# What is the basic math to understand particle physics?

• jonjacson
In summary, to understand books discussing quark-antiquark pairs, the Higgs field, and the Standard Model, you need a good knowledge of everything needed in the QM1 lecture, including Hilbert spaces and operators. It also helps to be familiar with the relativistic formulation of classical electrodynamics and a bit about the representation theory of groups. A good introductory textbook is M. D. Schwartz's "Quantum field theory and the Standard Model", while more advanced books include S. Weinberg's "The quantum theory of fields" (3 volumes). Additional topics that may be necessary include group theory, tensor mathematics, and the notion of a Hilbert space. A basic understanding of statistics and experimental methods may also be helpful.
jonjacson
Hello folks.

What math do I need to understand books talking about quark-antiquark pairs, the Higgs field, and the Standard Model?

You need a good knowledge of everything needed in the QM1 lecture (non-relativistic quantum mechanics), including Hilbert spaces, operators on Hilbert spaces. In addition it helps to be familiar with the relativistic formulation of classical electrodynamics and a bit about the representation theory of groups, but that should be covered also in a good textbook on relativistic QFT as far as it is needed to understand the underlying fundamentals concerning the Poincare group. A good introductory textbook is

M. D. Schwartz, Quantum field theory and the Standard Model, Cambridge University Press, Cambridge, New York, 2014.

For the finer details at the more advanced level, the best books are the three volumes by Weinberg:

S. Weinberg, The quantum theory of fields, 3 vols, Cambridge University Press.

jonjacson
Peter Woit recently discussed the subject in the context of a paper by Chad Orzel which he links to. http://www.math.columbia.edu/~woit/wordpress/?p=8940

The meat of the answer is as follows:

Chad Orzel has a very sensible piece at Forbes, headlined What Math Do You Need For Physics? It Depends, which addresses the question of what math a physicist like him (experimental AMO physics) really needs. I’m glad to see that he emphasizes the same basic courses my department offers aimed at non-majors:

• Multivariable calculus
• Differential equations
• Linear algebra
together with statistics (which here at Columbia is handled by a separate department). He disses complex analysis, for reasons that I can understand. That’s a beautiful subject, and the results you can get out of analytic functions and contour integration are often unexpected and seemingly magic, but they’re not of as general use as the other subjects.

One subject he doesn’t mention that I would argue for is Fourier Analysis, which is the class I’ll be teaching next semester. That’s an incredibly useful as well as profound subject which every physicist should know, but it is true some of its basics often gets taught in other courses (for example in ode courses as a method for solving differential equations).

For what it is worth, I studied Fourier Analysis in Differential Equations, and there are at least a few topics that I think are necessary which he doesn't discuss (perhaps due to the experimentalist focus):

* Group theory (often as part of a class on abstract algebra);
* Tensor mathematics (which is often omitted from all of the courses listed above); and
* the notion of a Hilbert space, which is also often omitted from an introductory differential equations class.

Most multivariable calculus classes would discuss path integrals, but if they don't you absolutely need to understand those as well.

One also needs to learn physics notation for concepts that often have equivalents in mathematics, but are usually notated differently (e.g. bra ket notation), and physics shorthand abbreviations (e.g. writing masses as 10 TeV rather than 10 TeV/c^2 even when the latter is meant).

Last edited:
jonjacson
jonjacson
it depends what kind of article you are reading about those stuff... for example the Standard Model, Higgs field don't really need a lot of maths to follow... just being used to Lagrangians (and techniques associated with them) and a basic idea of group theory and special relativity you can easily deal with several of the SM or Higgs aspects.
For several calculations (related to propagators and stuff) you'll need complex analysis.
Now the deeper you want to get, the more understanding you must have on maths (because you look at abstractness). Maths is like a tool, the better you understand the tools you are working with, the better your job will be done. Afterall maths is not something you can say "I learned it, so I can move to that subject"... I believe they only build your way of thinking, so that when you come across new mathematical challenges you'll be able to think right. So from that aspect even calculus helps.

ohwilleke said:
and physics shorthand abbreviations (e.g. writing masses as 10 TeV rather than 10 TeV/c^2 even when the latter is meant).
That is not an abbreviation. Particle physicists often work with unit systems where c=1. Those two expressions are exactly the same.
BvU said:
(But in particle physics there are also experimental physicists, machine physicists, computer wizards etc etc)
Exactly. For experimental particle physicists, statistics is much more important, while the mathematics of the standard model is rarely important.

vanhees71
mfb said:
For experimental particle physicists, statistics is much more important
I know a little bit of statistics and even that little bit helped a lot in understanding how physics works. So I think theoretical physicists should at least be familiar with the statistical methods, or other experimental methods, that experimental physicists use.

vanhees71
If they are in contact with experimental physicists, sure. Pure model-building or attempts to combine QFT and gravity is usually far away from experiments, everything else has some contact to experimental physics.

jonjacson said:
Hello folks.

What math do I need to understand books talking about quark-antiquark pairs, the Higgs field, and the Standard Model?
The most important is to have a good grasp of group theory. If you want to do actual calculations of physical processes, complex analysis and variational calculus are also important.

jonjacson
This thread has an underlying, implicit fallacy that one can simply pick up bit and pieces of something, and be sufficient to understand a particular subject.

The same way one can't simply study physics in bits and pieces, the same thing applies to mathematics. As Mary Boas had stated in her book, sometime a physics major needs more math than even a math major! You need to study the same set of mathematics that is needed by any physics major. There is no shortcut around this. If you pick up Boas's text "Mathematical Methods in the Physical Science", that's the starting point of all the different topics in mathematics that one will need for ANY subject area in physics.

