6-dimensional representation of Lorentz group

[Silviu]

Hello! I understand that the vector formed of the scalar and vector potential in classical EM behaves like a 4-vector ($A^\nu=\Lambda^\nu_\mu A^\mu$). Does this means that the if we make a vector with the 3 components of B field and 3 of E field, so a 6 components vector V, will it transform as $V^\nu=M(\Lambda^\nu_\mu) V^\mu$, with M being a 6-dimensional representation of the Lorentz group? So can we build all the tensorial formalism of EM using this 6-dimensional representation, in the same way we did with a 4-vector representation?

 

[Dale]

). Does this means that the if we make a vector with the 3 components of B field and 3 of E field, so a 6 components vector V, will it transform as Vν=M(Λνμ)VμVν=M(Λμν)VμV^\nu=M(\Lambda^\nu_\mu) V^\mu,

Not quite. The relativistic object which contains the components of both the non relativistic E and B field is called the Faraday tensor. It is a rank 2 tensor in 4D spacetime, so it has 16 components. But the Faraday tensor is antisymmetric, so it only has 6 independent components (the diagonal elements are 0 and the upper triangle is the opposite of the lower triangle)

 

[Silviu]

Not quite. The relativistic object which contains the components of both the non relativistic E and B field is called the Faraday tensor. It is a rank 2 tensor in 4D spacetime, so it has 16 components. But the Faraday tensor is antisymmetric, so it only has 6 independent components (the diagonal elements are 0 and the upper triangle is the opposite of the lower triangle)

So the vector I just described, doesn't transform in a relativistic from, it must be through a tensor?

 

[Dale]

So the vector I just described, doesn't transform in a relativistic from, it must be through a tensor?

Well, technically what you described is not even a vector. A vector has 4 components in relativity, so something with 6 components cannot be a vector.

 

[Silviu]

Well, technically what you described is not even a vector. A vector has 4 components in relativity, so something with 6 components cannot be a vector.

Oh, so what is the point of a n-dimensional representation of the Lorentz group, if it makes sense only for n=4?

 

[vanhees71]

There are two convenient ways to get the transformation of the electromagnetic-field components. First you can use the Faraday tensor components,
$$F_{\mu \nu}=\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu},$$
which transform as any 2nd-rank-tensor components
$$\bar{F}^{\mu \nu}(\bar{x}) = {\Lambda^{\mu}}_{\rho} {\Lambda^{\nu}}_{\sigma} F^{\rho \sigma}(x).$$
The second posibility is to use the Riemann-Silberstein vector. In terms of the field components in the 1+3-formalism it reads
$$\vec{F}=\vec{E}+\mathrm{i} \vec{B}.$$
The two invariants one can form from the $F_{\mu \nu}$ are $F_{\mu \nu} F^{\mu \nu}$ and $\epsilon_{\mu \nu \rho \sigma} F^{\mu \nu} F^{\rho \sigma}$, which are proportional to $\vec{E}^2-\vec{B}^2$ and $\vec{E} \cdot \vec{B}$. Now using the usual scalar product for the complex RS vector, shows that under Lorentz transformations the scalar product
$$\vec{F} \cdot \vec{F} = (\vec{E}^2-\vec{B}^2) +2 \mathrm{i} \vec{E} \cdot \vec{B}$$
stays constant. Thus the representation of the proper orthochronous Lorentz group for the RS vector is $\mathrm{SO}(3,\mathbb{C})$. It turns out that the subgroup $\mathrm{SO}(3,\mathbb{R})$ is the representation of the rotations as it must be since $\vec{E}$ and $\vec{B}$ are real three-vectors. Rotations around a fixed direction with imaginary angles, represent the boosts in the corresponding direction. They mix electric and magnetic components.

 

[Orodruin]

Not quite. The relativistic object which contains the components of both the non relativistic E and B field is called the Faraday tensor. It is a rank 2 tensor in 4D spacetime, so it has 16 components. But the Faraday tensor is antisymmetric, so it only has 6 independent components (the diagonal elements are 0 and the upper triangle is the opposite of the lower triangle)

The set of anti-symmetric rank two tensors form a 6-dimensional vector space so: Yes, indeed the Faraday tensor transforms according to a 6-dimensional representation of the Lorentz group.

Well, technically what you described is not even a vector. A vector has 4 components in relativity, so something with 6 components cannot be a vector.

It is not a vector in the typical nomenclature of relativity, but it is an element of a 6-dimensional vector space (the anti-symmetric rank two tensors) that allows a representation of the Lorentz group.

I would not use the regular space-time indices to denote the components of this vector space though.

Oh, so what is the point of a n-dimensional representation of the Lorentz group, if it makes sense only for n=4?

The Faraday tensor indeed transforms under a 6-dimensional representation of the Lorentz group.