Relativity & Quantum Field Theory: Explaining Cosmological Constant

[somy]

Can anyone explain the relation between quantum field theory and relativity.
I mean the "cosmological constant".
Tanks a lot.

 

[hellfire]

The cosmological constant is firstly a geometric modification of the action of gravity (a modification of the Einstein-Hilbert action). This means that the way spacetime reacts against energy and momentum is postulated to be different whether a cosmological constant is introduced in the theory or not.

In general relativity it is described how spacetime is influenced by matter. The Einstein equations contain geometric terms on the left hand side (as a function of the metric of spacetime) and a description of matter on the right hand side (energy and momentum of matter). Originally the cosmological constant is term on the left hand side.

Nevertheless, one can shift this term to the right hand side, by defining an associanted energy-momentum (an energy associated to the cosmological constant). The origin of this energy is assumed to be the sum of the ground states of all mattter fields (the vacuum).

I believe both descriptions are equivalent, but the physical interpretation is different.

This second description relies on quantum field theory, where every matter field has a ground state of energy which may be different from zero energy. The actual value for the ground state of all matter fields predicted by QFT is too high to fit with the cosmological observations (the cosmological constant leads to an accelerated Hubble expansion).

Why it is assumed that the sum of the ground state of all matter fields has the same behavior as a cosmological constant (a scalar field), is an interesting question I cannot answer (may be someone can).

 

[cosmoboy]

This is quite right, that an extra source term is added in Einstein's equation which produces repulsive gravity or acceleration in place of deacceleration. It is well known that if some thing is moving away from a massive body (like galaxies are moving away from each other) then this massive body will try to slow down the motion of that body. This was what we expected from our universe also. We thought its motion will get slow down due to gravity, but what we found is just opposite. In 1998 from the observation of some objects known supernova Ia we found that in place of slowing down, expansion of the universe is speeding. It seems some there is some repulsive force behind it. Now the question is how to produce a repulsive gravity. All seansiable type of material produce attractive gravity, the clue is as follows:

Einstein's general theory of relativity predicts that not only matter but pressure also can produce gravity. Positive pressure produce attractive pressure and negative pressure repulsive. Nature of any type of matter or energy is determined by its equation of state
"p = w rho ", here "p" is the pressure , "rho", density and "w", a number. Any type of energy for which "w < 0", produce negative pressure, which means repulsive gravity. For the cosmological constant "w=-1".

Although it has been postulated that this cosmological constant is the weight of vacuum, but still there are some problems about it which are not clearly understood:

1. Why there is a large mismatch between its theoretical and observational value.

2. Why we are living in epoch when cosmological constant is starting to dominate.
http://arxiv.org/abs/hep-th/0212290

 

[Nereid]

Just to amplify cosmoboy's first problem ... the mismatch is indeed large; 120 orders of magnitude (though I note that Greene says this can be reduced to 'only' some 66 OOM through a bit of theoretical jiggery-pokery). Just to put the size of this mismatch into context - I don't know of any theory-observation mismatch ever that's been so large! Oh, and the number of baryons in the whole universe is 'only' approx 10^80, in case you were wondering how big a number 120 OOM is.

 

[marcus]

Can anyone explain the relation between quantum field theory and relativity...

From context, I assume that by relativity you mean General Relativity.

I guess the answer is that there is a disconnect.

Space, in GR, has a dynamic unpredetermined geometry which is constantly be influenced by the flow of matter.

One special solution of the GR equation is the static flat solution you get when the universe is perfectly empty----this is Minkowski space and it was the space of the earlier (1905) theory of special rel.

Quantum field theory has been constructed on Minkowski space.

QFT and GR do not even share the same space, which is one reason for the disconnect.

One widely used test of whether a theory is compatible with GR is
to see whether it is "background independent"-----that it does not rely on a prior rigid geometrical framework (like Minkowski space) but can be defined without precommitting to a background geometry.

