Fractal thermodynamics and ninionic statistics of coherent rotational states realization via imaginary angular rotation in imaginary time formalism M. N. Chernodub

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Fractal thermodynamics and ninionic statistics of coherent rotational states:

realization via imaginary angular rotation in imaginary time formalism

M. N. Chernodub

Institut Denis Poisson UMR 7013, Universit´e de Tours, 37200, Tours, France

(Dated: October 26, 2022)

We suggest the existence of systems in which the statistics of a particle changes with the quan-

tum level it occupies. The occupation numbers in thermal equilibrium depend on a continuous

statistical parameter that interpolates between bosonic or fermionic and ghost-like statistical dis-

tributions. We call such particle states “ninions”: they are diﬀerent from anyons and can exist

in 3+1 dimensions. We suggest that ninions can be associated with coherent angular momentum

states. In the Euclidean imaginary-time formalism, the ninionic statistics can be implemented via

the rotwisted boundary conditions, which are associated with the rigid global rotation of the system

with an imaginary angular frequency. The imaginary rotation is characterized by a P T -symmetric

non-Hermitian Hamiltonian and possesses a well-deﬁned thermodynamic limit. The physics of nin-

ions in thermal equilibrium is accessible for numerical simulations on Euclidean lattices. We provide

a no-go theorem on the absence of analytical continuation between real and imaginary rotations in

the thermodynamic limit. The ground state of ninions shares similarity with the θ-vacuum in QCD.

The ninions can produce negative pressure and energy, similar to the Casimir eﬀect and the cosmo-

logical dark energy. In the thermodynamic limit, the dependence of thermal energy of free ninions

on the statistical parameter is a fractal.

I. INTRODUCTION

The spin-statistics theorem implies that, in three spa-

tial dimensions, local particle ﬁelds can possess only

two types of statistics. Integer spins are associated

with bosonic particles characterized by commuting ﬁelds,

while half-integer spins describe fermionic particles de-

scribed by anti-commuting ﬁelds [1]. In two spatial di-

mensions, the spin-statistics theorem does not work, and

the value of particle spin can take any value [2] thus sup-

porting the existence of the third kind of particles, the

anyons [3]. The statistics of particles is of crucial impor-

tance as it plays a principal role in the physical properties

of any many-particle systems.

The rotation is at the heart of the statistical proper-

ties of particles. Under the full 2πrotation, the boson

wavefunction stays the same; the fermion wavefunction

picks a minus sign while the anyon wavefunction gets

multiplied by the phase eiθ which interpolates between

the bosonic, θ= 0 and fermionic, θ=πcases (and there-

fore one often says that the anyons possess the fractional

statistics). The same phases appear under the exchange

of two indistinguishable particles.

This paper considers imaginary rotation, which corre-

sponds to quantum systems that rotate with imaginary

angular frequency. The imaginary rotation can naturally

be formulated in the Euclidean imaginary-time formal-

ism, which describes thermodynamic systems residing in

thermal equilibrium1. The imaginary rotation serves as

1The Euclidean formulation of a theory is usually obtained after a

Wick transformation which is also called “the Wick rotation”. To

avoid confusion with the main topic of this article, we use the term

“the Wick transformation”.

an analytical tool often used in description of thermody-

namics of real rotating quantum systems [4–10].

Here we concentrate on the eﬀects of the rotation with

imaginary frequency on particle statistics. In Section II

we describe how the imaginary rotation can be formu-

lated in terms of a simple rotwisted boundary condi-

tion in the compactiﬁed imaginary time direction (Sec-

tion II A). We also show that the imaginary rotation

introduces the statistical parameter χ, which stipulates

the surprising fractal properties of thermodynamics (Sec-

tion II B). The latter fact leads to a no-go theorem for

analytical continuation from imaginary to real angular

frequencies (Section II C) if the rotation is understood as

a boundary condition. Section III demonstrates that the

imaginary rotation leads to the statistical transmutations

of particles and the appearance of a new, ninionic-type of

statistics, which is diﬀerent from anyons. In Section IV,

we argue that the ninions are the coherent states of the

angular momentum operator, which share similarity with

the coherent spin states and coherent angular momentum

stats [11] that, in turn, are similar to the coherent states

of the linear harmonic oscillator [12]. The coherent spin

states play an important role in quantum optics [13]. The

last Section is devoted to our conclusions.

II. IMAGINARY ROTATION:

THERMODYNAMICS

A. Imaginary rotation as a boundary condition

Consider a quantum-mechanical system of bosonic or

fermionic particles which rotates rigidly with the con-

stant angular velocity Ω= Ωnabout the axis n. Due to

the rigid nature of rotation, the maximal spatial size Rof

the system must be bounded in the xy plane, |Ω|R < 1,

arXiv:2210.05651v2 [quant-ph] 25 Oct 2022

in order to avoid a clash with causality [14, 15]. The

system resides in thermal equilibrium characterized by

temperature T= 1/β, deﬁned, along with the chemi-

cal potential µ, in the frame, which co-rotates with the

system. In the co-rotating frame, the thermal parts of

bosonic and fermionic free energies take, respectively, the

following forms:

F(b)

β=V

2βX

α,m X

c,r=±1

ln 1−e−β(ωα,m−rµ−cmΩ),(1)

F(f)

β=V

2βX

α,m X

c,r=±1

ln 1 + e−β(ωα,m−rµ−c(m+1

2)Ω)−1

,(2)

where Vis the spatial volume of the system, ω=ωα,m is

the energy spectrum of the particles, and αis a collective

notation of quantum numbers other than the projection

of angular momentum, m∈Z, on the axis of rotation n.

