Deformation quantization of the Pais–Uhlenbeck fourth order oscillator

Abstract

We analyze the quantization of the Pais-Uhlenbeck fourth order oscillator within the framework of deformation quantization. Our approach exploits the Noether symmetries of the system by proposing integrals of motion as the variables to obtain a solution to the ⋆-genvalue equation, namely the Wigner function. We also obtain, by means of a quantum canonical transformation the wave function associated to the Schrödinger equation of the system. We show that unitary evolution of the system is guaranteed by means of the quantum canonical transformation and via the properties of the constructed Wigner function, even in the so called equal frequency limit of the model, in agreement with recent results.

The Unruh effect for higher derivative field theory

“…However, by taking the equal frequency limit in our formulation we will have some insights that the Unruh effect will not emerge in this case. Further, as discussed in [17], even though for the classical equal frequency limit the canonical transformation diverges, at the quantum level the canonical transformation results well-defined in this limit allowing to pass the wave functions from the discrete to the continuous case, thus supporting our claims about the Unruh effect in the following sections. We will include appropriate comments whenever necessary in order to clarify this issue.…”

“…From now on, and without loss of generality, we will consider ω 1 > ω 2 . As we will see, the equal frequency limit results inadequately defined when obtained from the general perspective ω 1 = ω 2 as the quantum behaviour results completely different [16], [17]. As shown in these references, in the equal frequency limit case the quantum Hamiltonian may be decomposed into a part proportional to an angular momentum term with discrete spectrum and a part proportional to a vector norm term with continuous spectrum, in opposition to the case of our interest for which only a discrete spectrum is obtained.…”

“…Finally, we note that the term ∆ = ω 2 1 − ω 2 2 appearing in (10) becomes singular in the equal frequencies limit, this limit corresponds to the case of equal masses for a given mode. As stated in [16], [17], in the equal frequencies case, the behaviour of the Pais-Uhlenbeck system results completely different as the spectrum of the Hamiltonian is composed of a discrete spectrum part coming from an angular momentum term, and of a continuous part originated from a different term given by the squared norm of the position variables.…”

Unruh effect detection through chirality in curved graphene

“…Indeed, even at the classical level the Hamiltonian for the Pais-Uhlenbeck oscillator results different in nature whenever we consider different or equal frequencies from the beginning. As discussed in [25] (see also [22] and [26]) this inequivalence at the Hamiltonian level reflects in different spectrum for the quantum Hamiltonian operators corresponding to either the different or equal frequencies cases. Thus, as mentioned before, we will only consider here the different frequencies (ω 1 > ω 2 ) case, and from this perspective we will only provide some arguments pointing towards the absence of the Unruh effect in the equal frequencies limit.…”

Nonlocal dynamics and infinite nonrelativistic conformal symmetries

“…As a result the commutation rules (4.25) are converted into the following 27) in which we recognize two representations of the Virasoro algebra acting on C ± 's (with the nonhomogeneous part ± 1 4 , see, e.g., [50]). At this point, let us recall that the PU oscillator with the odd frequencies enjoys the (N − )-conformal non-relativistic symmetry which, for fixed N, possesses infinitedimensional extensions (see, [42]- [49]).…”