The classical and quantum formalism for a harmonic oscillator with time-dependent frequency has attracted significant interest through past decades. In [1] general formulae for real, $p$-adic and adelic propagators for this system were obtained. Here we consider a rather general system - a time-dependent harmonic oscillator. Instead of a canonical Lagrangian our consideration began with a system described by a DBI Lagrangian, motivated by certain applications found in the cosmological framework, more precisely in the theory of inflation [2,3]. This approach leads to a class of nonlinear systems.
We will present new results and a rather general formula for a time-dependent harmonic oscillator, motivated by the idea of "locally equivalent" Lagrangians, which enables us to present the propagator in a quadratic form (see references [2,3], as well as references therein). We conclude with a discussion on further applications in the theory of inflation, as well as in $p$-adic and adelic quantum mechanics.
References
[1] Goran S. Djordjevic, Branko Dragovich, $p$-Adic and Adelic Harmonic Oscillator with Time-Dependent Frequency, Theor. Math. Phys. 124 (2000) 1059 [2] G. S. Djordjevic, D. D. Dimitrijevic and M. Milosevic, On Canonical Transformation and Tachyon-Like "Particles" in Inflationary Cosmology, Romanian Journal of Physics, Vol 61, No 1-2 (2016) 99 [3] D. D. Dimitrijevic G. S. Djordjevic and M. Milosevic, Classicalization and quantization of Tachyon-Like matter on (non)Archimedean spaces, Romanian Reports in Physics, Vol. 68 No. 1 (2016), 5