Another question from St. Paul, the reader from New Zealand:

I hope that you are well and that you are having fun with your work. I see that you posted our email exchange on your blog, it was a great answer and much appreciated.

I actually have another question I would like to ask, (although I realise that you may well be planning on writing about it already) but as always I completely understand if you don’t have the time!

I’ve been reading about the recent detection of gravitational waves and how they confirm the theory of inflation. What I have found interesting is that there have been quite a few articles reporting that most models of inflation imply the existence of a multiverse, with quotes from Alan Guth, etc. I realise that the term “multiverse” can be used for several quite different situations, but they seem to be referring to one with variation of the laws of physics, meaning the anthropic principle can be invoked.

I was wondering what you make of this new discovery and what your take on the idea of the multiverse is? I have always felt that the fine-tuning argument was a helpful pointer to God, so I am curious about the implications of confirming inflation (although the existence of multiple universes certainly doesn’t rule Him out).

Thank you for your time,
Paul

A quick explanation concerning “fine-tuning” and the “multiverse”.   Fine-tuning refers to the observation that the fundamental constants of Nature seem to take special values which appear to be necessary to the existence of life.   The fine-tuning argument is a theistic argument which claims that this is good evidence for the existence of God.

One common atheistic retort is to say that maybe there are lots and lots of universes—with different laws of physics in each universe—and that any observers would therefore have to live in the universes which permit life.  This idea is a called the multiverse.

This may sound like crazy science fiction thinking, but I actually think it is the most plausible naturalistic response, given what we now know about physics.  Although there is no really good reason to believe in the multiverse, it seems much more plausible then any of the attempts to construct physical mechanisms to account for this fine-tuning.

However, it is not really clear to me that the multiverse is the sort of thing that ought to count as an explanation for fine-tuning.  In some moods it seems to me like cheating.  Science normally works by postulating theories to fit the observed data, not by postulating (new and unobservable) data to make the theories we have seem less weird.

In fact, there are in fact some serious controversies as to how to properly do Bayesian reasoning in the context of a multiverse.  Pretty much all viewpoints lead to some horrendous paradoxes.  Since the proper way to do probabilistic reasoning in this context is unclear, it is also unclear to what extent the multiverse would be an explanation for fine-tuning.  But this is a complicated question I don’t have time to go into right now.

Instead, Paul asks the different question of to what extent the multiverse is supported by real, actual Science.  In particular, the very recent results from last March about inflation.  For those of you who have been living under a rock, there was a recently announced experimental result in cosmology.  The BICEP2 experiment claims to have seen the gravitational waves resulting from inflation, a very early period in our universe’s history where the size of the universe expanded at an extremely quick, exponential pace.

[Update: the BICEP2 results have since been discredited.]

I wrote to Paul roughly as follows:

Most models of inflation predict “eternal inflation”, meaning not that there wasn’t a beginning, but that in some regions of the universe, inflation continues forever towards the future.

In order to have a multiverse of the sort that might be conceivably relevant to fine-tuning, you need to meet two criteria: (a) a mechanism for producing gazillions of different universes (at least \(10^{150}\) without supersymmetry, or \(10^{60}\) with supersymmetry), and (b) in these different universes, there are an equally large number of different effective parameters describing the low energy physics in each of the universes.

Eternal inflation is conducive to (a) insofar as it would result in widely separated regions which can never causally communicate with each other even at the speed of light.  But it does not by itself do anything to meet condition (b).  The best argument for (b) is probably string theory, which seems to have gazillions of different types of metastable vacua, but there is currently no successful experimental predictions for string theory.  (String theory does seem to imply the existence of gravity, but that’s more of a retrodiction, and isn’t unique to string theory…)

A friend of mine from St. John’s College, who was recently accepted to a physics doctoral program at Penn State, asked me what my opinion of Loop Quantum Gravity is.  I replied be email, and then I decided, why not tell the world!

Now, Loop Quantum Gravity is the main rival to String Theory as an attempt to quantize gravity, although it only commands about a tenth of the resources that String Theory does.  The people who work on it tend to have more of a General Relativity background than a Particle Physics background, and this tends to influence what types of problems they are trying to solve.

Warning: Unlike my other physics posts, I have made no attempt to make my commentary here accessible to non-physics people.  (Yes, that means every other time I wrote a physics post and nobody understood it, I was the one to blame for not making it accessible enough!)

