`International Journal of Mathematics and Mathematical SciencesVolume 2010, Article ID 890523, 26 pageshttp://dx.doi.org/10.1155/2010/890523`
Research Article

Contractions of Product Density Operators of Systems of Identical Fermions and Bosons

1Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
2School of Mathematics, West Pomeranian University of Technology, Szczecin, al. Piastów 17, 70-310 Szczecin, Poland

Received 17 May 2010; Accepted 5 December 2010

Abstract

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.

1. Introduction

This paper (see also preprint [1]), presenting the results of a part of the author's thesis [2], 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 is a separable Hilbert space of a single fermion (boson), then the space of the -fermion (resp. -boson) system is the antisymmetric (resp. symmetric) subspace (resp. ) of . Density operators of -fermion (resp. -boson) systems are identified with those defined on and concentrated on (resp. ).

Recall that the expectation value of an observable represented by a bounded self-adjoint operator on given Hilbert space in a state described by a density operator equals . If is an unbounded self-adjoint operator on a dense subspace of given Hilbert space, instead of one can consider its spectral measure (which is a bounded operator) of a Borel subset of the spectrum of . Then is the probability that the result of the measurement of the observable in question belongs to [3].

-particle observables of -fermion and -boson systems () are represented, respectively, by operators of the form (multiplied by ), where and are projectors of onto and , respectively, is the identity operator on and is a self-adjoint operator on (see [4]). Operators (1.1) are called antisymmetric and symmetric expansions of . In view of the earlier remark it is assumed that is bounded. The expectation values of observables represented by and in states represented by -fermion and -boson density operators and , respectively, can be expressed as (see [4, equations (1.7), (3.19)]), where -particle density operators and are -contractions of and (see Definition 2.1), called also reduced density operators. Such operators were investigated by Coleman [5], Garrod and Percus [6], and Kummer [4] (see also, e.g., [79] and references therein). A presentation of the basic ideas concerning reduced density operators and their applications can be found in [10].

In the present paper particular interest is taken in the case when and are product density operators, that is, they are of the form where , , and 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 and for and being, respectively, antisymmetric and symmetric products of identical trace class operators, including operators (1.3). The explicit form of the operators and proves to be quite complex. However, they can be replaced by operators with simpler structure if only the limiting values of expectations (1.2), in the sense of the thermodynamic limit, are of interest. The second objective of this paper is to find that simpler forms of contractions and for product density operators (1.3), equivalent to the complete expressions in the thermodynamic limit.

The problems described above have been solved for by Kossakowski and Maćkowiak [11], and Maćkowiak [12]. The formulae they derived were exploited in calculations of the free energy density of large interacting -fermion and -boson systems [11, 12], as well as in the perturbation expansion of the free energy density for the -impurity Kondo Hamiltonian [13]. In the case of investigation of many-particle interactions of higher order [1417], or higher order perturbation expansion terms of the free energy density, the expressions for and with can be used 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.

2. Preliminaries

In this section notation and terminology are set up.

2.1. Basic Notation

Let be a separable Hilbert space over or . The following notation is used in the sequel. : the identity operator on ,: the space of bounded linear operators on with the operator norm , : the space of trace class operators on with the trace norm ,: the space of bounded self-adjoint operators on , : the set of nonnegative definite bounded self-adjoint operators on , : the set of density operators (matrices) on , that is,

Set and denote by the group of permutations of the set . Let be the projectors such that for every . The closed linear subspaces and of are called the antisymmetric and symmetric subspace, respectively.

The antisymmetric and symmetric product of operators , are defined as and , respectively. It is assumed , , and . Clearly, if then , , and if then .

Set and . The product of measures, is denoted by and stands for . In subsequent sections use is made of product integral kernels, described in the appendix.

2.2. Contractions of Operators

The definition and basic properties of contractions of operators are now recalled for the reader's convenience. They were studied in [47]. A discussion of properties of reduced density operators can also be found in [10].

Let be a separable Hilbert space over the field or .

Definition 2.1. Let , , and . Then the -contraction of is the operator such that It is also assumed .

Remark 2.2. The operator always exists and is defined uniquely by (2.3). is a partial trace of with respect to the component of . If , where the measure is separable and -finite, and is a product integral kernel of (see the appendix) then has an integral kernel given by formula (A.4), according to Lemma A.5 and Corollary A.6.

Under the assumptions of Definition 2.1 one has , and if , , then . Moreover, if then , and if then .

Contractions of density operators are called reduced density operators. Contractions preserve the Fermi and the Bose-Einstein statistics of the contracted operator, that is, for and one has and . For such and (1.2) hold.

The following theorem is a part of Coleman's theorem [4, 5].

Theorem 2.3. Let , . For every (-fermion) density operator , , one has .

3. 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 over the field or , where the measure μ is separable and -finite.

The following theorem in the case of was proved in [11, 12].

