What does it mean to be Covariant?

[AlphaNumeric]

I don't see why you couldn't do this in quantum field theory as well- you just replace the derivatives with GR-style covariant derivatives (the two are equivalent in flat space), and that would tell you how your theory changes under general, smooth coordinate transforms. I guess the difference compared to GR is that GR ascribes certain physical properties to certain coordinate systems that might not have any inherent physical meaning in a conventional, flat space QFT.

There's a couple (as you'll see, no pun intended) of problems. The first is that spinors don't naturally couple (ha! genius!) to curved space, you have to use things like vierbeins and spin connections. The second is that at various points in things like renormalisation and loop calculations you do things like trace over the metric or contract indices or consider boson propogators, which involve the metric. Not to mention the fact expressions like $$p^{a}p^{b}g_{ab} = -m^{2}$$ are made more complicated and the energy-mass-momentum relationship of SR is regularly used to simplify integrals, such as how $$\frac{d^{3}p}{2E}$$ is Lorentz invariant (if memory serves). Heck, even Lorentz invariance must be viewed differently. No longer is the Lorentz invariance globally applicable, you have to do it point by point and then you get back to the vierbeins again.

Basically think of every point in QFT you use $$E^{2}-p^{2} = m^{2}$$, Lorentz invariance or anything to do with propogators and pretty much every one of those steps must be either justified again or completely reworked. And QFT in flat space gives enough people trouble.....

 

[Guest254]

I read a book by Fulling a little while ago about QFT in curved space-time. With regards the difficulties of field theories on curved backgrounds, the impression I got from this book is that the major obstacle is time: on your background you want a union of the notions of space and time, but you want your fields to behave as state vectors on some Hilbert space that evolve according to a "special" coordinate. I.e, GR is all about the $$x^\mu$$, whereas quantum evolution requires a $$(t,\mathbf{x})$$ type picture. In Minkowski space, there's no problem (obviously), but the waters are made murky on non-flat pseudo-Riemannian manifolds.

Spin, for instance, can be dealt with as long as we can construct a suitable spin bundle over the underlying spacetime. There are plenty of classification theorems for manifolds that admit such structures (I don't have my copy of Penrose and Rindler to hand, but I'm sure the relevant results will be in there somewhere).

Aside from all this. I feel daft replying to this thread. The OP is an idiot and a fraud, as has been demonstrated over and over again. Each time he posts a thread on "X", he gets to play pretend for that thread and will no doubt reply to responses with "thanks, I'll start looking into that...".

Green Destiny, if you want to learn about physics, then do what everyone else has done, and jump through some hoops. They are fun hoops. It is satisfying to precisely calculate how long it takes for a ball to roll down a hill if you take into account inertia. It is satisfying to show that Newton's theory of gravity predicts elliptic orbits. It is revealing to compute the spectra of the operator's associated with simple scattering problems. Ask genuine questions, and you'll find this forum beneficial. Personally, I love good old Newtonian dynamics, and it would be great to see you post a thread asking how long it would take a ball to roll down a hill.

 

[AlphaNumeric]

This is the best link available to get the ''just'' of why time ceases to exist as we know it. It has no flow. Nothing changes. Time ceases to tick.

http://www.fqxi.org/community/essay/winners/2008.1

As you will see, frozen time was a big topic for many of the authors.

You mean 'jist'. Besides, there's a difference between a reformulation of how you interpret various parameters and degrees of freedom in a system and saying they don't exist. Time, in the sense of a dimension of the universe, exists. Whether or not it can be said to 'flow' or whether it can be formulated to be entirely equivalent to spacial directions are different issues.

Does GR have solutions to the Einstein field equations where the metric signature has terms of different sign? Yes. Do physically viable solutions have such a thing? Yes. Does one of the parameters generally allow for a consistent parametrisation of dynamics? Yes.

You said that GR is plagued by a time problem from QM. Your links do not support that assertion in the literal sense, as GR doesn't include quantum mechanics. What some of the people in that link talk about is how to view space-time metric properties on ultra-short scales where quantum effects cannot be neglected and GR must be replaced with a quantum gravity model. At such scales many things, not just the notion of time, become blurred, like distance, causal structure and even what 'metric' means. GR, as described from the Einstein field equations, doesn't have this issue as it is, almost by definition, an effective theory which cannot be applied to such situations.

I recommend reading them all if you don't know of this amazing feature of relativity, and is a consequence of the Wheeler de Witt equation and is a solution of pure relativity, as you will see.

