[action-point-particle-relativity] Action
The result is a geodesic
Using the spacetime 's metric volume form restricted to a one-dimensional path, we obtain the length , which uses the square root of the quadratic form, rather than the quadratic form alone
For a path, in the "time coordinate" , let . Action
[equation-point-particle-relativity] let . Similar to the non-relativistic case, the equation of action
[point-particle-relativity-approximate-to-non-relativity] The relativistic action "approximates" to the non-relativistic action
Then the constant value will vary to zero
This non-relativistic approximation limit method is coordinate-dependent. On a curved manifold, since multiple coordinates may be needed to cover the entire manifold, the definition problem of the non-relativistic approximation limit will be more difficult.
Symmetry and conserved quantities
The symmetry group of spacetime is the isometry alias Poincare group
- Translation
Using time coordinates. Similar to the non-relativistic case, the relativistic versions of energy and momentum are [energy-momentum-point-particle-relativity]
Denoted as 4-momentum
The relativistic Lagrangian is invariant under , but the boost still changes the time and space endpoints of the path i.e. changes the action
- Rotation
Similar to the non-relativistic case, the relativistic version of momentum-point-particle-non-relativity is [rotation-momentum-point-particle-relativity]
- boost
boost by hyperbolic angle
So ฮด boost by hyperbolic angle, is
In a coordinate of , let the spatial vector , , corresponding to ฮด boost, define the hyperbolic cross product
Similar to the case of energy, boost also changes the action
The calculation result of boost momentum will have 4-momentum, thus energy will appear
[boost-momentum-point-particle-relativity]
Note that the spacetime metric has a negative definite spatial metric
Spatial vector
Also called boost momentum
Because coordinates are used to separate time and space, although rotational momentum and boost momentum are invariant, the representations and boost momentum are not invariant
Combined, it can be written as angular momentum
Particle system
potential
potential
point particle in Lorentz-manifold
Example
Coupling of relativistic point particles and gauge fields. Action
- Question
Hidden gauge symmetry
The gauge transformation used in field interaction leads to a transformation of the connection . For the action of a point particle and an electromagnetic field, is a divergence quantity , with the boundary being zero, the variation is zero.
Although what is invariant is the equation, not the action
This is different from, for example, the case of a scalar field, where the action is also invariant, and the invariance of the equation is achieved through the definition of covariant derivatives.
[current-gauge-particle] Can this hidden gauge symmetry give a conserved 4-current for a point particle?
Example
Coupling of relativistic point particles and gauge fields. Action
- Question
Hidden gauge symmetry
The gauge transformation used in field interaction leads to a transformation of the connection . For the action of a point particle and an electromagnetic field, is a divergence quantity , with the boundary being zero, the variation is zero.
Although what is invariant is the equation, not the action
This is different from, for example, the case of a scalar field, where the action is also invariant, and the invariance of the equation is achieved through the definition of covariant derivatives.
[current-gauge-particle] Can this hidden gauge symmetry give a conserved 4-current for point particles?