Inflation and the false vacuum

[friend]

I'm trying to understand more about how our present universe is supposed to be the result of a false vacuum falling to the present vacuum energy.

I've been told (correct this if it's wrong), the universe initially underwent a kind of hyperinflation, expanding exponentially due to a much higher vacuum energy than now. But at some point this high vacuum energy somehow fell to a lower vacuum energy which resulted in a much slower expansion rate. Is this much true, and can someone please supply a few more words about this? For example, was it the inflaton field that decayed into the present particles as the vacuum energy fell? Is this the point where the higgs mechanism gave particles mass? What is meant by vacuum energy? Is this the same concept that the fields had a zero-point-energy. Did the fields of nature during inflation still obey the Heisenberg Uncertainty Principle such that the energy of the field(s) times the duration they existed could be larger than today? Thanks.

 

[PeterDonis]

the universe initially underwent a kind of hyperinflation, expanding exponentially due to a much higher vacuum energy than now. But at some point this high vacuum energy somehow fell to a lower vacuum energy which resulted in a much slower expansion rate.

Basically, yes. But the term "vacuum energy" here might be misleading. A better term is "energy in the inflaton field", to keep it distinct from the dark energy we observe in the universe now. See below.

was it the inflaton field that decayed into the present particles as the vacuum energy fell?

More precisely, the energy in the inflaton field was converted into energy contained in matter and radiation. This process is called "reheating" by cosmologists. At the end of the process, the universe contained a very high density of matter and radiation, plus a very low (by comparison at that time) density of dark energy. The dark energy, at least in the model that AFAIK is currently accepted, was not produced from the energy in the inflaton field; it was there all the time. (This is because "dark energy" in this model is actually just a cosmological constant, which was "built in" to the structure of spacetime from the start, whereas the inflaton field is a separate field.)

Is this the point where the higgs mechanism gave particles mass?

No. That came later, during the electroweak phase transition.

What is meant by vacuum energy? Is this the same concept that the fields had a zero-point-energy.

Not really. See above.

Did the fields of nature during inflation still obey the Heisenberg Uncertainty Principle such that the energy of the field(s) times the duration they existed could be larger than today?

If they obeyed the HUP, which as far as we know they did, then the energy times the duration was the same as today--at least, heuristically (the actual quantum field theory involved is more complicated than that).

 

[nikkkom]

I'm trying to understand more about how our present universe is supposed to be the result of a false vacuum falling to the present vacuum energy.

I've been told (correct this if it's wrong), the universe initially underwent a kind of hyperinflation, expanding exponentially due to a much higher vacuum energy than now. But at some point this high vacuum energy somehow fell to a lower vacuum energy which resulted in a much slower expansion rate. Is this much true, and can someone please supply a few more words about this? For example, was it the inflaton field that decayed into the present particles as the vacuum energy fell? Is this the point where the higgs mechanism gave particles mass? What is meant by vacuum energy? Is this the same concept that the fields had a zero-point-energy. Did the fields of nature during inflation still obey the Heisenberg Uncertainty Principle such that the energy of the field(s) times the duration they existed could be larger than today? Thanks.

My understanding of the current situation with our knowledge is as follows:

General Relativity satisfactorily explains "classical" (non-quantum) behavior of space-time, including curvature (gravity). It includes a possibility of empty space having a property of self-expanding (lambda term in GR equations), and observations seem to indicate that lambda in observable Universe has a ery small, but non-zero value.

However, GR doesn't explain *why* lambda has this value.

Standard Model, the quantum theory of particles and forces, satisfactorily explains behavior of particles. Their interactions. Their decays.

However, these two theories don't mesh fully with each other. For example, in SM one can calculate the energy of empty space. It's non-zero because of vacuum fluctuations. And the result is *vastly* larger than tiny observed value of lambda. Basically, if we use SM'd predictions on energy density of Universe, then GM predicts that such Universe must have collapsed very soon after creation.

SM and/or GR needs modifying so that their predictions match observations. We know that this must be done.

Inflaton field is one way to extend SM. Roughly, it can work as follows: in "unbroken", symmetric state, vacuum energy is high and Universe expands very rapidly. This solves a few difficulties in early Big Bang scenarios.

