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, Université de Tours, 37200, Tours, France (Dated: December 13, 2022)

We suggest the existence of systems in which the statistics of a particle changes with the quantum 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 distributions. 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 ninions 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 cosmological 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 spatial 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 described by anti-commuting fields [1]. In two spatial dimensions, the spin-statistics theorem does not work, and the value of particle spin can take any value [2] thus supporting the existence of the third kind of particles, the anyons [3]. The statistics of particles is of crucial importance as it plays a principal role in the physical properties of any many-particle systems.

The rotation is at the heart of the statistical properties of particles. Under the full 2π rotation, the boson wavefunction stays the same; the fermion wavefunction picks a minus sign while the anyon wavefunction gets multiplied by the phase e iθ which interpolates between the bosonic, θ = 0 and fermionic, θ = π cases (and therefore 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 corresponds to quantum systems that rotate with imaginary angular frequency. The imaginary rotation can naturally be formulated in the Euclidean imaginary-time formalism, which describes thermodynamic systems residing in thermal equilibrium 1 . The imaginary rotation serves as an analytical tool often used in description of thermodynamics of real rotating quantum systems [4][5][6][7][8][9][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 formulated in terms of a simple rotwisted boundary condition in the compactified imaginary time direction (Section II A). We also show that the imaginary rotation introduces the statistical parameter χ, which stipulates the surprising fractal properties of thermodynamics (Section 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. 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 chemical 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 2β α,m c,r=±1
ln 1 -e -β(ωα,m-rµ-cmΩ) ,

(1)

F (f) β = V 2β
α,m c,r=±1 ln 1 + e -β(ωα,m-rµ-c(m+ 1 2 )Ω)

-1 , (2)
where V is the spatial volume of the system, ω = ω α,m is the energy spectrum of the particles, and α is 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 n is directed along the z axis while using from time to time n to keep the generality of the expressions. For Dirac fermions, the index α includes also spin polarizations, s z = ±1/2. The particle/anti-particle branches are represented by the index r. Throughout the paper, we work with units

ℏ = c = k B = 1.
The free energies also contain the zero-point contributions,

F (b) 0 = V f 0 for bosons and F (f) 0 = -2V f 0 for fermions, with f 0 = 1 2 Σ α ω α,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 t by 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( ĤΩ -µ N ) , which includes the Hamiltonian in the co-rotating frame:

ĤΩ = Ĥ0 -Ω • Ĵ , ( 3
)
where Ĥ0 is 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 corresponding 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 = iΩ I . (4)
The imaginary angular frequency Ω I ̸ = 0 corresponds to the uniform rotation of the whole spatial timeslice of the Euclidean spacetime [7]. As the imaginary time variable τ advances for a full period from τ = 0 to τ = β, the space experiences the rotation by the angle The rotation with the imaginary frequency corresponds to the Hamiltonian

χ = βΩ I , (5)
ĤΩ I = Ĥ0 -i ΩI • Ĵ , (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 :

Ω I = 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 Rχ x, τ + β , (8a)
ψ(x, τ ) = -Λχ ψ Rχ x, τ + β , (8b)
where the 3 × 3 matrix Rχ rotates rigidly the whole spacial Euclidean subspace, x → x ′ = Rχ x, by the angle (5) χ = βΩ I around the axis n = χ/χ = Ω I /Ω I which corresponds to the axis of the real rotation, as shown in Fig. 1. The matrix Λχ represents the rotation in the spinor space. A similar factor should appear for a vector boson. The rotwisted (from "rotation" and "twist") boundary conditions ( 8) can be implemented in the Euclidean lattice simulations of field theories [8,10].

For definiteness, we consider below the rotation around the z axis, which can be written in the cylindrical coordinates as follows: (ρ, φ, z, τ ) → (ρ, φ -Ω I β, 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 thermodynamic properties and host exotic excitations with occupation numbers different from those for bosons and fermions.

B. Fractal thermodynamics of imaginary rotation

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 Ω I β 2π ≡ χ 2π = p q , ( 9
)
where β is the length of the lattice in the imaginarytime 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 behavior of systems under imaginary rotation. First, it suggests that temperature changes in a non-analytical way2 with the imaginary frequency Ω I thus prohibiting an analytical continuation from imaginary to real-valued angular frequencies. Second, it stresses the significance of rational numbers which is particular for fractal structures [16]. Thus, Eq. ( 9) provides us with a signature of a fractal behavior of imaginary rotation, which we explore further below.

Free bosons at imaginary rotation

Let us now consider free bosons in thermal equilibrium in the thermodynamic limit. In the cylindrical coordinates, the Hamiltonian for a scalar particle possesses the following eigenfunctions, ϕ kρ,kz,m (ρ, z, φ) = N 1 e imφ e ikzz J m (k ρ ρ) , (10) where J m is the Bessel function and N 1 is a normalization factor. The corresponding energy eigenvalues are

ω kρ,kz = k 2 ρ + k 2 z + M 2 , (11)
where k ρ ⩾ 0 and k z ∈ R are the momenta along the radial direction ρ and the z axis, respectively. The projection of the angular momentum on the z axis, m ∈ Z, does not enter the expression for the eigenenergy (11) thus corresponding to a degeneracy factor of the eigenstates. The integration measure in Eq. ( 1) takes the form:

α,m ≡ ∞ 0 k ρ dk ρ 2π ∞ -∞ dk z 2π m∈Z . ( 12
)
For the rational angular momentum Ω I β/(2π) = p/q, we combine the elements of the sum over the angular quantum number m in Eq. ( 12) in the groups of q:

m∈Z f (m) = 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 β ∞ 0 k ρ dk ρ 2π ∞ -∞ dk z 2π m∈Z r=±1
× ln 1 -e -qβ(ω kρ ,kz -rµ) . (

) 14
where we have used the identity (taking γ > 0):

1 2 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 m and m which require a regularization. Introducing the ultraviolet regulator in Eq. ( 13), e -ϵ|m| with ϵ > 0, one finds in the limit ϵ → 0 that the sums over m and m are not equivalent but differ by the factor of 1/q, implying m∈Z = (1/q) m∈Z . Thus, free bosonic gas subjected to the rational imaginary angular frequency (9) at temperature T in the absence of background potential has the same free energy as the non-rotating gas at lower temperature T /q ≡ 1/(qβ):

F (b) β Ω I = 2π β p q = 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 p of the irreducible fraction p/q.

The equivalence ( 16) is easy to understand. The rotation with the angular frequency Ω I = 2π β p q rotates the xspace by the angle χ = 2π p q per one period β of the imaginary time τ . After q periods, the spatial x-space returns to the original position, thus signaling the qβ periodicity in the imaginary time 3 . This periodicity length provides us with the temperature of the system, T = 1/(qβ). While the extended imaginary time contains q copies of original space, the number of particles per one copy remains the same. In short, the rotation of a bosonic thermal gas with an imaginary angular frequency reduces the temperature of the gas by a q-factor related to the rational frequency (9).

Fractal thermodynamics for bosons

The thermodynamics of the particles under imaginary rotations is curious. In the absence of rotation, Ω I = Ω = 0, the density of free energy f = F/V of massless (for simplicity) neutral real-valued scalars at temperature T = 1/β is as follows:

f (b) 0 ≡ f (b) β Ω=0 = - π 2 90β 4 . (17)
Therefore, for the black-body scalar gas, the energy density ε 0 = ∂(βf 0 )/∂β, the pressure P 0 = -f 0 , and the entropy density s 0 = β(ε 0 + P 0 ), are:

ε 0 = 3P 0 = π 2 30β 4 , s 0 = 2π 2 45β 3 , (18)
where we drop the superscript "(b)" for brevity. We also set the chemical potential to zero, µ = 0, in the rest of the article. The bosonic equivalence (16) implies that the thermodynamic quantities (18) are affected by the rotation with the imaginary angular momentum Ω I as follows:

ε Ω I ε 0 1 4 = P Ω I P 0 1 4 = s Ω I s 0 1 3 = f T βΩ I 2π , (19)
where

f T (x) = 1 q if x = p q ∈ Q, with p, q ∈ N coprimes, 0 if x / ∈ Q , ( 20
)
is the Thomae function. It is equal to 1/q if its argument is a rational number given by an irreducible fraction, x = p/q ∈ Q, and zero otherwise. We would like to stress that the scaling behavior of thermodynamic quantities (19) is determined solely by the denominator q of the rational ratio βΩ I /(2π) and not by its numerator. In Fig. 2, we show the energy density of a massless boson gas as a function of the statistical angle χ, which is related to the imaginary angular velocity Ω I in Eq. (5). According to Eq. (19), this thermodynamic quantity is normalized to the non-rotating boson gas value at a fixed temperature (18). Thermodynamics exhibits an explicit fractal behavior characterized by a self-repeating structure at all temperature scales. If plotted on the logarithmic scale, the visual appearance of thermodynamic quantities resembles (fractal) water fountains. The selfsimilarity patterns in this fractal structure, generated by the Thomae function (20), are given by the denominators of the successive Farey sequences of fractions [17].

We have also found the fractal-like behavior of freefermion thermodynamics, similar to the bosonic fractal fountains shown in Fig. 2.

C. A no-go theorem for analytical continuation

The absence of analytical continuation is evident from the non-analytical nature of the fractal behavior of thermodynamics (19) at imaginary angular frequencies Ω I . The fractal is generated by the Thomae function (20) which is discontinuous at every rational point because irrational numbers come infinitely close to any given rational number. Thus, at every value of Ω I , the thermodynamic quantities are non-differential, non-analytical functions of the imaginary angular frequency Ω I .

The no-go statement is illustrated in Fig. 3 where the energy density is shown as the function of the imaginary angular momentum Ω I for various sequences p/q = P m /P n , with m > n, of the prime numbers P i . The slopes of these sequences are different, thus demonstrating the ambiguity of the analytical continuation (or invalidity of any analytical method based, for example, on a Taylor expansion). Since the fractal properties of imaginary rotations are common for bosons and fermions, the same no-go argument is also valid for both types of particles.

In concluding this section, we would like to add four comments. First of all, we notice that although our no-go arguments are based on the behavior of free theories and the properties of classical solutions, they should also work in interacting models (at least, in perturbation theory over classical backgrounds).

Second, the analytical continuation can exist for nonlocal quantities. For example, the Polyakov loop in gauge field theories [10] allows us to formulate a Euclidean analog of the Tolman-Ehrenfest law in rotating systems. The Minkowski and Euclidean Tolman-Ehrenfest laws appear to be connected by an analytical continuation between real and imaginary angular frequencies. One can show that this analytical continuation has a pure kinematic origin.

FIG. 2.

Fractal thermodynamics under imaginary rotation in thermodynamic limit: the characteristics of free massless boson gas as function of the statistical angle χ related to imaginary angular frequency ΩI in Eq. ( 5). We show only the energy density as it is connected to other thermodynamic quantities via Eq. (19). The normalization to the non-rotating gas is implied (18). The plots show various zoom scales from 10 0 to 10 -3 which show the self-similar pattern generated by the Thomae function (20).

