Is a Grand Unified Theory Really Necessary for Understanding Our Universe?

[Lelan Thara]

This question touches on both relativity and quantum theory, but it is general enough that I think this is the place for it.

I am aware that the Holy Grail of modern physics is a "Grand Unified Theory", or "Theory of Everything", that will combine general relativity and quantum theory into a cohesive whole. The fundamental issue, as I understand it, is trying to establish a model that successfully quantizes gravity.

My questions are:

Are general relativity and the Standard Model of quantum mechanics contradictory in some way? Do they make contrary predictions where we must assume that one or the other (or both theories) are false unless we accomplish a GUT?

Is there some fundamental reason why we must believe gravity is quantized? Why couldn't we just accept at face value that gravity differs from the other 3 fundamental forces?

Alternately - is the search for a GUT based on something more ephemeral? Is it really a search for theoretical "elegance" - conceptual "beauty"? Do we look for a GUT simply because it feels right that there should be one?

My thanks in advance to anyone who can help clear this up for me.

 

[DaveC426913]

Are general relativity and the Standard Model of quantum mechanics contradictory in some way?

Yes. In a nutshell, GR requires a smooth continuum to space. QM says space is quantized, and further, that Uncertainty increases as your scale decreases. What this means is that if you try to apply to GR at arbitrarily small distances, you basically get energy levels that are arbitrarily large.

QM is an equation, GR is an equation. If you do a substitution, putting GR into QM, the numbers that pop out are infinities. It means that the smallest iota of empty space can contain an infinitely large amount of energy.

Thus, we know that our understanding is incomplete.

 

[Lelan Thara]

That's a remarkably succinct and clear explanation, DaveC - thanks.

Thus far attempts at a GUT have focused on trying to quantize GR, right? Has anyone tried the other approach - getting QM to work in a continuous spacetime?

To phrase it another way - both GR and QED are very successful, correct? Yet the approach towards a GUT seems to be to try to "fix" GR.
Is there a reason for that which can be easily explained?

 

[Lelan Thara]

Another quick question, Dave, that I hope won't sound facetious.

I have read that the results gotten from the equations of QED also result in infinities that require "renormalization" (division by infinity, as I understand it) to provide useful results.

Are these the same infinities that you are referring to that arise when GR is incorporated into QM?

If it's a different set of infinities - why can't they be renormalized like QED infinities are?

Thanks.

 

[selfAdjoint]

Another quick question, Dave, that I hope won't sound facetious.

I have read that the results gotten from the equations of QED also result in infinities that require "renormalization" (division by infinity, as I understand it) to provide useful results.

Are these the same infinities that you are referring to that arise when GR is incorporated into QM?

If it's a different set of infinities - why can't they be renormalized like QED infinities are?

Thanks.

Let me just jump in here quickly to tell you that renormalization is NOT "division by zero". Divisions by zero are precisely what renormalization removes. Basically you redefine the scale of your physics by saying that it doesn't - can't be expected to - describe what happens at really tiny scales, which corresponds to really high energies. So you represent this unknown physics in your integrals by constants. You carry these undefined constants through the mathematical developments as you compute your probabilities, and then at the last step you find (if your theory is a renormalizable one) that you can assume your particles have the known experimentally defined mass and so on, and you can "absorb" the constants into these measured numbers. The infinities in quantizing GR are of the same kind as in other field theories, but you have to use an infinite set of constants to do the renormalization trick, and it's impossible to absorb all of them together. So GR is said to be non-renormalizable.

 

[topovrs]

Are general relativity and the Standard Model of quantum mechanics contradictory in some way? Do they make contrary predictions where we must assume that one or the other (or both theories) are false unless we accomplish a GUT?

GR and QM have different, incompatible concepts of time. A QM state must be defined for the whole universe at one time. In GR, except in special situations, it is not possible to invariantly define such global instants. These conflicting concepts of time prevent both GR and QT from being used simultaneously in situations where non-trivial aspects of both are relevant.