But this state is not a state with lowest energy. "Eventually" inflaton field finds a state with energy minimum (similar to how "symmetric" Higgs field is not stable and finds its own minimum), in which vacuum energy is very low (matches observed lambda).

Since currently all "inflaton" theories are very young, they can't yet give predictions testable with experiment (e.g. they can't predict current lambda). Giving time, we can start making predictions, comparing them to observations and see how well it goes.

 

[PeterDonis]

"Eventually" inflaton field finds a state with energy minimum (similar to how "symmetric" Higgs field is not stable and finds its own minimum), in which vacuum energy is very low (matches observed lambda).

I'm not sure this is an accurate description of all inflation models. In at least some of them, as I understand it, the dark energy (or lambda) that we currently observe is a different field (or cosmological constant) from the inflaton field.

 

[friend]

Was a field (e.g. inflaton field) necessary to cause inflation? Or could inflation occur without any fields being involved? It does seem necessary, don't you think, that something with energy must exist everywhere (thus a "field") to cause change (expansion) everywhere all at once? If a field was involved, was it necessary to be a quantum field, by which I mean did it have a conjugate momentum and a non-zero commutator between the field and its conjugate momentum? If it had a non-zero commutator, then doesn't this imply a non-zero uncertainty principle between the expectation value of the field times the expectation of the conjugate momentum, and wouldn't that result in a vacuum energy? Then doesn't this all imply that there are virtual particles (of inflatons to begin with)? And if these virtual inflatons are not charged, then how would these virtual particles recombine and what would they recombine into? (This last question might be asked about higgs bosons and gluons, if such particles can exist as virtual particles) Is the inflaton field the same thing as the unified field that is supposed to exist from which all the other fields came from?

I appreciate any answers. You don't have to answer all the questions in order to answer one. Thanks.

 

[PeterDonis]

Was a field (e.g. inflaton field) necessary to cause inflation?

Yes. More precisely, from a GR point of view, inflation requires a nonzero stress-energy tensor with particular properties. The only ways we know of to produce such a stress-energy tensor are a cosmological constant or a scalar field. But a cosmological constant is, well, constant, so if that had caused inflation it would never have stopped. So the only possibility left is a scalar field.

If a field was involved, was it necessary to be a quantum field, by which I mean did it have a conjugate momentum and a non-zero commutator between the field and its conjugate momentum?

I don't know that the quantum aspects of the field would come into play in predicting inflation itself; a simple classical scalar field can do that. But quantum fluctuations in the field are required to explain the "wrinkles" that are being found in, for example, the CMB, by the WMAP satellite and now by the Planck satellite.

doesn't this all imply that there are virtual particles (of inflatons to begin with)?

No. Inflation is not a perturbative phenomenon, i.e., the perturbation theory formalism is not used to describe it. So the virtual particle idea isn't useful in trying to visualize how the inflaton field works.

Is the inflaton field the same thing as the unified field that is supposed to exist from which all the other fields came from?

No. The "unified field" idea applies to the fields in the Standard Model (plus, in some versions, gravity). The inflaton field is a separate field, not included in any of that.

 

[friend]

No. The "unified field" idea applies to the fields in the Standard Model (plus, in some versions, gravity). The inflaton field is a separate field, not included in any of that.

What about reheating? Isn't this where the inflation field decays to ordinary particles (perhaps without mass yet)? That would suggest that the inflaton field is the unified field, right?

 

[PeterDonis]

What about reheating? Isn't this where the inflation field decays to ordinary particles (perhaps without mass yet)? That would suggest that the inflaton field is the unified field, right?

No. In reheating, the potential energy stored in the inflaton field gets transferred to the fields corresponding to the ordinary Standard Model particles (and yes, they are all massless at this temperature--they don't gain mass until the electroweak phase transition, which comes later, when the temperature has fallen far enough for the Higgs field to condense out and take on a vacuum expectation value). It's just an ordinary transfer of energy from one field to another.

 

[friend]

No. In reheating, the potential energy stored in the inflaton field gets transferred to the fields corresponding to the ordinary Standard Model particles (and yes, they are all massless at this temperature--they don't gain mass until the electroweak phase transition, which comes later, when the temperature has fallen far enough for the Higgs field to condense out and take on a vacuum expectation value). It's just an ordinary transfer of energy from one field to another.

