Taming Infinities: Examining Quantum Theory and Spacetime with Penrose

[Cerenkov]

Hello.

I have some questions concerning this extract from, 'The Nature of Space and Time' the Isaac Newton Institute series of Lectures and debate between Stephen Hawking and Roger Penrose in 1994, published by Princeton University Press in 1996. I have crudely reproduced fig 4.1, using MS Paint.

Chapter 4. Quantum Theory and Spacetime. R. Penrose.

"The great physical theories of the twentieth century have been quantum theory (QT), special relativity (SR), general relativity (GR), and quantum field theory (QFT). These theories are not independent of each other: general relativity was built on special relativity, and quantum field theory has special relativity and quantum theory as inputs (see fig. 4.1)."

"Although these four theories have been remarkably successful, they are not without their problems. QFT has the problem that whenever you calculate the amplitude for a multiply-connected Feynman diagram, the answer is infinity. These infinities must be subtracted away or scaled away as part of the process of renormalization of the theory. GR predicts the existence of spacetime singularities."

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Penrose then goes on to briefly describe how it was hoped that QFT might “smear” out the singularities of GR in some way. He also writes that the divergence problems of QFT could be solved in part by an ultraviolet cutoff from GR.

Now to my questions.

1.
As well as renormalization, what other techniques are used in physics to eliminate or minimize infinities?

2.
Besides the infinities mentioned here where else in physics do infinities occur?

3.
Was Planck's solution to the Ultraviolet Catastrophe an example of how an infinity was 'tamed'?

4.
Is this the kind of 'taming' Penrose refers to when he talks about... 'an ultraviolet cutoff from GR'?

5.
I Googled 'taming infinities' and this link came up.
https://warwick.ac.uk/fac/sci/maths...2014/statmech/adk/abstracts/martin-taming.pdf
Relevant much?

6.
(Going out on a limb now, so please be gentle. My level of understanding is Basic, with a capital B.)
If I read this Wiki page right... https://en.wikipedia.org/wiki/Semiclassical_gravity ...should combine GR and QFT. Which might be illustrated in an extension of figure 4.1 by two lines emanating from the GR and QFT boxes, converging on a new box entitled, Semiclassical Gravity. Like this. Am I even warm?

7.
If so, won't the fundamental problems of each - singularities for GR and infinities for QFT - rear their ugly heads in Semiclassical gravity too?

Any help given at my Basic level would be greatly appreciated.

Thank you.

Cerenkov.

 

[LURCH]

1.
As well as renormalization, what other techniques are used in physics to eliminate or minimize infinities?
.

When I was first introduced to that word, It was my understanding that “renormalization” is a blanket term referring to all of the techniques used to reduce or remove infinities.

 

[kimbyd]

When I was first introduced to that word, It was my understanding that “renormalization” is a blanket term referring to all of the techniques used to reduce or remove infinities.

I don't think this is accurate. I only ever heard it refer to a very specific method used in QFT:

1. Cut off the calculations before the infinites are reached.
2. Replace the remaining bits that aren't being calculated with parameters.
3. Measure the values of those parameters experimentally.

As to the OP's question of how infinities are typically dealt with, usually it's by drawing a circle (or other shape) around the infinity and restricting your calculations to things that happen outside. You can still have some parameter to describe the properties of the region inside, as with renormalization, but you can't describe what's going on in there.

For example, in the interior of a black hole, General Relativity won't work properly if you try to do calculations including the center of the black hole. But you can draw a sphere around it, and have the boundary of that sphere act as if there was some mass with some electric charge and some angular momentum inside the boundary. Nothing about the shape of the mass, what kinds of particles make up the charge, or how things in there are actually moving can be known. Only those three properties.

This process of avoiding infinities is basically how it's always done. Things are taken to the limit of the location where the infinity occurs, but not actually including it. There are lots of techniques and methods that are applicable in different scenarios, but that basic idea seems to be part of pretty much all of them.

 

[Cerenkov]

My thanks to LURCH and kimbyd for their responses.