Without loss of generality, we will assume that nis di-

rected along the zaxis while using from time to time

nto keep the generality of the expressions. For Dirac

fermions, the index αincludes also spin polarizations,

sz=±1/2. The particle/anti-particle branches are rep-

resented by the index r. Throughout the paper, we work

with units ~=c=kB= 1.

The free energies also contain the zero-point contri-

butions, F(b)

0=V f0for bosons and F(f)

0=−2V f0for

fermions, with f0=1

2Σα

Rωα,m. These quantities depend

neither on chemical potential, temperature, or angular

frequency Ω; therefore, we will ignore them below.

In the standard imaginary time formalism, the Wick

transformation substitutes the time variable tby the

imaginary time τ=it. Thus, the quantum theory in

thermal equilibrium is formulated in Euclidean space

with coordinates (x, τ). At ﬁnite temperature T, the

imaginary time direction is compactiﬁed to a circle of

the length β= 1/T .

The partition functions (1) and (2) can be represented

as traces over all quantum states of the statistical density

matrix, Z= Tr e−β(ˆ

HΩ−µˆ

N), which includes the Hamil-

tonian in the co-rotating frame:

HΩ=ˆ

H0−ˆ

Ω·ˆ

J,(3)

where ˆ

H0is the Hamiltonian in the laboratory frame.

The spectrum of the Hamiltonian (3), both for fermionic

and bosons systems, is bounded from below provided the

causality constraint is respected, |Ω|R < 1. The corre-

sponding partition function, at real angular frequency, is

a well-deﬁned quantity in the Euclidean imaginary time

formalism.

Since the angular frequency, Ω has the dimension of

angle per unit of time, one can also map it, under the

Wick transformation, to a purely imaginary quantity:

Ω = iΩI.(4)

The imaginary angular frequency ΩI6= 0 corresponds

to the uniform rotation of the whole spatial timeslice of

the Euclidean spacetime [7]. As the imaginary time vari-

able τadvances for a full period from τ= 0 to τ=β,

the space experiences the rotation by the angle

χ=βΩI,(5)

about the same axis n=ΩI/ΩI. The imaginary Eu-

clidean rotation modiﬁes the boundary conditions of the

ﬁelds as illustrated in Fig. 1.

O′

A′

O′

A′

A′ ′

(a)

(b)

τ= 0

τ=β

FIG. 1. Illustration of (a) standard (7) and (b) rotwisted (8)

boundary conditions with the segments O0A0at τ= 0 and

OA at τ=βidentiﬁed. The axis of rotation zis not shown.

The value of the rotwisted angle χdepends on the imaginary

angular frequency (5) and plays a role of the statistical pa-

rameter.

The rotation with the imaginary frequency corre-

sponds to the Hamiltonian

HΩI=ˆ

H0−iˆ

ΩI·ˆ

J,(6)

which follows directly from Eqs. (3) and (4). Despite

the resemblance of real and imaginary rotations, they

correspond to diﬀerent physical environments [7, 8].

In the absence of rotation, the bosonic (fermionic)

wavefunction φ(ψ) is a periodic (anti-periodic) function

of the imaginary time τ:

ΩI=0: φ(x, τ)=+φ(x, τ +β),

ψ(x, τ) = −ψ(x, τ +β).(7)

Under the imaginary rotation, the boundary conditions

become as follows:

φ(x, τ)=+φˆ

Rχx, τ +β,(8a)

ψ(x, τ) = −ˆ

Λχψˆ

Rχx, τ +β,(8b)

where the 3 ×3 matrix ˆ

Rχrotates rigidly the whole spa-

cial Euclidean subspace, x→x0=ˆ

Rχx, by the angle (5)

χ=βΩIaround the axis n=χ/χ =ΩI/ΩIwhich cor-

responds to the axis of the real rotation, as shown in

Fig. 1. The matrix ˆ

Λχrepresents the rotation in the

spinor space. A similar factor should appear for a vec-

tor boson. The rotwisted (from “rotation” and “twist”)

boundary conditions (8) can be implemented in the Eu-

clidean lattice simulations of ﬁeld theories [8, 10].

For deﬁniteness, we consider below the rotation around

the zaxis, which can be written in the cylindrical coordi-

nates as follows: (ρ, ϕ, z, τ )→(ρ, ϕ −ΩIτ, z, τ +β). The

boundary conditions (8) are visualized in Fig. 1. The

imaginary rotation in the whole spacetime does not lead

to causality problems [4] and can be formulated in the

thermodynamic limit in the whole Euclidean space [8].

