Contractions of product density operators of systems of identical fermions and bosons

Recurrence and explicit formulae for contractions (partial traces) of antisymmetric and symmetric products of identical trace class operators are derived. Contractions of product density operators of systems of identical fermions and bosons are proved to be asymptotically equivalent to, respectively, antisymmetric and symmetric products of density operators of a single particle, multiplied by a normalization integer. The asymptotic equivalence relation is defined in terms of the thermodynamic limit of expectation values of observables in the states represented by given density operators. For some weaker relation of asymptotic equivalence, concerning the thermodynamic limit of expectation values of product observables, normalized antisymmetric and symmetric products of density operators of a single particle are shown to be equivalent to tensor products of density operators of a single particle. This paper presents the results of a part of the author's thesis [W. Radzki,"Kummer contractions of product density matrices of systems of $n$ fermions and $n$ bosons"(Polish), MS thesis, Institute of Physics, Nicolaus Copernicus University, Toru\'{n}, 1999].


Introduction
This paper, presenting the results of a part of the author's thesis [10], deals with contractions (partial traces) of antisymmetric and symmetric product density operators representing mixed states of systems of identical noninteracting fermions and bosons, respectively.
If H is a separable Hilbert space of a single fermion (boson) then the space of the n-fermion (resp. n-boson) system is the antisymmetric (resp. symmetric) subspace H ∧n (resp. H ∨n ) of H ⊗n . Density operators of n-fermion (resp. n-boson) systems are identified with those defined on H ⊗n and concentrated on H ∧n (resp. H ∨n ).
Recall that the expectation value of an observable represented by a bounded selfadjoint operator B on given Hilbert space in a state described by a density operator ρ equals Tr Bρ. If B is an unbounded selfadjoint operator on a dense subspace of given Hilbert space, instead of B one can consider its spectral measure E B (∆) (which is a bounded operator) of a Borel subset ∆ of the spectrum of B. Then Tr E B (∆)ρ is the probability that the result of the measurement of the observable in question belongs to ∆ [9].
k-particle observables of n-fermion and n-boson systems (k < n) are represented, respectively, by operators of the form H are projectors of H ⊗n onto H ∧n and H ∨n , respectively, I is the identity operator on H and B is a selfadjoint operator on H ⊗k (see [4]). Operators (1) are called antisymmetric and symmetric expansions of B. In view of the earlier remark it is assumed that B is bounded. The expectation values of observables represented by ∧ Γ n k B and ∨ Γ n k B in states represented by n-fermion and n-boson density operators K and G, respectively, can be expressed as (see [4,Eqs. (1.7), (3.19)]), where k-particle density operators L k n K and L k n G are (n, k)-contractions of K and G (see Definition 2.1), called also reduced density operators. Such operators were investigated by Coleman [1], Garrod and Percus [2], and Kummer [4].
In the present paper particular interest is taken in the case when K and G are product density operators, i.e. they are of the form K = 1 Tr ρ ∧n ρ ∧n , G = 1 Tr ρ ∨n ρ ∨n , where ρ ∧n = A H , and ρ is a density operator of a single fermion or boson, respectively. The first objective of this paper is to find the recurrence and explicit formulae for L k n K and L k n G for K and G being, respectively, antisymmetric and symmetric products of identical trace class operators, including operators (3). The explicit form of the operators L k n K and L k n G proves to be quite complex. However, they can be replaced by operators with simpler structure if only the limiting values of expectations (2), in the sense of the thermodynamic limit, are of interest. The second objective of this paper is to find that simpler forms of contractions L k n K and L k n G for product density operators (3), equivalent to the complete expressions in the thermodynamic limit.
The problems described above have been solved for k = 1, 2 by Kossakowski and Maćkowiak [3], and Maćkowiak [6]. The formulae they derived were exploited in calculations of the free energy density of large interacting n-fermion and n-boson systems [3,6], as well as in the perturbation expansion of the free energy density for the M -impurity Kondo Hamiltonian [8]. In the case of investigation of manyparticle interactions of higher order [15,14,7,13], or higher order perturbation expansion terms of the free energy density, the expressions for (Tr ρ ∧n ) −1 L k n ρ ∧n and (Tr ρ ∨n ) −1 L k n ρ ∨n with k ≥ 3 prove to be needed in the canonical and grand canonical ensemble approach, which is the physical motivation for the present paper.
The main results of this paper are Theorems 3.1, 3.4, 4.9, and 4.14.

