ajb

Okay, so correct me! Looking at the syllabus of a few universities it does not seem that a typical student leaving with an undergrad education will have been exposed to any high brow mathematics - the same is true of typical physics educations. After that things will depend a lot more on the tastes and interests of the individuals - chemistry research covers a lot of things including things boarding with theoretical physics.

Can you point to some results to back up what you are supporting?

I guess the only thing that can be done is to learn the mathematics and mathematical language needed. One may only need a working knowledge rather than a deep understanding.

Can you expand on that with some nice examples that really are beyond arithmetic, basic calculus, linear algebra and integral transforms? One thing that has been touched upon is 'mathematical chemistry', which does use some graph theory, combinatorics and topology. There is also quantum chemistry, but I think as dealing with more than a few interacting particles explicitly is impossible one resorts to numerical methods - I am not sure that many people in this field really need deep results from topological algebra, operator theory and spaces. Chemical engineering - as suggested in this thread - uses numerical modelling of various phenomena. There will also be the need for some basic statistics and data analysis - it depends on who you ask if this is really mathematics. I know that some group theory can also be useful, but I doubt many chemists are well-versed in groups. For sure group theory is needed in spectroscopy - but how much beyond basic group theory does one need here? I would be very happy to be corrected and shown how advanced mathematics is part of a typical chemists tool kit.

Chemistry is far from my interests - however I am sure you can find MSc programmes on mathematics and chemistry. You have have to leave Kosovo though.

So you are thinking of an MSc or PhD in `mathematical chemistry'? I accept that.

At what level? Generally, I would say that there is almost no mathematics in chemistry apart from basic numeracy. There are exceptions like quantum chemistry, molecular dynamics and similar. And even then, the level of mathematics will depend on personal preferences to a large extent.

Great suggestion - you really can picture what is going on.

Some of these questions you should be able to find reasonable answers to via wikipedia... But as to the point of calculus, well there are two at first seemingly different topics in calculus i) Differentiation ii) Integration The first as you say deals with the instantaneous rates of change of functions (and similar objects). For example, a straight line can be described by y(x) = mx +c. Taking the derivative gives dy/dx = m. This is like the `velocity'. Taking the derivative (w.r.t. x) gives zero - the rate of change of the gradient or slope of a straight line is zero, thus it does not change. There is then an inverse operation of differentiation - given up to an additive constant - this we call the antiderivative. Following the simple example, if I have a constant function m, then the antiderivative is y(x) = mx+c, but the c is arbitrary as there is no way to fix this without some further information. On to integration, this is usually introduced as a calculational tool for working out the areas under a curve y(x) between x0 and x1 (say). This then can be generalised to find volumes and so on. The loose idea is to cut up the area under the curve into thin strips and then add up all these strips. For a finite number of strips you get an approximation to this area. If you consider an infinite number of strips then you get the area - but to make sense of this you need limits. This gives us the definite integral as it is between two points. There is also the indefinite integral, where no bounding points are given. Now, the amazing thing is the fundamental theorem of calculus tells us that the antiderivative and the indefinite integral are the same thing. Thus differentiation and integration are tightly related.

I think that could have been a possibility. For sure special relativity via the Poincare transformations is `written into' Maxwell's equations, as are other important things in modern physics like conformal invariance, gauge invariance and electromagnetic duality (in vacuum). Maxwell's work really was the starting place of a lot of modern physics - so like Einstein and Lorentz's work on time dilation the philosophy of physics was changed by the understanding of the mathematics.

What time dilation tells us is that the old notion of time ticking away in the 'background' is not really the right way to view the Universe. A global time for everyone works okay for Newtonian physics - well this is actually written into the maths - but this is only an approximation and relativity gives us a deeper view of time. Still, lots of things we don't really understand about time and its direction...

I think that you understand this well, but Tim88 I am not so sure... Just for future reference, the Poincare group (Lorentz + translations) is a Lie group - that is both a smooth manifold and a group. Minkowski space-time can be considered as a homogeneous space of the Poincare group. In fact this approach is more used with the supersymmetric extensions or spaces like ADS and DS, but it is worth knowing.

Just to clarify something, one does not think of the manifold of space-time as a field - this is just nonsense - but the metric is a field. Classical fields are sections of various fibre bundles over a manifold. In slightly less technical language, fields are well defined mathematical objects that you attach to space-time.

The biggest problem is that in our standard canonical formulation of quantum theory time plays a special role and one that is different to space. The ethos of Einsteinian relativity is that we should treat space and time equally, but in a canonical quantum theory this is not easy. If you apply the ADM formulation of general relativity (which used a space-time cut) you see that one does not really have dynamics, but rather a Hamiltonian constraint.

