# Why is energy minimized over the string landscape?

As understand it, the 4D string landscape is a function that assigns an energy to every possible compactification of the 6 small spatial dimensions. We expect our universe to lie in a local energy minimum, and if there is a lower minimum at another compactification, then our universe would only be metastable, because we expect that it would eventually quantum tunnel into the lower minimum.

What is the process that prefers compactifications with lower energy? I can understand why we might expect total energy should be conserved, but not why it should be minimized. To me, the logical statistical ensemble with which to describe the universe would be the microcanonical ensemble, in which the total energy is arbitrary and not necessarily minimized.

Usually there are two processes that cause a system to lower its energy: (1) a dissipative system (e.g. one with friction or air resistance) continuously loses energy into an external sink, or (2) a system in thermal equilibrium at zero temperature minimizes its energy because it is coupled to a much larger external heat bath which gains so much entropy from its energy that the total entropy is maximized when all the energy leaves the system and goes into the bath. In both cases, the idea is that the entropically favored situation is the one in which all the energy leaves the system and goes into some external "environment." But by definition, the universe cannot be coupled to any external environment, so there is nowhere outside of the universe for the energy to go. So why do we expect low-energy configurations to be any more likely than high-energy ones?

## 2 Answers

As understand it, the 4D string landscape is a function that assigns an energy to every possible compactification of the 6 small spatial dimensions. We expect our universe to lie in a local energy minimum, and if there is a lower minimum at another compactification, then our universe would only be metastable, because we expect that it would eventually quantum tunnel into the lower minimum.

That's a very strange description, and it is important to first get the terminology right:

• Perturbative string theory is defined by a conformal non-linear $\sigma$-model taking values in a ten-dimensional target manifold $M$, on which it couples to background fields $G,B,A,\Phi$, which are the target space metric, the Kalb-Ramond field, an ordinary background gauge field and the dilaton, respectively. Conformal symmetry of the model imposes the vanishing of all $\beta$-function and yields "string equations of motion" for the VEVs of the background fields. Solutions to these equations together with a fixed geometry for the target $M$ are called vacua of string theory, since they correspond to classical solutions of the effective QFT on the target space that approximates string theory at low energy. Most strikingly, the string e.o.m. for the background metric is, to first order, the Einstein equation.

• The landscape of string theory is now the collection of all these vacua, or, more restrictively if we want a nice 4D phenomenology, the collection of all vacua witha target $M$ that decomposes as $M^{(4)}\times M^{(6)}$ where $M^{(6)}$ is a compact "internal" manifold and $M^{(4)}$ Minkowski, deSitter or anti-deSitter space. There is intrinsically no "function" that assigns energies to any of these vacua.

Pure string theory itself has no notion of dynamics for these vacua - they are not states in its quantum theory, and they don't tunnel. The background in the standard string theory description is arbitary, but fixed, since it is a starting point for string perturbation theory. However, much in analogy to the vacua in usual QFT, one might conceive of these vacua themselves being stable/meta-stable/unstable (e.g. like the zero-VEV vacuum for the Higgs field) and changing dynamically. I don't know whether some more detailed description of such dynamics is known.

However, the "energy function" you are likely referring to is "obvious": All the effective SUGRA theories on some background have a cosmological constant. It's the value of the cosmological constant - possibly together with a function of overall flux of the higher gauge fields - that one could take as the "energy" of the chosen vacuum. Our standard heuristic understanding of general physics then suggests that this be minimized for stable vacua.

Finally, let me remark that "compactification" is a somewhat misleading name. We are not starting from some non-compact space and compactifying it. We are from the outset searching for soutions to the string background equations of motion, and one such solution, if the internal space is compact, is called a "compactification", but there is generally no process involved that would obtain that manifold from a general non-comapct manifold. This can be seen particularly clearly for fluxless backgrounds on whom preservation of SUSY is imposed - the internal space there is a Calabi-Yau manifold, and those are studied in their own right, not obtained by an actual compactification "process".