Preliminaries
In this section notation and terminology are set up.
2.1. Basic notation. Let (H, ·, · ) be a separable Hilbert space over C or R. The following notation is used in the sequel.
for every ψ 1 , . . . , ψ n ∈ H. The closed linear subspaces H ∧n = A The antisymmetric and symmetric product of operators The product of measures µ, µ 1 is denoted by µ ⊗ µ 1 and µ ⊗n stands for µ ⊗ · · · ⊗ µ n . In subsequent sections use is made of product integral kernels, described in Appendix A.

2.2.
Contractions of operators. The definition and basic properties of contractions of operators are now recalled for the reader's convenience. A discussion of these properties was carried out by Kummer [4,5].
Let H be a separable Hilbert space over the field K = C or R.
Definition 2.1. Let k, n ∈ N, k < n, and K ∈ T (H ⊗n ). Then the (n, k)-contraction of K is the operator L k n K ∈ T (H ⊗k ) such that It is also assumed L n n K = K.
Remark 2.2. The operator L k n K always exists and is defined uniquely by Eq. (4). L k n K is a partial trace of K with respect to the component where the measure µ is separable and σ-finite, and K is a product integral kernel of K (see Appendix A) then L k n K has an integral kernel K 0 given by formula (60), according to Lemma A.5 and Corollary A.6.
Under the assumptions of Definition 2.1 one has Tr H ⊗k L k n K = Tr H ⊗n K, and if p ∈ N, k < p < n, then L k p (L p n K) = L k n K. Moreover, if K ∈ B * (H ⊗n ) then L k n K ∈ B * (H ⊗k ), and if K ∈ B * ≥0 (H ⊗n ) then L k n K ∈ B * ≥0 (H ⊗k ). Contractions of density operators are called reduced density operators. Contractions preserve the Fermi and the Bose-Einstein statistics of the contracted operator, i.e. for K ∈ A (n) H . For such K and G Eqs.

Recurrence and explicit formulae for contractions of products of trace class operators
In this section recurrence and explicit formulae for contractions of antisymmetric and symmetric powers of single particle operators are derived.
In the whole section use is made of the Hilbert space H Y := L 2 (Y, µ) over the field K = C or R, where the measure µ is separable and σ-finite.
The following theorem in the case of k = 1, 2 was proved in [3,6].
Let T jk ∈ S k denote the transposition j ↔ k for j < k (then (−1) k+j (−1) −k−j+1 = (−1) = sgn T jk ) and the identity permutation for j = k (with sgn T kk = 1). Expression (10) can be written as The function P 1 : Y k × Y k → K, such that P 1 (x 1 , . . . , x k , y 1 , . . . , y k ) is µ ⊗2k -a.e. equal to expression (11), is an integral kernel of the operator which appears on the r.h.s. of Eq. (5). Consider now the second term of the sum on the r.h.s. of Eq. (9). One can change the indices of the integral variables x k+1 , . . . , x j in all summands of n j=k+1 except the first one, according to the rule x j → x k+1 → x k+2 → · · · → x j for the jth summand, and simultaneously change the order of the columns of the determinant inversely (which changes the sign by (−1) (j−1)−k = (−1) (k+1)−j ). The resulting sum n j=k+1 then contains n − k terms identical to the one with j = k + 1. Thus the second term of sum (9) equals The function equal to expression (12), is an integral kernel of the operator which occurs on the r.h.s. of Eq. (5). One concludes that the kernel L of the operator on the l.h.s. of Eq. (5) is µ ⊗2k -a.e. equal to the kernel P 1 + P 2 of the operator on the r.h.s. of Eq. (5), which proves the equality of both operators. The proof of Eq. (5) for n = k + 1 and the proof of Eq. (7) proceed analogously. Similarly, the proof of Eqs. (6), (8) is accomplished by changing the product ∧ into ∨ and replacing determinants in all formulae by pernaments, defined for every Notice that signs of permutations are omitted in this case, similarly as the multipliers ±1 in the Laplace expansions.
. . , m} , and HY and S (k) HY . The proof of the above lemma consists in demonstrating the invariance of R under permutations of factors in the tensor products. To this end it suffices to observe that R is invariant under transpositions of neighbouring factors.
(For p = 1 the only summation index is j 1 and the summation runs over the operators ρ j1 .) and Proof. Eq. (13) will be first proved for p > 2. One has The first and the third term after the last of equalities (15) yield The sum of expressions (16) and (17) is equal to the r.h.s. of Eq. (13) for p > 2.
After simplifications the proof also applies to the case of p = 2. The proof of Eq. (14) is analogous to that of Eq. (13).
The next theorem provides the explicit form of (n, k)-contractions of product operators. The proof for k = 1, 2 was given in [3,6]. The author of [6] emphasized that formula (18) for k = 2 was derived by S. Pruski in 1978.
One has Thus, according to the inductive hypothesis for n ∈ {2, . . . , m − 1} , Assuming validity of formula (20) for k ∈ {1, . . . , p − 1} (and every n > k), where p ∈ N, p > 1, its validity will be shown for k = p. For arbitrarily fixed p the proof will be carried out by induction with respect to n > p. a) (n = p + 1) By the inductive hypothesis with respect to k and Lemma 3.3, HY .
According to the inductive hypothesis for n ∈ {p + 1, . . . , m − 1} one thus obtains which, in view of Theorem 3.1, yields m p L p m ρ ∧m = Π ∧p m (ρ). This completes the inductive proof for Eq. (20) with respect to n > p and with respect to k. Now turn to the second of equalities (18). For k = 1 it is identity. Let k ≥ 2.
one checks that both sides of the equality in question are equal to The proof of Eq. (19) is analogous to that of Eq. (18).

