Electromagnetic_tensor

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The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system in Maxwell\'s theory of electromagnetism. The field tensor was first used after the 4-dimensional tensor formulation of special relativity introduced by Hermann Minkowski. The tensor allows some physical laws to be written in a very concise form.

Contents 1 Details 1.1 Properties 2 Derivation of tensor 3 Relation to classical electromagnetism 4 Significance of the Field Tensor 5 The field tensor and relativity 6 Role in Quantum Electrodynamics and Field Theory 7 See also 8 References

Details

Mathematical note: In this article, the abstract index notation will be used.

The electromagnetic tensor $F\_\{\backslash alpha\backslash beta\}$ is commonly written as a matrix:

$F\_\{\backslash alpha\backslash beta\}\; =\; \backslash begin\{bmatrix\}$

0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & B_z & -B_y \\ E_y/c & -B_z & 0 & B_x \\ E_z/c & B_y & -B_x & 0 \end{bmatrix}

where

E is the electric field,

B the magnetic field, and

c the speed of light.

Properties

From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties:

• antisymmetry: $F^\{\backslash alpha\backslash beta\}\; \backslash ,\; =\; -\; F^\{\backslash beta\backslash alpha\}$ (hence the name bivector).
• six independent components.

If one forms an inner product of the field strength tensor a Lorentz invariant is formed:

$F\_\{\backslash alpha\backslash beta\}\; F^\{\backslash alpha\backslash beta\}\; =\; \backslash \; 2\; \backslash left(\; B^2\; -\; \backslash frac\{E^2\}\{c^2\}\; \backslash right)\; =\; \backslash mathrm\{invariant\}$

The product of the tensor $F^\{\backslash alpha\backslash beta\}\; \backslash ,$ with its dual tensor gives the pseudoscalar invariant:

$\backslash epsilon\_\{\backslash alpha\backslash beta\backslash gamma\backslash delta\}F^\{\backslash alpha\backslash beta\}\; F^\{\backslash gamma\backslash delta\}\; =\; \backslash frac\{2\}\{c\}\; \backslash left(\; \backslash vec\; B\; \backslash cdot\; \backslash vec\; E\; \backslash right)\; =\; \backslash mathrm\{invariant\}\; \backslash ,$

where   $\backslash \; \backslash epsilon\_\{\backslash alpha\backslash beta\backslash gamma\backslash delta\}\; \backslash ,$ is the completely antisymmetric unit pseudotensor of the fourth rank or Levi-Civita symbol. Notice that

$\backslash det\; \backslash left(\; F\; \backslash right)\; =\; \backslash frac\{1\}\{c^2\}\; \backslash left(\; \backslash vec\; B\; \backslash cdot\; \backslash vec\; E\; \backslash right)\; ^\{2\}$

More formally, the electromagnetic tensor may be written in terms of the 4-vector potential $A^\{\backslash alpha\}\; \backslash ,$:

$$

F_{ \alpha\beta } \ \stackrel{\mathrm{def}}{=}\ \frac{ \partial A_{\beta} }{ \partial x^{\alpha} } - \frac{ \partial A_{\alpha} }{ \partial x^{\beta} } \ \stackrel{\mathrm{def}}{=}\ \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha} Where the 4-vector potential is:

$A^\{\backslash alpha\}\; =\; \backslash left(\; \backslash frac\{\backslash phi\}\{c\}\; ,\; \backslash vec\; A\; \backslash right)$ and its covariant form is found by multiplying by the Minkowski metric $\backslash eta\; \backslash ,$:

$A\_\{\backslash alpha\}\; \backslash ,\; =\; \backslash eta\_\{\backslash alpha\backslash beta\}\; A^\{\backslash beta\}\; =\; \backslash left(\; -\backslash frac\{\backslash phi\}\{c\},\; \backslash vec\; A\; \backslash right)$

Derivation of tensor

To derive all the elements in the electromagnetic tensor we need to define the derivative operator:

$\backslash partial\_\{\backslash alpha\}\; =\; \backslash left(\backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; \backslash right)\; =\; \backslash left(\backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\},\; \backslash vec\{\backslash nabla\}\; \backslash right)\; \backslash ,$

and the 4-vector potential:

$A\_\{\backslash alpha\}\; =\; \backslash left(-\backslash frac\{\backslash phi\}\{c\},\; A\_x,\; A\_y,\; A\_z\; \backslash right)\; \backslash ,$

where

$\backslash vec\; A\; \backslash ,$ is the vector potential and $\backslash left(A\_x,\; A\_y,\; A\_z\; \backslash right)$ are its components

$\backslash phi\; \backslash ,$ is the scalar potential and

$c\; \backslash ,$ is the speed of light.

Electric and magnetic fields are derived from the vector potentials and the scalar potential with two formulas:

$\backslash vec\{E\}\; =\; -\backslash frac\{\backslash partial\; \backslash vec\{A\}\}\{\backslash partial\; t\}\; -\; \backslash vec\{\backslash nabla\}\; \backslash phi\; \backslash ,$

$\backslash vec\{B\}\; =\; \backslash vec\{\backslash nabla\}\; \backslash times\; \backslash vec\{A\}\; \backslash ,$

As an example, the x components are just

$E\_x\; =\; -\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; t\}\; -\; \backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; x\}\; \backslash ,$

$B\_x\; =\; \backslash frac\{\backslash partial\; A\_z\}\{\backslash partial\; y\}\; -\; \backslash frac\{\backslash partial\; A\_y\}\{\backslash partial\; z\}\; \backslash ,$

Using the definitions we began with, we can re-write these two equations to look like:

$E\_x\; =\; -c\; \backslash left(\backslash partial\_0\; A\_1\; -\; \backslash partial\_1\; A\_0\; \backslash right)\; \backslash ,$

$B\_x\; =\; \backslash partial\_2\; A\_3\; -\; \backslash partial\_3\; A\_2\; \backslash ,$

Evaluating all the components results in a second-rank, antisymmetric and covariant tensor:

$F\_\{\backslash alpha\backslash beta\}\; =\; \backslash partial\_\{\backslash alpha\}\; A\_\{\backslash beta\}\; -\; \backslash partial\_\{\backslash beta\}\; A\_\{\backslash alpha\}\; \backslash ,$

Thus, for example,

$F\_\{12\}=\backslash partial\_1A\_2-\backslash partial\_2A\_1=\backslash partial\_xA\_y-\backslash partial\_yA\_x=B\_z,$

and

$F\_\{13\}=\backslash partial\_1A\_3-\backslash partial\_3A\_1=\backslash partial\_xA\_z-\backslash partial\_zA\_x=-B\_y.$

Compare with the matrix above.

Relation to classical electromagnetism

Classical electromagnetism and Maxwell\'s equations can be derived from the action defined:

$\backslash mathcal\{S\}\; =\; \backslash int\; \backslash left(\; -\backslash begin\{matrix\}\; \backslash frac\{1\}\{4\; \backslash mu\_0\}\; \backslash end\{matrix\}\; F\_\{\backslash mu\backslash nu\}\; F^\{\backslash mu\backslash nu\}\; \backslash right)\; \backslash mathrm\{d\}^4\; x\; \backslash ,$

where

$\backslash mathrm\{d\}^4\; x\; \backslash ;$   is over space and time.

