On Non Ergodic Property of Bose Gas with Weak Pair Interaction

In this paper we prove that Bose gas with weak pair interaction is non ergodic system. In order to prove this fact we consider the divergences in some nonequilibrium diagram technique. These divergences are analogous to the divergences in the kinetic equations discovered by Cohen and Dorfman. We develop the general theory of renormalization of such divergences and illustrate it with some simple examples. The fact that the system is non ergodic leads to the following consequence: to prove that the system tends to the thermal equilibrium we should take into account its behavior on its boundary. In this paper we illustrate this thesis with the Bogoliubov derivation of the kinetic equations.

The accurate consideration for the quantum case can be found in section 10. Suppose that our system is ergodic, i.e. there are no first integrals of the system except energy. Then, the distribution function depends only on energy. We can represent the distribution function f (E) as follows: where the sum can be continuous (integral). Let 1 be some enough large but finite subsystem of our system. Let 2 be a subsystem obtained from 1 by translation on the vector l of sufficiently large length parallel to the x-axis. Let 12 be a union of the subsystems 1 and 2. Let ρ 1 , ρ 2 and ρ 12 be distribution functions for the subsystems 1, 2 and 12 respectively. Let Γ 1 , Γ 2 and Γ 12 be points of the phase spaces for the subsystem 1, 2 and 12 respectively. By the same method as the method used for the derivation of the Gibbs distribution we find: in the obvious notation. But the weak cluster property implies that Therefore all the coefficients c α are equal to zero except one. We find that for some constants c and E 0 . So each finite subsystem of our system can be described by Gibbs formula and we obtain a contradiction. Non-ergodic property means that there is no thermalization in infinite Bose-gas system.
This fact implies to prove that the system tends to thermal equilibrium we should take into account the behavior of the system on its boundary. Indeed if a system has no boundary the system is infinite.
To illustrate this fact we will study Bogoliubov derivation of kinetic equations [6]. When one derives BBGKI-chain one neglects some boundary terms. If one takes into account this boundary terms and uses the Bogoliubov method of derivation of the kinetic equations one finds that these boundary terms compensate the scattering integral. I think that the dependence of behavior of the system of boundary can be observed for small systems such as nanosystems or biological systems.
Note that our main result is closely related with so-called the Prigogin hypothesis which states that the infinite dimensional Liuville dynamics can not be derived from the Hamilton dynamics. The Prigogin hypothesis is proven in [7] The paper is organized as follows. In section 2 we introduce the notion of the algebra of canonical commutative relations and develop an useful representation for some class of the states on this algebra. In section 3 we describe the von Neumann dynamics for the states. In section 4 we describe an useful representation for the von Neumann dynamics -the dynamics of correlations. In section 5 we describe the decomposition of the kinetic evolution operator by so-called trees of correlatios. In section 6 we describe the general form of the counterterms which subtract the divergences in the nonequilibrium perturbation theory. In section 7 we describe so-called Friedrichs diagrams. In section 8 we describe the Bogoliubov-Parasiuk prescriptions and formulate our main theorem. In section 9 we prove our main theorem. In section 10 we derive the non-ergodic property of Bose gas with weak pair interaction from our main result. In section 11 we consider one example related to our general theory. In section 12 we reconsider the Bogoliubov derivation of the Boltzmann equation. This example illustrates the main thesis of this paper: to prove that the system tends to the thermal equilibrium one has to take into account its behavior on its boundary. Section 13 is a conclusion.

The Algebra of Canonical Commutative Relations
Let S(R 3 ) be a Schwatrz space of test functions (infinitely-differentiable functions decaying at infinity faster than any inverse polynomial with all its derivatives). The algebra of canonical commutative relations C is an unital algebra generated by symbols a + (f ) and a(f ) f ∈ S(R 3 ) satisfying the following canonical commutative relations: [a(f ), a + (g)] = f, g , (8) where f, g is a standard scalar product in L 2 (R 3 ), f, g := d 3 xf * (x)g(x).
Let ρ 0 be a Gauss state on C defined by the following correlator where n(k) is a real-valued function from the Schwartz space. In the case then where µ ∈ R, µ < 0, ρ 0 is called the Plank state. Here ω(k) = k 2 2 . Let C ′ be a space of linear functionals on C, and C ′ +,1 be a set of all states on C. Let us make the GNS construction corresponding to the algebra C and the Gauss state ρ 0 . We obtain the set (H, D,ˆ, ) consisting of the Hilbert space H, the dense linear subspace D in H, the representationˆof C by means of the linear operators from D to D, and the cyclic vector ∈ D, i.e. the vector such thatĈ = D. This set satisfies the following condition: ∀a ∈ C â = ρ 0 (a). Below we will omit the symbolˆ, i.e. we will write a instead of a.
Let us introduce the field operators: We say that the state ρ on C satisfies the weak cluster property if where δ i ∈ {1, 0}, i = 1, 2...n and f (x 1 , ..., x n ) is a test function. e 1 is an unit vector parallel to the x-axis. Definition The vector of the form is called a finite vector. The finite linear combination of the vectors of the form (15) is also called a finite vector. Let f (x 1 , ..., x k |y 1 , ..., y l |v 1 , ..., v m |w 1 , ..., w n ) be a function of the form where g is a function from Schwartz space. Consider the following functional on C ×f (x 1 , ..., x n |y 1 , ..., y l |v 1 , ..., v m |w 1 , ..., w n ) Here the symbol means that when one transforms the previous expression to the normal form according to the Gauss property of ρ 0 one must neglect all correlators ρ 0 (a ± (x 1 )a ± (x n ) such that a ± (x 1 ) and a ± (x n ) both do not come from A.
Let C ′ be a subspace in C ′ spanned on the functionals just defined. Now let us introduce an useful method for the representation of the states just defined.
Let C 2 = C + ⊗C − , where C + and C − are the algebras of canonical commutative relations. The algebras C ± are generated by the generators a ± (k), a + ± (k) respectively satisfying the following relations: Here we put by definition a − ± := a ± . Let us consider the following Gauss functional ρ ′ 0 on C 2 defined by its two-point correlator One can prove that the functional ρ ′ 0 is a state. Let us make the GNS construction corresponding to the state ρ ′ 0 and the algebra C 2 . We obtain the set (H ′ ,D,ˆ, ) consisting of the Hilbert space H ′ , the dense linear subspaceD in H ′ , the representationˆof C 2 by means of the linear operators fromD toD, and the cyclic vector ∈D, i.e. the vector such thatĈ =D. This set satisfies the following condition: ∀a ∈ C 2 â = ρ ′ 0 (a). Below we will omit the symbolˆ, i.e. we will write a instead ofâ. Now we can rewrite the functional, defined in (17) ρ f as follows where A ′ is an element of C 2 such that it contains only the operators a − , a + − and can be represented through a − , a + − in the same way as A can be represented through a, a + . S f is an element of C 2 of the form ×f (x 1 , ..., x n |y 1 , ..., y l |v 1 , ..., v m |w 1 , ..., w n ) Here the symbol : ... : is a normal ordering with respect to the state ρ ′ 0 . Denote byD ′ the space dual toD. We just construct the injection from C ′ intoD ′ . Denote its image byH ′ .
By definition the space C ′′ is a space of all functionals on C which can be represented as finite linear combinations of the following functionals Here A ′ is an element of C 2 such that it contains only the operators a − , a + − and can be represented through a − , a + − in the same way as A can be represented through a, a + and S f i are the elements of the form (22). Denote byH ′′ the subspace inD ′ spanned on the vectors : S f 1 ...S fn : (in obvious sense).

