Markus Hanke

Thanks all

You surprise me How did you stumble across this relic, I haven't seen this mentioned in a long time !

Thank you all, I am going to have a bit of reading to do A quick glance through the sources seems to indicate though that my understanding is largely in-line, but definitely still incomplete. Not too shabby for three weeks in though...

Ok, that makes sense. So essentially I am not the only one who finds this stuff hard-going...I'm glad to hear that

Thank you all My point is precisely that I want to learn about the various possible representations of that group ( and I can see even from the Wiki article that there is more going on that just simple tensors and spinors ), and how they relate to spacetime physics and particle physics. I know only very general group theory, but haven't dived into the details of that particular group yet. At least now I know where to look for further research.

Ok, thanks for the pointer, I haven't studied that specific topic yet. Goes on my reading list now https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group Holy crap...it seems that this is quite a complex topic. This is going to need more than just a fleeting read before bedtime Have you any recommendations for a math textbook that treats this topic in some detail, at undergrad level ?

As we all know there is a correspondence between the tensor rank of a field representation, and the spin of the associated particle - so for example, a spin-½ is represented by a spinor field, a spin-0 by a scalar field, a spin-1 by a vector field, and so on. This seems to just be taken for granted in all of the texts I have seen, but is never really explained. So my question is, what is the deeper reason for this correspondence ? Is there a mathematical reason for it, specifically in terms of group theory and/or differential geometry ? Any ideas, anyone ?

