Undergrad Inflation and the false vacuum

[friend]

I'm trying to trace down my mistunderstandings here. Perhaps I'm getting confused because I'm relying on what I think I understand about the SM fields. The vacuum expectation value of each of those fields, a.k.a. vacuum energy, as I recall is ONLY made up of the energy in the zero point energy of the lowest frequency oscillation. And this is due to the fact that that field obeys the Heisenberg Energy-time Uncertainty Principle. I'm going on memory here. Is this much right? Or is it a sum of the zero-point energy of more than one frequency mode? But in any case, you're saying that the vacuum expectation value for the inflaton and the higgs is not calculated this way because there is a constant value that must be added to it, right?

 

[PeterDonis]

The vacuum expectation value of each of those fields, a.k.a. vacuum energy, as I recall is ONLY made up of the energy in the zero point energy of the lowest frequency oscillation.

No. The vacuum expectation value of all SM fields except the Higgs is zero. The vacuum state of a quantum field theory is a state with, on average, no "oscillations" (more precisely, no excitations of the field--excitations do not have to be describable as "oscillations"). Quantum fluctuations in the vacuum state are fluctuations in the value of the field around the vacuum expectation value (which, again, is zero for all SM fields except the Higgs). The vacuum expectation value is not "made up of" fluctuations; it's the average value around which the field fluctuates. As I said before, the fluctuations do not help at all in determining what the average value is around which they fluctuate; the average value comes first, then the fluctuations are fluctuations around it.

Also, the vacuum expectation value of the field is not the same as the vacuum energy. Energy is an operator, not a field; the vacuum energy is the expectation value of the energy operator when the field is in its vacuum state, which is a separate thing from the expectation value of the field itself.

you're saying that the vacuum expectation value for the inflaton and the higgs is not calculated this way because there is a constant value that must be added to it, right?

No. You're getting things backwards. The vacuum expectation value, as I said above, comes first; it is determined by the equations that govern the field. It is not "added to" a value determined by the fluctuations; it determines the value around which the field fluctuates.

 

[friend]

No. The vacuum expectation value of all SM fields except the Higgs is zero.

What are you talking about? I read on wikipedia for the vacuum expectation value for the SM fields is, "an observable physical effect that results from the vacuum expectation value of an operator is the Casimir effect." So if the vacuum expectation value were zero, there would be no Casimir effect.

It's been more than 20 years since I was in school. But I do remember the professor saying that there is an infinite amount of energy at every point in space. I remember because this disturbed me a little when I considered it. As I recall he was adding up all the zero-point energies in all the vibrational modes in QFT. He was saying that the way around this was to realize that only differences in energy are relevant. And as I recall, one thing that reduces this calculation to something finite is to place a ultraviolet limit on the frequency modes. And then we have only 120 orders of magnitude more than that which is observed. And this is called the vacuum catastrophe. But if the VEV were zero, there would be no vacuum catastrophe. So I don't know what you are talking about.

 

[PeterDonis]

I read on wikipedia

Wikipedia is not a trustworthy source. Please consult an actual textbook on quantum field theory. This is a complicated subject and you can't expect to get a good understanding of it by reading Wikipedia articles or discussions on the web.

Here are the basics of what you will find if you work through a QFT textbook: any vacuum state of a field must be locally Lorentz invariant; i.e., it can't pick out a preferred direction in space, a preferred point in spacetime, or a preferred state of inertial motion. Such a state must also be invariant under other symmetry transformations, such as charge conjugation. For any field that has nonzero spin or charge, any state of the field with a nonzero value is not invariant under these transformations; i.e., the only possible vacuum state for such fields is the one with a zero value for the field. Only an uncharged scalar field (i.e., zero spin) can have a nonzero vacuum expectation value without breaking invariance under these transformations.

I do remember the professor saying that there is an infinite amount of energy at every point in space.

What he meant was, if you do a "naive" calculation based on quantum field theory (by "naive" I mean "without actually taking into account what the fields mean, physically") of how much energy there is in a vacuum state of a quantum field, you get the answer "infinity". The way to fix that is to not do a "naive" calculation.

the way around this was to realize that only differences in energy are relevant.

