
Register  Sign In 

QuickSearch
EvC Forum active members: 52 (9184 total) 
 
paulwilliam  
Total: 918,407 Year: 5,664/9,624 Month: 70/619 Week: 3/56 Day: 3/35 Hour: 0/0 
Thread ▼ Details 
Junior Member (Idle past 5264 days) Posts: 24 From: Chorley, Lancs, UK Joined: 

Thread Info



Author  Topic: Missing Matter  
Straggler Member (Idle past 201 days) Posts: 10333 From: London England Joined: 
How familiar are you with Lagrangian mechanics? Well I would be motivated to brush up on this long forgotten knowledge in order to take part in the Cavediver/Son Goku EvC String theory "correspondence" course.


NosyNed Member Posts: 9008 From: Canada Joined: 
Funny you should ask.
Once, long, long ago I did actually have some idea and a bit of group theory too. However, even then I had not gotten to what you are asking for. I finished an honours BSc in physics and realized that I couldn't handle any more math. And I've forgotten what I did know anyway. Once upon a time I remember telling a doctoral student that some things had to be expressed as math. He insisted that it is possible to at least explain the concepts without it. I happen to disagree with him. I'm wondering how far we can go without jumping headfirst into the deep end to start with? A good example is the dimensions thing: I can understand how such a term as a d10 may appear and need to be handled. (btw, I did try to read a post grad text on GR on my own some years ago. It was not a great success I'm afraid. )


Son Goku Inactive Member 
Well I've an idea on how to explain things.
Let's start with quantum mechanics and the Schrodinger equation. First of all, what happens when one tries to describe a single particle moving through space in classical mechanics? Usually it is done by finding what the position of the particle q is at time t. This information is encapsulated in the function q(t). You can find q(t) for a specific situation by using the Lagrangian or the Hamiltonian. (However I won't discuss them much further.) In quantum mechanics however the particle doesn't have a specific position q at a time t, so we can't use something like q(t). Instead we have the probability amplitude that the particle is found at q. The probability amplitude that a particle is to be found at q is labelled (q). The probability amplitude can change over time so we denote it (q,t) instead to indicate this. Obviously we need a way of finding what the value of (q,t) is for some choice of q,t. This is provided by the Schrodinger equation. Now for fields things move up a notch. Classically a field, like the electromagnetic field, is an object spread over space and has a different value, indicating its strength, at every point in space. The strength at a point in space x is labelled (x). If one knows all these values for every point you know everything about the field. Assuming you have this total knowledge you just call the field . Total knowledge of field is called a configuration. Here are four different configurations of a field:If the field was the ocean, then the configuration is the specific form of the ocean at the current time. Now just like the case of quantum mechanics, if you add in quantum considerations and move to quantum field theory then a field can't have precise configuration, labelled . Instead all you can have is the probability amplitude that the field is in the configuration . This is labelled (). Just as before the probability amplitude can change and so we have to include time (,t). The values of (,t) are found using a kind of "super" Schrodinger equation that is almost impossible to solve. We solve it approximately using diagrams known as Feynman diagrams to keep track of terms in the approximation. Does this make any sense?


NosyNed Member Posts: 9008 From: Canada Joined: 
I am fine with that so far. Nice step by step addition of complications Son. Thank you,
Don't stop.


cavediver Member (Idle past 3779 days) Posts: 4129 From: UK Joined: 
Well I've an idea on how to explain things. I think it depends on what you're trying to explain If we're doing some string theory, then we definitely don't want to be introducing quantum theory this way. And talking about the wavefunction of the field is just destined for confusion, as it now seems that we have two fields!! It is far better to forget all about and just deal with the operator valued quantum field. But the diagrams were nice


Straggler Member (Idle past 201 days) Posts: 10333 From: London England Joined: 
Well Cavediver seems unconvinced but in the absence of knowing where it is we are trying to end up and thus no idea of the best way to get there, your description so far seemed perfectly clear to me.
I don't know how you typed the symbols you did but my understanding is that the quantum field equation (of your last paragraph) tells us the probability of the field being in any particular state at time t in the same way that the normal Schrod equation tells us the probability of a particles position at a time t. Where the state of the field is itself a description of it's strength at any position x. I realise that my descriptions probably add little but they are my way of checking that I have understood and allow for my interpretation to be corrected if obviously wrong. Feel free to do so.


Son Goku Inactive Member 
And talking about the wavefunction of the field is just destined for confusion, as it now seems that we have two fields!!
I understand what you mean, the field wavefunctional is rarely discussed outside the ultrarigour style approaches of Glimm and Jaffe, Frohlich, etc. However without it I can't go from QM to QFT the way I'm trying to. I'm just going to see how it goes, maybe it won't work. Although I'm doing this from an observation that field theory seems to make more sense to people after they can explicitly connect it back to QM, even if that's not how field theory is done in practice. Usually it clicks after one hears of the wavefunctional and then one can say "Oh Yeah!, just like QM". For instance
operator valued quantum field.
I've heard many times "an operator on what?" The answer of course is the s, "just like QM". I think it is helpful to talk about the s pedagogically, but of course different fields (in the academic sense) will require different pedagogical approaches. I'm not going to go near String theory, as I said I don't know the subject. Hopefully though I can give a good summary of QFT.