Pure GR has nothing to do with quantum mechanics, it is a classical theory. And I think it's a little rich you making comments like "I recommend reading them..." and trying to pretend you know GR, given you clearly don't and you don't follow anyone else's advice to do some reading yourself. You're the guy who doesn't know polar coordinates and yet you're trying to pretend you've understood things relating to quantum gravity and
the nature of metrics on non-trivial manifolds!

 

[CptBork]

There's a couple (as you'll see, no pun intended) of problems. The first is that spinors don't naturally couple (ha! genius!) to curved space, you have to use things like vierbeins and spin connections.

I remember briefly learning about vierbeins, I remember it has something to do with locally transforming coordinates to simplify the metric, and that there's a bunch of algebra you crunch through in order to guarantee results equivalent to the normal Christoffel symbol method. Then I promptly forgot most of it after turning in the homework

So, what's the deal with spinors? I figured something would come up with them and was just re-reading Srednicki's derivation of the finite group representations for SU(2)XSU(2). I suppose then there's no obvious way to define their general transformation rules without vierbeins? I never thought about this too much until recently, as everything I've seen in GR involves strictly scalars, vectors and generalizations thereof.

Not to mention the fact expressions like $$p^{a}p^{b}g_{ab} = -m^{2}$$ are made more complicated and the energy-mass-momentum relationship of SR is regularly used to simplify integrals, such as how $$\frac{d^{3}p}{2E}$$ is Lorentz invariant (if memory serves).

Well on the first point, $$p_\mu p^{\mu}+m^2=0$$ being more complicated to solve, that doesn't on its own stop you from generalizing QFT coordinates, does it? But when you mention integrals of the form $$\int\frac{d^{3}p}{2E_p}$$, in Peskin & Schroeder (and probably Srednicki too) they re-express it in the form:

$$\int d^4p\delta(p^2-m^2)$$

I believe the delta function portion of the integral is a scalar under any coordinate transformation, but I'm pretty sure the volume elemental d^4p doesn't stay constant for general transformations. I suppose that could make the whole quantization procedure pretty complicated... One of these days I'm gonna start learning about quantum gravity, probably within the next two years.

Spin, for instance, can be dealt with as long as we can construct a suitable spin bundle over the underlying spacetime. There are plenty of classification theorems for manifolds that admit such structures (I don't have my copy of Penrose and Rindler to hand, but I'm sure the relevant results will be in there somewhere).

I Gotta learn some topology too, that's the only well-known area of math I haven't touched yet (although I do know Euler's formula for planar figures)...

Aside from all this. I feel daft replying to this thread. The OP is an idiot and a fraud, as has been demonstrated over and over again. Each time he posts a thread on "X", he gets to play pretend for that thread and will no doubt reply to responses with "thanks, I'll start looking into that...".

Yeah I don't buy Green Destiny's claims to be ready for any of this stuff either, based on what I've seen so far, but I figure it's still good to reply and have the real truth out there for everyone to see, with references and justifications. I don't think any of the real scientists here came for the sake of attention or admiration, we just have a huge driving passion in common and love to share and trade our nuggets of wisdom, however much we might have. I'm not in a race to see who's smarter than who or who knows more in what subject, or who has the biggest... Few of the knowledgeable people here really are. Faking credentials just makes a person look stupid, and it'll show more and more over time if it keeps happening, like in Green Destiny's case.

 

[kurros]

I think the idea of covariance is that any object written in covariant form can be defined on some manifold (i.e. flat Minkowski spacetime Cartesian coordinates), and then if you know how that manifold maps to some other manifold under a change of coordinates, the rules of covariance automatically define how you write that object down in your new coordinate system. The change of coordinates could come from a boost, translation or rotation, a switch to a non-inertial frame, or it could even be something like a switch to polar coordinates where the physical situation remains identical.

I'm not really sure what you mean here. A manifold is still the same manifold whatever coordinates you use to describe it. I haven't done geometry for a while though so I can't really remember how this goes... I guess this is just (pseudo-)Riemannian manifolds we are talking about, and we only have to preserve the inner product... so I guess there is some set of coordinate transformations which are compatible with this. Of course the inner product doesn't make sense until we define the metric tensor so this is getting circular... maybe I'll go for a walk and find some mathematicians...

Although we can describe a manifold with whatever screwed up coordinates we like, so I guess the metric tensor just has to transform appropriately as we change coordinates to make sure the inner product stays the same... which seems to be something very general, i.e. there is no reason to restrict to some particular coordinate transformations... So in QFT if you start in flat space you can bend your coordinates however you like but the space will stay just as flat... right? It just won't be obvious that it is flat and the theory would start to look pretty screwed up if you pick stupid coordinate systems... which motivates the inclusion of gravity somehow I guess so that the physical laws can look the same under arbitrary coordinate transformations? Ok forget QFT for the moment let us just think of classical field theory. We can make up an action that is the normal Einstein-Hilbert action and stick in some yang mills fields with the derivatives replaces by covariant derivatives, and I assume we can stick in some matter fields as well and use vielbeins and spin connections to build the appropriate covariant derivative for the spinors, and then we have a nice classical theory with gravity and various other fields and everything is fine, maybe.
I'm not sure that little rant of mine really helped me clarify anything though...

