Casimir effect in polymer scalar field theory

Abstract

In this paper, we study the Casimir effect in the classical geometry of two parallel conducting plates, separated by a distance L, due to the presence of a minimal length λ arising from a background independent (polymer) quantization scheme. To this end, we polymer-quantize the classical Klein-Gordon Hamiltonian for a massive scalar field confined between the plates and obtain the energy spectrum. The minimal length scale of the theory introduces a natural cutoff for the momenta in the plane parallel to the plates and a maximum number of discrete modes between the plates. The zero-point energy is calculated by summing over the modes, and by assuming λ L, we expressed it as an expansion in powers of 1/N , being N = L/λ the number of points between the plates. Closed analytical expressions are obtained for the Casimir energy in the cases of small and large scalar mass limits.

Quantum Groups and Polymer Quantum Mechanics

“…For other approaches see also [17]. Despite the rich literature, the general issue remains still completely open, and part of the focus is shifting on the so-called polymer quantization (PQ) [18,19] (that is a slightly simplified quantization inspired by LQG, that is reliable and heavily used [20][21][22][23][24][25][26]) and to its application to finite degrees of freedom systems, polymer quantum mechanics (PQM) [27][28][29]. PQ is based on the polymer representation of the Weyl-Heisenberg (WH) algebra, which is a non-regular representation, inequivalent to the standard Schrödinger or Fock-Bargmann representations [30].…”

Scalar Casimir effect for a conducting cylinder in a Lorentz violating background

“…Also, the Casimir effect stands as a potential handle to distinguish between Lorentz-invariant and Lorentz-violating formulations of quantum field theory. In this regard, the paradigmatic Casimir effect between two parallel conductive plates has been extensively studied in different scenarios, including spacetimes with nontrivial topologies [21,22], non-Euclidean spacetimes [23][24][25], string theory [26][27][28], and theories with minimal length [29][30][31], just to name a few. Concerning Lorentz-violating field theories, the Casimir effect has been attracted great attention as well.…”

Justification for zeta function regularization

“…One of the well-known examples of the application of zeta function regularization in physics is the Casimir force [5][6][7][8][9] , which is observed in the cavity between two metallic plates 10,11 . The Casimir force between two perfectly conducting plates at zero temperature (T = 0 K) is given by zeta function 12,13 to regularize infinite summation of zero-point energy 14 , in which a slowly decaying function is introduced to avoid divergence in the summation 5,12,13,15 . Another example of zeta function regularization is magnetization of graphene 16,17 , where the summation of the Landau levels (LLs) is diverging.…”