So I have recently ( 2-3 weeks ago ) begun teaching myself QFT in earnest. Truth be told, it is hard going - while the basic concepts are straightforward and easy enough to understand, the finer details are most definitely not. The trouble I am having is that a lot of texts seem to focus much on the mathematical details ( for obvious reasons ), at the expense of the bigger picture. I am currently struggling through the Weinberg-Salam part of the SM, and I pretty much had to piece together the bigger picture from all the maths involved in it. I wonder if someone here can confirm whether or not my understanding is correct : So the basic idea is that we have a Lagrangian that has a leptonic part, a gauge part, and an interaction part. The leptonic part is just a sum of Dirac fields for the electron, muon, and tau, together with their associated neutrinos; since the neutrino is taken as massless, there is a bit of an asymmetry here in that the right-handed fields have no neutrino part. The spinor fields are hence of the form [latex]\displaystyle{\Psi =\binom{v_e}{e_L}+\binom{0}{e_R}}[/latex] and likewise for muon and tau. We then introduce the electroweak charges ( electric, isospin, hyper charge ), and assign them to the various particles in such a way that charged currents couple to left-handed particles, and to right-handed anti-particles. The neutrino interacts weakly, but not with the photon; the left-handed electron/tau/muon interact weakly and electromagnetically, and their right-handed versions couple to hyper charge and electric charge, but not isospin. So far so good. Here's where it gets a little tricky now. We have an overall symmetry group for the Lagrangian which is [latex]\displaystyle{SU(2)\otimes U(1)}[/latex] This means we need four gauge fields, three to cover SU(2), and one to cover U(1), corresponding to the three components of isospin, plus hypercharge. Those fields are [latex]\displaystyle{U(1)\Rightarrow B_{\mu}}[/latex] and [latex]\displaystyle{SU(2)\Rightarrow W_{\mu}^{1},W_{\mu}^{2},W_{\mu}^{3}}[/latex] We introduce field strength tensors for these, compute the corresponding kinetic terms in the Lagrangian, and introduce gauge covariant derivatives to ensure covariance of the Lagrangian under the respective symmetry transformations. I understand how these things are done, and how to arrive at the final leptonic Lagrangian : [latex]\displaystyle{L_{lepton}=i\overline{\psi }_R\gamma ^{\mu}\left ( \partial _{\mu}+\frac{1}{2}ig_BB_{\mu} \right )\psi _R+i\overline{\psi }_L\gamma ^{\mu}\left ( \partial _{\mu}+\frac{1}{2}ig_BB_{\mu}+\frac{1}{2}ig_W\vec{\tau}\cdot \vec{W}_{\mu} \right )\psi _L}[/latex] [latex]\displaystyle{L_{gauge}=-\frac{1}{4}f_{\mu \nu}f^{\mu \nu}-\frac{1}{8}Tr\left ( F_{\mu \nu}F^{\mu \nu} \right )}[/latex] Looks horrible, but I understand the meaning of the various terms, and where they come from, so I suppose I'm ( mostly - see below ) good with this. It does show quite nicely the asymmetry of the weak interaction between left- and right-handed particles. What I am less clear about - and that is never explicitly explained in my text - is the actual status of these fields. My understanding is that these are not really to be taken as physical fields yet; instead, you perform a rotation of the B and W(3) field about the Weinberg angle, and arrive at A and Z fields, which then are physical ( their excitations are just the photon and the Z-boson ) : [latex]\displaystyle{\binom{A_{\mu}}{Z_{\mu}}=R(\theta _W)\binom{B_{\mu}}{W_{\mu}^{3}}}[/latex] So you basically take the original fields as purely mathematical entities, which need to be rotated by some angle ( to be determined experimentally ) in order to match physical results in the real world. The remaining two ( not yet physical ) W-fields are then combined according to electric charge, such that [latex]\displaystyle{W_{\mu}^{\pm }=\frac{1}{\sqrt{2}}\left ( W_{\mu}^{1}\pm iW_{\mu}^{2} \right )}[/latex] These are again physical fields, the excitations of which are just precisely the two charged W-bosons. Is this understanding correct ? The next step is to break the symmetry via the usual Higgs mechanism, so that the leptons and gauge boson can acquire mass. The Higgs field in this case is a complex scalar field, and the gauge is chosen such that [latex]\displaystyle{\varphi =\binom{0}{\varphi _0+\frac{h(x)}{\sqrt{2}}}}[/latex] Now, this is again never really explained in my text, but I presume this gauge choice is largely arbitrary, and made simply for mathematical convenience ( what happens if I make a different choice ?? ). You then go through the usual motions - you compute the Yukawa interaction term ( and find that all neutrino terms drop out, so they are massless, while the field still couples to the electron/tau/muon ), and the gauge covariant derivative for the Higgs field ( which introduces mass terms for the W/Z gauge bosons ). Again, I understand how to do that calculation in principle, even though in practice it seems extremely tedious and complicated. By comparison with the generic quadratic mass term, and using the Weinberg angle, you can get the particle masses as functions of the coupling constants and the Higgs field vacuum. Does all this sound correct, or am I way off the wall here somewhere ? I am definitely unclear on the relationship between the various fields ( B/W, A/Z ) and the Weinberg angle - my understanding is that this angle is an external parameter in the SM that needs to be determined empirically, and thus far I really have not fully grasped its meaning at all, as this is never explained at all in my text. They just basically say "now we need to perform a rotation by this angle..." without a deeper explanation - I am deducing the above from the maths only. I just need a heads-up on whether what I have written above is right. I am also somewhat confused on what the trace operator is doing in the term ; [latex]-\frac{1}{8}Tr\left ( F_{\mu \nu}F^{\mu \nu} \right )[/latex] The summation of the tensor indices should leave a scalar as a result, so what's with the trace ?? Apologies if this is elementary stuff and the answers obvious, but they aren't to me. I have to start somewhere, and unless I keep checking that my understanding is correct, I will never be able to get my head around this properly. Learning GR was child's play compared to all this QFT stuff

Absolutely no offence intended towards the OP, but I find it hard to fathom that in the year 2016 there are still people out there who genuinely believe these ridiculous conspiracy theories ( moon landing hoax being only one example amongst many others ). I can only surmise that it is somehow part of human nature; I'm not a psychologist, but it must be because thinking you know something that the vast majority doesn't makes people feel superior and "relevant". I don't know, it is just really beyond me, and I'm not sure whether I should laugh or cry over this. Possibly both at the same time.

