Why do the $Gamma$ matrices behave like vectors and tensors in the spinor representation of SO groups?
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One of the things that confuse me most when I study group theory and quantum field theory is that I constantly run into the situations where $psi C Gamma_Mchi$ are treated like vectors, $psi CGamma_MGamma_N Gamma_N chi$ are treated like tensors, where $psi$ and $chi$ are spinors in the SO group's spinor representation.
But I never have been formally introduced that tensors and vectors in this way. I currently feels that maybe many traditional vectors (like what we learned since high school) can be represented in this way by the contraction with $gamma$ matrices?
So back to the topic, what does it mean when some author says that "they are like vectors and tensors"? Would these objects completely satisfy the definition of tensors? How do we manipulate the transformations of them; would they transform like tensors?
P.S.: I'm a physics student, so I would really appreciate it if your answer not heavily relies on math terms. Thank you in advance.
group-theory lie-algebras quantum-field-theory spin-geometry
edited Mar 3 at 23:22
Andrews
1,2812422
asked Dec 10 '18 at 19:50
CollinCollin
1447
$endgroup$
add a comment |
$begingroup$
One of the things that confuse me most when I study group theory and quantum field theory is that I constantly run into the situations where $psi C Gamma_Mchi$ are treated like vectors, $psi CGamma_MGamma_N Gamma_N chi$ are treated like tensors, where $psi$ and $chi$ are spinors in the SO group's spinor representation.
But I never have been formally introduced that tensors and vectors in this way. I currently feels that maybe many traditional vectors (like what we learned since high school) can be represented in this way by the contraction with $gamma$ matrices?
So back to the topic, what does it mean when some author says that "they are like vectors and tensors"? Would these objects completely satisfy the definition of tensors? How do we manipulate the transformations of them; would they transform like tensors?
P.S.: I'm a physics student, so I would really appreciate it if your answer not heavily relies on math terms. Thank you in advance.
group-theory lie-algebras quantum-field-theory spin-geometry
edited Mar 3 at 23:22
Andrews
1,2812422
asked Dec 10 '18 at 19:50
CollinCollin
1447
$endgroup$
$begingroup$
Thank you I will try the physics SE also!
$endgroup$
– Collin
Dec 10 '18 at 19:59
$begingroup$
Unless it's something super mathy they'll probably have an answer for you. I wouldn't recommend cross posting though.
$endgroup$
– Matt Samuel
Dec 10 '18 at 20:03
add a comment |
$begingroup$
One of the things that confuse me most when I study group theory and quantum field theory is that I constantly run into the situations where $psi C Gamma_Mchi$ are treated like vectors, $psi CGamma_MGamma_N Gamma_N chi$ are treated like tensors, where $psi$ and $chi$ are spinors in the SO group's spinor representation.
But I never have been formally introduced that tensors and vectors in this way. I currently feels that maybe many traditional vectors (like what we learned since high school) can be represented in this way by the contraction with $gamma$ matrices?
So back to the topic, what does it mean when some author says that "they are like vectors and tensors"? Would these objects completely satisfy the definition of tensors? How do we manipulate the transformations of them; would they transform like tensors?
P.S.: I'm a physics student, so I would really appreciate it if your answer not heavily relies on math terms. Thank you in advance.
group-theory lie-algebras quantum-field-theory spin-geometry
edited Mar 3 at 23:22
Andrews
1,2812422
asked Dec 10 '18 at 19:50
CollinCollin
1447
$endgroup$
One of the things that confuse me most when I study group theory and quantum field theory is that I constantly run into the situations where $psi C Gamma_Mchi$ are treated like vectors, $psi CGamma_MGamma_N Gamma_N chi$ are treated like tensors, where $psi$ and $chi$ are spinors in the SO group's spinor representation.
But I never have been formally introduced that tensors and vectors in this way. I currently feels that maybe many traditional vectors (like what we learned since high school) can be represented in this way by the contraction with $gamma$ matrices?
So back to the topic, what does it mean when some author says that "they are like vectors and tensors"? Would these objects completely satisfy the definition of tensors? How do we manipulate the transformations of them; would they transform like tensors?
P.S.: I'm a physics student, so I would really appreciate it if your answer not heavily relies on math terms. Thank you in advance.
group-theory lie-algebras quantum-field-theory spin-geometry
group-theory lie-algebras quantum-field-theory spin-geometry
edited Mar 3 at 23:22
Andrews
1,2812422
asked Dec 10 '18 at 19:50
CollinCollin
1447
edited Mar 3 at 23:22
Andrews
1,2812422
asked Dec 10 '18 at 19:50
CollinCollin
1447
edited Mar 3 at 23:22
Andrews
1,2812422
edited Mar 3 at 23:22
Andrews
1,2812422
edited Mar 3 at 23:22
Andrews
1,2812422
1,2812422
asked Dec 10 '18 at 19:50
CollinCollin
1447
asked Dec 10 '18 at 19:50
CollinCollin
1447
asked Dec 10 '18 at 19:50
CollinCollin
1447
1447
$begingroup$
Thank you I will try the physics SE also!
$endgroup$
– Collin
Dec 10 '18 at 19:59
$begingroup$
Unless it's something super mathy they'll probably have an answer for you. I wouldn't recommend cross posting though.
$endgroup$
– Matt Samuel
Dec 10 '18 at 20:03
add a comment |
$begingroup$
Thank you I will try the physics SE also!
$endgroup$
– Collin
Dec 10 '18 at 19:59
$begingroup$
Unless it's something super mathy they'll probably have an answer for you. I wouldn't recommend cross posting though.
$endgroup$
– Matt Samuel
Dec 10 '18 at 20:03
$begingroup$
Thank you I will try the physics SE also!
$endgroup$
– Collin
Dec 10 '18 at 19:59
$begingroup$
Thank you I will try the physics SE also!
$endgroup$
– Collin
Dec 10 '18 at 19:59
$begingroup$
Unless it's something super mathy they'll probably have an answer for you. I wouldn't recommend cross posting though.
$endgroup$
– Matt Samuel
Dec 10 '18 at 20:03
$begingroup$
Unless it's something super mathy they'll probably have an answer for you. I wouldn't recommend cross posting though.
$endgroup$
– Matt Samuel
Dec 10 '18 at 20:03
add a comment |
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$begingroup$
Thank you I will try the physics SE also!
$endgroup$
– Collin
Dec 10 '18 at 19:59
$begingroup$
Unless it's something super mathy they'll probably have an answer for you. I wouldn't recommend cross posting though.
$endgroup$
– Matt Samuel
Dec 10 '18 at 20:03