How can you tell the spin of a particle by looking at the Lagrangian?

[vanhees71]

You look how the fields transforms under Lorentz Transformations.

Scalars - spin 0, transforms as scalars.
Fermions - spin ½, transforms as spinors (½,0) or (0,½) representation (depending if you have a "left-handed" or a "right-handed" spinor).
Vectors - spin 1, transforms as vectors (½,½) representation.

http://sharif.edu/~sadooghi/QFT-I-96-97-2/LorentzPoincareMaciejko.pdf

Note, this is applicable to classical fields as well.

Try to get one book and read it from start to finish instead of scattered notes etc. I can recommend the book by Mark Srednicki, you can get a free draft version on his homepage. The book is divided into several sections depending on spin and discusses at some lenght what is "special" about the various cases in terms of their Lorentz Transformations http://web.physics.ucsb.edu/~mark/qft.html

Concerning the representation theory of the Poincare group, I think you should refer to Weinberg, QT of fields vol. 1, which is very detailed and complete.

A very good somewhat simpler treatment, restricted to the special cases needed for the standard model, i.e., spin 0, 1/2, and 1, can be found in Quantum Field Theory Lectures by Sidney Coleman. This book also nicely works out in detail, why the first-quantization approach doesn't work in relativistic QT.

 

[malawi_glenn]

Quantum Field Theory Lectures by Sidney Coleman

I have still failed to receive my copy :(

 

[vanhees71]

Don't order it from World Scientific directly. It's awful. I never got my copy from them and just had to write several times for getting my payment back. Then I ordered it from Amazon, and also for them it took pretty long, but it finally arrived, and they take the money from the credit card only after they really shipped it!

 

[BiGyElLoWhAt]

? replacing operators?

Should read "replacing with operators". So $p \to \nabla$ and $E \to \frac{\partial}{\partial t}$ or $E^2 - p^2 \to \Box^2$
$\frac{p^2}{2m} + V = E \to \frac{1}{2m}\nabla^2 + V = \partial_t$
$E^2 - p^2 = m^2 \to \Box^2 = m^2$

I do see though that those are just wave equations.

Edit
I think I have (since last night) answered my own question.

 

[arivero]

It will be much clearer if you wrote down the spinor indices. See Srednicki ch 36.

And for a student, restoration of hbars is also a helpful practice.

 

[BiGyElLoWhAt]

And for a student, restoration of hbars is also a helpful practice.

I'm fairly confident in my unit analysis. I've been working with c=1 for a couple years now.

 

[Orodruin]

I'm fairly confident in my unit analysis. I've been working with c=1 for a couple years now.

There comes a moment in a professor’s life where he is so used to c=1 that he starts using c to represent other things in his relativity lectures …

 

[malawi_glenn]

There comes a moment in a professor’s life where he is so used to c=1 that he starts using c to represent other things in his relativity lectures …

Pure evil!

 

[BiGyElLoWhAt]

Also, @malawi_glenn , I looked into the "replacing with operators" thing I was rambling about to try to be more precise. Apparently what I'm talking about is called first quantization. That term never got mentioned in my Quantum class, and I only found out what it meant yesterday. Unfortunately.

 

[Orodruin]

Also, @malawi_glenn , I looked into the "replacing with operators" thing I was rambling about to try to be more precise. Apparently what I'm talking about is called first quantization. That term never got mentioned in my Quantum class, and I only found out what it meant yesterday. Unfortunately.

That’s probably because second quantization is rarely mentioned in introductory quantum physics classes.

 

[BiGyElLoWhAt]

We definitely covered the concept in a round-about way. We did the (probably standard) photoelectric effect experiment, and talked about how light behaved like a particle. Granted, I think this was modern lab, not QM. There wasn't any real quantum done in modern, only the very basic principles were covered.

 

[vanhees71]

Light never behaves like a particle. It's always behaving like a (quantized) electromagnetic field. There is no consistent relativistic quantum theory of interacting "quantum stuff" in first-quantization formulation. The only way we know is to formulate it as a local relativistic quantum-field theory. That's also plausible from the empirical evidence: Whenever you consider scatterings of particles at relativistic energies, with some probability you create new particles and/or destroy the incoming particles, and that cannot be described by first-quantization QM but only with a QFT ("second quantization").

 

[robphy]

possibly useful:

https://projecteuclid.org/journals/...pin-in-curved-space-times/cmp/1104254357.full
Massive fields of arbitrary spin in curved space-times
Reinhard Illge
Comm. Math. Phys. 158(3): 433-457 (1993).

https://articles.adsabs.harvard.edu/pdf/1988AN....309..253I
On Spinor Field Equations of Buchdahl and Wunsch
Illge, R.
Astronomische Nachrichten, Vol. 309, Issue 4, p. 253, 1988

https://royalsocietypublishing.org/doi/10.1098/rspa.1939.0140
On relativistic wave equations for particles of arbitrary spin in an electromagnetic field
M. Fierz and Wolfgang Ernst Pauli
Published:28 November 1939
https://doi.org/10.1098/rspa.1939.0140

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1936.0111
Relativistic wave equations
Paul Adrien Maurice Dirac
01 July 1936
Proc Roy Soc A - Volume 155 Issue 886
https://doi.org/10.1098/rspa.1936.0111

 

[vanhees71]

Very important is this one:

E. P. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, Annals of Mathematics 40, 149 (1939),
https://doi.org/10.1016/0920-5632(89)90402-7

 

[BiGyElLoWhAt]

possibly useful:

This is huge, actually. I'm trying to follow through the development pseudo-historically, so Pauli and Dirac original papers are perfect for this. Also the generally relativistic and spinor paper, but especially the Dirac and Pauli papers.

 

[BiGyElLoWhAt]

Very important is this one:

For some reason the name Wigner seems familiar, but wasn't someone I was immediately aware of. From his wiki, he seems like someone I should probably be more familiar with.

 

[malawi_glenn]

For some reason the name Wigner seems familiar, but wasn't someone I was immediately aware of. From his wiki, he seems like someone I should probably be more familiar with.

Perhaps Breit-Wigner distribution, and/or Wigner-Eckart theorem

 

[robphy]

Possibly interesting...

Geroch's Quantum Field Theory 1971 notes has sections on spinors
http://home.uchicago.edu/~geroch/Course Notes
(Some of these were cleaned-up and published by http://www.minkowskiinstitute.org/mip/books/ln.html )but maybe off track...

This is huge, actually. I'm trying to follow through the development pseudo-historically, so Pauli and Dirac original papers are perfect for this. Also the generally relativistic and spinor paper, but especially the Dirac and Pauli papers.

DIrac: https://diginole.lib.fsu.edu/island...rge_ms:Dirac,\ Paul,\ 1902\-1984\ \(Creator\)
Pauli: https://cds.cern.ch/collection/Pauli Archives?ln=en

For some reason the name Wigner seems familiar, but wasn't someone I was immediately aware of. From his wiki, he seems like someone I should probably be more familiar with.

Wigner and Dirac are also brothers-in-law.
https://ysfine.com/dirac/wigsis.html

https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
https://www.google.com/search?q="th...ics+in+the+natural+sciences"+by+eugene+wigner

 

[kparchevsky]

View attachment 317504

and there are still some suppressed indices …

In the very first term (and in the whole expression), don't indices mu and nu should be on different levels like -1/2 d_\nu g_\mu^a d^\nu g^{\mu a}?

 

[Orodruin]

In the very first term (and in the whole expression), don't indices mu and nu should be on different levels like -1/2 d_\nu g_\mu^a d^\nu g^{\mu a}?

Yes. There are however some authors that consider it so basic that they state that is obvious and subtextual. I know that Schwartz does this in his QFT book for example (it is explicitly stated in the introduction).