# Which maths should I need for QFT?

## Main Question or Discussion Point

Thanks for all the help on my QFT question I posted earlier. I have a new question I am very much interested in understanding QFT at a much higher level without going back to college and was wondering which mathematics' in particular are needed to really understand QFT. I did attend Ohio State University for a year Honors level physics and did well with Calculus 1,2,and 3 b average let me know your thoughts. Thanks much.
Sussan

George Jones
Staff Emeritus
Gold Member
In terms of mathematics, I would say a little complex analysis (e.g., contour integration), and the physicists' version of operators on Hilbert spaces.

What about physics prerequisites? Have you some seen stuff on Lagrangians and Hamiltonians? These are used to formulate, and do calculations in, quantum field theories. And quantum mechanics? It would helpful to know quantum mechanics at the level of "Introduction to Quantum Mechanics" by Griffiths.

Which maths should I need for QFT.

In terms of mathematics, I would say a little complex analysis (e.g., contour integration), and the physicists' version of operators on Hilbert spaces.

What about physics prerequisites? Have you some seen stuff on Lagrangians and Hamiltonians? These are used to formulate, and do calculations in, quantum field theories. And quantum mechanics? It would helpful to know quantum mechanics at the level of "Introduction to Quantum Mechanics" by Griffiths.
Hi,

Yes I am pretty much up on QM and a working knowledge of Feynman diagrams. As for Lagrangians and Hamiltonians not so much I have 'heard" of them. My IQ is 135 so "can" if need be learn fairly quickly.
Sussan

MathematicalPhysicist
Gold Member
In terms of mathematics, I would say a little complex analysis (e.g., contour integration), and the physicists' version of operators on Hilbert spaces.

What about physics prerequisites? Have you some seen stuff on Lagrangians and Hamiltonians? These are used to formulate, and do calculations in, quantum field theories. And quantum mechanics? It would helpful to know quantum mechanics at the level of "Introduction to Quantum Mechanics" by Griffiths.
The Physicists' version differs from the mathematicians'?!

Maybe different notation, but it's the same mathematics.

jtbell
Mentor
it's the same mathematics.
...but sloppier!

George Jones
Staff Emeritus
Gold Member
The Physicists' version differs from the mathematicians'
Most certainly. The mathematics of quantum theory is based on functional analysis, and I STRONGLY recommend that a person learning quantum theory not get bogged down with stuff like domains of self-adjointness of unbounded operators.

jgens
Gold Member
The Physicists' version differs from the mathematicians'?!
Hopefully someone more knowledgeable can chime in and answer this definitively, but my impression is that most operators appearing in QM and QFT are unbounded. On the other hand, while mathematicians certainly study unbounded operators, unless you really dig into the analysis literature most texts deal only with bounded operators. So aside from notation there are genuine differences in emphasis.

George Jones
Staff Emeritus
Gold Member
my impression is that most operators appearing in QM and QFT are unbounded.
If two self-adjoint operators satisfy the canonical commutation relation of quantum theory, then at least one of the operators must be unbounded.

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bhobba
Mentor
...but sloppier!
Exactly - its the difference between Dirac's Principles of QM and Von-Neumanns mathematical foundations.

Its only of recent times have Dirac's methods been given the necessary rigor via Rigged Hilbert Spaces.

To the OP you need to learn QM first.

My suggested sequence is:

Susskinds Theoretical Minimum to get you up to speed of Hamiltonians and such:
https://www.amazon.com/dp/046502811X/?tag=pfamazon01-20

The Quantum Mechanics version:
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

Quantum Mechanics Dymystified:
https://www.amazon.com/dp/0071765638/?tag=pfamazon01-20

Then you need complex variables
http://rutherglen.science.mq.edu.au/wchen/lnicafolder/lnica.html

Quantum Field Theory Demystified
https://www.amazon.com/dp/0071543821/?tag=pfamazon01-20

Quantum Field theory In A Nutshell:
https://www.amazon.com/dp/0691140340/?tag=pfamazon01-20

Then you MIGHT be in a position to study a 'proper' textbook like Banks:
https://www.amazon.com/dp/0521850827/?tag=pfamazon01-20

BTW - forget this IQ 135 stuff. I have an IQ over that as well as an honors degree in math and QFT is HARD - really HARD.

Thanks
Bill

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WannabeNewton
Aha! Thank you very much for this link. I've been looking for a resource on contour integration for ages that isn't Boas or Arfken and these notes seem to fit the bill perfectly. Most of the other resources I've come across waste my time with proofs that I really don't care about.

