Lagrangian density functions in tri-space mechanics

These scalar functions have the dimensions of energy density (based on the Born probability density ψ^*ψ, where ψ^* is complex conjugate of wave function ψ) and the total Lagrangian is the sum of contributions from each of the connected spaces. They depend only on the wave function and its first derivatives, in each space. If we choose the tempospace (driving) term to be positive, then the real space and modal space terms are both negative.

The calculus of variations can be applied in each space separately, taken over any driven phase interval in tempospace and throughout real space, giving the Euler-Lagrange equation shown below. Variations in tempospace describe possible paths of the particle: variations in real space describe possible projected shapes. The two metrics require the condition of inter-connection (dr_s .dr_t =0) throughout the accessible manifold (e.g. for integrations).

This result is conditional on the vanishing of the WF towards the boundaries of integration, in real space. That condition, combined with the E-L equation for ψ^*, describes vector current densities in tempospace (j) and in real space (y). Temporal currents are directional vectors; spatial currents are either rotational or outward-spreading, always with zero averge. The E-L equation expresses the overall conservation of currents, which are shown here as real vectors:

Here t₀ is time in the rest (centre of mass) frame, where the total energy is E₀. The real space term is always missing, when working in the Minkowski metric. That difference can be patched over, but only for point objects, e.g. the Dirac equation for electrons.

The example below shows a simple boson, with scalar wave function ψ, driven in time t with internal mass M, projected into real space r_s and in the spatial potential is V(r_s):

Differential operators here act in the directions arrowed. The double ended arrow denotes the resultant - positive to the right and negative to the left.

Applying the E-L equation (top) to this Lagrangian density function gives the GEC shown on Home page 2. The corresponding vector current densities, in tempospace and in real space:

Lagrangian density functions can be constructed corresponding to all posible GECs in tri-space. Particles with intrinsic spin require multiple amplitudes of connection, leading to matrix connection algebra and this method is easily extended (in tri-space) to describe objects with any half-integer amount of internal spin. However, the description of the elementary particles must also involve the structure of modal space.

Tempospace driving functions are either fermion or boson, depending on the number of elementary fermions within the driven object being odd or even. Fermion drivers are 'odd', so that the wave function changes sign during a 360 degree rotation in tempospace, which is possible for any even number of driving amplitudes.

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Robert Herrod
Örkelljunga, Sweden, February 2022