Particle Physics in Curved Spacetime (Background Independent)

[rogerl]

Special Relativity is just a special case where spacetime is flat and
fixed. Meaning it is not really a correct picture of reality because
nature chooses curved spacetime that is background independent. So
quantum particles shouldn't be occurring in a flat fixed background but
a dynamic one. But how come Dirac Equation which is simply a marriage
of Quantum Mechanics and Special Reality works so well and can explain
1/2 spin particles as well as predict antimatter when Special Reality
or flat minkowski spacetime is not even the correct background?? What
would happen if Dirac Equation is adjusted to occur in a curved
background independent spacetime??

 

[Bill_K]

What would happen if Dirac Equation is adjusted to occur in a curved
background independent spacetime??

Hawking radiation.

 

[rogerl]

Hawking radiation.

Not in the sense of curved spacetime in a black hole. What I meant to ask was what if particles were not modeled as moving in flat spacetime background but particles part of dynamical curved spacetime itself.

Is this related to Quantum Gravity. I mean. Does quantum gravity only involve inside Planck scale physics or is it about particles occurring in dynamical curved spacetime or background independent?

If the latter, then why does the Dirac Equation work when it occurs in the backdrop of flat spacetime which is not what nature chooses?

 

[Fra]

particles shouldn't be occurring in a flat fixed background but
a dynamic one

A complete answer would involve a full unificaiton of GR and QM, which is a still open question.

The NOTION of "particle" depends on the minkowski space. If you throw that away, then how do you DEFINE a particle? This is non-trivial.

Also, to understand how come this apparently flawed QFT works, when we at the same time know that spacetime at large isn't flat minkowski, I think one can simply note this:

QFT and the SM of particle physics as we know it, works and it tested for subsystems only. This means that in order to make a measurement, including preparations etc (ie. to construct the "observer") you NEED a controlled environment, in which the system we study is embedded.

And then since any reasonable differentiable curved space is of course locally flat if you just constrain your study to a sufficiently small subsystem, then there we have the flat background. Ie. any obserer, with all what it means you have for many practical circumstances a flat reference frame.

If you understand it this way, it's not trivial either to just throw away this reference, because it derives from the observer. The minkowski background spacetime, is of course only a small subset of the full observer-context. This is something under debate. There is no consensus here.

There problem is how the formalism of measurement theory itself deforms, if you try to remove the observer choice.

The background independence debate is IMHO, just a special case of a much larger problem that at its EXTREME is "observer independent" measurement theory, and one can certainly question what exactly that even means.

/Fredrik

 

[bapowell]

Not in the sense of curved spacetime in a black hole. What I meant to ask was what if particles were not modeled as moving in flat spacetime background but particles part of dynamical curved spacetime itself.

There is an entire field of study dedicated to the development of quantum field theories in general relativistic -- including time dependent -- backgrounds, called Quantum Field Theory in Curved Spacetime. In general you find that each quantum oscillator mode is time dependent and are therefore not modes of definite frequency. It is a semiclassical theory -- quantized fields are placed on a dynamic background. I'm not sure what you mean by particles being 'part of dynamical curved spacetime itself'. You'll need to be more precise with your meaning here.

If the latter, then why does the Dirac Equation work when it occurs in the backdrop of flat spacetime which is not what nature chooses?

Keep in mind that the flat spacetime of SR is still vitally important as the tangent space in general relativity. We require a fully formulated field theory to exist there on account of the equivalence principle. The Dirac Equation in flat space is in fact a direct consequence of the Lorentz symmetry of SR, so, it sort of has to work in flat space by construction. But that's not to say that a curved-space version of the Dirac Equation doesn't exist -- it does. See Wald or Birrel and Davies for some good discussion on these topics.

 

[rogerl]

There is an entire field of study dedicated to the development of quantum field theories in general relativistic -- including time dependent -- backgrounds, called Quantum Field Theory in Curved Spacetime. In general you find that each quantum oscillator mode is time dependent and are therefore not modes of definite frequency. It is a semiclassical theory -- quantized fields are placed on a dynamic background. I'm not sure what you mean by particles being 'part of dynamical curved spacetime itself'. You'll need to be more precise with your meaning here.

Oh. For a moment I remembered Einstein search for a Unified Theory wherein he tried to model particle was some kind of geometry and part of spacetime. I thought background independent particle physics meant the particle didn't have any background but part of the background. But now I understand it means simply that the background is dynamic and not static.. meaning as the particle moves, the background oscillates too or something like that.. right?

Anyway. Is there any unification programme now where the particle is part of the background of spacetime itself somewhat like Einstein original attempt but now the particle being quantum particle that doesn't need any spacetime background at all or the particle part of quantum spacetime itself?

 

[bapowell]

Background independence simply means that there is no prior or preferred geometrical structure in the theory. I think you might be using the term differently from this. QFT in curved space considers quantized fields on a classical background. The background can be general -- expanding, static, contracting, flat, curved, what have you. The quantum fields are distinct from the background -- they are analogous to quantum fields in flat space and belong to some Hilbert (or Fock) space. However, because all fields feel gravity, the fields couple to the classical background and therefore respond to its behavior. In general, one finds that the nontrivial spacetime dynamics manifests itself in the time-dependence of the quantum field modes. In flat space, while a positive frequency mode will remain positive frequency for all time, in curved space an initially positive frequency mode will in general become a combination of positive and negative frequency. This has a profound impact on the physics. For example, Hawking radiation is the result of placing quantum fields in the vicinity of an event horizon. The mixture of positive and negative frequency modes is manifested as particle creation at the horizon. The same thing happens at the de Sitter horizon in spacetimes undergoing accelerated expansion. There are many more examples.