As this thread is in the philosophy section we should discuss some philosophy... When people discovered that Maxwell's equations give the speed of electromagnetic radiation as being c, this was interpreted as meaning that Maxwell's equations really only hold in the rest frame of the aether. There was then a lot of theoretical work to understand this, which leads to a more and more 'magical' aether with less and less reasonable properties. The whole idea was to make sense of this canonical inertial frame - the philosophy was that one should have some quasi-mechanical aether and that one must have some singled out inertial frame. A lot of effort in the late 19th was in this direction. Later, with Einstein and others, it was realised that singling out some canonical rest frame was unnatural and not needed - this is one of the core ideas of special relativity. This was a big philosophical change and the one that meant that the aether was just not needed. Also, quantum mechanics was developed and the dual nature of light as both particles and waves became apparent. The need to understand light as classical waves in some medium was abandoned. Again, this is a big philosophical change.

The rules of this forum say that you should not simply link to pdf files - you should present some of the theory here and use pdfs and so on as additional supporting information. So, with that in mind, could you sketch the theory and make it clear what you want to discuss?

We call this a Wick rotation... it is often used in quantum field theory as analysis works better with a positive definite metric. One then uses what is called analytic continuation - loosely it means that we can rotate between 't' and 'it' (c=1 as ever!). This is generally not okay, but in physics the kinds of functions encountered in quantum field theory are okay for this to make sense.

One should use ct as this is what appears in the metric and is essential in the Poincare transformations which mix x and ct.

I like the way you mathematical physics researchers that they are almost right! Lorentz for sure tried to understand the mathematics in terms of a aether. Einstein also worked with this idea, but I think he did abandon the idea as it was once understood. I know that Einstein tried to introduce the idea that space-time is the 'aether', but this language did not catch on. For one, we do not think of space-time as being material, so the language is not really correct. I disagree. Field theory, both classical and quantum is the bedrock of modern physics. Field theory gives us the most comprehensive and unified understanding of the Universe. I don't understand your claim here. The block universe is a natural concept in special and general relativity, though as we like dynamics it is common and sometimes necessary to cut space-time into space and time. My own philosophical thoughts here - and many also hold the same thoughts - are that this cut should be avoided. Einstein tells us that space and time should be treated on equal footing. This I do agree with. Though, in all honesty, most people working in modern physics are less worried with the philosophical implications than they are with matching theory with observation. Special and general relativity work well and for many people that is enough - also I am not convinced that metaphysics can say a lot more with any confidence. Why is one reason I think that one cannot really advance much in metaphysics.

Right, so the modern understanding of the Lorentz, or really the Poncare transformations are as isometries of Minkowski space-time. As for any older and no longer used understandings, I know less about. Lorentz and others who were working with the aether hypothesis tried to understand the invariance of Maxwell's equations in terms of the rest frame of the aether and so on... If there is some notion of the rest frame of the aether - whatever that aether is - then we have a canonical inertial frame to work with (at least ignoring gravity). One could then formulate everything in terms of this rest frame and look at what happens in other (inertial) frames. But again, our modern understanding is that there is no aether and that field theory is the best understanding we have. There are no true canonical frames to work with and such frames are not needed in modern formulations. You care of course free to pick a frame (inertial or otherwise) to work with, and that is what one usually does in practice. But someone else is free to pick another frame and the physics is not truly dependent on this choice - though the physics may look a bit different at first. That is a slightly different situation that I was think of - I was thinking of the space-time itself rather than the motion of test particles. If you are thinking of back reaction then I can imagine that the question of reversibility is more complex. I am not sure.

You are hinting at space-time cuts. That is making a meaningful cut of space-time into space and time. While this is not a problem locally, doing in globally in a nice way is problematic, unless your space-time has some nice properties - in particular the space-time is globally hyperbolic. In a loose sense this means that you can cut you space-time into Cauchy surfaces that evolve in time. In particular, all the information about the theory is contained on each surface and you can evolve this. This is needed in the standard formulation of quantum field theory on curved backgrounds (there is some work on space-times that are not globally hyperbolic with in a algebraic framework). So, I would say that 'pure GR' tells us that the block universe is the 'right' point of view, but as soon as you want other fields about a space-time cut is needed, or at least for now.

Special relativity is a model itself - you may be thinking of interpretations or analogies. The model of special relativity is not very deep once you start to think geometrically, but for sure some of the results are not so intuitive from our everyday Newtonian perspective.

Well, particles with negative mass might exist, but we don't have any evidence of such thing - they would break the various energy conditions imposed on general relativity. Interestingly, negative energy densities are common in quantum field theories on curved backgrounds, one may be able to exploit quantum effects and manufacture the necessary conditions. However, it is not at all clear that this is possible as subtle effect in quantum theory may render wormholes and so on unphysical. That could avoid free will issues, but causality is a problem whatever you send through time machines. The speed limit is that of the local speed of light. By using non-trivial geometry and maybe topology we can beat a light ray by taking another path. A soon as you do this you have questions about causality and time travel. However, none of this violates what we know of general relativity.