The von Neumann Dynamics
Suppose that our system is described by the following Hamiltonian where H 0 = d 3 k(ω(k) − µ)a + (k)a(k) and Here the kernel v(p 1 , p 2 |q 1 , q 2 ) belongs to the Schwartz space of test functions.
To point out the fact that H is represented through the operators a + , a − we will write H(a + , a − ). The von Neumann dynamics takes place in the spaceH ′′ and is defined by the following differential equation: where the von Neumann operator has the form where we put by definition: dq j v(p 1 , ..., p n |q 1 , ..., q m ) : a + (p 1 )...a + (p n )a(q 1 )...a(q n ) :) † = n i=1 dp i m j=1 dq j v(p 1 , ..., p n |q 1 , ..., q m ) * : a + (p 1 )...a + (p n )a(q 1 )...a(q n ) : . (31) Let us divide the von Neumann operator into the free operator L and the interaction L int , L = L 0 + λL int , where Note that the operators L 0 and L 1 are real (with respect the involution ⋆).
Let us introduce kinetic evolution operator (in the interaction representation) After differentiating with respect to t we find the differential equation for U(t, t ′ ). where So the state ρ under consideration in the spaceH ′′ in the interaction representation has the form where T is the time-ordering operator. Note that we have a linear map fromH ′′ into C ′ . It is easy to see that the von Neumann dynamics is in agreement with the Heizenberg dynamics in C ′ .

Dynamics of Correlations
Let us construct some new representation of the von Neumann dynamics useful for the renormalization program. This representation is called the dynamics of correlations. The ideas of the dynamics of correlations belongs to I. Prigogin [8]. The dynamics of correlations takes place in the space Now let us define L c int . Let |f ∈ H c , belongs to the subspace ⊗ nH′ and has the form: where f i j has the form : n j=1 f i j : to the normal form. Let us denote by h l the sum of all the terms in the previous expression such that exactly l operators f i j couple with L int . We find that h l has the following form for some k. Here g i k has the form of right hand side of (40). Now let us consider the following vector where we define symmetrization operator as follows (S n -the group of permutation of n elements.) Put by definition One can prove that this definition is correct. Analogously, in the following expression let us keep only the terms such that L int does not couple with any of f i j . Let us write the sum of such terms as follows Here h i j has the form of right hand side of (40). Let |h be a vector in sym⊗ n+H′ defined as follows Put by definition We have the evident linear map F : H c →H ′′ which assigns to each vector sym : f  : ⊗ ...⊗ : f n : the vector : f 1 ...f n : . Denote by U c the evolution operator in interaction representation in the dynamics of correlation. The following statement describes the relation between the von Neumann dynamics and the dynamics of correlations.
Statement. The following relation holds:

The tree of correlations
The useful representation of dynamics in H c is a decomposition by so called trees of correlations. Definition. A graph is a triple T = (V, R, f ), where V , R are finite sets called the set of vertices and set of lines respectively and f is a map: where Remark. We use this unusual definition of graphs only in purpose of this section to simplify our notations.
Definition. The graph Γ is called connected graph if for two any vertices v, v ′ there exists a sequence of vertices v = v 0 , v 1 , ..., v n = v ′ such that ∀ i = 0, ..., n − 1 the vertices v i and v i+1 are connected by some line.
By definition we say that the line r is an internal line if f (r) = (v 1 , v 2 ) for some vertices v 1 and v 2 .
For each graph Γ we define its connected components by the obvious way. Definition. We say that the graph Γ is a tree or an acyclic graph if the number of its connected components increases after removing an arbitrary line.
Definition. The elements of the set where T is a tree and Φ v and Φ sh are the following maps: Definition. We will consider the following two directed trees (T, as identical if we can identify the sets of lines R and R ′ of T and T ′ respectively and identify the sets of vertices V and V ′ of T and T ′ respectively such that after these identification the trees T and T ′ become the same, the functions Φ v and Φ ′ v become the same and the functions Φ sh and Φ ′ sh become the same. Denote by r(T ) the number of roots of T and by s(T ) the number of shoots of T . Below, we will denote each directed tree (T, Φ v , Φ sh ) by the same symbol T as a tree omitting the reference to Φ v , Φ sh and write simply tree instead of the directed tree.
We say that the connected directed tree T is right if there exists exactly one line from f −1 (V × {+}).
We say that the tree T is right if each its connected component is right.
To point out the fact that some object A corresponds to a tree T we will often write A T . For example we will write Definition. For each connected right tree T there exists an essential partial ordering on the set of its vertices. Let us describe it by induction on the number of its vertices. Suppose that we have defined this relation for all right trees such that the number of their vertices is less or equal than n − 1. Let T be a right tree such that the number of its vertices is equal to n. Let v max be a root vertex of T . Put by definition that the vertex v max is a maximal vertex. Let v 1 , ..., v k be all of its children i.e. the vertices connected with v max by lines. By definition each vertex v i < v max , i = 1, ..., k. We can consider the vertices v 1 , ..., v k as a root vertices of some directed trees T i , i = 1, ..., k. By definition the set of vertices of T i consists of all vertices v which can be connected with v i by some path .., l. The incident relations on T i are induced by incident relations on T . Put by definition that ∀(i, j), i, j = 1, ..., k, i = j and for any two vertices in the sense of ordering on T i . We put also v < v max for every vertex v = v max . These relations are enough to define the partial ordering on T .
If the tree T has several connected components we define a partial ordering at each its connected components as previously and put v 1 ≯ v 2 if v 1 and v 2 do not belongs to the same connected component of T .
Below without loss of generality we suppose that for each tree of correlation T and its line r the pair and ϕ is a map which assigns to each vertex v of T an element of a space of linear maps from r→vH ′ toH ′ .
In (r→v)H ′ the tensor product is taken over all lines r such that r → v.
Let v be a vertex of the tree we say that the line comes into the vertex v and write r → v. Definition. Let (T, ϕ, τ ) be a tree of correlations such that for each vertex v ϕ(v) = L c,lv int , where l v is a number of lines coming into v. We call this tree the von Neumann tree and denote it by T τ . We also say that ϕ is a von Neumann vertex function.
Definition. To each tree of correlations (T, ϕ, τ ) we assign an element by the following way: If T is disconnected then Here the number of connected components of T is equal to n, and connected components of T are denoted by CT . Cϕ and C τ are the restrictions of ϕ and τ to the sets of vertices and lines of CT respectively. R sh (CT ) is a set of shoots of CT . Now let T be a connected tree. To define by induction it is enough to consider the following two cases. case 1). The tree T has no shoots. a) Suppose that the tree T has more than one vertex. Let v min be some minimal vertex of T and v 0 be a vertex such that an unique line r 0 comes from v min into v 0 . Let T ′ be a tree obtained from T by removing the vertex v min of T . Let τ ′ be a restriction of τ to R \ {r 0 }. Let ϕ ′ be a function, defined on V \ {v min } as follows: where Put by definition b) The tree T has only one vertex v min . Then Case 2.) The tree T has a shoot r 0 coming into the vertex v 0 . In this case instead of the tree (T, ϕ, τ ) we consider the tree (T ′ , ϕ ′ , τ ′ ), where the tree T ′ has the same vertices as T , the set of lines of T is obtained by removing the line r 0 from the set of lines of T ′ , the function τ ′ is a restriction of the function τ to the set of lines of T ′ and the function ϕ ′ is defined as follows: Here we put where the sum is taken over all lines r ′ which forms decreasing way coming from + to v 0 . Put by definition where Let (T, ϕ, τ ) be some tree of correlations. We can identify the tensor product and the tensor product Using these identifications let us consider an operator V t T,ϕ ( τ ) : H c → H c defined by the following formula where P sh(T ) is a projection of H c to sym Remark. If (T, ϕ, τ ) is a von Neumann tree of correlations then we will shortly denote the operators U t (T,ϕ) and V t (T,ϕ) by U t T and V t T respectively. The following theorem holds: Theorem. The following representation for the evolution operators holds (in the sense of formal power series on coupling constant λ).
Here n T is a number of vertices of the directed tree T .
6 The general theory of renormalization of In the present section we by using the decomposition of correlations dynamics by trees describe the general structure of counterterms of U(t, −∞) , which subtract the divergences from U(t, −∞) . We will prove in the section 10 below that there exist divergences in the theory. Note that the structure of R-operation for the processes at large times for some class of systems has been considered at [9] Let T be a tree. Let us give a definition of its right subtree.
The set R Tv 1 ,...,vn of all lines of the tree The Bogoliubov -Parasiuk renormalization prescription. Let us define the following operator: where by definition, We say that the amplitudes {A T,ϕ } are time -translation invariant amplitudes if for each tree T and for each its root line r 0 For each set of amplitudes A T,ϕ put by definition: where T is an arbitrary tree without shoots. Now let us formulate our main result.
Theorem. There exists a procedure called renormalization which to each tree T without shoots assign the amplitudes Λ T,ϕ satisfying to the following properties a)-f): a) If the tree T is not connected and {CT } is a set of its connected components, while {Cϕ} is a set of its restriction of ϕ to CT in obvious notations.
c) The amplitude Λ T,ϕ satisfies the property of time-translation invariance.
It has been proven that In the last formula the summation is taken over all trees T without shoots. Let T be a tree without shoots and T ′ be a right subtree of T in the described before sense. Let us define the amplitude Let by definition T \T ′ be a tree obtained by removing from the set V T all the vertices of T ′ and from the set R T all the internal lines of T ′ . In (79) τ is a map from R T \T ′ into R + . We can consider the amplitude U t T \T ′ as a map By using this identification we simply put Now let us define the renormalized amplitudes, by means of the counterterms Λ T by the following formula: d) The renormalized amplitudes (R Λ U)(t, −∞) are finite.
The renormalized amplitudes satisfy to the following properties: This property simply follows from the definition of (R Λ U)(t, −∞) and means that the functional (R Λ U)(t, −∞) is a stationary state. Property 2.
This property follows from the following representation of (R Λ U)(t, −∞) . where and the sum in the last formula is taken over all von Neumann trees without shoots. Property 2 means that the functional (R Λ U)(t, −∞) satisfies to the von Neumann dynamics.
Remark. The existence of the stationary translation invariant functional satisfying to the weak cluster property follows from the previous theorem and the properties 1, 2.

