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Did God have any choice in creating the world? So asked Albert Einstein. He was
being poetic. What he really meant, was whether the universe could have been any other way. Could
it have had different laws of physics, driven by different fundamental constants. Or is this one
vast and complex universe the inevitable result of an inevitable and unique underlying principle,
perhaps expressible as a supremely elegant Theory of Everything. It certainly seems
that Einstein thought this should be the case … that God had no choice in whether
or how to create the world. It seems like a pretty arm-chair philosophical
and perhaps unanswerable question, but the modern “problem” of naturalness may lead to an answer.
time caps, one with a white hole and one with an all black hole. Links in the description.
Naturalness or fine-tuning problems in general are those in which some property of the universe seems
oddly specific—as though whatever process decided on the parameters of the natural laws cared that
this particular property has the value it does. We’ve been talking about this a bit, including
in two recent episodes where we discussed why many physicists think that the small value of the mass
of the Higgs boson seems unnatural—this is called the hierarchy problem. The idea, if you recall,
is that there should be processes influencing the Higgs mass that operate at very high energies,
which should in turn lead to a very high Higgs mass unless there’s some precision suppression
of those influences. Those influences are interactions with quantum fields,
and the suppression is supposed to be that they cancel each other out. No “natural” mechanism has
yet been discovered to achieve such high precision and coordinated cancelling, but the alternative to
coordinated canceling is fortuitous cancelling. If the latter is true then it looks like the various
cancellations were “finely tuned” to achieve that small Higgs mass, which feels unnatural.
Another supposed fine-tuning problem is the apparently very small value of
the cosmological constant. So we observe that the expansion of the universe is accelerating,
and a possible culprit is vacuum energy—a faint energy density in the quantum fields that fill
all of space. We call this influence dark energy and describe it with the cosmological constant
in the Einstein equations. And for basically the same reason that physicists are confused by the
low Higgs mass, many are also confused that dark energy isn’t extremely strong
due to the contributions of high-energy components of those fields. And again,
quantum cancellations could reduce this energy. It’s not such a stretch to imagine a perfect
cancellation if there’s a high level of symmetry in those contributions. But to
cancel almost but not quite perfectly seems an oddly specific result. Finely tuned, in fact.
So far we’ve been talking about these issues in very mechanistic terms.
That’s a potential source of confusion, because in a sense the mechanisms don’t
matter. There’s a much more general way to express the problem that gets to the
heart of the questions we started with: Is our very particular universe inevitable?
I’m going to get to that after a few more words to put the mechanistic picture to bed.
Quantum field theory describes essentially everything in terms of myriad interactions
between quantum fields, expressed in this theory in terms of virtual particle interactions. Many,
perhaps most physicists understand this virtual stuff to be just a mathematical tool representing
the messy interactions of quantum fields. But even if the Feynman sums in QFT are mathematical
fictions, we can still consider the complex field interactions they represent as being “real” in
some sense, whatever real really means. The astonishing success of the standard model of
particle physics, itself a QFT, means we kind of have to take quantum field interactions seriously.
That includes ideas like field interactions contributing to particle mass and even the
idea of field interactions canceling each others’ influence. There’s also independent evidence that
this “virtual” activity is in a sense real—for example the Casimir effect, in which a measurable
force is induced by excluding supposed virtual field modes from between two conducting plates.
So, the clockwork behind the Standard Model—the mechanism of quantum field interactions and
cancelations—make very sensible predictions in some cases, and nonsensical ones in other cases.
In particular, prior to being fixed, the QFT behind the standard model predicted the Higgs
mass and dark energy should be enormous. Can we just ignore those predictions?
In the case of the standard model, the answer was yes. At least, if all we want to do is to
use the theory in its domain of applicability. The supposed exploding masses of the particles
can be dealt with if we just input by hand their true masses as measured in the lab.
The process is called renormalization, and it involves adding some made-up canceling
terms to get the exploding masses back down to where we know they should be. With this change,
those particles' masses are no longer predicted by the theory,
but rather become free parameters. In this way the Standard Model became internally self-consistent.
