Spectral actions for q-particles and their asymptotics Fabio Ciolli Francesco Fidaleo

2025-05-03 0 0 477.08KB 23 页 10玖币

侵权投诉

Spectral actions for q-particles

and their asymptotics

Fabio Ciolli, Francesco Fidaleo

ciolli@mat.uniroma2.it, ﬁdaleo@mat.uniroma2.it

Dipartimento di Matematica, Universit`a di Roma Tor Vergata,

Via della Ricerca Scientiﬁca, 1, I-00133 Roma, Italy

Abstract

For spectral actions consisting of the average number of particles

and arising from open systems made of general free q-particles (in-

cluding Bose, Fermi and classical ones corresponding to q=±1 and

0, respectively) in thermal equilibrium, we compute the asymptotic

expansion with respect to the natural cut-oﬀ. We treat both relevant

situations relative to massless and non relativistic massive particles,

where the natural cut-oﬀ is 1/β =kBTand 1/√β, respectively. We

show that the massless situation enjoys less regularity properties than

the massive one. We also treat in some detail the relativistic massive

case for which the natural cut-oﬀ is again 1/β. We then consider the

passage to the continuum describing inﬁnitely extended open systems

in thermal equilibrium, by also discussing the appearance of conden-

sation phenomena occurring for Bose-like q-particles, q∈(0,1]. We

then compare the arising results for the ﬁnite volume situation (dis-

crete spectrum) with the corresponding inﬁnite volume one (continu-

ous spectrum).

2020 Mathematics Subject Classiﬁcation. 82B30, 58B34, 58J37, 46F10.

Key words and phrases. Thermodynamics of grand canonical ensemble, q-

particles, Dirac operator, Spectral action, Bose Einstein Condensation, Distribu-

tions, Asymptotic analysis.

1 Introduction

Very recently, the investigation of the fascinating topic called Connes’ noncom-

mutative geometry had a formidable growth for many potential applications to

arXiv:2210.06395v1 [math-ph] 12 Oct 2022

mathematics and physics. Among those, we mention the possible role in the at-

tempt to solve the, still open, long-standing problem of the Riemann conjecture

involving the zeroes of the zeta function (e.g. [12]). We also mention the crucial

problem concerning the attempt to provide a model, still not available, which unify

the gravitation with the three remaining elementary interactions (e.g. [7]). The

reader is referred to the monograph [13] for a quite complete and understandable

treatise on noncommutative geometry.

The main ingredient arising in noncommutative geometry is the so-called Dirac

operator, which is a kind of square root of the Laplace operator. The Dirac operator

is supposed to encode the main properties of the associated, commutative or non

commutative, manifold under consideration. The Dirac operator enters in the

deﬁnition of the so-called spectral triples (e.g. [13, 14] and the references cited

therein), and the spectral action (e.g. [7]).

Concerning the spectral triples, most of the investigation is devoted to com-

mutative manifolds, and noncommutative ones equipped with a canonical tracial

state (i.e. the easiest noncommutative generalisation of the Lebesgue/Liouville

measure), and thus providing von Neumann factors of type II1. Only recently

in [23, 22, 11], it was considered the possibility to investigate examples based

on the noncommutative 2-tori, for which the involved Dirac operator is suitably

deformed by the use of the Tomita modular operator, the last being unbounded

and not “inner” in all interesting cases. Therefore, such deformed spectral triples,

called modular since the modular data might play a crucial role, naturally arise

from type III representations.

The spectral action Swas introduced in [7] with the aim to provide a set of

consistent equations in which all fundamental interactions in nature are uniﬁed.

Also in this case, the Dirac operator Dplays a crucial role because the spectral

action is supposed to have the form S(D, Λ) = Trf(|D|/Λ). Here, the function

fis associated to an extensive quantity, and Λ is a natural cut-oﬀ determined by

the properties of the underlying physical system. It should be also noted that the

asymptotic expansion of the heat kernel associated to the (powers of the) Dirac

operator Dand the spectral action S(D, Λ) with respect to the cut-oﬀ Λ for Λ ↑ ∞,

is an important tool in most approaches to spectral geometry. For further details

on this crucial point, the reader is referred to [7, 8, 17, 18, 16] and the references

cited therein.

