49. Cylindrical Wigner Measures

  • Marco Falconi Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Keywords: Infinite Dimensional Semiclassical Analysis, CCR algebra, Constructive Quantum Field Theory, Wigner measures

Abstract

In this paper we study the semiclassical behavior of quantum states acting on the C*-algebra of canonical
commutation relations, from a general perspective. The aim is to provide a unified and flexible approach
to the semiclassical analysis of bosonic systems. We also give a detailed overview of possible
applications of this approach to mathematical problems of both axiomatic relativistic quantum field
theories and nonrelativistic many body systems. If the theory has infinitely many degrees of freedom, the
set of Wigner measures, \emph{i.e.}\ the classical counterpart of the set of quantum states, coincides
with the set of all cylindrical measures acting on the algebraic dual of the space of test functions for
the field, and this reveals a very rich semiclassical structure compared to the finite-dimensional
case. We characterize the cylindrical Wigner measures and the \emph{a priori} properties they inherit from
the corresponding quantum states.

References

\bibitem[Acerbi et~al.(1993)]{acerbi1993jmp}
F.~Acerbi, G.~Morchio, and F.~Strocchi.
Infrared singular fields and nonregular representations of canonical
commutation relation algebras.
\emph{J. Math. Phys.}, 34(3), 899--914,
1993}.
DOI 10.1063/1.530200;
zbl 0792.46055;
MR1207957

\bibitem[Acerbi et~al.(1993)]{acerbi1993rmp}
F.~Acerbi, G.~Morchio, and F.~Strocchi.
Nonregular representations of {CCR} algebras and algebraic fermion
bosonization.
In \emph{Proceedings of the {XXV} {S}ymposium on {M}athematical
{P}hysics ({T}oru\'n, 1992)}, volume~33, pages 7--19, 1993.
DOI 10.1016/0034-4877(93)90036-E;
zbl 0816.46077;
MR1268670

\bibitem[Ammari and Falconi(2014)]{ammari2014jsp}
Z.~Ammari and M.~Falconi.
{W}igner measures approach to the classical limit of the {N}elson
model: {C}onvergence of dynamics and ground state energy.
\emph{J. Stat. Phys.}, 157(2):330--362, 2014.
DOI 10.1007/s10955-014-1079-7};
zbl 1302.82009;
MR3255099;
arxiv 1403.2327

\bibitem[Ammari and Falconi(2017)]{ammari2017sima}
Z.~Ammari and M.~Falconi.
{Bohr's correspondence principle for the renormalized Nelson model}.
\emph{SIAM J. Math. Anal.}, 49(6):5031--5095,
2017.
DOI 10.1137/17M1117598;
zbl 1394.35383;
MR3737034

\bibitem[Ammari and Liard(2018)]{ammari2016barxiv}
Z.~Ammari and Q.~Liard.
On the uniqueness of probability measure solutions to {L}iouville's
equation of {H}amiltonian {PDE}s.
\emph{Discrete Contin. Dyn. Syst.}, 38(2): 723--748, 2018.
DOI 10.3934/dcds.2018032;
MR3721874
arxiv 1602.06716

\bibitem[Ammari and Nier(2008)]{ammari2008ahp}
Z.~Ammari and F.~Nier.
Mean field limit for bosons and infinite dimensional phase-space
analysis.
\emph{Ann. Henri Poincar\'e}, 9(8): 1503--1574, 2008.
DOI 10.1007/s00023-008-0393-5;
zbl 1171.81014;
MR2465733;
arxiv 0711.4128

\bibitem[Ammari and Nier(2009)]{ammari2009jmp}
Z.~Ammari and F.~Nier.
Mean field limit for bosons and propagation of {W}igner measures.
\emph{J. Math. Phys.}, 50(4): 042107, 16 p., 2009.
DOI 10.1063/1.3115046;
zbl 1214.81089;
MR2513969;
arxiv 0807.3108

\bibitem[Ammari and Nier(2011)]{ammari2011jmpa}
Z.~Ammari and F.~Nier.
Mean field propagation of {W}igner measures and {BBGKY} hierarchies
for general bosonic states.
\emph{J. Math. Pures Appl. (9)}, 95(6): 585--626, 2011.
DOI 10.1016/j.matpur.2010.12.004;
zbl 1251.81062;
MR2802894;
arxiv 1003.2054