How is this different than what I was thinking? Was it my use of the term "decay"? Of course I meant the energy of the inflaton field was "transfered" to the SM fields, which is what you said. The same thing happens in the unified field, right, the energy of the unified field gets transferred to the SM fields? I fail to see the distinction.

 

[PeterDonis]

How is this different than what I was thinking?

You asked if the inflaton field "is" the unified field. It isn't.

Of course I meant the energy of the inflaton field was "transfered" to the SM fields, which is what you said.

That's not the same as saying the inflation field "is" the unified field. "Is" means "is", not "can transfer energy to".

The same thing happens in the unified field, right, the energy of the unified field gets transferred to the SM fields?

No, the unified field is the SM fields. More precisely, what we call "the SM fields" are all just different low-energy states of the unified field. They're not different fields that the energy of the unified field gets transferred to. (Note that this all assumes that we have a grand unification theory that incorporates all the Standard Model fields, which we don't; we have candidates for such a theory, but no good experimental data to test them.)

 

[friend]

No, the unified field is the SM fields. More precisely, what we call "the SM fields" are all just different low-energy states of the unified field. They're not different fields that the energy of the unified field gets transferred to. (Note that this all assumes that we have a grand unification theory that incorporates all the Standard Model fields, which we don't; we have candidates for such a theory, but no good experimental data to test them.)

Is the unified field the same as when energies are so high that the coupling constants for charge and mass become equal so that we can not tell the difference between the forces and particles?

 

[nikkkom]

At example of a theory which extends SM to explain inflation:

http://arxiv.org/abs/1307.1848

 

[nikkkom]

Is the unified field the same as when energies are so high that the coupling constants for charge and mass become equal so that we can not tell the difference between the forces and particles?

Unified field is a field which contains (in case of electroweak and strong unification) leptons and quarks as its components, showing that they are not unrelated things. Components should change into each other under a suitable symmetry group, such as SU(5).

 

[PeterDonis]

Is the unified field the same as when energies are so high that the coupling constants for charge and mass become equal

What's called the "grand unification energy" is the energy at which the coupling constants for the three Standard Model interactions (strong, weak, and electromagnetic) become equal. Charge is one of these (the electromagnetic coupling constant). Mass is not; the Higgs mechanism for giving the SM particles mass is separate from all this.

The "unified field" is a hypothesis for why the coupling constants become equal at the grand unification energy--because all three SM interactions are really just aspects of the same interaction of the unified field.

so that we can not tell the difference between the forces and particles?

You can still tell the difference between "forces" (interactions, i.e., gauge bosons) and "particles" (fermions) even at energies at or above the grand unification energy.

 

[friend]

What's called the "grand unification energy" is the energy at which the coupling constants for the three Standard Model interactions (strong, weak, and electromagnetic) become equal. Charge is one of these (the electromagnetic coupling constant). Mass is not; the Higgs mechanism for giving the SM particles mass is separate from all this.

So what is supposed to cause the forces to distinguish themselves? Is this where symmetry breaking comes in? Is that accomplished by the higgs mechanism? Or is it something else? Thanks.

 

[PeterDonis]

what is supposed to cause the forces to distinguish themselves? Is this where symmetry breaking comes in?

Yes. At least, that is the hypothesis on which all of the current candidates for a grand unified theory are based. But as I said, we don't really have a way to experimentally test any of them. The Standard Model does not include any of this.

Is that accomplished by the higgs mechanism?

No. The Higgs mechanism is associated with electroweak symmetry breaking, which is included in the Standard Model, and which is a different symmetry breaking process than the ones hypothesized in connection with grand unification. Electroweak symmetry breaking happens at a much lower energy than the hypothesized grand unified symmetry breaking.

 

[friend]

Yes. At least, that is the hypothesis on which all of the current candidates for a grand unified theory are based.

What is supposed to break then the unified field symmetry if not the higgs mechanism? Is there supposed to be a different field like the higgs, but not the higgs, that breaks this greater symmetry? Or perhaps more basic, what generally breaks a symmetry? Is this where something turns on or off or changes the value of the structure constants of the group associated with that symmetry? Thanks again.