 

[temur]

Are there any physics that can be done with e.g., the Schrodinger equation on a Riemannian manifold? I imagine this should be easier than setting up a quantum field theory on a curved space.

 

[prometheus]

This thread has taken a sharp improvement.

Are there any physics that can be done with e.g., the Schrodinger equation on a Riemannian manifold? I imagine this should be easier than setting up a quantum field theory on a curved space.

The Schrodinger equation is not even relativistic, so it wouldn't tell you too much about QFT in curved spaces to put it on a curved background. There have been papers that put the Schrodinger equation on a curved space, (for example) and also some that look at the Klein Gordon equation (the equation for a relativistic spin zero particle) in curved spaces (another example).

 

[Green Destiny]

You said that GR is plagued by a time problem from QM.

I am getting the impression you never bothered now to read any of it, maybe a casual glance.

http://www.fqxi.org/data/essay-contest-files/Markopoulou_SpaceDNE.pdf

''In quantum gravity, the timelessness of general relativity clashes with time in quantum
theory and leads to the \problem of time" which, in its various forms, is the main obstacle
to a successful quantum theory of gravity.''

Pure GR has nothing to do with quantum mechanics, it is a classical theory. And I think it's a little rich you making comments like "I recommend reading them..." and trying to pretend you know GR, given you clearly don't and you don't follow anyone else's advice to do some reading yourself. You're the guy who doesn't know polar coordinates and yet you're trying to pretend you've understood things relating to quantum gravity and
the nature of metrics on non-trivial manifolds!

I disagree, and I believe the link does as well.

 

[Green Destiny]

I advise you to be careful. Listening to certain individuals here will make you spout bullshit, like this:

Few of the knowledgeable people here really are. Faking credentials just makes a person look stupid, and it'll show more and more over time if it keeps happening, like in Green Destiny's case.

Never once, ever, have I made any claims on crudentials. If anything, I\ve admitted the lack of them.

 

[CptBork]

Never once, ever, have I made any claims on crudentials. If anything, I\ve admitted the lack of them.

You've claimed to understand certain advanced concepts and then shown a very basic lack of understanding when it comes to the fundamentals underlying those concepts. Example: your thread on EM vorticities, anyone who's actually done vector calculus would spot major nonsense in there right away, even if they didn't know the first thing about electromagnetism. I don't know anyone here who would object to you asking questions about advanced topics, but if you want to ask and then try to discuss these things on a mathematical level where you need the background, and you insist that you have that background while making very basic mistakes (far beyond simple typos and memory lapses), you'll get hammered for it.

 

[Green Destiny]

You've claimed to understand certain advanced concepts and then shown a very basic lack of understanding when it comes to the fundamentals underlying those concepts. Example: your thread on EM vorticities, anyone who's actually done vector calculus would spot major nonsense in there right away, even if they didn't know the first thing about electromagnetism. I don't know anyone here who would object to you asking questions about advanced topics, but if you want to ask and then try to discuss these things on a mathematical level where you need the background, and you insist that you have that background while making very basic mistakes (far beyond simple typos and memory lapses), you'll get hammered for it.

That is not a statement of crudentials. That was a thread, and no where in it did I claim I knew more than who was replying.

 

[CptBork]

That is not a statement of crudentials. That was a thread, and no where in it did I claim I knew more than who was replying.

Well then, to eliminate any confusion for the future, why don't you tell us what level you think you're at right now? What kind of topics are you presently studying in serious detail at a mathematical level? We don't need details about your personal life or whether you're currently taking classes, we just want to know what level you want others to believe you're at. Then if someone accuses you of misrepresenting yourself, you can point them here and tell them you've been clear you don't have the background on whatever subject, or vice versa if you claim to have a level of advancement well beyond whatever's indicated in your posts, others will refer to this when they hammer you.

 

[Green Destiny]

Well then, to eliminate any confusion for the future, why don't you tell us what level you think you're at right now? What kind of topics are you presently studying in serious detail at a mathematical level? We don't need details about your personal life or whether you're currently taking classes, we just want to know what level you want others to believe you're at. Then if someone accuses you of misrepresenting yourself, you can point them here and tell them you've been clear you don't have the background on whatever subject, or vice versa if you claim to have a level of advancement well beyond whatever's indicated in your posts, others will refer to this when they hammer you.

