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stub for relativistic particle (in order to record a good reference kindly pointed out by Igor Khavkine to me)
I hope that idea will once become a section called “Hi brow approach” and we will have down to Earth approach first. I can do it once, taking the point of view of say Landau-Lifschitz textbook which is very clear and simple. Of course the basic idea is the same: the action depends just on Lorentz invariant feature of the path, namely the invariant length.
the action depends just on Lorentz invariant feature of the path, namely the invariant length.
I don’t understand. Isn’t that precisely what the entry says?
But, sure, feel free to work on the entry.
Isn’t that precisely what the entry says?
That is why I said that the basic principle is the same. But the entry starts by saying in hi-brow that the basic principle all stems from is that it is a special case of sigma model. This is like defining a group as a special case of infinity-group. This interpretation is nice, but not basic, simple and fundamental, rather an additional bonus.
You know, I am wondering if not the opposite is true. Eventually it would be good to get some feedback here from mathematicians who have not been brought up as physicists and want to learn what it is that physicists mean when they speak of things like “the relativistic particle”. So we need a first-principle conceptual description of what that is supposed to mean.
I am thinking the very first step in such a description must be: it is a certain functional on a space of maps from the line into some pseudo-Riemannian manifold. That’s precisely what “$\sigma$-model” means. So I am thinking that this is the first information to give.
Then we can specify which functional exactly,
And then we can talk about what it would mean to pass to the quantization of that functional.
it is a certain functional on a space of maps from the line into some pseudo-Riemannian manifold
Aha. That part I agree, and that is very simple and not different from Landau-Lifschitz (except for need for “certain” – one can have any which is (pseudo)isometry invariant – and reduced to the simplest of equivalent choices). Sigma model in minds of most people however usually presuposes messy condition of belonging to certain class of integral formulas for the functional and allows often for additional internal degrees of freedom; the notion of sigma model is rather late in history and not taught to students who will read this article. In standard derivation of the action, one does not specify by hands the functional but reduces the degrees of freedom to Lorentz invariant characteristics of the path and finally shows it must be a function of the path length. By the chain rule for the variation for the action one can take the length itself.
Eventually it would be good to get some feedback here from mathematicians who have not been brought up as physicists and want to learn what it is that physicists mean when they speak of things like “the relativistic particle”.
That being OK, I still think that most weight should be to have the basic ideas understandable to physics undergraduates (edit: I hope my concern does not look imposing).
Since the issue will come up in many examples, I tend to think that we should work on improving the entry sigma-model, so that whoever feels the need can go there to find details and exposition.
The issue is not a problem if treated in order. In idea section not using advanced words and constructions (unless the entry itself is about something advanced or recent), and then putting wider scope in later paragraphs of the entry. I do not think that it is the best to use lots of language with the excuse that each word has its own entry/link. Maybe it is overreaction in this entry but I hope you agree in general.
Hi there,
how about “a relativistic particle is a timelike path in a spacetime” as a zeroth order approximate definition?
Hi Tim,
good to see you back!
Sounds good. Go ahead.
(I get the feeling it is not clear, so let me say it explicitly: when I say I created a “stub entry” I mean that I created it just so the link exists and points to whatever little bit of informaton there is. So we don’t need to have a long discussion about whether there should be more information in the entry: for sure we should have,)
Alright, did it.
good to see you back!
I’m sticking around, but have only time and energy to contribute to one project, which at the moment is helping JB to get Azimuth started :-)
Tim, I have edited your addition a bit. I don’t think it’s good to say “the relativistic particle is a curve”. The curve is instead one confguration/trajectory of the system called the relativistic particle. So I moved that statement instead to the Properties-section. Then I wrote an Idea-section and a Definition-section. You and Zoran should have a look.
There is a gap there currently in the discussion: for the massless case it is not true that the action functiona is the length functional. Instead one has to pass to the Polyakov-type action functional. If anyone feels like it, that would be good to describe. Me, I need to do more $A_\infty$-operads now…
but have only time and energy to contribute to one project, which at the moment is helping JB to get Azimuth started
Hm, secondary effects of climate change are begginning to impact the quality of the $n$Lab…
I have added the detailed computation of the equations of motion by variation, at relativistic particle
Thanks, Jim, that’s nice to hear.
I should clarify: the gravitational field is not given by a circle bundle with connection, but by the tangent bundle with connection. The circle bundle with connection encodes the genuine gauge background field: the electromagentic field.
I have made a little addition to the entry in order to clarify this.
You were mentioning in mechanics something that there is Hamiltonian in non-relativistic systems, what did you mean ? Maybe that it is more convenient to do manifestly covariant formalism with tensor of energy-momentum or there is some thing which can not work even non-covariantly for relativistic system of classical particles.
