Abstract
Canonical quantization relies on Cartesian, canonical, phase-space coordinates to promote to Hermitian operators, which also become the principal ingredients in the quantum Hamiltonian. While generally appropriate, this procedure can also fail, e.g., for covariant, quartic, scalar fields in five-and-more spacetime dimensions (and possibly four spacetime dimensions as well), which become trivial; such failures are normally blamed on the ‘problem’ rather than on the ‘quantization procedure’. In Enhanced Quantization the association of c-numbers to q-numbers is chosen very differently such that: (i) there is no need to seek classical, Cartesian, phase-space coordinates; (ii) every classical, contact transformation is applicable and no change of the quantum operators arises; (iii) a new understanding of the importance of ‘Cartesian coordinates’ is established; and (iv) although discussed elsewhere in detail, the procedures of enhanced quantization offer fully acceptable solutions yielding non-trivial results for quartic scalar fields in four-and-more spacetime dimensions. In early sections, this paper offers a wide-audience approach to the basic principles of Enhanced Quantization using simple examples; later, several significant examples are cited for a deeper understanding. An historical note concludes the paper.
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Notes
- 1.
The simple form of \(d\sigma ^2\) owes much to the equation \((Q+iP)|0\rangle =0\) that defines the fiducial vector, but using a general, normalized, fiducial vector \(|\eta \rangle \), it follows that \(d\sigma ^2=(2/\hbar )[Adp^2+Bdpdq+Cdq^2]\), where \(A=\langle (\varDelta Q)^2\rangle \), \(B=\langle (\varDelta Q\varDelta P+\varDelta P\varDelta Q)\rangle \), and \(C=\langle (\varDelta P)^2\rangle \), with \(\langle (\cdot )\rangle \equiv \langle \eta |(\cdot ) |\eta \rangle \) and \(\varDelta X\equiv X-\langle X\rangle \). Clearly, a suitable linear change of the coordinates would lead to Cartesian coordinates.
- 2.
The term \(\mathcal{{O}}(\hbar ;p,q)\) could (i) significantly modify the nature of classical behavior, or (ii) in other systems in which different fiducial vectors occur, it could account for some ambiguity in the enhanced classical dynamical behavior; but since such terms arise at the quantum level, they are negligible for any macroscopic system.
- 3.
To appreciate the effect of motion in shifting the Fourier transform, one may recall that Christian Doppler hired a train, put a musical band on a flat car, and ran the train through a terminal where spectators clearly heard the band have a higher pitch as they approached and a lower pitch as they departed.
- 4.
This example is a toy model of gravity where \(q>0\) plays the role of the metric tensor with its positivity constraint while p plays the role of minus the Christoffel symbol [30]; note: this reference uses different notation.
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Acknowledgements
The author thanks T. Adorno, J. Ben Geloun, and G. Watson for their contributions to the enhanced quantization program and its consequences.
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Klauder, J.R. (2018). Enhanced Quantization: The Right way to Quantize Everything. In: Antoine, JP., Bagarello, F., Gazeau, JP. (eds) Coherent States and Their Applications. Springer Proceedings in Physics, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-319-76732-1_1
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