Actually, no. The MM experiment just happens to be the most well-know, and in itself it doesn't really tell us anything much. However, you need to remember that there has been a very large number of different experiments over the past two centuries or so, all of which have attempted to find some indication of an aether via a myriad of different methods and approaches. This includes rather crude classical experiments, as well as more modern high-precision quantum set-ups. All of those experiments, without exception, have come out negative. While the outcome of a single, isolated experiment such as MM can always be subject to debate, the consistent failure of a large number of different experiments taken together to turn up any trace of an aether is a very significant finding. That is one of the reasons why modern physics has abandoned the concept; the MM experiment played only a small role in that, even though it is very well known. The other main reason then is of course that an aether is quite simply not needed - so if it isn't needed, requires impossible properties to be consistent, and no trace of it can be empirically found, then Occam's razor indicates that it more than likely doesn't exist. One of the immediate consequences of the existence of an aether would be a violation of local Lorentz invariance, so many of the aforementioned experiments fall into that category : https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_violation Take careful note though that while the existence of a detectable (!) aether would imply a Lorentz violation, the reverse is not true - Lorentz violations do not automatically imply an aether, but can result from other mechanisms as well.

Allow me to point out that the speed of light is not constant at all, it is invariant - these are not always the same !

I am sorry if my post came across as saying that there should be no questioning of existing models, and hence no further breakthroughs and developments. It certainly was not meant that way, rather it seems I didn't explain myself carefully enough. Existing models must always be questioned and continuously tested, and precisely because of that there will be further developments in our understanding; any other state of affairs would be tantamount to total stagnation, which is not what we want. However, what I was trying to point out is that not all questions that can potentially be asked will lead somewhere meaningful in the context of physics - specifically the "why" type of question lends itself to infinite regress ( as imatfaal has pointed out ), since every answer you give can be followed by another "why", ad infinitum. There may well be a few layers of "why" questions that have meaningful answers, but you need to cut it off somewhere, or else you are no longer within the domain of physics.

I should reiterate again here that what physics does is make descriptive models of the universe; it does not concern itself with the question of whether these models are the "ultimate truth" or not, only with whether or not they are valid in the sense of being good descriptions. Physics is to the universe what Ordnance Survey is to Ireland - a way to find as accurate a description ( map ) as possible. So, if String Theory turns out to be "true", then that means only that it is a good model. If LQG turns out to be "true", then that means only that it is good model. And so on, I think you get my drift. Of course it is permissible to go on and keep asking in what way a description is "the truth", but then you are no longer doing physics. Such questioning should be encouraged of course, if for no other reason than philosophical and metaphysical curiosity, but I also think it is fairly important to keep the boundaries of ( and delineations between ) the various domains of enquiry in mind. Certain domains permit certain types of questions to be asked, but will fail to answers others outside that domain - e.g. you can ask a physicist how clocks in different places are related, and he will be able to answer you, but if you ask him why an ideal clock reads time, then you will likely not receive a satisfactory answer, because in physics that is simply how it is defined to be. Likewise, a philosopher can write you an entire treatise on the question of what time is, yet he will generally be unable to calculate for you even the most simplistic of clock relationships. That is because these two domains of enquiry have different aims, ask different questions, and use different methodologies. Likewise, for the discussion at hand, there needs to be a cut-off point where we quite simply have to say "it's a valid model because it agrees with all available empirical data", and be content with that, because it achieves exactly what physics sets out to do in the first place. If we keep asking "but why is that so", then eventually there will come a point where no further answer can be given. That is not a failing of physics, but rather a manifestation of the fact that all domains of enquiry are limited in scope. Just my own two cents' worth P.S. I do not mean to suggest that we shouldn't question models in physics - far from it, since that is an essential part of the scientific method. All I really mean to say is that it is important to realise that there are questions that physics quite simply cannot answer within its own domain.

It would still be the same, but note that 9.8m/s^2 is coordinate acceleration, not proper acceleration, so its numerical value depends on how you set up the coordinate system. It is also only valid on the surface of the Earth, not anywhere else.