Ω/Ω 0 (E/E 0 ) 1/4 𝖯 m 𝖯 n n>m⩾1 2 𝖯 n 3 𝖯 n 5 𝖯 n 7 𝖯 n 11 𝖯 n (ε Ω I /ε 0 ) 1/4 Ω I /Ω 0 FIG. 3.
Absence of analytical continuation from imaginary to real rotation for free bosonic system: the energy density vs. the imaginary angular frequency for different prime-number sequences, ΩI = 2πβ -1 Pm/Pn in units of Ω0 = 2πT .

Third, the absence of analytical continuation between real and imaginary rotations for local thermodynamic quantities implies that the imaginary rotation is principally different and disconnected from the rotation with real-valued angular momentum. This conclusion is valid in the thermodynamic limit. Below, we show that the imaginary rotation leads to a new type of particle statistics.

Fourth, our conclusions are valid in the strict thermodynamic limit where the system under the imaginary rotation is considered in the infinite volume.

III. STATISTICAL TRANSMUTATION A. Examples of statistic transmutations

Transmutation of fermions to bosonic ghosts

To illustrate the concept of our idea, let us start with a few examples. In Ref. [9], we have shown that the rotation with the imaginary frequency Ω I = 2π/β converts fermions into ghost-like particles that behave as bosons but contribute to the free energy with the wrong, for bosons, sign. Here, we explore this property further for a generic rational frequency by repeating the above arguments for fermionic particles. We skip most of the derivation, which is trivial. Instead of the bosonic identity (15), we use the following relation for fermions:

1 2 c=±1 q-1 m=0 ln 1 + e -γ+2πic(m+ 1 2 ) p q = ln 1 -(-1) p+q e -qγ , (21)
which leads us to the identities:

F (f) β Ω I = 2π β p q =      F (f) qβ Ω=0 , p + q ∈ odd, -2F (b) qβ Ω=0 , p + q ∈ even. ( 22
)
The effect of the imaginary angular frequency on temperature and degeneracy is exactly the same as for the bosons: an imaginary-rotated fermion ensemble gets temperature T = 1/(qβ), as in Eq. ( 9). The mapping [START_REF] Milton | The Casimir effect: physical manifestations of zero-point energy[END_REF] shows that for odd numbers p + q, the rotating gas is described by fermions, as expected. However, for even p + q, the rotation produces ghosts out of fermions. The ghosts obey bosonic statistics but have free energy with the wrong (fermionic) sign:

F (gh) β ≡ -F (b) β . ( 23
)
The factor 2 in Eq. ( 22) at even p + q corresponds to a spin degeneracy: under the imaginary rotation, one Dirac fermion produces two bosonic ghosts.

The thermodynamics of fermions under the imaginary rotation also has the "fractal fountain" structure similar to the one of Fig. 2. The fractal fountains, however, acquire a two-color pattern due to even/odd partitioning of the free energy [START_REF] Milton | The Casimir effect: physical manifestations of zero-point energy[END_REF]. The relations ( 22) can be understood along the same analysis that we performed in the case of bosons, taking into account the anti-periodicity of fermions in the τ direction.

For fermions, the rotation with the imaginary frequency Ω I = 2π/β corresponds to an exceptional case due to anti-periodicity of fermionic fields. Therefore, for fermions, the irreducibility of p/q is allowed for p = q = 1. A full rotation of the system at the angle χ = 2π after one period changes the sign of the fermionic wavefunction, which, together with the original anti-periodicity (8b), gives the periodic boundary conditions for the fermionic fields at temperature T = 1/β. Thus, the imaginary rotation at the frequency Ω I = 2π/β transmutes fermions into bosonic ghosts that have bosonic statistics but enter the free energy with the wrong, "fermionic" sign.

Transmutation of bosons to fermionic ghosts

The ghosts can emerge not only in fermionic [START_REF] Lamoreaux | [END_REF] but also in bosonic systems. Let us consider, for example, a massless boson ϕ in the space with two crossing infinitely large walls as shown in Fig. 4. The walls are oriented in the xz and yz planes and intersect along the z axis. We impose that these xz and yz walls set the Dirichlet and Neumann (DN) boundary conditions, respectively:

DN : ϕ(x, τ ) y=0 = 0, ∂ϕ(x, τ ) ∂x x=0 = 0 , (24)
In the cylindrical system of coordinates (ρ, φ, z), the wavefunctions consistent with the boundary conditions (24) are as follows:

ϕ kρ,kz,m (ρ, z, φ) = N 2 sin(mφ)e ikzz J m (k ρ ρ) . ( 25
)
where N 2 is a normalization factor. The energy spectrum takes the form (11) with one important modification: the walls diminish the level of the degeneracy of the states by (24). The potential set by the walls is consistent with the imaginarytime boundary conditions (8a) set by the imaginary rotation ΩI = π/β (the statistical parameter χ = π). Subjected to imaginary rotation, this system converts a boson into a fermionic ghost with the free energy (27) and thermodynamic characteristics given in Eqs. (28).

restricting the angular momentum to odd positive numbers: m = 2m + 1 with m = 0, 1, . . . . This restriction leads to a factor of "1/4" in thermodynamic quantities. In the absence of rotation, the energy density, pressure, and entropy of the crossed wall system of Fig. 4 (denoted by "✛") can be derived from the generic expression for the bosonic free energy (1). One gets:

ε (✛) Ω I =0 = 3P (✛) Ω I =0 = π 2 120β 4 , s (✛) Ω I =0 = π 2 90β 3 . ( 26
)
These quantities are, expectedly, four times smaller than their counterparts in the system without walls (18). Consider now the imaginary rotation about the axis z with the imaginary angular frequency Ω I = π/β. Setting Ω = iπ/β in the expression for the free energy (1), one gets e βcmΩ = -1 for odd angular momentum m. This sign flip changes ("transmutes") effectively the statistics in the bosonic free energy (1):

F (b) β Ω I = π β = 2V β α ∞ m=0 ln 1 + e -βωα,2m+1 . (27)
The free energy of the imaginary rotating system (27) becomes similar to the free energy of fermions (2) but with the wrong, "bosonic" sign. Thus, according to our definition above [START_REF] Lamoreaux | [END_REF], the bosons have been transmuted into fermionic ghosts. Apart from the transmutation of statistics, the rotation with the imaginary angular frequency Ω I = π/β leads, according to Eq. ( 9), to the decrease of temperature by the factor of 2 with β → 2β. Finally, taking into account the mentioned reduction factor 1/4 due to the presence of the walls, we obtain that the bosonic gas under the cross potential of Fig. (4) subjected to the imaginary rotation with the frequency Ω I = π/β becomes the gas of fermionic ghosts characterized by the exotic thermodynamics:

ε (✛) Ω I = π β =3P (✛) Ω I = π β = - π 2 1920β 4 , s (✛) Ω I = π β = - π 2 1440β 3 . (28)
Notice that for the statistical transmutation of free bosons to the fermionic ghosts, one needs the presence of both a background potential and an imaginary rotation. The previous subsection shows that free fermions can be transmuted into bosonic ghosts only under an imaginary rotation without a background potential.

Thermodynamics of ghosts: examples

According to relations ( 17), ( 18), ( 22) and ( 23), the ghosts possess rather exotic thermodynamic features characterized by negative energy density, negative pressure, and negative entropy. These quantities should decrease with the temperature increase (while being negative, they increase in absolute value).

The unusual properties of the thermodynamic ensemble, generated by the rotwisted boundary conditions (8), can be probed in first-principle Monte Carlo calculations. One can mention simulations of Yang-Mills theory at the imaginary angular momentum Ω I = 1 4 × 2π/β which is consistent with symmetries of the hypercubic lattice [10]. The theoretical considerations on the same imaginary frequency have also been presented in Ref. [8].

In principle, the lattice field theory can also be formulated at a triangular lattice which also admits the value of the imaginary angular frequency Ω I = 1 3 × 2π/β. According to Eq. ( 22), such imaginary rotation leads to the transmutation of one free Dirac fermion into two bosonic ghosts at temperature T = 1/(3β) with the energy density, pressure and entropy becoming negative quantities:

Ω I = 2π 3β →    ε (gh) Ω I =3P Ω I = -π 2 1215β 4 , s (gh) Ω I = -4π 2 1215β 3 . (29)
The meaning of negative energy and entropy remains to be understood. One of the interpretations is that these ghosts could be interpreted as collective excitations in a many-body system. While, formally, the entropy of specific excitations can take negative values, the whole many-body system should possess a positive entropy. These bosonic and fermionic ghosts can also play a distinctive role as a candidate of the cosmological dark energy used to explain the observed acceleration of the expansion of the Universe [18].

The negative values of thermodynamic quantities (29) can be interpreted as a signature of the object to carry a negative number of degrees of freedom. Therefore, we employ the term "ghost" in our paper. The same terminology has been used in a different context but with a similar purpose. These are the Faddeev-Popov ghosts that appear in a gauge-fixing procedure in gauge theories [19] and are utilized to subtract the unphysical degrees of freedom from the gauge bosons.

The negative values of the thermodynamic quantities for the ghost particles (29) do not imply the emergence of any (tachyonic) instability in the system. For example, in systems with a finite spatial size, the vacuum pressure can take negative values while the system is perfectly stable. This phenomenon is associated with the Casimir effect [20] which appears due to the presence of physical objects that affect fluctuations of quantum fields around them. The latter effect changes both energy and pressure of zero-point vacuum fluctuations [START_REF] Bordag | Advances in the Casimir effect[END_REF][START_REF] Milton | The Casimir effect: physical manifestations of zero-point energy[END_REF] and produces experimentally observable forces [START_REF] Lamoreaux | [END_REF].

Our ghosts have the same effect on the vacuum energy as the Casimir phenomenon. At particular statistical parameter values, the ghosts can make energy density and pressure negative. Moreover, the ghosts also contribute negatively to the entropy of the system (29). This negative entropy phenomenon has also been noticed to appear in certain Casimir systems [24][25][26][27].

B. Level-dependent statistics under imaginary rotation

Ninion thermodynamics

The fermionic and bosonic ghosts represent particular cases of a more general excitation which we call a "ninion" 4 . Similarly to the ghost particle, the signature of a ninion statistics can be seen in its thermodynamic contribution to the free energy of the system.

Let us consider a boson or a fermion with free energy, Eq. ( 1) or (2), at the imaginary angular frequency (4). We subject the particle to a background potential V (x), which (partially) lifts the degeneracy of the energy spectrum on the angular momentum m. We do not specify the form of the potential but require that it should (i) be consistent with the rotwisted boundary conditions (8) corresponding to the imaginary frequency Ω I , and

(ii) admit the quantization of the wavefunctions in terms of the angular quantum number m. 5 The corresponding free energy (1) takes the following form:

F (a) β = ± V 2β α,m r=±1 ln 1 ∓ 2e -β(ωα,m-rµ) cos ξ (a) m +e -2β(ωα,m-rµ) , a = b, f, (30)
where upper (lower) sign corresponds to bosons, a = b (fermions, a = f ). The auxiliary statistical parameters,

ξ (b) m = mχ , ξ (f) m = m + 1 2 χ , (31)
4 From "neither bosons nor fermions", with the use of the negative conjunction construction "ni . . . ni . . . " (neither . . . nor . . . ) which exists in French and certain Slavic languages. 5 The latter requirement does not imply the invariance of the potential V (x) under a group of continuous rotations about the z axis. An example is given by the non-axially-symmetric background (24), visualized in Fig. 4, which possess eigenfunctions (25) labeled by the angular momentum m.

depend on the angle χ ≡ βΩ I ∈ (-π, π] set by the rotwisted boundary condition (5). We remind that the collective index α in Eq. ( 30) includes all labels of the quantum states except for the angular momentum m and that the Dirac fermionic index α also contains a sum over the spin projection s z = ±1/2. In free space, the particle energy does not depend on the angular momentum m. Therefore, all levels sum up in the free energy (30), which reduces, via the identities ( 15) and ( 21), to its bosonic (16) and fermionic [START_REF] Milton | The Casimir effect: physical manifestations of zero-point energy[END_REF] representations, that exhibit fractal thermodynamics. If the particles interact with a background potential, the contributions of different levels at a nonvanishing statistical angle χ ̸ = 0 become distinguishable.

