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I don't think there is a difference.

[math] 1/ 0[/math] is not infinity, it is undefined. [math] lim_{x\rightarrow 0}\frac{1}{x}[/math] is infinity. Infinities are just difficult things to think about and don't seem very natural. How would you measure an infinite quantity, what does a quantity being infinite physically mean? One should be careful about mathematical infinities in a theory and physical infinities in nature. No example of the latter has ever been found.

Did you want to know just the formula or how to derive the formula? If the latter, then consult mathworld

It can be viewed in many different ways. The most easy way to see it is via Fujikawa's argument that the path integral measure is not (generally) invariant under all classical symmetries. Roughly, the anomaly appears as the Jacobian. More accurately, the regularisation scheme employed does not respect the symmetry. This method directly shows the relation between anomalies and the index theorem. They also can be explained using BRST cohomology , K-theory or you can calculate the anomalous current directly using Feynman diagrams.

That is a shame as I would have been very happy to see if it is of any use to anyone! I have not used it for anything "useful" yet. I have written it with physics in mind, it will do some basic but useful manipulations on superfunctions like take derivatives, return the body and soul, return the even and odd parts, set the variables to zero, take the Berezin integral, expand and simplify products of superfunctions etc... It could also be useful for dealing with differential forms and multivectors. I am thinking about witting another package that uses my Superalgbera packages to deal with differential forms.

A theory is said to be anomalous if a classical symmetry of the theory does not survive quantisation (or regularisation). The anomaly itself is the failure of the expectation value of the conserved current to be (covariantly) conserved. [math]D_{\mu}\langle j^{\mu} \rangle_{a} = \mathcal{A}_{a}[/math] The calculation of anomalies is very interesting and involves geometry, topology, homological algebra and the such. I did my MSc project on the subject. You can find a copy of my thesis on my website if you are interested.

Just put the latest version on my website. Some bugs are fixed and some new functions added. However.... does not work any more. You get the wrong grading. I may look into this in the future.

Time dilation and length contraction can easily be seen by considering the invariant space-time interval [math]\Delta s^{2} = -\Delta t^{2} + \Delta x^{2} + \Delta y^{2} + \Delta z^{2} [/math]. Basically this defines a distance of a path in space-time. Consider two different paths in space-time such that the invariant space-time intervals agree. This means they have travelled the same distance in space-time, but not necessarily the same distance in space or time! Only the overall distance in space-time agrees. This is in essence time dilation and length contraction.

As has been said, most of the "strange behaviour" is due to the wave nature of particles. Also on a technical point, the word "anomalies" has a very specific meaning in quantum theory.

I am well aware of the classical origin of the cosmological constant. From a modern point of view, you should include all terms possible in a classical Lagrangian that are consistent with the symmetries you want and quantum field theory, (we ignore remormalisation of general relativity for now). Thus, one should include the cosmological constant (and topological terms) and explain why they are zero, if indeed they are. Just setting things to zero by hand is not natural. Indeed as I said, quantum effects may push these values from zero anyway. Maybe more familiar is the [math] F \wedge F[/math] term in Yang Mills theory. Even if you set this topological term to zero classically, instantons will generate this term in the quantum action. So, by analogy one would expect a non-zero cosmological constant.

That would be my definition if we have a Lorentz or just a time invariant system. This may not be the case generally, i.e. you could write down a lagrangian that is not (quasi-)invariant under time translations. I am not sure how you would define the energy of such a system? It is not a conserved quantity. However, you can still write down a Hamiltonian, (assuming the Legendre transformation is not singular) that would be the best definition of energy but it will depend on time. (just consider one particle non-relativistic machanics with a time dependent potential). It's not something I have thought much about. Usually we deal with Lorentz or Galilian invariant actions in which energy as a conserved quantity associated to (space-)time translations is well defined. In more general situations where we don't have this we still have the frame work to work with as Hamiltonian or Lagrangeian mechanics does not really require the notion of energy anyway, just a Lagrangian or Hamiltonian. (Introduced in terms of energy, but this is not fundamental. All we require is a Lagrangian or Hamiltonian that produces the classical equations of motion. The notion of an Lagrangian or Hamiltonian is more general than that of energy). If you forget about energy and just work with a Lagrangian or Hamiltonian, then the question of "what is energy" never comes back to bit you.

