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

In summary, people generally omit some indices when writing out spinor indices, which makes the spinor transform as a vector.
  • #36
arivero said:
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.
 
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  • #37
BiGyElLoWhAt said:
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 … 🤭
 
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  • #38
Orodruin said:
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!
 
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  • #39
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.
 
  • #40
BiGyElLoWhAt said:
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.
 
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  • #41
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.
 
  • #42
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").
 
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  • #43
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
 
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  • #45
robphy said:
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.
 
  • #46
vanhees71 said:
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.
 
  • #47
BiGyElLoWhAt said:
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
 
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  • #48
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...
BiGyElLoWhAt said:
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

BiGyElLoWhAt said:
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
 
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  • #49
Orodruin said:
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}?
 
  • #50
kparchevsky said:
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).
 
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