Far from an expert, but I believe that there have been efforts to incorporate particles into Loop Quantum Gravity as elements of the geometry (braids maybe?)

 

[rogerl]

Background independence simply means that there is no prior or preferred geometrical structure in the theory. I think you might be using the term differently from this. QFT in curved space considers quantized fields on a classical background. The background can be general -- expanding, static, contracting, flat, curved, what have you. The quantum fields are distinct from the background -- they are analogous to quantum fields in flat space and belong to some Hilbert (or Fock) space. However, because all fields feel gravity, the fields couple to the classical background and therefore respond to its behavior. In general, one finds that the nontrivial spacetime dynamics manifests itself in the time-dependence of the quantum field modes. In flat space, while a positive frequency mode will remain positive frequency for all time, in curved space an initially positive frequency mode will in general become a combination of positive and negative frequency. This has a profound impact on the physics. For example, Hawking radiation is the result of placing quantum fields in the vicinity of an event horizon. The mixture of positive and negative frequency modes is manifested as particle creation at the horizon. The same thing happens at the de Sitter horizon in spacetimes undergoing accelerated expansion. There are many more examples.

Far from an expert, but I believe that there have been efforts to incorporate particles into Loop Quantum Gravity as elements of the geometry (braids maybe?)

Lee Smolin wrote in Trouble With Physics:

"A quantum theory of gravity should also be background-independent. Space and time should arise from it, not serve as a backdrop fo the actions of strings"

In this context, does it mean QFT that is background-indepedent means that space and time should arise from the actions of fields or particles?? This is different from QFT merely on a curved spacetime.

What would happen if we make our QFT such that space and time should arise from the actions of fields or particles? and why isn't this done?

 

[marcus]

...Anyway. Is there any unification programme now where the particle is part of the background of spacetime itself somewhat like Einstein original attempt but now the particle being quantum particle that doesn't need any spacetime background at all or the particle part of quantum spacetime itself?

...
Far from an expert, but I believe that there have been efforts to incorporate particles into Loop Quantum Gravity as elements of the geometry (braids maybe?)

Particle physics on fixed curved spacetime backgrounds is an important and distinct research area---bapowell was just getting into it and it would be efficient for your thread to focus on that. It's extremely interesting. (horizons, Hawking radiation, the temperature of deSitter space that arises simply due to its geometry.)

Rogerl, I would actually suggest that you have two threads, one about particle physics on a fixed spacetime geometry (dependent on specifying a prior geometry which remains fixed.)
And another thread about background independent theories---ones where no prior geometry is specified and geometry interacts with matter (its all one combined interactive thing.)

Loop Gravity could be mentioned here. Brian Powell mentioned efforts to incorporate matter. Braiding is not the primary way matter is included, but was tried by a few people until around 2008. Very little since AFAIK. Interest in braid matter could spring up again, who can say? But the prevailing way is exemplified by the December paper "Spinfoam Fermions" http://arxiv.org/abs/1012.4719 .

There is also the followup January paper http://arxiv.org/abs/1101.3264
"Spinfoam Fermions: PCT Symmetry, Dirac Determinant, and Correlation Functions"

These papers have no connection with the 2005-2008 braid matter gambit. I can tell you the basic idea for combining matter and geometry, but the development is still extremely rudimentary. At this point the Loop effort to combine matter with geom. should not be compared with a more thoroughly elaborated approach like Noncommutative Geometry which does manage to be a single theory (not an infinite number one can't choose among) and does recover something like the standard particle model. I'm not saying Loop is wrong (or right) here, only that the enterprise of including matter in it is just getting started and is still quite primitive.

I will try to say what the basic idea is---then you could could continue with your discussion of NON-backgroundindependent particle physics on fixed curved spacetime geometries. (There is more to say about that than there is at this point about Loop, in the matter department.)

The basic idea is simple and at the same time can be difficult to grasp. It is that a graph (also the higher dimensional analog called a cell-complex) provides a TRUNCATION or arbitrary restriction of the variables one decides to consider. It gives a way to temporarily restrict the "degrees of freedom" of a problem.

You put geometry and matter in by painting the graph. Penrose, who seems to have helped originate the idea, called it "coloring" the graph. In less colorful language it simply comes down to labeling the links and nodes of the graph with convenient mathematical critters which can represent matter, or area, or volume or angle.

The graph by itself has no geometry---everything is in the labels. In the conventional (pure gravity, matterless) spin network case the nodes are labeled with volume and the links are labeled with areas which are like the areas where adjacent volumes touch. But these are quantum---indefinite---volumes and areas. The labeled graphs belong to a Hilbertspace of quantum states. You can have superpositions. There are operators corresponding to measurements made on the geometry. So you work some uncertainty and fuzziness in this way. Adding matter basically involves more paint of different colors.

The underlying graph is a finite vehicle for getting labeled with geom.+matter.

