Geometry of the transport equation in multicomponent WKB approximations
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The semi-classical projection operator expression can be further solved to arrive at ε(x,p) = ± sqrt((cp - eA(x))^2 + m^2 c^4).
“Which can be further solved (Emmrich and Weinsein, 1996) to arrive at 2 2 4 ( , p) : ( ( )) x cp eA x m c (41)”
From Assessment of Zitterbewegung Interpretation for Free Particle Solution Using the Concept of Relativistic Wave (2024) · cites this paper as Emmrich and Weinsein, 1996
Section 2 (pp. 4–6) develops projection matrices for general Hamiltonians, and Section 3 (p. 8) discusses a general eigenvalue function λ. Searches for the electromagnetic vector potential, Dirac energy, and the claimed square-root expression found no passage in our copy deriving ε(x,p) = ±√((cp − eA(x))² + m²c⁴). (LLM, conf=0.73)
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References 20 · Cited by 89
Research works printed but not linked: 11
- [2][2] Berry, M.V., Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London A 392 (1984), 45-57. -
- [3][3] Bishop, R.L., and Crittenden, R.J., Geometry of Manifolds , Academic Press, New York, 1964. -
- [4][4] Braam, P.J., and Duistermaat, J.J., Normal forms of real symmetric systems with multiplicity, Indag. Math., New Series. 4 (1993), 407-421. -
- [5][5] Cordes, H.O., A pseudo-algebra of observables for the Dirac equation, Manuscripta Math. 45 (1983), 77-105. -
- [6][6] Dencker, N., On the propagation of polarization sets for systems of real principal type, J. Funct. Anal. 46 (1982), 351-372. -
- [10][10] Karasev, M.V., New global asymptotics and anomalies for the problem of quantization of the adiabatic invariant, Funct. Anal. Appl. 24 (1990), 104-114. -
- [11][11] Karasev, M.V., and Maslov, V.P., Operators with general commutation relations and their applications. I. Unitary-nonlinear operator equations, J. Sov. Math. 15 (1981), 273-368. -
- [12][12] Kobayashi, S., Differential Geometry of Complex Vector Bundles , Princeton University Press, Princeton, 1987. -
- [14][14] Littlejohn, R.G., and Flynn, W.G., Phase integral theory, coupled wave equations, and mode conversion. Chaos 2 (1992), 149-158. -
- [17][17] Simon, B., Holonomy, the quantum adiabatic theorem, and Berry's phase, Phys. Rev. Lett. 51 (1983), 2167-2170. -
- [19][19] Yan, D., Yang-Mills Theory on Symplectic Manifolds , PhD Thesis, Berkeley (in preparation).
- Authors
- Emmrich, C., & Weinstein, A.
- Affiliations
- University of California, Berkeley (US)
- Venue
- Communications in Mathematical Physics
- Published
- 1996-03
- Volume
- 176
- Issue
- 3
- Pages
- 701-711
- ISSN
- 0010-3616 (ISSN-L)
- 1432-0916
- OpenAlex
- W2019635950
- Cited by
- 89
- References
- 20
- Citations by year
- 2026: 3 · 2025: 1 · 2024: 5 · 2023: 2 · 2022: 2 · 2021: 3 · 2020: 2 · 2018: 1 · 2017: 2 · 2016: 2 · 2015: 1 · 2014: 7 · 2013: 1 · 2012: 4
- FWCI
- 3.94
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Communications in Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Physical Sciences › Mathematics › Mathematical Physics › Mathematical and Theoretical Analysis
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Mathematical Physics Problems
- Keywords
- WKB approximation, multicomponent systems, transport equation, principal symbol, second fundamental form, Poisson tensor, quantization condition, geometric derivation
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