 

[PeterDonis]

What is supposed to break then the unified field symmetry if not the higgs mechanism? Is there supposed to be a different field like the higgs, but not the higgs, that breaks this greater symmetry?

I believe that's one of the candidate hypotheses, yes.

Or perhaps more basic, what generally breaks a symmetry?

A good general way to look at symmetry breaking--or more precisely, "spontaneous symmetry breaking", which is what the various mechanisms we've been talking about are, is that it happens in systems which have two dynamic regimes. In the "high energy" regime, solutions to the equations describing the system have the same symmetry as the equations themselves. But in the "low energy" regime, solutions no longer have that same symmetry; they may have a lesser symmetry or they may have none at all. Instead, there will be families of solutions that, taken all together, manifest the symmetry of the equations.

So what breaks the symmetry is simply a lowering of energy of the system; for example, the average temperature of the universe decreases as it expands, and when the temperature of the early universe became lower than the threshold for electroweak symmetry breaking, the electroweak phase transition happened, and the symmetry of the underlying electroweak equations was no longer manifested directly in the solution describing the particles in the universe. Instead, you would have to look at a whole family of possible solutions (only one of which was actually realized in our universe) to see the underlying symmetry of the equations.

Is this where something turns on or off or changes the value of the structure constants of the group associated with that symmetry?

No. The structure constants are just that: constants. They never change. The underlying group structure is still there; it just manifests differently.

 

[friend]

A good general way to look at symmetry breaking--or more precisely, "spontaneous symmetry breaking", which is what the various mechanisms we've been talking about are, is that it happens in systems which have two dynamic regimes. In the "high energy" regime, solutions to the equations describing the system have the same symmetry as the equations themselves. But in the "low energy" regime, solutions no longer have that same symmetry; they may have a lesser symmetry or they may have none at all. Instead, there will be families of solutions that, taken all together, manifest the symmetry of the equations.

I was thinking more mathematically. As I understand it, symmetry means that the fields involved have a non-zero commutator that defines the Lie algebra for those fields, and there are certain structure constants associated to those commutators. So the only way for symmetry to break or change would be to change the structure constants of that commutator that defines the Lie algebra of the Lie group involved. This makes me think that somehow the structure constants have to change in order to break the symmetry. Does any of this sound right?

 

[PeterDonis]

As I understand it, symmetry means that the fields involved have a non-zero commutator that defines the Lie algebra for those fields, and there are certain structure constants associated to those commutators.

No, that's not what symmetry means. Symmetry in general means that there is some group of transformations that leaves something unchanged. In physics, what is left unchanged is the laws or equations that apply to something. Some examples:

(1) Lorentz invariance is a symmetry in SR that applies to physical laws: all valid physical laws in SR must remain unchanged under the group of Lorentz transformations.

(2) Gauge invariance is a symmetry that applies to the laws of electromagnetism; those laws remain unchanged under the group of gauge transformations. For electromagnetism, that group is the group U(1).

(3) Electromagnetism is just one example of a gauge theory; there is a more general class of similar theories where the group of gauge transformations is some other group besides U(1). Since those other groups are not Abelian (i.e., the group operation does not commute), these theories are often called non-Abelian gauge theories. Another name for them is Yang-Mills theories, after the two physicists who first discovered them. The Standard Model is an example of such a theory; the gauge group in the SM is SU(3) x SU(2) x U(1). (Actually, that's not precisely true, but it's close enough for this discussion.)

The connection with Lie algebras, commutators, and structure constants is simply that all of the groups I mentioned above are Lie groups, and any Lie group has an associated Lie algebra, which can be thought of as the algebraic structure of the group in an infinitesimal neighborhood of the identity. For an Abelian group such as U(1), since the group operation commutes, all commutators of the generators of the Lie algebra (the "generators" are the set of linearly independent operators that form the basis of the Lie algebra) are zero. (Note that this shows that a symmetry does not need to be associated with nonzero commutators of anything.)