The Friedrichs diagrams
Now let us start to give a constructive description of the counterterms Λ T such that the amplitude R(U)(t, −∞) is finite, and the counterterms Λ T satisfy the properties a) -f) from the previous section.
At first we represent U t T,ϕ ( τ ), where T is some tree without shoots, as a sum taken over all so-called Friedrichs graphs Φ concerned with T .
Definition. A Friedrichs graph Φ T concerned with the directed tree T without shoots is a set (Ṽ , R, Or, f + , f − , g), whereṼ is a union of the set of vertices of T and the set {⊕}. Recall that there is a partial order on V T . We define a partial order on the setṼ if we put ∀v ∈ V T ⊕ > v. f + and f − are the maps f + , f − : R → V such that f + (r) > f − (r). Or is a map R → {+, −} called an orientation. g is a function which to each pair (v, r), v ∈ V T , r ∈ R such that f + (r) = v or f − (r) = v assigns + or −. The graph (Ṽ , R, Or, f + , f − , g) must satisfy the property: if we consider ⊕ as a vertex, the obtained graph is connected.
If we want to point out that the object B concerned with the graph Φ we will write B Φ . For example we will write V Φ and R Φ for the sets of vertices and lines of Φ respectively.
At the picture we will represent the elements of V by points and the element ⊕ by ⊕. We will represent the elements of R by lines. The line r connects the vertices f + (r) and f − (r) at the picture. We will represent orientation Or(r) by arrow on r. If Or(r) = + the arrow is directed from f − (r) to f + (r). If Or(r) = − the arrow is directed from f + (r) to f − (r). To represent the map g : (r, v) → {+, −} we will draw the symbol g((r, v)) (+ or −) near each shoot (r, v). At the picture a shoot (r, v) is a small segment of the line r near v.
Definition. The Friedrichs diagram Γ is a set (T, Φ, ϕ, h), where T is a tree, Φ is a Friedrichs graph, ϕ is a map which assigns to each vertex v of T a function of momenta {p r |r ∈ R Φ } of the form where ψ v is a test function of momenta coming into (from) the vertex v.
It will be clear that it is enough to consider only the diagrams Γ such that for each its vertex v and set S i ∈ {S i } nv i=1 there exists a line r such that (r, f − (r)) ∈ S i .
Here p rext are momenta of external lines, i.e. such lines r that f + (r) = ⊕. We choose the lower index of a ± ± (p rext ) by the following rule. Let v be a vertex such that f − (r ext ) = v. If g((r, v)) = + we choose + as a lower index, and if g((r, v)) = − we choose − as a lower index. We choose the upper index of a ± ± (p rext ) by the following rule. If the lower index of a ± ± (p rext ) is {-} then the upper index is equal + if the corresponding line comes from the vertex v and this index is equal − if the corresponding line comes into the vertex v.  r)) dp r r∈R G(Or(r), g((r, f + (r))), g((r, f − (r))))(p).
Let us describe the elements of this formula. R Γ is a set of all lines of diagram Γ. Symbol r ⇆ v denotes that the line r comes into (from) the vertex v. In the expression we take the upper sign + if the line r comes into the vertex v and we take lower sign − in the opposite case. The symbol R T denotes the set of lines of the tree T from the triple (T, Φ, ϕ) and symbol r T means the line from R T . The symbol V r denotes the set of all vertices v such that f + (r) ≥ v ≥ f − (r). The symbol (R T ) r denotes the set of all lines r T of R T such that the increasing path coming from f − (r) into f + (r) contains r T . G(Or(r), g(f + (r)), g(f − (r)))(p) is a factor defined as follows G(Or(r), g(f + (r)), g(f − (r)))(p)δ(p − p ′ ) = ρ ′ 0 (a sgn(−Or(r)g((r,f + (r))) g((r,f + (r))) (p), a sgn(Or(r)g((r,f − (r))) g((r,f − (r))) (p ′ )).
Below we will simply write G r (p) instead of G(Or(r), g(f + (r)), g(f − (r)))(p). It is evident that we can represent U 0 T ( τ ) as a sum taken over some Friedrichs diagrams Γ corresponding to the tree T of the quantities U 0 Γ ( s). Now let us define the quotient diagrams. Definition. Let Γ = (T, Φ, ϕ, h) be a Friedrichs diagram and A ⊂ R T be a subset of the set R T of lines of T and τ be a map from R T into R + .
We define the quotient diagram Γ A τ := (T A , Φ A , ϕ A τ , h A ) in the following way. To obtain the tree T A we must tighten all lines from A into points. To obtain Φ A we must remove all loops obtained by tightening all lines from A into the point. Now let us define ϕ A τ . Joint all the vertices of T to A. We obtain a tree denoted by A T . Let {C A T } be a set of all connected components of A T . Let v 0 be a vertex of Φ A corresponding to the connected component C A T of A T . Put by definition: Let us point out the notations in the previous formula. R in is a set of all lines of Φ A such that f + (r) and f − (r) are the vertices of C A T . (R T ) r denotes the set of all lines r T of R T such that the increasing path coming from f − (r) into f + (r) contains r T . The symbol V r denotes the set of all vertices v such Definition. Let Γ be a Friedrichs diagram. Let F Γ be a space of all functions of external momenta of the diagram Γ of the form: where ψ(...p ext ...) is a test function of external momenta. The convolution of the amplitude A Γ ( τ )(...p ext ...) with the function f ∈ F Γ we denote by A Γ ( τ )[f ].