That's one way, but it only works in flat spacetime (i.e., it doesn't work in the presence of gravity), and it doesn't solve all of the problems with "naive" QFT.

one thing that reduces this calculation to something finite is to place a ultraviolet limit on the frequency modes. And then we have only 120 orders of magnitude more than that which is observed.

Yes. But many QFTs also have problems with infrared divergences, and this method does not fix those.

if the VEV were zero, there would be no vacuum catastrophe.

No, that is not correct. Once again, you are confusing the energy operator with the field itself. The infinite sum that you get when you do the "naive" calculation is for the Hamiltonian--the energy operator--when applied to the field in its vacuum state. As I explained above, for any field with nonzero spin or charge, the field value is zero in this state, so there is no contribution to the value of the Hamiltonian from the field value itself. The problem is that there is an extra term in the Hamiltonian which is independent of the value of the field; it's just a constant. When you integrate this constant over an infinite range of frequencies, you get an infinite answer. Again, if you consult a QFT textbook, you should find all of this worked through.

 

[friend]

Again, if you consult a QFT textbook, you should find all of this worked through.

What book would you recommend for details?

Also, John Baez' rendition of the problem is here. This is what I seem to remember, and also seems to agree with what you're saying.

 

[PeterDonis]

What book would you recommend for details?

If you ask a dozen people who are familiar with QFT that question, you will probably get at least thirteen answers. I find Anthony Zee's Quantum Field Theory In A Nutshell to be a fairly good source, but it presumes that you have a good understanding of ordinary quantum mechanics. Weinberg's classic Quantum Theory Of Fields, in three volumes, seems to me to be kind of the equivalent of Misner, Thorne, & Wheeler in GR: a comprehensive reference, but not necessarily the best introduction to the subject (although at the beginning of his first volume, Weinberg gives an excellent survey of the history of the development of QFT).

 

[friend]

Have we been able, then, to explain inflation in terms of micro-physics or quantum physics? I thought that was the whole point of the inflaton as a quantum field. It seems obvious that global effects must also be explainable in microscopic terms because macro-properties are ONLY explainable as an accumulative effect of micro physics. How, then, does the inflaton cause inflation at the micro scale? Is it possible, for example, that virtual pairs splitting apart and coming together requires more space than if that did not happen?

 

[PeterDonis]

Have we been able, then, to explain inflation in terms of micro-physics or quantum physics?

The current explanation (which, btw, is, AFAIK, still speculative, since we don't really have any way of testing it against evidence) involves a quantum scalar field (the inflaton field), but quantum fluctuations in the field do not play a part in the explanation of why inflation happens; only the field's vacuum expectation value (i.e., its average value in its vacuum state) does. So the explanation is really a classical one; it only makes use of the stress-energy tensor of a classical scalar field, which has an equation of state that causes accelerating expansion of the universe.

(Note that, at least as I understand the model, quantum fluctuations in the field do play a part in explaining how inflation ends; quantum fluctuations are what trigger the phase transition that transfers the energy in the inflaton field to ordinary matter and radiation. But this is different from explaining how the inflaton field causes accelerated expansion before the phase transition occurs.)

Is it possible, for example, that virtual pairs splitting apart and coming together requires more space than if that did not happen?

No; at least, virtual particles/quantum fluctuations play no part in the current explanation of how the inflaton field causes inflation. See above.

 

[friend]

So the explanation is really a classical one; it only makes use of the stress-energy tensor of a classical scalar field, which has an equation of state that causes accelerating expansion of the universe.

Thanks for the reply. Your use of the term equation of state tells me that you are explaining things in terms of global variables, and not in terms of what's happening at the micro level to create or stretch out space so that the overall effect is inflation. Perhaps this is where quantum gravity comes in. Although I'm not sure it's right to talk of gravity during inflation.

 

[PeterDonis]

Your use of the term equation of state tells me that you are explaining things in terms of global variables, and not in terms of what's happening at the micro level to create or stretch out space so that the overall effect is inflation.

The equation of state relates pressure and energy density. Those are not global variables; they are local variables. They lead to inflation via the Einstein Field Equation, which is also local.

Perhaps this is where quantum gravity comes in.

If by the "micro level" you mean something underlying spacetime itself, then yes, this would involve some kind of quantum gravity theory.

 

[friend]

The equation of state relates pressure and energy density. Those are not global variables; they are local variables. They lead to inflation via the Einstein Field Equation, which is also local.

