Covariant Quantum Fields

[mieral]

I just read Carlo Rovelli new book "Reality is Not What It Seems: The Journey to Quantum Gravity" in one sitting. I'd like to know about the following:

"Fields that live on themselves, without the need of a spacetime to serve as a substratum, as a support, and which are capable by themselves of generating spacetime, are called "covariant quantum fields." The substance of which the world is made has been radically simplified in recent years. The world, particles, light, energy, space, and time - all of this is nothing but the manifestation of a single type of entity: covariant quantum fields."Are all quantum gravity theories based on covariant quantum fields where the fields don't live on spacetime but created space and time?

 

[julian]

In classical GR, the resolution of Einstein's hole argument implies that physical entities are located with respect to one another only and not with respect to space-time. In GR the "space-time" manifold is a mathematical construct devoid of physical meaning. Quoting Rovelli's book "Quantum Gravity", in particular the part on classical general relativity: "The world is made up of fields. Physically, these do not live on spacetime. They live, so to say, on one another. No more fields on spacetime, just fields on fields."

Rovelli would argue the quantum theory should not be based on notions devoid of physical meaning in the classical theory.

 

[julian]

I wrote up Einstein's hole argument a while back. In fact I was doing a search for it earlier on today but the thread goes back so many years it doesn't seem to exist anymore. I'll write it up again tomorrow. It doesn't require an extensive knowleadge of GR, but it would be good if you understand how to combine the metric tensor function in space-time $x-$coordinates, $g_{\mu \nu} (x)$, together with an infintesimal coordinate separation $dx^\mu$ in order to obtain the proper time $ds$ interval:

$ds^2 = \sum_{\mu , \nu} g_{\mu \nu} (x) dx^\mu dx^\nu$

which describes a particular space-time geometry.

 

[MathematicalPhysicist]

I think Rovelli is confused between what is reality and what does the maths represent?

Fields are defined as real or complex (I think you can extend it to hypercomplex, quarternions, etc; but these aren't necessarily mathematical number fields) functions from some mathematical space to another; in the context of calculus 3 if you will, we have real fields as functions from $\mathbb{R}^n$ to $\mathbb{R}^m$.

So if we take Max Tegmark's metaphysical stance, then fields are still defined on some "space", just it's not called spacetime (which is a 4-dim Pseudo Euclidean space), and not the confusing idea of "Fields living on themselves", whatever that means...

 

[haushofer]

I'm also curious what exactly is meand by such fields.

 

[MathematicalPhysicist]

I'm also curious what exactly is meand by such fields.

Never shall I read pop sci books, I haven't read a single pop sci book since I entered university (I tried reading Barrow's book on "nothing", but got distracted by my demanding studies.

If someone really wants to know how things are done in science then there is no other way but reading the technical literature; otherwise it's just reading mambo jambo for all my account.

 

[mieral]

I think Rovelli is confused between what is reality and what does the maths represent?

Fields are defined as real or complex (I think you can extend it to hypercomplex, quarternions, etc; but these aren't necessarily mathematical number fields) functions from some mathematical space to another; in the context of calculus 3 if you will, we have real fields as functions from $\mathbb{R}^n$ to $\mathbb{R}^m$.

So if we take Max Tegmark's metaphysical stance, then fields are still defined on some "space", just it's not called spacetime (which is a 4-dim Pseudo Euclidean space), and not the confusing idea of "Fields living on themselves", whatever that means...

But if as Max Tegmark wrote in the Mathematical Universe which I also happened to read. What if the software that created this interactive simulation was written or used the language of General Relativity and Quantum Mechanics as well as Loop Quantum Gravity. Then classicality and conventional space is all illusion. What is wrong with thinking in terms of GR, QM and LQG when trying to think of our world.

In this case. The map is really the territory.. because it is all map in the software or programme.. there is no real world or physical objects are also just effects or illusions.. so how can you justify to retain the word "space"?

 

[Demystifier]

Are all quantum gravity theories based on covariant quantum fields where the fields don't live on spacetime but created space and time?

No.

 

[Denis]

Are all quantum gravity theories based on covariant quantum fields where the fields don't live on spacetime but created space and time?

No. As far as we can talk about "all quantum gravity theories" (one can argue that this set is empty) all I know about have a spacetime background.

Given that it is not allowed here to refer to theories which are not mainstream, all what I'm allowed to mention would be string theory, which is a theory with some background, even if this background has some non-standard dimension. I'm not sure if LQG already pretends to the status of being a theory of quantum gravity, instead of a vague hope to create some in some future, then it would have to be a covariant one. There is the quantization of Newtonian gravity, and this is clearly a non-covariant theory with background.

There is a simple argument that covariant theories of QG are impossible. Namely, consider a simple superpositional state like in a double slit experiment of some heavy particle. Use a test particle to measure the position via gravitational interaction. Is the interaction strong enough to measure the position, and therefore to destroy the superpositional state? The answer is trivial in Newtonian QG. But if you try a generalization to GR, with the test particle moving on GR solutions created by the heavy particle, you see that the answer depends on the scalar product of the wave functions computed on the two different GR solutions, which correspond to the heavy particle being in one or the other of the two slits.

