action principle for spacetime events

[AntonioLao]

The usual understanding of action principle is through the study of calculus of variations. For our particular discussion, we can sidestep all the classical details and jump over directly to spacetime descriptions asserting that the following contact transformation integral exists.

[math]A=\\\\int L(\\\\psi , \\\\dot{\\\\psi} , \\\\phi) d\\\\phi [/math]

The general form of this integral is the multiple integrals over the entire domain of [math] T(\\\\phi_i) [/math] given by:

[math] A^i = \\int_i L^i (\\psi_i , \\phi_i , \\dot{\\psi_i} , \\dot{\\phi_i} ) d\\phi_i [/math] , where [math] i=1, 2, 3, \\\\ldots , \\\\infty [/math]

and

[math]\\\\dot{\\\\psi_i}=\\\\frac{\\\\partial \\\\psi_i}{\\\\partial \\\\phi_i} [/math] have the dimensions of velocities.

One particular multiple action integral of interest is the double action integral, which is equivalent to the quantization of local infinitesimal spacetime.

[math]A^2 = \\\\int_1 \\\\int_2 L^2 (\\\\psi_1,\\\\psi_2,\\\\phi_1,\\\\phi_2) d\\\\phi_1 d\\\\phi_2[/math]

or

[math]A^2 = \int_1 \int_2 L^2 (\psi_1,\psi_2,\phi_1,\phi_2) dt_1 dt_2[/math]
where [math]dt_1[/math] and [math]dt_2[/math] are now differential time intervals.

The unassuming function [math]L^2 [/math] is now defined as the square of energy if and only if the [math]\\\\psi_i[/math] have units of length and the [math]\\\\phi_i[/math] have units of force and the following statement is true:

[math]E^2=\\\\psi_1 \\\\times \\\\phi_1 \\\\cdot \\\\psi_2 \\\\times \\\\phi_2 [/math]

In this formulation, all time elements of the set [math]T(\\\\phi_i)[/math] have the physical attribute of direction. This directional property makes time equivalent to force. Furthermore, it gives meaning to the existence of time axes for all spacetime events.

[mkirkpatrick]

These are all presumally relative events within the physical continuum?



regards michael.

[AntonioLao]

relative events within the physical continuum?

Physical continuum uses the principle of least action for all mechanical systems see
http://en.wikipedia.org/wiki/Principle_of_least_action
For space quantization without time parameter, the principle of least action must be extended into a spacetime continuum but only being described in 3D continuum.

[mkirkpatrick]

Physical continuum uses the principle of least action for all mechanical systems see
http://en.wikipedia.org/wiki/Principle_of_least_action
For space quantization without time parameter, the principle of least action must be extended into a spacetime continuum but only being described in 3D continuum.

Thanks will check it out later.




regards michael.