? Pressure is a "local" variable? I thought that was a description of an accumulative effect, an average over many particles.

 

[PeterDonis]

Pressure is a "local" variable? I thought that was a description of an accumulative effect, an average over many particles.

"Local" just means "has a value at each point of spacetime". I think the word you are looking for is "microscopic" as opposed to "macroscopic". Pressure is a macroscopic variable, as is energy density. In the ordinary case of fluid matter, statistical mechanics shows how pressure arises from the microscopic motions of particles (and in at least some idealized cases, the equation of state relating pressure and energy density can also be derived in this way). But not all pressure arises in this way; only what is called "kinetic pressure" does. The pressure due to dark energy is not kinetic pressure; neither is the pressure that appears in the equation of state for a scalar field. It is possible that those kinds of pressure are indeed explainable as arising from statistical mechanics over some set of microscopic degrees of freedom; but if so, nobody currently knows what those microscopic degrees of freedom are.

 

[nikkkom]

How, then, does the inflaton cause inflation at the micro scale? Is it possible, for example, that virtual pairs splitting apart and coming together requires more space than if that did not happen?

My understanding is that in any vacuum, virtual pairs production/annihilation can cause vacuum to have nonzero "zero-point energy", and this translates to nonzero lambda. (My understanding is that currently SM has a problem here: its predicted zero-point energy is way too high, and physicists expect that future developments in (B)SM will fix this: new/fixed theory will predict a tiny nonzero zero-point energy for today's vacuum.)

Inflaton field per se is not required to cause inflation. Inflation happens if vacuum is in a state with large zero-point energy. *Any* mechanism which achieves such result will explain inflation. Not only a mechanism via a new field.

I already posted a link to one of proposed theories, and it happens to _not_ use any inflaton fields:

http://arxiv.org/abs/1307.1848

 

[friend]

My understanding is that in any vacuum, virtual pairs production/annihilation can cause vacuum to have nonzero "zero-point energy", and this translates to nonzero lambda. (My understanding is that currently SM has a problem here: its predicted zero-point energy is way too high, and physicists expect that future developments in (B)SM will fix this: new/fixed theory will predict a tiny nonzero zero-point energy for today's vacuum.)

PeterDosis' point seems to be that these virtual particles are fluctuations necessarily about an average. In the case of the SM (not the higgs or inflaton), this average is zero. In the case of the inflaton field the average is not zero. However, if virtual particles are not responsible for accelerated expansion, then I do not see how a constant average value can be responsible for any kind of dynamics. How can a constant anything be involved with the moving parts of any dynamic process. If nothing is moving or changing (like with virtual particles) then how can things change? What dynamics can there be with a constant? That seems like a contradiction of terms.

In order to answer this question I think somehow quantum processes needs be linked to spacetime itself (and not just have quantum processes occurring in spacetime).

 

[Jimster41]

This has been a great thread to follow and think about, w/respect to something I'm reading, and I've learned from it. It's my default to ask questions in context, rather than start another thread that buries active ones, but if it feels off topic feel free to move itand let me know.

Some questions:

1. Are vacuum scalar fields "observable", or are particles (excited states of those fields) the only observables? I'm confused about whether or not the fundamental fabric of reality is quanta, or something else. If they are considered real what is it scalar fields are thought to be made of?
2. What is the scalar field responsible for the a(t) in the FRLW solution(s) to GR? (this conversation has helped my disambiguate it from the cosmological constant). Since it is a solution to GR field eq. I gather relationships between matter and energy density are considered the "causes" of expansion? And that the "Inflation field" discussed here is really sort of the "first cause" setting the boundary conditions of the FRLW in a way that requires a(t)?

 

[PeterDonis]

My understanding is that in any vacuum, virtual pairs production/annihilation can cause vacuum to have nonzero "zero-point energy", and this translates to nonzero lambda.

First of all, "virtual pairs" is a heuristic description only. A better term would be "quantum fluctuations", and an even better term would be "an extra term in the Hamiltonian that is nonzero in the vacuum state". In other words, the nonzero energy of the vacuum state, in this "naive" theoretical view, is due to the fact that the energy operator, the Hamiltonian, has a term that is independent of the value of the field.

