A Is renormalization the ideal solution?

[bhobba]

The fact that we're talking about debatable math is crazy to me. Math is supposed to be the one thing that is black and white, but I guess we've reached the realm of theoretical math where that's no longer the case.

What was it the great polymath (and of course mathematician both pure ad applied) John Von-Neumann said: “Young man, in mathematics you don't understand things. You just get used to them.”

A bit sad really - but the truth. I did a math degree undergrad and was always asking questions with Hilbert's view in mind that was put on his gravestone - "We must know, we will know". Thing is, after more experience and lecturers that said - I knew you were going to ask that, I just knew it - forget it or you are letting yourself in for a whole heap of hurt - the answer at the rigor you are after is contained in tomes you simply would not read. Sometimes I have read such tomes eg Geflands 6 volume set on Generalized Functions and I do not recommend it even though in hindsight I did come out of it with many questions resolved.

That's one of the good things about a site like this - those that have been through the 'wars' can steer others away from pitfalls.

There is much more that can be said about what you wrote - but you have a math based bachelors, best if you research it yourself. If you can't find a satisfactory answer, then post here and we will help.

Thanks
Bill

 

[ftr]

No. it just means that when one renormalizes it one has much more freedom in choosing the details in the theory. it is somewhat analogous to the freedom in choosing functions analytic at zero. The renormalizable case corresponds to functions that are representable by quartics, while the nonrenormalizable case corresponds to functions that are representable by an arbitrary power series. In the latter case, many more parameters are to be chosen. For an effective theory, only the first few matter. The high order terms only affect the theory at energies too high to be deemed relevant.

Thus the standard model (being renormalizable) is completely fixed by fixing slightly over 30 constants, all of which are known to some meaningful accuracy. Whereas canonical gravity needs infinitely many constants to single out the unique true theory, and only the lowest order constant (the gravitational constant) is known. To determine the next constant we already need to observe quantum gravity effects, which is still beyond the capability of experimenters.

Two questions.
1. Are you saying(or might) that gravity needs the high energy components in the interaction which interact at huge distances, yet in scattering theory where particles collide and they are at "zero" distance can be ignored.
2. AFAIK the constants origins are not known , hence they are non calculable, So how are they energy dependent?(OK alpha is energy dependent from experiment, but still)

 

[A. Neumaier]

What are all of the ways that we can test the limitations of a quantum theory?
[...]
maybe it would be appropriate to make a new post regarding that question to avoid going too far off topic. Do you think this question is too broad for Physics Forums?

Fyi, if you don't ignore me at some point then I will never shut up. I have a divergent number of questions about physics (I also have subtle jokes), and a desire to avoid making myself seem uneducated in public in order to find the answers to them is nothing in comparison to my desire to find those answers.

Generally, you should stick within one thread to one questions and minor variations of it, and ask sgnificantly different questions in a new thread.

 

[A. Neumaier]

Two questions.
1. Are you saying(or might) that gravity needs the high energy components in the interaction which interact at huge distances, yet in scattering theory where particles collide and they are at "zero" distance can be ignored.
2. AFAIK the constants origins are not known , hence they are non calculable, So how are they energy dependent?(OK alpha is energy dependent from experiment, but still)

Every QFT needs a suitable high energy formulation to be consistent, but different effective versions of the same theory differ in the high energy details. Energy dependence can be formulated even when the values of the constants are unknown. It is one of the strength of mathematics that one can formulate relations even when they involve unknown quantities.

 

[Geonaut]

@bhobba I see what you're saying. I realize that there are many lines of thought that we can follow that won't lead to an answer without coming to understand ridiculously complex math, and even then, a definite answer likely won't be found. I'm not a mathematician and so I won't try to pursue that type of problem, but it is likely that my curiosity will lead me to learn about more complex theoretical math in order to find certain deeper understandings about physics. You seem to be advising me against diving in too deep, and I am trying to avoid that since I am not a talented mathematician. My only advantages as an aspiring physicist has been my extreme tenacity and curiosity. I've spent over a decade now (I started at 13) trying to come up with an idea for a research project and I finally found something that got me excited a year before I graduated with a B.S. I've spent the last 3 years trying to develop it into something useful. It has been rough without an advisor as I decided against going to grad school for various reasons. Despite that, I'm just about finished developing it, but there are still a few details left for me to work out. I am obviously an amateur in comparison to tenured professors with PhDs, but nevertheless, I might have something interesting to show you superior physicists. We'll see, I'm sure I'll end up talking to you gentlemen about the details some time very soon. I can't express how eager I am to do that as I have not discussed it with anyone yet... Hopefully it can withstand the scrutiny, my heart tells me that it will as it appears to be elegant, but if not then I'll shed a tear and move on... After all, humility is important if you actually care about finding the truth.

My goal has been to learn only the things that I need to know in order to progress, but the problem with that is that there might be something important that I'm missing. Fortunately, after talking to you gentlemen, I'm no longer extremely suspicious of renormalization... I'm only mildly suspicious, but that's all I need to sleep at night.

@A. Neumaier thank you for taking the time to give me your opinion. I was worried that the question might be too broad. If I do post about it then I'll be sure to write my own answer to the question first and just leave the blanks for other members to fill in.

 

[bhobba]

You seem to be advising me against diving in too deep,

I am advising you to dive in as deep as you want - or do not want. Do not get too worried about things that are unclear, but have known answers, if they are not really germane to what you are studying. If answering those kind of questions interests you by all means study them - but you do not have to. An extreme example is Wittgenstein. He was a good aeronautical scientist and to advance his knowledge decided to do a PhD in math at I think Manchester University. Here he came into contact with Bertrand Russell who interested him in the foundations of mathematics and he switched from studying math relevant to aeronautical science to the foundations of math and then onto philosophy. This also happened to Russell himself - he was 7th Wrangler at Cambridge and a career in math seemed inevitable. But he, like Wittgenstein, became interested in its foundations and the rest is history. Still it was well known that the one thing that would be sure to engage Russell was math - he particularly liked talking to Ramanujan, but Hardy and Littlewood were the main mathematicians working with Ramanujan - it was just an interest for Russell.

Thanks
Bill

 

[Geonaut]

Do not get too worried about things that are unclear, but have known answers, if they are not really germane to what you are studying.

I see, thank you for the reassurance/advice.

I've often asked myself the question, especially when I was a kid - how do people shape themselves into good scientists? There's some obvious answers to the question such as constantly studying, but I always felt that the full answer was more complicated and extremely important to think about. I think your advice is definitely part of the right answer if you wish to use your time studying as effectively as possible.

Thanks again for the input @everyone.

 

[Elias1960]

Is renormalizability considered a physical quality of a theory or merely a mathematical property? If it's merely mathematical why was it such a crucial guideline in forming the SM?

I would say it is, in the concept of effective field theory, a natural property of the large distance approximation. The non-renormalizable terms decrease faster with increasing distance. So what remains are the renormalizable terms. As long as there are any - for gravity, there are none, so the surviving term is also non-renormalizable. And, therefore, for the same reason, very small in comparison with the other forces. So, this has some explanatory power too.