{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,19]],"date-time":"2026-10-19T20:50:00Z","timestamp":1792443000284,"version":"4.5.0"},"reference-count":8,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2015,2,9]],"date-time":"2015-02-09T00:00:00Z","timestamp":1423440000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>There is a well-known analogy between statistical and quantum mechanics. In statistical mechanics, Boltzmann realized that the probability for a system in thermal equilibrium to occupy a given state is proportional to \\(\\exp(-E\/kT)\\), where \\(E\\) is the energy of that state. In quantum mechanics, Feynman realized that the amplitude for a system to undergo a given history is proportional to \\(\\exp(-S\/i\\hbar)\\), where \\(S\\) is the action of that history. In statistical mechanics, we can recover Boltzmann's formula by maximizing entropy subject to a constraint on the expected energy. This raises the question: what is the quantum mechanical analogue of entropy? We give a formula for this quantity, which we call ``quantropy''. We recover Feynman's formula from assuming that histories have complex amplitudes, that these amplitudes sum to one and that the amplitudes give a stationary point of quantropy subject to a constraint on the expected action. Alternatively, we can assume the amplitudes sum to one and that they give a stationary point of a quantity that we call ``free action'', which is analogous to free energy in statistical mechanics. We compute the quantropy, expected action and free action for a free particle and draw some conclusions from the results.<\/jats:p>","DOI":"10.3390\/e17020772","type":"journal-article","created":{"date-parts":[[2015,2,9]],"date-time":"2015-02-09T10:17:04Z","timestamp":1423477024000},"page":"772-789","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Quantropy"],"prefix":"10.3390","volume":"17","author":[{"given":"John","family":"Baez","sequence":"first","affiliation":[{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore"},{"name":"Department of Mathematics, University of California, Riverside, CA 92521, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Blake","family":"Pollard","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,2,9]]},"reference":[{"key":"ref_1","unstructured":"Peat, F., and Hiley, B. Negative probability. Available online: http:\/\/cds.cern.ch\/record\/154856\/files\/pre-27827.pdf."},{"key":"ref_2","unstructured":"Feynman, R.P., and Hibbs, A.R. (1965). Quantum Mechanics and Path Integrals, McGraw-Hill."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hawking, S., and Penrose, R. (1996). The Nature of Space and Time, Princeton University Press.","DOI":"10.1038\/scientificamerican0796-60"},{"key":"ref_4","unstructured":"Jaynes, E.T. Available online: http:\/\/omega.albany.edu:8008\/JaynesBook.html."},{"key":"ref_5","unstructured":"Lisi, G. (Quantum mechanics from a universal action reservoir, 2006). Quantum mechanics from a universal action reservoir, arXiv:physics\/0605068."},{"key":"ref_6","unstructured":"M\u00f6rters, P., and Peres, Y. Available online: http:\/\/www.stat.berkeley.edu\/~peres\/bmbook.pdf."},{"key":"ref_7","first-page":"93","article-title":"Canonical relational quantum mechanics from information theory","volume":"8","author":"Munkhammar","year":"2011","journal-title":"Electron. J. Theor. Phys"},{"key":"ref_8","unstructured":"Reif, F. (1965). Fundamentals of Statistical and Thermal Physics, McGraw-Hill."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/2\/772\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:42:29Z","timestamp":1760215349000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/2\/772"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,9]]},"references-count":8,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,2]]}},"alternative-id":["e17020772"],"URL":"https:\/\/doi.org\/10.3390\/e17020772","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,2,9]]}}}