The total particle density,

N (a) = - 1 β ∂F (a) ∂µ ≡ α,m n (a) ωα,m (ξ (b,f) m ), a = b, f, ( 32
)
is determined by the sum over the occupation numbers over energy levels characterized by the angular momentum m and the collective quantum number α. Using the expression for the free energy (30) with the rotwisted boundary conditions, we get the following expression for the "ninionic" occupation numbers:

n (a) ω (ξ) = e β(ω-µ) cos ξ ∓ 1 1 ∓ 2e β(ω-µ) cos ξ + e 2β(ω-µ) , a = b, f. (33)
The upper and lower signs indicate that the ninion statistics originate from bosonic (a = b) and fermionic (a = f) particles, respectively. The continuous parameter ξ, defined for bosons and fermions in Eq. [START_REF] Kapusta | Finite-temperature field theory: Principles and applications[END_REF], controls the population of states and interpolates between different kinds of statistics. For example, using the notation ε = β(ω -µ), for brevity:

• at the angle ξ = 0, the population numbers [START_REF] Mustafa | [END_REF] correspond to standard bosonic, n (b) = 1/(e ε -1), and fermionic, n (f) = 1/(e ε + 1), distributions.

• at the angle ξ = π/2, the bosons become fermionic ghosts n (b) = -1/(e 2ε + 1) while the fermions remain fermions n (f) = 1/(e 2ε + 1), albeit, in both cases, at twice lower temperature, T = 1/(2β);

• at the angle ξ = π, the bosons become fermionic ghosts n (b) = -1/(e ε + 1) and the fermions become bosonic ghosts n

(f) = -1/(e ε -1) at T = 1/β.
The examples of the particle occupation numbers (33) are shown in Fig. 5.

The occupation numbers (33) can take negative values. For example, the ninionic occupation number, which originates from bosons, tends to the universal value in the high-temperature limit: This property is seen in Fig. 5(a). On the other hand, under the imaginary rotation, the fermions give rise to the ninions that have a negative occupation number just above the Fermi surface. This effect is visible in Fig. 5(b).

lim β→0 n (b) ω (ξ) = - 1 2 , ξ ̸ = 0 mod 2π . (34
We suggest that ninions can appear in the many-body systems where the negative occupation number from a ninion is accompanied by a larger positive contribution from other states so that the total particle density remains non-negative. The physical circumstances of the negative density of ninions should be similar to the physics of holes that live below Fermi surfaces in electronic systems. In the latter case, removing an electron from a fermionic sea does not lead to the "negative density of electrons". Instead, this process creates a positive density of the holes. In other words, the absence of a negatively charged electron can be understood as the presence of a positively charged hole. In the particle physics context, a similar analogy is given by Dirac's positron, which can be associated with an electron removed from the Dirac sea. We expect that the same idea also applies to the ninions.

Ninions, anyons, and spin-statistics

For a non-trivial value of parameter ξ, the ninionic statistical distribution [START_REF] Mustafa | [END_REF] is different from occupation numbers of bosons and fermions. Since the thermal distribution of particle states is inherently connected to the type of statistics of the particle, one could attempt to associate the ninions with particles possessing unconventional statistics. However, this attempt faces a contradiction with the spin-statistics theorem in 3 + 1 dimensions.

Indeed, the existence of ninions is a generic property of any system that admits spatial rotations; therefore, the ninions can emerge in d ⩾ 2 spatial dimensions. In particular, the ninion can exist in d = 3 spatial dimensions where the spin-statistics theorem forbids any statistics different from bosonic or fermionic [1]. Therefore, let us reiterate that the ninion cannot be associated with a particle possessing a definite statistics different from the one of bosonic or fermionic particles.

Our results demonstrate that the ninionic property is associated with the energy level of the system rather than with the particle itself. While the occupation number of a ninion (33) carries a trace of information about the original statistics of this particle, the form of the statistical distribution depends on the quantum level occupied by the particle. For example, a fermionic particle can be a ghost occupation number at one quantum level, and it can again acquire a fermionic occupation number at the other level.

To illustrate this property, let us consider the occupation number (33) for a bosonic particle at the statistical parameter χ = π. Let us subject the particle to a potential that lifts the degeneracy of the energy levels with respect to the angular momentum (a remaining degeneracy of reflections, m → -m, does not play any role). Then, according to our discussion after Eq. ( 33), the thermal occupation number corresponds to (i) a boson at temperature T = 1/β for m = 4k;

(ii) a fermionic ghost at T = 1/(2β) for m = 2k + 1, (iii) a fermionic ghost at T = 1/β for m = 4k + 2, with k ∈ Z. The form of thermal distribution depends on the energy level occupied by the particle.

The non-trivial fractional statistics is often attributed to anyons which are particle-like objects associated with charge-flux composites [28] that also have various possible realizations in solid-state systems (for a recent review, see Ref. [29]). However, the anyons are different from the ninions due to the following three reasons:

1. The anyons, as locally defined fields, exist only in two spatial dimensions since the spin-statistics theorem does not apply to (2+1)d, and particles can be of any spin. The ninions, on the contrary, exist in d ⩾ 2 spatial dimensions.

2. An anyon is a local object that remains the anyon at any quantum level of the system. However, the statistics of ninions -seen via the thermal distributions-depends on the level of the system.

3. The commutation relations of a particle field reflect its statistics. For an anyon, the commutation relation differs from its bosonic and fermionic counterparts. In short, for an anyonic field c, one has c(x)c(x ′ ) = e iθ c(x ′ )c(x) with θ ∈ (-π, π]. However, as we show below, the commutation relation for the ninion fields corresponds to conventional bosonic or fermionic relations, thus excluding an identification of a ninion with an anyon.

IV. COHERENT ROTATIONAL STATES A. Rotwisted boundary in path-integral formalism

In the standard approach to a field theory in thermal equilibrium, bosonic and fermionic fields should satisfy the periodic and anti-periodic boundary conditions in the imaginary time direction. What is the thermodynamic sense behind the rotwisted boundary conditions ( 8) that differ from both mentioned boundary conditions? Here we follow Ref. [30] where the same question has been addressed with the translationally shifted boundaries.

Consider the grand canonical ensemble of a finite-temperature finite-density spin-0 bosonic system described by the Hamiltonian Ĥ with the particle number operator N . The relative contribution to the partition function Z coming from the states with the definite projection of the total angular momentum m on a fixed axis n is given by the following expectation value:

R n β, µ; m = P(m) n = Tr{e -β( Ĥ-µ N ) P(m) n } Z(β, µ) , (35)
where P(m) n is the projector onto these states and Z(β, µ) = Tr{e -β( Ĥ-µ N ) } is the path integral of the system. The traces are taken over all the states of the Hilbert space. Inessential volume factors are neglected.

The generating function K n , associated with the angular momentum distribution around the axis n, is defined by the Fourier transform of the relative contribution ( 35):

e -Kn(β,µ;χ) = m∈Z e iχm R n (β, µ; m) , (36)
where the compact quantity χ ∈ (-π, π] is the conjugated variable with respect to the momentum m. Reversing Eq. ( 36) we get the relative contribution coming from the states with the fixed angular momentum m:

R n (β, µ; m) = π -π dχ 2π Z(β, µ; χn/β) Z(β, µ) e -iχm , (37)
where

Z(β, µ; Ω I ) = Tr e -β( Ĥ-µ N -i ΩI • Ĵ ) , ( 38
)
is a partition function in which the states carrying the angular momentum m are weighted by the phase e iβΩ I • Ĵ .

Here Ĵ is the total angular momentum operator which, for a particle with a nonzero spin, also includes the spin contribution. For our spin-0 bosonic states |m⟩ with the definite value of m, one has e iβΩ I • Ĵ |m⟩ = e iβΩ I m |m⟩ and, therefore, we recover the phase factor in Eq. (37).

The "rotated" partition function

Z(β, µ; Ω I ) = m∈Z e iβΩ I m Z m (β, µ) , (39)
is the path integral in Euclidean time with the rotwisted boundary conditions in the imaginary time (8). Notice that the angle βΩ I ≡ χ, which determines the phase e iχm attributed to different m sectors in the path integral [START_REF] Rubakov | Classical Theory of Gauge Fields[END_REF], is nothing but the statistical angle (5)! Moreover, we immediately arrive to the conclusion that the partition function ( 39) can be represented as a sum over contributions Z m (β, µ) coming from different "topological sectors" characterized by the total angular momentum with respect to the axis n = Ω I /Ω I . Evidently, Z(β, µ) ≡ Z(β, µ; Ω I = 0). Consequently, the generating function for the relative contributions of the angular-momentum states can be written as the ratio of two partition functions,

e -Kn(β,µ;Ω I ) ≡ e iβ ΩI • Ĵ = Z(β, µ; Ω I ) Z(β, µ) , (40)
corresponding to the rotwisted (8) and standard (7) boundary conditions of the same theory. The quantity (40) has a well-defined thermodynamic limit. Therefore, at the level of the path integral, the rotwisted boundary conditions (8) produce the generating functional [START_REF] Belavin | [END_REF] for the states with the definite projection of the angular momentum number m. The statistical angle χ, related to the imaginary angular rotation frequency Ω I via Eq. ( 5), plays a role of the conjugate variable with respect to the angular momentum m.

The partition function for a field theory can also be formulated as a path integral over the fields [START_REF] Kapusta | Finite-temperature field theory: Principles and applications[END_REF]. For a scalar field theory in d + 1 dimensional time, the finitetemperature partition function (38) can be written as:

Z(β, µ; Ω I ) = Dπ rotwisted(Ω I ) Dϕ exp β 0 dτ × d d x iπ(x) ∂ϕ(x) ∂τ -H π, ϕ +µN π, ϕ . (41)
The term "rotwisted(Ω I )" indicates that the integration is performed over the scalar field ϕ(x) subjected to the rotwisted boundary condition (8a) with the angle χ = βΩ I . The integration over the conjugate momentum π(x) is not restricted. For a fermionic system, the angular momentum in the phase factors of Eqs. ( 36), (37), and (39) should be shifted, m → m + 1/2. This change reflects the anti-periodicity of the fermionic fields under 2π spatial rotations which also enters the corresponding rotwisted boundary conditions (8b). The partition function can be formulated similarly to Eq. ( 41).