Only after you've mastered this can you diverge into more in-depth and advanced topics in mathematics that certain areas of physics may require.

Zz.

In principle I agree; a certain standard repertoire in mathematics is mandatory. In the German curriculum at universities that's Analysis 1-4, Linear Algebra, Functional Analysis. However one should also encourage an interested student just to start to read an introductory textbook and then fill the math gaps when necessary with additional math books. I can't say anything about Boas since I don't know this book very well.

How could I forget this comic (click on the panels to continue)?

ZapperZ said:
This thread has an underlying, implicit fallacy that one can simply pick up bit and pieces of something, and be sufficient to understand a particular subject.

The same way one can't simply study physics in bits and pieces, the same thing applies to mathematics. As Mary Boas had stated in her book, sometime a physics major needs more math than even a math major! You need to study the same set of mathematics that is needed by any physics major. There is no shortcut around this. If you pick up Boas's text "Mathematical Methods in the Physical Science", that's the starting point of all the different topics in mathematics that one will need for ANY subject area in physics.

Only after you've mastered this can you diverge into more in-depth and advanced topics in mathematics that certain areas of physics may require.

Zz.

That is not true.

There is not any fallacy, and I don't expect to learn something extremely complex "easily".

But you must agree with me that the standard model uses some mathematical methods, and I am absolutely sure I am correct. Don't they? Is particle physics a phylosophical thing without formalism? I guess the answer is NO. They use or differential equations, or algebra or whatever, anyway they use very precise methods, I am sure about that. And if I pick up a book and I start reading, I want to be sure that the formalism is completely clear to me. Because without a good math background reading a physics book is useless.

OK, judging the fact that SM (QFT) is (most of the times) a grad-course, I suppose you should be confident with the following:
1. linear algebra (matrices, bases, eigenstates/values)
2. calculus (integrals, sequencies, derivatives)
3. group theory (mainly continuous groups, algebras etc)
4. complex analysis
5. be used with special relativity (I wouldn't say differential geometry, however it CAN help)
6. be used to Lagrangians/Hamiltonians
and finally be able to do some taylor expansions (but they are everywhere)
And you'll be able to go through a lot of textbooks with those. In some cases topology is also important.

jonjacson said:
That is not true.

There is not any fallacy, and I don't expect to learn something extremely complex "easily".

But you must agree with me that the standard model uses some mathematical methods, and I am absolutely sure I am correct. Don't they? Is particle physics a phylosophical thing without formalism? I guess the answer is NO. They use or differential equations, or algebra or whatever, anyway they use very precise methods, I am sure about that. And if I pick up a book and I start reading, I want to be sure that the formalism is completely clear to me. Because without a good math background reading a physics book is useless.

And that is what I've been trying to tell you! I don't know what you read in my post, but this is what you need to do to learn the physics - MATHEMATICS. ChrisVer has listed practically most of the topics covered in Boas's text. You have to learn the basic coverage of mathematics that any typical physics undergraduate will have to know. You don't learn just bits and pieces of math, the way you seem to be asking.

Zz.

mfb said:
That is not an abbreviation. Particle physicists often work with unit systems where c=1. Those two expressions are exactly the same.Exactly. For experimental particle physicists, statistics is much more important, while the mathematics of the standard model is rarely important.

Abbreviation, notation convention; tomato, tomahto. The point is that someone not familiar with the conventional way of writing this would see inconsistent units which are confusing, unless you are aware of some unstated assumptions about what that means.

OCR
ZapperZ said:
And that is what I've been trying to tell you! I don't know what you read in my post, but this is what you need to do to learn the physics - MATHEMATICS. ChrisVer has listed practically most of the topics covered in Boas's text. You have to learn the basic coverage of mathematics that any typical physics undergraduate will have to know. You don't learn just bits and pieces of math, the way you seem to be asking.

Zz.
If that is what you understood maybe I didn't express it in the better way, but I don't mean bits or pieces, I meant the basic tools.

jonjacson said:
If that is what you understood maybe I didn't express it in the better way, but I don't mean bits or pieces, I meant the basic tools.

Then, as I've said numerous times on here, pick up Mary Boas's text "Mathematical Methods in the Physical Sciences". That should answer this question.

Zz.

jonjacson
For the basics I like Byron and Fuller, Mathematics of Classical and Quantum Physics. They start with vectors in classical physics and eventually arrive at Hilbert space and group theory. They show how group theory is applied to physical problems.

ohwilleke and jonjacson

## What is particle physics?

Particle physics is the branch of physics that studies the fundamental building blocks of matter and the forces that govern their interactions.

## What is a particle?

In particle physics, a particle is a tiny unit of matter that cannot be broken down into smaller components. This includes both elementary particles, such as electrons and quarks, and composite particles, such as protons and neutrons.

## What is the basic math used in particle physics?

The basic math used in particle physics includes calculus, linear algebra, and statistics. These mathematical tools are used to describe and analyze the behavior of particles and their interactions.

## What is the Standard Model?

The Standard Model is the current theoretical framework that describes the fundamental particles and their interactions. It is based on the principles of quantum mechanics and special relativity.

## What is a particle accelerator?

A particle accelerator is a scientific instrument that uses electromagnetic fields to accelerate and collide particles at high energies. These collisions can reveal new particles and interactions, providing insights into the fundamental nature of matter.

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