Is there some fundamental reason why we must believe gravity is quantized? Why couldn't we just accept at face value that gravity differs from the other 3 fundamental forces?

There is no such fundamental reason. Many theorists believe that QT is more fundamental, and that gravity must thus be quantized, but it is still quite reasonable to seek a fundamental theory in which gravity / spacetime geometry is distinctly different from the quantum particle fields. Various recent approaches to emergent spacetime take this latter path.

 

[Hurkyl]

A QM state must be defined for the whole universe at one time.

That's not a requirement for any quantum theory. To wit, LQFT explicitly defines, for any open set U of Minkowski space-time, a set of states over U.

Why couldn't we just accept at face value that gravity differs from the other 3 fundamental forces?

We have empirical evidence of quantization in gravitational contexts.

 

[DaveC426913]

Let me just jump in here quickly to tell you that renormalization is NOT "division by zero". Divisions by zero are precisely what renormalization removes.

Lelan didn't say "division by zero". Lelan said "division by infinity".

I think the division term is used loosely in this thread, certainly the math is beyond me. But I get the analogy that, whatever renormalization does, the effect is that infinities magically disappear.

 

[DaveC426913]

Why couldn't we just accept at face value that gravity differs from the other 3 fundamental forces?

Keep in mind that any complete theory of the universe will include an understanding of the 3 forces and gravity. It doesn't necessarily mean they must be the same thing.

Imagine we developed a theory of, say, light. We have a model for all light from red, orange, yellow and green, through to blue light. Our model does not explain violet light, so we have another theory for violet. The two models could be completely different - no problem with that. But if we don't understand why violet is different from the other kinds of light, then you must concede that we don't have a complete model of light. In fact, once we find out how violet is different, that difference becomes part of our theory - our unified theory. See?

So: a complete GUT does not necessarily have to make gravity the same as the other forces, it merely has to include an explanation for how it is different.

One theory; answers all questions; doesn't leave any relevant threads hanging.

 

[vanesch]

We have empirical evidence of quantization in gravitational contexts.

Yes, but this has nothing to do with quantization of gravity itself. Although it is quite an experimental feat and I respect the work of my colleague quite a lot, what is quantized is simply the neutron, and gravity enters as an entirely classical potential. In fact, the principle is no surprise and was already established by neutron interferrometry (as for instance explained in Sakurai), where the classical gravitational potential enters into the Schroedinger equation.
What is spectacular with Nesvizhevsky's work (apart from the experimental challenge! I can tell you that about every single aspect of the experiment needed some clever tricks), is:
1) the fact that we now have a bound state (at least in one direction)
2) the length scale on which this operates (tens of micrometers).

You could, for instance, compare this to calculating the spectrum of the hydrogen atom (bound state of a quantized charged particle in a Coulomb potential). Although it indicates that the EM field enters into the Schroedinger equation as a potential, it tells you nothing about about any quantization of the EM field itself.

 

[vanesch]

Are general relativity and the Standard Model of quantum mechanics contradictory in some way? Do they make contrary predictions where we must assume that one or the other (or both theories) are false unless we accomplish a GUT?

Both quantum theory and GR have internal difficulties: quantum theory has "renormalization" problems, and GR has singularities, which somehow indicate that they cannot be "the final word".

Apart from that, GR and quantum theory offer totally different "world ontologies".

However, the fundamental difficulty resides IMO in 2 aspects which are related. The first, as pointed out here before, is "the problem of time". Quantum theory has a rather "traditional" structure where there is a "state" and an "evolution". This can be adapted to special relativity, where the "state" and the "evolution" transform according to special rules (representations of the Lorentz transformation). However, it will be difficult in this way to make time a dynamical quantity, which is required by GR. It is only when the "background is given" that one can split things in "a state" and "an evolution".