There is a philosophical idea lurking here which is that any experiment with nature involves only a finite number of measurements and detections of events. Our information about geom+matter is finite. So working with a graph reflects this----instead of working with a continuum.

Then when you want graph-defined states to evolve you get colored foams--a foam is a "2-cell-complex" just like a graph is a "1-cell-complex". A foam is made of vertices, edges and faces. It is the one-dimension-higher analog of a graph. So it is the path along which a graph evolves from its initial to its final state.

A foam also is finite. A finite number of events and measurements and boundary restrictions on the process.

After the model is defined on these finite truncations (these simplified worlds of colored graph and foam) then to finish defining the theory there is a way to let the size of the graph go to infinity. Like letting N --> infty in an ordinary power series in calculus.
You try to do your calculation with a "cutoff" and then you remove the artificial restriction and let the complexity grow indefinitely.

It's background independent in the same way that Gen Rel is. In GR there is no fixed prior metric, the metric interacts with matter and evolves. In Loop there is no fixed prior labeling with volumes, areas etc. The labels are variable.
Also there is no standard space in which the graph is embedded. It is an abstract graph or you can picture it as flexible and stretchy so it can be mooshed around according to "diffeomorphism invariance". But I think it is cleaner to have no embedding, just a pure abstract graph. It is not "in" space, but instead it "is" space.

So much for the philosophical Loop outlook. Sorry for the interruption, since I think it is more of a footnote to the main discussion.

 

[Fra]

Lee Smolin wrote in Trouble With Physics:
"A quantum theory of gravity should also be background-independent. Space and time should arise from it, not serve as a backdrop fo the actions of strings"
...
What would happen if we make our QFT such that space and time should arise from the actions of fields or particles? and why isn't this done?

I think it helps a lot to pull these arguments apart, and see exactly why this is difficult.

Looking at the history of observer invarance, as it lead to SR and QG, we see that the argument behind required lorentz, poincare and ultimately diff invariance from the laws of physics, is that it would be a contradiction if different observers, see different laws.

But let's not forget that these theories are realist theories. These theories make statements about reality (often as relations or correlcations between observers).

At the advent of QM, we are working with a measurement theory. Meaning that the theory makes statements about outcomes of specific measurements. And each measurements obvious needs a backdrop or context (of some kind; NOT necessarily classical spacetime).

In relativity, the choice of background, actually corresponds to observer frames. Thus there is a very subtle relation here between background independence in the sense of GR, and observer invariance.

I think it's obvious here that there seems to be a paradox here. HOW can we formulated a measurement theory (which requires an observer) in a way that is observer independent?

I think Smolin in his paper also mentions that background independence, like I suggested here, is in fact MORE thatn JUST avoiding a spacetime background. The spacetime background is part of the observer context.

Maybe it's the case that a proper measurement teory, can not be made B/I. But then the next question comes, and that's what it means for different observers to make the same predictions? Does this mean we don't have unique laws? Does the laws evolve?

The STRICT application of the strongest form of B/I (as observer independence) is somewhat at clash with the whole idea of measurement theory. Conceptually how is this paradox resolved? This is a difficult question so no wonder it's an open one.

Measurement theory in a fixed, but curved and dynamical classical background, is a semiclassical description that doesn't address this.

/Fredrik

 

[A. Neumaier]

I'm not saying Loop is wrong (or right) here, only that the enterprise of including matter in it is just getting started and is still quite primitive. [...]

The basic idea is simple and at the same time can be difficult to grasp. It is that a graph (also the higher dimensional analog called a cell-complex) provides a TRUNCATION or arbitrary restriction of the variables one decides to consider. It gives a way to temporarily restrict the "degrees of freedom" of a problem. [...]

After the model is defined on these finite truncations (these simplified worlds of colored graph and foam) then to finish defining the theory there is a way to let the size of the graph go to infinity. Like letting N --> infty in an ordinary power series in calculus.
You try to do your calculation with a "cutoff" and then you remove the artificial restriction and let the complexity grow indefinitely.

It's background independent in the same way that Gen Rel is. In GR there is no fixed prior metric, the metric interacts with matter and evolves. In Loop there is no fixed prior labeling with volumes, areas etc. The labels are variable.
Also there is no standard space in which the graph is embedded.

Just curious: How does one get the dimension of space-time out of this limit? Doesn't it have to be encoded already into the finite graph or triangulation?

 

[rogerl]

Particle physics on fixed curved spacetime backgrounds is an important and distinct research area---bapowell was just getting into it and it would be efficient for your thread to focus on that. It's extremely interesting. (horizons, Hawking radiation, the temperature of deSitter space that arises simply due to its geometry.)

Rogerl, I would actually suggest that you have two threads, one about particle physics on a fixed spacetime geometry (dependent on specifying a prior geometry which remains fixed.)
And another thread about background independent theories---ones where no prior geometry is specified and geometry interacts with matter (its all one combined interactive thing.)

When I wrote the initial message and thread. I didn't even know there's the the important difference between particle physics on a fixed spacetime geometry and where geometry interacts with matter in one combined internative thing as you said. I thought the latter is just an Einstein fail attempt for Unfied Field Theory. When you guys replied. You didn't emphasize there are those 2 important differences. So I thought the answer why particle physics were not modeled in curved spacetime is because in a small local portion of spacetime, curvature is not very important.. and only in black holes where extreme curvature can cause hawking radiation for example. I thought background independence was this meaning. Then I came that passage in Lee Smolin book about "A quantum theory of gravity should also be background-independent. Space and time should arise from it, not serve as a backdrop fo the actions of strings", then I realized it is different to your answers, that it is not just about the local area being flat. But space and time arising out of it.

