Number theory derivation from infinity; speculations on equations that are derived in terms of the Field

[NTuft]

1 hour ago, joigus said:

Sorry I can't comment much on that. I know next to nothing about Polyakov loops. From what I gather, those are Wilson loops applied to a QFT upon which you've performed a Wick rotation. I suppose they give you topological invariants of the corresponding thermal theory?

What I had in mind was a discrepancy in the trace -- a Wilson loop is functional if the perimeter measure is correct, whereas a Polyakov loop implies the area between confined states is what's needed (fuzzy re-hash). I think it's related to confinement and chiral symmetry breaking. I'll follow up on what you've mentioned.

I shouldn't have asked the last leading question, Glueballs. Would you explain what about it makes you think it "is considered to be the toughest problem around concerning physics?" I thought it was the conceptual or theoretical mathematical description of what has been pretty well established physically. So the back and forth between Physico-Mathematics, Mathematical-Physics; it does exist!

[studiot]

On 2/11/2023 at 8:01 PM, joigus said:

I think you're still confused as concerns 'fields' in algebra vs 'fields' in physics. Very different things.

It's like 'dog' in 'hot dog' vs 'dog' in 'I'm walking my dog.'

That's why I have given up on this thread a long time ago.   +1

Edited February 12, 2023 by studiot

[joigus]

1 hour ago, NTuft said:

I shouldn't have asked the last leading question, Glueballs. Would you explain what about it makes you think it "is considered to be the toughest problem around concerning physics?"

Here. I didn't remember very well. Mathematicians seem to have proved it's unsolvable, on account of incompleteness theorems:

https://www.eurekalert.org/news-releases/490733

[NTuft]

59 minutes ago, studiot said:

That's why I have given up on this thread a long time ago.   +1

I'll repeat the distinction between the two.

Quote

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure [...]
Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. [...]

Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties. Division by zero is, by definition, excluded.^[2] In order to avoid existential quantifiers, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two nullary operations (the constants 0 and 1). These operations are then subject to the conditions above. Avoiding existential quantifiers is important in constructive mathematics and computing.^[3] One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two constants 1 and −1, since 0 = 1 + (−1) and −a = (−1)a.^{[nb 1]}
[...]

So addition and multiplication, multiplicative inverse, and effectively two nullary operations would be an equivalent way to define the field. I think I posited those, though I called addition non-commutative for the reasons I explained.

Quote

----
In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time.^[1][2][3

You seemed to argue earlier that  wasn't a field--was it a problem with the operations I was proposing?

Feel free to give up. -1

[NTuft]

9 hours ago, joigus said:

Sorry I can't comment much on that. I know next to nothing about Polyakov loops. From what I gather, those are Wilson loops applied to a QFT upon which you've performed a Wick rotation. I suppose they give you topological invariants of the corresponding thermal theory?

In TQFT, for Wilson loops the expectation value does not change under smooth deformations..They are gauge invariant.
By Wick rotations obects from thermal physics, exp(βH), are related to quantum physics, exp(−iH T).. Polyakov loop is "thermal analogue" to Wilson loop.. 
An imaginary temporal compactification, length of β=1/T(emp.), leads to topologically nontrivial loops around the compact direction known as Polyakov loops. Those loops (assuming the group center change of basis is not trivial) are gauge dependent, 

6 hours ago, joigus said:

Here. I didn't remember very well. Mathematicians seem to have proved it's unsolvable, on account of incompleteness theorems:

https://www.eurekalert.org/news-releases/490733

Ok. I read your questions on TQFT, and those and the leads to "functorials", etc., look interesting. Interesting formulation. Thanks for this, article, too, but they make statements with caveats, so I wouldn't conclude it's unsolvable.

Edited February 13, 2023 by NTuft

[joigus]

7 hours ago, NTuft said:

Ok. I read your questions on TQFT, and those and the leads to "functorials", etc., look interesting. Interesting formulation. Thanks for this, article, too, but they make statements with caveats, so I wouldn't conclude it's unsolvable.

I'm OK with that. Every time mathematicians publish a proof, there's a time period during which the jury is still out.

Another caveat on terminology though. Mathematically-minded people like to say 'QFT is a functor' --thereby 'functorial.' This is a concept in category theory. People also say 'action is a 'functional' --a function of function that produces a number. Those are very different concepts with very similar terms, so it's necessary to tread carefully.