Asymptotic form for contractions of product states
The explicit forms of the contractions of product states given by Theorem 3.4 are quite complex. In the present section they are replaced by simpler operators, equivalent in the thermodynamic limit. The main results in this section are Theorems 4.9 and 4.14.
Let M(Ω) be a fixed family of measurable subsets of Ω such that 0 < µ(Y ) < +∞ for every Y ∈ M(Ω) (it can be the family of all such subsets). Fix d ∈ R, d > 0, and assume that there exists a sequence {Y n } n∈N ⊂ M(Ω) such that n µ(Yn) → d as n → ∞. Special attention will be given to the families of complex numbers of the form Tr (L k n K Y,n )C Y , where k, n ∈ N, n > k, K Y,n ∈ T (H ⊗n Y ), and C Y ∈ B(H ⊗k Y ). Definition 4.1 does not guarantee the convergence of families {b Y,n } of interest in physics. To obtain such a convergence, additional conditions (such as conditions of uniform growth [11]) are usually imposed on the sequence {Y n } n∈N in question. However, those additional conditions do not affect considerations in this paper.
Expression of expectation values of observables in mixed states by using trace, mentioned in Introduction, is the motivation for the following definition.   uniformly bounded trace norms Tr |A Y,n | and for every sequence {a n } n∈N ⊂ C convergent to a ∈ C one has a n A Y,n ≈ aA Y,n .
Moreover, if the operators A Y,n , B Y,n are selfadjoint then Proof. Implication (22) follows from Definition 4.2 and the estimate Now assume that A Y,n ≈ B Y,n , which is equivalent to the condition where D Y,n := A Y,n − B Y,n . The operators D Y,n have the spectral representations where P ϕi(Y,n) are the projectors onto orthogonal one dimensional subspaces of satisfy the condition which, in view of implication (22) proved and condition (24), yields F Y,n ≈ D Y,n ≈ 0. In particular, where Observe that Tr F Y,n C Y,n = Tr |F Y,n | , hence condition (26) gives Since Tr |D Y,n | ≤ Tr |D Y,n − F Y,n | + Tr |F Y,n | , conditions (25) and (27) yield Tr |D Y,n | = 0, which proves implication (23).
The following lemma follows from Lemma 4.5.
In the sequel {ρ Y } Y ∈M(Ω) denotes a family of nonnegative definite selfadjoint operators ρ Y ∈ T (H Y ) , and for every (Y, n) ∈ M(Ω) × N it is assumed that The objective of this section is to find density operators of the most simple form which are asymptotically equivalent to the operators Y,n+1 and the reals s ∨ Y,n ρ Y , (Y, n) ∈ M(Ω) × N, are uniformly bounded then If, additionally, s ∨ Y,n ρ Y ≤ ǫ for some ǫ < 1 and every (Y, n) ∈ M(Ω) × N then Proof. By Theorem 3.1 and the assumption Since Tr The explicit form of The proof of relations (30), (31) runs parallel to that of (28), (29). Notice that in this case the expression (I + s ∧ Y,n+1 ρ Y ) −1 = 1 from estimate (33) is replaced by The following theorem for k = 2 (with the reservation of Remark 4.3) was obtained in [3,6]. The author of [6] gave also arguments that can be used to check the assumptions of this theorem.
By Theorem 3.1 for n ≥ q one has Assumption (34) implies hence, in view of Eq. (43), Remark 4.4, and the assumption Thus, by relation (28) from Theorem 4.8, Lemma 4.6, and Remark 4.4, one has since the trace norms of the operators on the r.h.s. of (44) are uniformly bounded, by assumption (34). Furthermore, in view of Lemma 4.6 and the inductive hypoth- HY .
Furthermore, according to assumption (36), The rest of the proof of (37) is by induction with respect to k ≥ 2 and proceeds analogously to the proof of (35) with condition (39) replaced by (46) and the operators I ∓ s ∧ Y,n+1 ρ Y replaced by I ± s ∨ Y,n+1 ρ Y (inversion of signs).
Theorem 4.9 allows to replace (n, k)-contractions of antisymmetric and symmetric product density operators by antisymmetric and symmetric products of 1-particle contractions, respectively, if the number n of particles in the system is large. Further simplification, consisting in replacement of antisymmetric and symmetric products by tensor products, will be now proved possible. To this end weaker conditions on the asymptotic equivalence relation will be imposed. Definition 4.11. Let k ∈ N, k ≥ 2. Fix π ∈ S k . A set X ⊂ {1, . . . , k} is called a cyclic set of the permutation π, if X = {l 1 , . . . , l q } for some l 1 , . . . , l q ∈ {1, . . . , k} , q ∈ {2, . . . , k} , such that π(l s ) = l s+1 for every s ∈ {1, . . . , q − 1} , and π(l q ) = l 1 . A singleton {l} ⊂ {1, . . . , k} such that π(l) = l is also called a cyclic set of the permutation π.
Note that the set {1, . . . , k} from the above definition can be represented as the union of disjoint cyclic sets of π.
Notice that Eq. (50) can be also proved analogously to Eq. (51) under the additional assumption Y,n . The proof of the next theorem for k = 2 was given in [3,6].
The proof of relation (55), after discarding the permutation signs and replacing ∧ by ∨, proceeds analogously.