B. Commutation relations for ninions

The standard imaginary-time formalism implies that the bosonic and fermionic) fields are periodic (antiperiodic) functions of the imaginary time (7). This requirement is closely related to the spin-statistics theorem, which implies, in 3+1 dimensions, that bosonic fields ϕ (fermionic fields ψ) are represented by commuting (anticommuting) operators, respectively:

[ϕ(x), ϕ(x ′ )] = 0, {ψ(x), ψ(x ′ )} = 0 . (42)
Here equal times (t = t ′ ) and different spatial coordinates (x ̸ = x ′ ) are implictly imposed. Since the spin-statistical properties in (3+1)d determine the boundary conditions (7) in the 4d Euclidean space with imaginary time, one could naturally ask ourselves whether the rotwisted boundary conditions (8) bring a modification to the commutation relations between the field operators, and thus determine a new statistics similar to one of the anyons in (2+1)d? There are two possible ways how to answer this question.

Standard commutation relations with complex-valued Hamiltonian

The first option is to assume that the path integral (38) corresponds to the imaginarily rotating system is governed by the Hamiltonian (6). Then, the thermal Green's function is [START_REF] Das | Finite temperature field theory[END_REF][START_REF] Mustafa | [END_REF],

G β (τ, τ ′ ) = ⟨T [Φ H (τ )Φ H (τ ′ )]⟩ β (43) = Tr (ρ T [Φ H (τ )Φ H (τ ′ )]) ,
where we introduced, for brevity, the statistical operator:

ρ = Z -1 (β)e -β ĤΩ I , (44)
where Z = Tr e -β ĤΩ I . The Green's function (43) also involves the imaginary-time ordering operator,

T [Φ H (τ )Φ H (τ ′ )] = Θ(τ -τ ′ )Φ H (τ )Φ H (τ ′ ) (45) ±Θ(τ ′ -τ )Φ H (τ ′ )Φ H (τ ) ,
where the upper and lower signs correspond to bosonic and fermionic field Φ, respectively. For simplicity, we consider hereafter a zero-density limit, µ = 0. In Eq. ( 43), the imaginary-time evolution of the field,

Φ H (τ ) = e τ ĤΩ I Φ e -τ ĤΩ I , (46)
is determined by the complex-valued Hamiltonian (6). The imaginary time variables τ and τ ′ are restricted to the principal interval: 0 < τ, τ < β.

The standard chain of the Kubo-Martin-Schwinger (KMS) relation [START_REF] Das | Finite temperature field theory[END_REF][START_REF] Mustafa | [END_REF] gives us:

G β (x, x ′ ; τ, τ ′ ) τ >τ ′ = Tr (ρ T [Φ H (x, τ )Φ H (x ′ , τ ′ )]) = Tr [ρ Φ H (x, τ )Φ H (x ′ , τ ′ )Θ(τ -τ ′ )] = Tr [Θ(τ -τ ′ )Φ H (x ′ , τ ′ ) ρ Φ H (x, τ )] = Tr Θ(τ -τ ′ )e -β ĤΩ I e β ĤΩ I Φ H (x ′ , τ ′ ) ρ Φ H (x, τ ) = Tr Θ(τ -τ ′ ) ρ Φ H ( Rχ x ′ , τ ′ + β)Φ H (x, τ ) = ±Tr ρ Θ(τ -τ ′ )Φ H (x, τ )Φ H ( Rχ x ′ , τ ′ + β) = ±Tr ρ T Φ H (x, τ )Φ H ( Rχ x ′ , τ ′ + β) ≡ ± G β x, Rχ x ′ ; τ, τ ′ + β , (47)
where we inserted the unit operator 1 = e -β ĤΩ I e β ĤΩ I , took into account the commutation relations (42), and used the cyclic properties of the trace. For definiteness, we took τ > τ ′ (see also Ref. [START_REF] Ambrus | [END_REF] for an accurate discussion of the case of real rotation). In deriving Eq. ( 47), we also used the imaginary-time evolution of the state:

Φ H ( Rχ x, τ + β) = e β ĤΩ I Φ H (x, τ )e -β ĤΩ I , (48)
which follows directly from the form of the Hamiltonian (6). The evolution (48) translates the state for one step along the imaginary time, τ → τ + β, and simultaneously rotates it by the angle (5), x → Rχ x about the spatial axis n = Ω I /Ω I . For fermions, Eq. (48) also includes rotation in the spinor space (8b) which is not shown explicitly.

The KMS relations set the boundary conditions for the fields along the compactified imaginary-time direction [START_REF] Das | Finite temperature field theory[END_REF][START_REF] Mustafa | [END_REF]. In our case, the chain of the relations (47) for the rotwisted boundary conditions (8) gives us the standard commutation relations (42) for bosonic, Φ = ϕ and fermionic, Φ = ψ, fields. Therefore, we conclude that under imaginary rotation, the bosons remain bosons, and the fermions remain fermions: no spin-statistical transformation for these particles occurs. However, as we already demonstrated for a generic system with imaginary rotation, the occupation numbers at different energy levels do not correspond, at thermal equilibrium, to bosonic or fermionic statistics.

Exotic commutation relations with real Hamiltonian

The second option is to assume that the system evolves according to the real-valued Hamiltonian defined in the laboratory frame:

ρ0 = Z -1 0 (β)e -β Ĥ0 , Z 0 = Tr e -β Ĥ0 . (49)
Then, instead of Eq. ( 48), we arrive to the standard evolution rule:

Φ H (x, τ + β) = e β Ĥ0 Φ H (x, τ )e -β Ĥ0 , (50)
which, however, does not allow to close the KMS chain of relations (47) with the standard commutation relations.

In order to make the evolution (50) to be consistent with the rotwisted boundary conditions (8), one needs to accept the following commutation relations:

Φ H (x ′ , τ ′ +β)Φ H (x, τ )= ± Φ H (x, τ )Φ H ( Rχ x ′ , τ ′ +β), (51)
where the upper (lower) index corresponds to a ninion originated from a boson (fermion). Equation (51) implies that the ninion fields (let us denote them η) are subjected to the following exotic equal-time commutation relations:

η(x, t)η(x ′ , t) = ±η(x ′ , t)η( Rχ x, t) . ( 52
)
We interpret the commutation relations (52) as follows.

The statistics of particles reveals itself in the evolution of the phase of a many-particle state when two of the identical particles swap their positions. Another manifestation of statistics emerges in the phase gained by a wavefunction of a single particle subjected to rotation. The commutation relation (52) incorporates both these concepts: as particles exchange their places, one of them rotates about the axis n = χ/χ -including the spin factor for fermions (8b) -and "misses" the correct place, x → Rχ x.

Regardless of the realization of the ninion statistics, the ninion system has an exotic statistical property that is different from the standard fractional statistics: the form of the occupation numbers for bosonic and fermionic particles depend on the level they occupy. In general, the level-dependent thermal distribution [START_REF] Mustafa | [END_REF] is related, via Eq. ( 31), to a continuous statistical parameter χ which interpolates between different types of statistical distributions. In the limiting cases, the thermal distribution takes either the standard bosonic or fermionic form.

C. Non-Hermiticity, P T -invariance, and unitarity of imaginary rotation

As we have already noticed, the form of the statistical operator (44) implies that the evolution of the imaginary rotating system is described by the "rotwisted" Hamiltonian (6). One can easily prove that, due to the presence of the purely imaginary term, this Hamiltonian is a non-Hermitian operator, Ĥ † Ω I ̸ = ĤΩ I , which could potentially lead to dissipation or instability in the system. However, it is somewhat amusing to recognize that so far we did not find any trace of these unwanted artificial features in the imaginary-time formulation of thermodynamics.

The reason behind this unusual property can be rooted in the invariance of the Hamiltonian (6) under the common action of the parity transformation, P : x → -x, and the time reversal, T : t → -t. Indeed, the total angular momentum Ĵ is a parity even (P Ĵ P = Ĵ ) and time-reversal odd (T Ĵ T = -Ĵ ) operator. Given the anti-linearity of the time reversal, T i T = -i, one finds that the rotwisted term of the Hamiltonian ( 6) is a P T -invariant operator:

P T (i ΩI • Ĵ )P T = i ΩI • Ĵ . (53)
In the P T -unbroken regime, the P T -symmetric non-Hamiltonians share the properties of the usual Hermitian Hamiltonians: they have a real-valued energy spectrum, describe the unitary evolution system and exhibit a welldefined thermodynamic limit [35]. Perhaps, we observe a somewhat similar effect in the imaginary time formalism: despite the non-Hermiticity of the Hamiltonian ( 6), the imaginary rotation appears to be a well-defined concept in thermodynamics.

D. Analogies with Quantum Chromodynamics

1. θ-angle in QCD in the path-integral formulation

The last term of the rotwisted Hamiltonian (6) shares similarity with the θ term in the theory of strong fundamental interactions, Quantum Chromodynamics (QCD). The θ term in QCD has a topological origin,

J θ QCD = -θq(x) , (54)
where

q(x) = g 2 16π 2 Tr G µν Gµν , (55)
is the topological charge density expressed via the gluonic field strength tensor

G µν = ∂ µ A ν -∂ ν A µ + i[A µ , A ν ]
and its dual Gµν = 1 2 ϵ µνρσ G ρσ . The integral over the topological density (55),

ν[A] = d 4 x w(x) ∈ Z , (56)
is an integer which determines the topological winding number ν of the gluonic configuration A µ . The potential existence of the θ term in the theory leads to "strong CP -problem" (for a review, see Ref. [36]). The vacuum structure of QCD depends on the distribution of the topological winding number (56) which determines important properties of QCD vacuum [37,38]. The Euclidean action of QCD with the θ term is

S QCD (θ) = S QCD 0 -iθν[A] , (57)
where

S QCD 0 = S G + d 4 xq -i / D + M q , (58)
is the standard Euclidean action which contains the gauge Yang-Mills term,

S G = d 4 x 1 2 Tr [G µν G µν ] , (59)
and the quark sector with the spinor fields q, the mass matrix M , and the covariant derivative / D = γ µ D µ (A). The QCD partition function involves the sum over all gauge-field configurations which are characterized by the winding number ν. Therefore, the partition function is a sum over different topological sectors ν weighted with the phase which depends on the θ angle:

Z QCD (θ) = ν∈Z e iθν Z QCD ν . (60)
In the partition function,

Z QCD ν = [DA µ ] ν det -i / D + M e -S G , (61)
the integration over the gluons A µ restricted to the sector with the fixed topological winding number ν. The functional determinant in Eq. ( 61) comes as a result of the integration over the quark fields q.

One notices the striking similarity between the partition function of QCD (60) and the partition function of the generic field theory [START_REF] Rubakov | Classical Theory of Gauge Fields[END_REF] subjected to the rotwisted boundary conditions (8): both functions are determined by the sums of the respective topological sectors. The similarity is enforced by the observation that the topological term in Eq. ( 57) is an analog of the last term in the rotwisted Hamiltonian (6) which determines the imaginary rotation. The θ-angle parameter in QCD is analogous to the statistical parameter χ which defines the rotwisted boundary conditions. Moreover, the integervalued topological winding number ν corresponds to the angular momentum quantum number m (in a certain sense, the analogy is even more expressive in the physical sense since the angular momentum counts how fast the system "winds" about the given axis).

θ-vacuum in QCD

The analogy between topology in QCD and imaginary rotation goes further. Below we consider the QCD θvacuum encoded in the topological properties of the gluon fields. We consider Yang-Mills theory which describes the gluonic sector of the theory.