It doesn’t predict very large masses, but that’s by construction. It’s the unconstrained,
un-renormalized quantum field theory underneath the standard model that predicts large masses.
During the development of the Standard Model, some physicists had hoped that the quantum field
theory on which the model is built would be able to fully explain the masses of the particles. They
were not pleased that this feature didn’t make it into the release version of the theory. Richard
Feynman, who led the development of quantum electrodynamics—the electromagnetic part of
the standard model—called renormalization a “shell game” and a “dippy process”.
He said “such hocus-pocus has prevented us from proving that the theory of
quantum electrodynamics is mathematically self-consistent”. Paul Dirac said he was,
“very dissatisfied with the situation because this so-called 'good theory' does involve
neglecting infinities which appear in its equations, ignoring them in an arbitrary way.”
There are two takeaways here. One is that the underlying mechanics of the Standard Model does
lead to confusing stuff about the particles masses if you remove the hocus-pocus of renormalization.
And some would argue that sans-hocus-pocus, if you take this mechanism seriously, there
appear to be finely tuned cancellations in order to get the masses reasonable.
The other takeaway is that the Standard Model fails to realize the dream of Einstein, Feynman
and Dirac. They, and perhaps we, want a singular ultimate theory, one from which every property
of the world should be derivable in a non-dippy, non-arbitrary, and mathematically self-consistent
way. The Standard Model isn’t that, and so we can imagine a more true, more general theory—perhaps a
theory of quantum gravity—of which the Standard Model, general relativity, and everything else
are slightly crappy approximations, each with only a limited range of validity.
We talked about this stuff last time—how we have these layers of theory, with theories
describing larger scales and lower energies being “course-grainings” of deeper theories.
But all of those so-called effective theories are entirely defined by the theory at the bottom,
at least that’s how the world works in the reductionist paradigm. We like to call the
more course-grained, more zoomed-out, more emergent theory the infrared or IR theory,
while the deeper, more fundamental theory is the ultraviolet or UV theory.
The reason is by analogy with the ultraviolet catastrophe, which was in the last episode.
So let’s say we have an IR theory that’s a course-graining of a UV theory. Both
are defined by their own set of parameters. For example, there are 19 free parameters in
the Standard Model that seem arbitrary in the context of that theory alone. The free
parameters of the IR theory are not free in the context of the UV theory from which it comes.
Instead they are “calculable” in principle from the parameters of the UV theory.
What are the parameters of the deepest UV theory? We don’t know. Knowing that fully means knowing
the Theory of Everything. But we can at least imagine that the UV theory of our universe lives
in some sort of platonic space of all imaginable such theories. The UV theory that generates our
universe has a specific location in that space defined by its specific parameters. If we locate
that specific UV theory and you should be able to calculate the precise parameters of the IR theory
that inevitably emerges from it. At least if Einstein has his way. Einstein would also like the
UV theory itself to be inevitable, but really we have no idea how those parameters were “chosen”.
OK, how does naturalness play into this picture? Imagine making a new universe by choosing a random
spot in theory-space. The resulting UV theory defines the IR parameters like the Higgs mass
etc. You haven’t chosen your IR theory with any intention to get specific IR parameters,
so you probably wouldn’t expect them to be unusual or interesting in any way. If
they are, it would seem unnatural … not as random as you imagined.
Let’s try an analogy. You shoot an arrow into a barn wall. Blindfolded. You look to see where
you hit—just some random patch of wood. Do you act all surprised, like, “What are the chances
that I hit this square inch in particular? How lucky!” No, you had to hit somewhere,
and that’s where you hit. Perfectly natural. On the other hand, imagine if there was a target
painted on the wall with a very tiny bullseye. You hit the bullseye by chance without even knowing
it was there. You’d be justified in thinking it a crazy fluke, and not “just some spot on the wall”.
In case you missed it, in this analogy the barn wall is the space of possible UV theories.
