---
_id: '2133'
abstract:
- lang: eng
text: "Let ℭ denote the Clifford algebra over ℝ\U0001D45B, which is the von Neumann
algebra generated by n self-adjoint operators Q j , j = 1,…,n satisfying the canonical
anticommutation relations, Q i Q j + Q j Q i = 2δ ij I, and let τ denote the
normalized trace on ℭ. This algebra arises in quantum mechanics as the algebra
of observables generated by n fermionic degrees of freedom. Let \U0001D513 denote
the set of all positive operators \U0001D70C∈ℭ such that τ(ρ) = 1; these are the
non-commutative analogs of probability densities in the non-commutative probability
space (ℭ,\U0001D70F). The fermionic Fokker–Planck equation is a quantum-mechanical
analog of the classical Fokker–Planck equation with which it has much in common,
such as the same optimal hypercontractivity properties. In this paper we construct
a Riemannian metric on \U0001D513 that we show to be a natural analog of the classical
2-Wasserstein metric, and we show that, in analogy with the classical case, the
fermionic Fokker–Planck equation is gradient flow in this metric for the relative
entropy with respect to the ground state. We derive a number of consequences of
this, such as a sharp Talagrand inequality for this metric, and we prove a number
of results pertaining to this metric. Several open problems are raised."
author:
- first_name: Eric
full_name: Carlen, Eric
last_name: Carlen
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
citation:
ama: Carlen E, Maas J. An analog of the 2-Wasserstein metric in non-commutative
probability under which the fermionic Fokker-Planck equation is gradient flow
for the entropy. *Communications in Mathematical Physics*. 2014;331(3):887-926.
doi:10.1007/s00220-014-2124-8
apa: Carlen, E., & Maas, J. (2014). An analog of the 2-Wasserstein metric in
non-commutative probability under which the fermionic Fokker-Planck equation is
gradient flow for the entropy. *Communications in Mathematical Physics*.
Springer. https://doi.org/10.1007/s00220-014-2124-8
chicago: Carlen, Eric, and Jan Maas. “An Analog of the 2-Wasserstein Metric in Non-Commutative
Probability under Which the Fermionic Fokker-Planck Equation Is Gradient Flow
for the Entropy.” *Communications in Mathematical Physics*. Springer, 2014.
https://doi.org/10.1007/s00220-014-2124-8.
ieee: E. Carlen and J. Maas, “An analog of the 2-Wasserstein metric in non-commutative
probability under which the fermionic Fokker-Planck equation is gradient flow
for the entropy,” *Communications in Mathematical Physics*, vol. 331, no.
3. Springer, pp. 887–926, 2014.
ista: Carlen E, Maas J. 2014. An analog of the 2-Wasserstein metric in non-commutative
probability under which the fermionic Fokker-Planck equation is gradient flow
for the entropy. Communications in Mathematical Physics. 331(3), 887–926.
mla: Carlen, Eric, and Jan Maas. “An Analog of the 2-Wasserstein Metric in Non-Commutative
Probability under Which the Fermionic Fokker-Planck Equation Is Gradient Flow
for the Entropy.” *Communications in Mathematical Physics*, vol. 331, no.
3, Springer, 2014, pp. 887–926, doi:10.1007/s00220-014-2124-8.
short: E. Carlen, J. Maas, Communications in Mathematical Physics 331 (2014) 887–926.
date_created: 2018-12-11T11:55:54Z
date_published: 2014-11-01T00:00:00Z
date_updated: 2021-01-12T06:55:30Z
day: '01'
doi: 10.1007/s00220-014-2124-8
extern: '1'
intvolume: ' 331'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: 'http://arxiv.org/abs/1203.5377 '
month: '11'
oa: 1
oa_version: Submitted Version
page: 887 - 926
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4901'
quality_controlled: '1'
status: public
title: An analog of the 2-Wasserstein metric in non-commutative probability under
which the fermionic Fokker-Planck equation is gradient flow for the entropy
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 331
year: '2014'
...