The lowest-energy state of the classical Yang-Mills theory is an infinitely degenerate state due to the presence of the infinitely many classical minima [START_REF] Rubakov | Classical Theory of Gauge Fields[END_REF]. The minima are topologically inequivalent gluon configurations labeled by the topological number n ∈ Z. This number is usually associated with the difference between so-called Chern-Simons numbers n CS ≡ N CS [A] of the given vacuum state and a "fiducial" vacuum state.

In a semi-classical Yang-Mills theory, the transition between vacuum states is described by instantons [START_REF] Belavin | [END_REF] which are gluonic configurations A µ that are interpreted as tunneling events. An instanton event changes the topological number of the state, n CS → n CS +ν, by the integer-valued topological number (56) ν = ν[A] of the instanton.

In quantum theory, the tunneling between neighboring vacuum states removes the classical degeneracy of the ground state. The quantum ground state is now described by the θ vacuum:

|θ⟩ = ν∈Z e iθν |ν⟩ . (62)
The value of θ labels different vacuum states and, thus, removes the classical degeneracy in the quantized non-Abelian Yang-Mills gauge theory. The θ angle serves as an independent physical parameter of the theory [START_REF] Rubakov | Classical Theory of Gauge Fields[END_REF]. This situation, of course, is not pertinent only to Yang-Mills theory. Examples include quantum mechanics of a particle in a periodic potential, where the role of θ is played by a quasimomentum in the Bloch phase, and a charged particle on a ring, where the particle states feel the Aharonov-Bohm phase [START_REF] Rubakov | Classical Theory of Gauge Fields[END_REF]. In analogy with quantum mechanical properties of a harmonic oscillator, the θ-vacuum state (62) can be represented, up to a global phase, as a sum of the extended coherent states [41]:

|ν⟩ θ = e iθ NCS |ν⟩ , (63)
with distinct vacua |ν⟩ serving as fiducial vectors. Below, we will show that the ninions are the coherent states of the rotation operators similar to the topological coherent states in QCD (63).

E. Coherent states and imaginary rotation

Coming back to the rotwisted boundary conditions, we define the coherent rotational "χ-state" in similarity with the θ-vacuum in QCD (62): where |α, m⟩ n is a vector in the Hilbert space representing a state which carries the angular momentum m with respect to a fixed axis n and Ĵn ≡ n • Ĵ is the operator of projection of the angular momentum onto the axis n. As usual, the collective index α denotes the quantum numbers other than m. The set M α in the sum (64) incorporates all values of the angular momenta m with the same value of energy, ε α,m1 = ε α,m2 = ε α for m 1,2 ∈ M α . Therefore, the state (64) is a coherent state given by the superpositions of all energy-degenerate angular momentum modes:

Ĥ |α; χ⟩ n = ε α |α; χ⟩ n . (65)
In the absence of degeneracy, the state (64) is represented by a coherent state of the rotation operator:

|α; χ⟩ n = e iχ Ĵn |α, m⟩ n , (66)
where the state |α, m⟩ n serves as a fiducial vector in the Hilbert space.

In quantum optics, the vectors (66) are known as the "coherent spin states" [12,13]: the quantum number m is associated with the spin of a particle s z and the operator Ĵn becomes the operator of the spin projection Ŝz on the z axis, with n ≡ e z . In the context of imaginary rotation, the state (66) is an elementary ninion state in which the coherence parameter χ coincides exactly with the statistical parameter χ encoded in the rotwisted boundary conditions along the imaginary time direction (8).

Consider now the case of free quantum theory which we studied earlier. The eigenenergies of a free boson (11) are infinitely degenerate with respect to the angular momentum m similarly to the infinitely degenerate ground states of classical Yang-Mills theory that are labeled by the topological index ν. Thus, the rotational χ-state (64) for the free scalar theory can be written in the following suggestive form:

|p ρ , p z ; χ⟩ n = m∈Z e iχm |m; p ρ , p z ⟩ n , (67)
where the vector |m; p ρ , p z ⟩ n represents the state (10) and the vector n points along the z axis. For the ground state with a vanishing momentum, p ρ = p z = 0, the ground state can be called the χ-vacuum state (64) in analogy with the θ-vacuum in QCD (62).

Extending further the analogy with the physics of QCD, let us notice that the instantonic transition from one vacuum |ν⟩ to the other, |ν + 1⟩, can be represented as a change of the phase of the θ-vacuum by e iθ . Therefore, this phase factor is the eigenfunction of the corresponding instanton transition operator [START_REF] Rubakov | Classical Theory of Gauge Fields[END_REF].

To maintain the above analogy for the imaginary rotation, let us introduce the operator T which increases the angular momentum of the rotating system by one, T|m⟩ n = |m + 1⟩ n . In the free theory, the infinite degeneracy of the energy states (11) implies that the Hamiltonian and the shift operator commute with each other, [ Ĥ, T] = 0. Therefore, these operators share the eigenstates. The statistical parameter χ defines the eigenvalue of the angular-momentum-increasing operator, T |p ρ , p z ; χ⟩ n = e iχ |p ρ , p z ; χ⟩ n .

(68)

It is the pure phase, e iχ , similarly to the eigenvalue e iθ of the instanton operator acting on the QCD θ-vacuum [START_REF] Rubakov | Classical Theory of Gauge Fields[END_REF].

If the field is interacting with a background potential, then the infinite degeneracy of the states with respect to the angular momentum m is partially lifted out (an example is given in Section III A 2 on page 6). Therefore, for interacting systems, the sum in Eq. (67) goes only over the m-degenerate eigenenergies, m ∈ M α , leading to the series of the degenerate coherent χ-states (64).

In summary, the ninions can be associated with the coherent angular momentum states (66). In a free theory, such states possess an infinite degeneracy which makes their ground state similar to the θ-vacuum of QCD.

V. CONCLUSIONS

We discussed quantum field theories in thermal equilibrium in the modified Euclidean imaginary time formalism. The standard (anti-)periodicity of fields in the compactified imaginary time direction is generalized to the "rotwisted" boundary condition (8) which supplements the advance along the imaginary time with a global spatial rotation (illustrated in Fig. 1). We demonstrated that the fields satisfying such a condition -which formally corresponds to rotation with an imaginary angular momentum-evolve according to a P T -symmetric non-Hermitian Hamiltonian, which points to the unitary evolution of the underlying physical system.

The rotwisted boundary conditions describe new excitations that we call "ninions" (the etymology of this word is given in footnote 4 on page 7). The ninions have intriguing statistical features as the functional dependence of the ninion occupation number on temperature, and chemical potential depends on the quantum eigenstate that this particle occupies. As the functional form of the occupation number determines the statistical distribution of particles in the ensemble, the ninion possesses variable, level-dependent statistics, different from bosons and fermions. The ninions exist in three spatial dimensions and are also different from anyons. We gave an explicit example of a ninionic system for which, at a certain eigenstate, the particle occupation number is given by the Bose-Einstein statistics, while at another eigenstate, the same particle becomes a fermionic ghost. Such a ghost is described by the Fermi-Dirac statistics but enters the system's free energy with the wrong, "bosonic" sign.

The ninionic occupation number [START_REF] Mustafa | [END_REF] is labeled by the continuous number related to the statistical angle χ. This angle enters the rotwisted boundary condition [START_REF] Kapusta | Finite-temperature field theory: Principles and applications[END_REF] and can be incorporated in the path integral formalism (41). The ground state of free ninions shares similarity with the θ-vacuum in Quantum Chromodynamics, with the topological θ angle playing the role of the statistical parameter χ while the topological QCD charge is associated with the angular momentum. We argue that the rotwisting (statistical) angle χ is an independent coupling in thermodynamics in the same sense as the θ angle is an independent coupling in QCD.

Interestingly, certain ninionic states appear to be ghosts that possess negative pressure (29). This property could make a ninionic state an exciting candidate for cosmological dark energy, which could play a role in the observed acceleration of the Universe's expansion.

We also demonstrated that the dependence of thermodynamic characteristics of noninteracting ninions on the statistical angle χ in thermodynamic limit is determined by the Thomae function (20) which is discontinuous at every point. This non-analyticity implies a no-go theorem on the absence of analytical continuation between real and imaginary rotation if the latter is introduced via the rotwisted boundary.

A concrete experimental realization of ninions might exist in systems hosting coherent angular momentum states [11] and coherent spin states [12] that play a significant role in quantum optics [13]. The exotic occupation numbers of ninions may point to the possibility of nonequilibrium but steady states. Furthermore, properties of ninions are also accessible for numerical simulations on Euclidean lattices using standard Monte Carlo methods.

FIG. 1 .

 1 FIG.1. Illustration of (a) standard (7) and (b) rotwisted (8) boundary conditions with the segments O ′ A ′ at τ = 0 and OA at τ = β identified. The axis of rotation z is not shown. The value of the rotwisted angle χ depends on the imaginary angular frequency(5) and plays a role of the statistical parameter.

  FIG.4. Intersecting Dirichlet and Neumann walls(24). The potential set by the walls is consistent with the imaginarytime boundary conditions (8a) set by the imaginary rotation ΩI = π/β (the statistical parameter χ = π). Subjected to imaginary rotation, this system converts a boson into a fermionic ghost with the free energy(27) and thermodynamic characteristics given in Eqs.(28).

)ξ = π / 8 ξ = π / 4 ξFIG. 5 .

 845 FIG.5. The examples of the ninionic (33) thermal occupation numbers corresponding to (a) bosons and (b) fermions in the thermal equilibrium for a set of the parameters ξ. The latter are related to the imaginary angular frequency and the statistical parameter χ via Eqs. (5) and[START_REF] Kapusta | Finite-temperature field theory: Principles and applications[END_REF]. The case ξ = 0 corresponds to the standard Bose-Einstein and Fermi-Dirac distributions, respectively.

  |α; χ⟩ n = m∈Mα e iχm |α, m⟩ n ≡ m∈Mα e iχ Ĵn |α, m⟩ n , (64)

The 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".

For example, two close values of the imaginary frequency, Ω I β/(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.

Here, we do not consider irrational imaginary frequencies Ω I for which the periodicity does not exist. Irrational frequencies (in units of 2π/β) correspond to a zero temperature according to our discussion below, based on the Thomae function(20). The analogy is similar to dynamics of an electron on a square lattice exposed to rational and irrational magnetic fields that have very different effects on the electron dynamics[16].

ACKNOWLEDGMENTS

The author is grateful to Victor Ambrus , , Vladimir Goy, and Stam Nicolis for useful and helpful comments.


Pierre Sauleau

Pascal Retailleau

Jean Vacelet

Marie-Lise Bourguet-Kondracki

New polychlorinated pyrrolidinones from the Red Sea marine sponge Lamellodysidea herbacea

Keywords: Lamellodysidea herbacea, Sponge, Dysidamide, Polychlorinated pyrrolidinones

Chemical investigation of the dichloromethane extract of the Red Sea marine sponge Lamellodysidea herbacea led to the isolation of seven new polychlorinated derivatives 3-6 and 8-10 in addition to the known dysidamide 1, dysidamide B 2 and dysidamide C 7. Their structures were established by extensive NMR spectroscopic data. The absolute configuration of compound 9 was determined by X-ray crystallographic diffraction analysis. Dysidamide 1 exhibited neurotoxic effects towards both mesencephalic and cortical murine neurones at 0.8 mg/ml.

