Origin of Universe

Question

The universe is believed to have originated from absolutely nothing, and we know it is still expanding. Physics says "something can arise from nothing". I understand how mass and energy are related and one can take the form of other. But, how can everything arise from absolutely nothing?

Answer 1

These types of theories that physicists such as Krauss espouse of a "Universe Coming From Nothing" are quite flawed, as by no means are they talking about nothing! Further, the concepts of particles, mass, and energy are not even well-defined when talking about the universe in general.

I wrote a paper on this (excuse the shameless self-promotion), it can be found on the arXiv: http://inspirehep.net/record/1298212

The paper itself came out of many interesting conversations I had last summer with my colleague George Ellis, who has commented at length about Krauss' book and other such "universe from nothing" theories.

Here, however, are the important points behind the paper:

Krauss' main claim is that "in quantum gravity, universes can, and indeed always will, spontaneously appear from nothing", which he bases on the Wheeler-DeWitt (WDW) version of quantum gravity. In fact, Krauss explicitly cites two formalisms built upon the WDW approach to quantum gravity, the Hartle-Hawking no boundary proposal and Vilenkin's tunnelling idea.

It is important to note that in the WDW formalism, one represents the quantum state of the universe as $\Psi(h_{ij})$ on superspace, where $h_{ij}$ is a spatial metric, which itself is subject to the Hamiltonian constraint. In the literature, the Wheeler-DeWitt approach has only been applied to minisuperspaces. Superspace is the space of all spatial metrics, and each point in superspace corresponds to a spatial metric $h_{ab}$. DeWitt's supermetric is defined as \begin{equation} G^{abij} = \frac{1}{4} \sqrt{h} \left(h^{ai}h^{bj} + h^{aj}h^{bi} - 2h^{ab} h^{ij}\right). \end{equation} One obtains a minisuperspace model by working with universe models that have finite degrees of freedom such as the FLRW or Bianchi models. With the aforementioned wave functional, the Wheeler-DeWitt equation takes the form (in the case of minisuperspace models) \begin{equation} \mathcal{H} \Psi[h_{ij}] = 0, \end{equation} and to solve such an equation, one needs conditions on the spatial geometry usually given in terms of boundary conditions on $\Psi$. One typically solves the Wheeler-DeWitt equation using a path integral formulation. In the Hartle-Hawking no-boundary proposal which is also mentioned in Krauss' book, we have \begin{equation} \Psi[h_{ij}] = \int \mathcal{D}g_{uv} \exp(-I[g_{uv}]), \end{equation} where $\mathcal{D}g_{uv}$ is a measure on the space of 3-geometries, and $I[g_{uv}]$ is the Euclidean action, which has a $S^{3}$ geometry as its boundary. This methodology leads to a "beginning of time", where a classical description then becomes valid.

On the other hand, Vilenkin's method [1] produced a variant of the Wheeler-DeWitt equation for FLRW universes as \begin{equation} \Psi''(a) - \left[a^2 - a_{1}^{-2} a^{4}\right]\Psi(a) = 0. \end{equation} This equation gives tunnelling probabilities for the wave function "from nothing" to a closed universe of radius $a_{1}$.

These methods are essentially the quantum gravity approaches Krauss refers to to show that it is plausible that a universe can come from nothing, but the nothing that Krauss refers to is in fact no space and no time. The Wheeler-DeWitt approach entails a universe coming from no space for the reason that the entire DeWitt formalism relies on an underlying superspace, which as we mentioned above, is the space of all spatial metrics, $h_{ij}$. Since both approaches above are Wheeler-DeWitt equations, they also exist on some space, namely, this superspace/minisuperspace. In particular, for such a proposal to be considered as a valid physics-based proposal, it has to be at least in principle, testable. Namely, one would have to show that preceding the big bang, or the creation of our universe, that there really was such a superspace in existence. It is not clear how at the present time that we could even begin to consider how this could be accomplished.

Notwithstanding the previous point, there are significant problems with both approaches, many types of divergences occur, namely that the path integral itself is ultraviolet divergent, and in fact, cannot be renormalized. In the Hartle-Hawking no-boundary proposal in particular, conformal modes lead to the Einstein-Hilbert action not being bounded from below, which in turn implies that the sum over all 4-geometries leads to a sum over topologies that cannot be computed.

In Vilenkin's approach, there is also a problem of time. There are some approaches that try to treat $a$ in the Vilenkin equation as an effective time variable, but, as Barbour [2] has pointed out, it is very difficult to make this work from a practical sense. One essentially has from these approaches that $\dot{|\Psi(t)\rangle} = 0$, which implies a static solution, and the concept of a time-evolving universe is thus difficult to see. Another important point is how exactly one interprets the concept of a "wavefunction" and probabilities when there is only one object. Can one even give any meaningful definition of the wavefunction of a universe in these contexts?

In Krauss' book, the concept of superspace is not mentioned a single time, even though this is the entire geometric structure for which the proposal he is putting forward of a universe coming from nothing is based upon. There is also a deep philosophical issue that cannot be ignored. Superspace is the space of all possible 3-geometries, and the question is, what types of universes should be considered as part of a particular superspace. For example, one can consider the Bianchi cosmologies which are spatially homogeneous and anisotropic cosmological models, which have three degrees of freedom in the mini-superspace sense, of which the FLRW cosmologies are special cases. The existence of this structure is not predicted by the WDW formalism, it is assumed to exist, which itself goes back to the theme of this paper. How can one claim that something arises from nothing, when this nothing is at minimum, minisuperspace?