The other aspect is the fundamental incompatibility between a classical theory and a quantum theory. The quantum theory allows for superpositions of states. So the quantum state where the Earth is both at its current place, and twice as far from the sun, exists in principle. There's no way to implement this in GR, because in GR one has to "decide" where it is. Only in a hypothetical quantum version of gravity can you have a smooth connection between the quantum theory allowing for superpositions of energy/mass/momentum states and the gravitational interaction. However, this will then allow for a kind of "superposition" of "times", so which time do we now take as the time in the Schroedinger equation ?

In usual applications of quantum theory, this doesn't play any role, because the gravitational effects of superpositions needed are utterly neglegible (the superposition of the positions of electron positions in a molecule have absolutely no observational consequence gravitationally). But if we apply quantum theory all the way, then it is thinkable to have superpositions of states which are "gravitationally different".

Quantum evolution needs some kind of "classical time" for its unitary evolution, but this classical time is not compatible with GR and a superposition of mass-energy tensors as allowed for in quantum theory.

There are different suggestions to get out of this mess, up to date none really totally successful. But here are the reasons I guess, why people think that the current state of affairs needs a fix.

 

[selfAdjoint]

What Patrick just said is absolutely right, and I think all the various contending schools of Beyond the Standard Model would agree. One other constraint is that bothe the pure quantum theory of the standard model and the classical general relativity have passed many many tests and been as successful as any theories in history. So people don't really want to give them up or develop some unification that contradicts them at the low energies they've been tested at. This is a big constraint on any proposed GUT.

 

[Lelan Thara]

Let me just jump in here quickly to tell you that renormalization is NOT "division by zero". Divisions by zero are precisely what renormalization removes. Basically you redefine the scale of your physics by saying that it doesn't - can't be expected to - describe what happens at really tiny scales, which corresponds to really high energies. So you represent this unknown physics in your integrals by constants. You carry these undefined constants through the mathematical developments as you compute your probabilities, and then at the last step you find (if your theory is a renormalizable one) that you can assume your particles have the known experimentally defined mass and so on, and you can "absorb" the constants into these measured numbers. The infinities in quantizing GR are of the same kind as in other field theories, but you have to use an infinite set of constants to do the renormalization trick, and it's impossible to absorb all of them together. So GR is said to be non-renormalizable.

As DaveC pointed out, I did say, "division by infinity", not division by zero. But that aside, you explanation helps me understand renormalization a bit better, and I accept what you're saying - that GR is non-renormalizable.

The most fundamental thing I am taking away from this discussion is that the search for a GUT is not a search for "elegance" - there are real problems to be solved. That's the major thing I wanted to know.

But please, carry on - this is very interesting!

 

[turbo]

Yes, but this has nothing to do with quantization of gravity itself. Although it is quite an experimental feat and I respect the work of my colleague quite a lot, what is quantized is simply the neutron, and gravity enters as an entirely classical potential. In fact, the principle is no surprise and was already established by neutron interferrometry (as for instance explained in Sakurai), where the classical gravitational potential enters into the Schroedinger equation.
What is spectacular with Nesvizhevsky's work (apart from the experimental challenge! I can tell you that about every single aspect of the experiment needed some clever tricks), is:
1) the fact that we now have a bound state (at least in one direction)
2) the length scale on which this operates (tens of micrometers).

You could, for instance, compare this to calculating the spectrum of the hydrogen atom (bound state of a quantized charged particle in a Coulomb potential). Although it indicates that the EM field enters into the Schroedinger equation as a potential, it tells you nothing about about any quantization of the EM field itself.

Could this experiment be viewed as a measurement of the fine structure of space-time? In other words, perhaps gravity can be continuous, not quantized, but neutrons have a higher probability to exist at discrete intervals defined by the texture of space.

 

[bananan]

Both quantum theory and GR have internal difficulties: quantum theory has "renormalization" problems, and GR has singularities, which somehow indicate that they cannot be "the final word".

Apart from that, GR and quantum theory offer totally different "world ontologies".

