Abstract
The metaplectic correction in Kähler quantization is a result of the introduction of the half-form structure when considering geometric quantization over Kähler manifolds. This correction is a very interesting topic, since no mathematical argument can prove or reject its necessity, and only a physical consideration can point to an answer. As a result, no universally accepted reason for considering such a structure has been identified in the context of quantization theory. In this letter, we view this topic not in a quantization context, but as a means to understand an exact connection between the canonical and path integral formulation of quantum mechanics. More specifically, we investigate whether or not the contribution of the metaplectic correction is related to the recent results found in the context of coherent-state path integrals, regarding the “correct” identification of the continuum limit. We then use the theory of coherent-state path integrals as a template, and our results indeed point to the necessity of this correction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and material
The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.
Notes
In many cases, the differences may just be overall unphysical phases, but in more complicated systems the differences are bound to be more significant. Thus, if one seeks an exact equality between these two quantities, a specific \(H(x^\nu (t))\) must be identified.
This representation can be identified by acting on the holomorphic coherent states, which are defined as \({|z\rangle _b=\sum _{n=0}^{\infty }\frac{z^n}{\sqrt{n!}}|n\rangle }\) and \({|z\rangle _s=\sum _{j=-s}^{s}\left[ \frac{(2s)!}{(s-j)!(s+j)!}\right] ^\frac{1}{2}z^{s-j}|s,j\rangle }\) for the bosonic and spin systems, respectively. The bases used in this construction are the usual mode basis \({\{|n\rangle \}}\), \(n=0,1,2,\dots \) of the harmonic oscillator and the z-direction component of spin basis \({\{|s,j\rangle \}}\), \(j=-s/2,\dots ,s/2\).
References
Woodhouse, N.M.J.: Geometric Quantization, Second Edition, 2nd edn. Oxford University Press Inc., New York (1997)
Blau, M.: Lecture Notes: Symplectic Geometry and Geometric Quantization (1992)
Nair, V.P.: arXiv:1606.06407 [hep-th] (2016)
Hall, B.C.: Quantum Theory for Mathematicians. Springer, New York (2013)
Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics; and Financial Markets. World Scientific Publishing Co. (2006)
Feynman, R.P.: Space–Time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367 (1948)
Feynman, R.P.: An operator calculus having applications in quantum electrodynamics. Phys. Rev. 84, 108 (1951)
Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766 (1963)
Berezin, A.: Feynman path integrals in a phase space. Sov. Phys. Usp. 23, 763 (1980)
Junker, G., Klauder, J.R.: Coherent-state quantization of constrained fermion systems. Eur. Phys. J. C 4, 173 (1998)
Klauder, J.R.: Path integrals and stationary-phase approximations. Phys. Rev. D 19, 2349 (1979)
Wilson, J.H., Galitski, V.: Breakdown of the coherent state path integral: two simple examples. Phys. Rev. Lett. 106, 11401 (2011)
Kordas, G., Mistakidis, S.I., Karanikas, A.I.: Coherent-state path integrals in the continuum. Phys. Rev. A 90, 032104 (2014)
Kordas, G., Kalantzis, D., Karanikas, A.I.: Fermionic path integrals and correlation dynamics in a 1D XY system. Ann. Phys. 372, 226 (2016)
Gosson, M. de .: The classical and quantum evolution of Lagrangian half-forms. Ann. Inst. Henri Poincaré 70(6) (1999)
Gosson, M. de .: The quantum motion of half-densities and the derivation of Schrödinger’s equation. J. Phys. A: Math. Gen. 31(2) (1998)
Gosson, M. de.: On half-form quantization of Lagrangian manifolds and quantum mechanics in phase space. Bull. Sci. Math. 121(4) (1997)
Funding
This research is co-financed by Greece and the European Union (European Social Fund—ESF) through the Operational Programme \(<<\)Human Resources Development, Education and Lifelong Learning\(>>\) in the context of the project ”Strengthening Human Resources Research Potential via Doctorate Research” (MIS-5000432), implemented by the State Scholarships Foundation (IKY).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lyris, I., Lykourgias, P. & Karanikas, A.I. The importance of the metaplectic correction in Kähler quantization: a coherent-state path integral perspective. Lett Math Phys 111, 25 (2021). https://doi.org/10.1007/s11005-021-01368-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11005-021-01368-3