Equivalence between classical and quantum dynamics. Neutral kaons and electric circuits

Abstract

An equivalence between the Schrödinger dynamics of a quantum system with a finite number of basis states and a classical dynamics is presented. The equivalence is an isomorphism that connects in univocal way both dynamical systems. We treat the particular case of neutral kaons and found a class of electric networks uniquely related to the kaon system finding the complete map between the matrix elements of the effective Hamiltonian of kaons and those elements of the classical dynamics of the networks. As a consequence, the relevant ǫ parameter that measures CP violation in the kaon system is completely determined in terms of network parameters.

Demonstration of ${\rm K^0}{\overline{\rm K}}{}^0$, ${\rm B^0}{\overline{\rm B}}{}^0$ and ${\rm D^0}{\overline{\rm D}}{}^0$ transitions with a pair of coupled pendula

“…Examples for coupled and damped oscillations with a CPT-symmetric and T-violating equation of motion are given in the literature; a Foucault pendulum with damping in one direction [24,25], a ball rolling in a bowl on a turntable [23], and an electrical setup in which the non-reciprocity in the coupling between two resonant circuits is achieved by a gyrator [23,26]. To our knowledge, none of the proposed demonstration setups has been built so far.…”

The quantum CP-violating kaon system reproduced in the electronic laboratory

“…One can eventually take two identical classical systems but prepared with different initial conditions. We see that is possible establish a bridge between this two systems of two states (|1 , |2 ) and (q 1 , q 2 ) via the isomorphism Φ presented in [6]. This bridge can be established to translate (as a dictionary) two systems with any number of denumberable states.…”

“…where K ∈ C 2x2 , with elements K ij = −ı i|H|j . In order to correctly state the equivalence with a classical system it is necessary to perform a decomplexification [6]. Consequently, the vector decomplexification map D : C 2 −→ R 4 , gives rise to D(ψ) = (ϕ ϕ ϕ 1 , ϕ ϕ ϕ 2 ) with ϕ ϕ ϕ 1 = ( (ψ 1 ), (ψ 2 )) , ϕ ϕ ϕ 2 = ( (ψ 1 ), (ψ 2 )) and denotes the matrix transposition.…”

“…Consequently, the vector decomplexification map D : C 2 −→ R 4 , gives rise to D(ψ) = (ϕ ϕ ϕ 1 , ϕ ϕ ϕ 2 ) with ϕ ϕ ϕ 1 = ( (ψ 1 ), (ψ 2 )) , ϕ ϕ ϕ 2 = ( (ψ 1 ), (ψ 2 )) and denotes the matrix transposition. We use the decomplexification introduced by Arnold [12] that is equivalent to the process presented in [6]. Equation (1) can be written as…”

“…To complete the physical description it is necessary to give the initial condition for the evolution. All the information on the decay channels is contained in (6) as is clear from the matrix elements of M − ıΓ. The CP T −symmetry implies that M 11 = M 22 and Γ 11 = Γ 22 , while if T (CP ) would be also a symmetry, then M 12 = M * 12 and Γ 12 = Γ * 12 , where z * is conjugate of the complex number z.…”