Parity conservation and the Field-Strength Tensor‏

In summary: Parity is not violated, but the information is lost in the transformation.On page 558 Jackson states "Transformation (11.149) shows that E and B have no independent existence. A purely electric or magnetic field in one coordinate system will appear as a mixture of electric and magnetic fields in another coordinate frame."Isn't he ignoring the fact that the electro-magnetic field-strength tensor obliterates the parity designations implicit in the separation of E and B fields?In summary, the usage apparently originated with Minkowski, and the receivers of that usage justified its acceptance by saying that it preserved parity information. However, Jackson's equations show that this is not actually the case, and parity information
  • #1
PhilDSP
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In reexamining chapter 11 of Jackson's Classical Electrodynamics, especially equations 11.148, it seems obvious that in placing the E and B transformation values into the electro-magnetic field-strength tensor one is ignoring the standard rules which do not allow combining polar vectors and axial vectors, or in this case, scalars and pseudoscalars. The result is that one loses parity information.

That usage apparently originated with Minkowski. How did he motivate that apparent lapse of rigor? How did the receivers of that usage justify its acceptance?

On page 558 Jackson states "Transformation (11.149) shows that E and B have no independent existence. A purely electric or magnetic field in one coordinate system will appear as a mixture of electric and magnetic fields in another coordinate frame."

Isn't he ignoring the fact that the electro-magnetic field-strength tensor obliterates the parity designations implicit in the separation of E and B fields?

Pauli, on p. 78 and 79 of his monograph on relativity delves a little deeper and seems to say that one may choose one of two different forms for the electro-magnetic field-strength tensor, one of which he calls the dual tensor. But the issue remains. The parity conservation rules from classical EM evaporate leaving an indeterminacy, don't they?
 
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  • #2
Look at the transformation carefully and you will see that parity is NOT violated by the transformation.
 
  • #3
I didn't say it was violated, but rather becomes indeterminate. But yes, further details in its handling seem necessary.
 
  • #4
No, it doesn't become indeterminate. Look harder.
 
  • #5
PhilDSP said:
in placing the E and B transformation values into the electro-magnetic field-strength tensor one is ignoring the standard rules which do not allow combining polar vectors and axial vectors, or in this case, scalars and pseudoscalars. The result is that one loses parity information.

No, one doesn't, because B is not really a vector; it's an antisymmetric 2nd-rank tensor, which can simply be incorporated directly into the EM field tensor. Parity only comes into play when you insist on working with axial vectors instead of antisymmetric 2nd-rank tensors, because converting the latter into the former forces you to make a parity choice, corresponding to which direction the standard "normal" vector to a plane points. In other words, when you work with axial vectors, the vector by itself doesn't contain all the information that was in the corresponding antisymmetric 2nd-rank tensor; some of that information now resides in the definition of the standard "normal" to a plane. If you work with the tensors directly, all the information is in the tensor.
 
  • #6
For reference, here are Jackson's equations (11.149) [tex]{\bf E'} = \gamma ({\bf E} + \boldsymbol{\beta \times} {\bf B} ) - \frac{\gamma ^2}{\gamma+1}\boldsymbol{\beta}(\boldsymbol{\beta} \cdot {\bf E}),[/tex] and [tex]{\bf B'} = \gamma ({\bf B} - \boldsymbol{\beta \times} {\bf E}) - \frac{\gamma ^2}{\gamma+1}\boldsymbol{\beta}(\boldsymbol{\beta} \cdot {\bf B}).[/tex]
 
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  • #7
dauto said:
For reference, here are Jackson's equations (11.149) [tex]{\bf E'} = \gamma ({\bf E} + \boldsymbol{\beta \times} {\bf B} ) - \frac{\gamma ^2}{\gamma+1}\boldsymbol{\beta}(\boldsymbol{\beta} \cdot {\bf E}),[/tex] and [tex]{\bf B'} = \gamma ({\bf B} - \boldsymbol{\beta \times} {\bf E}) - \frac{\gamma ^2}{\gamma+1}\boldsymbol{\beta}(\boldsymbol{\beta} \cdot {\bf B}).[/tex]

PhilDSP, note a key property of these equations:

In the first equation, ##\bf{B}## only appears in a cross product; while in the second equation, ##\bf{E}## only appears in a cross product. The cross product converts polar vectors to axial vectors and vice versa; so all the terms in each equation are the same kind of vector, hence parity information is not lost.

In other words, taking a cross product involves making a parity choice, just as converting a 2nd-rank antisymmetric tensor to an axial vector does (since a cross product actually *is* a 2nd-rank antisymmetric tensor "under the hood", so writing it as a vector requires exactly such a conversion).
 
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  • #8
Okay thanks! That's pretty obvious from 11.149. And in 11.148 a lot of terms from the cross product drop out because the motion is specified to be along the x axis.

I need to ponder the subtleties of this a bit more.
 

1. What is parity conservation?

Parity conservation is a fundamental principle in physics that states that the laws of physics should be the same if a system is reflected in a mirror. In other words, the physical properties of a system should be unchanged if it is viewed from a different perspective.

2. What is the Field-Strength Tensor?

The Field-Strength Tensor is a mathematical object used in physics to describe the strength and direction of a vector field. It contains information about the electric and magnetic fields in a given system.

3. How is parity conservation related to the Field-Strength Tensor?

Parity conservation is closely related to the Field-Strength Tensor because it is used to describe the symmetry of a physical system. The tensor is used to determine whether a system is symmetric under a mirror reflection, which is essential for determining whether parity conservation holds in a given situation.

4. What happens if parity conservation is violated?

If parity conservation is violated, it means that the laws of physics are not the same when a system is reflected in a mirror. This can have significant implications for our understanding of physical laws and may require new theories to explain the observed phenomena.

5. How is the Field-Strength Tensor used in practical applications?

The Field-Strength Tensor is used in a variety of practical applications, including electromagnetism, quantum mechanics, and general relativity. It plays a crucial role in understanding the behavior of electric and magnetic fields and is essential for the development of new technologies, such as computer chips and medical imaging devices.

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