So, you have an observable (if interaction with a test particle destroys a superposition or not is observable), but the formula you need to compute it depends on a scalar product which is undefined in a covariant theory of gravity.

 

[julian]

You do realize that Witten wants to formulate a background-independent theory:

“Finding the right framework for an intrinsic, background independent formulation of string theory is one of the main problems in string theory, and so far has remained out of reach.” ... “This problem is fundamental because it is here that one really has to address the question of what kind of geometrical object the string represents.”

E Witten: “Quantum background independence in string theory” hep-th/9306122. “On Background independent open-string field theory” hep-th/9208027.

 

[julian]

String/M-theory, at the moment, is not really a theory - it is a collection of ideas instead - with the hope of a underlying theory.

Canonical and covariant LQG have a defining set of equations. In fact Rovelli makes a point of stating the equations of covariant LQG on page 166 of the book mentioned by the OP. Canonical and covariant LQG are formulated in terms of Hilbert spaces and hence have inner product structures. Doing practical calculations of general transitions amplitudes may be difficult (but not impossible however).

You said "LQG already pretends to the status of being a theory of quantum gravity" ...but

LQG has been applied to physically relevant situations - black holes and the early universe.

Also, systematic procedures for calculating background-independent scattering amplitudes have been developed by Rovelli et al. These are to make contact with the usual background-dependent formulism. See for example the book "Covariant loop quantum gravity". See the quote in particular:

"Therefore the semiclassical limit of the theory gives back usual linearised gravity, indicating that the theory is General Relativity in an appropriate limit."

but together with caveats spelled out there. Much more needs to be computed. There is a draft version of the book that can be found at:

http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf

(By the way classical general relativity is background-independent; the Einstein-Hilbert action is formulated in a way that makes no reference to an a-priori given background geometry. A subtle point is that a solution of Einstein's equations is not a particular background spacetime but an equivalence class of spacetimes related to each other through diffeomorphisms - and by a diffeomorphism I mean the type of transformation that for example turns a doughnut-shaped manifold into a coffee-cup shaped copy. On page 168 of the book referred to by the OP, the diagram there indicates that "covaraint fields" are part of what is required in interpretating the classical theory! Classical GR requires a relational interpretation where there is no meaning to localization with respect to spacetime coordinates...Background-dependent classical Newtonian gravity is an appropriate limit of general relativity!)

 

[Denis]

Why should one care what Witten wanted to do since 1992 but has not succeeded?

If LQG would find a way to answer the argument above, that would be interesting. If, as claimed, the semiclassical limit is fine, this should not be a problem.

 

[atyy]

Why should one care what Witten wanted to do since 1992 but has not succeeded?

If LQG would find a way to answer the argument above, that would be interesting. If, as claimed, the semiclassical limit is fine, this should not be a problem.

The semiclassical limit in Rovelli's favoured version of LQG is not fine. And because of a direction of research within LQG called group field theory, there are links between Razvan Gurau's LQG-related work and recent work of Witten's - a search for "Gurau-Witten" will turn up interesting things.

https://arxiv.org/abs/1303.4636:
- "That these extra terms spoil the classical limit of the theory can be seen by looking at spin foams on triangulations with more than one 4-simplex."
- "it also allows the resulting spin foam theory to be recast in terms of something called a group field theory"

 

[atyy]

Also, systematic procedures for calculating background-independent scattering amplitudes have been developed by Rovelli et al. These are to make contact with the usual background-dependent formulism. See for example the book "Covariant loop quantum gravity". See the quote in particular:

"Therefore the semiclassical limit of the theory gives back usual linearised gravity, indicating that the theory is General Relativity in an appropriate limit."

but together with caveats spelled out there. Much more needs to be computed. There is a draft version of the book that can be found at:

http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf

This is not agreed on in the LQG community. I believe Rovelli favours a proposal usually called EPRL. There is agreement about extra terms, but Rovelli thinks they are harmless whereas Jonathan Engle thinks they are not desired. Engle has proposed a correction to EPRL to remove the undesired terms, but as far as I understand, Rovelli and Vidotto refer to the original EPRL proposal.

https://arxiv.org/abs/1303.4636:
"That these extra terms spoil the classical limit of the theory can be seen by looking at spin foams on triangulations with more than one 4-simplex."

https://arxiv.org/abs/1107.0709
The Plebanski sectors of the EPRL vertex
Modern spin-foam models of four dimensional gravity are based on a discrete version of the Spin(4) Plebanski formulation. Beyond what is already in the literature, we clarify the meaning of different Plebanski sectors in this classical discrete model. We show that the linearized simplicity constraints used in the EPRL and FK models are not sufficient to impose a restriction to a single Plebanski sector, but rather, three Plebanski sectors are mixed. We propose this as the reason for certain extra `undesired' terms in the asymptotics of the EPRL vertex analyzed by Barrett et al. This explanation for the extra terms is new and different from that sometimes offered in the spin-foam literature thus far.

 

[julian]

Engle may be right. But I think systematic procedures for calculating background-independent scattering amplitudes will be similar for other models.