Background
Let qA and qB represent unit, orientation quaternions of grain A and grain B of a grain boundary in the lab reference frame, respectively. Let qm be the misorientation quaternion of qA and qB. Quaternion inversion (qinv) and quaternion multiplication (qmult) are discussed in Quaternion...
@JohnRennie It’s said that quantum fields do not fluctuate. Does that mean eternal inflation is not valid because it relies on quantum fluctuations making the field’s value fluctuates?
We need to clarify a few points first. If I don't do this I'll get shouted at by other members of the chat room :-)
In quantum field theory the "quantum field" is a purely mathematical object that we can use to calculate what happens. It is not a real physical field.
But we assume that there is some real physical "quantum field" that our maths is describing. And when we talk about the inflaton field it's this real physical field we are talking about.
But it's important to be clear that no-one knows what this real physical field is or even if it exists at all.
In eternal inflation we talk about an inflaton field that is driving the inflation. Then at random points the inflaton field decays to form a region of normal spacetime like our universe. Yes?
And like any superposition this can collapse to give either a decayed or a not decayed state.
The mechanism for collapse of superpositions is not understood, or rather there are many conflicting theories and we don't know which if any is correct.
What happens in eternal inflation is that the state of the inflaton field randomly collapses at some point to create a bubble of normal spacetime at that point.
It's convenient to talk about the field "fluctuating" and we all do this, but what we actually mean is that the field exists in a superposition of different states. It isn't really fluctuating.
OK. As a layman apologies if i’m talking nonsense.And how does that the fact that the field’s value is in superposition of states, not that that the field’s value “actually” fluctuates
And how does that the fact that the field’s value is in superposition of states, not that that the field’s value “actually” fluctuates make inflation eternal?
At any particular point in space there is some probability that the inflaton field will decay to produce a bubble of normal spacetime.
If it does decay we get a bubble universe like ours. But if it doesn't decay the space where the inflaton field hasn't decayed keeps on inflating.
If the probability of decay is small enough the inflation of the undecayed parts of spacetime "outruns" the decayed parts and the inflation keeps going forever. Hence the eternal inflation.
By existing in a superposition of different states, does it mean value of the field at a point is not precisely determined, and hence doesn’t have a well-defined value?
It's one of the hardest ideas for students learning QM to grasp that quantities we take for granted, like position or energy, can be undefined for a quantum system.
sure, it quite literally shatters our classical notion of reality, but perhaps it's simpler to start accepting it for tiny particles before accepting it for an all-permeating field
@JohnRennie But the inflaton field is rolling down the potential. By collapsing into a not decayed state, does it mean the field’s value lift upward to a higher potential energy?
You need to be careful about taking pictures like the inflaton field rolling down a potential too literally.
Quantum fields can exchange energy with each other. In fact it is precisely this exchange of energy that allows particles to be created at colliders like the LHC.
When we talk about the inflaton field decaying we mean energy is transferred from the inflaton field to other quantum fields.
When we talk about rolling down a potential we mean it becomes more and more energetically favourable for energy to be transferred from the inflaton field to other fields like the electron and quark quantum fields.
The field continues to roll down the potential. So how can some points in space still collapse into an undecayed state which means increasing potential energy?
It's not "the field" that rolls down a potential, it's the expectation value of the field that changes from the "top" of the potential to the minimum at the bottom. The expectation value can be different in different areas of space(time)
as John explained, there is no single value for "the field" in a quantum theory
This "rolling down a potential" describes what happens to the field as it is diluted by the expansion of space.
If the original idea of inflation, not eternal inflation, the field is treated as being the same everywhere.
As spacetime expands the field is diluted and this affects the field so the decay probability becomes higher and higher as the spacetime expands. Eventually the probability becomes so so high that the field decays everywhere at the same time.
Well, approximately at the same time, there are small variations from place to place but basically we get a single decay everywhere at once.
If the field rolls down the potential everywhere, how can some parts of space increase its potential energy and stuck on the top of the potential, continue inflating?
As spacetime expands the field is diluted the potential energy decreases. And by some parts of space collapsing into an undecayed state means increased potential energy, right?
In EI the field at any point in spacetime has some probability of the decay starting. Once the decay starts then I guess you could describe the decay process at that point, but only at that point, as rolling down a potential.
