Digesting a Paper on Flatness (Part 3)

 See Part 1 to understand what this is about.

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The partition function of a statistical ensemble can be represented by a path integral over configurations that are periodic in imaginary time.

The mention of a path integral indicates that we are talking about the partition function in relation to quantum field theory here rather than statistical mechanics, or as Wikipedia helpfully puts it:

In quantum field theory, the partition function {\displaystyle Z[J]}Z[J] is the generating functional of all correlation functions, generalizing the characteristic function of probability theory.

Um, ok.  That doesn’t really help.

I think it’s entirely possible to get hung up on this sentence overly long.  For the moment, therefore, suffice it to say that we are talking about a partition function.  A partition function is the normalising constant applied to the probability equation for a microstate.

Think about statistical mechanics of, for example, a gas.  A gas consists of a multitude of particles (molecules usually, atoms if it’s a noble gas).  Knowing the microstate would require an exhaustive cataloguing of all the molecules, how many there are, what they are, where they are and what momentum they all have.  Knowing that would give you other characteristics of the gas (pressure and temperature for example).  But we never know all that and the microstate is changing all the time.  Instead, we can think of the likelihood of an ensemble of microstates (a bunch of them).  The sum of all probabilities has to be 1 – hence the normalising factor.

What I think they are getting at here is that some microstates are more likely than others and as you integrate more microstates, you arrive at a more likely overall state (or “path”) for the system to be in an eventually you can say “I don’t know what the precise microstate is at any one time, but the temperature of the gas is this and it’s pressure is that”.

Go even deeper and think about things at the quantum rather than molecule level, and you have the same sort of situation.  It is not possible to know the precise microstate (quantum state, maybe) of a system, especially when you have superposition of different states that don’t manifest until there’s been a measurement, but you can integrate the likelihood of states to arrive at things that you can say about the overall state.  The sum of the probabilities of all quantum states is going to be 1 again, and you therefore need a normalising factor, which would be the partition function.  The partition function would, therefore, tell you something about the likelihood of each ensemble of states (or configurations).

What I expect happens is that there is a peak of probability at what constitutes reality (which is a bit fuzzy across a relatively narrow range) – other states are not impossible, but they are very unlikely.

Why this occurs in imaginary time is not immediately obvious.  It might be worth noting what imaginary time is though.

There is what is called a Wick rotation, which converts the geometry of spacetime from Lorentzian to Euclidean.  This converts the metric for spacetime from ds² = - (dt)² + dx² + dx² + dx² to ds² = (i.dt)² + dx² + dx² + dx² (because i² = -1).  Note that this sort of metric is implied in On Time where I talk about an invariant “spacetime speed” (although I deliberately collapsed down the spatial components to just one x dimension for ease of calculation).

The phrase “periodic in imaginary time” is a reference to the alignment of temperature (in statistical mechanics) with oscillations in imaginary time (in quantum mechanics) – in that the related equations are of the same form.

Hawking and collaborators first used these methods to investigate black hole thermodynamics, a topic which has since burgeoned into holographic studies [10, 11] and experimental tests using analog systems [12, 13]. We shall follow Hawking et al.'s original approach here [14, 15, 16, 17].

This is a reference to the idea that the entropy of a black hole relates to its surface and not its volume (this seems rather obvious when it’s clear that time and space go to zero at the surface of a black hole – there isn’t anything “inside” a black hole in terms of our universe, everything that is going on with it is going on either at or above the surface [where particles, if there are any, are ripped apart and thus give off radiation]).

In the cosmological setting, it is an excellent approximation to treat the radiation as a relativistic fluid, in local thermal equilibrium at a temperature declining inversely with the scale factor.

I think this is just talking about radiation (composed of photons) being conceptually similar to a fluid that moves very fast.  If you’ve ever heard a term like “awash in photons” (probably in science fiction), then you have the idea of a relativistic fluid.  It should be noted that there are equations for fluid flow, which are analogous to equations for electricity, because electrons in a wire act a bit like molecules of oil in a hydraulic system and the same analogy can be made to radiation.

The instantons we present are saddle points of the path integral for gravity: real, Euclidean-signature solutions to the Einstein equations for cosmologies with dark energy, radiation and space curvature.

We’re talking about four-dimensional manifolds again.  I do not believe that the saddle points mention are in those manifolds but are rather points of high likelihood (and thus reality) among the possible (micro or quantum) states and the argument is that those states manifest gravity of the sort that we observe in our universe.

The associated semiclassical exponent iS/ħ is real, large and positive, and may be interpreted as the gravitational entropy.

The exponent iS/ħ appears in the equation for the “generating functional Z(J)”, where a generating function is a way of representing the sum of an infinite sequence of numbers, but in this case, we are representing a space of functions – it’s basically a description of the machine (think of a black box with an input and an output) that does something to something.  For what it’s worth, this is that equation (taken from here) – note that they have an erroneous close bracket after the final x, but I've removed it for the replication below:

or

It’s written slightly differently here (with no erroneous close bracket):

Note that e^ix cycles eternally between 1, i, -1 and -i as x increases in magnitude from zero (1 where x = 2nπ, i where x = (2n+1/2)π, -1 where x=(2n+1)π and -i where x = (2n+3/2), where n is any integer).  Therefore, with either equation, iS/ħ is a phase offset.

In these equations, Φ refers to a scalar field – which is our manifold again (the d⁴x tells us that we are using a four dimensional scalar field) – and the DΦ is telling us to “integrate over all possible classical field configurations Φ(x) with a phase given by the classical action S[Φ] evaluated in that field configuration”.  J or J(x) is an auxiliary function, referred to as a current.

But what does all this mean?

In the words of Vivian Stanshall, about three o’clock in the morning, Oxfordshire, 1973: In layman's language, “It's blinking well baffling. But to be more obtusely, buggered if I know. Yes, buggered if I know.”

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A couple of clarifications:

Z(J) is a functional integral (space of functions) while S is an action functional.  An action functional is related to the stationary-action principle, which when put very, very simply (perhaps to the extent of being simplistic) is about things not doing more than they have to.  If an object is moving in a particular direction at a particular speed (so, with a velocity) it will continue to move at that velocity unless acted upon by a force.  We know about a lot of forces that act on objects to change an object’s velocity, gravity, resistance (air, friction), other objects colliding into it, etc, etc.  Digging into the concepts of relativity, each object has its own frame (of reference) in which it is stationary (with respect to itself).

In a sense, every single object in the universe has a spacetime speed (note, not velocity) which is invariant.  The actions between objects could be worked out by considering how the spacetime speed is distributed between their relative temporal speed and their relative spatial speed (see On Time).  Effectively, every single object in the universe could be considered as stationary (different types of “stationary”, admittedly, but stationary all the same).

The statement “(t)he associated semiclassical exponent iS/ħ is real, large and positive” is interesting in itself.  Given that i is the imaginary marker and ħ is a real, small and positive constant (about 10^-34J/K), that would make S itself imaginary (not complex), large (or even just average) and negative.  Negative is understandable, if it’s representing gravitational entropy which is understood to be negative … but imaginary?  Not quite sure about that.

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Hopefully it’s quite obvious that this series is more of an open pondering session rather than any statement of fact about what the authors of Gravitational entropy and the flatness, homogeneity and isotropy puzzles intended to convey.  If I have misinterpreted them, then I’d be happy to hear about it.