What is called "relativistic quantum field theory" is not, in fact, background independent and is therefore not relativistic in the GR sense.

A major longterm theoretical objective that has been identified, is to construct a truly relativistic quantum physics---a general relativistic quantum field theory. The first requirement is to provide a background independent quantum theory of space and time within which venue, or on top of which foundation, such a background independent QFT can be built.

Increasingly, I would say, this is how quantum gravitists see the problem of uniting QFT with GR.
===========

by comparison, the talk about the 120 order of magnitude mismatch is just scratching the surface.

we have a clear indication that there is a small positive cosmological constant-----which is by definition a small positive curvature term in the einstein equation----we do not know that this is produced by an invisible constant energy.
But if we SUPPOSE that it is produced by a dark energy or vacuum energy then that constant energy density must be

0.6 joules per cubic kilometer

or, if you know Planck units, there is a Planck energy unit and a Planck volume unit, the cube of the Planck length, and the same energy density
is
1.3 x 10^-123 Planck energy units per Planck volume

But thinking of it that way could be inappropriate and in any case is jumping the gun---the thing might just be an intrinsic curvature that space has for some other reason, or simply a correction term in the Einstein equation.

the cosm. constant is something that comes up in Gen Rel, and in astronomical observation, and which (special relativistic) QFT doesn't say anything about. And I can't say that I would expect it to.

Some people hype the CC situation into a big scandal, but I think it is better seen as just one aspect of a larger crisis in physics.
they hastily assume that the CC arises from a energy density
and then they presume that (effectively non-relativistic) late-20th century QFT should be able to explain it as a quantum vacuum energy, and
then they make scandalized noises when QFT predicts a vacuum energy density that is many OOM off----say 120 orders of magnitude---or this or that, things get a bit vague because the QFT predictions of vacuum energy are absurd and would have caused space to collapse long ago if they were real. So this is headlined by the attentiongetters and being the biggest-ever scandal in theoreticial physics. But all it means is that
(non-background independent) QFT is beginning to go out of date.

Well somy, that is my personal take on it, and with luck you will get a few more viewpoints that you can contrast with this one! wish the best of luck in hunting answers.

 

[Nereid]

Thanks marcus.

From another perspective, the 'mismatch' doesn't seem so severe ... both QM and GR are highly successful theories of physics - AFAIK, there are no good observational or experimental results which are inconsistent with these theories, within their domains of applicability. Indeed, some aspects of QED are among the most accurately tested in all of science (QED comes through with flying colours). GR hasn't been tested so stringently, but it has passed all tests so far, some to one part in nearly 100,000.

What gives? Well, for one, there's a huge gulf between domains where QM and GR would seem to be wildly inconsistent (crudely, the Planck regime) and the best we can do today - whether size, time, or energy - so maybe all will become clear if we can just get another order of magnitude or 10 out of our labs (energy-wise), or watch quarks and gluons in super^5 slowmo.

 

[turbo]

It is possible that the absolute energy of the quantum vacuum fields (as envisioned in QFT) cannot be easily translated to GR. It may be that our universe exists at a ground state from which we cannot sense the absolute energy of the quantum vacuum.

 

[hellfire]

Space, in GR, has a dynamic unpredetermined geometry which is constantly be influenced by the flow of matter.

One special solution of the GR equation is the static flat solution you get when the universe is perfectly empty----this is Minkowski space and it was the space of the earlier (1905) theory of special rel.

Quantum field theory has been constructed on Minkowski space.

QFT and GR do not even share the same space, which is one reason for the disconnect.

One widely used test of whether a theory is compatible with GR is
to see whether it is "background independent"-----that it does not rely on a prior rigid geometrical framework (like Minkowski space) but can be defined without precommitting to a background geometry.

What is called "relativistic quantum field theory" is not, in fact, background independent and is therefore not relativistic in the GR sense.

From the perspective of general relativity it is known that at cosmological scales spacetime can be approximated in a good accuracy with the expanding Robertson-Walker metric. A Quantum Field Theory can be indeed defined on this expanding background.