The Bogoliubov -Parasiuk renormalization prescriptions
Let for each Friedrichs diagram Γ = (T, Φ, ϕ) A Γ ( τ )(...p ext ...) be some amplitude. Fix some diagram Γ and let T ′ be some right subtree of the tree T corresponding to Γ. Let Γ T ′ be a restriction of the diagram Γ on T ′ in obvious sense. Define the amplitude A Γ T ′ ⋆ U Γ (...p ext ...) by the following formula: In this formula V ′ is a set of all vertices v such that v is not a vertex of V T ′ , R ′ is a set of all lines r of Φ Γ such that f + (r) is not a vertex of T ′ . (R ′ T ) r is a set of all lines r T of T such that r T is not a line of T ′ and there exists an increasing path on T coming from f − (r) into f + (r) such that this path contains r T . V ′ r is a set of all vertices v of T such that v is not a vertex of T ′ and f + (r) ≥ v ≥ f − (r).
Let A Γ ( τ )(p) be some amplitude. Put by definition: where n is a number of lines of T Γ . Below we will consider the amplitudeŝ A Γ ( s)[f ] as distributions on (R + ) n i.e. as an element of the space of tempered distributions S ′ ((R + ) n ). Let ψ( s) be a test function from S((R + ) n ). The convolution of the amplitudeÂ Γ ( s)[f ] and the function ψ( s) we denote by: The Bogoliubov -Parasiuk prescriptions. It will be clear below, that we can take into account only the diagrams Γ such that for each line r T of the corresponding tree of correlations T ♯R r T ≥ 3. Here R r T is a set of all lines r of Γ such that the increasing path on T which connects f − (r) and f + (r) contains r T . Below we will consider only such diagrams. Other diagram can be simply subtracted by some counterterms Λ T .
According to the Bogoliubov -Parasiuk prescriptions we must to each diagram Γ (corresponding to the connected tree) assign the counterterm am-plitudeĈ Γ ( s)[f ] f ∈ F Γ satisfying the following properties. a) (Locality.)Ĉ Γ ( s)[f ] is a finite linear combination of δ functions centered at zero and their derivatives. b) Let Γ be a Friedrichs diagram and T be a corresponding tree of correlations. Let A ⊆ R T and T ′ is some right subtree of T such that: 1) all lines r T of T such that r T is not a line of R T ′ belong to A, 2) All the root lines of T ′ do not belongs to A. ThenĈ where where τ = (τ 1 , ..., τ n ) = ( 1 s 1 , ..., 1 sn ), the symbol ⊂ means here the strong inclusion and T is some subtract operator.
d) The amplitudesĈ Γ ( s)[f ] satisfy to the property of time-translation invariance, i.e.
e) Let Γ be a Friedrichs diagram. Let The amplitudesR Γ ( s) are well defined distributions on (R + ) n . f) The amplitudesR Γ ( s) satisfy the weak cluster property. This property means the following. Let f (...p ext ...) be a test function. Then as a → ∞. Here p 1 r is a projection of p r to the x-axis. Put by definition for each diagram Γ: where ′ in the sum means that all the root lines of T Γ do not belong to A. Put where the symbol Γ ∼ T means that the sum is taken over all diagrams corresponding to T with suitable combinatoric factors. Suppose that the properties a) -f) are satisfied. Then Λ T are the counterterms needed in the section .
Theorem -Construction. It is possible to find such a subtract operator T such that there exist countertermsĈ Γ satisfying the properties a)f).
Note that it is not necessarily for us use not real counterterms. Indeed the evolution operator is real, so after renormalization we can simply take Re (RU)(t, −∞) .