Einstein’s theory of general relativity is background free, meaning that it does not start with any absolute background space or time, but instead allows the spacetime geometry to be dynamically constructed from the evolution of the metric.  A theory of quantum gravity ought to be similar—it ought to be expressed in a way which doesn’t depend on the prior specification of any spacetime metric.  I think this is really important, but no one really knows how to do this.  There are many ideas, but they all have various difficulties.

In principle, I think the idea of LQG—to build spacetime out of a discrete, quantum structure—is a very elegant and moving idea.  (I first got interested in quantum gravity by reading the online writings of John Baez, who used to work on LQG.)  Also, the LQG people have a very beautiful quantization of space at one time, in terms of spin networks.  Essentially, by doing a step-by-step quantization of GR at one time (minus the dynamics), making only a few arbitrary choices, they were able to obtain spin networks.  I’m sure you [i.e. the friend I was writing to—AW] know what these are, but let me assure you that they are beautiful and have some deep connections to geometrical ideas.

The next step in the construction of LQG is to decide what the dynamics are.  Technically, this is done either (A) by choosing a “Hamiltonian constraint” in parallel with the Hamiltonian formulation of GR, or (B) in the spin-foam formalism, by postulating some sort of sum over histories assigning an action to each spin foam.  It is here which we encounter the major problem: There is no agreement over how to implement the dynamics!  There are many ideas, but no consensus on what to do.  Implementing dynamics seems to involve some arbitrary choices.  Some of the proposed solutions seem to me obviously wrong (e.g. see Smolin’s criticism of Thiemann’s Hamiltonian constraint: arXiv:gr-qc/9609034).  There is also a serious danger that by choosing the wrong dynamics, one breaks the diffeomorphism invariance of the theory.  In the Hamiltonian approach this manifests itself in so-called “anomalies in the constraint algebra”, while in the spin foam approach it is unclear whether the inner product obtained from the sum over histories really has the necessary gauge invariance.  I summarized these problems in passing, with citations, in the Introduction to this article of mine: arxiv:1201.2489.

Thus—even leaving aside the critical hard problem of whether and how a continuum spacetime can emerge from a discrete description (a problem aggravated by the fact that it is difficult to see how any discrete model of spacetime besides causal sets could possibly preserve Lorentz invariance, see arXiv:gr-qc/0605006)—I would say that LQG really doesn’t exist yet as a well-defined theory.  Unless you consider dynamics to be an unimportant part of a theory.  And finding sensible dynamics is a really hard problem, perhaps impossible.

Yet, despite the lack of dynamics, there’s no end of papers where people do specific applications, like count black hole entropy, or even attempt to do quantum cosmology (basically by truncating the theory to a finite number of degrees of freedom, and then quantizing those degrees of freedom in a way which is “loopy” in spirit).  But all of these things are totally provisional until one can embed them in an actual theory with dynamics.   People used to be really interested in solving these hard problems, but I feel like a lot of them have now given up and are seeking more limited goals.  This is a shame, since I think progress can only come by facing the hard issues head on.  And maybe by showing some flexibility in how the theory is formulated.

Once one has the dynamics, again one can say nothing about the real world until one has identified the correct vacuum state.  An arbitrarily constructed “weave” state that happens to look like some Riemannian geometry doesn’t cut it.  You have to figure out how to identify the right vacuum state—the one with lowest energy (once you figure out how to define that!).  Many deep questions here!  I think most people in LQG are asking all the wrong questions.

One can put too much emphasis on quantizing gravity—really that’s backwards, we need the classical theory to emerge from the quantum theory, not vice versa.  When people calculate discrete area and volume spectra for spin network edges and vertices, they’ve got things backwards.  These are just some operators at the Planck scale.  The really interesting question is not, how much “area” is associated with each spin, but how many of each type of spin crosses a given area of the vacuum state (if such a thing even exists).

I despise the ignorant bigotry which most string theorists show towards LQG, even though LQG barely exists as a theory.  Their contempt is undeserved.  The LQG people are trying to do something genuinely harder—to reconstruct spacetime from first principles.  We don’t know how to formulate string theory except by means of strings propagating in some background spacetime, or via dualities like AdS/CFT.  Since the theory has gravitons, with a diffeomorphism gauge symmetry, it’s clear to me there has to be some background free formulation of string theory, but no one has any idea what this would look like.  And most string theorists don’t even understand why it is important.