Appendix A. Product integral kernels of trace class operators
In this section theorems concerning product integral kernels, exploited in Section 3, are formulated.
Fix the Hilbert space H Y := L 2 (Y, µ) over the field K = C or R, where the measure µ is separable and σ-finite. For every n ∈ N the space H ⊗n Y is identified with L 2 (Y n , µ ⊗n ). Unless otherwise stated, elements of L 2 spaces are identified with their representatives and denoted by the same symbols.
Let K ∈ L 2 (Y 2 , µ ⊗2 ). In the case of the integral operator K : H Y → H Y defined for every ϕ ∈ H Y and µ-a.a. x ∈ Y by (Kϕ)(x) = Y K(x, y)ϕ(y) dµ(y) both K regarded as an element of L 2 (Y 2 , µ ⊗2 ) as well as its arbitrary representative is called an integral kernel of K. The kernel K is unique as an element of L 2 (Y 2 , µ ⊗2 ) but a representative of K of a special form, given in Lemma A.3 and Definition A.4, is useful in computations of the trace of K.
Let HS(H Y ) be the space of Hilbert-Schmidt operators on H Y with the inner product defined by A, B HS(HY ) := Tr A * B and the induced norm denoted by · HS(HY ) . In the sequel use is made of the following theorem, the proof of which can be found in [12].
Recall that K ∈ B(H Y ) is a trace class operator iff there exist operators K 1 , K 2 ∈ HS(H Y ) such that K = K 1 K 2 . Moreover, Tr K = K * 1 , K 2 HS(HY ) . This fact, Theorem A.1, and Corollary A.2 imply the following lemma, in which elements of the L 2 space are distinguished from their representatives. The element of the L 2 space represented by a function f is denoted by [f ].
Lemma A.3. Let K ∈ T (H Y ) , K = K 1 K 2 , where K 1 , K 2 ∈ HS (H Y ) . Let [K 1 ], [K 2 ] ∈ L 2 (Y 2 , µ ⊗2 ) be integral kernels of K 1 , K 2 . Then for any choice of rep- is µ ⊗2 -square integrable and it is an integral kernel of K. The function L : Y → K defined for µ-a.a. x ∈ Y by L(x) = K(x, x) is µ-integrable. Moreover, In the following lemma, which follows from Lemma A.3, the function K 0 need not be a product integral kernel of K 0 but the integral formula for the trace of K 0 still holds for K 0 .
Lemma A.5. Let k, n ∈ N, k < n, and let K be a product integral kernel of K ∈ T H ⊗n Y ≡ T L 2 (Y n , µ ⊗n ) . Then the function K 0 : Y k × Y k → K defined for µ ⊗2k -a.a. (x ′ , y ′ ) ∈ Y k × Y k by is µ ⊗2k -square integrable and the integral operator K 0 with the kernel K 0 belongs to T H ⊗k Y . For every χ, ϕ ∈ H ⊗k Y and every orthonormal basis {ψ i } i∈N of H ⊗(n−k) Y one has Corollary A.6. Under the assumptions of Lemma A.5, if C ∈ B H ⊗k Y then Tr CK 0 = Tr (C ⊗ I ⊗(n−k) )K. prepared also, on his own initiative, the English translation of appropriate parts of the author's thesis, which was useful for the author in editing of the present paper.