OP, check out the following notes and see for yourself how much of it you can and can't understand in order to gauge what you're missing: http://www.damtp.cam.ac.uk/user/dt281/qft/qft.pdf

Exactly - its the difference between Dirac's Principles of QM and Von-Neumanns mathematical foundations.

Its only of recent times have Dirac's methods been given the necessary rigor via Rigged Hilbert Spaces.

To the OP you need to learn QM first.

My suggested sequence is:

Susskinds Theoretical Minimum to get you up to speed of Hamiltonians and such:
https://www.amazon.com/dp/046502811X/?tag=pfamazon01-20

The Quantum Mechanics version:
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

Quantum Mechanics Dymystified:
https://www.amazon.com/dp/0071765638/?tag=pfamazon01-20

Then you need complex variables
http://rutherglen.science.mq.edu.au/wchen/lnicafolder/lnica.html

Quantum Field Theory Demystified
https://www.amazon.com/dp/0071543821/?tag=pfamazon01-20

Quantum Field theory In A Nutshell:
https://www.amazon.com/dp/0691140340/?tag=pfamazon01-20

Then you MIGHT be in a position to study a 'proper' textbook like Banks:
https://www.amazon.com/dp/0521850827/?tag=pfamazon01-20

BTW - forget this IQ 135 stuff. I have an IQ over that as well as an honors degree in math and QFT is HARD - really HARD.

Thanks
Bill
Thanks a bunch Bill,

I will take every item you listed and study what I may try first and go from there. I do agree IQs though important to some extent are not everything.
In the two slit experiment I kind of "felt" the answer was that being the objects were so small that the results were from the "looking" thus changing the outcome. Also had this "gut" feeling the "particles" were just waves not solid in any way "real" as it were just wave like functions that appear particle only in appearance and pretty much everything "real" was simply waves interacting in fashions that brought about reality as we know it. Then I heard about superposition and was surprised that this as well made total sense again and now have this "feeling" all "things" are in superposition again might be totally wrong but just feel this is "it".
Sussan

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Learn math, mainly complex analysis and linear algebra by using Hassani. Learn Lagrangian and Hamiltonian mechanics using Landau. Learn Quantum Mechanics using Shankar or Sakurai. Learn special relativity using Schutz. Then you'll be ready for QFT. Enjoy the ride!

Learn math, mainly complex analysis and linear algebra by using Hassani. Learn Lagrangian and Hamiltonian mechanics using Landau. Learn Quantum Mechanics using Shankar or Sakurai. Learn special relativity using Schutz. Then you'll be ready for QFT. Enjoy the ride!
THANKS!! That's more than a mouth full, however will give it a go. Talk about cutting to the chase. Again thanks a load.
Sussan

ps Will this book
Mathematical Methods in the Physical Sciences by Mary L. Boas work on it's own?

THANKS!! That's more than a mouth full, however will give it a go. Talk about cutting to the chase. Again thanks a load.
Sussan

ps Will this book
Mathematical Methods in the Physical Sciences by Mary L. Boas work on it's own?
I haven't read that book but it should be alright as long as

1. It covers complex analysis, including contour integrals
2. Fourier series and transforms
3. Greens functions
4. Linear algebra, with emphasis on inner product spaces.

Hassani has a pretty decent coverage of these topics, and does so using QM and QFT notation.

Also, the choice of Landau for Mechanics may be a little difficult for a first go, but I don't know of any good undergraduate level text covering Lagrangian and Hamiltonians that also has a good coverage of symmetries, conservation laws and the Noether procedure, i.e the tools that are useful for QFT. Maybe Hand & Finch. Taylor is a decent and easy going text. It doesn't really emphasize the previously stated topics, but may still be worth going through.

I haven't read that book but it should be alright as long as

1. It covers complex analysis, including contour integrals
2. Fourier series and transforms
3. Greens functions
4. Linear algebra, with emphasis on inner product spaces.

Hassani has a pretty decent coverage of these topics, and does so using QM and QFT notation.

Also, the choice of Landau for Mechanics may be a little difficult for a first go, but I don't know of any good undergraduate level text covering Lagrangian and Hamiltonians that also has a good coverage of symmetries, conservation laws and the Noether procedure, i.e the tools that are useful for QFT. Maybe Hand & Finch. Taylor is a decent and easy going text. It doesn't really emphasize the previously stated topics, but may still be worth going through.
Thanks a bunch again!!
My dad was or should I say still is though he has been gone a long, long, time a Nobel Laureate and Physics was his "thing" wish I would of asked all these questions when he was alive (only if one knows what the future would bring ). Thanks Sussan