Anyway, I was thinking. This space and time arising out of it. I think this is only important inside the Planck scale where space/time and quantum fluctuations are meaningful, no? In QFT, the particles are thousands of times larger than the Planck scale.. so there is no sense that space and time should arise from the particles. Or is the QFT scale also important that space and time should also arise from the particles (and not just a within the Planck scale thing)??

 

[atyy]

Spacetime doesn't serve purely as a backdrop for strings. In perturbative string theory, although there is always some part of spacetime that is fixed, there is also a fluctating part that arises from the vibrations of strings.

In the AdS/CFT correspondence, which describes a part of string theory, only the boundary of the spacetime is fixed, while the bulk spacetime arises from the vibrations of particles. This part of string theory probably doesn't describe our universe, but it is the closest thing we have now to a background independent quantum gravity, and hopefully studying it will give us clues for generalizing it.

 

[rogerl]

Spacetime doesn't serve purely as a backdrop for strings. In perturbative string theory, although there is always some part of spacetime that is fixed, there is also a fluctating part that arises from the vibrations of strings.

In the AdS/CFT correspondence, which describes a part of string theory, only the boundary of the spacetime is fixed, while the bulk spacetime arises from the vibrations of particles. This part of string theory probably doesn't describe our universe, but it is the closest thing we have now to a background independent quantum gravity, and hopefully studying it will give us clues for generalizing it.

Strings are smaller than the Planck scale.. is space and time arising from the strings only relevant in the Planck scale. What I want to know is. Aren't particles like quarks and electrons affected by it too. In QFT, the particles are thousands of times larger than the Planck scale.. so there is no sense that space and time should arise from the particles like electrons or quarks too? Or is the QFT scale also important that space and time should also arise from the electron and quark particles for instance (and not just a within the Planck scale thing)??

 

[atyy]

The sorts of QFTs that describe electrons, photons and quarks are defined on a fixed 4D spacetime background, and they are great theories at the energy levels of eg. the LHC.

 

[rogerl]

The sorts of QFTs that describe electrons, photons and quarks are defined on a fixed 4D spacetime background, and they are great theories at the energy levels of eg. the LHC.

Why, can't space and time arise out of the fermions too? Why shouldn't Smolin context of background independence work in QFT too?

Also isn't it that beneath the Planck scale, space and time may not even exist. So the reason QFT is not modeled in an interactive dynamic spacetime where space and time arise from the particles is because we lack the model?

 

[atyy]

QFT on a fixed 4D spacetime is not expected to work at the Planck scale. The QFT's we have that describe photons, electrons and quarks are only approximate theories. Why do they work, even though they are wrong? Just the same way that Newton's laws work even though they are wrong - in a limited range of energies. At the Planck scale, a theory that is not QFT on a fixed 4D spacetime must take over, maybe string theory, maybe loop quantum gravity, or something else we don't yet know about.

 

[rogerl]

QFT on a fixed 4D spacetime is not expected to work at the Planck scale. The QFT's we have that describe photons, electrons and quarks are only approximate theories. Why do they work, even though they are wrong? Just the same way that Newton's laws work even though they are wrong - in a limited range of energies. At the Planck scale, a theory that is not QFT on a fixed 4D spacetime must take over, maybe string theory, maybe loop quantum gravity, or something else we don't yet know about.

You still don't get my question. A particle like electron or quark is thousand of times larger than the Planck scale. Yet since mass-stress-energy couple to the stress tensor in Einstein Field Equations. Shouldn't space and time also arise from the electron or quark since they are all dynamical? Why is this concern only in the Planck scale?

 

[marcus]

A. Neumaier asked a very interesting question. How do you determine the dimension of space in Loop Gravity. Presumably it could be a quantum observable obtained by comparing measurements of area and volume---or radius and volume. Tom Stoer has discussed LQG dimensionality in another thread. I can't claim expertise but I would like to try to answer Neumaier's question, and also give anyone else who has ideas about it a chance to contribute.

Prof. Neumaier's post contained a condensed version of one of mine, which I will reproduce to provide context.

I'm not saying Loop is wrong (or right) here, only that the enterprise of including matter in it is just getting started and is still quite primitive. [...]

The basic idea is simple and at the same time can be difficult to grasp. It is that a graph (also the higher dimensional analog called a cell-complex) provides a TRUNCATION or arbitrary restriction of the variables one decides to consider. It gives a way to temporarily restrict the "degrees of freedom" of a problem. [...]

After the model is defined on these finite truncations (these simplified worlds of colored graph and foam) then to finish defining the theory there is a way to let the size of the graph go to infinity. Like letting N --> infty in an ordinary power series in calculus.
You try to do your calculation with a "cutoff" and then you remove the artificial restriction and let the complexity grow indefinitely.

It's background independent in the same way that Gen Rel is. In GR there is no fixed prior metric, the metric interacts with matter and evolves. In Loop there is no fixed prior labeling with volumes, areas etc. The labels are variable.
Also there is no standard space in which the graph is embedded.