For a non-Abelian group, however, such as SU(2) or SU(3) (or the tensor product group that is the gauge group for the SM), the commutators of the generators of the Lie algebra are nonzero. In such cases, it turns out that each commutator of a pair of generators can be written as a linear combination of the generators; the structure constants are simply the coefficients in the linear combinations. (Note that these are commutators of operators, not fields.)

the only way for symmetry to break or change would be to change the structure constants of that commutator that defines the Lie algebra of the Lie group involved.

The structure constants of the Lie algebra of any Lie group, as I said before, cannot change; they are inherent properties of the group. When spontaneous symmetry breaking occurs, what happens, as I said before, is that, instead of the full symmetry of the equations being manifest in a solution, only a reduced symmetry (or no symmetry at all) is manifest in a solution--the full symmetry is only manifest in a family of solutions. If a reduced symmetry is still present, the reduced symmetry group must be a subgroup of the full symmetry group.

For example, if we just look at the electroweak sector of the Standard Model, the full symmetry group is SU(2) x U(1). At energies above the electroweak phase transition energy, this full symmetry group is manifest in the actual physical solution. This is usually described as the symmetry transformations--i.e., the SU(2) x U(1) gauge transformations--leaving invariant the vacuum state--the state of lowest energy. When electroweak symmetry breaking occurs, the full set of SU(2) x U(1) gauge transformations no longer leave the vacuum state invariant; instead, the vacuum state is left invariant only by a reduced set of U(1) gauge transformations. The reduced symmetry, U(1), is a subgroup of the full symmetry group SU(2) x U(1). (These U(1) gauge transformations are in fact just the gauge transformations of electromagnetism.)

 

[friend]

No, that's not what symmetry means. Symmetry in general means that there is some group of transformations that leaves something unchanged. In physics, what is left unchanged is the laws or equations that apply to something. Some examples:

Thank you. That was a very well written answer. You've given me a lot to think about. I can not think of any more questions.

 

[friend]

Thanks for your responses. But I guess what I was really after, then, is HOW the inflaton field cause spacetime to inflate. Is there a word description of how this occurred?

 

[PeterDonis]

is HOW the inflaton field cause spacetime to inflate.

The inflaton field, dynamically, works just like dark energy; it behaves like a cosmological fluid with an equation of state of $p = - \rho$, so that it causes accelerating expansion. Since the energy density of the inflaton field was very high, much higher than the density of dark energy in our current universe, the rate of acceleration of the expansion was correspondingly much larger.

 

[friend]

The inflaton field, dynamically, works just like dark energy; it behaves like a cosmological fluid with an equation of state of $p = - \rho$, so that it causes accelerating expansion. Since the energy density of the inflaton field was very high, much higher than the density of dark energy in our current universe, the rate of acceleration of the expansion was correspondingly much larger.

OK, so you seem to be describing inflation in terms of global variables, $p = - \rho$ . But I'm wondering about it in microphysical terms. Since it's called an inflaton, I'm assuming it is a quantum field. Does this suggest particles or virtual particles of inflatons? I know we don't have a quantum theory of gravity yet. I don't know if we don't have a quantum theory of inflation, though.

 

[PeterDonis]

Since it's called an inflaton, I'm assuming it is a quantum field.

Strictly speaking, yes, but quantum fluctuations in the field don't really play a role in the dynamics. A classical scalar field can produce inflation by the mechanism I described. As I understand it, quantum fluctuations in the field are necessary to trigger the phase transition that ends inflation, though.

Does this suggest particles or virtual particles of inflatons?

Not necessarily. Not all states of a quantum field have particle interpretation.

 

[friend]

Not all states of a quantum field have particle interpretation.

I suppose I can imagine this. Perhaps it's like the higgs field that does not necessarily have boson yet even though the field is not zero. Do all the other SM fields necessarily exhibit particles when the field is not zero?

 

[PeterDonis]

Perhaps it's like the higgs field that does not necessarily have boson yet even though the field is not zero.

The Higgs nonzero vacuum expectation value is one example of a quantum field state that does not have a particle interpretation, yes. But note that this is a vacuum state, i.e., it is the state of lowest energy of the field. Even in the particle interpretation, vacuum states are states of zero particles--more precisely, they are eigenstates of the particle number operator with eigenvalue zero.