However, as has already been noted in this thread, the "naive" theoretical view gives the answer "infinity" to the question of how much energy is in the vacuum. Even if we adjust the calculation to have a cutoff at some finite frequency, we still get an answer that is more than 120 orders of magnitude larger than the actual observed value of $\Lambda$. So the best answer of all, as the Baez article linked to says, is that we do not know why $\Lambda$ has the value it actually has.

these virtual particles are fluctuations necessarily about an average. In the case of the SM (not the higgs or inflaton), this average is zero. In the case of the inflaton field the average is not zero.

Yes, although, as noted above, "virtual particles" is not the term I would use to describe what is going on. Also, it's worth noting, once again, that the average value of the field in the vacuum state is different from the expectation value of the energy (i.e., of the Hamiltonian) in the vacuum state; the latter can be nonzero even if the former is zero.

if virtual particles are not responsible for accelerated expansion, then I do not see how a constant average value can be responsible for any kind of dynamics.

Why not? I don't understand this at all. The constant average value of any field doesn't just sit there. It has physical effects; for example, the constant nonzero vacuum expectation value of the Higgs field gives mass to the other SM particles.

 

[friend]

I'm wondering if I can re-focus this thread. It appears we may not have really addressed the relationship between inflation and the false vacuum. So let me ask some basic questions. What is the vacuum they are referring to when they say false vacuum? Is this the vacuum energy? Is this the zero point energy of all the SM quantum fields? Does the tunneling to a lower value cause inflation? Or does the lowering of this vacuum energy cause a bubble nucleation of present physics within the larger universe of eternal inflation? And of particular interest, if this vacuum energy is a lowering of the zero point energy of all SM quantum fields, then what quantum properties need to change to cause a lowering of the zero point energy? I thought that was fixed by the Heisenberg Uncertainty principle. Thanks.

 

[PeterDonis]

What is the vacuum they are referring to when they say false vacuum?

In the simplest model of inflation, the inflaton field has two possible states with two different vacuum expectation values: one is nonzero and the other is zero. The "false vacuum" is the state in which the VEV is nonzero; the nonzero VEV causes exponential expansion, i.e., inflation. The "true vacuum" is the state in which the VEV is zero; the end of inflation is a state transition of the inflaton field from the false vacuum state to the true vacuum state (and an associated transfer of energy from the inflaton field to the Standard Model fields).

The inflation models actually under consideration have various complications added, but their basic structure is still the same as the above.

Does the tunneling to a lower value cause inflation?

No; the transition of the inflation field from the false vacuum to the true vacuum (which can be thought of as "tunneling" although that is not the only possible model), as I said above, is the end of inflation. The nonzero VEV of the inflaton field in the false vacuum state is what causes inflation.

Or does the lowering of this vacuum energy cause a bubble nucleation of present physics within the larger universe of eternal inflation?

This is one model being considered, yes.

if this vacuum energy is a lowering of the zero point energy of all SM quantum fields

It isn't. The zero point energy of all the SM fields is constant; it doesn't change.

 

[friend]

Thanks for your answer, PeterDonis.

Is this inflationary lowering of the vacuum of a different quality than that posed to possibly happen in the future? There they also talk of a bubble nucleation with a fall from the presently supposed false vacuum to a lower vacuum energy. I don't know what field energies could fall even lower than today?

 

[PeterDonis]

Is this inflationary lowering of the vacuum of a different quality than that posed to possibly happen in the future?

It's the same general mechanism.

I don't know what field energies could fall even lower than today?

The hypothesis, as I understand it, is that what we currently observe as dark energy--i.e., whatever it is that is causing the current accelerating expansion of the universe--is itself a nonzero vacuum expectation value of some scalar field, i.e., a false vacuum state. If that is true, then it should be possible in principle for that field to transition to a true vacuum state with a zero VEV, and we would expect that process to, at the very least, transfer the energy to some other field, and possibly also change the observed laws of physics. This is all speculative, and I'm not aware of any data, other than the existence of dark energy itself, that supports such a hypothesis. It's basically the idea that, if it happened once, perhaps it can happen again.

(Actually, the Higgs field would be another possibility, since it is a scalar field. This is at least open to some investigation, since we can measure the mass of the Higgs boson and try to figure out theoretically what range of masses would correspond to a false vacuum state. AFAIK nothing along these lines so far has indicated that the observed Higgs mass is in an unstable range.)