Proof of the theorem-construction
In this section we prove the theorem-construction from the previous section. Note that to prove our theorem we will use some ideas of the papers [10][11][12].
Before we prove our theorem let us prove the following Lemma 1. Let L 1 = S(R k ), L 2 = S((R + ) n ), k, n = 1, 2, .... Let A(p) be some nonzero quadratic form on R k . Let T 1 t , t ≥ 0 be an one-parameter semigroup acting in L 1 drfined as follows: and for each i = 1, ..., l, t ≥ 0 for some coefficients a i−1 , ..., a 1 and the element f ∈ M.
Let g be a functional on L 1 ⊗ M such that g is continuous with respect to the topology on S(R k ) × S((R + ) n ). Suppose that ∀f ∈ L 1 ⊗ M and ∀t > 0 g, Then, there exists an continuous extensiong of g on S(R k ) × S((R + ) n ) such that ∀f ∈ L 1 ⊗ L 2 and t > 0 g, By definition we say that the functional h on Proof of the lemma 1. At first we extend our functional g to the invariant functionalg on L 1 ⊗ L 2 and then we prove thatg is continuous.
Let N be a subspace of L 1 of all functions of the form, Let k be a continuous functional on N defined as follows: Letk be an arbitrary continuous extension of k on whole space L 1 . The existence of such continuation follows from Malgrange's preparation theorem [13]. Now we define the continuous functionalg 1 on L 1 ⊗ M 1 as follows: According to (113) we find thatg 1 is an invariant extension of g on Step by step we can extend by the same procedure the functional g to the functionalsg 2 , ....g l on .., f l }} respectively. Just constructed functional is separately continuous so it is continuous. The lemma is proved.
Sketch of the proof of the theorem. We will prove the theorem by induction on the number of lines of the tree of correlations T Γ corresponding to the diagram Γ. It is evident that it is enough to consider only the diagrams with connected tree of correlations.
The base of induction is evident. Suppose that the theorem is proved for all diagrams of order < n. (Order is a number of lines of the tree of correlations.) Let us give some definitions. Let ξ(t) be a smooth function on [0, +∞) such that 0 ≤ ξ(t) ≤ 1, ξ(t) = 1 in some small neigborhood of zero and ξ(t) = 0 if t > 1 3n . Let us define a decomposition of unit {η A ( s)|A ⊂ {1, ..., n}} by the formula Let ψ(x) be some test function on real line such that ψ(t) ≥ 0, ψ(t)dt = 1 and ψ(t) = 0 if |t| > 1 10 . Put by definition: We have The inner integral in (118) converges according to the inductive assumption. Therefore if Ψ( s) ∈ S N ((R + ) n ) and N is large enough the integral at the right hand side of (118) converges. So defines a separately continuous functional on S(R 3f ) ⊗ S N ((R + ) n ). f = l − 1, where l is a number of external lines of Γ. To define a subtract operator T we must extend the functional R Γ ( s)[f ], Ψ( s) to the space S(R 3f ) ⊗ S((R) n ) such that extended functional will satisfy to time-translation invariant property. To obtain this extension we use the lemma. In our case L 1 = S(R 3f ), Or(r)p 2 r . T 2 t is an operator acting in the S((R + ) n ) as follows.
The basis {f 1 , ...f l } from the lemma is {s m 1 1 ....s mn n η ∅ ( s)}, m 1 , ...m n = 1, 2, 3..., m 1 + m 2 + ... + m n ≤ N lexicographically ordered. We can now apply our lemma directly. Now let us prove the weak cluster property. Let p ∈ R 3 . Denote by p 1 , p 2 , p 3 the projections of p to the x, y, z-axis respectively. To prove the weak cluster property it is enough to prove the following statement: for each connected diagram Γ the function is a distribution of variables ...p 2 ext ...p 3 ext ... (constrained by momentum conservation law) which depends on ...p 1 ext ... (constrained by momentum conservation law) by the continuously differentiable way. We will prove this statement by induction on the number of lines of the corresponding tree of correlations. The base of induction is evident. Suppose that the statement is proved for all the trees of correlations such that the number of their lines < n. Let Γ be a diagram such that the number of the lines of the corresponding tree of correlations is equal to n. It is evident that if Ψ( s) has a zero of enough high order at zero then F Γ ( s)(...p ext ...), Ψ( s) belongs to the required class (its enough to use our construction with decomposition of unit). Therefore we need to solve by induction the system of equations of the form: According to Malgrange's preparation theorem [13] we can choose the solution F Γ ( s)(...p ext ...), Ψ( s) such that it belongs to the required class if F Γ ( s)(...p ext ...), d dt T 2 t Ψ( s) belongs to the required class. Therefore the statement is proved. So our theorem is proved.

Derivation of non ergodic property from main result
Let us prove (more accurately as in introduction) that our system (Bose gas with weak pair interaction in thermodynamical limit) is non-ergodic system. Let us recall definition of ergodicity [15]. Definition. Consider a quantum system described by Hamiltonian H. This system is said to be ergodic if the spectrum of H is simple.
This definition is equivalent to the following Definition. A quantum system described by Hamiltonian H is said to be ergodic if each bounded operator commuting with H is a function of H.
The generalization of this definition to the case when where exists some additional commuting first integrals is obvious [15].
It is to difficult to define a Hilbert space and Hamiltonian (as a selfadjoint operator in Hilbert space of states) of the system, in thermodynamical limit. So we give some new definition of ergodicity for this case which can be considered as some variant of last definition. Let us introduce some useful notations. Let V be an algebra of all Wick monomials with kernels from the Schwartz space, i.e V is a linear space of all expressions of the form w(p 1 , ..., p n |q 1 , ..., q m ) where the multiplication is defined by canonical commutative relations. Let V ′ be an algebraically dual of V . We say that the functional ρ ∈ V is a stationary functional if ∀v ∈ V ρ([H, v]) = 0. Here H is a Hamiltonian of our Bose gas. We say that the functional ρ ∈ V is a translation-invariant functional if ∀v ∈ V ρ([ P , v]) = 0, where P is an operator of momentum of our system.
. Now let us introduce a notion of Gibbsian states. Let β ∈ R, β > 0, v ∈ R 3 . We define Gibbsian state on V formally by the following formula: where a ∈ V , and Z is so called statistical sum: This states corresponds to canonical distributions. Note that one of the basis statement of statistical mechanics states that there no difference which distribution we use: canonical or micro-canonical distribution. Bellow we will omit v, P at all formulas to simplify or notations. Let V ′ G be a subspace spanned by all Gibbsian states, i.e. V ′ G is a set of all functionals · on V of the form: where the sum is understood in some generalized sense, for example it may be continuous (integral). It is evident that V ′ G ⊆ V s . Now we can give the definition of ergocity for Bose gas in thermodynamical limit.
Definition. We say that our system is ergodic if each translation invariant stationary state can be represented as a superposition of Gibbsian states, i.e V ′ s = V ′ G . After these previous discussion let us start to prove our statement. Recall that we find non Gibbsian real stationary translation invariant functional · constructed as a formal power series on coupling constant λ satisfying to the weak cluster property. This functional can be represented as follows: where · 0 is a functional of zero order of coupling constant λ, · 0 is a functional of first order of coupling constant λ ant e.c.t.. Suppose that our system is ergodic. We do not suppose that the series · 0 + · 1 + ... converges, but we will work with it formally as with convergent series and find explicit formulas for · under the assumption of ergodicity. Let us illustrate formal manipulation that we will use by several examples. x i . We do not suppose that |x| < 1. Denote by S the sum of this series. We have Therefore Consider the function sin x x = a 0 x + a 1 x + a 2 x 2 + ... as a polynomial of infinite degree. Let us use the Viete theorem for this "polynomial". The roots of sin x x are x i = iπ, i ∈ Z \ {0}. According to the Viete theorem we have i∈Z\{0} x i = C, j∈Z\{0} i∈Z\{0,j} We find from these equations that But x n = πn, so we finally have Such formal manipulation was widely used by Euler and others. Suppose that the set of such formal rules is enough large from one hand and does not contain a contradiction from other hand. These rules we call the Euler rules. If we can find the "sum" of some series by using the Euler rules then this series is called convergent in Euler sense. The "sum" of this series is called a sum in Euler sense. Let us prove that our functional · can be represented as follows (under the assumption of ergodicity): where c α are the "sums" of probably divergent series. The convergence (in the Euler sense) of this series will be proven below (under the assumption of ergodicy). The sum can be continuous (integral). Let {e α , α ∈ A} be a Hamele basis of V s , V s = Lin{e α , α ∈ A}. Let {e β , β ∈ B}, A ∩ B = ∅ be a completion of {e α , α ∈ A} to the Hamele basis of V , i.e. {e α , α ∈ A} ∪ {e β , β ∈ B} be a Hamele basis of V . ∀γ ∈ A ∪ B let f γ be an element of V ′ such that f γ (e γ ) = 1 and f γ ( An arbitrary functional ρ from V ′ now can be represented as a sum where l α , l β are arbitrary numbers. Note that for arbitrary l α , l β the right hand side of last equation is well defined because ∀v ∈ V f γ (v) = 0 only for finite number of elements γ ∈ A ∪ B. It is obvious now that an arbitrary element f ∈ V ′ s ⊆ V ′ G (ergodicity) can be represented as follows: where l β are arbitrary numbers. ∀i = 0, 1, 2, ... we have the following representations because e α ∈ V s and · is a translation invariant stationary functional.
in Euler sense. Finally where we put · ′ i = β∈B s i β f β ∈ V G and our statement is proved. Let 1 be some enough large but finite subsystem of our system. Let 2 be a subsystem obtained from 1 by translation on the vector l of sufficiently large length parallel to the x-axis. Let 12 be a union of the subsystems 1 and 2. Let U 1 , U 2 and U 12 be density matrices for the subsystems 1, 2 and 12 respectively (which correspond to · ).
Let {ϕ 1 n , n = 1, 2, ...} be a basis of eigenvectors of Hamilton operator for subsystem 1. Let{ϕ 2 m , m = 1, 2, ..} be a basis of eigenvectors of Hamilton operator for subsystem 2. Then the basis of eigenvectors of Hamiltonian for subsystem 12 is {ψ n,m = ϕ n ⊗ϕ m , m, n = 1, 2, ...}. According to the assumption that the systems 1,2 are enough large we find the following expression for density matrix of subsystems 1,2 in obvious notations. We have also the following expression for the density matrix of subsystem 12.
But if l = +∞ the weak cluster property implies that U 12 = U 1 ⊗ U 2 . This leads to the following relations: Here {E n , n = 1, 2, ...} be a set of eigenvalues of H 1,2 . But the set of sequences is linear independent. So for each α we have According to the linear independence of {e − En Tα } we find that ∀α So the series representing c α are convergent in the Euler sense and ∀α c α = 0, 1.
We see that · is a canonical Gibbsian distribution for some inverse temperature β.
Let us calculate now a(k)a + (k ′ ) . We have constructed · by some Gauss state ρ 0 described by some test function n(k). It follows from our construction of · that: But if momentum k is sufficiently large we can neglect by potential energy and find If we chose n(k) such that n(k) tends to zero as k → ∞ slowly than each Gauss function we obtain a contradiction. This contradiction proves non ergodic property of our system. Now let us discuss so called the Boltzmann ergodic hypothesis (1871). Let · be a translation invariant stationary functional on V such that ∀t ∈ R the functionals e itH (·)e −itH are well defined. The Boltzmann hypothesis states that for each such functional there exists an element We see that according to V s = V G the Boltzmann hypothesis does not hold.