However, the fundamental difficulty resides IMO in 2 aspects which are related. The first, as pointed out here before, is "the problem of time". Quantum theory has a rather "traditional" structure where there is a "state" and an "evolution". This can be adapted to special relativity, where the "state" and the "evolution" transform according to special rules (representations of the Lorentz transformation). However, it will be difficult in this way to make time a dynamical quantity, which is required by GR. It is only when the "background is given" that one can split things in "a state" and "an evolution".

The other aspect is the fundamental incompatibility between a classical theory and a quantum theory. The quantum theory allows for superpositions of states. So the quantum state where the Earth is both at its current place, and twice as far from the sun, exists in principle. There's no way to implement this in GR, because in GR one has to "decide" where it is. Only in a hypothetical quantum version of gravity can you have a smooth connection between the quantum theory allowing for superpositions of energy/mass/momentum states and the gravitational interaction. However, this will then allow for a kind of "superposition" of "times", so which time do we now take as the time in the Schroedinger equation ?

In usual applications of quantum theory, this doesn't play any role, because the gravitational effects of superpositions needed are utterly neglegible (the superposition of the positions of electron positions in a molecule have absolutely no observational consequence gravitationally). But if we apply quantum theory all the way, then it is thinkable to have superpositions of states which are "gravitationally different".

Quantum evolution needs some kind of "classical time" for its unitary evolution, but this classical time is not compatible with GR and a superposition of mass-energy tensors as allowed for in quantum theory.

There are different suggestions to get out of this mess, up to date none really totally successful. But here are the reasons I guess, why people think that the current state of affairs needs a fix.

How well do you think string theory as a theory of quantum gravity addresses these fundamental issues, as opposed to lqg?

How does the graviton s-matrix in string theory provide a quantum theory of gravity and what sort of ontology does it address, as opposed to LQG's background independence?

 

[Lelan Thara]

Could this experiment be viewed as a measurement of the fine structure of space-time? In other words, perhaps gravity can be continuous, not quantized, but neutrons have a higher probability to exist at discrete intervals defined by the texture of space.

How could it be that this texture would affect neutrons but not texturize gravity, since the mass of the neutrons is contributing to the gravity?

 

[topovrs]

How could it be that this texture would affect neutrons but not texturize gravity, since the mass of the neutrons is contributing to the gravity?

Certainly, the neutrons must contribute to gravity. But the product of Newton's constant and the neutron mass is so small that any contribution to spacetime curvature is unmeasurable. By contrast, the affect of the gravitational potential of the entire Earth on a neutron is certainly measurable, and was demonstrated in the cited experiment to be no different than any other classical potential. The observed discrete quantum steps confirm the quantum nature of the neutron, but tell us nothing about the quantum structure of space.

With no empirical data, how a non-local quantum state contributes to gravity remains a subject of theoretical speculation, apart from the clear requirement that general relativity must emerge in the classical limit. Claims that gravity must be quantized, and particular approaches to such quantization, are based on theoretical models whose viability remain to be demonstrated.

 

[vanesch]

Certainly, the neutrons must contribute to gravity. But the product of Newton's constant and the neutron mass is so small that any contribution to spacetime curvature is unmeasurable. By contrast, the affect of the gravitational potential of the entire Earth on a neutron is certainly measurable, and was demonstrated in the cited experiment to be no different than any other classical potential. The observed discrete quantum steps confirm the quantum nature of the neutron, but tell us nothing about the quantum structure of space.

Exactly, that's what I wanted to say too

 

[vanesch]

How well do you think string theory as a theory of quantum gravity addresses these fundamental issues, as opposed to lqg?

How does the graviton s-matrix in string theory provide a quantum theory of gravity and what sort of ontology does it address, as opposed to LQG's background independence?

I could say some things, but I don't consider myself qualified enough to do so (although I noticed that talking with assurance about things one is not qualified is a key to success... )

 

[arivero]

Let me just jump in here quickly to tell you that renormalization is NOT "division by zero". Divisions by zero are precisely what renormalization removes. .

Indeed. A metaphor : suppose you define naively a "percentual variation of a function" as f(x+d)/d. Fine, but this quantity diverges when d approaches zero. The solution is to substract a counterterm f(x)/d.

