hardest known theory

[`hýsøŕ]

hopefully when i'm older and more experienced I'll know more about the existing theories in physics in depth, but until then, I'm interested to see which theory you all think is the hardest to learn (for any reason, hard maths, hard concepts, etc).

From what i've heard it's probably string theory, but GR and QFT look pretty intense too, lots of unfamiliar maths required to understand them.

Edited May 3, 2014 by `hýsøŕ

[mathematic]

String theory made be harder than the others, but it may be a waste of time, since no experimental information is available for testing.

[`hýsøŕ]

yeah, im not convinced its worth the time going into string theory

[ajb]

...I'm interested to see which theory you all think is the hardest to learn

 

...string theory, but GR and QFT ...

Getting a grounding in the basics of three of these should be okay, I covered that during my masters degree. String theory and QFT once you know some quantum mechanics are easy enough to get the basic ideas, you don't need much advanced mathematics for that. GR you can build up a little differential geometry along the way.

 

 

Getting new cutting edge results in these areas requires a lot more and depending on what you want to do could require more advanced mathematics.

[rktpro]

When I was little, I thought factorising polynomials is difficult, the simple quadratic ones. Now, it is no issue. Hardness and difficulty are relative. For a layman, elementary physics textbook with pulleys and springs might look hard.

Edited May 4, 2014 by rktpro

[`hýsøŕ]

thanks for the replies;

@ajb i guess that seems reasonable, the idea that getting used to the beginnings of the theory is relatively doable but the deeper you go the harder it gets. that seems to be the case everywhere anyway lol

@rktpro true, i hope that applies to all mathematics though, i've been trying to learn what tensors are for a while now with only some success

[studiot]

 

i hope that applies to all mathematics though, i've been trying to learn what tensors are for a while now with only some success

 

 

Tensors are quite intimidating at first since they were essentially developed as 'shorthand' for something the authors already knew in 'longhand'.

 

A good way to approach any part of maths is to (laboriously) write out the longhand until your own mind says to you

 

"Can we shorten this?"

 

Then you are ready for the more compact notation.

 

Tensors also suffer from their relationship to vectors, since they are also a type of general mathematical vector, but are often introduced as an extension to 'vectors', where it is not made clear that the vector system being extended is a particular type of vector.

 

You need a good grounding in the algebra of linear spaces to cope with tensors ( and many other things besides).

 

Modern theory is tending towards the use of differential forms and you might find these easier to study than tensors. (You also need some linear algebra for this).

 

go well

Edited May 5, 2014 by studiot

[`hýsøŕ]

thanks for the info, so far im kinda getting used to the notation with all the contraction and the summation convention, still a bit unusual though. doesn't help that this is my exam period where tensors aren't on the syllabus so i don't have time to study it in depth xD

 

you say i need a grounding in the algebra of linear spaces and other things, do you mean like topology and differential geometry? thats what i've seen people reccomend when i google tensors

[studiot]

No, sorry if I wasn't clear, the grounding is the algebra of linear spaces. The other things refer to the fact that this algebra spreads far and wide to many different (and important) branches of applied mathematics.

 

Theoretical physicists learn tensors as an exercise in formality.

 

I learned them from the practical viewpoint of stress and strain. This approach gives the student something tangible to grab hold of.

Not only did I know what and why I needed something more than vectors. but I also knew what the results of the manipulation would bring before I started.

 

But linear algebra is used in diverse applications as the solutions of large sets of simultaneous equations, including differential ones.

Laplace transforms.

Approximation theory and curve fitting.

Fourier series

 

 

A good modern book to have on your shelves is

 

An intorduction to Linear Analysis

 

by Kreider, Kuller, Ostberg and Perkins

 

Addison Wesley.

 

Tensors are not mentioned, but it is a good precursor.

 

A digestible introduction to tensor and associated methods is to be found in

 

Advanced Mathematical Methods for Engineering and Science Students

 

Stephenson & Radmore

 

Cambridge University Press

 

The traditional tensor is best introduced in

 

Coordinate Geometry with Vectors and Tensors

 

E A Maxwell

 

Oxford University Press

Edited May 5, 2014 by studiot

[`hýsøŕ]

ah, many thanks for the resources those will definetly come in handy, before i wasn't really sure what textbooks to get when the time comes where i get some time to study tensors in depth.

 

also i think i'll have a look at the stress strain approach. right now i don't find it clear why something so ..strange.. would come in so useful. eh maybe its not strange when im used to it.

[Orodruin]

Theoretical physicists learn tensors as an exercise in formality.

 

As a theoretical physicist (well, high-energy phenomenologist, but people tend to put us in the TP box), I am not sure I am ready to sign this statement ... On the other hand, what you and I see as a formality may be different. If I remember correctly, Einstein's 1905 paper on electromagnetism (the special relativity one) contains the Lorentz transformations of the Maxwell equations in component form ... I would describe my relationship with tensors as an appreciation of a general framework and amazement over how useful it is in different branches of science.

[studiot]

Here are some ideas that may help you.

 

Mathematically we collect together all the objects that have some property(ies) of interest in a set we call a 'space'.

These object obey desired rules of combination.

 

We do this because we want our maths to benefit from any useful common properties

For instance 'closure' on our space means that for some operation, F, between any members (A, B and C) of this set or space F(A,B) yields another member of the set.

For example the integers are closed under addition.

Adding any two integers will always result in another integer, not a fraction or any other sort of number.

 

This may seem trivial, but it is an incredibly powerful idea. It is what we use to prove the existence and uniqueness of solutions to equations.

 

If our set contains functions, we can select functions that are solutions to an equation of interest and combine them to find other solutions.

 

If we have a second set of constants (a, b, c etc) and our objects combine according to the rule a*A + a*B = a*(A+B), then our set is called a 'vector space' and A and B are called vectors. Actually there are about 8 or 9 rules in all for vector spaces.

 

These are the rules of 'linear algebra' and include 'free' vectors in physics such as forces, directions, momenta, accelerations and many more.

 

Mathematically the term vector also includes definite integrals, solutions to many kinds of equations, differential or otherwise, matrices, in fact any mathematical object that obeys the above rule.

Tensors also obey this rule so they are a type of vector in this sense.

 

Linear algebra or linear analysis is all about this type of mathematical behaviour.

 

Hence my reference to Kreider

 

This terminology has one unfortunate aspect, however.

 

If we consider the straight line y=mx+C, this is a straight line, but it is not 'linear' in the above sense.

The addition of the constant causes a problem that introduces a new type of mathematics we call 'affine'

 

Now here is where the physicists view of a vector as a simple type of tensor comes into play.

 

There are 'constant' tensors we can add to copy the affine structure of the straight line.

 

But there are no constant 'vectors' in physics available for this.

 

I'm sorry if this was a bit rambling but it was rather dashed off to get something down whilst you were online.

[ajb]

As a theoretical physicist (well, high-energy phenomenologist, but people tend to put us in the TP box)...

With some jest we have a discrete spectrum here;

i) Phenomenologists don't know what they are doing, but know how it is related to nature.

ii) Theoretical physicists don't know what they are doing, nor how it is related to nature.

iii) Mathematical physicists know what they are doing, but not how it is related to nature.

[`hýsøŕ]

sounds almost like an interlude into abstract algebra xD thanks for the info though. and with the help of one of these video series' on youtube, im getting a bit more used to the notational rules of tensors and things, hardest part it seems is remembering the rules. this video series also explains why they're useful, when you combine the contravariant and covariant together, their transformation properties between coordinate systems cancel and you get an invariant.

(i can send you a link to the series if you want to show it to somebody also struggling with tensors, so far i've found it very helpful)