Personally (and unexpectedly for me) I’ve found that as someone who studies black hole theormodynamics, I can interface better with string theorists than with LQG people—the ones who are really interested in fundamental concepts, like Don Marolf and others at UCSB, for example—even though I don’t really consider myself a string theorist.  This may be a bit of a conceit at this point, since I’ve now written multiple papers on AdS/CFT.  My heart is more strongly devoted to the types of ideas LQG people explore, but my mind recognizes that they really haven’t made all that much progress.

Suppose you want to write down the laws of physics.  How would you go about it?

What?  You want to do some experiments first?  Forget about that.  This is theoretical physics.  Let’s not worry about pedantic things like what the actually correct laws of physics are.  Instead, let’s try to ask what they should look like more generally.  What are the ground rules for trying to construct laws of physics?

(Of course, in reality we do get these ground rules from experiment.  The way it works is, we make up rules to describe lots of specific systems which we actually measure, and then eventually we get some idea of what the meta-rules are, i.e. the rules for constructing the rules.  But let’s just try to make something up here, and see how close we get to reality.)

Let’s try to do this step by step.  Let’s take for granted the existence of a spacetime.  In the first step, we need to decide what kind of entites there are moving around in this spacetime.  Since we’re on the hook for giving an exact description, we’d better start with something which is mathematically simple.  For example, we could postulate that there are a bunch of point particles flying around.  If there are \(N\) particles, and space is 3 dimensional, then we can describe all of their positions with \(3N\) parameters.  (We can then think of the universe as a point moving around in a \(3N\) dimensional space, called configuration space.)

Or maybe there’s a bunch of strings wiggling around.  Or perhaps there are fields, whose values are defined at each point of space.  (In these cases, we will need an infinite number of parameters to describe what is going on at each moment of time!   But don’t worry—since we won’t be doing any actual calculations, this won’t necessarily make things any harder.)

All right.  Now that we’ve decided what kind of stuff we have, we need to know how it changes with time.  For this we need to write down equations of motion.

We could write down an equation involving one derivative of positions with respect to time.  This would determine the velocity of each piece of particle/string/field/whatever in terms of its position.  But that won’t be like real physics since real physical objects have inertia.  Stuff keeps on trucking until a force acts on it.  This means that the future motion of an object doesn’t just depend on where it is right now, but also on how fast it is going.

So instead we need to write down an equation involving two derivatives of the position with respect to time.  This will determine the acceleration of each entity, as a function of its position and/or velocity.  That’s a bit more like real life.   (In other words, to work out what happens we need to know about both the positions and velocities.  If we have \(N\) particles, this is a \(6N\) dimensional space called phase space.)

So you could just sit down and write down some second-order differential equation equation involving acceleration, and call that the laws of physics.  But most of these would still be qualitatively different from the fundamental laws of actual physics.  For example, nothing would stop you from including friction terms which would cause the motion of objects to slow down as time passes.  For example, if we have a particle moving along the x-axis, we could write down an equation like this:$$\ddot{x} = -\dot{x}.$$This would cause the particle to slow down as time passes.  But in reality, friction only ever happens when some object rubs up against another object.  The motion doesn’t disappear, it just goes into the other object.  This is related to Newton’s Third Law, a.k.a conservation of momentum.

So physics has more rules then one might think are really necessary.  You can’t just write down any old equations of motion.  They have to be special, magical equations, which satisfy certain properties.

We could just make some giant list of desired laws.  But that would be rather ad hoc.  Instead, physicists try to derive all of the magic from some simple framework.  We’ve just seen that just writing down equations of motion is not the best framework since it doesn’t guarantee basic physics principles like conservation laws.

There are two particularly simple frameworks which can be used.  For most systems these are equivalent, and you can derive one framework from the other.  I’m just going to summarize these at lightening speed here:

• Lagrangian mechanics:  Here the fundamental concept is the action, a number $$S = \int L(x,\,\dot{x})\,dt$$ obtained by integrating some function \(L(x,\,\dot{x})\) of the positions and velocities over all moments of time.  (\(L\) is called the “Lagrangian”, and is normally equal to the kinetic energy minus the potential energy).  The basic rule is that a small change \(\delta x(t)\) in the paths of particles/strings/fields/whatever in any finite time interval \( t_\mathrm{initial} < t < t_\mathrm{final}\) should leave the action unchanged, to first order (i.e. up to terms linear in \(\delta x(t)\)).  In other words: $$\frac{\delta S}{\delta x(t)} = 0.$$Here \(x\) can be any of the position parameters in the theory.  Once you write down a single equation specifying \(S\), all of the equations of motion for all entities are determined by this rule.
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  As a simple example, consider a point particle moving along a 1-dimensional coordinate \(x\), with a potential \(V(x)\) which depends on your position.  This might describe a train sliding frictionlessly along a roller coaster track, where \(x\) is the length measured along the track and \(V(x)\) is proportional to its height measured from the ground. The Lagrangian is kinetic energy minus potential energy: $$\frac{m\dot{x}^2}{2} – V(x).$$The rule here is that given the initial and final locations of the train in some short time interval, the train moves in a way that minimizes the total action of its trajectory—which implies by basic principles of calculus that small variations of the path have to leave the action unchanged.
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  Imagine if you are walking from your house to a shop.  You leave your house at 9 am, and you need to be at the shop at exactly 10 am.  You don’t like walking too quickly, because it expends too much energy.  On the other hand, if it’s a bit chilly you might also prefer to spend more time in sunny areas, and less time in shady areas.  What would you do?  If you want to maximize your happiness (or minimize your unhappiness), you would compromise by walking more quickly in the shade than in the sun.  Similarly, if we fancifully suppose that the train had a soul and that it preferred to spend more time up high (so long as it gets to its destination on time), we would then have an explanation for why the train lingers at the higher parts of the track.  More generally, when the potential energy is higher the kinetic energy is less—one can prove that the total energy is conserved.

• Hamiltonian mechanics: The fundamental concept here is that all parameters in physics come in “conjugate” pairs.  For example, for a regular particle, the conjugate variable to the position \(x\) is the momentum \(p = m\dot{x}\), while the conjugate variable to momentum is minus the position, \(-x\).  (That minus sign is important: without it conservation laws wouldn’t work properly.)  The variable which is conjugate to time is known as the “Hamiltonian” \(H\)—this turns out to be nothing other than the total energy of the system (kinetic plus potential).   It turns out that if you know the Hamiltonian \(H(x,\,p)\) as a function of the positions and their conjugate momenta, you can work out everything that happens.  You work out the equations of motion with the rule (called “Hamilton’s equations” that the change of a parameter with respect to time, equals the change of the energy with respect to the conjugate variable.  In other words: $$\frac{dx}{dt} = \frac{\partial H}{\partial p},\qquad\frac{dp}{dt} = -\frac{\partial H}{\partial x}.$$The minus sign in the second equation means that position is to momentum as momentum is to minus position, just like I told you.
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  A consequence of “Hamilton’s equations” is that, assuming \(H\) does not depend on some particular position coordinate \(x\), \(\partial H / \partial x = 0\) and so \(p\) is conserved.  More generally, Hamilton’s second equation says that the “force” \(\dot{p}\) is zero when the gradient (i.e. derivative) of \(H\) with respect to position is zero.  Similarly, if the gradient of \(H\) with respect to the \(p\) coordinate is zero, then \(\partial H / \partial p = 0\), and Hamilton’s first equation says that the velocity \(\dot{x}\) is zero.  If the fomula for kinetic energy is the usual nonrelativistic formula \(p^2 / 2m\) (written as a function of the momentum \(p\) instead of \(\dot{x}\) since this formulation of physics is all about \(p\)’s), this tells us that the “velocity” is zero when the momentum is zero.
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  More generally, Hamilton’s equations tell you that if you graph out the 2 dimensional phase space of a particular pair of x-p coordinates, the trajectory of the system in the x-p plane is at right angles to the direction of the gradient of \(H\), and equal in size to the gradient.  This means that the system always moves along a direction where \(H\) isn’t changing, and so \(H\) is conserved (unless we make it an explicit function of time, in which case we would have to write it as \(H(x,\,p\,t)\)).

From either of these two equivalent formulations of physics, there is a famous theorem first proved by Emmy Noether.  She showed that any time \(H\) or \(L\) has a symmetry which shifts some parameter, its conjugate parameter is conserved (it doesn’t change with time).  I’ve already shown you some specific examples (symmetry with respect to \(x\) shifts makes \(p\) be conserved, symmetry with respect to \(t\) shifts makes \(H\) be conserved).  This is the most important theorem in all of theoretical physics.

If you just start by trying to write down equations of motion for your laws of physics, you can’t prove Noether’s theorem.  It just doesn’t work.  Since you don’t have a notion of conjugate quantities, you can’t even get started.  Many important physical concepts such as energy, momentum, mass, force, and so on won’t even be defined.  So there’s a lot more to life than the equations of motion.