[studiot]

18 hours ago, NTuft said:

You seemed to argue earlier that  wasn't a field--was it a problem with the operations I was proposing?

Feel free to give up. -1

Did I ?

Please quote the passage since this forum insists on not numbering the posts.

 

In order to show that some set S constitutes a mathematical Field it is necessary to prove that S and its elements satisfy each and every one of the 11 Field axioms.

https://www2.math.upenn.edu/~kazdan/202F13/notes/FieldAxioms.pdf

 

If it is a field with extra properties these are extra axioms that will also need proof.

But remember each and every axiom must be proven satisfied.

 

[NTuft]

23 hours ago, studiot said:

Did I ?

Please quote the passage since this forum insists on not numbering the posts.

 

In order to show that some set S constitutes a mathematical Field it is necessary to prove that S and its elements satisfy each and every one of the 11 Field axioms.

https://www2.math.upenn.edu/~kazdan/202F13/notes/FieldAxioms.pdf

 

If it is a field with extra properties these are extra axioms that will also need proof.

But remember each and every axiom must be proven satisfied.

 

I suppose I quibbled,
"Accept this definition from wikipedia."
And your stance is,
"That is unacceptable."
It was on the issue that I'd missed a point you were emphasizing about Fields, but I didn't see the point, and wanted to wave it away and make do with a "group", though I didn't define the group operation well, either.

I grant your recent echoed point. I am out walking the dog in the... p^-1ark!... and uncertain on which end is the wiener, in natural casing, and which end is gauging the walking. So I think it is a confusion of the mathematical theory and matematical modeling, which is nuanced, but I think it is good.

I had premised that I wanted to take as granted the sets and their field axioms. In going in to describe the problem in situ, I think there needs to be a failure of the commutator. My work-up is not formally acceptable or explicit.

I think that commutative multiplication can describe energy being added to a particle. I think the failure of the commutator to allow addition can describe the phenomena discussed. I do not think it is too far afield to have your wiener and walk it, too, if on a lattice gauge the spacing in the physics field is bringing together the interactions that are described by the operations that characterize the mathematical field.

The 

book recommended Thursday is in the mail. +1 over there. Ought to walk before you run, and a group is more basic than a field, so maybe start there? Perhaps you have some thoughts on the set  to generate a group? 

 

  

On 2/13/2023 at 6:16 AM, joigus said:

I'm OK with that. Every time mathematicians publish a proof, there's a time period during which the jury is still out.

Another caveat on terminology though. Mathematically-minded people like to say 'QFT is a functor' --thereby 'functorial.' This is a concept in category theory. People also say 'action is a 'functional' --a function of function that produces a number. Those are very different concepts with very similar terms, so it's necessary to tread carefully.

I need to study the Lagrangian formalism, among other things. And be more precise in my terms and work-up to be taken seriously... Tread carefully.

Edited February 14, 2023 by NTuft

[studiot]

Yes, a groups is simpler and more basic than a field.

If you can demonstrate the set elements forming a group, your propsed field can inherit group properties to satisfy field axioms.

Field, group, algebra are some of the many words that combine with others into specialist phrases.

Here are some useful definitions for you in algebra and number theory.

[NTuft]

On 2/13/2023 at 11:59 AM, studiot said:

Please quote the passage since this forum insists on not numbering the posts.

couldn't add edit to #1229842, so for what it's worth here, if you hover over the time stamp on a post the last # in the url (e.g. in your post above #1229844) looks like the post number over the whole forum.

[studiot]

7 minutes ago, NTuft said:

couldn't add edit to #1229842, so for what it's worth here, if you hover over the time stamp on a post the last # in the url (e.g. in your post above #1229844) looks like the post number over the whole forum.

All I see is a yellow (ish) box with a date and time popping up.

But if you really have found a numbering system that would be marvelous.

[NTuft]

5 minutes ago, studiot said:

All I see is a yellow (ish) box with a date and time popping up.

But if you really have found a numbering system that would be marvelous.

chromebook Shenanigans displays url on webpage lower left-hand side on my end. Perhaps right-click, copy link address, paste it and it'll be at end of url.

Edited February 14, 2023 by NTuft

[studiot]

2 hours ago, NTuft said:

chromebook Shenanigans displays url on webpage lower left-hand side on my end. Perhaps right-click, copy link address, paste it and it'll be at end of url.

