Are mathematicians and physicists obsessed with index notation?

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In summary, the speaker is studying mathematics and has to choose modules for the next semester. They are interested in physics and are considering taking partial differential equations, stochastic processes, geometry, and topology. Other options include numerical analysis, algebra, and discrete math. Suggestions for additional classes are given, including differential geometry, functional analysis, and group theory. There is a discussion about the differences between mathematics and physics, with suggestions for which subjects may be more relevant for different fields of physics. The importance of geometry, topology, and group theory in physics is also mentioned.
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Forever_searcher
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I am studying mathematics bachelor and next semester I have to choose some modules. I am really interested in physics, especially particle physics and quantum mechanics. So I am taking partial differential equations and Stochastic processes or Geometry and topology.
There are also numercal analysis, algebra and discrete math classes.
Any suggestions would be very appreciated 🙂
 
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:welcome:
 
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Forever_searcher said:
I am studying mathematics bachelor and next semester I have to choose some modules. I am really interested in physics, especially particle physics and quantum mechanics. So I am taking partial differential equations and Stochastic processes or Geometry and topology.
There are also numercal analysis, algebra and discrete math classes.
Any suggestions would be very appreciated 🙂
Differential equations and stochastics are probably better than geometry and topology. I would add differential geometry and functional analysis.

But in any case, the difference between mathematics and physics is in my opinion more of a linguistic one than of a content one. The tech speak is different, not the theorems. You need some inclination to work with coordinates.
 
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fresh_42 said:
Differential equations and stochastics are probably better than geometry and topology. I would add differential geometry and functional analysis.

But in any case, the difference between mathematics and physics is in my opinion more of a linguistic one than of a content one. The tech speak is different, not the theorems. You need some inclination to work with coordinates.
Why is that?
If here were to specialize in string theory or QFT I guess geometry and topology would be preferable than Stochastics. DEs is always important.
 
  • #5
fresh_42 said:
Differential equations and stochastics are probably better than geometry and topology. I would add differential geometry and functional analysis.

But in any case, the difference between mathematics and physics is in my opinion more of a linguistic one than of a content one. The tech speak is different, not the theorems. You need some inclination to work with coordinates.
I've heard that group theory is also very important. I am really confused because there are this many choices and all seem to have a connection with physics.
 
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MathematicalPhysicist said:
Why is that?
If here were to specialize in string theory or QFT I guess geometry and topology would be preferable than Stochastics. DEs is always important.
Does geometry have to do with anything? What you need from topology is mainly metric spaces. The few necessary basics can be read on the side.

QFT uses Hilbert spaces, and Hilbert spaces are the main subject of functional analysis. Differential geometry is the background for GR (and via Noether of QFT), and stochastics is the backbone of many specific fields in physics: Bose-Einstein, thermodynamics, plasma physics, etc. Not to mention that wave functions are interpreted as probabilities.
 
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Forever_searcher said:
I've heard that group theory is also very important. I am really confused because there are this many choices and all seem to have a connection with physics.
I must disagree with my esteemed colleague @fresh_42. There is a world of difference between physics and mathematics. Group Theory provides a good example. If you majored in mathematics, you would spend a lot of time on group theory before you got to many of the group theoretic ideas that are important in physics: Lie Groups and Group Representation Theorems etc. The priority for a physicist is to extract as much practical value from what mathematics provides as possible.

There is a thread about Group Theory for the physicist here:

https://www.physicsforums.com/threads/should-i-take-a-group-theory-course-before-qft.1005489/
 
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fresh_42 said:
Does geometry have to do with anything? What you need from topology is mainly metric spaces. The few necessary basics can be read on the side.

QFT uses Hilbert spaces, and Hilbert spaces are the main subject of functional analysis. Differential geometry is the background for GR (and via Noether of QFT), and stochastics is the backbone of many specific fields in physics: Bose-Einstein, thermodynamics, plasma physics, etc. Not to mention that wave functions are interpreted as probabilities.
Well, you can give a look at books called Geometry,Topology and Physics or some variation of them.
They hardly end in only mentioning metric spaces...
If he has the money and time he should take them both.
 
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Forever_searcher said:
I've heard that group theory is also very important. I am really confused because there are this many choices and all seem to have a connection with physics.
That depends on what you mean by group theory! What you need to know about groups from classical group theory is a handful of definitions. Wikipedia will do. The groups used in physics are linear algebraic groups and Lie groups. The former are matrix groups (linear algebra) and the latter analytic (differentiable) manifolds; neither are finite groups (finite fields as scalars of no interest to physics aside). (Classical) Group theory is about finite groups, field extensions, and Galois theory.

If a mathematician says metric, then he thinks about the triangle inequality and topology; if a physicist says metric, then he thinks about a ##2##-form and differential geometry.
 
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PeroK said:
I must disagree with my esteemed colleague @fresh_42. There is a world of difference between physics and mathematics.
And where did I state something different?
 
  • #11
fresh_42 said:
And where did I state something different?
fresh_42 said:
But in any case, the difference between mathematics and physics is in my opinion more of a linguistic one than of a content one.
 
  • #12
See my example above about metrics. Generator is an example, too, covariance another. The language is a different one. But I have never said something as if Galois theory was important for physicists. Your objection, however, could have been read this way. There is a language issue and in my opinion a severe one. I have never seen so many indices in such a density as on PF. Even numerical algorithms look pale in comparison.

My personal opinion is, that physicists are index fetishists. Mathematicians try to avoid them as long as possible.
 
  • #13
fresh_42 said:
See my example above about metrics. Generator is an example, too, covariance another. The language is a different one. But I have never said something as if Galois theory was important for physicists. Your objection, however, could have been read this way. There is a language issue and in my opinion a severe one. I have never seen so many indices in such a density as on PF. Even numerical algorithms look pale in comparison.

My personal opinion is, that physicists are index fetishists. Mathematicians try to avoid them as long as possible.
In my opinion mathematicians are index-free notation fetishists 😜
 

Related to Are mathematicians and physicists obsessed with index notation?

1. Are mathematicians and physicists the only ones who use index notation?

No, index notation is also commonly used in engineering and other scientific fields. It is a useful tool for representing and manipulating multi-dimensional quantities.

2. What is index notation and why is it important?

Index notation is a way of representing and manipulating multi-dimensional quantities using indices or subscripts. It is important because it allows for concise and efficient notation in complex mathematical and physical equations.

3. Is index notation necessary for understanding advanced mathematical and physical concepts?

While index notation is commonly used in advanced mathematics and physics, it is not necessary for understanding these concepts. However, it can greatly simplify and streamline calculations and representations.

4. Are there any drawbacks to using index notation?

One potential drawback of index notation is that it can be difficult to read and understand for those who are not familiar with it. Additionally, it may not be the most efficient notation for certain types of calculations.

5. Can index notation be used for all types of mathematical and physical equations?

Index notation is most commonly used for representing and manipulating multi-dimensional quantities, but it can also be used for other types of equations. However, there may be other notations that are more suitable for certain types of equations.

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