Do all the other SM fields necessarily exhibit particles when the field is not zero?

"Field not zero" is not necessarily a good way to look at it; as above, a state where the field is not zero could be a vacuum state.

A better way to look at it is to ask what kind of phenomena we are dealing with; some phenomena are traditionally associated with particles, others are not. For example, when we do scattering experiments, we deal with phenomena that are traditionally associated with particles: tracks in cloud chambers or bubble chambers, clicks in detectors, dots on a screen where the particle hits it. But when we look at the inflaton, we are looking at something that isn't anything like that; it's something that "looks like" empty space, but empty space that expands rapidly in accelerating fashion. Trying to force something like that into a particle interpretation is, IMO, pointless; it's just nothing like the phenomena we traditionally associate with particles.

In the case of the other SM fields, they can certainly produce particle-like phenomena; that's how we discovered them, after all. But they can also produce phenomena that aren't particle-like. Look up, for example, the Aharonov-Bohm effect.

 

[nikkkom]

I suppose I can imagine this. Perhaps it's like the higgs field that does not necessarily have boson yet even though the field is not zero.

Particles are excitations of some field. Say, electrons are excitations of electron field, each with integral over all space equal to 1. This means that a field which is constant everywhere is not an excitation.

If this constant field is nonzero, you can't write it down as a sum of finite number of "particles" - you need an infinite number of them.

However, actual equations of SM don't know anything about particles per se. They express how *fields* interact. In SM, even a constant field interacts with other fields, not only "particles" (excited field) do that.

If some field somehow ends up with nonzero constant value everywhere, then this field will still have particles. But now these "particles" are excitations *on top* on this nonzero constant value.

There is an excellent simplified mathematical explanation here:

http://profmattstrassler.com/articl...higgs-field-works-with-math/1-the-basic-idea/

Do all the other SM fields necessarily exhibit particles when the field is not zero?

All SM fields except Higgs field have nonzero spin. If any of them would acquire nonzero constant value everywhere, vacuum would not be Lorentz-invariant. There would be a preferred direction in space.

 

[friend]

Even in the particle interpretation, vacuum states are states of zero particles--more precisely, they are eigenstates of the particle number operator with eigenvalue zero.

Sure, but I'm wondering about virtual particles of the inflaton field. When I hear you say that the vacuum state of the field does not have particles, I wonder if you mean real particles (higgs bosons, or inflatons) as opposed to virtual particle. It is the virtual particles (as I understand it) of whatever fields (SM or Inflatons) that actually cause cosmic acceleration. So I think my question remains open: Does the vacuum state of a field (higgs, inflaton, or SM) have VIRTUAL particles? Aren't virtual particles guaranteed by the uncertainty principle, and doesn't the inflaton field at all times also obey the uncertainty principle? Would the virtual particles be the "condensate" of the higgs field and perhaps the inflaton field? Are the virtual particles of any fields that which contributes to cosmic acceleration? Thanks.

 

[PeterDonis]

I'm wondering about virtual particles of the inflaton field.

You can't model the state of the inflaton field during inflation using virtual particles if you want to explain how it causes inflation. See below.

It is the virtual particles (as I understand it) of whatever fields (SM or Inflatons) that actually cause cosmic acceleration.

That is not correct. It is the vacuum expectation value of the inflaton field that causes inflation. Similarly, it is the current vacuum value of dark energy density (which could be a cosmological constant or could be the vacuum state of some field) that causes the expansion of the universe to accelerate. Virtual particles are not involved in this at all.

Does the vacuum state of a field (higgs, inflaton, or SM) have VIRTUAL particles?

The vacuum state of any field includes quantum fluctuations. If by "virtual particles" you just mean "quantum fluctuations", then yes, the vacuum state of a field has virtual particles.

However, if you're trying to understand what caused inflation, or what is causing the expansion of the universe to accelerate now, quantum fluctuations of the vacuum state of the field are irrelevant. The key point is that the average value of the field in its vacuum state, i.e., its vacuum expectation value, is not zero. Virtual particles/quantum fluctuations can only describe fluctuations around the average value of the field; they are no help at all in explaining why the average value of the field is what it is.

 

[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.