Yes, on small scales (high energy) gravity will need to be explained in some quantum frame work. This will presumably, get rid of the infinite curvature. The point of my post is that you have top be careful trying to extract "physics" from infinities. Infinite curvature is a signiture of general relativity failing on some scale. So I would be very careful about trying to attatch real physics to this near or at this singularity. I also think (within general relativity) you have to be clear about if you are talking about "naked" singularities or hidden ones. As an outside observer we never see the singularity inside a black hole, it it hidden from us by the event horizon. Also if we send a probe towards the event horizon we never see it reach the horizon, although from the probes point of view it does cross the horizon. So, in some sence what you said is correct, we never see anything passing through the blackhole. Of course, there are huge tidal forces near the singularity that will rip apart our probe, but we will never know about it. As for naked singularites (if they are possible in GR) I don't know. As these don't have event horizons maybe we can see a little more? See what google says!

Zero is not very natural in physics. If the constant is set to zero classically, then quantum effects are likely to push this from zero. This is generally a fine-tuning problem. That is you would have to set your classical value very precisely to get the observed value of zero. Unless you have a symmetry or some other mechanism to stop this happening. So I think that an exact value of zero would be very hard to explain, but a small non-zero value would be much more natural. So as swansont has pointed out, trying to see if it is zero or just very small is hard to do.

I don't really have a clear picture of energy, other than what swansont said. Sometimes it is useful to think of it as a variable (or coordinate) conjugate to time or as a conserved quantity that arrises through your system being invariant in time. (you can make what I said more precise and meaningfull). But really, the only "universal" definition is what swansont gave.

No body thinks that we have infinite curvature in a black hole. It is just showing the break down of general relativity and telling us that we need some new physics. Throughout the development of modern physics the presence of infinities has lead to new developments. For example, the electron's self energy led to quantum field theory. Why should we expect any difference in general relativity? Most people think that on the small scale general relativity will no longer hold and that some quantum theory of gravity will "smooth out" the infinite curvature. We also have time dilation effects to take into account as a distant observer. We never see the particle cross the event horizon. Time dilation effects means that from our point of view, it takes infinite time for the particle to reach the event horizon. (again another infinity, new physics?) But as far as the particle is concerned, it does cross the event horizon in finite time. So you need to make your statments more precise and state which observer you are talking about.

The Schrodinger's and Heisenberg's formulations of quantum mechanics (with finite degrees of freedom) are equivalent. This is the so called Stone-von Neumann theorem. Both pictures are in fact equivalent representations of the canonical commutation relations (CCR). What Dirac did was construct a hybrid of these two picture suitable for discussing scattering. This is the so called Dirac or Interaction picture. First point, the Stone-von Neumann theory does not apply when we have infinite number of degrees of freedom and so does not extend to quantum field theory. Second point, Hagg showed that in relativistic quantum field theory, the interaction picture is not well defined. However, it does seem to work! Also, the standard formulation of quantum field theory is based on the Heisenberg picture. You can construct a Schrodinger picture, but as I said I don't think it is obvious that these two formulations are the same. You then use the interaction picture to construct scattering states and calculate S-matricies, scattering amplitudes etc. All these ideas can be found on google if you care to search.

Some basic lessions in probabilty theory might help. and what is 1/infinity? LOL

My Ex-girlfriend did a degree in genetics, her maths and physics training as almost non-existent. She had to do a maths course in her first year and that was it! I don't know how you can say you understand a biological system/process with out some basic physics. The ideas of conservation of mass, energy or anything like that were not stressed at all. Maybe things are different elsewhere...

Nope, but I do some teaching at university if that was what you were asking? The notion of a professor here in the UK is slightly different to that in Europe and the US.

I don't think that "relativistic theormodymanics" is well understood. I don't think it is clear how temperature should transform under a Lorentz transformation or that it should be Lorentz invariant. But that is not what I was hinting at. What i do know, is that a noninertial observer will appear to himself to be in a warm bath of radiation proportinal to his proper acceleration. This is due to the lack of a well defined vacuum for all observers. This is the Unruh effect. Thus the notion of a temperature of the vacuum is ambiguous.

I am very exited about the LHC, it should show us new physics; the Higgs particle(s) and supersymmetry. As for the balck holes, they would be very small and so evapourate very quickly with no harmful effects to the world. They would appear as exotic particles. Don't worry about it.

My highest qualification at the moment is an MSc in particle theory and right now I am half way through my PhD in mathematical physics. My research area is classical and quantum field theory and (super)geometry. I also hope to get something published soon. I would consider myself to be on the road to being a professional scientist.

Yes, Maxwell's equations. They describe light as waves in the electromagantic field. This description of light as a wave is fine for a lot of phenomena, things like antenna theory. However, as soon as you start to look a how light interacts with atoms and molecules you need to use quantum mechanics which tells us that light is a particle. So, I think it depends on what you are trying to describe/calculate to how you think about light/photons.

Two words "Unruh radiation". The temperature of "empty space" depends on the observer!

It is known as a period of inflation. It looks like the universe is going through a second stage of inflation.