Maxwell Tensor Identity [duplicate]Expanding electromagnetic field Lagrangian in terms of gauge fieldNoether current for the Yang-Mills-Higgs LagrangianRiemann tensor in 2d and 3dDerivation of the quadratic form of the Dirac equationEnergy-momentum tensor for dustDielectric tensor vs. conductivity tensor in (cold) plasmasExpanding electromagnetic field Lagrangian in terms of gauge fieldHow can I see where this formula for a general vertex factor comes from?Equation of Motion for non-linear sigma model (WZW)What is meant by the coupling term $g_munuT^munu$ in Supergravity?Electromagnetic energy stress tensor with non zero current

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Maxwell Tensor Identity [duplicate]


Expanding electromagnetic field Lagrangian in terms of gauge fieldNoether current for the Yang-Mills-Higgs LagrangianRiemann tensor in 2d and 3dDerivation of the quadratic form of the Dirac equationEnergy-momentum tensor for dustDielectric tensor vs. conductivity tensor in (cold) plasmasExpanding electromagnetic field Lagrangian in terms of gauge fieldHow can I see where this formula for a general vertex factor comes from?Equation of Motion for non-linear sigma model (WZW)What is meant by the coupling term $g_munuT^munu$ in Supergravity?Electromagnetic energy stress tensor with non zero current













1












$begingroup$



This question already has an answer here:



  • Expanding electromagnetic field Lagrangian in terms of gauge field

    1 answer



In Schawrtz, Page 116, formula 8.23, he seems to suggest that the square of the Maxwell tensor can be expanded out as follows:



$$-frac14F_mu nu^2=frac12A_musquare A_mu-frac12A_mupartial_mupartial_nuA_nu$$



where:



$$F_munu=partial_mu A_nu - partial_nuA_mu$$



For the life of me, I can't seem to derive this. I get close, but always with an extra unwanted term, or two.



Anyone have a hint on the best way to proceed?










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This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • 4




    $begingroup$
    Does Schwartz include an integral? If so, he might be integrating out certain terms to the boundary and setting them to zero.
    $endgroup$
    – Aditya
    12 hours ago










  • $begingroup$
    He does not, and I thought I had derived this in the past sans integral. I'll try that out, though. At the very least, I learn a new way of deriving this
    $endgroup$
    – EthanT
    12 hours ago






  • 2




    $begingroup$
    Possible duplicate of Expanding electromagnetic field Lagrangian in terms of gauge field
    $endgroup$
    – knzhou
    12 hours ago











  • $begingroup$
    Yeah, this was really easy keeping it under the integral of S. However, I thought there was a way to achieve the same thing, w/ just tensor manipulation. Maybe I am not remembering correctly, though
    $endgroup$
    – EthanT
    11 hours ago










  • $begingroup$
    Products of zeroth and second derivatives are not generally equal to products of first derivatives in any identity... What you need is a context where a derivative of a product is zero, as $d(x~dx)=dx~dx + x~d^2x.$ Getting the left hand side to vanish in this context might be possible with antisymmetry but looks non-trivial—maybe it amounts to a boundary term in some integral though?
    $endgroup$
    – CR Drost
    11 hours ago















1












$begingroup$



This question already has an answer here:



  • Expanding electromagnetic field Lagrangian in terms of gauge field

    1 answer



In Schawrtz, Page 116, formula 8.23, he seems to suggest that the square of the Maxwell tensor can be expanded out as follows:



$$-frac14F_mu nu^2=frac12A_musquare A_mu-frac12A_mupartial_mupartial_nuA_nu$$



where:



$$F_munu=partial_mu A_nu - partial_nuA_mu$$



For the life of me, I can't seem to derive this. I get close, but always with an extra unwanted term, or two.



Anyone have a hint on the best way to proceed?










share|cite|improve this question











$endgroup$



marked as duplicate by knzhou, John Rennie electromagnetism
Users with the  electromagnetism badge can single-handedly close electromagnetism questions as duplicates and reopen them as needed.