However, this does not yield a correct prediction for the cosmological constant anyway. I agree that in view of the discrepancy, something is probably wrong with the definition of vacuum in QFT (assuming that cosmological constant = ground state of all fields), but I do not see how it necessarily follows from this that a background independent theory is needed.

In my understanding, a cosmological background independent theory would be needed if quantum effects at cosmological scales would be important in a way that they might modify the background. To me this seams not to be realistic.

One may require a background independent field theory due to other reasons, but not necessarily to explain the cosmological scenario...

Am I missing something?

 

[Chronos]

I think there is something fundamentally wrong with QT predictions of ZPF energy density. Unless GR is horribly flawed, which is not evident, QT appears more suspect at this point. While QT has been enjoyed some huge successes, it has also demonstrated some profound deficiencies - notably that certain particles that should have appeared at certain energy levels, have not, and cutoff energy limits predicted have been grossly violated.

 

[marcus]

Can anyone explain the relation between quantum field theory and relativity.
I mean the "cosmological constant".
Tanks a lot.

hello somy, Nereid, hellfire, Chronos...
just to clarify:
what Nereid says about the mismatch between QFT and the astronomically measured CC----if you interpret it as being due to a "dark energy" which is one perfectly OK interpretation----is quite right. there is some 120 OOM mismatch which people can play around with and change but remains ridiculously huge. I have nothing to add to what's already said here.

My point refers to how somy put the question:
Can anyone explain the relation between quantum field theory and relativity.

I think there is a split between the picture of nature you get in QFT and in GR which goes much deeper than the failure of QFT to make a sensible vacuum energy. Somy's question sounds like somy ASSUMES that the problem between the two theories is confined to disagreement about the Cosmological Constant.

Indeed that is the impression you may sometimes get when you listen to scientists, especially when for example they are making science sound interesting to lay-folk or just dramatizing the most accessible topics for entertainment. Maybe some people seriously believe that this problem with QFT predicting a huge vacuum energy is the crux of the problem between QFT and GR.

But I am anxious to challenge that assumption because I think the problem is deeper and is not just about the CC

Well, somy assumes that explaining "relation between QFT and GR" means talk about the CC, so he or she asks about the CC, and that is fine, so everybody should discuss the CC and the QFT estimate of vacuum energy.

I am just pointing out that the disfunction between them is that they live in different ideas of spacetime. In other words, they are built on incompatible foundation.

the resolution which I see in progress is to complete the job of making QFT relativistic------it is already special, but it is still not general relativistic.
so it has gotten to the 1905 point (when special was formulated) but it has not yet gotten to the 1915 point (when gen. rel. was formulated)
completing the job looks like it is going to take more than just fixing the CC

 

[marcus]

I think there is something fundamentally wrong with QT predictions of ZPF energy density. Unless GR is horribly flawed, which is not evident, QT appears more suspect at this point. While QT has been enjoyed some huge successes, it has also demonstrated some profound deficiencies - notably that certain particles that should have appeared at certain energy levels, have not, and cutoff energy limits predicted have been grossly violated.

I agree
the embarrassment about the QFT vacuum energy goes back decades.
Steven Weinberg made some famous remarks about it in the dim past.
I can't remember. but long long before 1998 which was when "dark energy" hit the news.

It was only in 1998 that by supernova ranging they found that the CC was a small positive amount of curvature, instead of zero.
Before that, most people assumed it was zero, which naively is even MORE at odds with the QFT estimate

but the raw QFT vacuum energy estimate has always been known to be insanely wrong----it is on the order of energy densities that people estimate may have existed around the time of the big bang, nobody seems able to agree on a valid way to cancel it out by fiddling with the theory.

so this is a longstanding QFT problem which astronomers merely called public attention to in 1998 with the accelerated expansion business, but had been there already.

I think I am just corroborating what Chronos just said.

 

[somy]

Thanks marcus.