| double time integrals -
08-22-2007, 04:19 PM
Single space integral transforms a point function into a curve function. Double space integrals transform a point function into a surface function. The single time integral of energy is called the action S and S=òEdt between two fixed times t1 and t2. However, this is not a closed time loop unless t1=t2 such that the variation of the action is stationary with a value of zero as required by a principle of least action.
On the other hand, what then would be the double time integrals of square of energy: òòE²dtdt? Can this be the square of the action S²? Then S²=òEdtòEdt and combining integrands it becomes S²=òòE²dtdt. This would imply a closed time surface but obviously lack any physical meaning. Nevertheless, what conditions would then allow the physical meaning of square of energy?
These conditions became the transition from nonrelativistic to relativistic mechanics when Schrödinger proposed his quantum mechanical extension from the former to the latter in 1926. This proposal extends the nonrelativistic relations between energy and momentum of a free elementary particle: E=p²/2m (which is another expression for its kinetic energy as a function of momentum with constant mass) to the corresponding relativistic relation: E²=c²p²+(mc²)² where the right-hand side now includes the square of the rest-mass energy mc².
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
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