claimed to exist. However, there are indications in the original work of Everett that he may have had a genuine physical split in mind.¹⁶ This idea of a literal splitting was popularized by DeWitt.¹⁷ Another matter of controversy is what type of quantum measurement causes the ‘splitting off’ into new branches or under what circumstances an entire universe splits (as opposed to a restricted region of the universe acted on by the measurement operator). DeWitt seems to have had in mind any microscopic interaction which produces entanglement (whether or not there are any macroscopic objects involved), whereas Everett thought a splitting would only occur in the specific case when a measurement transforms a product state for a system and a macroscopic measuring apparatus into a maximally entangled state.¹⁵

A key point is that the universes which branch away from ours also continue splitting and diverging further over time. It is obviously impossible to even attempt to quantify the splittings which would occur in the divergent parallel universes which we do not directly observe. For this reason, we assume a simplified scenario in which it becomes possible to estimate the number of worlds in the current superposition of parallel worlds. First, we assume that the observable being measured can always be expressed in terms of an orthonormal basis $\{|a_1〉,|a_2〉\}$ of a Hilbert space with dimension two. This implies that at each split, two slightly different universes branch away from the previous one. To deal with the fact that we do not know when other parallel universes split, we assume that a physical branching of an entire universe (not a local region) only occurs when all of the parallel universes in the superposition also split simultaneously in exactly the same way due to an identical measurement interaction. This results in the schematic picture shown in Fig. 1. In this scenario, the total number of universes after n splittings is $2^n-1$, whereas the number of universes in the nth stage is $2^n-2^{n-1}$.

Once the number of universes in the superposition becomes sufficiently large, the chance of the same splitting occurring simultaneously by exactly the same measurement interaction in all of the universes is extremely small. We will assume on average that this only happens once every $1.1\times 10^{15}$s. The age of the universe since the Big Bang is around $4.4\times 10^{17}$s, implying that 407 full splittings of all parallel universes have occurred in total. Inputting the value $n=407$ for the formula estimating the number of universes in a superposition after n branchings, we obtain a figure of $1.65\times 10^{122}$ universes. We will not discuss the mathematical details which would be involved in considering a splitting of many parallel universes or how it would fit into the usual geometric framework. The space involved would presumably be non-Hausdorff or one could have an additional dimension in which each splitting lives.

We note that this very large number would be of the correct order of magnitude to cover the discrepancy between the theoretical and experimentally measured values for the vacuum energy density of the Universe. This coincidence leads us to speculate that the huge theoretical value for the vacuum energy due to zero-point fluctuations could correspond to the value which occurs across the entire superposition of universes and that the fluctuations are distributed between all the parallel worlds, leading to a correspondingly tiny value measured for our Universe. In effect, the theoretical prediction for the vacuum energy density is the value for all worlds.

As an example, consider the usual position wave function. This wave function is spread throughout all of space in an infinite range, but during a given measurement the wave function is confined to a certain small spatial volume. Similarly, one could have a single initial Big Bang and then separation from there into many parallel worlds. We suggest that if a many worlds interpretation represents physical reality and there exists a superposition of a vast number of parallel universes, the extremely large theoretical value for ${\rho }_V$ in our Universe should be modified experimentally by the fact that we only measure ${\rho }_V$ in one of those universes.

3. Conclusion

To conclude, we have suggested that the enormous discrepancy between the experimental and theoretical values for the vacuum energy density could be ameliorated by assuming the existence of a superposition of many parallel universes such that what we normally consider to be the theoretical value for ${\rho }_V$ in our Universe is actually the value for all universes. Our proposal is speculative, but we note that other well-known suggestions in the literature involving wormholes are equally outlandish.¹¹ Since no compelling resolution of the vacuum energy problem has so far been suggested, it is of interest to form speculative proposals which could at least cover the magnitude of the discrepancy. We emphasize again that unlike other multiverse-inspired approaches to the cosmological constant, the argument which we are suggesting here is $not$ anthropic.