Introduction

Marine sponges of the genus Dysidea have yielded secondary metabolites of a great structural diversity such as sesqui-and sesterterpenes, [1][2][3] polybrominated derivatives of diphenyl ether 1 and dibenzo-p-dioxin types 4,5 and polychlorinated derivatives 6 including amino acids 7 as well as peptides. 8 Some of them exhibited interesting pharmacological activities such as inhibition of iodide transport in thyroid cells, 9 neurotoxicity 10 and inhibition of protein phosphatase. 11 In our ongoing chemical studies on marine sponges, we have examined a sample of Lamellodysidea herbacea (formerly Dysidea herbacea) collected in the Red Sea. Chemical investigations of the dichloromethane extract led to the isolation of seven new polychlorinated derivatives 3-6 and 8-10 in addition to the known dysidamide 1, dysidamide B 2 and dysidamide C 7,p r e v i o u s l y reported. 12,13 This paper describes the isolation and structure elucidation of these new polychlorinated derivatives.

Results and discussion

The marine sponge Lamellodysidea herbacea was collected by hand using SCUBA (K20 m) in the Red Sea and was airdried. Silica gel flash chromatography (CH 2 Cl 2 /MeOH) of the dichloromethane extract followed by Sephadex LH-20 and/or reversed-phase HPLC yielded compounds 1-10 in pure form. The known compounds dysidamide 1, dysidamide B 2 and dysidamide C 7, originally isolated from the marine sponge Dysidea herbacea 12,13 were rapidly identified by comparison with literature values. An X-ray diffraction analysis 14 of dysidamide 1 showed the presence of two conformers in the unit cell, in which all asymmetric carbon centres exhibited (S) configuration, as previously reported. 15 Compounds 3-6 and 8-10 were identified through extensive 1D and 2D NMR spectroscopic data.

Compound 3 was isolated as a colourless glass, [a] D ZK4.0 (c 0.15, CH 2 Cl 2 ). Electrospray mass spectrum of compound 3 exhibited a cluster ion at m/z 440, 442, 444, 446, 448 characteristic of the presence of five chlorine atoms in the molecule. The molecular formula C 15 H 22 NO 3 Cl 5 was deduced from HREIMS and 13 C NMR spectrum. In the IR spectrum, absorption bands at 3606, 1741 and 1697 cm K1 indicated the presence of an hydroxyl (or NH) and two carbonyl groups, respectively. NMR spectral data were similar to those of dysidamide 1, pointing out the presence of a pyrrolidinone ring system (see Table 1). In comparison with dysidamide 1, we observed in the 1 H NMR spectrum the presence of one additional signal at d 5.95 (1H, d, JZ 3.3 Hz). The 13 C NMR spectrum showed the loss of the quaternary carbon signal at d 104.9 and the presence of one additional methine carbon signal at d 77.8. Furthermore, we noticed that the methine proton at d 50.7 was shifted at d 40.1. This information suggested the replacement of one CCl 3 group by a CHCl 2 functionality in 3. This hypothesis was strengthened by COSY spectrum. Starting from the additional proton at d 5.95 (H4 0 ), correlations could be observed to the methine proton at d 2.78 (H3 0 ), which in turn coupled to both methylene protons at d 3.19-3.03 (H2 0 ) and methyl protons at d 1.18 (H5 0 ), confirming the presence of a Me-CH(CHCl 2 )-CH 2 -side chain in 3. A second spin system was observed in the COSY spectrum with correlations between the methyl doublet at d 1.36 (H11) to the methine proton at d 2.98 (H7), which in turn coupled to the methylene protons at d 2.16 (H6). These protons further coupled to the methine proton at d 4.36, which coupled to the methine proton at d 4.14 (H4), as in dysidamide 1. Key HMBC correlations between both methylene protons at d 3.19-3.03 (H2 0 ) and methine proton at d 4.36 (H5) with the carbonyl C1 0 at d 172.9, unambiguously located each chain on the pyrrolidinone ring as depicted in Figure 2. NOESY correlations between the methyl protons at d 1.21 in position 10 and both methine protons at d 4.36 (H5) and 4.14 (H4) and between the methyl protons at d 1.19 in position 9 with the methylene protons at d 2.16 indicated that the relative stereochemistry of the new compound 3 was identical to that of dysidamide 1. We propose the name dysidamide D for this compound 3.

Compound 4 was isolated as a colourless glass, [a] D ZK3.5 (c 0.3, CH 2 Cl 2 ). The molecular formula, C 15 H 22 NO 3 Cl 5 , deduced from HREIMS indicated that compound 4 was isomer of compound 3. The 1 H NMR spectrum rather matched that of dysidamide B 2 by the presence of the non equivalent methylene protons at d 2.03-1.88. However, we noticed in the 1 H NMR spectrum the loss of one doublet signal at d 5.95 (see Table 1). The COSY spectrum delineated the presence of two spin systems, whose position was secured by HMBC correlations, as presented in Figure 3. NOESY correlations between the methyl protons Me-10 at d 1.21 and the methine protons at d 4.31 (H5) and 4.12 (H4) and between the methyl protons at d 1.22 and the methylene protons at d 2.03-1.88 were observed, assigning the stereochemistry of 4 identical to that of dysidamide 1. Thus, combined 2D NMR data allowed to propose the structure depicted in Figure 1 for compound 4 (see also Tables 1 and5), we named dysidamide E.

Compound 5 was obtained as white needles, mp 173-174 8C, [a] D ZK41.8 (c 1.1, CH 2 Cl 2 ). The molecular formula C 16 H 25 NO 4 Cl 6 was obtained by combined HREIMS and 13 C NMR spectral data. The spectral data of 5 were similar to those obtained for dysidamide 1 (see Table 2). The significant difference in 1 H NMR spectrum was the presence of two additional signals at d H 5.84 and d 3.70. This last signal gave a carbon resonance in 13 CNMR spectrum at d 52.4, indicating the presence of a methoxyl group in the molecule. HMBC correlations between these methoxyl protons at d 3.70 to the carbonyl at d 178.9 suggested the presence of a methyl ester. HMBC correlations between the two singlet methyl protons at d 1.19 and 1.28 to the carbonyl at d 178.9, placed the methyl ester function in position 2 on the molecule. We also observed the presence of two resonances at d 106.1 and 105.0 due to two CCl 3 groups. Also in this case, the COSY spectrum confirmed the presence of the two chains identical to those of dysidamide 1. In addition, the deshielded proton at d 5.84, which did not matched any carbon resonance in the HSQC spectrum, showed an HMBC correlation to the carbonyl at d 169.7, showing the presence of an amide function. At last, key HMBC correlations between the methine proton at d 2.48-1.45 (H5) to the carbonyl at d 169.7, suggested the arrangement reported in Figure 1 for compound 5.

Compound 6 was obtained as white needles, mp 95-96 8C;

[a] D ZK29.6 (c 0.44, CH 2 Cl 2 ), that analysed for C 16 H 23 NO 4 Cl 6 by HREIMS. The IR spectrum showed absorptions due to the presence of three carbonyl groups (l 1749, 1723 and 1683 cm K1 ). The spectral data of 6 were similar to those of compound 5 (see Table 2). The only noticeable difference in the 1 H NMR spectrum was the loss of the methine proton at d 3.59. We observed in the 13 C NMR spectrum an additional signal at d 206.9. These spectral differences were readily accommodated by the replacement of the secondary alcohol group in 5 by a ketone function in position 3 of the molecule. This proposal was confirmed by interpretation of combined 2D NMR spectra (see Table 2). ). The 1 HN M R spectrum recorded in CD 3 OD was similar to that of compound 8 but showed that all 1 H signals were split into two in a 3 ⁄ 4 : 1 ⁄ 4 proportion, suggesting the presence of two diastereoisomers (see Table 3). Compound 9 was crystallized by slow evaporation from a mixture of MeOH/H 2 O (7/3) for X-ray studies. The crystals belong to the triclinic system, space group P1, chiral. Anomalous dispersion from the chlorine atoms of the molecule permitted us to deduce the relative stereochemistry of the asymmetric carbons, namely C5 and C7.

The crystal is made of two pairs of molecules with occupancy factorZ0.5 in the unit cell. In each pair, one of the two molecules, (Ia), exhibits the S-configuration at both quaternary carbons C5(S) and C7(S) (see Fig. 4). In the hydrogen bonds, involving N1a and O2 00 or O2b (dZ 2.02 A ˚, angle N-H-OZ151.88) and O2a and N1 00 or N1b (dZ2.11 A ˚, angle N-H-OZ171.08), respectively (see Fig. 5).

In the unit cell, molecule (Ia) adopts a N4-envelope conformation as deduced from the torsion angles (see Table 4). The second molecule (Ib) adopts a half-chair conformation and also exhibits S-configuration at both quaternary carbons. The third molecule (II) appears to be C5(R), C7(S) and adopts a half-chair conformation (see Fig. 4).

Hence, the crystal is made of two diastereoisomers in the following proportion 3 ⁄ 4 : 1 ⁄ 4 which is somewhat consistent with the 1 H NMR results aforementioned. We propose the name dysidamide G for compound 9a and 5-epi-dysidamide G for 9b (see Fig. 1). Previous intensive studies of the same species, referred to as Dysidea herbacea demonstrated that large quantities of the filamentous cyanobacterium Oscillatoria spongeliae are responsible for the occurrence of the isolated chlorinated compounds. 16,17 Our work also illustrates and re-enforces the chemical diversity within the genus Lamellodysidea.

Dysidamide 1 was tested towards both mesencephalic and cortical murine neurones. In these two neuronal cultures, dysidamide 1 provoked the entire and rapid death of neurones, even at doses of 0.8 mg/ml. No significant effect was observed at 100 ng/ml. Further investigations are in progress to evaluate the neurotoxic effect of the other compounds in the series.

Experimental

General experimental procedures

Silica gel column chromatographies were carried out using ) can be H-bonded either to its conformer (Ib) or its diastereoisomer (II). Note that in the crystallographic refinement, N1b is equivalent to N1 00 , C2b to C2 00 , C3b to C3 00 , C4b to C4 00 , C9b to C9 00 , C10 to C10 00 , O2b to O2 00 , O4b to O4 00 , and so are the corresponding attached hydrogens.

Kieselgel 60 (230-400 mesh, E. Merck). Fractions were monitored by TLC using aluminium-backed sheets (Si gel 60 F254, 0.25 mm thick). Analytical reversed-phase HPLC (Kromasil RP18 column K2185, 4.6!250 mm, MeOH/H 2 O) were performed with a L-6200A pump (Merck-Hitachi) equipped with a UV-vis detector (lZ 210 nm) L-4250C (Merck-Hitachi) and a chromatointegrator D-2500 (Merck-Hitachi).

IR spectra were recorded on a Nicolet IMPACT 400D FT-IR spectrophotometer.

Mass spectra were recorded on an API Q-STAR PULSAR I of Applied Biosystem and on a JEOL MS 700BE for low and high-resolution, respectively. 13 C NMR spectra were obtained on a Bruker AC300 at 75.47 MHz, 1 H NMR spectra 1D and 2D (COSY, HSQC, HMBC, NOESY) were obtained on a Bruker AVANCE 400.