 

[friend]

Your answer, PeterDonis, makes no mention of the vacuum energy that caused inflation being equivalent to any kind of zero point energy. I suppose that is because zero point energy is only concerned with harmonic oscillators, whereas the inflationary vacuum is caused by some non-zero constant field (or changing slowly). Is there no zero point energy (as such) involved with initial inflation?

It seems we do have a zero point energy now associated with the SM fields, and the naive calculation is like 120 orders of magnitude greater than what is observed. Although, I've heard a few people say that there is some cancellation going in with the effects of nearby oscillating fields so that the net effect could be zero, or near zero. Is it even true that the zero point energy of all the SM fields equivalent to the cosmological constant or dark energy. If so, how are we ever going to know whether this zero point energy or dark energy can decrease to a true vacuum energy when we presently have such a terrible disagreement with measurement? And what would that do to the SM fields if that zero point energy did decrease?

But if the present dark energy value is the only scalar field whose energy could fall and cause another phase transition, then I suppose that means we don't have to worry about another round of inflationary expansion. Expansion could only slow down from its present value since the vacuum energy would be less than today.

Just curious: do you know of any physical eschatological theories that could cause the universe to experience another round of rapid expansion?

 

[PeterDonis]

Your answer, PeterDonis, makes no mention of the vacuum energy that caused inflation being equivalent to any kind of zero point energy.

That's because what caused inflation is not vacuum energy, it's a nonzero vacuum expectation value for the inflaton field. They are not the same thing. Vacuum energy is the expectation value of the energy operator--the Hamiltonian--when the field is in a vacuum state. The vacuum expectation value of the inflaton field is the expectation value of the field operator when the field is in a vacuum state. The field operator is not the same as the Hamiltonian operator.

I suppose that is because zero point energy is only concerned with harmonic oscillators

No, it isn't, it's there for any quantum field. "Zero point energy" is just another way of saying that the expectation value of the Hamiltonian operator when the field is in a vacuum state is not zero; or, to put it another way, the Hamiltonian operator has an extra term in it that is independent of the state of the field, it's just a nonzero constant, which is referred to as "zero point energy".

It seems we do have a zero point energy now associated with the SM fields

Yes, as I said above, it's there for any quantum field. But all of the SM fields, except for the Higgs, have nonzero spin, so they cannot have a nonzero vacuum expectation value, even though they have nonzero zero point energy. A nonzero vacuum expectation value is required for a field to be able to cause inflation (but even then it's not sufficient; the Higgs has a nonzero VEV but does not cause inflation).

Is it even true that the zero point energy of all the SM fields equivalent to the cosmological constant or dark energy.

That's one hypothesis, but we don't really know.

do you know of any physical eschatological theories that could cause the universe to experience another round of rapid expansion?

I'm not aware of any, no.

 

[friend]

I feel your explanation in post #48 may not be consistent with your last post. In post #48 you seem to be saying that there are two values of the VEV of the field (not the energy of the field). But as I recall, The inflaton field is a plot of field strength (on the x-axis) vs. energy (on the y-axis). The inflaton energy is not zero when the inflaton field is zero. Instead it has a somewhat constant energy value and there is a slow slope as the field increases. Then it rather sharply decreases at some level of the field (this is where inflation stops and energy is transferred to the SM fields). Then with slightly more field strength, the energy reaches a local minimum and begins to increase again. (The Mexican hat potential, as I recall). So it would then not be the field strength that causes or ends inflation, it's the energy of the field that causes all of this. In fact the field strength itself increases a bit at the end of inflation where the energy falls dramatically IIRC. So perhaps we need to take a look again at whether it is the "vacuum energy"=zero-point-energy that is causing inflation, etc.

 

[PeterDonis]

In post #48 you seem to be saying that there are two values of the VEV of the field (not the energy of the field).

Yes. More precisely, there are two states of the inflaton field, one with a nonzero VEV and another with a zero VEV; the "false vacuum" and "true vacuum" states, respectively.

as I recall, The inflaton field is a plot of field strength (on the x-axis) vs. energy (on the y-axis).