\bibitem[Ammari and Nier(2015)]{ammari2015asns}
Z.~Ammari and F.~Nier.
Mean field propagation of infinite-dimensional {W}igner measures with
a singular two-body interaction potential.
\emph{Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)}, 14(1): 155--220, 2015.
DOI 10.2422/2036-2145.201112_004;
zbl 1341.81037;
MR3379490;
arxiv 1111.5918

\bibitem[Ammari and Zerzeri(2014)]{ammari2014hok}
Z.~Ammari and M.~Zerzeri.
On the classical limit of self-interacting quantum field
{H}amiltonians with cutoffs.
\emph{Hokkaido Math. J.}, 43(3): 385--425, 2014.
DOI 10.14492/hokmj/1416837571;
zbl 1304.81114;
MR3282640;
arxiv https://arxiv.org/abs/1210.5441

\bibitem[Ammari et~al.(2016)Ammari, Falconi, and Pawilowski]{ammari2016cms}
Z.~Ammari, M.~Falconi, and B.~Pawilowski.
On the rate of convergence for the mean field approximation of
bosonic many-body quantum dynamics.
\emph{Commun. Math. Sci.}, 14(5): 1417--1442, 2016.
DOI 10.4310/CMS.2016.v14.n5.a9;
zbl 1385.81013;
MR3506807;
arxiv 1411.6284


\bibitem[Ammari et~al.(2019)Ammari, Breteaux, and Nier]{ammari2017arxiv}
Z.~Ammari, S.~Breteaux, and F.~Nier.
Quantum mean field asymptotics and multiscale analysis.
\emph{Tunisian J. Math.}, 1(2): 221--272, 2019.
DOI 10.2140/tunis.2019.1.221;
arxiv 1701.06423

\bibitem[Amour et~al.(2015)Amour, Jager, and Nourrigat]{amour2015jfa}
L.~Amour, L.~Jager, and J.~Nourrigat.
On bounded pseudodifferential operators in {W}iener spaces.
\emph{J. Funct. Anal.}, 269(9): 2747--2812, 2015.
DOI 10.1016/j.jfa.2015.08.004;
zbl 1328.47052;
MR3394620;
arxiv 1412.1577

\bibitem[Amour et~al.(2016)Amour, Lascar, and Nourrigat]{amour2016amrex}
L.~Amour, R.~Lascar, and J.~Nourrigat.
Beals Characterization of Pseudodifferential Operators in Wiener
Spaces.
\emph{Applied Mathematics Research eXpress}, AMRX, no. 1: 242--270, 2017.
\url{http://amrx.oxfordjournals.org/content/early/2016/02/07/amrx.abw001.abstract}.
DOI 10.1093/amrx/abw001;
zbl 06943997;
MR3748494;
arxiv 1507.02567

\bibitem[Amour et~al.(2017)Amour, Lascar, and Nourrigat]{amour2017jmp}
L.~Amour, R.~Lascar, and J.~Nourrigat.
Weyl calculus in {QED} {I}. {T}he unitary group.
\emph{J. Math. Phys.}, 58(1): 013501, 24, 2017.
DOI 10.1063/1.4973742;
zbl 1355.81158;
MR3595177;
arxiv 1510.05293

\bibitem[Benedikter et~al.(2016)Benedikter, Porta, and
Schlein]{benedikter2016sbmp}
N.~Benedikter, M.~Porta, and B.~Schlein.
\emph{Effective evolution equations from quantum dynamics}, volume~7
of \emph{SpringerBriefs in Mathematical Physics}.
Springer, Cham, 2016.
DOI 10.1007/978-3-319-24898-1;
zbl 1396.81003;
MR3382225;
arxiv 1502.02498

\bibitem[Berezin(1971)]{berezin1971ms}
F.~A. Berezin.
Wick and anti-{W}ick symbols of operators.
\emph{Mat. Sb. (N.S.)}, 86(128): 578--610, 1971.
DOI 10.1070/SM1971v015n04ABEH001564;
zbl 0247.47018;
MR0291839

\bibitem[Bohr(1947)]{bohr1947apf}
H.~Bohr.
\emph{Almost {P}eriodic {F}unctions}.
Chelsea Publishing Company, New York, N.Y., 1947.
MR0020163