How exactly you interpret the effect of the wave depends on how you set up your coordinate system, but in the transverse traceless gauge the key to this is that the light traces out a geodesic in spacetime, whereas the constituent particles making up the arms of the LIGO setup do not, since they are part of a rigid structure. Furthermore, there are two perpendicular arms, and since a gravitational wave is quadrupole radiation, a passing wave will induce a phase difference since the two arms are effected differently. One way to think of it is a time-varying Shapiro delay - while the coordinate distance the light has to travel remains constant, the metric ( =proper ) distance does not, so in essence you are getting a fluctuation in the coordinate speed of light along one arm of the setup, but not along the other ( due to the polarisation of the wave ). This is detectable.

The motion of ( uncharged, non-spinning ) test bodies is independent from any measurable property other than its initial position and momentum. Therefore, it does not make any difference how massive they are, when it comes to relative linear acceleration between them. Do take note though that the 9.8m/s^2 figure is valid only on the surface of the Earth, and when both bodies are of comparable mass, then to obtain the figure one has to take into account the motion of both of them - i.e., we have a full 2-body problem, instead of just an isolated test particle in free fall. The numerical value itself will not be affected though.

Indeed. And the problem reaches even further than initially meets the eye, because if you eliminate local Lorentz invariance, you inadvertently also eliminate the CPT invariance of quantum field theory. That means that, not only would you bring down relativity in the classical realm, but you would also bring down the entire Standard Model with it.

Nothing is "getting curved", that is the point. It just so happens that the relationships between measurements taken at different events in spacetime are the same relationships as those between points on an appropriately curved manifold. Hence, curved manifolds are good models to describe gravity. There is no "fabric" here that gets mechanically distorted in any way.

Because they have energy-momentum. For ordinary bodies, their mass would be what makes up most of the contribution, but if they have a substantial amount of angular momentum or net electric charge, or if they carry a magnetic field, then that would play a role as well ( albeit a small one ). What's more, if the bodies have spatial extension ( i.e. if they can't be considered point-like ), then their shape and internal composition would also matter.

I can think of a good few places where you might find those. Sorry, couldn't help myself. I'm too old now to pretend I still care about being politically correct

The term has a mathematically precise definition, so I'm not sure what you are getting at. The word "curvature" is actually just an umbrella term for a number of related, but quite distinct mathematical concepts. Yes, pretty much. I'm not sure I would agree with that. Even something as simple as a sheet of paper is intrinsically flat to a pretty high degree of accuracy.

That is not what happens. Time dilation is a relationship between different clocks in spacetime, it is not a physical "change" of some sort that "happens" to a clock. False. Two photon clocks at different places, when compared, are subject to time dilation just like any other type of clock. There's nothing particularly special about light at all. You can't eliminate time out of the equation. In fact, it is the principle of extremal ageing which determines most of the dynamics of GR, i.e. the behaviour of test particles under the influence of gravity. The geodesic equation is nothing other than a mathematical statement of that principle. Also, remember that locally your theory must reduce to Special Relativity, so time is an integral part. That doesn't make any sense, I'm afraid. The curvature in your example is just the relationship between different events along the light beam. There is no mechanical "bending" involved, since the light is not subject to acceleration at any point on its trajectory. I have tried this for literally years when I first learned GR, and I can assure you that this won't work, because all analogies are necessarily limited and severely flawed. GR seems counterintuitive to many people precisely because they get stuck thinking in terms of analogies, not because the model itself is in any way extraordinary. Ultimately - and I might make myself unpopular by saying this - the only way to understand GR is to actually abandon all visualisation aids and analogies, and go and learn the mathematics behind it. The point here is that there is no need to master those mathematics - you need to only get to a level where you can understand their meaning, so complete mastery isn't required. That is perfectly achievable for the vast majority of people, even those who aren't mathematically inclined, but it does take patience, effort, and perseverance. The answer is - the metaphor you are looking for simply does not exist. You cannot capture all relevant aspects of a non-trivial 4-dimensional concept by drawing pictures on a screen; sometimes even I wish that was possible, but the raw truth is that it's not. Yes, you can come up with certain visualisation aids for certain aspects of the model, but it is crucially important to understand the limitations of those analogies - and that's where a lot of people fall down, and where the "counter-intuitiveness" comes in. They confuse the analogy with the model itself. GR is not a rubber sheet, it is not an expanding loaf, and it is not ripples on a pond. Full GR is simple geometry, but it is not the geometry we are used to from everyday human experience, but a generalisation of it. I should explicitly point out here that the "usual sense of the word" curvature is not the type of curvature that is used in GR at all. When the average person on the street mentions curvature, then the mental image that is invoked is one of some kind of embedded surface bending within a higher-dimensional space - like the surface of a 2-sphere bending in 3-space for example ( that's where the rubber-sheet analogy comes from ). This type of curvature is called extrinsic curvature. However, GR uses an entirely different notion, called intrinsic curvature - this is defined by what happens when you transport tangent vectors along closed curves. This type of curvature is called "intrinsic", because it makes no reference to any embedding into higher-dimensional spaces, or any measurements that are external to the manifold itself. Instead, it is entirely determined by the measurements within the manifold. There is no good way to visualise this, and if one tries to force the issue, then that's where the counter-intuitiveness comes in. For example, people see a cylinder, and they intuitively say "oh, that's curved !"; and so far as extrinsic curvature is concerned, that is correct. However, GR considers only the intrinsic geometry of the surface, and it turns out that the surface of cylinder has no intrinsic curvature at all - it is perfectly flat, and that is true whether or not the cylinder is embedded anywhere. People find that counter-intuitive, but only because they do not know the correct meaning of "curvature". Hence, the only way to really understand GR is to go away from the analogies and visualisations, and learn the actual meaning behind it.