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7 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • 4




    $begingroup$
    Does Schwartz include an integral? If so, he might be integrating out certain terms to the boundary and setting them to zero.
    $endgroup$
    – Aditya
    12 hours ago










  • $begingroup$
    He does not, and I thought I had derived this in the past sans integral. I'll try that out, though. At the very least, I learn a new way of deriving this
    $endgroup$
    – EthanT
    12 hours ago






  • 2




    $begingroup$
    Possible duplicate of Expanding electromagnetic field Lagrangian in terms of gauge field
    $endgroup$
    – knzhou
    12 hours ago











  • $begingroup$
    Yeah, this was really easy keeping it under the integral of S. However, I thought there was a way to achieve the same thing, w/ just tensor manipulation. Maybe I am not remembering correctly, though
    $endgroup$
    – EthanT
    11 hours ago










  • $begingroup$
    Products of zeroth and second derivatives are not generally equal to products of first derivatives in any identity... What you need is a context where a derivative of a product is zero, as $d(x~dx)=dx~dx + x~d^2x.$ Getting the left hand side to vanish in this context might be possible with antisymmetry but looks non-trivial—maybe it amounts to a boundary term in some integral though?
    $endgroup$
    – CR Drost
    11 hours ago













1












1








1





$begingroup$



This question already has an answer here:



  • Expanding electromagnetic field Lagrangian in terms of gauge field

    1 answer



In Schawrtz, Page 116, formula 8.23, he seems to suggest that the square of the Maxwell tensor can be expanded out as follows:



$$-frac14F_mu nu^2=frac12A_musquare A_mu-frac12A_mupartial_mupartial_nuA_nu$$



where:



$$F_munu=partial_mu A_nu - partial_nuA_mu$$



For the life of me, I can't seem to derive this. I get close, but always with an extra unwanted term, or two.



Anyone have a hint on the best way to proceed?










share|cite|improve this question











$endgroup$





This question already has an answer here:



  • Expanding electromagnetic field Lagrangian in terms of gauge field

    1 answer



In Schawrtz, Page 116, formula 8.23, he seems to suggest that the square of the Maxwell tensor can be expanded out as follows:



$$-frac14F_mu nu^2=frac12A_musquare A_mu-frac12A_mupartial_mupartial_nuA_nu$$



where:



$$F_munu=partial_mu A_nu - partial_nuA_mu$$



For the life of me, I can't seem to derive this. I get close, but always with an extra unwanted term, or two.



Anyone have a hint on the best way to proceed?





This question already has an answer here:



  • Expanding electromagnetic field Lagrangian in terms of gauge field

    1 answer







homework-and-exercises electromagnetism lagrangian-formalism






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 11 hours ago









Qmechanic

106k121961222




106k121961222










asked 12 hours ago









EthanTEthanT

382110




382110




marked as duplicate by knzhou, John Rennie electromagnetism
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marked as duplicate by knzhou, John Rennie electromagnetism
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7 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









  • 4




    $begingroup$
    Does Schwartz include an integral? If so, he might be integrating out certain terms to the boundary and setting them to zero.
    $endgroup$
    – Aditya
    12 hours ago










  • $begingroup$
    He does not, and I thought I had derived this in the past sans integral. I'll try that out, though. At the very least, I learn a new way of deriving this
    $endgroup$
    – EthanT
    12 hours ago






  • 2




    $begingroup$
    Possible duplicate of Expanding electromagnetic field Lagrangian in terms of gauge field
    $endgroup$
    – knzhou
    12 hours ago











  • $begingroup$
    Yeah, this was really easy keeping it under the integral of S. However, I thought there was a way to achieve the same thing, w/ just tensor manipulation. Maybe I am not remembering correctly, though
    $endgroup$
    – EthanT
    11 hours ago










  • $begingroup$
    Products of zeroth and second derivatives are not generally equal to products of first derivatives in any identity... What you need is a context where a derivative of a product is zero, as $d(x~dx)=dx~dx + x~d^2x.$ Getting the left hand side to vanish in this context might be possible with antisymmetry but looks non-trivial—maybe it amounts to a boundary term in some integral though?
    $endgroup$
    – CR Drost
    11 hours ago