Just curious: How does one get the dimension of space-time out of this limit? Doesn't it have to be encoded already into the finite graph or triangulation?

I recall there was a Loop Gravity paper around 2008 by Leonardo Modesto in which he studied the dimensionality of the LQG space and its dependence on scale. the LQG spacetime is 4d at macroscopic scale. But if you probe dimensionality at very small scale you find it is less than 4d. I think Modesto determined dimensionality by comparing the Length and Volume operators---or anyway some comparison of quantum observables.

I don't remember the details so I will hunt for the paper.

BTW I don't think one has to go to some kind of "continuum limit" just to measure the dimensionality. I could be wrong though. Let's see how Modesto does it.

 

[marcus]

Here is Modesto's first paper about spacetime dimensionality in LQG. This was published in Classical and Quantum Gravity in 2009. Let's see what method he uses.

http://arxiv.org/abs/0812.2214
Fractal Structure of Loop Quantum Gravity
Leonardo Modesto
5 pages, 5 figures
(Submitted on 11 Dec 2008)
"In this paper we have calculated the spectral dimension of loop quantum gravity (LQG) using simple arguments coming from the area spectrum at different length scales. We have obtained that the spectral dimension of the spatial section runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decrees from high to low energy. We have calculated the spectral dimension of the space-time also using results from spin-foam models, obtaining a 2-dimensional effective manifold at high energy. Our result is consistent with other two approach to non perturbative quantum gravity: causal dynamical triangulation and asymptotic safety quantum gravity."



There is also a later paper in case anyone is interested in following it up.
Unfortunately it uses the older (Barrett-Crane) spinfoam model which has now been superseded.
http://arxiv.org/abs/0911.0437
Fractal Space-Time from Spin-Foams
Elena Magliaro, Claudio Perini, Leonardo Modesto
5 pages, 2 figures
(Submitted on 2 Nov 2009)
"In this paper we perform the calculation of the spectral dimension of spacetime in 4d quantum gravity using the Barrett-Crane (BC) spinfoam model. We realize this considering a very simple decomposition of the 4d spacetime already used in the graviton propagator calculation and we introduce a boundary state which selects a classical geometry on the boundary. We obtain that the spectral dimension of the spacetime runs from ~2 to 4, across a ~ 1.5 phase, when the energy of a probe scalar field decreases from high E < E_Planck/25 to low energy. The spectral dimension at the Planck scale E ~ E_Planck depends on the areas spectrum used in the calculation..."

 

[Fra]

You still don't get my question. A particle like electron or quark is thousand of times larger than the Planck scale. Yet since mass-stress-energy couple to the stress tensor in Einstein Field Equations. Shouldn't space and time also arise from the electron or quark since they are all dynamical? Why is this concern only in the Planck scale?

I think it was unclear exactly what you were asking. But now I think you are asking, why the particles of QFT and atomic physics doesn't influence the geometry, even outside Planckscale?

First of all, the spacetime background in a QFT experiment is defined by the laboratory frame. This is in fact where all the "clocks" and "rules" live so to speak. When you considering scattering experiments, you effectively refer to asymptotic observables that are defined in the massive labframe. This backdrop is so overly massive that normal energy changes in any atom does not even close to deform the reference frame. So the approximation holds perfect FAPP.

Statements about spacetime in the microscopic domain (not directly accesible) are merely inferences based on asymtotic data, based on a battery of premises, such as renormalization of interactions.

If this fails, we say that the backdrop makes sens, if not, it doesn't and we need something better.

If we focus energy in the detectors to the point where small black holes are created then this procedure fails to make sense and we aren't able to make sene of fixed context in the renormalization scheme. This is where the QFT/RG/fixed backdrop scheme breaks down.

As far I understand, the whole battery of baggage here in QFT, backgound + RG really are inseparable. So it's not a priori obvious if the entire game fails, wether it's due to failure of assuming a background or if the renormalization procedure is wrong.

I'm tempted to say this: In a cosmological scenario ie (measurement theory at cosmo scale) then the fixed background seems an obvious failing premise. But at the small subsystem scale, I'm more inclined to blame they way interactions are scaled. I mean, even if spacetime is something twisted from the inside view, the asymptotic context of laboratory may still be perfectly well defined. And after all, this is where our measurment device are.

/Fredrik

 

[rogerl]

I think it was unclear exactly what you were asking. But now I think you are asking, why the particles of QFT and atomic physics doesn't influence the geometry, even outside Planckscale?

/Fredrik

Who says the particles of QFT doesn't influence the geometry. Our theory just ignored it. Or rather. We still don't have the theory of it. Anyway. Pls. go to the thread "True Background-Independent Particle Physics" for more questions about it. This is to make distinction between the more basic concept of particles being in fixed flat or curved spacetime... which is different from the true meaning of background-independent, which I only realized days after I wrote the initial message. I thought the meaning of background-indepedent was just particles being in fixed curved spacetime, but found out it was something more profound.

 

[member 11137]

Not in the sense of curved spacetime in a black hole. What I meant to ask was what if particles were not modeled as moving in flat spacetime background but particles part of dynamical curved spacetime itself...

Just curious: How far has been that hypothesis explored? I mean: if any particle is a part of the dynamical spacetime itself, it does no more make sense to build a theory separating the background and the particles. In that case (a particle is a part of the spacetime in evolution) the pertinent question seems to be: how do we identify, what is the signature, of a given type of particle?