Yes I see it at the end of the address bar.

But how would my browser know which post I was looking at ?

For instance it shows blah-blah-blah/6/#comment-1229858 for every post on page 6.

[joigus]

4 hours ago, NTuft said:

I need to study the Lagrangian formalism, among other things. And be more precise in my terms and work-up to be taken seriously... Tread carefully.

You definitely should. The very reason why there are gauge fields is better understood in terms of the Lagrangian formalism. It's a beautiful language to express every fundamental physical theory we know. It also makes hard problems look simple. The downside is perhaps that it's far less intuitive than thinking about forces of different kinds. Another downside --and a very big one, mind you-- is that systems with dissipation are not possible to describe by means of Lagrangians. That's because friction is an emergent behaviour.

 

Edited February 14, 2023 by joigus
minor correction

[studiot]

29 minutes ago, joigus said:

You definitely should. The very reason why there are gauge fields is better understood in terms of the Lagrangian formalism. It's a beautiful language to express every fundamental physical theory we know. It also makes hard problems look simple. The downside is perhaps that it's far less intuitive than thinking about forces of different kinds. Another downside --and a very big one, mind you-- is that systems with dissipation are not possible to describe by means of Lagrangians. That's because friction is an emergent behaviour.

well said. +1

 

@NTuft

Cowan has a very easy transition from Newton to Eulerian to Lagrangian mechanics.

Hamill is much more red loodedas the whole book is as its title suggests. But still relatively easy going.

[NTuft]

@swansont,

On 7/28/2022 at 2:41 PM, NTuft said:

want to equate mass and charge,

 

On 6/30/2022 at 3:59 PM, swansont said:

How else do you get it to look like gravity, which has monopoles?

On 7/3/2022 at 12:49 PM, swansont said:

We want science. This being a science discussion board.

Which all beside the point, since GR reduces to Newtonian gravity and quantum gravity is similarly only important for small scales.

And doesn’t look like magnetism.

  Using Weber's electrodynamics we can substitute masses for charges to get a description of gravitational force. By my read, it wouldn't need monopoles or point charges then -- the masses act like point charges. It does reduce to Newtonian gravity, and with Mach's principle the system is extended as Relational Dynamics by A.K.T. Assis.
  Any thoughts on that? I'm pro-particle. You want to discuss the science I brought up on that? It was an overture to your particle leanings, I thought you'd like to discuss the cosmology development and why we can't find magnetic monopoles.

@studiot,
  I think you had the technical issue explained to you by someone else, too? You have it figured now? Thanks to you and @joigus for more input here, and keywords.. I'll try to follow up. You guys all sure know a lot.

Edited March 23, 2023 by NTuft
style correction, added address

[swansont]

2 hours ago, NTuft said:

Using Weber's electrodynamics we can substitute masses for charges to get a description of gravitational force. By my read, it wouldn't need monopoles or point charges then -- the masses act like point charges. It does reduce to Newtonian gravity, and with Mach's principle the system is extended as Relational Dynamics by A.K.T. Assis.
  Any thoughts on that? I'm pro-particle. You want to discuss the science I brought up on that? It was an overture to your particle leanings, I thought you'd like to discuss the cosmology development and why we can't find magnetic monopoles.

That is not the same as “gravity and Magnetism are synonymous” which was my objection.

[NTuft]

25 minutes ago, swansont said:

That is not the same as “gravity and Magnetism are synonymous” which was my objection.

Fair enough. Objection warranted.

[studiot]

6 hours ago, NTuft said:

@studiot,
  I think you had the technical issue explained to you by someone else, too? You have it figured now? Thanks to you and @joigus for more input here, and keywords.. I'll try to follow up. You guys all sure know a lot.

No idea what you mean by this ?

[NTuft]

21 hours ago, studiot said:

No idea what you mean by this ?

The issue with post numbering--there was a thread where you, Genady, and some others were discussing how you can put links to specific posts within threads inside a reply as a way to refer back to prior material. Discussion there about the issue with posts not being numbered, and you mentioned someone else had tried to explain it to you. This should probably be PM material, but, anyway, did you get around to where you can identify specific posts by their forum-wide post number? You can see that number is contained within a URL/web address that is attached to each post?? The time stamp at the top of a post is a hot-link on my computer. If you copy the link address and paste it the post number will be displayed.
Other than that just saying thanks for reading and participating in my rambling speculation here.