Right, good idea.

Well, I guess I would class myself under as a freshman at best, perhaps a second year college student. Mathematically, I am very poor, but I think I have a good enough understanding to whizz through most popularized physics textbooks for collegers and highschool students.

I suppose this is why I think I have only scratched the surface; but I'll never stop learning, I don't believe anyone just stops learning.

My mathematics probably needs a lot of attention. I have pretty good knowledge of the preliminaries of LVS's, but that is only because I took time out to understand things like Hilbert spaces, ect, what I've already explained.

I think overall though, whilst I have a long way to go, I can understand physics very well, and you might not be giving me the credit or potential required to learn these subjects. There are admittedly many things required to know, but I think I would do quite well at ensuring that I do.

So, there you have it. That's how I view myself.

Back on topic, I noticed alphanumeric you never answered me. Perhaps you are admitting you were wrong yes?

 

[Tach]

Edit: In fact, in a covariant theory you can write down the same equations in the same general form for any coordinate system you want, as long as the variables you plug into the equation are given with respect to that coordinate system.

Yes, this is by far the best definition. To illustrate, divB=0 (one of the Maxwell's laws) has the same exact form in all inertial frames of reference within the scope of SR.
By the same token, we would be very hard pressed to produce a covariant form of , say. Hooke's law in SR. (to my best knowledge there is no such formulation, I think Gron tried such a thing and failed miserably.).

 

[prometheus]

Yes, this is by far the best definition. To illustrate, divB=0 (one of the Maxwell's laws) has the same exact form in all inertial frames of reference within the scope of SR.
By the same token, we would be very hard pressed to produce a covariant form of , say. Hooke's law in SR. (to my best knowledge there is no such formulation, I think Gron tried such a thing and failed miserably.).

There is a discussion on a covariant formulation of Hooke's law here, which you may find interesting.

 

[CptBork]

Yes, this is by far the best definition. To illustrate, divB=0 (one of the Maxwell's laws) has the same exact form in all inertial frames of reference within the scope of SR.
By the same token, we would be very hard pressed to produce a covariant form of , say. Hooke's law in SR. (to my best knowledge there is no such formulation, I think Gron tried such a thing and failed miserably.).

Here's how I would try it out, starting with Hooke's law in a frame at rest w.r.t. the spring, located in a locally flat region:

$$\frac{d^2x^i}{dt^2}=-\left(\frac{k}{m}\right)(x-x_0)^i$$ (i runs from 1 to 3 over spatial indices, x[sub]0[/sub] is the spatial vector pointing to the spring's center)

Then you would do what they do in G.R., write $$\frac{d}{dt}=\frac{\partial}{\partial t}+\frac{dx^j}{dt}\frac{\par}{\par x^j}$$

So after substituting in covariant derivatives, the expression for Hooke's law would become

$$\left[\frac{\partial}{\partial t}+\frac{dx^j}{dt}\nabla_j\right]\left(\frac{dx^i}{dt} \right)+\left(\frac{k}{m}\right)(x-x_0)^i=0$$

Or at least, I think it should be covariant, for arbitary smooth coordinate transforms that only affect the space components, but not time. I haven't heard of a relativistic formulation, nor do I see why one should expect Hooke's law, which is only a simple classical approximation in any case, to be written down in the same form for any observer, when not all observers will be at rest w.r.t. the spring's center.

 

[Tach]

There is a discussion on a covariant formulation of Hooke's law which you may find interesting.

Thank you, the user CE1 seems to have accomplished the covariant formulation under SR. Great link, thank you!

 

[Tach]

Or at least, I think it should be covariant, for arbitary smooth coordinate transforms that only affect the space components, but not time.

But this would defeat the purpose of generating a Lorentz-covariant formulation. Seems that user CE1 has managed exactly that in the link pointed by prometheus.

 

[CptBork]

But this would defeat the purpose of generating a Lorentz-covariant formulation. Seems that user CE1 has managed exactly that in the link pointed by prometheus.

Ah, but that link involves describing a relativistic spring, in such a way that the law reduces to Hooke's law at low velocities. I thought someone was looking for a covariant formulation of Hooke's law itself, or the next best thing you could do with it, but obviously my version wouldn't be valid in relativity.

 

[Tach]

Ah, but that link involves describing a relativistic spring, in such a way that the law reduces to Hooke's law at low velocities. I thought someone was looking for a covariant formulation of Hooke's law itself, or the next best thing you could do with it, but obviously my version wouldn't be valid in relativity.

This underscores the difficulty of the problem. Maxwell's laws come covariant "out oif the box", the other empirical laws may require some very serious rework in order to make them covariant.