- #1
- 13,388
- 4,035
Since there's another thread on the same subject in the GR forum, but on this forum about 8 months ago an interesting discussion on the same subject took place, https://www.physicsforums.com/showthread.php?t=114620, i want to draw everyone's attention on the post \#24 in that thread in which the author of that post attempts to give a proof of the fact that for an arbitrary classical field theory which admits rigid continuous symmetries, the charges associated with this symmetries are Lorentz invariant (i.e. numbers which are reference frame independent) on the stationary surface of field equantions.
In particular this discussion is valid for global U(1) symmetry (which we know that after gauge-ing and quantization of the full interacting theory describes the em. interaction at quantum level btw electrically charged particles) and its associated charge
Q=\int_{\mathbb{R}^{3}} d^3 x J^{0} \left(x^{0},\vec{x}\right) (1).
where the integration is made on arbitrary constant time hypersurface in \mathbb{M}_{4} which we know that is topologically \mathbb{R}^{3}, the latter being the ordinary space of classical Newtonian physics.
The proof is, unfortunately, incomplete as the derivation below will show.
We want to compute the TOTAL variation of the charge (1) under infinitesimal Lorentz transformations (in the sequel ''LT's'')
\delta Q=\frac{\partial Q}{\partial x^mu}} \delta x^{\mu} + \delta\{0\}Q =\TEXTsymbol{\backslash}delta\_\{0\}Q=\TEXTsymbol{\backslash}delta\_\{0\} \TEXTsymbol{\backslash}left[\TEXTsymbol{\backslash}int\_<br /> <br /> <br /> \{\TEXTsymbol{\backslash}mathbb\{R\}\symbol{94}\{3\}\} d\symbol{94}\{3\}x \TEXTsymbol{\backslash} \TEXTsymbol{\backslash} J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\{0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right) \TEXTsymbol{\backslash}right] (2),
038<p type="texpara" tag="Body Text" >,where \TEXTsymbol{\backslash}delta\_\{0\} stands for the INTRINSIC variation of the charge under infinitesimal LT's (i used the notation in Pierre Ramond's ''Field Theory: A Modern Primer''). We definitely know that
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}frac\{\TEXTsymbol{\backslash}partial Q\}\{\TEXTsymbol{\backslash}partial x\symbol{94}\{\TEXTsymbol{\backslash}mu\}\} =0\smallskip (3)
038<p type="texpara" tag="Body Text" >,because Q is time-independent on the stationary surface of the field eqns. and Q doesn't depend on the space variables, as it is integrated wrt them in its very definition (1).
038<p type="texpara" tag="Body Text" >Therefore we need
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta Q=\TEXTsymbol{\backslash}int\_\{\TEXTsymbol{\backslash}mathbb\{R\}\symbol{94}\{3\}\} \TEXTsymbol{\backslash}delta\_\{0\}\TEXTsymbol{\backslash}left[ d\symbol{94}\{3\}x <br /> <br /> <br /> \TEXTsymbol{\backslash} J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\{0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslash}right] =<br /> <br /> 038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}int\_\{\TEXTsymbol{\backslash}mathbb\{R\}\symbol{94}\{3\}\} d\symbol{94}\{3\}x \TEXTsymbol{\backslash} delta\_\{0\}J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\<br /> <br /> {0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslash}right] (4)
038<p type="texpara" tag="Body Text" >,since the intrinsic variation of the volume element under local/infinitesimal LT's is zero.
038<p type="texpara" tag="Body Text" >(4) means that we must calculate
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\{0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right) = \TEXTsymbol{\backslash} ... \TEXTsymbol{\backslash} ? (5)
038<p type="texpara" tag="Body Text" >\smallskip We know that J\symbol{94}\{\TEXTsymbol{\backslash}mu\}, the current, is, on the stationary surface of the field eqns. , a conservative Lorentz vector field and therefore, under arbitrary infinitesimal LT's, it transforms like
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta J\symbol{94}\{\TEXTsymbol{\backslash}mu\} = \TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\}\{\}\_\{\TEXTsymbol{\backslash}nu\} <br /> <br /> J\symbol{94}\{\TEXTsymbol{\backslash}nu\}=\TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{\TEXTsymbol{\backslash}mu\}+\TEXTsym<br /> <br /> bol{\backslash}frac\{\TEXTsymbol{\backslash}partial J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\}\{\TEXTsymbol{\backslash}partial x\symbol{94}\{\TEXTsymbol{\backsl<br /> <br /> ash}nu\}\}\TEXTsymbol{\backslash}delta x\symbol{94}\{\TEXTsymbol{\backslash}nu\} (6)
038<p type="texpara" tag="Body Text" >,where the infinitesimal antisymmetric parameters \TEXTsymbol{\backslash}epsilon are constant wrt ''x''.
038<p type="texpara" tag="Body Text" >It's important to know that (3) provide the total variation of the field (which is a vector) under infinitesimal LT's and part of it comes from the fact that the current itself depends on ''x'' which in turn undergoes infinitesimal LT's. We know that
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta x\symbol{94}\{nu\}=epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\}\{\}\_\{lambda\}x\symbol{94}\{\TEXTsymbol{\backslash}lambda\} (7)
038<p type="texpara" tag="Body Text" >So
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{\TEXTsymbol{\backslash}mu\}=\TEXTsymbol{\backslash}epsilon\symbol{94}\{mu\}\{\}\_\{\TEXTsymbol{\backslash}nu\}J\symbol{94}\{\TEXTsy<br /> <br /> mbol{\backslash}nu\}-\TEXTsymbol{\backslash}left(\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\} J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslash}<br /> <br /> epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}eta\_\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}J\symbol{94}\{\TEXTsymbol{\backslash}nu\}-\TEXTsymbol{\backslas<br /> <br /> h}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >+J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}x\_\{\TEXTsymbol{\backslash}lambda\}<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{<br /> <br /> \backslash}lambda\}\TEXTsymbol{\backslash}left(\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)J\<br /> symbol{94}\{\TEXTsymbol{\backslash}nu\}- \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}le<br /> <br /> ft(J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTs<br /> ymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >+J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsy<br /> mbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}eta\_\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backsla<br /> <br /> sh}mu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)-\TEXTsymbol{\backslash}left(\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}J\symbol{94}\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslas<br /> <br /> h}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}<br /> <br /> 038<p type="texpara" tag="Body Text" >- \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J \symbol{94}\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}<br /> <br /> lambda\}\TEXTsymbol{\backslash}right) - \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right) (8)
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{0\}= \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{\TEXTsymbol{\bac<br /> <br /> kslash}nu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right) - \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}partial\_\{0\}\TEXTsymbol{\backslash}left(J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right) + \TEXTsymbol{\backslash}partial<br /> <br /> \_\{i\}\TEXTsymbol{\backslash}left(J\symbol{94}\{i\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >- \TEXTsymbol{\backslash}partial\_\{0\}\TEXTsymbol{\backslash}left(J\symbol{94}\{0\}\TEXTsymbol{\ba<br /> <br /> ckslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)- \TEXTsymbol{\backslash}partial\_\{i\}\TEXTsymbol{\backslash}left(J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\sym<br /> bol{94}\{i\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right) <br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}partial\_\{i\}\TEXTsymbol{\backslash}left[\TEXTsymbol{\backslash}left( J\symbol{94}\{i\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}-J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\{i\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right] (9)
038<p type="texpara" tag="Body Text" >\smallskip Plugging (9) into (4) we get
038<p type="texpara" tag="Body Text" >\smallskip \TEXTsymbol{\backslash}delta Q=\TEXTsymbol{\backslash}int\_\{\TEXTsymbol{\backslash}mbox\{R\}\symbol{94}\{3\}\} d\symbol{94}\{3\}x \TEXTsymbol{\backslash} \TEXTsymbol{\backslash}partial\_\{i\}\TEXTsymbol{\backslash}left[\TEXTsymbol{\backslash}left(J\symbol{94}\{i\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}-J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\{i\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right] =0 ,
038<p type="texpara" tag="Body Text" >\smallskip by virtue of using classical fields, namely the J's components vanish at infinity faster that any power of ''x'' (we can safely consider them test functions from the Schwarz space over \TEXTsymbol{\backslash}mbox\{R\}\symbol{94}\{4\}) and thus the final equality follows from the Gauss-Ostrogradski's theorem in \TEXTsymbol{\backslash}mathbb\{R\}\symbol{94}\{3\}.
038<p type="texpara" tag="Body Text" >The proof is now complete.
In particular this discussion is valid for global U(1) symmetry (which we know that after gauge-ing and quantization of the full interacting theory describes the em. interaction at quantum level btw electrically charged particles) and its associated charge
Q=\int_{\mathbb{R}^{3}} d^3 x J^{0} \left(x^{0},\vec{x}\right) (1).
where the integration is made on arbitrary constant time hypersurface in \mathbb{M}_{4} which we know that is topologically \mathbb{R}^{3}, the latter being the ordinary space of classical Newtonian physics.
The proof is, unfortunately, incomplete as the derivation below will show.
We want to compute the TOTAL variation of the charge (1) under infinitesimal Lorentz transformations (in the sequel ''LT's'')
\delta Q=\frac{\partial Q}{\partial x^mu}} \delta x^{\mu} + \delta\{0\}Q =\TEXTsymbol{\backslash}delta\_\{0\}Q=\TEXTsymbol{\backslash}delta\_\{0\} \TEXTsymbol{\backslash}left[\TEXTsymbol{\backslash}int\_<br /> <br /> <br /> \{\TEXTsymbol{\backslash}mathbb\{R\}\symbol{94}\{3\}\} d\symbol{94}\{3\}x \TEXTsymbol{\backslash} \TEXTsymbol{\backslash} J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\{0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right) \TEXTsymbol{\backslash}right] (2),
038<p type="texpara" tag="Body Text" >,where \TEXTsymbol{\backslash}delta\_\{0\} stands for the INTRINSIC variation of the charge under infinitesimal LT's (i used the notation in Pierre Ramond's ''Field Theory: A Modern Primer''). We definitely know that
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}frac\{\TEXTsymbol{\backslash}partial Q\}\{\TEXTsymbol{\backslash}partial x\symbol{94}\{\TEXTsymbol{\backslash}mu\}\} =0\smallskip (3)
038<p type="texpara" tag="Body Text" >,because Q is time-independent on the stationary surface of the field eqns. and Q doesn't depend on the space variables, as it is integrated wrt them in its very definition (1).
038<p type="texpara" tag="Body Text" >Therefore we need
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta Q=\TEXTsymbol{\backslash}int\_\{\TEXTsymbol{\backslash}mathbb\{R\}\symbol{94}\{3\}\} \TEXTsymbol{\backslash}delta\_\{0\}\TEXTsymbol{\backslash}left[ d\symbol{94}\{3\}x <br /> <br /> <br /> \TEXTsymbol{\backslash} J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\{0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslash}right] =<br /> <br /> 038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}int\_\{\TEXTsymbol{\backslash}mathbb\{R\}\symbol{94}\{3\}\} d\symbol{94}\{3\}x \TEXTsymbol{\backslash} delta\_\{0\}J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\<br /> <br /> {0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslash}right] (4)
038<p type="texpara" tag="Body Text" >,since the intrinsic variation of the volume element under local/infinitesimal LT's is zero.
038<p type="texpara" tag="Body Text" >(4) means that we must calculate
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\{0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right) = \TEXTsymbol{\backslash} ... \TEXTsymbol{\backslash} ? (5)
038<p type="texpara" tag="Body Text" >\smallskip We know that J\symbol{94}\{\TEXTsymbol{\backslash}mu\}, the current, is, on the stationary surface of the field eqns. , a conservative Lorentz vector field and therefore, under arbitrary infinitesimal LT's, it transforms like
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta J\symbol{94}\{\TEXTsymbol{\backslash}mu\} = \TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\}\{\}\_\{\TEXTsymbol{\backslash}nu\} <br /> <br /> J\symbol{94}\{\TEXTsymbol{\backslash}nu\}=\TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{\TEXTsymbol{\backslash}mu\}+\TEXTsym<br /> <br /> bol{\backslash}frac\{\TEXTsymbol{\backslash}partial J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\}\{\TEXTsymbol{\backslash}partial x\symbol{94}\{\TEXTsymbol{\backsl<br /> <br /> ash}nu\}\}\TEXTsymbol{\backslash}delta x\symbol{94}\{\TEXTsymbol{\backslash}nu\} (6)
038<p type="texpara" tag="Body Text" >,where the infinitesimal antisymmetric parameters \TEXTsymbol{\backslash}epsilon are constant wrt ''x''.
038<p type="texpara" tag="Body Text" >It's important to know that (3) provide the total variation of the field (which is a vector) under infinitesimal LT's and part of it comes from the fact that the current itself depends on ''x'' which in turn undergoes infinitesimal LT's. We know that
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta x\symbol{94}\{nu\}=epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\}\{\}\_\{lambda\}x\symbol{94}\{\TEXTsymbol{\backslash}lambda\} (7)
038<p type="texpara" tag="Body Text" >So
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{\TEXTsymbol{\backslash}mu\}=\TEXTsymbol{\backslash}epsilon\symbol{94}\{mu\}\{\}\_\{\TEXTsymbol{\backslash}nu\}J\symbol{94}\{\TEXTsy<br /> <br /> mbol{\backslash}nu\}-\TEXTsymbol{\backslash}left(\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\} J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslash}<br /> <br /> epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}eta\_\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}J\symbol{94}\{\TEXTsymbol{\backslash}nu\}-\TEXTsymbol{\backslas<br /> <br /> h}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >+J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}x\_\{\TEXTsymbol{\backslash}lambda\}<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{<br /> <br /> \backslash}lambda\}\TEXTsymbol{\backslash}left(\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)J\<br /> symbol{94}\{\TEXTsymbol{\backslash}nu\}- \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}le<br /> <br /> ft(J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTs<br /> ymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >+J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsy<br /> mbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}eta\_\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backsla<br /> <br /> sh}mu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)-\TEXTsymbol{\backslash}left(\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}J\symbol{94}\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslas<br /> <br /> h}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}<br /> <br /> 038<p type="texpara" tag="Body Text" >- \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J \symbol{94}\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}<br /> <br /> lambda\}\TEXTsymbol{\backslash}right) - \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right) (8)
038<p type="texpara" tag="Body Text" >\TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{0\}= \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{\TEXTsymbol{\bac<br /> <br /> kslash}nu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right) - \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}partial\_\{0\}\TEXTsymbol{\backslash}left(J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right) + \TEXTsymbol{\backslash}partial<br /> <br /> \_\{i\}\TEXTsymbol{\backslash}left(J\symbol{94}\{i\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)<br /> <br /> 038<p type="texpara" tag="Body Text" >- \TEXTsymbol{\backslash}partial\_\{0\}\TEXTsymbol{\backslash}left(J\symbol{94}\{0\}\TEXTsymbol{\ba<br /> <br /> ckslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)- \TEXTsymbol{\backslash}partial\_\{i\}\TEXTsymbol{\backslash}left(J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\sym<br /> bol{94}\{i\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right) <br /> <br /> 038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}partial\_\{i\}\TEXTsymbol{\backslash}left[\TEXTsymbol{\backslash}left( J\symbol{94}\{i\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}-J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\{i\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right] (9)
038<p type="texpara" tag="Body Text" >\smallskip Plugging (9) into (4) we get
038<p type="texpara" tag="Body Text" >\smallskip \TEXTsymbol{\backslash}delta Q=\TEXTsymbol{\backslash}int\_\{\TEXTsymbol{\backslash}mbox\{R\}\symbol{94}\{3\}\} d\symbol{94}\{3\}x \TEXTsymbol{\backslash} \TEXTsymbol{\backslash}partial\_\{i\}\TEXTsymbol{\backslash}left[\TEXTsymbol{\backslash}left(J\symbol{94}\{i\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}-J\symbol{94}\{0\}\TEXTsymbol{\backslash}epsilon\{i\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right] =0 ,
038<p type="texpara" tag="Body Text" >\smallskip by virtue of using classical fields, namely the J's components vanish at infinity faster that any power of ''x'' (we can safely consider them test functions from the Schwarz space over \TEXTsymbol{\backslash}mbox\{R\}\symbol{94}\{4\}) and thus the final equality follows from the Gauss-Ostrogradski's theorem in \TEXTsymbol{\backslash}mathbb\{R\}\symbol{94}\{3\}.
038<p type="texpara" tag="Body Text" >The proof is now complete.
Last edited:
