A current algebra approach to the equilibrium classical statistical mechanics and its applications

The non-relativistic current algebra approach is analyzed subject to its application to studying the distribution functions of many-particle systems at the temperature equilibrium and their stability properties. We show that the classical Bogolubov generating functional method is a very effective tool for constructing the irreducible current algebra representations and the corresponding different generalized measure expansions including collective variables transform. The effective Hamiltonian operator construction and its spectrum peculiarities subject to the stability of equilibrium many-particle systems are discussed.


Introduction
It is well known [1][2][3][4] that a complete physical theory, both relativistic and non-relativistic, can be described entirely in terms of current algebra operators, such as current densities, rather than in terms of canonical field operators, which is motivated by the fact that the current densities are physically observable quantities contrary to the canonical non-observable field operators. Moreover, the current algebra approach appeared to be also very effective in studying both quantum and classical statistical problems of many-particle systems by means of the Bogolubov generating functional, whose mathematical structure became a fruitful source of many approximation methods in modern statistical physics. Amongst them it is necessary to mention a very powerful collective variables transform suggested firstly by D. Bohm [5] and deeply developed by N. Bogolubov [6], D. Zubarev [7] and I. Yukhnovskii [8]. This transform has been reanalyzed in terms of the current algebra approach for classical many-particle systems in [9][10][11], where there was constructed a corresponding Bogolubov generating functional of distributions as a mathematical expectation of an infinite hierarchy of the non-interacting many-particle systems embedded into an external oscillatory potential field, with respect to a suitably defined infinite divisible Gauss type measure.
Based on these results and making use of some additional properties of the corresponding functional equations for the Bogolubov generating functional of many-particle distribution functions we have constructed, for the case of classical statistical mechanics, a new operator representation for an effective Hamiltonian operator defined in a suitable Hilbert space, whose ground state energy peculiarities make it possible to conceive the physical nature of the related phase transitions and to describe the behavior of multi-particle distribution functions.

Non-relativistic quantum and statistical mechanics: the current algebra approach
We assume a non-relativistic spinless many particle system of densityρ ∈ R + to be described by means of the non-relativistic quantum Hamiltonian operator yW (x, y)ψ + (x)ψ + (y)ψ(y)ψ(x) (2.1) acting in a suitable Fock space Φ, here we have denoted by 〈·, ·〉 the standard scalar product in the Euclidean space R 3 , W : R 3 × R 3 → R is a translation invariant interaction potential, and creation ψ + (x) : Φ → Φ , x ∈ R 3 and annihilation operators ψ(y) : Φ → Φ, y ∈ R 3 , satisfy the standard canonical commutation relationships: (2. 2) The current algebra representation of the Hamiltonian operator (2.1) is based on the following selfadjoint density operators: particle number density at any point x ∈ R 3 , satisfying the well known classical current Lie algebra commutator relationships: where we have defined the smeared [3,12,13] density operators for any Schwartz functions f ∈ S (R 3 ; R) and g ∈ S (R 3 ; R 3 ) and [g 2 , g 1 ] := 〈g 2 , ∇〉g 1 −〈g 1 , ∇〉g 2 for any g 1 , g 2 ∈ J (R 3 ; R 3 ).
The exponential current operators are as follows: for any t ∈ R and x ∈ R 3 , satisfy the current group for the semi-simple product G := S ⋊Diff(R 3 ) and the abelian Schwartz group S (R 3 ; R),where f 1 , f 2 and f ∈ S (R 3 ; R), ϕ 1 , ϕ 2 and ϕ ∈ Diff(R 3 ). The latter appeared to be very important in constructing the corresponding group G = S ⋊ Diff(R 3 ) representations in suitable Hilbert spaces and their physical interpretation as a classical generating Bogolubov functional [6,11,13,14] for the corresponding many-particle distribution functions. The Hamiltonian operator (2.1) permits the following current algebra representation
The current group G = S ⋊ Diff(R 3 ), as is well known, possesses many different irreducible unitary representations in suitable Hilbert spaces. In particular, in the standard N -particle Hilbert space H (N) := L (sym) 2 (R 3N ; C) for an arbitrary but fixed N ∈ Z + the particle density operator acts as f (x j ) ω (2.11) and the current density operator acts as for any f ∈ S (R 3 ; R), g ∈ S (R 3 ; R 3 ) and arbitrary vector ω ∈Ĥ (N) .
In the general case, the current group (2.7) possesses many different irreducible unitary representations in suitable Hilbert spaces H , which can be written down as where µ : 2 S ′ → R + is some cylindrical measure on the generalized space S ′ := S ′ (R 3 ; R), H F are marked by elements F ∈ S ′ (R 3 ; R) complex linear spaces, which for many physical applications [3,4,13] are one-dimensional. In the case dim H F = 1, one obtains from (2.13) that H ≃ L (µ) 2 (S ′ ; C). Now, if an element ω(F ) ∈ H is taken arbitrarily, from (2.7) one easily follows that where, by definition, (ϕ * F, f ) := (F, f • ϕ), dµ(ϕ * F )/dµ(F ) is the standard Radon-Nykodym derivative of the measure µ(ϕ * F ) with respect to the measure µ(ϕ * F ) and χ ϕ (F ) is a complex-valued factor of a unit norm, referred to as the co-cycle, satisfying the relationship for any ϕ 1 , ϕ 2 ∈ Diff(R 3 ) and arbitrary point F ∈ S ′ (R 3 ; R).
Now, we can interpret the functional (2.16) as a generating functional of the current group G = S ⋊ Diff(R 3 ) irreducible representations [9,13,14] in the physically proper Hilbert space H . This is based on the following theorem [1] owing to H. Araki.
for any a ∈ G.
Having applied theorem 2.1 to the functional (2.16), we derive that by means of constructing suitable generating functionals subject to the given Hamiltonian operator (2.8) one can find the corresponding operator representations of the current group G = S ⋊ Diff(R 3 ) and vice versa.

The current algebra representations and the Hamiltonian operator reconstruction
Based on the relationships (2.14) and the generating functional expression (2.16), one can easily calculate that for some suitably determined quasi-invariant measure µ : 2 S ′ → R + , that is an ergodic measure with respect to the diffeomorphism group Diff(R 3 ): for any Diff(R 3 )-invariant set Q ⊂ S ′ (R 3 ; R) either µ(Q) = 0 or µ(S ′ \Q) = 0. As a result of (3.1) one finds, as an example, that the standard quantum mechanical Nparticle representation of the current group G = S ⋊ Diff(R 3 ) is described [1,9,13] by the generalized singular measure Consider now the generating functional (2.16) and observe that the following quantities : for arbitrary n ∈ Z + represent the n-particle distribution functions of the quantum mechanical manyparticle system with the Hamiltonian (2.8), or equivalently, the functional (2.16) is respectively, the Bogolubov generating functional of many-particle distribution functions. Since the essence of the Bogolubov generating functional is held in the correspondingly derived [6] functional equation, we proceed now to determine its exact analytical form taking into account the structure of the related current group Following the standard operator construction, suggested in [13,14], one can define a selfadjoint oper-

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satisfied for any g ∈ S (R 3 ; R 3 ), where we put, by definition, To proceed further we need an important proposition concerning the matrix elements of the operators J (g ) and H : H → H for any g ∈ S (R 3 ; R 3 ).
Based now on simple enough but slightly cumbersome calculations, one can derive the following renormalized Hamiltonian operator expression: where, by definition, the operatorK for all x ∈ R 3 . Now, making use of (3.6), we can rewrite the defining condition (3.9) in the following functional equation form: where we put for any Similarly, one can calculate the matrix element values for the renormalized Hamiltonian operator (3.7) subject to the irreducible cyclic representation of the current group G = S ⋊ Diff(R 3 ): for all f 1 , f 2 ∈ S (R 3 ; R), meaning that two current algebra operator representations (2.8) and (3.7) of the initial Hamiltonian operator (2.1), defined in the canonical Fock space, are physically completely equivalent.

The generating Bogolubov functional equation for the temperature equilibrium states
Assume that a classical many-particle system is at a bounded inverse temperature β ∈ R + and its Gibbs statistical operator P := exp(−βH) tr exp(−βH) , for any f ∈ S (R 3 ; R). having imposed on the functional (4.2) the Araki's conditions of theorem 2.1, we can easily derive that there exists [1,3,4,11,14] an effective normalized cyclic vector Ω β ∈ H β , naturally corresponding to the effective Hamilton operator such that where we have put, by definition, for any x ∈ R 3 . As a result of the definition (4.2) and relationships (4.5), one easily finds [9], as the Planck constant ħ → 0, that where L 0 ( f ), f ∈ S (R 3 ; R), is the generating functional for the noninteracting equilibrium many-particle system and, by definition, we put Similarly to the above reasonings one also finds that the generating functional (4.6) satisfies [6,9] the Bogolubov type functional equation where the corresponding operator A β (x; ρ) : H β → H β for any x ∈ R 3 linearly depends on the binary interparticle interaction potential W : Hence, one easily infers that the generating functional L 0 ( f ), f ∈ S (R 3 ; R) for the unitary representation of the current group G = S ⋊ Diff(R 3 ) satisfies the reduced functional equation whose general non-normalized solution equals the integral with respect to some Radon measure µ β : 2 R → R on the real axis R. Thus, submitting (4.10) into (4.6), we obtain from (4.8) that for any x ∈ R and, respectively, : L ( f ) .

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The functional equation (4.12), being well known long ago owing to the classical results of Bogolubov [6], makes it possible, using the current algebra approach, to interpret it as an equation for the generating functional of irreducible current group G = S ⋊ Diff(R 3 ) representations in a suitable Hilbert space H β with a cyclic vector Ω β ∈ H β being the ground state vector for a respectively renormalized "effective" positive definite Hamiltonian operator (4.3) and satisfying the conditions (4.5). This gives rise to the following canonical current algebra representation of the Hamiltonian operator (4.3): where the effective inter-particle potentialsW R 3 j → R, n −2 ∈ Z + , non-locally depend on the initial interparticle potential W : R 3 × R 3 → R and on the inverse temperature parameter β ∈ R ÷ .
The Hamiltonian operator (4.13) can be respectively transformed to the canonical form (4.14) acting in the standard Fock space Φ. The latter can be used for determining the related canonical Nparticle representation of the Hamiltonian (4.14) by means of the following defining relationship:    4.17) and the inverse temperature parameter β ∈ R + . In particular, the condition (4.17) allows one to determine the above introduced measure dµ β , entering the functional (4.10). Now, it is important to recall that an equilibrium many-particle statistical system is stable [17,18], if its generating functional satisfies the well Kubo-Martin-Schwinger analycity condition. This condition, in particular, imposes a strong analytical dependence on the inverse temperature parameter β ∈ R + of the 23702-7 spectrum σ(H (N) β ) ⊂ R + as N → ∞ in such a way that the densityρ = lim N /Λ ∈ R + persists to be constant. One obtains another inference from the important fact that the number operator N β := R 3 d 3 xρ(x) is a conserved quantity, that is [H β , N β ] = 0 (4.18) for those parameters β ∈ R + , for which the equilibrium many-particle system is stable and does not pass a phase transition.