Animal material

Specimens of Lamellodysidea herbacea (Keller, 1889) (Order Dictyoceratida, Family Dysideidae) were collected by J. Vacelet in the Red Sea at a depth of 20 m in January 1985 during the Ardoukoba expedition. A voucher specimen is available from the Muse ´um d'Histoire Naturelle in Marseille as collection number MHNM 13660. In the new classification of Hooper and Van Soest, the species D. herbacea and D. chlorea have been transferred from Dysidea to the new genus Lamellodysidea. [START_REF] Cook | Systema Porifera: A Guide to the Classification of Sponges[END_REF] This new genus has been split from Dysidea because of the consistent presence in these sponges of an encrusting basal plate and the lack of orientation of the skeleton with respect to the surface. Many sponge specimens from the Indo-Pacific area have been previously identified to Dysidea herbacea, which has been originally described from the Red Sea. However, it is likely that this species has been confused with other species such as D. chlorea, [START_REF] Cook | Systema Porifera: A Guide to the Classification of Sponges[END_REF] which could explain the reported chemical diversity.

Extraction and isolation

The air-dried sponge (288 g) was extracted with 3L methanol at room temperature for 3 days. The concentrated extract (60 g) was partitioned into hexane, CH 2 Cl 2 , MeOH soluble fractions. The CH 2 Cl 2 soluble extract was then fractionated by rapid filtration on a silicagel (Merck silica gel 70-230 mesh) column using dichloromethane with increasing amounts of methanol as eluent. Fraction eluted with 2% MeOH (5.4 g) was subjected to successive reversed-phase HPLC using MeOH/H 2 O 7/3 as eluent (flow rate: 1 ml/min, det. wavelength of 210 nm) to yield in order of elution dysidamide B 2 (4.3 mg, at tZ12.66 mn), dysidamide D 3 (1.5 mg at tZ15.44 mn), dysidamide E 4 (3.0 mg, at tZ15.90 mn), dysidamine 1 (450 mg at tZ 17.99 mn), compound 5 (11.4 mg at tZ19.55 mn), compound 6 (4.4 mg at tZ19.56 mn), dysidamide H 10 (6.4 mg at tZ20.86 mn). Fraction eluted with 5% MeOH (5.5 g) was first separated on a Sephadex LH20 in 5 subfractions. Subfraction 3 (3.6 g) was successively subjected to reversed-phase column and RP-18 HPLC using MeOH/ H 2 O 7/3 as eluent to yield dysidamide C 7 (1 mg), dysidamide F 8 (10 mg) and dysidamide G 9a with 5-epidysidamide G 9b (26 mg). 13 C NMR data recorded in CDCl 3 : see Tables 1 and5

X-ray crystallographic analysis

A total of 8577 (4051 unique) reflections were collected up to 2qmaxZ50.08 in 234 frames of 28 oscillation range using an Enraf-Nonius Kappa-CCD diffractometer with graphite monochromated Mo-K radiation (lZ0.71073 A ˚) at room temperature. The structure was solved by Patterson method to locate four over six chlorides, the rest of atoms was derived from successive Fourier difference syntheses. Structural parameters were refined based on F 2 using the programs from the SHELX97 package. [START_REF] Sheldrick | Program for the Refinement of Crystal Structures from Diffraction Data[END_REF] All non-hydrogen atoms were refined anisotropically, except for C11b and C11 00 . Hydrogen atoms were assigned idealized locations and given isotropic displacement parameters 1.

Figure 1 .

 1 Figure 1. Structures of compounds 1-10 isolated from the marine sponge Lamellodysidea herbacea.

Figure 2 .

 2 Figure 2. COSY and selected HMBC correlations used to determine the structure of 3.

Figure 3 .

 3 Figure 3. COSY and selected HMBC correlations used to determine the structure of 4.

  crystal packing, molecule (Ia) is connected to either its c o n f o r m e r( I b )o rt oi t sd i a s t e r e o i s o m e r( I I )b yt w o

  additional carbonyl signal at d 209.4. Extensive 1D and 2D spectra were consistent with structure presented in Figure1. In particular, HMBC correlations between the gem methyl protons at d 1.40, the methine proton at d 4.52 and the methylene protons at d 2.20 with the carbonyl at d 209.4 supported this structure. NOESY correlation between the methyl at d 1.31 in position 9 and the methylene protons at d 2.20 was observed. We propose the name dysidamide H for compound 10.

Figure 4 .

 4 Figure 4. ORTEP-plots showing the molecule Ia: C5(S), C7(S), Ib: C5(S), C7(S) and II: C5(R), C7(S) of compound 9. Atomic displacement ellipsoids drawn at the 30% probability level, hydrogen atoms drawn as spheres of arbitrary radius.

Figure 5 .

 5 Figure 5. Crystal packing of compound 9: interpretation of the unit cell content in the bc plane. Molecule (Ia) (left column) can be H-bonded either to its conformer (Ib) or its diastereoisomer (II). Note that in the crystallographic refinement, N1b is equivalent to N1 00 , C2b to C2 00 , C3b to C3 00 , C4b to C4 00 , C9b to C9 00 , C10 to C10 00 , O2b to O2 00 , O4b to O4 00 , and so are the corresponding attached hydrogens.

  ; HREIMS [MCH] C found at m/z 406.0500 (DK1.0 mmu) for C 15 H 24 NO 3 35 Cl 4 . 3.3.3. Compound 3. Dysidamide D, 1-(4,4-dichloro-3methyl-1-oxobutyl)-4-hydroxy-3,3-dimethyl-5-(3,3,3-trichloro-2-methylpropyl)-2-pyrrolidinone. 0.0015 g, 0.05!10 K2 % dry weight colourless glass; [a] D ZK4.0 (c 0.15, CH 2 Cl 2 ); IR n cm K1 : 3606, 1741, 1697, 1386, 1371, 1 H and 13 C NMR data recorded in CDCl 3 : see Tables 1

Cl 3 . 3 . 3 . 10 .

 33310 Compound 10. Dysidamide H, 3,3-dimethyl-4oxo-1-(4,4,4-trichloro-3-methyl-1-oxobutyl)-5-(3,3,3-trichloro-2-methylpropyl)-2-pyrrolidinone. 0.0064 g, 0.22!10 K2 % dry weight colourless glass; [a] D ZK62.8 (c 0.6, CH 2 Cl 2 ), IR n cm K1 : 1776, 1741, 1708, 1385, 1373; 1 H and 13 C NMR data recorded in CDCl 3 : see Tables 1 and 5;H R E I M S[ M CH] C found at m/z 471.9579 (DC0.5 mmu) for C 15 H 20 NO 3 35 Cl 6 .

Table 1 .

 1 1 H NMR data of compounds 1-4 and 10 recorded in CDCl 3 (d H , multiplicity, J in Hz)

	No	12341 0
	4	4.13 (d, 7.5)	4.10 (br d, 7.3)	4.14 (d, 7.7)	4.12 (d, 7.3)
	5	4.38 (dt, 7.5, 5.3)	4.28 (m)	4.36 (dt, 7.6, 5.2)	4.31 (ddd, 7.3, 4.5, 7.0)	4.52 (t, 6.5)
	6	2.17 (m)	2.04 (ddd, 14.1, 6.0, 7.0)	2.16 (m)	2.03 (brdd, 14.0, 6.9)	2.20 (dd, 6.0, 6.8)
			1.85 (m)		1.88 (ddd, 14.3, 5.9, 4.5)
	7	2.97 (m)	2.48 (m)	2.98 (m)	2.48 (m)	2.98 (m)
	8		5.93 (d, 2.9)		5.94 (d, 2.8)
	9	1.21 (s)	1.20 (s)	1.19 (s)	1.22 (s)	1.31 (s)
	10	1.22 (s)	1.20 (s)	1.21 (s)	1.21 (s)	1.40 (s)
	11	1.37 (d, 6.4)	1.17 (d, 6.7)	1.36 (d, 6.5)	1.19 (d, 6.0)	1.47 (d, 6.4)
	2 0	3.55 (dd, 18.1, 1.6)	3.20 (dd, 18.1, 5.9)	3.19 (dd, 18.1, 5.8)	3.58 (dd, 18.1, 2.6)	3.68 (dd, 18.0, 1.8)
		3.14 (dd, 18.1, 9.4)	2.98 (dd, 18.1, 6.9)	3.03 (dd, 18.1, 7.1)	3.11 (dd, 18.1, 9.3)	3.25 (dd, 18.0, 9.2)
	3 0	3.29 (m)	2.77 (m)	2.78 (m)	3.30 (ddq, 2.7, 9.2, 6.5)	3.31 (m)
	4 0		5.95 (d, 3.3)	5.95 (d, 3.3)
	5 0	1.34 (d, 6.4)	1.19 (d, 6.6)	1.18 (d, 6.7)	1.35 (d, 6.5)	1.39 (d, 6.6)

  Under mild basic conditions with 0.1% K 2 CO 3 in MeOH, dysidamide 1 gave rise to a compound, whose NMR data were in complete accordance with those of the natural product 8, suggesting (S) configuration for all asymmetric centres. We propose the name dysidamide F for compound 8.

	This compound was previously obtained after hydrolysis of
	dysidamide 1.
	This is the first report of this compound as a natural product.

Compound 8 was isolated as white needles, mp 171-172 8C, [a] D ZK36.0 (c 0.84, CH 2 Cl 2 ). Its ESI mass spectrum exhibited a cluster of peaks at m/z 288, 290, 292, 294 with a pattern characteristic of a trichlorinated compound. The molecular formula C 10 H 16 NO 2 Cl 3 was determined on the basis of HREIMS and NMR data. Comparison of the NMR data of 8 with those of dysidamide 1 indicated that 8 also contained a pyrrolidinone ring system. The 1 HN M R spectrum also showed the presence of a methylene group at d 2.33-1.73, one methine at d 2.79 and one methyl doublet at d 1.40, indicating that 8 possessed the same MeCH(CCl 3 )CH 2 -side chain as dysidamide 1 on C5. NOESY correlations between the methyl protons in position 10 at d 1.15 and protons H-4 and H-5 and between the methyl protons in position 9 at d 1.12 with the methylene protons at d 2.33-1.73 were also observed.

The structure of 8 was secured by hydrolysis of dysidamide 1.

12

Compound 9 was isolated as colourless needles, mp 109 8C, [a] D ZK38.5 (c 0.75, CH 2 Cl 2 ). Its ESI mass spectrum exhibited a characteristic cluster of three chlorine atoms for the [MCH] C peaks at m/z 286, 288, 290, 292. From the lower peak at m/z 286.0161 (DK0.8 mmu) by HREIMS, a molecular formula of C 10 H 14 NO 2 Cl 3 was deduced. Its IR spectrum showed the presence of hydroxyl or NH groups (l 3419 cm K1 ), two carbonyl functions (l 1775, 1711 cm K1 ), one gem dimethyl (l 1416, 1389 cm K1

Table 2 .