Ok, we've talked about dealing with mathematical and spacetime infinities to make them tractable, but I wonder...? What about infinities at the other end of the scale? The following links describe what appear to me to be efforts to 'tame' the infinities generated by inflationary cosmology. These require 'taming' too?

https://arxiv.org/abs/hep-th/0609095 The Measure Problem in Cosmology
https://dspace.mit.edu/handle/1721.1/62646 The Measure Problem in Eternal Inflation
https://www.quantamagazine.org/the-multiverses-measure-problem-20141103/
https://en.wikipedia.org/wiki/Measure_problem_(cosmology)
http://iopscience.iop.org/article/10.1088/1475-7516/2010/09/008

Please note that I am neither advocating the actual existence of an infinite multiverse nor declaring my support for any of the theories discussed in these articles. Instead, I cite them here because I'm exploring how science deals with infinities, no matter where they are found.

So, what's the deal with solutions to the Measure Problem? Are they similar (in principle) to renormalization and 'cutting' singularities out of spacetime? That is, they're ways of keeping the math finite and excluding infinity out of the calculations?

Thank you.

Cerenkov.

 

[kimbyd]

It's a little difficult for me to parse which infinities exactly you're referring to. Could you please elaborate?

With respect to the measure problem, the issue there is that if the universe is infinite in spatial extent, then there's no unique way to define a probability measure. And many of the more obvious ways of defining a probability measure come up with absurd results.

Much of this can be filed under, "It's hard to get the math right for the early universe."

 

[Cerenkov]

It's a little difficult for me to parse which infinities exactly you're referring to. Could you please elaborate?

With respect to the measure problem, the issue there is that if the universe is infinite in spatial extent, then there's no unique way to define a probability measure. And many of the more obvious ways of defining a probability measure come up with absurd results.

Much of this can be filed under, "It's hard to get the math right for the early universe."

Hello again Kimbyd.

My apologies for not responding to you sooner. Unfortunately a bout of ill health prevented me from doing so. But I'm on the mend now. Yes, from the vagueness of my posts it's a little difficult for you to parse which infinities I'm referring to. Sorry about that.

In a few days I'll reply to you more fully and try to convey my thinking more succinctly.

Thanks,

Cerenkov.

 

[Cerenkov]

Hello kimbyd.

Can we start here, please? https://www.physicsforums.com/insights/need-infinity-physics/
It was this article that started me thinking about how scientists 'tame' infinities. Daniel Ciobotu seems to be saying that infinities are purely mathematical entities. That they are idealizations or abstract concepts, with no concrete or physical counterparts in reality.
Am I on the right track here?

Thank you,

Cerenkov.

p.s.
Even though I'm replying to kimbyd, if anyone else would like to respond I'd be very grateful. Please note that I'm approaching this on a Basic level. Thanks.

 

[kimbyd]

Hello kimbyd.

Can we start here, please? https://www.physicsforums.com/insights/need-infinity-physics/
It was this article that started me thinking about how scientists 'tame' infinities. Daniel Ciobotu seems to be saying that infinities are purely mathematical entities. That they are idealizations or abstract concepts, with no concrete or physical counterparts in reality.
Am I on the right track here?

Thank you,

Cerenkov.

p.s.
Even though I'm replying to kimbyd, if anyone else would like to respond I'd be very grateful. Please note that I'm approaching this on a Basic level. Thanks.

It's honestly hard to know whether or not there are real infinities. Certainly he's right that infinities crop up all over the place in the math used to do these calculations. Usually the infinities themselves are not used directly, as they muck up the calculations. So the infinities are avoided in precise ways to keep the calculations moving.

In quantum mechanics, the basic idea here is that we don't have a way to solve these equations exactly. So we use an approximation. That approximation is where the infinities come from. We can't know if the infinities are fundamental to the equations, because the approximations are the only way we've figured out how to calculate the results.

