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WhatIsGravity

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In summary, the conversation discusses the concept of quantization in relation to gravity and time. The participants consider whether a quantum theory of gravity would also mean that time must be quantized. They also touch on the idea of discrete energy levels and how this relates to the smoothness of space-time. The conversation concludes with a recommendation to read a recent paper on the subject and a reflection on the vagueness of certain terms such as "quantized" and "energy."

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WhatIsGravity

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marcus

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A "quantum theory" of gravity does not mean something is permanently broken up into little bits, or "made" of little grains all the same size.

Think about energy. different atoms have all different energy levels, a bit of energy (involved in some interaction, transition, measurement, transaction) can be any size

Think about light: a photon of light can be any size depending on wavelength (which can vary continuously).

A quantum theory of gravity has to be a quantum theory of geometry (i.e. having to do with measurement of angles, areas, volumes distances etc., a theory of geometric observables and states of geometry.)

Space and time can be SMOOTH in a quantum theory. Space does not have to be thought of as broken up into little grains of space. However the observable that measures a given area can have discrete spectrum.

Talking about little grains of space is more something you might do in**popular media** and mass market books.

That said, google "compact phase discrete" and you get this recent paper as the first hit:

http://arxiv.org/abs/1502.00278

**Compact phase space, cosmological constant, discrete time**

Carlo Rovelli, Francesca Vidotto

(Submitted on 1 Feb 2015 (v1), last revised 4 Feb 2015 (this version, v2))

We study the quantization of geometry in the presence of a cosmological constant, using a discretiza- tion with constant-curvature simplices. Phase space turns out to be compact and the Hilbert space finite dimensional for each link. Not only the intrinsic, but also the extrinsic geometry turns out to be discrete,**pointing to discreteness of time,** in addition to space. We work in 2+1 dimensions, but these results may be relevant also for the physical 3+1 case.

Comments: 6 pages

Notice that this is a new result, it wasn't obvious that LQG or any other currently-worked on QG would have discrete time. They had to discover this in a particular they consider with a non-zero cosmological constant. It was NEWS that they found discrete time. It was not automatic.

Notice also that this does not mean that time is "made" of little grains or pebbles of time, all the same size, like one might naively imagine from the word "quantize". that is usually a bad way to think.

Safer to think in terms of MEASUREMENT. Maybe a time measurement observable has a discrete spectrum---a discrete set of possible values, something that happens when you put an observer in the picture.

Look at the very end of page 4:

==quote==

This is evident form the fact that Hilbert space is finite dimensional (for each link), and therefore** all local operators have discrete spectrum**. Therefore the extrinsic geometry is quantized as well.

The extrinsic curvature Kab determines the rate of change of the intrinsic geometry, because (in the Lapse=1, Shift=0 gauge) it is the proper-time derivative of the metric:...

... Since all these quantities have discrete spectrum, we expect**proper-time intervals, measured using gravitational observables, to be discrete **as well. ...

==endquote==

IN OTHER WORDS TIME CAN BE SMOOTH, but space and time geometry are QUANTIZED and the possible outcomes of any particular measurement are a discrete set of values. they are not saying that time itself is discrete. they are saying the spectrum or range of values resulting from a given measurement is discrete.

It's an important paper and much of it is comparatively easy to read. Have a look, it is 6 pages. Parts of it are non-technical and explain stuff in words without equations. You might get something from it. A first-hand taste of current research (and it corresponds to the*time discreteness* thing you were asking about.)

Think about energy. different atoms have all different energy levels, a bit of energy (involved in some interaction, transition, measurement, transaction) can be any size

Think about light: a photon of light can be any size depending on wavelength (which can vary continuously).

A quantum theory of gravity has to be a quantum theory of geometry (i.e. having to do with measurement of angles, areas, volumes distances etc., a theory of geometric observables and states of geometry.)

Space and time can be SMOOTH in a quantum theory. Space does not have to be thought of as broken up into little grains of space. However the observable that measures a given area can have discrete spectrum.

Talking about little grains of space is more something you might do in

That said, google "compact phase discrete" and you get this recent paper as the first hit:

http://arxiv.org/abs/1502.00278

Carlo Rovelli, Francesca Vidotto

(Submitted on 1 Feb 2015 (v1), last revised 4 Feb 2015 (this version, v2))

We study the quantization of geometry in the presence of a cosmological constant, using a discretiza- tion with constant-curvature simplices. Phase space turns out to be compact and the Hilbert space finite dimensional for each link. Not only the intrinsic, but also the extrinsic geometry turns out to be discrete,

Comments: 6 pages

Notice that this is a new result, it wasn't obvious that LQG or any other currently-worked on QG would have discrete time. They had to discover this in a particular they consider with a non-zero cosmological constant. It was NEWS that they found discrete time. It was not automatic.

Notice also that this does not mean that time is "made" of little grains or pebbles of time, all the same size, like one might naively imagine from the word "quantize". that is usually a bad way to think.

Safer to think in terms of MEASUREMENT. Maybe a time measurement observable has a discrete spectrum---a discrete set of possible values, something that happens when you put an observer in the picture.

Look at the very end of page 4:

==quote==

This is evident form the fact that Hilbert space is finite dimensional (for each link), and therefore

The extrinsic curvature Kab determines the rate of change of the intrinsic geometry, because (in the Lapse=1, Shift=0 gauge) it is the proper-time derivative of the metric:...

... Since all these quantities have discrete spectrum, we expect

==endquote==

IN OTHER WORDS TIME CAN BE SMOOTH, but space and time geometry are QUANTIZED and the possible outcomes of any particular measurement are a discrete set of values. they are not saying that time itself is discrete. they are saying the spectrum or range of values resulting from a given measurement is discrete.

It's an important paper and much of it is comparatively easy to read. Have a look, it is 6 pages. Parts of it are non-technical and explain stuff in words without equations. You might get something from it. A first-hand taste of current research (and it corresponds to the

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WhatIsGravity

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marcus

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I'm bothered by the vagueness of the WORDS. What does quantized mean? It may mean something different to you that what it means to someone who is working on quantizing the theory of GR---getting to a quantum theory of geometry---the goal of "quantum gravity". Or the two of you may mean the same.

In what sense is energy definitely quantized" There is no universal "penny" of energy so that you could say that all the amounts of energy you could measure are multiples of that universal smallest energy. Different atoms, different systems have different energies and they aren't commensurate, different observers SEE different amounts of energy depending on their motion.WhatIsGravity said:... if gravity is or can be quantized, does that mean that time must be quantized too? I guess I'm thinking of energy- definitely quantized. Gravitational energy... hmmm. Makes me wonder, if space-time truly is smooth (general relativity), then there must be a difference between potential and kinetic energy?

And yet energy IS quantized at the level of interactions/measurement--- if you specify one particular system and in a quantum mechanics context define an observable that measures some specific energy. Like energy levels of something definite.

WhatIsGravity said:... Space and time could be fully smooth, but our only ways to measure are discrete. ...I can see how the constant speed of light, no matter the reference frame thus the relativity of time, and Planck's constant too, could result from a space-time that is discrete.

Maybe I was wrong. I shouldn't have said space and time smooth. How can we know what it actually IS? All we can know is how Nature responds to measurements. Niels Bohr said something like that: physics is not about what the system IS, but concerns what we can say about it. Correlations among observations, predictions about future observations based on earlier ones.

I don't see how this involves the constancy of speed of light. Maybe I'm just extra dense today. You could explain your thinking about this. Someone more nimble witted might engage with your idea. You've raised some interesting questions. Could be a range of opinion.

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Helios

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marcus said:Maybe I was wrong. I shouldn't have said space and time smooth. How can we know what it actually IS? All we can know is how Nature responds to measurements. Niels Bohr said something like that: physics is not about what the system IS, but concerns what we can say about it.

This sounds like the philosophical issue of the Noumenon contrasted with the Phenomenon. The noumenon is the thing behind it all, but there's only the phenomenon we can know about.

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Jimster41

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Very interesting. Can't say I often hear Marcus hold forth so directly. I like it.

That said the idea of reality being made only of discrete quanta, as far as we can tell, does not bother me one bit. In fact it's the idea that there is anything smooth, perfectly linear, reversible, everywhere differentiable, that I find implausible. What evidence is there really for that? Evidence that is in the world first and formal thought second? Everything in the world is a thing, an event. A smooth manifold of qualitative continua seems more a platonic canonical hangover from Newton, who dreamed up a very effective approximation of reality, that we've been stuck with ever since. Of things we can touch, as far as I know there are only discrete bit-like events that define discrete moments. They have lots of freedom to to different things, but there is only them.

That a complex-featured seemingly continuous experience can be derived from such grains of sand, is a really interesting puzzle, but it seems a matter of fact to me. The evolution of species from genetic information seems a relevant example of how.

But I drink a lot of popular science.

That said the idea of reality being made only of discrete quanta, as far as we can tell, does not bother me one bit. In fact it's the idea that there is anything smooth, perfectly linear, reversible, everywhere differentiable, that I find implausible. What evidence is there really for that? Evidence that is in the world first and formal thought second? Everything in the world is a thing, an event. A smooth manifold of qualitative continua seems more a platonic canonical hangover from Newton, who dreamed up a very effective approximation of reality, that we've been stuck with ever since. Of things we can touch, as far as I know there are only discrete bit-like events that define discrete moments. They have lots of freedom to to different things, but there is only them.

That a complex-featured seemingly continuous experience can be derived from such grains of sand, is a really interesting puzzle, but it seems a matter of fact to me. The evolution of species from genetic information seems a relevant example of how.

But I drink a lot of popular science.

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WhatIsGravity

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With energy, when you're comparing falling apples to falling apples with someone in a different reference frame, my understanding is that a base unit of the energy of that falling apple is an 'integer' function of Planck's constant?

In my silly brain at least, I can see how a constant speed of light would result if nature is discrete. Light, massless energy, goes from 'here to there' exactly that fast- c. Zero passing of time; pure energy. The only place you can ever measure that speed is either 'here', or with relativity 'there', so no matter if you're moving with it, or against it, or at some calculable angle or whatever... you'll still measure the speed of light as it goes from 'here to there'- that discrete jump in spacetime.

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WhatIsGravity

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rootone

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In some theory there are quanta of gravity, they are called 'gravitons'.

I have my doubts though, I think gravity is more of an analog thing and has no quanta.

No gravity 'particles', just a 'field', which has a (scaler?) value, but that's just me.

I have my doubts though, I think gravity is more of an analog thing and has no quanta.

No gravity 'particles', just a 'field', which has a (scaler?) value, but that's just me.

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arivero

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The speed in such orbit is such that centripetal and centrifugal forces are equal: ;

Note from here that the energy depends on the radius E = 1/2 m v^2 = K/2r, so that 2 r E = K

To find the quantisation, multiply K/r^2 = m v^2 / r by mr^3 to get m K r = m^2 v^2 r^2 = L^2

where L = p x = m v r is the angular momentum, the quantity we know how to quantise: L = nh

To the radius has discrete solutions r = L^2 / mK = (n h)^2 / mK for any integer n

and the Energy has discrete solutions because of r; we have E = K/2r = mK^2/ 2 n^2 h^2

I would distinguish between having discrete values, as happens with the solutions of r and E, and being "quantised", as happens with L.

For gravity, some people quantise the product of time and space, xt, thus we have quantised areas and with some extra conditions we could imply discrete values of time and length.

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WhatIsGravity

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Probably beers, but that just seems like a very interesting way to get big G involved in quantized things.

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Ilja

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But clocks can, of course, show some quantum behaviour. Imagine an atom with myons instead of electrons so that the lifetime of the myon works as a clock. In the Bohr model, one could associate with each state some velocity, thus, also some different discrete lifetime. Thus, a "quantization" of proper relativistic time is something even accessible today to experimental physics. And in principle the gravitational field would be also different for different myon states, an effect which is, of course, completely negligable, but could be computed without any problem (and without need for a relativistic quantum theory of gravity, because the Newtonian approximation of gravity would be sufficient).

A short search was sufficient to find out that the life times of myonic atoms are known and measured, and myonic 1S atoms appear very short lived, but this is because of some interactions (inverse to radioactive decay) with the atomic kernel.

So, while such discrete lifetimes of myonic atoms would have, obviously, nothing to do with a "quantization of time", I would bet that similar effects in quantum gravity theories would be sold as "quantization of time", if not already by the physicists themself, then certainly by the science journalists.

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julian

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In quantum mechanics observables are represented as operator on some Hilbert space. Generally, the spectrum of an operator can be continuous or discrete. You get a discrete spectrum if the Hilbert space is finite or (in the case of infinite dimensional Hilbert spaces) if there is some type of compactness. For example, if an electron is subject to a confining potential its wavefunction is `concentrated' in a finite (compact) region of space and as a result the energy spectrum will be discrete. A free electron that can `roam' all of space has a continuous energy spectrum.

In LQG the intrinsic geometry (quantum geometry `on' a spatial slice) is described by the spectrum of area and volume operators. These turn out to have discrete spectra and what is directly responsible for this is the compactness of the SU(2) gauge group.

In GR the extrinsic geometry (how the intrinsic geometry is embedded in space-time) is related to `time-evolution'. In the paper marcus quotes "Compact phase space, cosmological constant, discrete time") the Hilbert space is finite, and not only is the intrinsic geometry discrete, but also the extrinsic geometry turns out to be discrete.I'll just note that energy is a tricky subject in GR, so trying to talk about a conjugate `time-operator' this way might be troublesome. In Rovelli's paper ("Compact phase space, cosmological constant, discrete time") they say "Therefore also the variable conjugate to the intrinsic geometry is compact." - they don't mention energy in the paper.

Should be noted that in ordinary quantum mechanics (i.e. not quantum gravity) defining the `time-operator' - an operator conjugate to the energy operator is tricky. Rovelli tries to address the problem in a paper called "Time-of-arrival in quantum mechanics" arXiv:quant-ph/9603021 "We derive

a well defined uncertainty relation between time-of-arrival and energy; this result shows that the well known arguments against the existence of such a relation can be circumvented."

In LQG the intrinsic geometry (quantum geometry `on' a spatial slice) is described by the spectrum of area and volume operators. These turn out to have discrete spectra and what is directly responsible for this is the compactness of the SU(2) gauge group.

In GR the extrinsic geometry (how the intrinsic geometry is embedded in space-time) is related to `time-evolution'. In the paper marcus quotes "Compact phase space, cosmological constant, discrete time") the Hilbert space is finite, and not only is the intrinsic geometry discrete, but also the extrinsic geometry turns out to be discrete.I'll just note that energy is a tricky subject in GR, so trying to talk about a conjugate `time-operator' this way might be troublesome. In Rovelli's paper ("Compact phase space, cosmological constant, discrete time") they say "Therefore also the variable conjugate to the intrinsic geometry is compact." - they don't mention energy in the paper.

Should be noted that in ordinary quantum mechanics (i.e. not quantum gravity) defining the `time-operator' - an operator conjugate to the energy operator is tricky. Rovelli tries to address the problem in a paper called "Time-of-arrival in quantum mechanics" arXiv:quant-ph/9603021 "We derive

a well defined uncertainty relation between time-of-arrival and energy; this result shows that the well known arguments against the existence of such a relation can be circumvented."

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julian

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On the trouble of defining a time operator conjugate to the Hamiltonian in ordinary QM:

"On the one hand, one imposes self-adjointness as a requirement for any observable; on the other hand, according to Pauli's argument, there is no self-adjoint time operator canonically conjugating to a Hamiltonian if the Hamiltonian spectrum is bounded from below"

"On the one hand, one imposes self-adjointness as a requirement for any observable; on the other hand, according to Pauli's argument, there is no self-adjoint time operator canonically conjugating to a Hamiltonian if the Hamiltonian spectrum is bounded from below"

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Gaz

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julian

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According to Einstein's hole argument - spatial/temporal relations in the context of GR only have meaning with respect to material reference systems. Since ALL material reference systems are quantum mechanical in nature we MUST take this into account. Pullin and Gambini have actually formulated a new interpretation of QM based in this - arguing that the decoherence argument is not just a `practical' (and so incomplete) resolution of QM interpretation problems but a theoretical resolution - called the Montevideo interpretation...recent review here: http://fr.arxiv.org/pdf/1502.03410

It must be noted that this is motivated by considerations of classical GR but only taking the quantum nature of material reference systems into account, and so NOT quantum gravity!

There is an old but classic paper by Rovelli called "Quantum reference systems" uploaded - which does discuss the quantum nature of reference systems in the context of quantum gravity.

But I don't think any of this leads to discrete time evolution by itself.

It must be noted that this is motivated by considerations of classical GR but only taking the quantum nature of material reference systems into account, and so NOT quantum gravity!

There is an old but classic paper by Rovelli called "Quantum reference systems" uploaded - which does discuss the quantum nature of reference systems in the context of quantum gravity.

But I don't think any of this leads to discrete time evolution by itself.

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Quantized gravity is a theory that attempts to reconcile the principles of quantum mechanics and general relativity by treating gravity as a quantized field. This means that gravity is made up of discrete particles, similar to how other fundamental forces like electromagnetism are described.

Currently, there is no definitive answer to this question. Some theories suggest that if gravity is quantized, then time should also be quantized. However, this is still a topic of debate among scientists and more research is needed to fully understand the relationship between quantized gravity and time.

If quantized gravity is proven to be true, it would have a significant impact on our understanding of the universe. It would provide a more complete picture of how the fundamental forces of nature work and could potentially help us solve some of the biggest mysteries, such as the nature of black holes and the origins of the universe.

So far, there is no direct experimental evidence for quantized gravity. However, there have been some observations in astrophysics and cosmology that are consistent with the predictions of quantized gravity theories. Further research and experiments are needed to confirm these findings.

It is possible for quantized gravity and the idea of a continuous, smooth space-time to coexist. Some theories, such as loop quantum gravity, propose a discrete structure for space-time while still maintaining the concept of continuous space-time. However, this is still a topic of research and debate within the scientific community.

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