No, it's a plot of the field's value ("strength" if you like) on the x-axis vs. the potential energy on the y-axis. The potential energy is a portion of the Lagrangian (or Hamiltonian), but it is not all of it; if nothing else, it leaves out the kinetic energy associated with the field. So the y-axis on those plots is not the same as the "energy", which is the expectation value of the complete Hamiltonian operator.

The reason the potential energy is important is that the field has a tendency to "move downhill" towards a region of lower potential energy. See further comments below.

The inflaton energy is not zero when the inflaton field is zero. Instead it has a somewhat constant energy value and there is a slow slope as the field increases. Then it rather sharply decreases at some level of the field (this is where inflation stops and energy is transferred to the SM fields). Then with slightly more field strength, the energy reaches a local minimum and begins to increase again. (The Mexican hat potential, as I recall). So it would then not be the field strength that causes or ends inflation, it's the energy of the field that causes all of this. In fact the field strength itself increases a bit at the end of inflation where the energy falls dramatically IIRC.

You're misdescribing the model somewhat, and you're also mixing up two different models of inflation, "old" and "new"--and you're also mixing in models of spontaneous symmetry breaking that have nothing to do with inflation (the "Mexican hat" potential). Here is a better description:

In the "old" inflation model (Guth's original formulation), the potential energy as a function of the state of the field had two minima. One, called the "false vacuum", corresponded to a state of the field with a nonzero vacuum expectation value. This was only a local minimum of the potential energy, i.e., it had a lower potential energy than other "nearby" field states, but a significantly higher potential energy than a more "distant" field state, called the "true vacuum" state. The true vacuum field state had a zero vacuum expectation value of the field.

In this model, the field was hypothesized to start out in the "false vacuum" state (more precisely, to be driven there by the natural dynamics of the field "moving downhill" towards states of lower potential energy, but then getting "stuck" in the local minimum, like a small valley in a mountain range). In this state, the nonzero VEV of the field drove exponential expansion. You could say, I suppose, that the reason the nonzero VEV drove exponential expansion is that it was equivalent to there being a large positive cosmological constant, which can be thought of as an "energy of empty space" that makes empty space exponentially expand. But this only happens because the field itself (i.e,. the field operator, not the energy operator) has a nonzero VEV; at least, that's my understanding of the underlying math. Also, the energy involved here is not well described as "zero point energy" in any case; see below.

In this model, the "false vacuum" state is metastable: classically, the field will stay there forever because it's a local minimum, but when we add quantum fluctuations, there is a nonzero amplitude for the field to quantum tunnel to the "true vacuum" state. When that happens, it causes two things: first, the field's VEV changes from nonzero to zero, which stops inflation; second, the expectation value of the energy operator for the field decreases drastically, because that value is much lower for the "true vacuum" state than for the "false vacuum" state. (Note, though, that it is not zero for the "true vacuum" state; see comments at the end on "zero point energy".) That energy has to go somewhere, and where it goes is into the ordinary SM fields, "reheating" them to a very high temperature and creating the hot, dense, rapidly expanding "Big Bang" state. So at the end of all this, we have the inflaton field in the "true vacuum" state with zero VEV, where it will then remain forever, and the SM fields at very high temperature in the "Big Bang" state.

This model is simple, but it turned out to have a number of issues, and to try and address them, Linde and others came up with a somewhat different model called the "new inflation" model. In this model, the "false vacuum" state is not a local minimum of the potential energy; it is "at the top of a hill", but the potential energy as a function of the field state has a very, very small slope in that region. So the field "moves downhill" very, very slowly when it starts from the "false vacuum" state. While it is "moving downhill" very, very slowly, the field's VEV is almost constant at some nonzero value, and drives inflation as discussed above. (This is called the "slow roll" model of inflation.)

However, in this model, as the field "moves downhill" away from the original "false vacuum" state, the "hill" gradually gets steeper, and so the field "moves downhill" faster. This process ends up at the "bottom of the hill", which is the "true vacuum" state, with zero VEV, and once there, the field stays there forever, just as in the "old inflation" model. While the field is "rolling downhill" faster, inflation is stopping and energy is being transferred from the inflaton field to the SM fields, but this is somewhat more gradual than in the old inflation model where the transition was due to quantum tunneling. In the new inflation model no tunneling is necessary; the ordinary classical dynamics of the inflaton field will take it from the "false vacuum" to the "true vacuum" state.