\bibitem[Bourbaki(1969)]{bourbaki1969int9}
N.~Bourbaki.
\emph{\'{E}l\'ements de math\'ematique. {F}asc. {XXXV}. {L}ivre {VI}:
{I}nt\'egration. {C}hapitre {9}: {I}nt\'egration sur les espaces topologiques
s\'epar\'es}.
\newblock Actualit\'es Scientifiques et Industrielles, No. 1343. Hermann,
Paris, 1969.
zbl 0189.14201;
MR0276436

\bibitem[Bourbaki(1970)]{bourbaki1970ens}
N.~Bourbaki.
\emph{\'{E}l\'ements de math\'ematique. {T}h\'eorie des ensembles}.
Hermann, Paris, 1970.
zbl 0282.04001;
MR0276101

\bibitem[Bourbaki(1981)]{bourbaki1981evt1-5}
N.~Bourbaki.
\emph{{\'E}l{\'e}ments de math{\'e}matique. {E}spaces vectoriels
topologiques. {C}hapitres 1 \`a 5}.
Masson, Paris, new edition, 1981.
{\'E}l{\'e}ments de math{\'e}matique. [Elements of mathematics].
zbl 0482.46001

\bibitem[Bratteli and Robinson(1987)]{bratteli1987tmp1}
O.~Bratteli and D.~W. Robinson.
\emph{Operator algebras and quantum statistical mechanics. 1. {C*}- and {W*}-algebras, symmetry groups, decomposition of states}.
Texts and Monographs in Physics. Springer-Verlag, New York, second
edition edition, 1987.
DOI 10.1007/978-3-662-02520-8;
zbl 0905.46046;
MR0887100

\bibitem[Bratteli and Robinson(1997)]{bratteli1997tmp2}
O.~Bratteli and D.~W. Robinson.
\emph{Operator algebras and quantum statistical mechanics. 2. Equilibrium states. Models in quantum statistical mechanics}.
Texts and Monographs in Physics. Springer-Verlag, Berlin, second
edition edition, 1997.
DOI 10.1007/978-3-662-03444-6;
zbl 0903.46066

\bibitem[Brunetti and Fredenhagen(2009)]{brunetti2009lnp}
R.~Brunetti and K.~Fredenhagen.
Quantum field theory on curved backgrounds.
In \emph{Quantum field theory on curved spacetimes}, volume 786 of
\emph{Lecture Notes in Phys.}, pages 129--155. Springer, Berlin, 2009.
DOI 10.1007/978-3-642-02780-2_5;
zbl 1184.81099;
MR2581535;
arxiv 0901.2063

\bibitem[Brunetti et~al.(2014)Brunetti, Fredenhagen, Imani, and
Rejzner]{brunetti2014rmp}
R.~Brunetti, K.~Fredenhagen, P.~Imani, and K.~Rejzner.
The locality axiom in quantum field theory and tensor products of
{C*}-algebras.
\emph{Rev. Math. Phys.}, 26(6): 1450010, 10, 2014.
DOI 10.1142/S0129055X1450010X;
zbl 1308.46063;
MR3228281;
arxiv 1206.5484

\bibitem[Buchholz and Grundling(2008)]{buchholz2008jfa}
D.~Buchholz and H.~Grundling.
The resolvent algebra: a new approach to canonical quantum systems.
\emph{J. Funct. Anal.}, 254(11): 2725--2779, 2008.
DOI 10.1016/j.jfa.2008.02.011;
zbl 1148.46032;
MR2414219;
arxiv 0705.1988

\bibitem[Colin~de Verdi{\`e}re(1985)]{colindeverdiere1985cmp}
Y.~Colin~de Verdi{\`e}re.
Ergodicit\'e et fonctions propres du laplacien.
\emph{Comm. Math. Phys.}, 102(3): 497--502, 1985.
\url{http://projecteuclid.org/euclid.cmp/1104114465}.
DOI 10.1007/BF01209296;
zbl 0592.58050;
MR0818831

\bibitem[Correggi and Falconi(2018)]{correggi2017ahp}
M.~Correggi and M.~Falconi.
{Effective Potentials Generated by Field Interaction in the
Quasi-Classical Limit}.
\emph{Ann. Henri Poincar{\'e}}, 19(1): 189--235,
2018.
DOI 10.1007/s00023-017-0612-z;
zbl 1392.81132;
MR3743758;
arxiv 1701.01317