(the example can be sophisticated in ways that even the initial definition seems to have sense. Suppose you want to study a variation of some object starting with a value unity at a given initial time. Then you define $$f(t_i+ \delta t) \over f(t_i) \delta t$$ and for infinitesimal time intervals you must remove the divergence with the counterterm $$-1 / \delta t$$. Of course you get $$f' / f$$ evaluated at $$t_i$$)

 

[Lelan Thara]

Certainly, the neutrons must contribute to gravity. But the product of Newton's constant and the neutron mass is so small that any contribution to spacetime curvature is unmeasurable. By contrast, the affect of the gravitational potential of the entire Earth on a neutron is certainly measurable, and was demonstrated in the cited experiment to be no different than any other classical potential. The observed discrete quantum steps confirm the quantum nature of the neutron, but tell us nothing about the quantum structure of space.

With no empirical data, how a non-local quantum state contributes to gravity remains a subject of theoretical speculation, apart from the clear requirement that general relativity must emerge in the classical limit. Claims that gravity must be quantized, and particular approaches to such quantization, are based on theoretical models whose viability remain to be demonstrated.

I see - thanks, topovrs.

Speaking on the level of intuition, or if you prefer, "elegance" - which is the only level I can contribute here, unfortunately - it seems that almost by definition, any quantized model is less complete than a model describing a continuum. A quantized model must have "gaps" between the quanta, right? Energies, masses that are disallowed because they do not fall within the quantized values? My big fascination lies in the question, "what lies in the gaps between quanta?"

If QM is formulated in flat Minkowski space, it sounds like what we really have is a "special theory of QM". Hopefully a more general theory will someday emerge.

 

[arivero]

My big fascination lies in the question, "what lies in the gaps between quanta?"

A noncommutative torus?

 

[Lelan Thara]

A noncommutative torus?

Or, perhaps, angels dancing on the heads of pins?

 

[arivero]

Or, perhaps, angels dancing on the heads of pins?

No, I am serius. Well, about angels on the heads of pins, I can not tell if it is related to infinitesimal calculus but I guess the original discussion was not (actually I have never found where the original discussion came from EDITED: on other hand, one can guess that the discussion about the sex of angels is very related to quantify the sin of Sodoma).

But if you are thinking on orbits, then a valid Bohr-Sommerfeld orbit is one which can be rotated a rational fraction of its length and it is still the same. So in general quantised orbits correspond to rational rotations of the circle. "in the middle" you would have irrational rotations, what happen to be denoted as "orbits in a non commutative torus". Fancy name, yep, but pretty well know object.

My big fascination lies in the question, "what lies in the gaps between quanta?"

Of course, your fascination could lie in not actually answering the question but keeping in awe about the inmensity of it. Or sort of. Then well, angels is good.

 

[Lelan Thara]

No, I am serius.

I didn't doubt that you were serious. "Angels on pins" was my admittedly sarcastic way of saying that I do not share your faith that pure mathematics is more real than reality.

But if you are thinking on orbits, then a valid Bohr-Sommerfeld orbit is one which can be rotated a rational fraction of its length and it is still the same. So in general quantised orbits correspond to rational rotations of the circle. "in the middle" you would have irrational rotations, what happen to be denoted as "orbits in a non commutative torus". Fancy name, yep, but pretty well know object.

I will have to take your word for that, I'm afraid.

Of course, your fascination could lie in not actually answering the question but keeping in awe about the inmensity of it. Or sort of. Then well, angels is good.

My fascination lies in a hope that we will find hidden variables. I can't accept that we create reality by looking at it.

If you answer a question in such a way that your answers are unprovable, unfalsifiable, unobservable - but internally consistent - have you answered the question?

I am drifting into the philosophy of science now. I've probably said as much as I should. I do appreciate the info you share, arivero, even when I take it with a grain of salt.