 2 1 H and13 C NMR data of compounds 5-6 recorded in CDCl 3

	N8		56
		d C	d H (m, J in Hz)	HMBC (H/C)	d C	d H (m, J in Hz)	HMBC (H/C)
	1	178.9			173.5
	2	45.8			40.5
	3	78.9	3.59 (d, 1.1)	1, 2, 5, 9, 10	206.9
	4	46.7	4.33 (ddd, 12.6, 11.7, 1.1)	5, 6, 11	51.6	5.09 (ddd, 11.8, 9.5, 2.3)	5, 6, 11
	5	39.5	2.48 (ddd, 12.8, 12.8, 0.8)	4, 6, 7, 11, 1 0	36.9	2.12 (brdd, 14.2, 0.7)	4, 6, 7, 11, 1 0
			1.45 (ddd, 13.9, 2.5, 10.1)			1.68 (ddd, 14.2, 10.2, 2.3)
	6	52.0	2.39 (ddq, 10.1, 1.7, 6.3)	4, 5, 11, 1 0	51.7	2.47 (m)	4, 5, 11, 1 0
	7	106.1			105.3
	8	52.4	3.70 (s)	1	52.9	3.73 (s)	1
	9	20.1	1.19 (s)	1, 2, 3,10	21.9	1.42 (s)	1, 2, 3,10
	10	24.0	1.28 (s)	1, 2, 3, 9	22.2	1.43 (s)	1, 2, 3, 9
	11	16.8	1.41 (d, 6.3)	5, 6, 7	16.3	1.42 (d, 4.2)	5, 6, 7
	1 0	169.7			170.0
	2 0	40.4	2.92 (dd, 14.8, 2.5)	1 0 ,3 0 ,5 0	40.3	2.97 (dd, 14.9, 3.1)	1 0 ,3 0 ,5 0
			2.13 (dd, 14.8, 10.2)			2.20 (dd, 14.9, 9.6)
	3 0	51.6	3.17 (ddq, 10.2, 2.5, 6.4)	1 0 ,2 0 ,4 0 ,5 0	51.6	3.17 (ddq, 3.1, 9.6, 6.5)	1 0 ,2 0 ,4 0 ,5 0
	4 0	105.0			104.8
	5 0	16.8	1.33 (d, 6.4)	2 0 ,3 0 ,4 0	16.9	1.34 (d, 6.5)	2 0 ,3 0 ,4 0
	NH1 0		5.84 (d, 9.8)	3, 4, 1 0		6.02 (d, 9.5)	3, 4, 1 0

Table 3 .

 3 1 H NMR data of compounds 8, 9a and 9b recorded in CD 3 OD (d H , multiplicity, J in Hz)

	N8	89 a9 b
	4	3.97 (d, 5.3)
	5	3.86 (ddd, 5.3, 4.5, 9.5)	4.24 (dd, 3.4, 10.9)	4.22 (dd, 5.3, 9.5)
	6	2.33 (ddd, 14.1, 10.2, 1.5)	1.91 (ddd, 13.9, 10.5, 3.4)	1.80 (ddd, 13.9, 9.5, 5.3)
		1.73 (ddd, 14.1, 10.2, 4.0)	2.13 (ddd, 13.9, 10.9, 2.3)	2.36 (m)
	7	2.79 (m)	2.87 (ddq, 10.5, 2.3, 6.5)	3.03 (brdq, 9.5, 6.5)
	8-	-	-
	9	1.12 (s)	1.20 (s)	1.21 (s)
	10	1.15 (s)	1.23 (s)	1.22 (s)
	11	1.40 (

d, 6.5) 1.41 (d, 6.5) 1.42 (d, 6.5)

Table 4 .

 4 List of selected torsion angles (8) for compound 9

	Diastereoisomer

Table 5 .

 5 13 C NMR data of compounds 1-4 and 10 recorded in CDCl 3 and compounds 7-9 recorded in CD 3 OD Cl 2 ), IR n cm K1 : 3414, 1749, 1723, 1683, 1512, 1427; 1389, 1376, 1 H and13 C NMR data recorded in CDCl 3 : see Table2; HREIMS [MCH] C found at m/z 503.9821 (DK1.5 mmu), calcd 503.9836 for C 16 H 24 NO 4 35 Cl 6 .

	N8	1	2	3	4	7	8	9a	9b	10
	2	178.9	178.8	178.9	178.9	183.9	183.9	180.2	180.2	176.1
	3	46.4	46.5	46.3	46.5	46.7	46.7	47.4	47.4	49.1
	4	72.9	73.0	73.1	72.9	77.9	77.9	215.1	215.3	209.4
	5	57.0	56.7	56.9	56.8	54.8	55.3	60.2	59.9	61.4
	6	33.1	31.9	33.2	31.7	33.6	34.6	37.7	37.7	35.1
	7	53.5	46.5	53.5	42.8	42.1	53.3	52.7	52.2	50.4
	8	105.8	78.4	105.9	78.3	80.5	107.6	106.6	106.6	105.1
	9	19.3	19.3	19.3	19.2	18.6	18.8	21.2	21.4	20.8
	10	23.9	24.1	24.1	24.0	23.5	23.7	21.3	21.4	22.9
	11	16.9	15.1	16.9	15.5	14.9	16.7	16.1	17.0	15.8
	1 0	172.3	172.7	172.9	172.2					171.0
	2 0	41.9	40.9	40.8	41.9					42.4
	3 0	50.7	40.0	40.1	50.7					50.4
	4 0	104.9	77.8	77.8	105.0					104.7
	5 0	17.2	15.4	15.2	17.3					17.4

Table 5 ;

 5 ESIMS [MCH] C found at m/z 254.0578 for C 10 H 18 NO 2 Cl 2 .

	3.3.8. Compound 8. Dysidamide F, 4-hydroxy-3,3-
	dimethyl-5-(3,3,3-trichloro-2-methylpropyl)-2-pyrroli-
	dinone. 0.010 g, 0.3 10 K2 % dry weight white needles;
	mp 171-172 8C; [a] D ZK36.0 (c 0.84, CH 2 Cl 2 ); IR n cm K1 : 3610, 3423, 1716, 1392, 1386; 1 H and 13 C NMR data
	recorded in CD 3 OD: see Tables 3 and 5; HREIMS [MC H] C found at m/z 288.0319 (DK0.6 mmu) for C 10 H 17 NO 2 35 Cl 3 .
	3.3.9. Compound 9. Dysidamide G with 5-epi-dysida-
	mide G, 3,3-dimethyl-4-oxo-5-(3,3,3-trichloro-2-methyl-
	propyl)-2-pyrrolidinone. 0.026 g, 0.9!10 K2 %d r y
	weight colourless needles; mp 109 8C, R f (CH 2 Cl 2 /acetone 1:1) 0.11; [a] D ZK38.5 (c 0.75, CH 2 Cl 2 ), IR n cm K1 : 3419, 1775, 1711, 1416, 1389; 1 H and 13 C NMR data recorded in
	CD 3 OD: see Tables 3 and 5; HREIMS [MCH] C found at m/z 286.0161 (DK0.8 mmu) for C 10 H 15 NO 2 35

  15! that of the bonded carbon atoms (but 1.2! for the hydrogens in methyl groups). Crystallographic data for C 10 H 14 Cl 3 NO 2 (FWZ286.57), colourless thin needles [the crystal size, 0.1!0.3! 0.35 mm 3 ], triclinic, P1 (#1), aZ6.010 (4) A ˚, bZ10.341 (3) A ˚, cZ11.103 (4) A ˚, aZ92.268, bZ94.12 (2)8, gZ 97.248, VZ681.9 (6) A ˚3, ZZ2, D calcd Z1.396 g/cm 3 , m (Mo-Ka)Z0.658 mm K1 , observed unique reflectionsZ 4047, 358 parameters refined using 11 restraints. For 3210 FO4s (F): R1Z0.0590 (based on F), wR2Z0.1614 (based on F 2 ) and goodness of fit factorZ1.051 (based on F 2 ). Dr max Z0.29!10 3 A ˚, Dr min ZK0.21 eA ˚3. Crystallographic data (excluding structure factors) for the structure in this paper have been deposited with the Cambridge Crystallographic Data Centre (CIF file) as supplementary publication number CCDC 244215. Copies of the data can be obtained free of charge, on application to CCDC, 12 Union Road, Cambridge CB2 1, EZ, UK [fax: C44(0)1223-336033 or e-mail: http:// deposit@ccdc.cam.ac.uk].

Acknowledgements

This work is part of Pierre Sauleau's Thesis, supported by the Leg Pre ˆvost of the MNHN of Paris. We thank Alain Blond, Christelle Caux, Jean-Paul Brouard and Lionel Dubost (MNHN, Paris) for NMR and MS measurements, Nicole Morin (ENS, Paris) for HRMS. We also thank Thomas Debeir of the Ho ˆp i t a ld el aS a l p e ˆtrie `re for neurological experiments and Miche `le Guyot for helpful discussion.


Landry Duguet

Guillaume Cerutti

Carlos Galvan-Ampudia

Etienne Farcot

Teva Vernoux

Christophe Godin

Revisiting PIN1 polarity interpretation at the SAM based on automated multi-level reconstruction of PIN1 polarities from confocal images

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Revisiting PIN1 polarity interpretation at the SAM based on automated multi-level reconstruction of PIN1 polarities from confocal images Landry Duguet 1 , Guillaume Cerutti 1 , Carlos Galvan-Ampudia 1 , Etienne Farcot 2 , Teva Vernoux 1 , and Christophe Godin 1 • P ij : PIN1 level in the membrane of cell i facing cell j.

• PIN1 total wall values

P T ij = P ij + P ji . • PIN1 net wall values ∆P ij = P ij -P ji .
• PIN1 net wall vectors

∆P ij = ∆P ij nij
• (P T ij ) are 30.6% lower in the central zone (CZ). • 73.4% (global), 63.2% (CZ), 80.8% (peripheral zone) of (∆P ij ) are oriented toward the center.

PIN1 cell vectors

PIN1 Net ∆P i = i∼j ∆P ij a ij PIN1 Internal ∆P i = i∼j P ij nij a ij
• 97.8% of PIN1 net cell vectors are oriented toward the center (in the peripheral zone).

• Only 47.8% of PIN1 internal cell vectors are oriented toward the center (in the CZ).

• PIN1 net cell vectors are more oriented toward the center than PIN1 internal cell vectors regarding both number and strength.

• Both cell vectors have a low intensity in the central zone, and a global convergence toward the center. • v: PIN1 abstracted vector field.

• Dynamics: a diffuses and is transported by v.

• v: uniformly pointing to the center, lower in the CZ.

• Localized upregulations of v creates maxima and minima of a.

v a Quantification of different PIN1 quantities at different scales leads to enrich our caracterization of the SAM. PIN1 level is upregulated close to primordia. It is sufficient to make auxin accumulate according to the modeling.

Future work: track PIN1 during primordia development, include temporal aspects to the PIN1 modeling.

1

  Laboratoire Reproduction et Développement des plantes, Lyon, France 2 School of Mathematical Sciences, Nottingham, UK A u t o m a t e d P I N 1 e x t r a c t i o n C e l l l e v e l a v e r a g i on CLV3 and qDII markers Aligned images (PI) Automated PIN1 extraction method [Galvan-Ampudia et al. (2020)] based on pPIN1::PIN1-GFP and PI (cell wall) markers. The method infers a PIN1 level to both sides of the cell wall. PIN1 at membranes PIN1 wall net vectors radial projections PIN1 wall mean values

•

  Primordia are located in the highest convergence zones. • Continuous maps allow to compare different signals, individuals, and time points. ∂a(x, t) ∂t = ∆a(x, t) Diffusion -∇ • (v(x)a(x, t)) Advection at speed v Modeling PIN1 polarity fields • a: auxin continuous concentration.