  • 4




    $begingroup$
    Does Schwartz include an integral? If so, he might be integrating out certain terms to the boundary and setting them to zero.
    $endgroup$
    – Aditya
    12 hours ago










  • $begingroup$
    He does not, and I thought I had derived this in the past sans integral. I'll try that out, though. At the very least, I learn a new way of deriving this
    $endgroup$
    – EthanT
    12 hours ago






  • 2




    $begingroup$
    Possible duplicate of Expanding electromagnetic field Lagrangian in terms of gauge field
    $endgroup$
    – knzhou
    12 hours ago











  • $begingroup$
    Yeah, this was really easy keeping it under the integral of S. However, I thought there was a way to achieve the same thing, w/ just tensor manipulation. Maybe I am not remembering correctly, though
    $endgroup$
    – EthanT
    11 hours ago










  • $begingroup$
    Products of zeroth and second derivatives are not generally equal to products of first derivatives in any identity... What you need is a context where a derivative of a product is zero, as $d(x~dx)=dx~dx + x~d^2x.$ Getting the left hand side to vanish in this context might be possible with antisymmetry but looks non-trivial—maybe it amounts to a boundary term in some integral though?
    $endgroup$
    – CR Drost
    11 hours ago







4




4




$begingroup$
Does Schwartz include an integral? If so, he might be integrating out certain terms to the boundary and setting them to zero.
$endgroup$
– Aditya
12 hours ago




$begingroup$
Does Schwartz include an integral? If so, he might be integrating out certain terms to the boundary and setting them to zero.
$endgroup$
– Aditya
12 hours ago












$begingroup$
He does not, and I thought I had derived this in the past sans integral. I'll try that out, though. At the very least, I learn a new way of deriving this
$endgroup$
– EthanT
12 hours ago




$begingroup$
He does not, and I thought I had derived this in the past sans integral. I'll try that out, though. At the very least, I learn a new way of deriving this
$endgroup$
– EthanT
12 hours ago




2




2




$begingroup$
Possible duplicate of Expanding electromagnetic field Lagrangian in terms of gauge field
$endgroup$
– knzhou
12 hours ago





$begingroup$
Possible duplicate of Expanding electromagnetic field Lagrangian in terms of gauge field
$endgroup$
– knzhou
12 hours ago













$begingroup$
Yeah, this was really easy keeping it under the integral of S. However, I thought there was a way to achieve the same thing, w/ just tensor manipulation. Maybe I am not remembering correctly, though
$endgroup$
– EthanT
11 hours ago




$begingroup$
Yeah, this was really easy keeping it under the integral of S. However, I thought there was a way to achieve the same thing, w/ just tensor manipulation. Maybe I am not remembering correctly, though
$endgroup$
– EthanT
11 hours ago












$begingroup$
Products of zeroth and second derivatives are not generally equal to products of first derivatives in any identity... What you need is a context where a derivative of a product is zero, as $d(x~dx)=dx~dx + x~d^2x.$ Getting the left hand side to vanish in this context might be possible with antisymmetry but looks non-trivial—maybe it amounts to a boundary term in some integral though?
$endgroup$
– CR Drost
11 hours ago




$begingroup$
Products of zeroth and second derivatives are not generally equal to products of first derivatives in any identity... What you need is a context where a derivative of a product is zero, as $d(x~dx)=dx~dx + x~d^2x.$ Getting the left hand side to vanish in this context might be possible with antisymmetry but looks non-trivial—maybe it amounts to a boundary term in some integral though?
$endgroup$
– CR Drost
11 hours ago










3 Answers
3






active

oldest

votes


















4












$begingroup$

Your expression is part of a Lagrangian. As the physics remains the same as long as the action remains the same, one can always do partial integration in the action integral over the Lagrangian to derive alternative Lagrangians describing the same physics.






share|cite|improve this answer








New contributor




Paul is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$




















    3












    $begingroup$

    Hint: Try introducing an integral to the expression so it becomes $$-frac14int F_munuF^munutextd^d x$$
    and take the total derivative terms to vanish at infinity. A much more careful argument can be made here in the presence of boundaries, but this should get you started.