 

[Lelan Thara]

RE angels on pinheads, several quotes that some may find enlightening, or at least amusing - then I will drop this isue, I promise.

This question opens up a difficult issue. Is space continuous or discrete? If the former, it is arguable that there is no positive lower bound on the size of dancing angels. If the latter, there must be some lower bound, namely, the size of the smallest unit of space. But clearly this is too small, since dancing involves movement not only of the body but movement within the body (such as flailing arms or shifting feet). We might claim that, assuming space is discrete, the smallest possible size of a dancing angel is three times the size of the smallest unit of space, since this is the smallest volume of space that can have different shapes and therefore can be considered bendable.

A potential ambiguity in the question is the notion of the head of a pin. There is little point in considering the size of each angel if it is unknown what the size of the head of the pin is. We can stipulate that an average head of a pin is 3.5 millimeters wide, assuming that the pin being referred to is not a pushpin. The surface area of this, assuming the head is spherical, is 12.25 pi square millimeters.

At this point, the strictly philosophical analysis of the question must stop. It is still unresolved whether space is continuous or discrete, and if space is discrete, there are no obvious philosophical reasons to claim some particular size as the smallest possible. Thus, a preliminary answer to the question ``How many angels can dance on the head of a pin?'' is: If space is continuous, the number is probably unlimited; if space is discrete, then the number is finite and is a function of the surface area of the head of the pin and of the amount of surface area taken up by a being three times the size of the smallest unit of space.

Does that sound at all reminiscent of our discussion here of a Theory of Everything?

And:

Fact is, Aquinas did debate whether an angel moving from A to B passes through the points in between, and whether one could distinguish "morning" and "evening" knowledge in angels. (He was referring to an abstruse concept having to do with the dawn and twilight of creation.) Finally, he inquired whether several angels could be in the same place at once, which of course is the dancing-on-a-pin question less comically stated. (Tom's answer: no.) So the answer to your question is yes, medieval theologians did get into some pretty weird arguments, if not quite as weird as later wise guys painted them.

Substitute "electrons" for "angels" above and you have electrons jumping between shells without traversing the intervening distance, and the Pauli Exclusion Principle.

My point is - I hope you will forgive this humble layman if I occasionally appear skeptical. :)

 

[arivero]

Any idea where the first quote comes from? Yep my feeling is also the same, that the discussion could be about if "angels" have a size. It is somehow related to the comparision between Lucretius-Epicurus atoms (which have a size) and [Cavalieri-]Democritus atoms (which separate chunks of size, aka vacuum, but are sizeless themselves)

Substitute "electrons" for "angels" above and you have electrons jumping between shells without traversing the intervening distance,

No no, it is not about Bloch waves. What we have here is Aquinas discussing Aristotle discussion of Zeno. Interesting, I did not remember where this discussion appears. But what about the pin? Someone should be the first mentioning the pin? Indeed they are called "bizantine discussions", not "Aquinean discussions".

 

[Lelan Thara]

arivera, here's a link to the first quote.

http://everything2.com/index.pl?node_id=523054&lastnode_id=0

I found it in a google search - it appears to be a forum. The saleint point is that it's a modern discussion, so the person in question seems to be be rephrasing an old debate in terms he learned from modern physics.

As far as I know, Aquinas' arguments on the topic come from his Summa Theologica.

(BTW, glad to see you have both a sense of humor and broad range of intellectual interests.)

Anyhow, let me return to an on-topic question.

What I'm being told here is that general relativity requires a continuous space - QM requires a quantized space. There's two possible ways to reconcile them - you can quantize general relativity, or you can try to formulate the quantum world as the observed part of a continuum, some parts of which are not directly observable.

Is there a fundamental reason why quantum physics can't be conceived of as part of a larger continuum? Has there ever been a serious attempt to model QM as part of a continuum?

 

[selfAdjoint]

Anyhow, let me return to an on-topic question.

What I'm being told here is that general relativity requires a continuous space - QM requires a quantized space. There's two possible ways to reconcile them - you can quantize general relativity, or you can try to formulate the quantum world as the observed part of a continuum, some parts of which are not directly observable.

