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By Ruark A.E.

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Extra resources for A Critical Experiment on the Statistical Interpretation of Quantum Mechanics

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S. 21) for x is used. The four-vectors ξα (x) = (ξα (x)), where α = 1 . . s, are called Killing four-vectors. Dα u b (x) is called Lie derivative of u b in the direction ξα at the point x. 20). By S [u] we denote the action functional calculated for the fields u a on the whole space R 3 in the time interval [t , t ]: S [u] = d 4 x L(u a (x), ∂μ u a (x); x), where = {(ct, x) : t ∈ [t , t ], x ∈ R 3 }. 19) acting on gives a new region : = f ( ; ω). The action functional calculated for the new functions u a (x ) in the new region has the form S [u ] = d 4 x L(u a (x ), ∂u a (x ) ; x ).

Is Kronecker delta, while the second one is Dirac four-dimensional delta. 26) implies that δS [u ] δu a (x ) u (x )=F(u(x);ω) = 0, but this means that u a (x ) obeys the Euler–Lagrange equations in the region . 25) we derive the so called Noether’s identity. s. of this identity gives an explicit formula for the integrals of motion. s. 25) with respect to ωα . 19), that is J = det ∂x μ . 27) where δx μ = ωα ξαμ (x). 28) Here and in the subsequent calculations, the multi-dots denote terms of the second ∂u (x ) or higher order in ωα .

On-shell’ would mean that u a (x) were solutions of the Euler–Lagrange equations. s. of this formula is called the surface term. With the help of Stokes theorem it can also be written as the four-dimensional volume integral ∂ d Sμ K μ (u; x; ω) = d4x dKρ , dxρ where d/d x ρ denotes the total derivative. 25) might seem quite strange. Similarly as in the case of stationary action principle, its origin lies in quantum mechanics. In particular, the surface term can be related to a change of phase factor of state vectors.

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