Klein-Gordon oscillators and Bergman spaces

• URL: http://arxiv.org/abs/2405.14349v3
• Date: Wed, 23 Oct 2024 09:52:55 GMT
• Title: Klein-Gordon oscillators and Bergman spaces
• Authors: Alexander D. Popov,
• Abstract summary: We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $mathbbR3,1$. The general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic functions on the K"ahler-Einstein manifold $Z_6$.
• Score: 55.2480439325792
• License:
• Abstract: We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $\mathbb{R}^{3,1}$. It is shown that for a non-zero frequency parameter $\omega$ the covariant phase space of the classical Klein-Gordon oscillator is a homogeneous K\"ahler-Einstein manifold $Z_6=\mathrm{Ad}S_7/\mathrm{U}(1)=\mathrm{U}(3,1)/\mathrm{U}(3)\times \mathrm{U}(1)$. In the limit $\omega\to 0$, this manifold is deformed into the covariant phase space $T^*H^3$ of a free relativistic particle, where $H^3=H^3_+\cup H_-^3$ is a two-sheeted hyperboloid in momentum space. Quantization of this model with $\omega\ne 0$ leads to the Klein-Gordon oscillator equation which we consider in the Segal-Bargmann representation. It is shown that the general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic (for antiparticles) functions on the K\"ahler-Einstein manifold $Z_6$. This relativistic model is Lorentz covariant, unitary and does not contain non-physical states.

Related papers

• Quantum charges of harmonic oscillators [55.2480439325792]
  We show that the energy eigenfunctions $psi_n$ with $nge 1$ are complex coordinates on orbifolds $mathbbR2/mathbbZ_n$. We also discuss "antioscillators" with opposite quantum charges and the same positive energy.
  arXiv  Detail & Related papers  (2024-04-02T09:16:18Z)
• Vacuum Force and Confinement [65.268245109828]
  We show that confinement of quarks and gluons can be explained by their interaction with the vacuum Abelian gauge field $A_sfvac$.
  arXiv  Detail & Related papers  (2024-02-09T13:42:34Z)
• Quantum connection, charges and virtual particles [65.268245109828]
  A quantum bundle $L_hbar$ is endowed with a connection $A_hbar$ and its sections are standard wave functions $psi$ obeying the Schr"odinger equation. We will lift the bundles $L_Cpm$ and connection $A_hbar$ on them to the relativistic phase space $T*R3,1$ and couple them to the Dirac spinor bundle describing both particles and antiparticles.
  arXiv  Detail & Related papers  (2023-10-10T10:27:09Z)
• Rigorous derivation of the Efimov effect in a simple model [68.8204255655161]
  We consider a system of three identical bosons in $mathbbR3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$.
  arXiv  Detail & Related papers  (2023-06-21T10:11:28Z)
• Heisenberg versus the Covariant String [0.0]
  A Poincar'e multiplet of mass eigenstates $bigl(P2 - m2bigr)Psi = 0$ cannot be a subspace of a space with a $D$-vector position operator $X=(X_0,dots X_D-1)$: the Heisenberg algebra $[Pm, X_n] = i deltam_n$ implies by a simple argument that each Poincar'e multiplet of definite mass vanishes.
  arXiv  Detail & Related papers  (2022-12-14T14:46:00Z)
• Systematics of quasi-Hermitian representations of non-Hermitian quantum models [0.0]
  This paper introduces and describes a set of constructive returns of the description to one of the correct and eligible physical Hilbert spaces $cal R_N(0)$. In the extreme of the theory the construction is currently well known and involves solely the inner product metric $Theta=Theta(H)$. At $j=N$ the inner-product metric remains trivial and only the Hamiltonian must be Hermitized, $H to mathfrakh = Omega,H,Omega-1=mathfrak
  arXiv  Detail & Related papers  (2022-12-07T20:10:58Z)
• Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions [68.8204255655161]
  We present a non-Hermitian PT-symmetric extension of the Nambu--Jona-Lasinio model of quantum chromodynamics in 3+1 and 1+1 dimensions. We find that in both cases, in 3+1 and in 1+1 dimensions, the inclusion of a non-Hermitian bilinear term can contribute to the generated mass.
  arXiv  Detail & Related papers  (2020-04-08T14:29:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.