Mathematical Physics

[Sato]

What exactly is mathematical physics in contrast to non-mathematical physics?

 

I would imagine it is physics done in the way of pure maths, as in designating axioms, definitions, some inference rules implied by the context, and then deriving theorems about the subject matter, or similarly constructing structures that correspond to physical phenomena. But then what is the difference between that and, say, solving some equation and deriving the result using the informal tools of calculus/algebra. Each step in the derivation might not be given a justification like a proof step, and there may be differences in formatting and fullness, but wouldn't it be the same application of the theorems of algebra, analysis, and geometry? Is the only difference in style?

 

I asked this to an astrophysics professor two years ago and he said (paraphrased) "it's just physics done more mathematically". At the time I think my impression was that mathematical physicists used more obscure mathematics, so after asking if they also used something like, say, differential geometry, he answered that both types of physicists use it. I don't think I understood at all what he was getting at then, but think the idea above matches that, but am still unsure so wondering.

 

Thanks,

Sato

[ajb]

There is no clear definition of mathematical physics. Generally it means one of two things

 

i) Doing physics as if it were mathematics. (Which is what your astrophysics professor said)

ii) Studying and generalising the mathematical structures found behind theoretical physics.

 

The two are not really distinct and there is going to be some crossover.

 

However, ii) is closer to my own interests and work. The hope is that properly understanding the mathematics will lead to further developments and understanding of the physics. That said, mathematical physics is closer to mathematics than physics.

[studiot]

Here is my simplified take on the subject, based on the trickle-down model.

 

At one end of the scale the pute mathematician like to generalise. This means she doesn't like to solve specific problems like : solve x+4 = 6, but would rather solve all such problems and say that The solution to equations of the form x+A = B is x = (B-A).

 

Another example would be the general statement that a function f(x) is maximum if f'(x) = 0.

 

Enter the Mathematical Physicist who says

 

Ahah that's interesting so if I have a specific formula such as the power transferred to a load R in an electrical circuit is givne by the equation

 

[math]P = \frac{{{E^2}R}}{{{r^2} + 2rR + {R^2}}}[/math]

 

 

 

Where E is the source voltage and r the source resistance

And I differentiate and set to zero

[math]\frac{{dP}}{{dR}} = 0[/math]

This leads to

[math]R = r[/math]

 

Which is a formula for a general property of electrical networks derived from an even more general formula in mathematics.

 

Long comes the Engineering mathematician and says.

 

Oh that's interesting I have a need to maximise the power transfer from magnetic pickup heads to an electronic circuit.
The value of r for my heads is between 20k ohms and 100k ohms so if I make the input R for my amplifier 47k ohms that will be good.

 

And that is reason, o best beloved, why magnetic pickup inputs on audio amps are (usually) 47k.

 

(Have you ever read Kipling?)

 

 

[ajb]

Here is my simplified take on the subject, based on the trickle-down model.

It also 'trickles-up'. Many concepts in pure mathematics have their origin in physics, or at least their application in physics spurred on research in pure mathematics .

[Casey Wood]

For me, it has always come down to rigor. For example, a "non-mathematical" physicist might be someone who is perfectly content leaving out certain proofs or definitions. Whereas a mathematical physicist desires those proofs and definitions, as they are essential for understanding what we're doing as our equations become more and more mathematically complex. I would say that mathematical physics is a subject for those who wish to pursue the theoretical implications of physics, whereas, non-mathematical physics is more appropriate for those who prefer empirical methods.

 

Here's an example: In standard physics textbooks we are often told about the position vector and its applications. In three dimensions, it is usually given as the vector r (t) = xi + yj +zk. In standard physics we'll often take the difference of two position vectors to determine a displacement vector for something that has moved. This works fine and solves many problems associated with three dimensional motion in cartesian coordinates.

 

Then, a mathematical physicist comes along and asks the question: "Yeah, but isn't the vector space defined for n-dimensions. And furthermore, if I do a simple linear transformation of my coordinate system, then wouldn't that change the magnitude of the "position vector".?" The answer is yes, it does. "So, can we even consider the "postion vector" a vector at all?" The answer to that is no, not really. That is because as a tensor of rank 1, a vector is an entity that must be invariant under coordinate transformation. So let us now redefine the "position vector". The "position vector" can be equated to a radius arrow that points to a location in space, and is useful for finding points in a given coordinate system. As our radius arrow will have to be defined for n-dimensions, let us use indicial notation:

 

r = sum from j=1 to n of X j upstairs times L j downstairs such that j = 1,2,3... n

 

Where X is a cartesian dimension for a single reference frame, and L is a linearly independent unit vector. Noting that: A scaling factor can be introduced to act on X j upstairs.

 

My explanation is rather basic, but I hope it can offer some insight into the difference between standard physics and mathematical physics.

Edited August 26, 2015 by Casey Wood

[ajb]

I would say that mathematical physics is a subject for those who wish to pursue the theoretical implications of physics, whereas, non-mathematical physics is more appropriate for those who prefer empirical methods.

Mathematical physics is much closer to mathematics than physics. Theoretical physics is allied, but not the same as mathematical physics.

 

Theoretical physics is about building models and making predictions based on those models. It does involve mathematics, but not always much rigour. Depending on what one is doing, you do not always need very high brow mathematics.

 

Mathematical physics is about putting some rigour into the ideas of theoretical physics and also studying the mathematical structures used in theoretical physics. Generally it is much more like mathematics in how one works and writes in the subject.

 

There is of course a lot of cross-over and the distinction can just be a matter of taste.

Edited August 26, 2015 by ajb

[Casey Wood]

Hi ajb,

 

This topic has made me consider just what it is I'm doing half the time. When I'm working with water and experimenting in the critical regions of the Reynolds number, I've always felt like I was doing empirical work, and my math has been simplified significantly to aid that work. The experiments usually just call for algebra, and sometimes a little calculus. (Standard Physics) I take my results back to the office, and begin playing with the equations to see if I can adjust the models in a way that will aid the next experiment. My office work almost always involves vector and tensor analysis, so I think we could classify it as theoretical.

 

Yet, when I'm doing mathematical physics, I am studying the definitions of the vector space, linear independence, tensor operations, transformations etc. I'm getting a real feel for the space my equations live in, and I come away with a much deeper understanding of what those equations mean. Of-course, that deeper understanding aids in my ability to manipulate further equations and explore additional options in my theories. Rigorous mathematics, associated with physical concepts, are typically the way I've chosen to describe mathematical physics. I agree with you completely though... the "cross-over" between the subjects may well just be "a matter of taste".

 

Out of curiousity, what would you classify as "high brow" mathematics?

Edited August 26, 2015 by Casey Wood

[ajb]

Out of curiousity, what would you classify as "high brow" mathematics?

In the context of this thread, things like category theory, more advanced notions in differential geometry, homological algebra, more advanced ideas from algebraic topology...

 

Things that are often quite hidden behind the physics at first glance.