Quantum stochastics: the passage from a relativistic to a non-relativistic path integral
Abstract
Feynman's path integral for the one-dimensional Dirac particle uses paths for which Delta x approximately Delta t. For non-relativistic path integrals, typical paths satisfy ( Delta x) 2 approximately Delta t (as in Brownian motion). The authors demonstrate the consistency of these two stochastic schemes and show how the relativistic formalism contains within itself the scale of the transition regime, namely Delta x approximately h(cross)/mc, the Compton wavelength.
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The initial condition most widely studied, with the walker localized at one position, has (∆x)_0 ∼ 1 < λ, leading to a relativistic quantum walk.
“Thus, the initial condition most widely studied [11], with the walker localized at one position, has (∆ x ) 0 ∼ 1 < λ , leading to a relativistic quantum walk.”
From Relativistic quantum walks (2006) · cites this paper as [11]
The Hadamard walk's dispersion relation sin ω(k)=sin k/√2 does not satisfy ω(k)≃ω(0)+k^2/(2m) for small k, but rather ω(k)≃k/√2 - k^3/(12√2).
“the Hadamard walk's dispersion relation sin ω ( k ) = sin k/ √ 2 [11] does not satisfy ω ( k ) ≃ ω (0)+ k 2 / (2 m ) for small k , but rather ω ( k ) ≃ k/ √ 2 -k 3 / (12 √ 2).”
From Relativistic quantum walks (2006) · cites this paper as [11]
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Bialynicki-Birula, I. (1994) · Physical Review D · doi:10.1103/physrevd.49.6920 · Cited by 195 · see citing works
Strauch, F. W. (2006) · Physical Review A · doi:10.1103/physreva.73.054302 · Cited by 115 · see citing works
Free-Dirac-particle evolution as a quantum random walk
Bracken, A. J., Ellinas, D., & Smyrnakis, I. (2007) · Physical Review A · doi:10.1103/physreva.75.022322 · Cited by 60 · see citing works
- Authors
- Jacobson, T., & Schulman, L. S.
- Affiliations
- Technion – Israel Institute of Technology (IL)
- ORCID
- Schulman, L S
- Venue
- Journal of Physics A: Mathematical and General
- Publisher
- IOP Publishing
- Published
- 1984-02-01
- Volume
- 17
- Issue
- 2
- Pages
- 375-383
- ISSN
- 0305-4470 (ISSN-L)
- 1361-6447
- OpenAlex
- W2091895004
- Cited by
- 132
- References
- 0
- Citations by year
- 2025: 1 · 2024: 3 · 2023: 2 · 2022: 5 · 2021: 4 · 2020: 2 · 2019: 1 · 2017: 5 · 2016: 1 · 2015: 2 · 2014: 5 · 2013: 8 · 2012: 6
- FWCI
- 1.84
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Physics A Mathematical and General (publishedVersion)
- Funders
- National Science Foundation
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- Dirac particles, Compton wavelength, Brownian motion, transition regime, path integral
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