Much of the motivation of universes spontaneously being emitted from nothing seems to be the phenomenon that Krauss describes as particles being spontaneously emitted from a vacuum state. This in itself is quite problematic, as it is based on the naive particle interpretation of quantum field theory. It is well understood that in the context of cosmological models and more general spacetimes, one simply does not have a time-translation symmetry because of a lack of timelike Killing vector field. Therefore, the very definition of particles is undefined for general curved spacetimes, and only defined in the context of general relativity for asymptotically flat spacetimes. The reason is that for the particle interpretation to work, one needs to be able to decompose the quantum field into positive and negative frequency parts, which in itself depends heavily on the presence of a such a time translation symmetry in either an asymptotically flat spacetime or a Minkowski spacetime. The problem of course is that, our universe, or any spatially homogeneous and non-static universe, that is, one that does not contain a global timelike Killing vector field is necessarily not asymptotically flat. This can be seen from the arguments given in Stephani's classic book for example. Namely, consider a spacetime $(\hat{M}, \hat{g}_{ab})$. Let this spacetime have the following three properties:

1. There exists a function $\omega \geq 0 \in C^3$, such that $g_{ab} = \omega^2 \hat{g}_{ab}$,
2. on the boundary $\omega = 0$, and $\omega_{,a} \neq 0$,
3. Every null geodesic intersects the boundary in two points.

These spacetimes are called asymptotically simple. However, if we now associate the metric tensor, $g_{ab}$ with the Einstein field equations, the existence of these three conditions implies that the spacetime is asymptotically flat. It is only under these three conditions, for which one can in a meaningful way talk about "particles". The idea of these particles being spontaneously emitted from a vacuum is also not correct for the following reason.

Consider a two-level quantum mechanical system which is coupled to a Klein-Gordon field, $\phi$ in a Minkowski spacetime, for simplicity. The combined system will have a total Hamiltonian of the form

$\mathcal{H} = \mathcal{H}_{\phi} + \mathcal{H}_{q} + \mathcal{H}_{int}$,

where $\mathcal{H}_{\phi}$ is the Hamiltonian of the free Klein-Gordon field. We will consider the quantum mechanical system to be an unperturbed two-level system with energy eigenstates $| x_{o} \rangle$ and $|x_{1} \rangle$, with energies $0$ and $\epsilon$ respectively, so we can define

$\mathcal{H}_{q} = \epsilon \hat{A}^{\dagger} \hat{A}$, where we define \begin{equation} \hat{A} |x_{0} \rangle = 0, \quad \hat{A} |x_{1} \rangle = |x_{0} \rangle. \end{equation} The interaction Hamiltonian is defined as \begin{equation} \mathcal{H}_{int} = e(t) \int \hat{\psi}(\mathbf{x}) \left(F(\mathbf{x}) \hat{A} + o\right) d^{3}x, \end{equation} where $F(\mathbf{x})$ is a spatial function that is continuously differentiable on $\mathbb{R}^{3}$ and $o$ denotes the Hermitian conjugate. One then calculates to lowest order in $e$, the transitions of a two-level system. In the interaction picture, denoting $\hat{A}_{s}$ as the Schrodinger picture operator, one obtains \begin{equation} \hat{A}_{I}(t) = \exp(-i \epsilon t) \hat{A}_{s}. \end{equation} Therefore, we have that \begin{equation} (\mathcal{H}_{int})_{I} = \int \left(e(t) \exp(-i \epsilon t) F(\mathbf{x}) \psi_{I}(t,\mathbf{x}) \hat{A}_{s} + o\right) d^{3}x. \end{equation} Using Fock space index notion, we can then consider for some $\Psi \in \mathbb{H}$, where $\mathbb{H}$ is the associated Hilbert space, and note that the field is in the state \begin{equation} |n_{\Psi} \rangle = \left(0, \ldots, 0, \Psi^{a_{1}} \ldots \Psi^{a_{n}}, 0, \ldots \right). \end{equation} The initial state of the full system is then given by \begin{equation} |\Psi_{i} \rangle = | x \rangle |n_{\Psi} \rangle. \end{equation} One then obtains the final state of the system as being \begin{equation} |\Psi_{f} \rangle = |n _{\Psi} \rangle |x \rangle + \sqrt{n+1} \| \lambda \| (\hat{A} |x \rangle) |(n+1)^{'}\rangle - \sqrt{n} (\lambda, \Psi) (\hat{A}^{\dagger} |x\rangle) |(n-1)_{\Psi}\rangle. \end{equation}

The key point is that if $|x \rangle = |x_{0} \rangle$, that is, the system is in its ground state, this shows explicitly that this two-level system can make a transition to an excited state, and vice-versa. Note that the probability of making a downward transition is proportional to $(n+1)$, and even when $n = 0$, this probability is non-zero. This in the particle interpretation is interpreted as saying that the quantum mechanical system can spontaneously emit a particle. However, the above calculation explicitly shows that it is the interaction of the quantum mechanical system with the quantum field that is responsible for the so-called spontaneous particle emission. This misleading picture of the vacuum state is precisely promoted by the particle interpretation of quantum field theory. As the work above shows, this is not spontaneous particle emission from "nothing" in any sense of the word. One must have both a well-defined quantum mechanical system interacting with a well-defined vacuum state for such spontaneous emission to occur, we emphasize that these are not nothing!

There are also further claims Krauss makes in the book that are quite problematic, which I refer you to the paper linked above for more details.

References

[1] A. Vilenkin: Creation of Universes from nothing. Physical Letters B, 117:25-28 1982

[2] J. Barbour: The End of Time: The next Revolution in Physics. Oxford University Press, first edition, 2001