Is there a fundamental reason why quantum physics can't be conceived of as part of a larger continuum? Has there ever been a serious attempt to model QM as part of a continuum?

The bolded text is not quite true. Here's the skinny: Ordinary QM is formulated in Newtonian space plus time. There is a quantum position operator, for the position of some system (like a particle) that has a wave function. The operator acts on the wave function and the eigenstates give you the various possible positions while the corresponding eigenvalues determine the respective probabilities of those positions. Only one of them is realized in space. But space itself is continuous, indeed classical. Dirac showed how to do a quantized electron in Minkowski space, but that again is continuous.

 

[marcus]

Hi Lelan,
I think you may trace your confusion back to post #2, which was misleading or not quite right

Yes. In a nutshell, GR requires a smooth continuum to space. QM says space is quantized,...

QM is built on a smooth continuum. Quantum Field Theory (QFT) is built on a smooth continuum. So also is GR. In that respect the theories do not differ.

Basically I am just repeating what selfAdjoint said just now.

There is a difference however. In GR no background geometry is assumed ahead of time.

The continuum in GR is defined so as to be "floppy" and stretchy. So the metric, or distance function on it, can be a dynamically determined OUTCOME of interaction with matter in motion.

QM and QFT are usually defined on some fixed rigid continuum geometry decided in advance. Typically one works on a rigid Euclidean framework or the "Minkowski" framework of special relativity. These geometries are often referred to as "flat". they don't do exanding space stuff, they don't bend light rays, they don't have stars collapsing to black holes-----they are just flat rigid geometries.
Sometimes QFT is also defined on fixed curved geometries, but not on "floppy".So the usual QFT is not defined on the floppy stretchy continuum of Gen Rel.

That is the chasm that seems so hard to bridge
If QFT (the theory of matter) were only just defined on the flexible continuum of Gen Rel, then we wouldn't be hearing so much about the mismatch between Gen Rel and quantum theory.

One technical term that is used a lot in this context is background independence

Gen Rel (already as formulated in 1915) is manifestly a background independent theory for the very reason that it is formulated without choosing a pre-determined rigid background geometric framework ahead of time.

QFT on the other hand is not background independent, because the formulation of it depends on initial choice of a stiff background geometry.

AFAIK this is the key to the mismatch people talk about

 

[arivero]

As far as I know, Aquinas' arguments on the topic come from his Summa Theologica.

(BTW, glad to see you have both a sense of humor and broad range of intellectual interests.)

No, actually I haven't got a sense of humor and my intellectual interests are incredibly narrow . It is a sort of paradox that most people understand the contrary, at least as for the second point it refers.

Now, here is a more complete answer:
http://yedda.com/questions/8621841671611/
http://www.philosophy.leeds.ac.uk/GMR/articles/angels.html
http://www.baronyofvatavia.org/articles/medcul/pangel112002as37.php

the interesting point is that the concept seems to be transmited orally well before D'Israeli, and that someone brings it to the attention of Leibnitz:

in acus cuspide innumerabiles posse esse animulas, nullam inter se extensionem facientes.

‘there can be innumerable little souls on the point of a needle without their generating any space among themselves’

Of course this concrete answer is wrong. They do not contain space, but generate space between them

EDIT: it seems that the concept of "dancing" is introduced by Joseph Glanville, FRS, in 1661

 

[Lelan Thara]

selfadjoint, marcus - you are right, I guess I did latch onto that one statement by DaveC. Thank you for explaining that background dependence in QM is very different than saying QM is dependent on a quantized space.

arivero - I observed empirically that you were able to shift effortlessly from a discussion of GUTs to a discussion of angels on pinheads - and so, this observer created a reality in which you have a sense of humor.

And, of course, since we can't say that your reference frame is privileged in any way compared to mine - I'm afraid you are stuck with it. In my particular little corner of SpaceTime - you have a sense of humor.

There are worse things, you know.