\bibitem[Correggi et~al.(2019)Correggi, Falconi, and
Olivieri]{correggi2017arxiv}
M.~Correggi, M.~Falconi, and M.~Olivieri.
{Magnetic Schr\"odinger Operators as the Quasi-Classical Limit of
Pauli-Fierz-type Models}.
\emph{J. Spectr. Theory}, 2019. To appear.
arxiv 1711.07413

\bibitem[Dappiaggi et~al.(2011)Dappiaggi, Pinamonti, and
Porrmann]{dappiaggi2011cmp}
C.~Dappiaggi, N.~Pinamonti, and M.~Porrmann.
Local causal structures, {H}adamard states and the principle of local
covariance in quantum field theory.
\emph{Comm. Math. Phys.}, 304(2): 459--498, 2011.
DOI 10.1007/s00220-011-1235-8;
zbl 1233.81033;
MR2795329;
arxiv 1001.0858

\bibitem[Donald(1981)]{donald1981cmp}
M.~Donald.
The classical field limit of {$P(\varphi )_{2}$} quantum field
theory.
\emph{Comm. Math. Phys.}, 79(2): 153--165, 1981.
\url{http://projecteuclid.org/getRecord?id=euclid.cmp/1103908960}.
MR0612245

\bibitem[Eckmann(1975)]{eckmann1975lmp}
J.-P. Eckmann.
Remarks on the classical limit of quantum field theories.
\emph{Lett. Math. Phys.}, 1(5): 387--394, 1975.
DOI 10.1007/BF01793952;
MR0438945

\bibitem[Falconi(2013)]{falconi2013jmp}
M.~Falconi.
Classical limit of the {N}elson model with cutoff.
\emph{J. Math. Phys.}, 54(1): 012303, 30, 2013.
DOI 10.1063/1.4775716;
zbl 1280.81159;
MR3059880;
arxiv 1205.4367

\bibitem[Falconi(2016)]{falconi2016bpmas}
M.~Falconi.
{Semiclassical Analysis in Infinite Dimensions: Wigner Measures}.
\emph{BPMAS}, 2016.
\url{https://mathematicalanalysis.unibo.it/article/view/6686}.
DOI 10.6092/issn.2240-2829/6686;
zbl 1371.60004;
MR3631061

\bibitem[Falconi(2018)]{falconi2017ccm}
M.~Falconi.
{Concentration of cylindrical Wigner measures}.
\emph{Commun. Contemp. Math.}, 20(5): 1750055, 2018.
DOI 10.1142/S0219199717500559;
zbl 1394.81125;
MR3833903;
arxiv 1704.07676

\bibitem[Fannes and Verbeure(1974)]{fannes1974cmp}
M.~Fannes and A.~Verbeure.
On the time evolution automorphisms of the {CCR}-algebra for quantum
mechanics.
\emph{Comm. Math. Phys.}, 35: 257--264, 1974.
\url{http://projecteuclid.org/euclid.cmp/1103859595}.
MR0334743

\bibitem[Fermanian-Kammerer and G\'erard(2002)]{fermanian2002bsmf}
C.~Fermanian-Kammerer and P.~G\'erard.
Mesures semi-classiques et croisement de modes.
\emph{Bull. Soc. Math. France}, 130(1): 123--168,
2002.
DOI 10.24033/bsmf.2416;
zbl 0996.35004;
MR1906196

\bibitem[Fredenhagen and Rejzner(2011)]{fredenhagen2011qftg}
K.~Fredenhagen and K.~Rejzner.
Local covariance and background independence.
In \emph{Quantum field theory and gravity: Conceptual and
Mathematical Advances in the Search for a Unified Framework}. Birkh\"auser
Verlag, 2011.
zbl 1246.83075;
MR3074844;
arxiv 1102.2376

\bibitem[Fredenhagen and Rejzner(2016)]{fredenhagen2016jmp}
K.~Fredenhagen and K.~Rejzner.
Quantum field theory on curved spacetimes: axiomatic framework and
examples.
\emph{J. Math. Phys.}, 57(3): 031101, 38, 2016.
DOI 10.1063/1.4939955;
zbl 1338.81299;
MR3470430;
arxiv 1412.5125