Examples, chain diagrams
In this section we consider by direct calculation some class of divergent diagrams in Keldysh diagram technique. At first let us introduce the basis notion of the Keldysh diagram technique. Let us introduce the Green functions for the system ρ(T (Ψ ± H (t 1 , x 1 ), ..., Ψ ± H (t n , x n ))).
Symbol H near Ψ ± means here that Ψ ± H are Heizenberg operators. We require in nonequilibrium diagram technique the following representation for the Green functions The symbol 0 near Ψ ± means here that Ψ ± 0 are operators in the Dirac representation (representation of interaction). The S-matrix has the form and T is a symbol of the antichronological ordering here. ρ 0 is some Gauss state defined by density function n(k) as usual. Let us recall the basic elements of nonequilibrium diagram technique. The vertices coming from T -exponent are marked by symbol −. The vertices coming fromT -exponent are marked by symbol +. There exist four types of propagators Let us write the table of propagators The ovals represent the sum of one-particle irreducible diagrams. These diagrams are called chain diagrams. Let us suppose that all divergences of self-energy parts (ovals) are subtracted. The divergences arise from the fact that singular supports of propagators coincide. At first we consider diagrams with one self-energy insertion (one-chain diagram). These diagrams are pictured at fig. 2. The aim of this section is to prove that the Green functions can be made finite by the following renormalization of the asymptotical state:

Divergences
where h = h(k)a + (k)a(k)d 3 k, h(k) is a real-valued function and for any positive α, β. Therefore the Green function is translation invariant. So the density matrix is an integral of motion. Let In zero order of perturbation theory ρ(k) = n(k). But if there are no divergences in Keldysh diagram technique it is possible (see [14]) to derive the following kinetic equation for ρ(k) The right hand side of this equation is equal to zero only if for some α, β (α > 0, β > 0). But n(k) = ρ(k) in zero order of perturbation theory, so n(k) has a Bose-Einstein form. This contradiction proves our statement.

Regularization
Let us now introduce regularization. Note that Let us introduce the similar matrix for the self-energy operator Dyson equations in Fourier representation have the form We have from these equations that or in the matrix form It follows from Lemma 1 that Therefore detG  , detG are real and we have the following lemma. Lemma 3.
The following Lemma holds. Lemma 4.
Proof. The statement of lemma follows from the Dyson equation (175) and the following two obvious equalities: 11.5 Calculation of the propagators in one-chain approximation Lemma 5. The following limit equality holds (in the sense of distributions): Here reg means some correct distribution.
Proof. Let f (x) be some test function with compact support. We have for some smooth bounded function ψ(x). We have But So So the first equality is proved. One can prove other three equalities in the same way.
Therefore we see from the Lemmas 1,2, that we can consider only the function G −− (t, x). But the function G −− (t, x) can be represented as a sum of chain diagrams. At first let us consider the diagrams with one self-energy insertion (one-chain diagram). We have G −− ε = i,j=± H ij ε , where the diagrams for H ij ε are presented at the fig. 2. We have the following representation for the divergent parts of these diagrams.
The left hand side of this equation can be rewritten as follows (in approximation used in [14]) h(p) = 1 + 2n(p) 2n(p)(1 + n(p)) St(p), where St(p) is a scattering integral. So h(p) = 0 for non-equilibrium matter. Analogously one can consider two-chain diagrams presented at fig. 5 by direct calculation and prove that the divergences can be subtracted by the counterterms of the asymptotical state see cite.

Notes on Bogoliubov derivation of Boltzmann equations
In this section we study the problem of boundary conditions in Bogoliubov derivation of kinetic equations [6]. Let us consider N particles in R 3 . Let q i be a coordinates of particle number i, and p i be a momentum of particle number i, i = 1, ..., N. Suppose that particles interact by means of the pair potential V (q i − q j ). We suppose that V belongs to the Schwartz space. Let x i = (p i , q i ) be a point in the phase space Γ. Let f (x 1 , ..., x n ) be a distribution function of N particles. If we want to point out that f (x 1 , ..., x N ) depends on t we will write f (x 1 , ..., x N |t). Let and be marginal distribution functions. Put by definition ρ 1 (x 1 ) = Nf 1 (x 1 ), ρ 2 (x 1 , x 2 ) = N 2 f 2 (x 1 , x 2 ).
If A is a function on the phase space Γ, Γ = R 6N and then Now if A is a function on the phase space The function h(p) can be found from the following equation But in zero order of gas parameter the particles are free and for all q 2 ∈ S R \O, where O is a small neighborhood of the point q 0 := p 2 |p 2 | R ∈ S R . Diameter of O is approximately equal to diameter of suppV . Therefore the integral I is not equal to zero and equal to I = d 3 p 2 p 2 m 2πbdb ×{ρ 1 (p ′ 1 ((p 2 , q 2 (b)), (0, 0))ρ 1 (p ′ 2 ((p 2 , q 2 (b)), (0, 0)) −ρ 1 (p 2 )ρ 1 (0)}, where b := q 2 − q 0 . But the right hand side of (216) is a usual scattering integral. Therefore if we keep boundary terms in BBGKI-chain we obtain the kinetic equations without scattering integral.

Conclusion
In the present paper we have developed the general theory of the renormalization of nonequlibrium diagram technique. To study this problem we have used some ideas of the theory of R-operation developed by N.N. Bogoliubov and O.S. Parasiuk.
We illustrate our ideas by simple example of one-and two-chain diagrams in Keldysh diagram technique.
We want to illustrate in this paper the following general thesis: to prove that the system tends to the thermal equilibrium one should take into account its behavior on its boundary. In the last section we have shown that some boundary terms in BBGKI-chain which are usually neglected in Bogoliubov derivation of kinetic equation compensate scattering integral in kinetic equation.