This means the Lagrangian is

$\backslash mathcal\{L\}\; \backslash ,$	$=\; -\backslash begin\{matrix\}\; \backslash frac\{1\}\{4\backslash mu\_0\}\; \backslash end\{matrix\}\; F\_\{\backslash mu\backslash nu\}\; F^\{\backslash mu\backslash nu\}\; \backslash ,$
	$=\; -\backslash begin\{matrix\}\; \backslash frac\{1\}\{4\backslash mu\_0\}\; \backslash end\{matrix\}\; \backslash left(\; \backslash partial\_\backslash mu\; A\_\backslash nu\; -\; \backslash partial\_\backslash nu\; A\_\backslash mu\; \backslash right)\; \backslash left(\; \backslash partial^\backslash mu\; A^\backslash nu\; -\; \backslash partial^\backslash nu\; A^\backslash mu\; \backslash right).\; \backslash ,$
	$=\; -\backslash begin\{matrix\}\; \backslash frac\{1\}\{4\backslash mu\_0\}\; \backslash end\{matrix\}\; \backslash left(\; \backslash partial\_\backslash mu\; A\_\backslash nu\; \backslash partial^\backslash mu\; A^\backslash nu\; -\; \backslash partial\_\backslash nu\; A\_\backslash mu\; \backslash partial^\backslash mu\; A^\backslash nu\; -\; \backslash partial\_\backslash mu\; A\_\backslash nu\; \backslash partial^\backslash nu\; A^\backslash mu\; +\; \backslash partial\_\backslash nu\; A\_\backslash mu\; \backslash partial^\backslash nu\; A^\backslash mu\; \backslash right).$

The far left and far right term are the same, because $\backslash mu$ and $\backslash nu$ are just dummy variables after all. The two middle terms are also the same, so the Lagrangian is

$\backslash mathcal\{L\}\; \backslash ,$

$=\; -\backslash begin\{matrix\}\; \backslash frac\{1\}\{2\backslash mu\_0\}\; \backslash end\{matrix\}\; \backslash left(\; \backslash partial\_\backslash mu\; A\_\backslash nu\; \backslash partial^\backslash mu\; A^\backslash nu\; -\; \backslash partial\_\backslash nu\; A\_\backslash mu\; \backslash partial^\backslash mu\; A^\backslash nu\; \backslash right).$

We can then plug this into the Euler-Lagrange equation of motion for a field:

$\backslash partial\_\backslash nu\; \backslash left(\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\; \backslash partial\_\backslash nu\; A\_\backslash mu\; )\}\; \backslash right)\; -\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; A\_\backslash mu\}\; =\; 0\; .\; \backslash ,$

The second term is zero, because the Lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes:

$\backslash partial\_\backslash nu\; \backslash left(\; \backslash partial^\backslash mu\; A^\backslash nu\; -\; \backslash partial^\backslash nu\; A^\backslash mu\; \backslash right)\; =\; 0.\; \backslash ,$

That term in the parenthesis is just the field tensor, so this finally simplifies to

$\backslash partial\_\backslash nu\; F^\{\backslash mu\; \backslash nu\}\; =\; 0.\; \backslash ,$

That equation is just another way of writing the two homogeneous Maxwell\'s equations as long as you make the substitutions:

$~E^i\; /c\; \backslash \; \backslash \; =\; -F^\{0\; i\}\; \backslash ,$

$\backslash epsilon^\{ijk\}\; B^k\; =\; -F^\{ij\}\; \backslash ,$

where $i\; \backslash ,$ and $j\; \backslash ,$ take on the values of 1, 2, and 3.

Significance of the Field Tensor

Hidden beneath the surface of this complex mathematical equation is an ingenious unification of Maxwell\'s equations for electromagnetism. Consider the electrostatic equation

$\backslash vec\{\backslash nabla\}\; \backslash cdot\; \backslash vec\{E\}\; =\; \backslash frac\{\backslash rho\}\{\backslash epsilon\_0\}$

which tells us that the divergence of the electric field vector is equal to the charge density, and the electrodynamic equation

$$

\vec{\nabla} \times \vec{B} - \frac{1}{c^2} \frac{ \partial \vec{E}}{\partial t} = \mu_0 \vec{J}

that is the change of the electric field with respect to time, minus the curl of the magnetic field vector, is equal to negative 4π times the current density.