• A quick Google search suggests that a standard part of the current understanding of the string landscape is that the compactification could indeed change over time and evolve toward compactifications which produce a lower vacuum energy. For example, this article says "The geometry of the small dimensions will try to adjust to minimize this energy, just as a ball placed on a slope will start to roll downhill to a lower position." This is the mechanism I would like explained. It looks like they flesh out their argument ... – tparker Jul 29 '16 at 1:52
• ... here, but this article is too technical for me to digest. – tparker Jul 29 '16 at 1:52
• @tparker: That article uses instanton contributions to change the background. Perturbative string theory does not see such contributions, and no one knows what the correct non-perturbative formulation of string theory is, so while this is a proposal of what it should contain, I'd mark that as an area of active research, not something we're 100% sure is part of string theory. In particular, they use 4-form field strengths that don't always exist in "standard"string theory. The point is that we want the background to be dynamical, so you'll find lots of proposals for that. – ACuriousMind Jul 29 '16 at 10:41
• Okay, then my question remains: in this paper and other proposals for making the background dynamical, what is the physical argument for tunneling from higher- to lower-energy vacua being more likely than the reverse? – tparker Jul 29 '16 at 13:10
• @tparker: That's a question that can only be answered in the specific models, not in general. In this case there seems to be "brane nucleation" happening that drives down the energy, but there's no mechanism for branes to vanish again. If you wish, the energy formerly in the cosmological constant now sits in the brane tension, and the brane won't let go of it. Please ask a new question about those specific papers if you want detailed explanations of the models therein. – ACuriousMind Jul 29 '16 at 13:16

I think your question can be answered without referring to the specifics of String Theory. What is the physical mechanism driving systems to a minimum of their potential energy? It is the principle of least action.

In these considerations of the String Landscape, one looks at the scalar potential of the 4d effective theory, since the values of the scalars (moduli) in the theory will determine key properties of the theory, e.g., the compactification volume, string coupling etc.

Now, we are interested in stable solutions where all the scalar fields have constant values. Their kinetic energy can dissipate through Hubble friction or decay into light, non-scalar degrees of freedom. Note: in a dynamical background spacetime, energy is not conserved! Energy conservation comes from the time translation invariance, which is broken through the expansion of the universe.

In the end, technically, one equates the Lagrangean to the scalar potential and has the action $$S = \int \mathrm d^4 x \mathcal L = -V.$$ Now, throughout the study of field theory, the stable points of a theory were always the extrema of the action. There is no reason to assume otherwise in this case, ergo, the stable points are those that extremise $V$.

Final note: since Lagrange we know that extremising the action gives the physical equations of motion. This is postulated and works well in all theories constructed so far. We have not figured out the deeper reason why the action should be maximised (i.e. potential energy minimised), though.

• There are several different parts of the answer, whose relationship I find unclear. (1) The principle of least action only holds in classical field theory, not quantum field theory. The action-minimizing field configurations contribute most strongly to the path integral, but the higher-action field configurations contribute as well, with a weight $e^{i S/\hbar}$. – tparker Oct 7 '16 at 2:38
• (2) I certainly agree that time independent, classical solutions to the equations of motion must minimize the potential energy, but assuming time independence is begging the question. Why don't we consider time-dependent field configurations that also extremize the action? – tparker Oct 7 '16 at 2:42
• (3) You mention "Their kinetic energy can dissipate through Hubble friction or decay into light, non-scalar degrees of freedom" in one sentence, which is the heart of my question. According to these notes, the friction term reverses sign in a contracting universe. What values would the fields take in such a universe? – tparker Oct 7 '16 at 2:44
• @tparker (1) The higher-action field configurations give the quantum corrections to our classical field theory, after integrating over all possible field configurations. Still, the equations of motion are defined through extremising the action. – Neuneck Oct 7 '16 at 14:47
• @tparker (2) If you allow time-dependent configurations, you should compute the time-scale on which significant changes to your background spacetime happen. This is why meta-stable vacua, which are in principle time-dependent as they can and possibly will decay, are considered as backgrounds, if the decay time is longer than the age of the universe. For any field configuration that is not a local minimum of the potential energy (possible with the exception of slow-roll) the timescale would be of the order of the inverse energy scale (compactification scale), in contrast to observation. – Neuneck Oct 7 '16 at 14:52