Note that, as I mentioned above, neither of the potential energy functions in these models (old or new) is of the "Mexican hat" type. That type of potential is associated with a different process, the spontaneous symmetry breaking process that, for example, broke electroweak symmetry and allowed the Higgs field to give other SM fields a nonzero mass. In this kind of process, the field state with a zero VEV (the one at the top of the "Mexican hat") has a higher potential energy than a family of field states with a nonzero VEV (the whole circle of states at the bottom of the trough of the Mexican hat). (Note that in the case of the inflaton field above, the zero VEV "true vacuum" state had a lower potential energy than the nonzero VEV "false vacuum" state.) So the natural dynamics of the field will carry it from the zero VEV state to one of the nonzero VEV states--but it will have to pick one nonzero VEV state out of a whole family of possible ones. Picking one state out of the family breaks the underlying symmetry of the field--SU(2) x U(1) electroweak symmetry, in the case of the Higgs field.

perhaps we need to take a look again at whether it is the "vacuum energy"=zero-point-energy that is causing inflation, etc.

As I noted above, the energy in the inflaton field in the "false vacuum" state is not well described as "zero point energy" in any case. Why not? There are at least two reasons. First, the "false vacuum" state is not a state of lowest energy globally; it is only a state of lowest energy locally (i.e., with respect to "nearby" field states). So it's not a "zero point" state, because that implies a state with globally lowest energy.

Second, as I've noted several times now, all quantum fields have "zero point energy", and this energy is independent of the state of the field; it's an extra term in the Hamiltonian that's just a constant, with no dependence on the field state. So the inflaton field in the "true vacuum" state also has this energy--yet it doesn't cause inflation in that state. So whatever it is that is causing inflation when the field is in the "false vacuum" state, it has to be something else, something that isn't there in the "true vacuum" state--something other than "zero point energy". The obvious difference is the nonzero VEV of the field itself in the "false vacuum" state, as compared with its zero VEV in the "true vacuum" state; as I said above, one could also, I suppose, associate this with the extra energy stored in the field, but this energy would also not be properly described as "zero point energy", as above. (Also, all of the SM fields are present while inflation is happening--they are all in their own vacuum states, all with zero VEV, and all having "zero point energy" associated with them as well--but none of them are causing inflation.)

 

[friend]

(i.e,. the field operator, not the energy operator) has a nonzero VEV;

Thank you for the effort you put into your response. I'm starting to hit the "like" button on your posts, whether I understand them or not.

In both scenarios that you describe, there is a move towards lower "potential" energy on the graph (whether it is a mexican hat or not). This in itself means the field "strength" is not zero by the time the potential energy reaches its true vacuum level. So I don't know what you mean VEV of the field operator goes to zero in the true vacuum state. Where is the VEV of the field operator on the graph of potential energy vs. field strength? Your use of the term VEV seems to imply that we are dealing with quantum theoretic calculation involving quantum fluctuations of the inflaton field. This only begs the question as to how these "fluctuation" are manifest. Are they virtual particles of some sort? Is there a zero-pont-energy associated with them; that would be the average value represented by the line on the graph, right?
Also, the videos I watched where Leonard Susskind lectures on this tells me that there is an upturn with increased field after the local minimum of energy is reached in the potential energy vs inflaton field strength. He explains that there may be some oscillations at the bottom of the hill and that this may be what dark matter is. IIRC.

 

[nikkkom]

whatever it is that is causing the current accelerating expansion of the universe--is itself a nonzero vacuum expectation value of some scalar field, i.e., a false vacuum state.

I think that "falseness" is not related to "nonzero". True vacuum is not necessarily vacuum where all fields are zero (all-zero is not even necessarily a vacuum). True vacuum state is the state with not only local minimum of energy, but with global minimum of energy - there are no "lower vacuums than this one".

 

[nikkkom]

... address them, Linde and others came up with a somewhat different model called the "new inflation" model. In this model, the "false vacuum" state is not a local minimum of the potential energy; it is "at the top of a hill"

IIUC such a state is not a vacuum. Vacuum is a state where potential energy has a local minimum.

 

[PeterDonis]

I think that "falseness" is not related to "nonzero". True vacuum is not necessarily vacuum where all fields are zero (all-zero is not even necessarily a vacuum).

Yes, it's easy to get muddled in the terminology in this area--see below.