In quantum electrodynamics, for instance, has been tested to better than one part in a billion for many of its predictions (https://en.wikipedia.org/wiki/Precision_tests_of_QED). The process of QED calculations is a very involved process. A super-rough sketch is:
1) First, write down the field equations. The field equations are not obviously infinite any any sense.
2) Break up the calculation into pieces in a specific way. Each individual piece can be computed (there's no known way to compute the solution all at once).
3) Notice that the results of some of the computations are infinite. Cut off those calculations at some high energy and measure the values experimentally. This is known as renormalization.
4) Now that all of the individual results are finite, add them up. The result from adding all these individual calculations starts to converge on an answer which, as you can see above, is incredibly accurate.
5) Stop adding new terms before you get to something like the 140th-order terms. Because if you do, the answer starts to diverge and becomes infinite.

That point (5) is really frustrating. The renormalization trick to make some of the integrals finite is reasonable: it means that there's some high-energy theory that we don't yet understand which should change the results of those calculations at high energies.

But the fact that the series starts to diverge if you add too many of the pieces suggests that the entire process may be fundamentally wrong somehow. We know it works. We've tested that it works. But there's something we're not understanding about quantum mechanics for sure.

 

[Cerenkov]

Thanks for getting back to me, kimbyd.

You wrote...

"It's honestly hard to know whether or not there are real infinities. Certainly he's right that infinities crop up all over the place in the math used to do these calculations. Usually the infinities themselves are not used directly, as they muck up the calculations. So the infinities are avoided in precise ways to keep the calculations moving."

Ok, so perhaps the title of Daniel Ciobotu's article is a bit of a misnomer? Maybe, "The Need (to Get Rid of) Infinities In Physics" would be more accurate? Seeing as these mathematical infinities are avoided in precise ways? Such as renormalization? May I ask if you know where I can find a list or a description of these various 'precise ways' besides renormalization, kimbyd?

I ask because I'm assuming that infinities crop up in many (most?) branches of physics? Is that so? Assuming that your reply is affirmative, what about theoretical physics? Where no experimental measurement is possible, what happens when infinities crop up in the math?

Thanks in advance for any help given,

Cerenkov.

 

[kimbyd]

Ok, so perhaps the title of Daniel Ciobotu's article is a bit of a misnomer? Maybe, "The Need (to Get Rid of) Infinities In Physics" would be more accurate? Seeing as these mathematical infinities are avoided in precise ways? Such as renormalization? May I ask if you know where I can find a list or a description of these various 'precise ways' besides renormalization, kimbyd?

I don't know of a good summary offhand. The main things I can think of off the top of my head are:
1) Limits used in calculus.
2) Residuals used in complex analysis.
3) Renormalization.
4) Singularities in General Relativity (which are not included in the manifold).
5) Non-converging series.

I ask because I'm assuming that infinities crop up in many (most?) branches of physics? Is that so? Assuming that your reply is affirmative, what about theoretical physics? Where no experimental measurement is possible, what happens when infinities crop up in the math?

Usually what is done is to find another way to do the calculation which avoids the infinity. Most ideas are identical to the concept of limits: take the equation, whatever it is, and see what it's value is really really close to the point where the infinity occurs. If the value can be shown to get closer and closer to a specific, finite number the closer we get to the infinity, then we take that value as the answer.

Super simple example:
$$\lim_{x\rarrow2} {x^2 - 4 \over x - 2}$$

The value of $(x^2 - 4)/(x - 2)$ at $x = 2$ can't be calculated: it involves a division by zero. But the value at $x = 1.9$ is $3.9$. The value at $x = 1.99$ is $3.99$. The value at $x = 3.99999899991$. It's not hard to prove that the above limit approaches $4$.

Lots and lots of infinities can be tamed this way: simply do the calculations as close to the infinity as we can, and see if we can get a finite result. If not, look for another way to do the calculation.

 

[Cerenkov]

This is all good stuff, kimbyd. Thank you.

What I'm taking away from our dialogue is that infinities are a kind of roadblock to making progress in our understanding of something. Yes?