These two equations for electricity reduce to

$\backslash partial\_\{\backslash alpha\}\; F^\{\backslash alpha\backslash beta\}\; =\; \backslash mu\_0\; J^\{\backslash beta\}\; \backslash ,$

where

$J^\{\backslash alpha\}\; =\; (\; c\; \backslash ,\; \backslash rho\; ,\; \backslash vec\{J\}\; )\; \backslash ,$ is the 4-current.

The same holds for magnetism. If we take the magnetostatic equation

$$

\vec{\nabla} \cdot \vec{B} = 0

which tells us that there are no "true" magnetic charges, and the magnetodynamics equation

$$

\frac{ \partial \vec{B}}{ \partial t } + \vec{\nabla} \times \vec{E} = 0

which tells us the change of the magnetic field with respect to time plus the curl of the Electric field is equal to zero (or, alternatively, the curl of the electric field is equal to the negative change of the magnetic field with respect to time). With the electromagnetic tensor, the equations for magnetism reduce to

$F\_\{\; \backslash alpha\; \backslash beta\; ,\; \backslash gamma\; \}\; +\; F\_\{\; \backslash beta\; \backslash gamma\; ,\; \backslash alpha\; \}\; +\; F\_\{\; \backslash gamma\; \backslash alpha\; ,\; \backslash beta\; \}\; =\; 0.\; \backslash ,$

The field tensor and relativity

The field tensor derives its name from the fact that the electromagnetic field is found to obey the tensor transformation law, this general property of (non-gravitational) physical laws being recognised after the advent special relativity. This theory stipulated that all the (non-gravitational) laws of physics should take the same form in all coordinate systems - this led to the introduction of tensors. The tensor formalism also leads to a mathematically elegant presentation of physical laws. For example, Maxwell\'s equations of electromagnetism may be written using the field tensor as:

$F\_\{[\backslash alpha\backslash beta,\backslash gamma]\}\; \backslash ,\; =\; 0$ and $F^\{\backslash alpha\backslash beta\}\{\}\_\{,\backslash beta\}\; \backslash ,\; =\; \backslash mu\_0\; J^\{\backslash alpha\}$

where the comma indicates a partial derivative. The second equation implies conservation of charge:

$J^\backslash alpha\{\}\_\{,\backslash alpha\}\; \backslash ,\; =\; 0$

In general relativity, these laws can be generalised in (what many physicists consider to be) an appealing way:

$F\_\{[\backslash alpha\backslash beta;\backslash gamma]\}\; \backslash ,\; =\; 0$ and $F^\{\backslash alpha\backslash beta\}\{\}\_\{;\backslash beta\}\; \backslash ,\; =\; \backslash mu\_0\; J^\{\backslash alpha\}$

where the semi-colon represents a covariant derivative, as opposed to a partial derivative. The elegance of these equations stems from the simple replacing of partial with covariant derivatives, a practice sometimes referred to in the parlance of GR as \'replacing partial with covariant derivatives\'. These equations are sometimes referred to as the curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime):

$J^\backslash alpha\{\}\_\{;\backslash alpha\}\; \backslash ,\; =\; 0$

Role in Quantum Electrodynamics and Field Theory

The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity from $\backslash mathcal\{L\}=\backslash bar\backslash psi(i\backslash hbar\; c\; \backslash ,\; \backslash gamma^\backslash alpha\; D\_\backslash alpha\; -\; mc^2)\backslash psi\; -\backslash frac\{1\}\{4\; \backslash mu\_0\}F\_\{\backslash alpha\backslash beta\}F^\{\backslash alpha\backslash beta\},$  to incorporate the creation and annihilation of photons (and electrons).

In quantum field theory, it is used for the template of the gauge field strength tensor. That is used in addition to the local interaction Lagrangian, nearly identical to its role in QED.

See also

• Application of tensor theory in physics
• Classification of electromagnetic fields

References

• Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 0-19-514665-4. 
• Peskin, Michael E.; Schroeder, Daniel V. (1995). An Introduction to Quantum Field Theory. Perseus Publishing. ISBN 0-201-50397-2. 

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