> As spacetime expands the field is diluted the potential energy decreases
@JohnRennie And why doesn’t the field decay everywhere in eternal inflation?
“ As spacetime expands the field is diluted and this affects the field so the decay probability becomes higher and higher as the spacetime expands. Eventually the probability becomes so so high that the field decays everywhere at the same time.”
Let me attempt an analogy. Suppose we consider a chunk of uranium (or any radioactive material). Any particular atom in this chunk has some probability of decaying.
The probability that an atom decays is not a function of time. The uranium does not change with time. What happens is at random points in the block of uranium an atom will decay.
Well the inflaton field is like this in eternal inflation. The field has some decay probability that is not changing with time i.e. the field is unaffected by the fact spacetime is inflating under its feet.
What happens is we get decays at random points in spacetime.
“As spacetime expands the field is diluted and this affects the field so the decay probability becomes higher and higher as the spacetime expands. Eventually the probability becomes so so high that the field decays everywhere at the same time.”
In an expanding spacetime energy is not conserved. So while it seems obvious that the expansion must reduce the energy density of the field this is not necessarily the case.
The problem with inflation is that we do not know what causes the inflaton field so we have no idea what its properties are.
So what theorists do is postulate some properties of the field then do the calculation to see what happens.
In regular inflation we postulate that the field is affected by the expansion and we dream up a potential curve that has the right form to give a universe like the one we see around us.
I must emphasise this - the potential curve is made up. It isn't the result of any fundamental principle.
In eternal inflation we postulate that the field is not affected by the expansion.
This is one of the biggest criticisms of the whole idea of inflation, i.e. that it is based on untested assumptions that were in effect just dreamed up by some theorist.
Yes, but slow roll inflation is not eternal inflation.
Once the decay starts at some point in space then the eternally inflating field at that point may transition to a slow roll, but the rest of the field does not change.
@Forge I think you should be very skeptical of any "predictions" made from inflation. The original theory of inflation predicted a universe wildly different to the one we see. Theorists figured out that if they introduced a "slow roll" this produced predictions that match the universe we see.
But this is all a bit of a fudge because this slow roll potential was not derived from any fundamental principle. Instead the form of the potential was messed about with until it produced reasonable predictions.
tbf that's not an unusual way to do physics - many of our theories started their life as phenomenologic guesses and only later we figured out more general principles behind them
Inflation seems a wonderful idea because it solves so many of the problems in cosmology. We'd all love it to be true.
But I'm not sure many physicists would say it has been proved to be true. Only that it is probably true or possibly true, where the strength of the conviction will vary between different physicists.
@Forge That potential curve shows the potential as a function of the energy of the inflaton field. But the field is not in an energy eigenstate, so it is a superposition of different energies. That means it is in superposition of different positions on the potential curve. OK so far?
The undecayed states will be in the plateau region of the curve while the decaying/decayed states will be on the part of the curve where is rolling downwards.
And in the plateau region the field is unaffected by the expansion.
So the field is in a superposition of being unaffected by the expansion and affected by the expansion.
Collapse of this superposition happens randomly throughout spacetime, so some parts of the field carry on being unaffected and expanding eternally while other parts enter the slow roll part of the curve and start decaying to regular spacetime.
The field exists in a superposition, but this can be a wide superposition that spans a large range of energies or a narrow superposition that spans only a small range of energies.
If the superposition is narrow then the field is spread out over only a small part of the curve so the entire field has roughly the same state i.e. if it starts to decay at one point it will start decaying everywhere.
if the superposition is wide then even when some parts of the field are decaying there will always be other parts that are so far to the left on the curve that they are unaffected and just carry on inflating.
So, how do we know if the superposition is narrow or wide?
And ... we don't. Theorists just assume some width for the superposition and crunch through the equations to see what happens.
Does collapsing of superposition into points of space where some are at higher potential energy and some are at lower potential energy violate local conversation of energy in QM?
“ The energy doesn't fluctuate from point to point or from time to time. Energy is locally conserved, it is exactly locally conserved, not on average”
QM is not statistical mechanics. It is not like in classical mechanics, when studying a gas of particles, where you may have thermal fluctuations. The fact that, in QM, you cannot simultaneously measure the position and momentum of a particle is radically different from the fact that, in statistical mechanics, the velocity of the gas fluctuates about the average value.”