 

[rogerl]

Just curious: How far has been that hypothesis explored? I mean: if any particle is a part of the dynamical spacetime itself, it does no more make sense to build a theory separating the background and the particles. In that case (a particle is a part of the spacetime in evolution) the pertinent question seems to be: how do we identify, what is the signature, of a given type of particle?

My description was vague because then I didn't know the following important distinctions:

1. Particles in a fixed flat spacetime
2. Particles in a fixed curved spacetime
3. Particles not in any background spacetime but space and time arising out of the dynamics of the ensemble. (this is the true meaning of Background Independent)

In the sentence you were quoting me from. I mixed 2 & 3.. I thought Background Independent meant particles in a fixed curved spacetime (which I thought was dynamical).

See the thread "True Background Independent Particle Physics" for discussions of pure case #3 only which was what I was initially inquiring but got confused and mixed up the first two.

 

[atyy]

You still don't get my question. A particle like electron or quark is thousand of times larger than the Planck scale. Yet since mass-stress-energy couple to the stress tensor in Einstein Field Equations. Shouldn't space and time also arise from the electron or quark since they are all dynamical? Why is this concern only in the Planck scale?

Background independence means that matter causes spacetime curvature. Background independence is not ignored above the Planck scale, we treat it using classical general relativity. What is ignored is quantum corrections to general relativity. We know enough about quantum gravity to say that spacetime=gravity can be treated classically at the length scales probed by particle accelerators, as well as any larger length scale. Although spacetime is classical, matter still causes it to curve. The spacetime curvature caused by matter over the length scales probed by particle accelerators is very small, so we can approximate spacetime as being flat for these experiments.

At the Planck scale, we expect quantum corrections to general relativity to be important. What we don't have is a theory that gives us those corrections accurately. We do not know if spacetime exists as a fundamental entity at those scales. Asymptotic Safety is an approach to quantum gravity which postulates that spacetime is a fundamental entity at the Planck scale, while string theory postulates otherwise.

 

[A. Neumaier]

Here is Modesto's first paper about spacetime dimensionality in LQG. This was published in Classical and Quantum Gravity in 2009. Let's see what method he uses.

http://arxiv.org/abs/0812.2214
Fractal Structure of Loop Quantum Gravity
Leonardo Modesto
5 pages, 5 figures
(Submitted on 11 Dec 2008)
"In this paper we have calculated the spectral dimension of loop quantum gravity (LQG) using simple arguments coming from the area spectrum at different length scales. We have obtained that the spectral dimension of the spatial section runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decrees from high to low energy. We have calculated the spectral dimension of the space-time also using results from spin-foam models, obtaining a 2-dimensional effective manifold at high energy. Our result is consistent with other two approach to non perturbative quantum gravity: causal dynamical triangulation and asymptotic safety quantum gravity."

Let us focus on this. The definition of the spectral dimension D_s seems to be (13), evaluated with P_g(T) as defined in (18).

I don't understand the meaning of P_g(T) and hence D_s in terms of LQG, but it seems that d=3 (the low energy dimension) is input to the theory; see the remark after (18). Thus the paper only says that given the low energy dimension, spacetime is d+1 dimensional at low energies, becomes more fractal as the energy increases, until its dimension is (around the Planck scale) reduced to only d/2, and approaches 2d/3 at energies much larger than the Planck scale. (Where the fraction might turn out to be dimension-dependent, too.)

 

[marcus]

A. Neumeier, thanks for your reply. There are two things to say. One is that I must try again to reply to your question of how dimensionality arises in the new LQG formulation (most complete: 1102.3660) because my initial pointer to Modesto's paper is a false lead.

It does not help answer your question because, as you point out, the largescale dimensionality is put in by hand! He only deals with small scale quantum corrections to the diffusion process.

The other thing to say is that now that I look at this business of dimensional reduction at small scale it seems quite interesting. The initial paper by Modesto is hard to understand (for me) and may have some weak points, but he continued to struggle with it and finally has a paper on this which was published in Physical Review D, in 2010, and has 18 citations. He wrote this with someone at Uni Frankfurt. It is much easier for me to understand and is more general. It is not specific to LQG, it applies to any QG with a minimal length scale (so in particular to LQG).

In case anyone is interested the paper is
http://www-library.desy.de/cgi-bin/spiface/find/hep/www?rawcmd=FIND+EPRINT+0912.0220
To me it seems more clean and solid than the ones preceding it. Equation (8) is a beautifully simple formula for the spectral dimension showing its dependence on the diffusion time.

Also there have been papers about dimensionality reduction from other QG approaches such as CDT, Asymptotic Safety, and others. Modesto's papers just happened to come at it from the LQG direction. Now he cuts loose from his specific starting point, and has something more general.
==========================

But I am now back to square one! How does one know the dimensionality that arises in LQG?
It has to be "put in" somehow and when I look at 1102.3660 the only place I can see something that might force the particular largescale value of 4 is the choice of SU(2) in the definition of the Hilbertspace.

If someone else with sharper eyes and wits can find another "input", please tell me what I am missing! All the input I see is the choice of SU(2) as the ground on which the graph Hilbertspace H_Gamma is defined. The Lebesgue space of square integrable complex valued functions on the cartesian product SU(2)^L, where L is the number of links in the graph. And then later there is a correspondence set up between SU(2) reps and SL(2,C) reps---so one can say that another choice is made, but I see no alternative once one begins with SU(2).