    share|cite|improve this answer









    $endgroup$




















      0












      $begingroup$

      The relation as you state it does not hold. Only the space time integral of both hands of the equation is equal under suitable boundary conditions. So this would be an error.






      share|cite|improve this answer









      $endgroup$



















        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        4












        $begingroup$

        Your expression is part of a Lagrangian. As the physics remains the same as long as the action remains the same, one can always do partial integration in the action integral over the Lagrangian to derive alternative Lagrangians describing the same physics.






        share|cite|improve this answer








        New contributor




        Paul is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.






        $endgroup$

















          4












          $begingroup$

          Your expression is part of a Lagrangian. As the physics remains the same as long as the action remains the same, one can always do partial integration in the action integral over the Lagrangian to derive alternative Lagrangians describing the same physics.






          share|cite|improve this answer








          New contributor




          Paul is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.






          $endgroup$















            4












            4








            4





            $begingroup$

            Your expression is part of a Lagrangian. As the physics remains the same as long as the action remains the same, one can always do partial integration in the action integral over the Lagrangian to derive alternative Lagrangians describing the same physics.






            share|cite|improve this answer








            New contributor




            Paul is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.






            $endgroup$



            Your expression is part of a Lagrangian. As the physics remains the same as long as the action remains the same, one can always do partial integration in the action integral over the Lagrangian to derive alternative Lagrangians describing the same physics.







            share|cite|improve this answer








            New contributor




            Paul is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.









            share|cite|improve this answer



            share|cite|improve this answer






            New contributor




            Paul is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.









            answered 10 hours ago









            PaulPaul

            513




            513




            New contributor




            Paul is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.





            New contributor





            Paul is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.






            Paul is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.





















                3












                $begingroup$

                Hint: Try introducing an integral to the expression so it becomes $$-frac14int F_munuF^munutextd^d x$$
                and take the total derivative terms to vanish at infinity. A much more careful argument can be made here in the presence of boundaries, but this should get you started.






                share|cite|improve this answer









                $endgroup$

















                  3












                  $begingroup$

                  Hint: Try introducing an integral to the expression so it becomes $$-frac14int F_munuF^munutextd^d x$$
                  and take the total derivative terms to vanish at infinity. A much more careful argument can be made here in the presence of boundaries, but this should get you started.






                  share|cite|improve this answer









                  $endgroup$















                    3












                    3








                    3





                    $begingroup$

                    Hint: Try introducing an integral to the expression so it becomes $$-frac14int F_munuF^munutextd^d x$$
                    and take the total derivative terms to vanish at infinity. A much more careful argument can be made here in the presence of boundaries, but this should get you started.






                    share|cite|improve this answer









                    $endgroup$



                    Hint: Try introducing an integral to the expression so it becomes $$-frac14int F_munuF^munutextd^d x$$
                    and take the total derivative terms to vanish at infinity. A much more careful argument can be made here in the presence of boundaries, but this should get you started.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 12 hours ago









                    AdityaAditya

                    354113




                    354113





















                        0












                        $begingroup$

                        The relation as you state it does not hold. Only the space time integral of both hands of the equation is equal under suitable boundary conditions. So this would be an error.






                        share|cite|improve this answer









                        $endgroup$

















                          0












                          $begingroup$

                          The relation as you state it does not hold. Only the space time integral of both hands of the equation is equal under suitable boundary conditions. So this would be an error.






                          share|cite|improve this answer









                          $endgroup$















                            0












                            0








                            0





                            $begingroup$

                            The relation as you state it does not hold. Only the space time integral of both hands of the equation is equal under suitable boundary conditions. So this would be an error.






                            share|cite|improve this answer









                            $endgroup$



                            The relation as you state it does not hold. Only the space time integral of both hands of the equation is equal under suitable boundary conditions. So this would be an error.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 10 hours ago









                            my2ctsmy2cts

                            5,5692718




                            5,5692718













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