With inflation we are doing quantum field theory in a curved spacetime and that complicates matters. I think AFT's statements apply to a flat spacetime but I'm not sure how things change in a curved spacetime.
Remember that I started out by saying that in QFT the "quantum field" is a mathematical object. In QFT we use this mathematical quantum field to calculate scattering processes e.g. what happens if we smash particle A and particle B together hard enough.
And in these scattering processes energy is always conserved, just as AFT says.
We expect this to be true even in a curved spacetime.
But considering the time evolution of the inflaton field is a rather different business. I don't know how this is done or whether we would expect energy to be conserved in the process.
Globally energy will not be conserved because the field violates a fundamental symmetry responsible for energy conservation, but I don't know enough to say anything more than that.
I don't know enough about to comment, though I would say that all models of inflation are pretty speculative.
You will find well respected physicists who believe inflation is wrong and are happy to say so publicly, though I think this is a minority view at present.
Can the fact that tiny fluctuations in the CMB is tiny point to the width of superposition being narrow?
As it decay at about the same time as shown in hot spots and cold spots in the CMB
“ If the superposition is narrow then the field is spread out over only a small part of the curve so the entire field has roughly the same state i.e. if it starts to decay at one point it will start decaying everywhere.”
@Forge well yes, but that only applies to our little bit of the universe i.e. the inflation field decayed at about the same time in the tiny bit of the universe we can see.
We cannot comment on the rest of the universe because we will never be able to observe it.
Remember that the decay only started when the superposition collapsed. So the bit of the universe we can see is a bit where the superposition collapsed.
There are two ways the fluctuations have been calculated. These two ways are based on completely different assumptions, though they manage to come up with the same results. The most widely accepted calculation is that the fluctuations arise from Hawking radiation emitted from an event horizon that is an inherent feature of inflation. In this model the fluctuations are nothing to do with the inflaton field decaying at different times in different places.
There is a small group, I think mostly centred around Sean Carroll, who have created models describing the fluctuations as due to the inflation decaying at different times in different places, but even in this model the decay happens at basically the same time everywhere. The spread in te decay times that the group assumes is very, very small.
You cannot observe a superposition. If you have a system in a superposition and you observe it it always collapses to a specific state.
e.g. suppose you have an electron in a superposition of positions and you observe its position. This collapses the superposition into a state with a definite position.
But actually you can never measure the position exactly. The uncertainty principle forbids that. So when you measure the position you don't collapse the system to a precise position - you collapse it to a superposition of positions, but a much narrower superposition of positions that it started out with.
In the same way, the decay of the inflaton field results from a collapse of the initial superposition of the field. But it cannot collapse to a precise energy. It can only collapse to an almost precise energy i.e. it is still ina superposition but in a much narrower superposition.
The collapsed field then completes the roll down the potential to give the universe we see around us.
But we can see only the end result, not the initial state of the field. So we have no way of measuring what the field state looked like before the collapse.
I need to go now I'm afraid. I'll be around again tomorrow morning if you want to pick this up.
@JohnRennie Is it possible that the field is in superposition and doesn’t collapse, and the inflaton field decays from some other process (e.g. symmetry breaking)?
There are sometimes rather low quality questions appearing in the bounty section. Today these are this and this. In my opinion both require quite a bit of work (such as a discussion in the comments) to make them understandable and answerable. In normal conditions they would be closed, but, since ...
I was thinking about something. So, obviously, the Earth is an oblate spheroid. However, I think we all know that there are people who believe it to be something a little...flatter.
That made me think about pandemic models on a disc versus on a sphere. Obviously, for short time the two should be indistinguishable. But is there a parameter/time range in which a pandemic would spread faster or slower on a disc relative to a sphere of equal area?
Or instead of a pandemic, maybe just diffusion of heat or molecules or something, if it does not make a difference.
@Adamant I'm not completely sure about the 2d case, but in 1d usually it's very similar, except you get some extra modes that "go around" instead of reflecting at the boundary
for the spread of COVID I would guess it's not that relevant, what matters is the network of travel between countries etc (most of the surface is water/not populated anyway)
@fqq Well, we need to think about an idealized case in which all the water is eliminated from the surface of the planet and humans are modeled as uniformly distributed hard spheres undergoing Brownian motion, clearly.