So let's say that is the "input" that ultimately determines the 4D dimensionality, or better, spatial 3D. Now the question is, if you suddenly find yourself in the world defined by 1102.3660, how do you tell you are in spatial 3D? How does a hypothetical inhabitant test the macro dimensionality of his surroundings?

We should be able to look in 1102.3660 and find area and volume operators defined. And it should be possible to discover that when we measure some largescale areas and volumes we shall find that the volume goes as the 3/2 power of the surface area.

Largescale means that the volume encloses many nodes of the graph, and the surface area therefore cuts many links. (We give the inhabitant some matter to define location and this is how volumes and surface areas are defined for any specified region.)

To each node corresponds a volume, to each link corresponds an area. One adds up the volumes of the nodes to get the macro volume, and one adds up the areas of the links to get the macro area.

Since there is no spacetime manifold, this is the only way I can think of, at the moment, to define the experience of some given dimensionality. Maybe a careful look at 1102.3660 would unearth more, there are other operators defined, socalled Penrose "metric" operators---see section D on page 4. Probably one can use these as observables to experience dimensionality. I see they can be interpreted as angle observables!
These are said behave in the way expected measuring angles in 3D space.

Well that seems like it might work out better than my initial suggestion of looking at Modesto's papers (which was not very satisfactory---although it had some side interest.)

If you (or anyone else) has a reaction to this idea please say.

 

[rogerl]

Background independence means that matter causes spacetime curvature. Background independence is not ignored above the Planck scale, we treat it using classical general relativity. What is ignored is quantum corrections to general relativity. We know enough about quantum gravity to say that spacetime=gravity can be treated classically at the length scales probed by particle accelerators, as well as any larger length scale. Although spacetime is classical, matter still causes it to curve. The spacetime curvature caused by matter over the length scales probed by particle accelerators is very small, so we can approximate spacetime as being flat for these experiments.

At the Planck scale, we expect quantum corrections to general relativity to be important. What we don't have is a theory that gives us those corrections accurately. We do not know if spacetime exists as a fundamental entity at those scales. Asymptotic Safety is an approach to quantum gravity which postulates that spacetime is a fundamental entity at the Planck scale, while string theory postulates otherwise.

No. Background independence doesn't just mean that matter causes spacetime curvature. It is much more. Read again Markus reply #9 on this thread where he stated:

"Particle physics on fixed curved spacetime backgrounds is an important and distinct research area---bapowell was just getting into it and it would be efficient for your thread to focus on that. It's extremely interesting. (horizons, Hawking radiation, the temperature of deSitter space that arises simply due to its geometry.)

Rogerl, I would actually suggest that you have two threads, one about particle physics on a fixed spacetime geometry (dependent on specifying a prior geometry which remains fixed.)
And another thread about background independent theories---ones where no prior geometry is specified and geometry interacts with matter (its all one combined interactive thing.)"

---------------

Atyy. Particle physics on fixed curved spacetime background is not background independent. Background Independent means space and time must arise out of the particles themselves. In the case of particle on fixed curve spacetime. It is just surfing at it curved meaning gravity is connected but space and time don't arise out of the paricles themselves. Here's the important distinction.

 

[atyy]

No. Background independence doesn't just mean that matter causes spacetime curvature. It is much more. Read again Markus reply #9 on this thread where he stated:

"Particle physics on fixed curved spacetime backgrounds is an important and distinct research area---bapowell was just getting into it and it would be efficient for your thread to focus on that. It's extremely interesting. (horizons, Hawking radiation, the temperature of deSitter space that arises simply due to its geometry.)

Rogerl, I would actually suggest that you have two threads, one about particle physics on a fixed spacetime geometry (dependent on specifying a prior geometry which remains fixed.)
And another thread about background independent theories---ones where no prior geometry is specified and geometry interacts with matter (its all one combined interactive thing.)"

---------------

Atyy. Particle physics on fixed curved spacetime background is not background independent. Background Independent means space and time must arise out of the particles themselves. In the case of particle on fixed curve spacetime. It is just surfing at it curved meaning gravity is connected but space and time don't arise out of the paricles themselves. Here's the important distinction.

Background independence is that matter causes spacetime curvature, ie. particles do not move on a fixed background. If they move, spacetime changes, because the particles change the curvature of spacetime, so the background is not fixed. This explanation is a bit muddled because I am mixing 4D and 3+1 views, but the initial definition I gave is correct.

If particles move on a curved background without changing its curvature, then that is not background independent. However, I am talking about particle movement always changing spacetime curvature.

 

[rogerl]

Particle physics on fixed curved spacetime backgrounds is an important and distinct research area---bapowell was just getting into it and it would be efficient for your thread to focus on that. It's extremely interesting. (horizons, Hawking radiation, the temperature of deSitter space that arises simply due to its geometry.)

Rogerl, I would actually suggest that you have two threads, one about particle physics on a fixed spacetime geometry (dependent on specifying a prior geometry which remains fixed.)
And another thread about background independent theories---ones where no prior geometry is specified and geometry interacts with matter (its all one combined interactive thing.)