Each spot on the wall is a different set of parameters for that theory—each a different
underlying master theory leading to a different universe with different IR
parameters. The bullseye is a very particular UV theory, and it's special because it happens
to lead to a very special IR theory … in our case a small Higgs mass and small cosmological
constant. And the blind archer? God, perhaps? Or perhaps we could think of it as representing the
mechanism by which the parameters of the UV theory are set. This could be random,
meaning the whole barn wall is fair game. Or those parameters could be set by some mechanism that’s
intrinsic to the UV theory—a mechanism that only permits a single outcome—only one spot on the barn
wall was ever possible in the first place. And once you know the UV theory, you see that no other
set of UV parameters are possible. That seems to be what Einstein preferred. God has no choice.
But whether the UV parameters are set randomly or are unique and inevitable, the archer is
still blind and we do still have the problem of fine tuning. How so? Well, the blindness of the
archer doesn’t really represent the freedom of the UV theory to have different parameters. What it
really represents is a combination of two things: 1) Our own lack of knowledge of those parameters.
We know nothing about them, and so they could be anything—the whole barn wall is open as far as
our prior knowledge is concerned. And 2) the fact that the UV theory is blind to the IR theory. So
even if the mechanisms that set the parameters of the UV theory have no choice in where they land,
they aren’t doing that with the emergent IR theory in mind. At least, that’s the default assumption
of reductionism. So that’s the situation we’re in—we find an arrow in the bullseye, and it seems
that the arrow was put in place by a process that had no idea that the bullseye existed.
We can talk about all of this in terms of Bayesian reasoning—by assessing likelihoods relative to our
prior knowledge. When physicists talk about a “prior distribution” for the parameter
space of the UV theory, they typically aren’t talking about some real process that sets the
parameters of that theory. Instead, they usually mean a probability distribution
that quantifies our degree of knowledge and ignorance about some aspect of the world.
It’s the possible range for the parameters given what little we know what we call a Bayesian prior.
In reality the true parameters of the UV theory are what they are,
presumably a set of well defined values. But with little prior knowledge, their possible
distribution with respect to that knowledge is vast. In our Bayesian reasoning we’d use
a wide Bayesian prior because its job is exclusively to represent our knowledge.
Now let's say we know that this wide-open UV theory-space maps to some as-yet unmeasured
infrared parameter. We go to measure the parameter and ask “how much of the theory
space in our Bayesian prior could produce an IR parameter like the one we just measured?”
If the answer is a vanishingly small fraction, then we might be suspicious. That’s the
case with the Higgs mass and cosmological constant. It’s the arrow being suspiciously
in the bullseye. We would be justified in assuming that the archer actually peeked,
or that there’s some powerful collusion in the UV theory that propagates down to the IR theory,
in this case working to make the IR theory the low-energy theory that it is.
And it doesn’t really matter if the mechanism for suppressing IR parameters such as the
Higgs mass is quantum cancellations or anything else. As long as our UV
theory natively lives at much higher energy than the IR theory, either collusion between
the high-energy mechanics or extreme chance is needed to yield that stable low energy theory.
So that’s why it is reasonable to wonder at things like the smallness of the Higgs
mass and cosmological constant. Not because they’re actually unnatural, but because they
seem unnatural given the mechanisms we think are at work behind them. Nature is obviously being
perfectly natural, and so it means we’re missing something. Either the archer peeked—that’s the UV
theory being influenced by the IR theory—it sees the target. Or the archer shot many,
many arrows and the whole barn wall is a pin-cushion of alternate universes and
of course we’re in one of the extremely rare bullseye universes with Survivable higgs mass
and so on. I keep saying we’ll come back to these ideas of UV-IR mixing and anthropic reasoning,
and we still will. But the choice between these two may be our answer to Einstein’s
question. Did God have a choice in creating the world? If no then there was only one arrow and
that seems to demand a connection between the fundamental and the emergent that we don’t fully
understand. If yes then the arrow could have landed anywhere on that wall and perhaps did,
solving naturalness but generating a multiverse. Of course, in that case we can just redefine the
multiverse as the world and say that it is singular and inevitable and Einstein gets
to be right again. Just like he was with all that stuff about curved spacetime.
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