Very recently, in the paper [9] it was investigated the spectral action associated

to the extensive quantity, as the entropy, of some quite general quantum systems

describing Fermi particles or, in other words, systems based on the Canonical

Anti-commutation Relations. In [15], the investigation of the asymptotics of some

spectral actions has been extended to particular Hamiltonians, also involving the

Bose case.

By following the mentioned paper [15], it is then natural to address the inves-

tigation of the asymptotics of the spectral actions associated to inﬁnitely extended

systems describing Bose/Fermi free gas, and also classical particles obeying to the

Boltzmann statistics. Since the properties of such free gases are now available for

the general cases q∈[−1,1], relative to q-particles or quons, after computing their

grand partition function (cf. [10]), we carry out such an investigation for all cases,

q=±1 and q= 0 being the Bose/Fermi and Boltzmann cases, respectively.

Indeed, for spectral actions arising from open systems made of such general

free q-particles in thermal equilibrium, we compute the asymptotic expansion with

respect to the natural cut-oﬀ associated to 1/β =kBT,Tbeing the absolute

temperature and kB≈1.3806488 ×10−23 JK−1the Boltzmann constant. For the

sake of simplicity, we deal with the spectral actions associated to the “average

number of particles”, by considering both relevant situations for non relativistic

massive, and massless cases. The spectral actions associated to other extensive

quantities, like average energy and entropy, can be analogously studied.

Actually, we show that the massless situation enjoy less regularity properties

than the massive one. We also investigate in some detail the relativistic massive

case, which appears more involved than the corresponding non relativistic one.

We also consider the passage to the continuum describing inﬁnitely extended

open systems in thermal equilibrium for which the Hamiltonian, being a suitable

function of the Dirac operator, has continuous spectrum.

In the context of the continuum, we brieﬂy discuss the appearance of conden-

sation phenomena, occurring for Bose-like q-particles where q∈(0,1], by pointing

out that the condensation provides an additional addendum in the asymptotics.

We also compare the results relative to the asymptotics for the ﬁnite volume sit-

uation (discrete spectrum) with inﬁnite volume one (continuous spectrum).

We would also like to mention the so-called anyons, which are particles satis-

fying, at least formally, a commutation relation similar to (2.1) enjoyed by quons,

where now qis any value in the unit circle T≡ {z∈C| |z|= 1}. We point out

that such particles might have a role in the, still open problem of the explanation

of the integer and fractional quantum Hall eﬀect, see e.g. [31].

At this stage, the q-deformed particles (q∈[−1,1]) seem to be related to

quantum groups and quantum algebras, which drew much attention decades ago.

In fact, such particles naturally emerge from exactly solvable models in statistical

mechanics which acquire the Yang-Baxter equation. We also point out that the

irreducible representations of q-deformed particles are substantial extensions of the

quantum algebra in connections to the braid group statistics. For such interesting

applications of these quons, the reader is referred to the monograph [3].

The standard notations and basic results relative to the topics involved in the

present note are borrowed by the current literature, partially reported at the end,

to which the reader is referred.

2 The grand-partition function for q-particles,

q∈[−1,1]

We start with a system whose Hamiltonian His a selfadjoint positive (i.e. σ(H)⊂

[0,+∞)) operator with compact resolvent, acting on a separable Hilbert space H,

called the one-particle space.

In such a situation, the spectrum σ(H) is made by isolated points, accumulat-

ing at +∞if His inﬁnite dimensional. In addition, the multiplicity g(ε) of each

eigenvalue ε∈σ(H) is ﬁnite. Summarising, by considering the resolution of the

identity of H, we have IH=Pε∈σ(H)Pε,

H=X

ε∈σ(H)

εPε,and g(ε) := dimRan(Pε)<∞.

We also suppose that at any inverse temperature β:= 1/kBT,e−βH is trace-

class for each β > 0, and deﬁne the partition function ζ:= Tre−βH .

