B Does a field's vacuum density violate conservation of energy?

[TheQuestionGuy14]

The vacuum density, or the zero point energy, of a field, doesn't change as space expands, it remains constant. But, aren't particles and virtual particles just fluctuations of these fields? Meaning as space expands, more and more particles are being created, violating conservation of energy?

 

[PeterDonis]

Global conservation of energy does not hold in an expanding universe. Global conservation of energy only holds in spacetime with a timelike Killing vector field, and an expanding universe does not have one.

See this article by Carroll:

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

 

[TheQuestionGuy14]

Global conservation of energy does not hold in an expanding universe. Global conservation of energy only holds in spacetime with a timelike Killing vector field, and an expanding universe does not have one.

See this article by Carroll:

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

Does this mean there is more particles in the universe today than there was yesterday? Or is the energy being created not related to particles?

 

[PeterDonis]

is the energy being created not related to particles?

The "energy being created" is due to a nonzero cosmological constant, which is not "related to particles".

 

[Demystifier]

Global conservation of energy does not hold in an expanding universe.

More precisely, global conservation of energy of matter does not hold in an expanding universe. But if energy of the gravitational field is added too, then total energy is conserved. The reason why such a view is not often used is the fact that such a view cannot be made general covariant, except in a trivial sense. http://de.arxiv.org/abs/1407.8028

 

[A. Neumaier]

But if energy of the gravitational field is added too, then total energy is conserved.

... and is always exactly zero, hence uninteresting.

 

[Demystifier]

... and is always exactly zero, hence uninteresting.

Well, the vanishing of the Hamiltonian is a consequence of the time reparameterisation invariance, which is very interesting to me.

 

[A. Neumaier]

Well, the vanishing of the Hamiltonian is a consequence of the time reparameterisation invariance, which is very interesting to me.

But time reparameterisation invariance is a pphnomenon quite different from a conservation law, which was the subject of my remark.

 

[Demystifier]

But time reparameterisation invariance is a pphnomenon quite different from a conservation law, which was the subject of my remark.

It's different, but very related.
time rep. invariance $\rightarrow$ vanishing total Hamiltonian $\rightarrow$ conserved total Hamiltonian

 

[A. Neumaier]

It's different, but very related.
time rep. invariance $\rightarrow$ vanishing total Hamiltonian $\rightarrow$ conserved total Hamiltonian

Sure, but the latter is as uninteresting as the conservation of total force following from Newton's third law. In both cases, only the premise is interesting, not the conclusion.

 

[TheQuestionGuy14]

The "energy being created" is due to a nonzero cosmological constant, which is not "related to particles".

Isn't the cosmological constant interpreted to be the zero point energy of all the quantum fields? Doesn't that mean as space expands, more electric field is being created, more electromagnetic field etc. Doesn't that mean more particles? Sorry if it's a dumb question.

 

[PeterDonis]

Isn't the cosmological constant interpreted to be the zero point energy of all the quantum fields?

That's one hypothesis, but we don't know if it's correct.

Doesn't that mean more particles?

No. The fields are in their vacuum state, which means no particles.

 

[TheQuestionGuy14]

That's one hypothesis, but we don't know if it's correct.
No. The fields are in their vacuum state, which means no particles.

Oh right, that makes sense. But don't virtual particles fluctuate from the vacuum state? So there's more virtual particles

Also, don't virtual particles affect real particles? Meaning if you had a closed system, and more and more virtual particles were appearing, the total energy of the real particles would change.

 

[PeterDonis]

don't virtual particles fluctuate from the vacuum state?

No. See this Insights article:

https://www.physicsforums.com/insights/vacuum-fluctuation-myth/

 

[TheQuestionGuy14]

No. See this Insights article:

https://www.physicsforums.com/insights/vacuum-fluctuation-myth/

Oh ok. Thanks. I read it, I understand where the myth came from, but I still don't quite understand, where virtual particles actually come from then?

Real particles are still high energy parts of a field though right?

 

[PeterDonis]

I still don't quite understand, where virtual particles actually come from then?

The other Insights articles linked to from that one explain what "virtual particles" actually are and how the concept is used in quantum field theory. Basically, "virtual particles" are internal lines in particular Feynman diagrams. They do not correspond to anything observable and are only used in the particular mathematical formulation, perturbation theory, that uses Feynman diagrams.

Real particles are still high energy parts of a field though right?

Real particles are excitations of the field above its ground state, yes.

 

[TheQuestionGuy14]

The other Insights articles linked to from that one explain what "virtual particles" actually are and how the concept is used in quantum field theory. Basically, "virtual particles" are internal lines in particular Feynman diagrams. They do not correspond to anything observable and are only used in the particular mathematical formulation, perturbation theory, that uses Feynman diagrams.
Real particles are excitations of the field above its ground state, yes.

Oh ok, thank you. Since virtual particles are purely mathematical, what really causes the electromagnetic force and other forces? People usually say the virtual photons are the force carriers.

 

[PeterDonis]

Since virtual particles are purely mathematical, what really causes the electromagnetic force and other forces?

The electromagnetic field. There is nothing wrong with the general idea that fields like the EM field transmit forces. You just have to be aware that the more specific idea that the way fields like the EM field transmit forces is by exchanging virtual particles has limited usefulness and can be misleading if taken too literally.

 

[TheQuestionGuy14]

The electromagnetic field. There is nothing wrong with the general idea that fields like the EM field transmit forces. You just have to be aware that the more specific idea that the way fields like the EM field transmit forces is by exchanging virtual particles has limited usefulness and can be misleading if taken too literally.

Thank you, I appreciate the help. Just one last question though, what exactly is a quantum fluctuation then? A quantum fluctuation is the temporary change of energy in an area of space, but the wiki says it's the creation of virtual particles that causes this energy fluctuation. What is it really then?

 

[haushofer]

Virtual particles are conceptual aids to describe quantum fluctuations. Quantum fluctuations are just stochastic fluctuations of the field values.

 

[haushofer]

Mmm, conceptual aids...pun non intended for those abhorred by the usage of virtual particles in popular literature.

 

[DarMM]

Thank you, I appreciate the help. Just one last question though, what exactly is a quantum fluctuation then? A quantum fluctuation is the temporary change of energy in an area of space, but the wiki says it's the creation of virtual particles that causes this energy fluctuation. What is it really then?

Quantum Mechanics is statistical, so quantities have average values but deviations from that average can be observed. Like rolling a dice the average is 3.5, but you can get 1 or 6. Similarly when you observe quantum fields you have an average/expected value but you can observe values quite a bit away from this average.

These deviations from the average are called "fluctuations". They aren't caused by virtual particles they're just an intrinsic element of the theory since it involves probabilities.

 

[A. Neumaier]

Thank you, I appreciate the help. Just one last question though, what exactly is a quantum fluctuation then? A quantum fluctuation is the temporary change of energy in an area of space, but the wiki says it's the creation of virtual particles that causes this energy fluctuation. What is it really then?

It is nothing temporal, Wikipedia is poor on this topic.

 

[PeterDonis]

what exactly is a quantum fluctuation then?

It is a sloppy and misleading way of describing the fact that, if you measure a quantum system using an observable that the system is not in an eigenstate of, the measurement result will be uncertain.

the wiki says

Wikipedia is not a good source for something like this.

 

[Heikki Tuuri]

I have been studying just this question in the past few weeks.

The fields which we know in our everyday life, for example, the photon field and the electron-positron field, have a constant number of quanta as the universe expands, if we ignore reactions with other particles.

A photon will gain a redshift when it moves in an expanding universe, because if it is emitted by an event A and absorbed in another event B far away, then B "sees" A receding at a great speed - B sees the photon redshifted. If we take the energy of the photon to be what B sees, then the photon has lost energy.

The redshift mechanism reduces the kinetic energy of a electron, too, when it moves in an expanding universe.

What about a scalar field which some people claim, is causing the acceleration of the expansion right now?

The first thing to note is that a field whose energy increases in an expanding universe is "exotic matter", that is, it has a negative pressure and negative gravity. It breaks various energy conditions of General Relativity. We do not know such matter from our everyday experience. It is highly speculative to assume that such matter could exist.

What about the energy content in the hypothetical scalar field? Is that energy contained in quanta of some kind? If yes, does the number of such quanta increase as the universe expands?

Our everyday experience is that energy in a weakly interacting system is, indeed, divided into well-defined quanta which can carry mass and kinetic energy.

The question is harder in a strongly interacting system, say, a crystal. Is the vibrational energy in a crystal divided into quanta of some kind? Certainly not in any unique way.

Suppose that we work in the Minkowski spacetime. The universe is not expanding. We want to excite the Higgs scalar field which has the famous Mexican hat potential. We use a huge particle collider to produce a vast density of Higgs particles. In that way we store a lot of energy in the Higgs field. Can we say that the excited Higgs field has its energy stored in quanta of some kind? It is a strongly interacting system.

We expect the excited Higgs field to release just the same amount of energy as we pumped into it. What if the universe is expanding at the same time? Can the released energy be bigger than the pumped energy?

That would be surprising. Rather, I would expect the Higgs field lose some of the kinetic energy of the Higgs particles as the universe expands.

The hypothesis that a scalar field can grow its energy in an expanding universe is highly speculative. It is at odds with what we know about other fields.

Literature seems to ignore the fact that the hypothetical scalar fields in inflation and dark energy would be exotic matter, and would quantum mechanically behave in a surprising way.

 

[PeterDonis]

The redshift mechanism reduces the kinetic energy of a electron, too, when it moves in an expanding universe.

If the electron is moving at relativistic speed relative to comoving observers, yes. But the number of electrons that were doing that for the vast majority of the universe's history were negligible. "Cold" matter, i.e., matter that is at rest relative to comoving observers, does not redshift as the universe expands.

What about a scalar field which some people claim, is causing the acceleration of the expansion right now?

We don't know that it is a scalar field. A cosmological constant, which is the simplest explanation for accelerating expansion, is not a scalar field; it's just a constant.

a field whose energy increases in an expanding universe is "exotic matter"

Correct.

It is highly speculative to assume that such matter could exist.

No, it isn't. We already know of one scalar field: the Higgs field. We also know that quantum vacuum fluctuations are exotic. And we know that only something with the characteristics of exotic matter (which includes a cosmological constant--that is also exotic) can account for accelerating expansion. So given that we observe accelerating expansion, we have no choice but to believe that something exotic exists.

What about the energy content in the hypothetical scalar field? Is that energy contained in quanta of some kind? If yes, does the number of such quanta increase as the universe expands?

Not as long as there are no interactions, which was the condition you imposed on the other fields. In the absence of interactions, the number of quanta is constant for any field.

The question is harder in a strongly interacting system

In a strongly interacting system, the number of quanta does not have to be constant. But you said earlier you wanted to ignore interactions. Which is it? And if you don't want to ignore interactions, what interactions do you think are happening in the vast empty regions of our expanding universe?

The hypothesis that a scalar field can grow its energy in an expanding universe is highly speculative.

No, it isn't. It's an obvious consequence of the stress-energy tensor for scalar fields plus the absence of interactions.

Literature seems to ignore the fact that the hypothetical scalar fields in inflation and dark energy would be exotic matter

You really should read the literature before making such a claim.

 

[Heikki Tuuri]

https://en.wikipedia.org/wiki/Mathematical_formulation_of_the_Standard_Model
The Higgs lagrangian contains the term which is quartic (fourth power) on the Higgs field strength. I would say the Higgs field interacts strongly with itself.

The question which I posed in my previous post: how does such a field behave in an expanding universe? Is it possible that the energy of the field might grow?

Specifically for the Higgs field: assume that we have excited the field so that it is no longer in the lowest energy point. Would the energy of the field grow when space expands? If yes, then there is a huge negative pressure present. Can that negative pressure accelerate the expansion of space locally? Can it even create a new "bubble universe"?

 

[PeterDonis]

I would say the Higgs field interacts strongly with itself.

"Strongly" is subjective; the quartic term is simply the lowest order interaction term possible for a scalar field, and how "strong" the interaction is depends on the coupling constant, not the order of the term.

assume that we have excited the field so that it is no longer in the lowest energy point

Doing that in a brief experiment in the LHC is not at all the same as doing it throughout the entire universe. Averaged over the universe, any field will be very, very close to its ground state, and how far it departs from the ground state will be measured by a temperature. For example, the temperature of the CMBR is 2.7 K.

Would the energy of the field grow when space expands?

You can read off the answer to this from the stress-energy tensor, as I said in my previous post. Basically, if you have a stress-energy tensor of perfect fluid form, and there is a positive energy density $\rho$ and a pressure $p \le - \rho / 3$, then the field will cause accelerated expansion and the "energy" of the field will grow as the universe expands. A scalar field has $p = - \rho$, just as a cosmological constant does; the only difference is that a cosmological constant means $\rho$ and $p$ are the same everywhere in spacetime, whereas for a scalar field $\rho$ and $p$ can vary.

Can that negative pressure accelerate the expansion of space locally?

See above.

Can it even create a new "bubble universe"?

I don't know where you're getting this from; accelerated expansion has nothing whatever to do with "bubble universes".

 

[Heikki Tuuri]

The logic goes like this:

Suppose that we have a box full of a matter field.

If the energy of the field inside the box grows as the box expands,

THEN

we say that there is negative pressure in the box.

For positive pressure, the field energy inside decreases as the box expands.

The question is if there can exist a matter field whose energy grows as space expands. That is the question I am looking at. Cosmologists seem to take for granted that a scalar field can grow its energy as space expands.

 

[PeterDonis]

The logic goes like this:

Suppose that we have a box full of a matter field.

If the energy of the field inside the box grows as the box expands,

THEN

we say that there is negative pressure in the box.

For positive pressure, the field energy inside decreases as the box expands.

Applying this to an expanding universe is based on a heuristic (and often misleading) analogy between "space expanding" and the expansion of a fluid in a box in flat spacetime.

Cosmologists seem to take for granted that a scalar field can grow its energy as space expands.

No, they don't "take it for granted". As I said, they look at the stress-energy tensor of the scalar field and read off the (positive) energy density and (negative, with equal absolute value) pressure from it.