Yes, that's right. The term you were looking for is "metric signature", and only manifolds with a signature of (+,-,-,-) or (-,+,+,+) are useful models for spacetime. Purely mathematically though, you can of course have many other types of manifolds - the Euclidean space we are all familiar with from high school geometry is an example. Or you could consider a manifold with two time directions, like (-,-,+,+). But those are not suitable models for spacetime, because they don't accurately reflect how events in the real world are related.

In real terms, a geodesic is a curve the geometry of which is such that at each and every point the proper acceleration vanishes ( in technical terms : it parallel-transports its own tangent vector ). That means quite simple that, if you hold an accelerometer in your hand and free-fall along a geodesic, then the accelerometer will read zero at all times. To put it even more simply - in free fall, you are weightless. [latex]a^{\mu}=0[/latex] That is precisely the geodesic equation, though at first glance it mightn't look the same as what Mordred has written. In actuality, what is there are events - they are points in space at a given instant in time. What GR does is describe the relationships between these events - how far they are apart, what is the shortest connection between them, etc etc. In real physical terms, these relationships are simply measurements taken with instruments such as clocks and rulers. Clocks measure distances in time, rulers measure distances in space, and accelerometers measure acceleration. Taken together, those give us relationships between events. In the simplest case, events in spacetime are related in the same way as points on a 4-dimensional flat manifold are; this corresponds to the situation in Special Relativity. However, if there are sources of energy-momentum present ( planets, stars, electromagnetic fields etc ), then it turns out that the relationship between events is more complicated - these relationships are now the same as the relationships between points on curved manifolds. Curvature is just a measure of how much measurements taken with rulers and clocks deviate from reference measurements taken on a flat background. As such, what GR does is recognise that the relationship between physical events in spacetime happens to be the same as the geometric relationship between points on certain curved 4-dimensional manifolds - hence, such manifolds are adequate models to describe gravity, and how it relates to its sources. They provide the map that describe the territory. Don't think of curvature as a physical distortion of some mechanical medium, but rather think of it as a change in relationships between events. These changes can be measured by comparing rulers and clocks in different places; it isn't just an abstract concept, but something very very physical.

Join two Krasnikov tubes with opposite orientation together - effectively you would get a CTC, without having to get singularities or event horizons involved. This is an interesting topological structure - I am sure that there is some principle of nature which would prevent this from actually occurring in the real world, but so far as pure classical GR is concerned, this seems like a doable ( and survivable ! ) concept.