For the relativistic particle the Hamiltonian is the generator of parameter translation and constrained to vanish, thus encoding the invariance under parameterization of the worldline: $H = p_\mu p^\mu - m^2 \stackrel{!}{=} 0$. Also called the Hamiltonian constraint . After quantization the corresponding constrained Schroedinger equation
$i \hbar \frac{d}{d \tau} \psi = (p_\mu p^\mu - m^2 )\pi = 0$is the Klein-Gordon equation.
Where it says
worldvolume is the real line $\Sigma = \mathbb{R}$ or the circle $\Sigma = S^1$;
how can a particle perform a circle in spacetime? Maybe the trace of two virtual particles?
where the second term is the holonomy of the circle bundle with connection around $\nabla$
Should this be “around (or perhaps ’along’) $\gamma$”?
how can a particle perform a circle in spacetime?
Central Force, Orbits.
That’s a circle in space rather than spacetime.
I read that quote to specify the space- not the -time. Time “sweeps” out a volume (or sheet) from those surfaces.
Thus you have “R” for linear motion… sweeping out a worldsheet in 1D motion (the traditional “fabric of spacetime”), worldvolume in 2D, and worldhypervolume in 3D
and
“S1” for harmonic/circular motion… sweeping out a worldcylinder in 1D, worldsphere in 2D, and worldhypersphere in 3D.
David is right, indeed when one computes the partition function of a relativistic particle, one considers trajectories that go around a loop in spacetime, not in space. This is related to the vacuum energy. And indeed, ever since Feynman this is equivalently thought of as the process where a particle and an antiparticle appear, each trace out a trajectory of the shape $[0,1]$, and then merge again.
As I understand it, your use of worldsheet is not in line with the physicists use of worldsheet, which describes the embedding of a string ( a 2D primative) into space-time.
It may be the case that it also applies to virtual particle interactions though I’ve never encountered worldsheets in that context before (granted, I don’t do string theory).
Further, it seems to me that since virtual particle production occurs in virtually “no time” compared to real particles that those process’ would not sweep out a worldline or sheet the same as a real particle that has a much longer lifetime.
A particle sweeps out a worldline in Minkowski space. A string sweeps out a worldsheet. We’ll parameterize this worldsheet by one timelike coordinate τ , and one spacelike coordinate σ.
Sometimes, the term world line is loosely used for any curve in spacetime. This terminology causes confusions. More properly, a world line is a curve in spacetime which traces out the (time) history of a particle, observer or small object. One usually takes the proper time of an object or an observer as the curve parameter along the world line.
As I understand it, your use of worldsheet is not in line with the physicists use of worldsheet,
Not sure what your “you” and “your use” is referring to in this sentence. If you are thinking of the $n$Lab article relativistic particle you’ll notice that there is says “the worldvolume” and behind that link the special cases of worldline, worldsheet etc. are made explicit.
which describes the embedding of a string ( a 2D primative) into space-time.
Notice that it’s in general not an embedding, just a map.
It may be the case that it also applies to virtual particle interactions though I’ve never encountered worldsheets in that context before
Closed curves in spacetime are the most basic examples of Feynman diagrams exhibiting virtual particles.The corresponding term is called the 1-loop vacuum energy. Same for strings and toroidal worldsheets.
(granted, I don’t do string theory)
You don’t seem to do particle physics either.
you’ll notice that there is says “the worldvolume” and behind that link the special cases of worldline, worldsheet etc. are made explicit.
Where you may notice in turn that in the diagram from (Schmidt-Schubert 94) in the worldline entry, worldline is exclusively applied to localized QFT (standard model of particle physics) and worldsheets exclusively to string theory (not standard model of particle physics).
So as I understand it, a worldline in QFT describes the motion of a 1D object (point) moving in a 3D universe which is parametrized as 1D thus sweeping out a worldline… while a worldsheet in string theory is a 2d object (string) moving in a nD universe parametrized by 2D thus sweeping out a sheet in time.
So when I said “worldsheets are not common usage to physicists”, I mean particle physicists working in the experimentally verified standard model since particle physicists do not in general consider string “math” to be string “physics”.
Hence, as a particle physicist and not a string mathematician, I am very familiar with tenets of the former and admittedly ignorant of the latter.
You don’t seem to do particle physics either.
Tsk, tsk; no need for such snide remarks just because I’m ignorant or incorrect on a subject and you are not. :(
Urs, please forgive me if it turns out that our disagreement is semantic in that, given the current lack of phenomenological evidence, I don’t consider string theorists to be physicists but rather mathematicians
What you consider string theory to be or not to be is quite irrelevant for the present thread not only because your comments don’t display command over the much simpler matters that are actually being discussed in this thread.
It seems that you were thrown by the term “worldvolume” in the entry relativistic particle, misreading it as “worldsheet”. To expand my explanation of that from before: Every $p$-brane has its $(p+1)$-dimensional worldvolume, for the case $p = 0$ it reduces to the world-1-volume of particles, known as worldlines.
If you have questions on basics of Feynamn diagrams, virtual particles, and vavuum amplitudes, maybe even on the first little basics of string theory, you could open a thread with a corresponding question, either here or on Physics.SE.
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