IIUC such a state is not a vacuum.

It isn't in the sense you give, yes; but the term "false vacuum" is still used to describe it, probably for historical reasons, because of the way inflation theory developed. Unfortunately this happens often in science: confusing terminology gets established for historical reasons and then can't be dislodged.

 

[PeterDonis]

In both scenarios that you describe, there is a move towards lower "potential" energy on the graph (whether it is a mexican hat or not).

Yes.

This in itself means the field "strength" is not zero by the time the potential energy reaches its true vacuum level.

Not necessarily; it depends on the specifics of how the potential energy varies with the field state (I prefer to use the word "state" rather than "strength" here, because, as you appear to agree since you put "strength" in quotes, the numerical value of the field variable doesn't necessarily have any physical meaning in terms of "field strength"). Also, the "field variable" in these graphs is probably better interpreted as the VEV of the field, not as the "raw value" of the field (see below). In the "Mexican hat" case, involved in spontaneous symmetry breaking, the field state with zero VEV has a higher potential energy than field states with nonzero VEV; but I don't know that this is true in inflationary models in cosmology.

Where is the VEV of the field operator on the graph of potential energy vs. field strength?

The "field strength" in these graphs, as I mentioned above, is really the VEV--at least, I think that's the best interpretation of the graphs. AFAIK the graphs themselves are heuristics and are not intended to be exact representations of the underlying models. And the general role that the "field strength" on the graph--i.e., the horizontal axis variable--appears to play is as modeling the VEV of the field. That's certainly the case in the "Mexican hat" graph that is used as a heuristic model of electroweak symmetry breaking.

Your use of the term VEV seems to imply that we are dealing with quantum theoretic calculation involving quantum fluctuations of the inflaton field.

That's not really a good description of what a VEV is. In quantum field theory, the "field" itself is an operator. The VEV is simply the expectation value of that operator when the operator is applied to a vacuum state. In this more precise terminology, the "state" is not a state of the "field", since the field is an operator; rather, the "state" is some vector in a Hilbert space over which the field operator is defined. Taking an expectation value doesn't involve any "fluctuations"; the state the operator is applied to doesn't change, it is constant.

This only begs the question as to how these "fluctuation" are manifest. Are they virtual particles of some sort? Is there a zero-pont-energy associated with them; that would be the average value represented by the line on the graph, right?

The virtual particle picture is not a good one to use when trying to understand inflation models. Virtual particles arise in perturbation theory, and inflation models are not based on perturbation theory.

Also, even when you are using perturbation theory and thinking in terms of virtual particles, the intuitive picture of zero point energy arising from "fluctuations due to virtual particles" is of limited usefulness.

the videos I watched where Leonard Susskind lectures on this tells me that there is an upturn with increased field after the local minimum of energy is reached in the potential energy vs inflaton field strength. He explains that there may be some oscillations at the bottom of the hill and that this may be what dark matter is.

This is a speculative hypothesis; it's not the same as the basic model of inflationary cosmology. In the basic model, once the inflaton field reaches the true vacuum state, it stays there forever, and any quantum fluctuations in it are assumed to be negligible. That might not be true in reality, of course, but it's the simplest model.

 

[friend]

That's not really a good description of what a VEV is. In quantum field theory, the "field" itself is an operator. The VEV is simply the expectation value of that operator when the operator is applied to a vacuum state. In this more precise terminology, the "state" is not a state of the "field", since the field is an operator; rather, the "state" is some vector in a Hilbert space over which the field operator is defined. Taking an expectation value doesn't involve any "fluctuations"; the state the operator is applied to doesn't change, it is constant.

OK. With the field being an operator that when acting on the vacuum state gives 0 (after inflation), that seems to be an extra step that needs more explanation. I've not seen that calculation before. I've only seen the potential energy vs field curve. So when you say VEV of the field operator being zero at the end of inflation, you seem to be indicating that this is different than the field "strength" (x-axis) that we see on the curve. Obviously that curve is monotonic; lower inflaton energies can only be reached at larger field (larger values on the x-axis). I'm missing the math to get from the field (x-axis) to the VEV that you get once you apply the field (operator) on the vacuum state. How does one get a field operator from the scalar on the x-axis? Where does one get the vacuum state on which the field operator acts upon? Thanks.