Marcus. I'm a bit confused about one thing now. In a fixed curved spacetime background, time is affected by the Einstein Field Equation as it interacts with mass-stress-energy. Now in a true Background Indepedent model, where space and time arise from the particle interaction. How does time here differ to the first case?

Hawking radiation is supposed to occur because of particles on a fixed curved spacetime, isn't it? Or is hawking radiation due to pure Background Independence where space and time are created from the particles interacting with a formless metric? But isn't this what the EFE do already? Pls. elaborate the important distinctions. Thanks.

 

[rogerl]

Background independence is that matter causes spacetime curvature, ie. particles do not move on a fixed background. If they move, spacetime changes, because the particles change the curvature of spacetime, so the background is not fixed. This explanation is a bit muddled because I am mixing 4D and 3+1 views, but the initial definition I gave is correct.

If particles move on a curved background without changing its curvature, then that is not background independent. However, I am talking about particle movement always changing spacetime curvature.

I think Background Independence means there is no spacetime in the background at all. It is when there are particles that spacetime somehow is created. We treat spacetime as like an entity. But spacetime is just a model. So in BI theory, the particles created the spacetime. Let's see if this is what the BI expert Marcus thinks.

 

[atyy]

I think Background Independence means there is no spacetime in the background at all. It is when there are particles that spacetime somehow is created. We treat spacetime as like an entity. But spacetime is just a model. So in BI theory, the particles created the spacetime. Let's see if this is what the BI expert Marcus thinks.

That is the most radical form of background independence, not found in GR. That is the form of background independence found eg. in string theory. In string theory, part of spacetime emerges as the excitations of strings. In string theory, there is also a conjectured model in which the entire volume of spacetime emerges as the excitations of particles with only the boundary of spacetime fixed - this is the celebated AdS/CFT conjecture, and is the best example of emergent spacetime we have.

 

[A. Neumaier]

A. Neumeier,

My name is spelled Neumaier.

The other thing to say is that now that I look at this business of dimensional reduction at small scale it seems quite interesting. The initial paper by Modesto is hard to understand (for me) and may have some weak points, but he continued to struggle with it and finally has a paper on this which was published in Physical Review D, in 2010,

and http://arxiv.org/pdf/0912.0220

But I am now back to square one! How does one know the dimensionality that arises in LQG?
It has to be "put in" somehow and when I look at 1102.3660 the only place I can see something that might force the particular largescale value of 4 is the choice of SU(2) in the definition of the Hilbertspace.

If someone else with sharper eyes and wits can find another "input", please tell me what I am missing!

The remark after (23) in 1102.3660 seems to suggest that the dimension indeed comes from there and forces space-time dimesion 4 - though I don't understand how. It should be in work of Penrose (or refs [7-9] mentioned there):

there are other operators defined, socalled Penrose "metric" operators---see section D on page 4. Probably one can use these as observables to experience dimensionality. I see they can be interpreted as angle observables!
These are said behave in the way expected measuring angles in 3D space.

 

[fzero]

My name is spelled Neumaier.

and http://arxiv.org/pdf/0912.0220

The remark after (23) in 1102.3660 seems to suggest that the dimension indeed comes from there and forces space-time dimesion 4 - though I don't understand how. It should be in work of Penrose (or refs [7-9] mentioned there):

The dimension of spacetime comes from the conjectured relationship between the spin network and the Ashtekar formalism. I believe that the states constructed in the spin network are interpreted as the holonomies of the Ashtekar gauge field. The base group $$G$$ in the spin network must match the gauge group for the Ashtekar variables. We also see that the spin network formalism is supposed to be the lattice gauge theory of the Ashtekar formalism. In the Ashtekar formalism, spacetime is assumed to be $$\mathbb{R}\times \Sigma$$ and the dimension of the space $$\Sigma$$ corresponds to the dimension of that adjoint representation of the Ashtekar gauge group.

 

[marcus]

The dimension of spacetime comes from the conjectured relationship between the spin network and the Ashtekar formalism. I believe that the states constructed in the spin network are interpreted as the holonomies of the Ashtekar gauge field. The base group $$G$$ in the spin network must match the gauge group for the Ashtekar variables. We also see that the spin network formalism is supposed to be the lattice gauge theory of the Ashtekar formalism. In the Ashtekar formalism, spacetime is assumed to be $$\mathbb{R}\times \Sigma$$ and the dimension of the space $$\Sigma$$ corresponds to the dimension of that adjoint representation of the Ashtekar gauge group.

This all makes sense, and is certainly good to know, if only as historical basis.

But what interests me is that if you look at the new formulation, which has appeared only since 2010---say you look at the pedagogical review 1101.3660 that A. Neumaier just referenced---then where is the $$\mathbb{R}\times \Sigma$$ manifold?

Where are the Ashtekar variables? Where are the holonomies? Where is the old configuration space of pre-2010 Loop Gravity?

This is the point I was trying to make by the way I responded to Neumaier's question. It is actually very interesting. We have this new very concise formulation, with little or no "extra baggage". It is expressed in just a few equations---with Hilbertspace and operators defined in a rather direct transparent way.

In this new formulation, the question is very relevant---how do we know the dimensionality?

A. Neumaier refers to what it says right after equation (23) on page 4 of http://arxiv.org/abs/1102.3660 . Where paper [8] by Penrose is cited.
I think this may be the right place to look.