In the present paper, we generally deal with the so-called q-particles, usually

named as quons,q∈[−1,1]. Such exotic q-particles are naturally associated to

the following commutation relations (e.g. [27, 4])

aq(f)a†

q(g)−qa†

q(g)a(f) = hg, fiIH, f, g ∈H,(2.1)

Hbeing the one-particle space, enjoyed by the creators and annihilators. Such

commutation relations can be viewed as an interpolation between that associated

to particles obeying to the Fermi statistics (i.e. q=−1) and that of particles

obeying to the Bose statistics (i.e. q= 1), passing for the value q= 0 describing

the classical particles, and so obeying to the Boltzmann statistics. They can be

realised by (unbounded in the case q= 1) operators acting on the corresponding

Fock spaces Fq, see e.g. [6, 4].

Concerning the grand partition function Z, it comes by considering open sys-

tems in thermodynamic equilibrium at inverse temperature βand chemical po-

tential µ. It is customary to express the grand partition function Z=Z(β, z)

in terms of the independent variables, which are the inverse temperature βand

the so-called activity z:= eβµ. When it causes no confusion, we omit to indicate

such dependences. We would like to point out that, since the activity zshould

be considered as an independent thermodynamic variable, it is not involved in the

computation of the expansion of the quantities w.r.t. β↓0. Since we deal with

the general cases q∈[−1,1], the partition function Z=Zq=Zq(β, z) is also a

function of such an additional parameter q.

The partition function is computed as Tr e−βK by using the second quantisa-

tion grand canonical Hamiltonians K:= dΓ(H)−µN acting on the corresponding

Fock space. Here, Nis the number operator, and the Fock space would depend on

the statistics to which the involved q-particles obey.

Unfortunately, it is ﬁrst seen in [32] and then in [10], that such a method

to compute the grand partition function works only for the Bose/Fermi cases

corresponding to q=±1, see e.g. [6]. Such a computation fails even for the

classical (or Boltzmann) case q= 0, where the statistics is however well-known

(e.g. [24], Section 2), but it is necessary to correct by the Gibbs factor as follows:

+∞

n=1 Tr e−βH n

n!zn,instead of

+∞

n=1 Tr e−βH nzn.

Concerning the remaining cases q∈(−1,0) S(0,1), the partition function

should be computed by a diﬀerent method because it is completely unknown what

should be the statistics enjoyed by such exotic particles. However, in [10] it was

computed the reasonable partition function for all q∈[−1,1], uniquely determined

up to a multiplicative constant.

Indeed, by using only the commutation relations (2.1) and imposing the Kubo-

Martin-Schwinger condition in the grand canonical ensemble scheme, in [1] it was

proven for such a free gas of quons, that the average population of particles for a

ﬁxed energy-level εin thermodynamic equilibrium satisﬁes the natural extension

νq(ε) = 1

z−1eβε −q,−1≤q≤1,(2.2)

of the well-known Planck distribution 1

z−1eβε−1.

By taking in account the degeneracy g(ε) of the single level ε, we get the

average population

nq(ε) = g(ε)νq(ε) = g(ε)

z−1eβε −q,−1≤q≤1.(2.3)

For particles living in the continuum Rnas for the analysis in [1], the degeneracy

g(ε) is automatically absorbed, see (4.2).

On the other hand, the same distribution (2.2) was recovered in [24], in the

scheme of microcanonical ensemble, by maximising the q-entropy functional

Sq({ν(ε)}) := kBX

ε∈σ(H)

g(ε)1 + qν(ε)

qln 1 + qν(ε)−ν(ε) ln ν(ε),(2.4)

w.r.t. the set of variables {ν(ε)}ε∈σ(H)under the constrains

ε∈σ(H)

εg(ε)ν(ε) = E , X

ε∈σ(H)

g(ε)ν(ε) = N . (2.5)

Here, Eand Nare the pre-assigned values of the total energy and the total number

of particles of the physical system under consideration.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Spectralactionsforq-particlesandtheirasymptoticsFabioCiolli,FrancescoFidaleociolli@mat.uniroma2.it,daleo@mat.uniroma2.itDipartimentodiMatematica,UniversitadiRomaTorVergata,ViadellaRicercaScientica,1,I-00133Roma,ItalyAbstractForspectralactionsconsistingoftheaveragenumberofparticlesandarisingfromop...

展开>> 收起<<

Spectral actions for q-particles and their asymptotics Fabio Ciolli Francesco Fidaleo.pdf

共23页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Spectral actions for q-particles and their asymptotics Fabio Ciolli Francesco Fidaleo

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: