Introduction

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In Einstein's Special Relativity space and time are unified into a four dimensional space-time in which observations as seen by observers moving at different relative velocities transform under the Lorentz group, SO(3,1). The three dimensional vectors (3-vectors) and 3x3 tensors of Euclidean space are extended to four dimensional vectors (4-vectors) and 4x4 tensors of Minkowski_space-time. Including translations in space and time yields the larger Poincaré group, ISO(3,1).

A manifestly covariant formulation of Electromagnetism means a formulation expressing the physical quantities in terms of their transformation representation under the Lorentz and Poincaré groups. When this is done...

  • the usual scalar and 3-vector potentials are recognized as components of a single 4-vector,
  • the usual 3-vector electric and 3-vector magnetic fields become components of a single rank 2 anti-symmetric electro-magnetic field tensor,
  • The Coulomb and Lorentz forces are unified into a single, 4-dimensional covariant Lorentz force equation.
  • All eight of Maxwell's Equations (two scalar and two 3-vector) may then be expressed as two 4-vector equations.


Covariant Quantities
Covariant Quantity Components SI Units
Space-Time Coordinates   meters
Four-Velocity   meter/second
Four-Momentum   kilogram•meter/second
Four-Potential   volts=joules/coulomb
Minkowski Metric   unitless
Electromagnetic Field   volts/meter=newtons/coulomb

Conventions

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e.g.  
  • The standard coordinates (and their differentials) will be expressed using raised greek indice:
 
  • The covector of partial derivatives will be expressed using subscripted partial symbol:
 
 
  • Indices are lowered using this metric,   and raised using the reciprocal metric.

The 4-Vector Potential

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Electromagnetism is a U(1) gauge theory which is to say it is derivable by assuming a complex phase degree of freedom for particles moving through space-time. The way this complex phase,   changes as a particle with charge   is translated defines the 4-vector potential as the gauge connection:

 

NOTE: Neither the phase nor the components of the phase connection are physically observable although differences in phase connection may be observed via interference experiments. (Ref: Aharanov-Bhom effect.) TODO: discuss gauge transformations and canonical momentum's role as generator of translations.

 ( volts in SI units)

NOTE: Gauge transformations:  

Test Particle Lagrangian and The Covariant Lorentz Force

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TODO: Make note of the fact that we cannot use proper-time parametrization as proper-time is not defined until after we impose the dynamic constraints a la Euler-Lagrange equations.

We chose an arbitrary (time-like) parametrization of a test particle's path and define an action:

 

where the dotted coordinates correspond to parameter derivatives:

 

The Lagrangian[1] is:

 

Using the standard variational methods we obtain the Euler-Lagrange equations:

 

The canonical momentum is:

 

or

 

where   is the proper-time 4-velocity of the particle and   is then the kinetic 4-momentum.


The E-L equations then take the form:

 

expanding

 

yields the E-L equations in the form of the covariant Lorentz force:

 

This defines the electro-magnetic field tensor[2].

 

Note that the electro-magnetic field tensor is anti-symmetric,  

The Covariant Lorentz Force

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The electromagnetic field tensor is defined[3] as:

  (in units of volts/meter)

Evaluating typical components in terms of the conventional scalar and vector potentials gives us these components in terms of the E and B fields:

 
 

Likewise expanding each term gives[4]:

 

and

 
        • Edit Line: Changing sign convention.

The covariant Lorentz force becomes:

 
 


 
 

hence

 

The other components are similarly calculated and we have the combined Lorentz and Coulomb forces:

 

We also have the energy component:

 

Thus

 

This is the work done on the particle by the Coulomb force.

The Canonical Hamiltonian

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The generally covariant Hamiltonian is...

 
 
 
 

This zero Hamiltonian is typical of generally covariant dynamics. The zero value however must be understood as resulting from the dynamic constraints and so we seek the constraint equation   which will define the Hamiltonian.

(Badly worded and reasoned... fix)

 

 

 
 
 
 

Note that on the mass shell   and thus

 

!!!! This is Horrible!!!! Try again...

Lagrangian Density of Electro-Magnetic Field in the presence of Currents

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Lagrangian Density:

 

See also

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Notes and references

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  1. ^ TODO: give citation Rindler with note on change of sign. Note that the form involving the proper time and phase derivatives is independent of sign conventions. Also need to check question of proper convention on sign of the phase.
  2. ^ There are a number of choices in convention and no uniform consensus as to the sign of the electromagnetic field tensor.
  3. ^ There is a choice of convention in the sign in the definition of the electro-magnetic field tensor which corresponds with the choice of metric convention. (MORE)
  4. ^ Using Frow column convention.
  • W. Rindler, Introduction to Special Relativity, 2nd edition, Oxford Science Publications, 1991, ISBN 0-19-853952-5.

Further reading

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[[:Category:Fundamental physics concepts]]
[[:Category:Electromagnetism]]
[[:Category:Special relativity]]