The free energy of bosonic and fermionic systems under

the imaginary rotation are straightforwardly obtained by

identifying the angular frequencies (4) in the free energy

in the co-rotating frame (1) and (2), respectively.

Below we consider free gases of bosonic and fermionic

particles subjected to imaginary rotation. The latter is

formulated in terms of the boundary condition (8) in the

imaginary time formalism. In the thermodynamic limit,

these gases possess a fractal structure in their thermo-

dynamic properties and host exotic excitations with oc-

cupation numbers diﬀerent from those for bosons and

fermions.

B. Fractal thermodynamics of imaginary rotation

1. Hints from classical bosonic solutions

Analysis of classical solutions in Yang-Mills theory

has shown that the imaginary rotation with the rational

nonzero values of the angular frequency (5) corresponds

to the thermal bath with uniform temperature [9]:

T=1

qβ for ΩIβ

2π≡χ

2π=p

q,(9)

where βis the length of the lattice in the imaginary-

time direction and the rational number p/q, with positive

integers p, q = 1,2, . . . , represents an irreducible fraction.

Equation (9) provides us with two hints on the be-

havior of systems under imaginary rotation. First, it

suggests that temperature changes in a non-analytical

way2with the imaginary frequency ΩIthus forbidding

the analytical continuation from imaginary to real-valued

angular frequencies. Second, it stresses the signiﬁcance

of rational numbers which is particular for fractal struc-

tures [16]. Thus, Eq. (9) provides us with a signature of a

fractal behavior of imaginary rotation, which we explore

further below.

2. Free bosons at imaginary rotation

Let us now consider free bosons in thermal equilibrium

in the thermodynamic limit. In the cylindrical coordi-

nates, the Hamiltonian for a scalar particle possesses the

2For example, two close values of the imaginary frequency,

ΩIβ/(2π)=1/2 and 999/2000, correspond, respectively, to the

temperature values (9) T= 1/(2β) and T= 1/(2000β) that diﬀer

from each other by three orders of magnitude.

following eigenfunctions,

φkρ,kz,m(ρ, z, ϕ) = N1eimϕeikzzJm(kρρ),(10)

where Jmis the Bessel function and N1is a normalization

factor. The corresponding energy eigenvalues are

ωkρ,kz=qk2

ρ+k2

z+M2,(11)

where kρ>0 and kz∈Rare the momenta along the

radial direction ρand the zaxis, respectively. The pro-

jection of the angular momentum on the zaxis, m∈Z,

does not enter the expression for the eigenenergy (11)

thus corresponding to a degeneracy factor of the eigen-

states. The integration measure in Eq. (1) takes the form:

α,m

≡Z∞

kρdkρ

2πZ∞

−∞

dkz

2πX

m∈Z

.(12)

For the rational angular momentum ΩIβ/(2π) = p/q,

we combine the elements of the sum over the angular

quantum number min Eq. (12) in the groups of q:

m∈Z

f(m) = X

m∈Z

q−1

a=0

f(qm+a), q = 1,2, . . . .(13)

Then the bosonic partition function (1) at the imaginary

angular frequency (4) can be identically rewritten as

F(b)

β=V

βZ∞

kρdkρ

2πZ∞

−∞

dkz

2πX

m∈ZX

r=±1

×ln 1−e−qβ(ωkρ,kz−rµ).(14)

where we have used the identity (taking γ > 0):

c=±1

q−1

m=0

ln 1−e−γ+2πicm p

q= ln 1−e−qγ .(15)

Due to the degeneracy of the energy eigenvalues (11) with

respect to the angular momentum m, this identity can be

applied to Eq. (14). Special care is needed to match the

inﬁnite sums over mand mwhich require a regulariza-

tion. Introducing the ultraviolet regulator in Eq. (13),

e−|m|with  > 0, one ﬁnds in the limit →0 that the

sums over mand mare not equivalent but diﬀer by the

factor of 1/q, implying Pm∈Z= (1/q)Pm∈Z.

Thus, free bosonic gas subjected to the rational imag-

inary angular frequency (9) at temperature Tin the ab-

sence of background potential has the same free energy as

the non-rotating gas at lower temperature T/q ≡1/(qβ):

F(b)

βΩI=2π

=F(b)

qβ Ω=0

.(16)

This relation complies with Eq. (9) obtained for classical

bosonic solutions. Notice that Eq. (16) does not depend

on the numerator pof the irreducible fraction p/q.

The equivalence (16) is easy to understand. The ro-

tation with the angular frequency ΩI=2π

qrotates the

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Fractalthermodynamicsandninionicstatisticsofcoherentrotationalstates:realizationviaimaginaryangularrotationinimaginarytimeformalismM.N.ChernodubInstitutDenisPoissonUMR7013,UniversitedeTours,37200,Tours,France(Dated:October26,2022)Wesuggesttheexistenceofsystemsinwhichthestatisticsofaparticlechangesw...

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Fractal thermodynamics and ninionic statistics of coherent rotational states realization via imaginary angular rotation in imaginary time formalism M. N. Chernodub

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