Remark on electric charge Lorentz invariance

In summary, the concept of electric charge Lorentz invariance states that the laws of physics related to electric charge are the same for all observers moving at constant velocities. This principle is important in the study of electromagnetism as it allows for accurate predictions and measurements regardless of the observer's motion. Lorentz invariance has been extensively tested and cannot be violated, and it is a fundamental aspect of Einstein's theory of special relativity. It also has practical applications in technologies like GPS and particle accelerators, as well as in understanding high-energy physics experiments.
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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

[tex] Q=\int_{\mathbb{R}^{3}} d^3 x J^{0} \left(x^{0},\vec{x}\right) [/tex] (1).

where the integration is made on arbitrary constant time hypersurface in [itex] \mathbb{M}_{4} [/itex] which we know that is topologically [itex] \mathbb{R}^{3} [/itex], 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'')

[tex] \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\_


\{\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] [/tex] (2),

038<p type="texpara" tag="Body Text" >,where [itex] \TEXTsymbol{\backslash}delta\_\{0\}[/itex] 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" >[tex] \TEXTsymbol{\backslash}frac\{\TEXTsymbol{\backslash}partial Q\}\{\TEXTsymbol{\backslash}partial x\symbol{94}\{\TEXTsymbol{\backslash}mu\}\} =0 [/tex]\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" >[tex] \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


\TEXTsymbol{\backslash} J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\{0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslash}right] =

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}\

{0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslash}right] [/tex] (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" >[tex] \TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{0\}\TEXTsymbol{\backslash}left(x\symbol{94}\{0\},\TEXTsymbol{\backslash}vec\{x\}\TEXTsymbol{\backslash}right) [/tex] = \TEXTsymbol{\backslash} ... \TEXTsymbol{\backslash} ? (5)

038<p type="texpara" tag="Body Text" >\smallskip We know that [itex] J\symbol{94}\{\TEXTsymbol{\backslash}mu\} [/itex], 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" >[tex] \TEXTsymbol{\backslash}delta J\symbol{94}\{\TEXTsymbol{\backslash}mu\} = \TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\}\{\}\_\{\TEXTsymbol{\backslash}nu\}

J\symbol{94}\{\TEXTsymbol{\backslash}nu\}=\TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{\TEXTsymbol{\backslash}mu\}+\TEXTsym

bol{\backslash}frac\{\TEXTsymbol{\backslash}partial J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\}\{\TEXTsymbol{\backslash}partial x\symbol{94}\{\TEXTsymbol{\backsl

ash}nu\}\}\TEXTsymbol{\backslash}delta x\symbol{94}\{\TEXTsymbol{\backslash}nu\} [/tex] (6)

038<p type="texpara" tag="Body Text" >,where the infinitesimal antisymmetric parameters [itex] \TEXTsymbol{\backslash}epsilon [/itex] 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" >[tex]\TEXTsymbol{\backslash}delta x\symbol{94}\{nu\}=epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\}\{\}\_\{lambda\}x\symbol{94}\{\TEXTsymbol{\backslash}lambda\} [/tex] (7)

038<p type="texpara" tag="Body Text" >So

038<p type="texpara" tag="Body Text" >[tex] \TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{\TEXTsymbol{\backslash}mu\}=\TEXTsymbol{\backslash}epsilon\symbol{94}\{mu\}\{\}\_\{\TEXTsymbol{\backslash}nu\}J\symbol{94}\{\TEXTsy

mbol{\backslash}nu\}-\TEXTsymbol{\backslash}left(\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\} J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}right)\TEXTsymbol{\backslash}

epsilon\symbol{94}\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}

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

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)

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\}

038<p type="texpara" tag="Body Text" >=\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{

\backslash}lambda\}\TEXTsymbol{\backslash}left(\TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)J\
symbol{94}\{\TEXTsymbol{\backslash}nu\}- \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}le

ft(J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTs
ymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)

038<p type="texpara" tag="Body Text" >+J\symbol{94}\{\TEXTsymbol{\backslash}mu\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{\TEXTsy
mbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}eta\_\{\TEXTsymbol{\backslash}nu\TEXTsymbol{\backslash}lambda\}

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

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

h}epsilon\symbol{94}\{\TEXTsymbol{\backslash}mu\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}

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)

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}

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) [/tex] (8)

038<p type="texpara" tag="Body Text" >[tex] \TEXTsymbol{\backslash}delta\_\{0\}J\symbol{94}\{0\}= \TEXTsymbol{\backslash}partial\_\{\TEXTsymbol{\backslash}nu\}\TEXTsymbol{\backslash}left(J\symbol{94}\{\TEXTsymbol{\bac

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)

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

\_\{i\}\TEXTsymbol{\backslash}left(J\symbol{94}\{i\}\TEXTsymbol{\backslash}epsilon\symbol{94}\{0\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)

038<p type="texpara" tag="Body Text" >- \TEXTsymbol{\backslash}partial\_\{0\}\TEXTsymbol{\backslash}left(J\symbol{94}\{0\}\TEXTsymbol{\ba

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
bol{94}\{i\TEXTsymbol{\backslash}lambda\}x\_\{\TEXTsymbol{\backslash}lambda\}\TEXTsymbol{\backslash}right)

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] [/tex] (9)

038<p type="texpara" tag="Body Text" >\smallskip Plugging (9) into (4) we get

038<p type="texpara" tag="Body Text" >\smallskip [tex] \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 [/tex] ,

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 [itex] \TEXTsymbol{\backslash}mbox\{R\}\symbol{94}\{4\} [/itex]) and thus the final equality follows from the Gauss-Ostrogradski's theorem in [itex] \TEXTsymbol{\backslash}mathbb\{R\}\symbol{94}\{3\} [/itex].

038<p type="texpara" tag="Body Text" >The proof is now complete.
 
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Sorry, what you se above is LaTex sabotage. I'll post everything again. Tomorrow. It actually annoys me that writing a long post (also with LaTex code) in a fairly long period of time automatically leads to disconnection, so i have to write the post in SW and copy-paste it here, but as you can see, the LaTex compiler from this website doesn't recognize the code used by SW. Un coup de merde.

Daniel.
 
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1. What is the concept of electric charge Lorentz invariance?

The concept of electric charge Lorentz invariance refers to the idea that the laws of physics, specifically those related to electric charge, are the same for all observers moving at constant velocities relative to each other. This principle is a fundamental aspect of Einstein's theory of special relativity.

2. Why is Lorentz invariance important in the study of electromagnetism?

Lorentz invariance is important in the study of electromagnetism because it helps us understand and predict the behavior of electrically charged particles, such as electrons, in different frames of reference. It allows us to make accurate measurements and calculations regardless of the observer's relative motion.

3. Can Lorentz invariance be violated?

No, Lorentz invariance has been extensively tested and has been found to hold true in all experiments. Violating this principle would mean that the laws of physics would be different for different observers, which goes against the fundamental principles of relativity.

4. How is electric charge Lorentz invariance related to the theory of special relativity?

The principle of Lorentz invariance is a fundamental aspect of Einstein's theory of special relativity. It is one of the key principles that allows us to reconcile the laws of physics in different frames of reference and explains the observed phenomena related to time dilation and length contraction.

5. Are there any practical applications of electric charge Lorentz invariance?

Electric charge Lorentz invariance has a wide range of practical applications, including in the development of technologies like GPS, particle accelerators, and nuclear reactors. It also helps us understand and predict the behavior of particles in high-energy physics experiments.

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