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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...
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 |
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.
NOTE: Gauge transformations:
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 electromagnetic field tensor is defined [3] as:
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
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 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:
{{Physics-footer}} [[:Category:Fundamental physics concepts]] [[:Category:Electromagnetism]] [[:Category:Special relativity]]
![]() | This
user page or section is in a state of significant expansion or restructuring. You are welcome to assist in its construction by editing it as well. If this
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This page was
last edited by
PrimeBOT (
talk |
contribs) 6 years ago. (
Update timer) |
Articles about |
Electromagnetism |
---|
![]() |
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...
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 |
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.
NOTE: Gauge transformations:
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 electromagnetic field tensor is defined [3] as:
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
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 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:
{{Physics-footer}} [[:Category:Fundamental physics concepts]] [[:Category:Electromagnetism]] [[:Category:Special relativity]]