Assuming that's so, now let me ask another question.
The initial singularity is considered to possesses an infinite amount of heat, density, spacetime curvature and is infinitely small - zero dimensions.
An infinite universe is considered to possesses an infinite amount (number of) stars, planets, galaxies, etc. and it's dimensions (size) is also infinite.
Would I be right in thinking that both ends of the spectrum, the infinitely small and the infinitely large, even though they are only theorized to exist, are both roadblocks to our understanding? That, regardless of scale, both kinds of infinity need to be avoided to further our understanding?

(Ok, that's two questions.)

Thanks,

Cerenkov.

 

[kimbyd]

The initial singularity is considered to possesses an infinite amount of heat, density, spacetime curvature and is infinitely small - zero dimensions.

That's not quite right. While there is an initial singularity in the math, it isn't expected to relate to anything real. The most accurate way of describing that early state is that before a certain time, our current laws fail to describe what's going on. Remember: the infinity is excluded from the equations in order to make things work.

If we kept using those same equations to extrapolate backward, yes, there would be an infinity there. So to avoid the problems that would cause, we don't extrapolate that far. The interpretation is generally that a more fundamental understanding of physics would allow us to extrapolate back further in time, without an infinity encountered.

In this case, the infinity isn't a roadblock so much as evidence that there is more to learn.

An infinite universe is considered to possesses an infinite amount (number of) stars, planets, galaxies, etc. and it's dimensions (size) is also infinite.
Would I be right in thinking that both ends of the spectrum, the infinitely small and the infinitely large, even though they are only theorized to exist, are both roadblocks to our understanding? That, regardless of scale, both kinds of infinity need to be avoided to further our understanding?

Sort of. For most questions, whether or not the universe is infinite in spatial extent is completely irrelevant. The only case I know of where it actually matters is with reference to probability calculations: when trying to calculate probabilities of things happening in the universe, the infinite spatial extent makes it so that there is no one good way to calculate probabilities, making it impossible to make certain kinds of predictions (this is the measure problem, mentioned earlier).

If the universe is finite, the measure problem disappears. But then you have the question of the specific way in which the universe is finite.

 

[Cerenkov]

Hello kimbyd.

Yes, thanks for this correction.

That's not quite right. While there is an initial singularity in the math, it isn't expected to relate to anything real. The most accurate way of describing that early state is that before a certain time, our current laws fail to describe what's going on. Remember: the infinity is excluded from the equations in order to make things work.
If we kept using those same equations to extrapolate backward, yes, there would be an infinity there. So to avoid the problems that would cause, we don't extrapolate that far. The interpretation is generally that a more fundamental understanding of physics would allow us to extrapolate back further in time, without an infinity encountered.
In this case, the infinity isn't a roadblock so much as evidence that there is more to learn.

And Yes, I take your point in that last sentence.

So, in both cases, at both ends of the spectrum so to speak, scientists apply techniques to exclude the infinities in question. The infinities of the initial singularity are excluded, as per your description. As far as I'm aware some kind of cut-off is used to do the same thing with the measure problem.

But this raises another question in my mind, kimbyd.
Of the three types Friedmann universes, two are assumed to be infinite. Therefore, won't the measure problem rear it's ugly head in those two?

Thank you,

Cerenkov.

 

[kimbyd]

But this raises another question in my mind, kimbyd.
Of the three types Friedmann universes, two are assumed to be infinite. Therefore, won't the measure problem rear it's ugly head in those two?

Yes! But the measure problem is only an issue with certain kinds of probability questions like, "Why is the universe this way instead of that way?" It doesn't come up much.

 

[Cerenkov]

Cerenkov said: ↑
But this raises another question in my mind, kimbyd.
Of the three types Friedmann universes, two are assumed to be infinite. Therefore, won't the measure problem rear it's ugly head in those two?

kimbyd replied:
Yes! But the measure problem is only an issue with certain kinds of probability questions like, "Why is the universe this way instead of that way?" It doesn't come up much.

What you've explained to me is very interesting and helpful, kinbyd. Thanks.
I think I'd like to draw a line under this, otherwise I can see us edging towards the fine-tuning controversy, with all of it's attendant philosophical and theological issues. This is a physics forum, after all.

Thanks again.

Cerenkov.