\bibitem[Fr{\"o}hlich et~al.(2017)Fr{\"o}hlich, Knowles, Schlein, and
Sohinger]{frohlich2017arxiv}
J.~Fr{\"o}hlich, A.~Knowles, B.~Schlein, and V.~Sohinger.
{A microscopic derivation of time-dependent correlation functions of
the $1D$ cubic nonlinear Schr{\"o}dinger equation}.
, 2017. Preprint.
arxiv 1703.04465

\bibitem[Fr\"ohlich et~al.(2017)Fr\"ohlich, Knowles, Schlein, and
Sohinger]{frohlich2017cmp}
J.~Fr\"ohlich, A.~Knowles, B.~Schlein, and V.~Sohinger.
Gibbs {M}easures of {N}onlinear {S}chr\"odinger {E}quations as
{L}imits of {M}any-{B}ody {Q}uantum {S}tates in {D}imensions {$d \leqslant
3$}.
\emph{Comm. Math. Phys.}, 356(3): 883--980, 2017.
DOI 10.1007/s00220-017-2994-7;
zbl 1381.81177;
MR3719544

\bibitem[Gallavotti and Verboven(1975)]{gallavotti1975nc}
G.~Gallavotti and E.~Verboven.
On the classical {KMS} boundary condition.
\emph{Nuovo Cimento B (11)}, 28(1): 274--286,
1975.
DOI 10.1007/BF02722820;
MR0449393

\bibitem[Genovese and Simonella(2012)]{genovese2012jsp}
G.~Genovese and S.~Simonella.
On the stationary {BBGKY} hierarchy for equilibrium states.
\emph{J. Stat. Phys.}, 148(1): 89--112, 2012.
DOI 10.1007/s10955-012-0525-7;
zbl 1253.82051;
MR2950759;
arxiv https://arxiv.org/abs/1205.2788

\bibitem[G{\'e}rard and Wrochna(2014)]{gerard2014cmp}
C.~G{\'e}rard and M.~Wrochna.
Construction of {H}adamard states by pseudo-differential calculus.
\emph{Comm. Math. Phys.}, 325(2): 713--755, 2014.
DOI 10.1007/s00220-013-1824-9;
zbl 1298.81214;
MR3148100;
arxiv 1209.2604

\bibitem[G{\'e}rard(1991)]{gerard1991cpde}
P.~G{\'e}rard.
Microlocal defect measures.
\emph{Comm. Partial Differential Equations}, 16(11): 1761--1794, 1991.
DOI 10.1080/03605309108820822;
zbl 0770.35001;
MR1135919

\bibitem[Ginibre and Velo(1979)]{ginibre1979cmp2}
J.~Ginibre and G.~Velo.
The classical field limit of scattering theory for nonrelativistic
many-boson systems. {II}.
\emph{Comm. Math. Phys.}, 68(1): 45--68,
1979.
\url{http://projecteuclid.org/getRecord?id=euclid.cmp/1103905266}.
MR0539736

\bibitem[Ginibre and Velo(1979)]{ginibre1979cmpI}
J.~Ginibre and G.~Velo.
The classical field limit of scattering theory for nonrelativistic
many-boson systems. {I}.
\emph{Comm. Math. Phys.}, 66(1): 37--76,
1979.
\newblock ISSN 0010-3616.
\newblock URL
\url{http://projecteuclid.org/getRecord?id=euclid.cmp/1103904940}.

\bibitem[Ginibre et~al.(2006)Ginibre, Nironi, and Velo]{ginibre2006ahp}
J.~Ginibre, F.~Nironi, and G.~Velo.
Partially classical limit of the {N}elson model.
\emph{Ann. Henri Poincar\'e}, 7(1): 21--43, 2006.
DOI 10.1007/s00023-005-0240-x;
zbl 1094.81058;
MR2205462

\bibitem[Glimm and Jaffe(1987)]{glimm1987qp}
J.~Glimm and A.~Jaffe.
\emph{Quantum physics. A functional integral point of view}.
Springer-Verlag, New York, second edition, 1987.
DOI 10.1007/978-1-4612-4728-9;
MR0887102

\bibitem[Gl{\"o}ckner(2003)]{glockner2003mams}
H.~Gl{\"o}ckner.
Positive definite functions on infinite-dimensional convex cones.
\emph{Mem. Amer. Math. Soc.}, 166(789): xiv+128,
2003.
DOI 10.1090/memo/0789;
zbl 1039.43009;

\bibitem[Golse(2016)]{golse2016lnamm}
F.~Golse.
On the dynamics of large particle systems in the mean field limit.
In \emph{Macroscopic and large scale phenomena: coarse graining, mean
field limits and ergodicity}, volume~3 of \emph{Lect. Notes Appl. Math.
Mech.}, pages 1--144. Springer, [Cham], 2016.
DOI 10.1007/978-3-319-26883-5_1;
MR3468297;
arxiv 1301.5494

\bibitem[Gross(1967)]{gross1967pfbsmsp}
L.~Gross.
Abstract {W}iener spaces.
In \emph{Proc. {F}ifth {B}erkeley {S}ympos. {M}ath. {S}tatist. and
{P}robability ({B}erkeley, {C}alif., 1965/66), {V}ol. {II}: {C}ontributions
to {P}robability {T}heory, {P}art 1}, pages 31--42. Univ. California Press,
Berkeley, Calif., 1967.
zbl 0187.40903;
MR0212152

\bibitem[Haag(1992)]{haag1992tmp}
R.~Haag.
\emph{Local quantum physics. Fields, particles, algebras}.
Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992.
zbl 0777.46037;
MR1182152

\bibitem[Haag and Kastler(1964)]{haag1964jmp}
R.~Haag and D.~Kastler.
An algebraic approach to quantum field theory.
\emph{J. Math. Phys.}, 5: 848--861, 1964.
DOI 10.1063/1.1704187;
zbl 0139.46003;
MR0165864

\bibitem[Helffer et~al.(1987)Helffer, Martinez, and Robert]{helffer1987cmp}
B.~Helffer, A.~Martinez, and D.~Robert.
Ergodicit\'e et limite semi-classique.
\emph{Comm. Math. Phys.}, 109(2): 313--326, 1987.
\url{http://projecteuclid.org/getRecord?id=euclid.cmp/1104116844}.
DOI 10.1007/BF01215225;
zbl 0624.58039;
MR0880418

\bibitem[Hepp(1974)]{hepp1974cmp}
K.~Hepp.
The classical limit for quantum mechanical correlation functions.
\emph{Comm. Math. Phys.}, 35: 265--277, 1974.
\url{http://projecteuclid.org/euclid.cmp/1103859623}.
MR0332046

\bibitem[Jech(2003)]{jech2003smm}
T.~Jech.
\emph{Set theory. The third millennium edition, revised and expanded}.
Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.
DOI 10.1007/3-540-44761-X;
zbl 1007.03002;
MR1940513

\bibitem[Kay and Wald(1991)]{kay1991pr}
B.~S. Kay and R.~M. Wald.
Theorems on the uniqueness and thermal properties of stationary,
nonsingular, quasifree states on spacetimes with a bifurcate killing horizon.
\emph{Phys. Rep.}, 207(2): 49--136, 1991.
\url{http://www.sciencedirect.com/science/article/pii/037015739190015E}.
DOI 10.1016/0370-1573(91)90015-E;
zbl 0861.53074;
MR1133130

\bibitem[Kr{\'e}e and Raczka(1978)]{kree1978aihpa}
P.~Kr{\'e}e and R.~Raczka.
Kernels and symbols of operators in quantum field theory.
\emph{Ann. Inst. H. Poincar\'e Sect. A (N.S.)}, 28 (1): 41--73, 1978.
zbl 0386.47015;
MR0482179

\bibitem[Leopold and Pickl(2016)]{leopold2016arxiv}
N.~Leopold and P.~Pickl.
{Derivation of the Maxwell-Schr{\"o}dinger Equations from the
Pauli-Fierz Hamiltonian}.
\emph{ArXiv e-prints}, 2016.
\url{http://arxiv.org/abs/1609.01545}.

\bibitem[Lewin et~al.(2015)Lewin, Nam, and
Rougerie]{lewin2015amrx}
M.~Lewin, P.~T. Nam, and N.~Rougerie.
Remarks on the quantum de {F}inetti theorem for bosonic systems.
\emph{Appl. Math. Res. Express. AMRX}, 1: 48--63, 2015.
DOI 10.1093/amrx/abu006;
zbl 1310.81169;
MR3335056;
arxiv

\bibitem[Lewin et~al.(2015)Lewin, Nam, and
Rougerie]{lewin2015jepm}
M.~Lewin, P.~T. Nam, and N.~Rougerie.
Derivation of nonlinear {G}ibbs measures from many-body quantum
mechanics.
\emph{J. \'Ec. polytech. Math.}, 2: 65--115,
2015.
DOI 10.5802/jep.18;
zbl 1322.81082;
MR3366672;
arxiv 1410.0335

\bibitem[Lewin et~al.(2017)Lewin, Th{\`a}nh~Nam, and Rougerie]{lewin2017arxiv}
M.~Lewin, P.~Th{\`a}nh~Nam, and N.~Rougerie.
{Gibbs measures based on 1D (an)harmonic oscillators as mean-field
limits}.
\emph{J. Math. Phys.}, 59, no. 4: 041901, 17 p., 2018.
DOI 10.1063/1.5026963;
zbl 1392.82023;
MR3787331;
arxiv 1703.09422
\bibitem[Liard(2017)]{liard2017jfa}
Q.~Liard.
On the mean-field approximation of many-boson dynamics.
\emph{J. Funct. Anal.}, 273(4): 1397--1442, 2017.
DOI 10.1016/j.jfa.2017.04.016;
zbl 1367.81168;
MR3661404:
arxiv 1609.06254

\bibitem[Liard and Pawilowski(2014)]{liard2014jmp}
Q.~Liard and B.~Pawilowski.
Mean field limit for bosons with compact kernels interactions by
{W}igner measures transportation.
\emph{J. Math. Phys.}, 55(9): 092304, 23, 2014.
DOI 10.1063/1.4895467;
zbl 1302.82075;
MR3390788;
arxiv 1402.4261

\bibitem[Lions and Paul(1993)]{lions1993rmi}
P.-L. Lions and T.~Paul.
Sur les mesures de {W}igner.
\emph{Rev. Mat. Iberoamericana}, 9(3): 553--618,
1993.
DOI 10.4171/RMI/143;
zbl 0801.35117;
MR1251718

\bibitem[Loomis(1953)]{loomis1953vn}
L.~H. Loomis.
\emph{An introduction to abstract harmonic analysis}.
D. Van Nostrand Company, Inc., Toronto-New York-London, 1953.
zbl 0052.11701;
MR0054173

\bibitem[Neeb(1998)]{neeb1998mm}
K.-H. Neeb.
Operator-valued positive definite kernels on tubes.
\emph{Monatsh. Math.}, 126(2): 125--160, 1998.
DOI 10.1007/BF01473583;
zbl 0956.22003;
MR1639387

\bibitem[Nelson(1964)]{nelson1964jmp}
E.~Nelson.
Interaction of nonrelativistic particles with a quantized scalar
field.
\emph{J. Math. Phys.}, 5: 1190--1197, 1964.
DOI 10.1063/1.1704225;
MR0175537

\bibitem[Nelson(1973)]{nelson1973jfa}
E.~Nelson.
The free {M}arkoff field.
\emph{J. Funct. Anal.}, 12: 211--227, 1973.
DOI 10.1016/0022-1236(73)90025-6;
zbl 0273.60079;
MR0343816

\bibitem[Pitaevskii and Stringari(2003)]{pitaevskii2003oup}
L.~Pitaevskii and S.~Stringari.
\emph{Bose-{E}instein condensation}, volume 116 of
\emph{International Series of Monographs on Physics}.
The Clarendon Press, Oxford University Press, Oxford, 2003.
zbl 1110.82002;
MR2012737

\bibitem[Radzikowski(1996)]{radzikowski1996cmpI}
M.~J. Radzikowski.
Micro-local approach to the {H}adamard condition in quantum field
theory on curved space-time.
\emph{Comm. Math. Phys.}, 179(3): 529--553, 1996.
\url{http://projecteuclid.org/euclid.cmp/1104287114}.
DOI 10.1007/BF02100096;
zbl 0858.53055;
MR1400751

\bibitem[Reed and Simon(1975)]{reed1975II}
M.~Reed and B.~Simon.
\emph{Methods of modern mathematical physics. {II}. {F}ourier
analysis, self-adjointness}.
Academic Press, New York, 1975.
zbl 0308.47002;
MR0493420

\bibitem[Rivi\`ere(2010)]{riviere2010dmj}
G.~Rivi\`ere.
Entropy of semiclassical measures in dimension 2.
\emph{Duke Math. J.}, 155(2): 271--336, 2010.
DOI 10.1215/00127094-2010-056;
zbl 1230.37048;
MR2736167;
arxiv 0809.0230

\bibitem[Roos(1970)]{roos1970cmp}
H.~Roos.
Independence of local algebras in quantum field theory.
\emph{Comm. Math. Phys.}, 16: 238--246, 1970.
\url{http://projecteuclid.org/euclid.cmp/1103842118}.
DOI 10.1007/BF01646790;
zbl 0197.26303;
MR0266539

\bibitem[Schwartz(1973)]{schwartz1973tirsm}
L.~Schwartz.
\emph{Radon measures on arbitrary topological spaces and cylindrical
measures}.
Published for the Tata Institute of Fundamental Research, Bombay by
Oxford University Press, London, 1973.
Tata Institute of Fundamental Research Studies in Mathematics, No. 6.
zbl 0298.28001;
MR0426084

\bibitem[Segal(1961)]{segal1961cjm}
I.~E. Segal.
Foundations of the theory of dyamical systems of infinitely many
degrees of freedom. {II}.
\emph{Canad. J. Math.}, 13: 1--18, 1961.
MR0128839

\bibitem[Streater and Wightman(2000)]{streater2000plp}
R.~F. Streater and A.~S. Wightman.
\emph{P{CT}, spin and statistics, and all that}.
Princeton Landmarks in Physics. Princeton University Press,
Princeton, NJ, 2000.
Corrected third printing of the 1978 edition.
zbl 1026.81027;
MR1884336

\bibitem[Takesaki(1979)]{takesaki1979I}
M.~Takesaki.
\emph{Theory of operator algebras. {I}}, volume 124 of
\emph{Encyclopaedia of Mathematical Sciences}.
Springer-Verlag, Berlin, 1979.
Reprint of the first (1979) edition, Operator Algebras and
Non-commutative Geometry, 5.
zbl 0436.46043;
MR1873025

\bibitem[Vakhania et~al.(1987)Vakhania, Tarieladze, and
Chobanyan]{vakhania1987ma}
N.~N. Vakhania, V.~I. Tarieladze, and S.~A. Chobanyan.
\emph{Probability distributions on {B}anach spaces}, volume~14 of
\emph{Mathematics and its Applications (Soviet Series)}.
D. Reidel Publishing Co., Dordrecht, 1987.
Translated from the Russian and with a preface by Wojbor A.
Woyczynski.
DOI 10.1007/978-94-009-3873-1;
zbl 0698.60003;
MR1435288

\bibitem[Verch(1994)]{verch1994cmp}
R.~Verch.
Local definiteness, primarity and quasiequivalence of quasifree
{H}adamard quantum states in curved spacetime.
\emph{Comm. Math. Phys.}, 160(3): 507--536, 1994.
\url{http://projecteuclid.org/euclid.cmp/1104269708}.
DOI 10.1007/BF02173427;
zbl 0790.53077;
MR1266061

\bibitem[Wightman and G{{\aa}}rding(1964)]{wightman1965afksv}
A.~Wightman and L.~G{{\aa}}rding.
{Fields As Operator-Valued Distributions In Relativstic Quantum
Theory}.
\emph{Arkiv Fysik, Kungl. Svenska Vetenskapsak}, 28: 129--189, 1964.

\bibitem[Zworski(2012)]{zworski2012gsm}
M.~Zworski.
\emph{Semiclassical analysis}, volume 138 of \emph{Graduate Studies
in Mathematics}.
American Mathematical Society, Providence, RI, 2012.
DOI 10.1090/gsm/138;
zbl 1252.58001
Published
2018-11-28
Section
Teufel, Stefan: Mathematical Physics, Quantum Mechanics, Semiclassical Analysis