The current algebra representation aspects of the collective variables transform
The collective variables transform [5,6,[8][9][10] allows one to consequently take into account and separate two different impacts of a binary interaction potential W := W (l) + W (s) into the many-particle distribution functions subject to its long distance W (l) and short distance W (s) parts. Since the long distance interaction potential responds for the so-called "collective" behavior of the many-particle at a bounded inverse temperature parameter β ∈ R + , the corresponding Bogolubov generating functional (4.2) can be formally rewritten in the operational form as where, by definition, The functional (5.2) can be quite easily calculated by means of the Fourier transform representation of the long distance interaction potential and the standard quasi-classical limit ħ → 0: where we denoted the measure D(ω) := k∈R 3 i 2 dω k ∧dω −k , the parameterz := z exp − β 2 R 3 d 3 kν (l) (k) , and the free particle system statistical operator is equal to P := exp(−βH 0 ) tr exp(−βH 0 ) ,  (5.5) 23702-8 and the measure kernels ("Jacobian") The formal series expansion owing to (5.3) and (5.7), right away gives rise to the approximation of the generating functional (5.2) by means of the so called "screened" long distance potential under the external effect of an infinite set of oscillatory potentials: Having substituted the functional expression (5.9) into (5.1) one can easily obtain the corresponding Bogolubov generating functional in the Ursell-Mayer type infinite expansion form, based on the following operator expression: where g (l) n : R 3n → R, n ∈ Z + are so-called n-particle "cluster" distribution functions.
Observe also that the Bogolubov type generating functional (5.3) can be rewritten in the integral Gauss
As a simple corollary from the expression (5.18), one obtains that the effective long-distance Hamiltonian operator contains an infinite hierarchy of multinary potential energy terms, which should be in due course taken into account when studying the peculiarities of the corresponding many-particle distribution functions. In particular, the energy spectrum of the N -particle canonical representation of the

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Hamiltonian operator (5.18) possesses an important information on the many-particle system stability at a special inverse temperature parameter β ∈ R + .
As an example demonstrating the effective Hamiltonian operator construction, we will consider a one-dimensional many-particle system of densityρ ∈ R + on an axis R at a finite inverse temperature β ∈ R + , described by the following operator expression in the canonical Fock space Φ: where λ ∈ R + is a positive parameter. The corresponding Bogolubov generating functional L ( f ), f ∈ S (R; R), satisfies, owing to (4.12), the following functional equation: : L ( f ) .
|x − y| 2 ψ + (x)ψ + (y)ψ(y), ψ(x), (5.24) describing an infinite set of particles on the axis R, binarily interacting to each other by means of the inverse square potentialW (ω,H β ω) (ω,Ñ β ω) = λ 2 β 2 π 2ρ2 /6 (5.26) holds, where we denoted byÑ β := R dxρ(x) the corresponding particle number operator in the Hilbert space H β . The least average energy per particle (5.26) analytically depends on the inverse temperature parameter β ∈ R + . The same can also be obtained for the other energy excitations of the Hamiltonian operator (5.24). Thus, we infer that the initial one-dimensional many-particle system with the Hamiltonian (5.20) and at the inverse temperature parameter β ∈ R + is completely stable and permits no phase transition. Moreover, at the temperature parameter β = 1/λ ∈ R + the effective Hamiltonian operator (5.24) describes a many-particle noninteracting system of the densityρ ∈ R + and the least average energy per particleε β = π 2ρ2 /6, depending only on the density.

Conclusion
The investigation of statistical properties of classical many-particle systems at a finite inverse temperature β ∈ R + and a fixed densityρ ∈ R + by means of the current algebra representations has two main reasons: firstly, it provides an interesting reformulation of the initial quantum statistical problem in terms of physical observables such as the particle number density and the particle flux density, rather than the corresponding second-quantized field creation and annihilation operators.
The second reason is related to a very rich structure of the current group G = S ⋊Diff(R 3 ) irreducible representations, according to the Bogolubov functional equation for the generating many-particle distribution functional, and whose analytical property subject to the temperature parameter β ∈ R + are responsible for the system stability as it follows from the Kubo-Martin-Schwinger approach, applied to the classical statistical mechanics. Moreover, a very rich functional-operator structure of solutions to the related Bogolubov functional equations allows one to make physically reasonable re-expansions of the general irreducible representation measure, as it was show for the case of the classical collective variables transform, and whose generating functional permits an additive Gauss type representation, based on an infinite set of free noninteracting many-particle systems embedded in an external oscillatory type potential field.
As a dual aspect of irreducible representations of the current group G = S ⋊ Diff(R 3 ), related to the Bogolubov functional equation, we need to mention the construction of associated effective Hamiltonian operators subject to the basic ground state cyclic representation of the current group, whose analytical properties are responsible for the many-particle system stability and possibly, for the phase transition behavior. We hope that the approach devised in the work will prove to be helpful in further gaining insight into the statistical clustering properties of many-particle systems and in developing new more powerful and specialized analytical techniques for solving other interesting problems in statistical physics.