Abstract
Physics explains the laws of motion that govern the time evolution of observable properties and the dynamical response of systems to various interactions. However, quantum theory separates the observable part of physics from the unobservable time evolution by introducing mathematical objects that are only loosely connected to the actual physics by statistical concepts and cannot be explained by any conventional sets of events. Here, I examine the relation between statistics and dynamics in quantum theory and point out that the Hilbert space formalism can be understood as a theory of ergodic randomization, where the deterministic laws of motion define probabilities according to a randomization of the dynamics that occurs in the processes of state preparation and measurement.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Y. Aharonov and L. Vaidman, J. Phys. A: Math. Gen. 24, 2315 (1991).
Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60 1351 (1988).
Y. Aharonov, S. Popescu, D. Rohrlich, and P. Skrzypczyk, New J. Phys. 15, 113015 (2013).
S.-Y. Baek, F. Kaneda, M. Ozawa, and K. Edamatsu, Sci. Rep. 3, 2221 (2013).
C. Bamber and J. S. Lundeen, Phys. Rev. Lett. 112, 070405 (2014)
J. S. Bell, Physics 1, 195 (1964).
P. Busch, P. Lahti, and R. F. Werner, Phys. Rev. Lett. 111, 160405 (2013).
P. Busch, P. Lahti, and R. F. Werner, Rev. Mod. Phys. 86, 1261 (2014).
T. Denkmayr, H. Geppert, S. Sponar, H. Lemmel, A. Matzkin, J. Tollaksen, and Y. Hasegawa, Nat. Commun. 5, 4492 (2014).
P. A. M. Dirac, Rev. Mod. Phys.17 195 (1945).
J. Dressel and F. Nori, Phys. Rev. A 89, 022106 (2014).
J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa, and Y. Hasegawa, Nat. Phys. 8, 185 (2012).
M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G.J. Pryde, Proc. Natl. Acad. Sci. U. S. A. 108 1256 (2011).
M. J. W. Hall, Phys. Rev. A 69, 052113 (2004).
L. Hardy, Phys. Rev. Lett. 68, 2981 (1992).
W. Heisenberg, Quantentheorie und Philosophie (Reclam, Stuttgart, 1979)
M. Hiroishi and H.F. Hofmann, J. Phys. A: Math. Theor. 46, 245302 (2013).
H. F. Hofmann, New J. Phys. 13, 103009 (2011).
H. F. Hofmann, Phys. Rev. Lett. 109, 020408 (2012).
H. F. Hofmann, New J. Phys. 14, 043031 (2012).
H. F. Hofmann, Phys. Rev. A 89, 042115 (2014).
H. F. Hofmann, New J. Phys. 16, 063056 (2014).
H. F. Hofmann, Quantum Stud. : Math. Found. 1, 39 (2014).
H. F. Hofmann, Phys. Rev. A 91, 062123 (2015).
H. F. Hofmann, Eur. Phys. J. D 70, 118 (2016).
M. Iinuma, Y. Suzuki, T. Nii, R. Kinoshita, and H. F. Hofmann, Phys. Rev. A 93, 032104 (2016).
A. N. Jordan, A. N. Korotkov, and M. Büttiker, Phys. Rev. Lett. 97, 026805 (2006).
S. Kino, T. Nii, and H. F. Hofmann, Phys. Rev. A 92, 042113 (2015).
J. G. Kirkwood, Phys. Rev.44, 31 (1933).
S. Kochen and E.P. Specker E P, J. Math. Mech. 17, 59 (1967).
A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985).
J. S. Lundeen and C. Bamber, Phys. Rev. Lett. 108, 070402 (2012).
J. S. Lundeen and A. M. Steinberg, Phys. Rev. Lett. 102, 020404 (2009).
J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature474, 188 (2011).
N. H. McCoy, Proc. Natl. Acad. Sci. U. S. A.18, 674 (1932).
T. Nii, M. Iinuma, and H. F. Hofmann, Quantum Stud.: Math. Found. 5, 229 (2018)
M. Ozawa, Phys. Rev. A 67, 042105 (2003).
K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Lett. A 324, 125 (2004).
M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard, and A. G. White, Phys. Rev. Lett. 112, 020401 (2014).
L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, Phys. Rev. Lett. 109, 100404 (2012).
J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, Nature Photon 7, 316 (2013).
Y. Suzuki, M. Iinuma, and H. F. Hofmann, New J. Phys. 14, 103022 (2012).
J. Tollaksen, J. Phys. A: Math. Gen. 40, 9033 (2007).
G. Vallone and D. Dequal, Phys. Rev. Lett. 116, 040502 (2016).
Y. Watanabe, T. Sagawa, and M. Ueda, Phys. Rev. A 84, 042121 (2011).
M. M. Weston, M. J. W. Hall, M. S. Palsson, H. M. Wiseman, and G. J. Pryde, Phys. Rev. Lett. 110, 220402 (2013).
N. S. Williams and A. N. Jordan, Phys. Rev. Lett. 100, 026804 (2008).
S. Wu, Sci. Rep. 3, 1193 (2013).
K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, New J. Phys. 11, 033011 (2009).
P. Zou, Z. Zhang, and W. Song, Phys. Rev. A 91, 052109 (2015).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Hofmann, H.F. (2018). Dynamics and Statistics in the Operator Algebra of Quantum Mechanics. In: Ozawa, M., Butterfield, J., Halvorson, H., Rédei, M., Kitajima, Y., Buscemi, F. (eds) Reality and Measurement in Algebraic Quantum Theory. NWW 2015. Springer Proceedings in Mathematics & Statistics, vol 261. Springer, Singapore. https://doi.org/10.1007/978-981-13-2487-1_8
Download citation
DOI: https://doi.org/10.1007/978-981-13-2487-1_8
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-2486-4
Online ISBN: 978-981-13-2487-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)