Son Goku Inactive Member 
That's all perfectly correct. Good to see I've written a solid first message to refer back to.


cavediver Member (Idle past 3779 days) Posts: 4129 From: UK Joined: 
I've heard many times "an operator on what?" The answer of course is the s, "just like QM". I would jut say it is acting on the fock space of states (from a perturbative perspective) Which in turn is just another representation of the operators in the field. In QM, we go from localised point particle (x,p) to infinitely extended and gain lasting impression that there is some reality to . But this is wrong  it is really just a hint that we should actually be using QFT and not QM. And in QFT, "" does not contain any information that is not aready contained in the field  which is why I think it is a confusing and largely redundant notion for explaining quantum field theory. Thoughts? [Of course, if we are working in quantum cosmology and want to emphasise the superspace approach, it is perfect < !UB >


Son Goku Inactive Member 
Thoughts?
Loads, this is one of my specialities. Unfortunately if I go in to it we'll both start sounding like jargon cannons. So I'll leave my technical comments hidden below.
All the information contained in the states is contained in the fields as well as you mention. This is to be expected because the fields are irreducible as operators and so the Hilbert space can always be constructed from them and so they define the information within it. However there still is a Hilbert space. The same observation can be made in QM. There is no information, as such in L^2(R), that isn't contained in the Q and P operators. So I'm usually hesitant in defining the field operators as a primary entity in QFT since I don't do the same for P and Q in QM. Maybe I'm backwards with my Hilbert space first point of view. Certainly the field operators can be viewed in abstract and the Hilbert space is just some representation space for them. Again though this can be done in QM.
I'll probably put up my next big post sometime in the next few days. See what you think!
I'd love to talk about this, as you're aware we'd have to start talking about the fact that the fields of the physical world do not live in the Fock rep, e.t.c.


New Cat's Eye Inactive Member 
I just want to bump this in hopes that Son Goku and cavediver will have the time someday to continue thier discussion. Obviously, a lot of us find this stuff interesting and informative.
Thanks guys for your contributions. They are invaluable. Its threads like these that put evc way above the bar.


Stile Member (Idle past 179 days) Posts: 4295 From: Ontario, Canada Joined: 
Just wanted to say I'm peeking in and lurking. I doubt I'll post any questions or comments since I think this will very quickly go beyond my ability to follow.
But I'll certainly have fun trying


Son Goku Inactive Member 
Just wanted to say I haven't forgotten this. I've a few posts lined up for the weekend. They basically cover how quantum field theory deals with particles and the issue of renormalization.


cavediver Member (Idle past 3779 days) Posts: 4129 From: UK Joined: 
Bringing your comments out of hiding I, for one, was far more inspired by reading stuff way above my comprehension than the toneddown popular accounts  certainly as a child, and even as late as my final year of undergrad, when I was moving into quantum gravity from an astrophsyics background. I'm sure there are some here who will enjoy the jargon...
All the information contained in the states is contained in the fields as well as you mention. This is to be expected because the fields are irreducible as operators and so the Hilbert space can always be constructed from them and so they define the information within it. However there still is a Hilbert space. The same observation can be made in QM. There is no information, as such in L^2(R), that isn't contained in the Q and P operators. So I'm usually hesitant in defining the field operators as a primary entity in QFT since I don't do the same for P and Q in QM. Maybe I'm backwards with my Hilbert space first point of view. Certainly the field operators can be viewed in abstract and the Hilbert space is just some representation space for them. Again though this can be done in QM. Yep, and my personal view is exactly as you suggest  I view this as backwards, even though it is the obvious chronological approach. I think we get tricked into reading too much into quantum mechanics by taking it before QFT.* I used to debate this for hours with a colleague who thought much as you did  he was very much into the preeminence of Hilbert space  or Pontryagin space in his case, as he loved his indefinite norms *ABE: just to be clear  I am not saying that P and Q are primary over the Hilbert Space  in my mind, QM is simply invalid as a description of reality. Edited by cavediver, : No reason given.


Agobot Member (Idle past 5666 days) Posts: 786 Joined: 
cavediver writes:  in my mind, QM is simply invalid as a description of reality. IMHO, this is the most fundamental issue that needs to be resolved before we can argue Creator/No Creator on EvC. That's why i made the thread about matter and reality, I wish we all had the QM/CM knowledge to participate as equals in such an epic discussion. Edited by Agobot, : No reason given.



Do Nothing Button
Copyright 20012023 by EvC Forum, All Rights Reserved
™ Version 4.2
Innovative software from Qwixotic © 2024