Theorem 3.1 (Recurrence formulae). Let . If , , then and if , , then

Proof. Let be a product integral kernel of . For every define the mapping by the formula Then the mapping given by is a product integral kernel of .
Equation (3.1) will be first proved for . In view of Remark 2.2, an integral kernel of can be given by for -a.a. . Performing permutations of the first rows and permutations of the first columns of the determinant defining and expanding that determinant with respect to the th column one obtains
Consider the first term on the r.h.s. of (3.8). In all summands of except the last one the th row of the determinant (containing the variable ) can be shifted into the th position, changing thereby the sign of the determinant by . Then the first term of sum (3.8) assumes the form Let denote the transposition for (then ) and the identity permutation for (with ). Expression (3.9) can be written as The function , such that is -a.e. equal to expression (3.10), is an integral kernel of the operator which appears on the r.h.s. of (3.1).
Consider now the second term of the sum on the r.h.s. of (3.8). One can change the indices of the integral variables in all summands of except the first one, according to the rule for the th summand, and simultaneously change the order of the columns of the determinant inversely (which changes the sign by ). The resulting sum then contains terms identical to the one with . Thus the second term of sum (3.8) equals The function , such that is -a.e. equal to expression (3.12), is an integral kernel of the operator which occurs on the r.h.s. of (3.1). One concludes that the kernel of the operator on the l.h.s. of (3.1) is -a.e. equal to the kernel of the operator on the r.h.s. of (3.1), which proves the equality of both operators.
The proof of (3.1) for and the proof of (3.3) proceed analogously.
Similarly, the proof of (3.2) and (3.4) is accomplished by changing the product into and replacing determinants in all formulae by pernaments, defined for every complex matrix as Notice that signs of permutations are omitted in this case, similarly as the multipliers in the Laplace expansions.

Lemma 3.2. Let , , , , and (for the only summation index is . Then and .

The proof of the above lemma consists in demonstrating the invariance of under permutations of factors in the tensor products. To this end it suffices to observe that is invariant under transpositions of neighbouring factors.

Lemma 3.3. Let , , for , , , and for , . (For the only summation index is and the summation runs over the operators .) If then

Proof. Equation (3.17) will be first proved for . One has The first and the third term after the last of equalities (3.20) yield for , , . By Lemma 3.2, the second term after the last of equalities (3.20) equals The sum of expressions (3.21) and (3.22) is equal to the r.h.s. of (3.17) for . After simplifications the proof also applies to the case of .
The proof of (3.18) is analogous to that of (3.17).

The next theorem provides the explicit form of -contractions of product operators. The proof for was given in [11, 12]. The author of [12] emphasized that formula (3.23) for was derived by S. Pruski in 1978.

Theorem 3.4 (Explicit formulae). Let , , , , for , and , . Then (For the only summation indices are and and the summation runs over the operators and , resp.)

Proof. For every , , let be defined as in Lemma 3.3. Then the first of equalities (3.23) can be written as The proof of (3.25) will be carried out by (double) induction with respect to and, for fixed , with respect to .() () This part of the proof is by induction with respect to .(a)() According to Theorem 3.1, .(b)Assuming validity of formula (3.25) (with for , where , , its validity will be shown for .
One has Thus, according to the inductive hypothesis for , which, in view of Theorem 3.1, yields .()Assuming validity of formula (3.25) for (and every , where , , its validity will be shown for . For arbitrarily fixed the proof will be carried out by induction with respect to .(a)() By the inductive hypothesis with respect to and Lemma 3.3, hence , according to Theorem 3.1.(b)Assuming validity of formula (3.25) for , where , , , its validity will be shown for .
By the inductive hypothesis for and Lemma 3.3 one has According to the inductive hypothesis for one thus obtains which, in view of Theorem 3.1, yields . This completes the inductive proof for (3.25) with respect to and with respect to .
Now turn to the second of equalities (3.23). For it is identity. Let . Setting , ,, or, equivalently, , , ,, , one checks that both sides of the equality in question are equal to
The proof of (3.24) is analogous to that of (3.23).

4. 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.

In what follows use is made of the Hilbert space (over or ), where the measure is separable, -finite, and satisfies the condition . For every-measurable subset it is assumed .

Let be a fixed family of measurable subsets of such that for every (it can be the family of all such subsets). Fix , , and assume that there exists a sequence such that as .

Definition 4.1. Fix , , and let be a family of complex numbers. A complex number is said to be the thermodynamic limit of this family if for every sequence such that the condition is fulfilled. In such a case is denoted by .

Special attention will be given to the families of complex numbers of the form , where , , , and .

Definition 4.1 does not guarantee the convergence of families of interest in physics. To obtain such a convergence, additional conditions (such as conditions of uniform growth [18]) are usually imposed on the sequence 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.

Definition 4.2. Fix and , . Families and of operators are said to be asymptotically equivalent (symbolically: , if for every family of operators with uniformly bounded operator norms one has

Condition (4.1) is required to hold in particular for families such that for all , .

Remark 4.3. The authors of [11, 12] used some different definition of asymptotic equivalence of families of operators, closer to Definition 4.10 in this paper.

Remark 4.4. For fixed and , , the relation is an equivalence relation. If then for every family of operators as in Definition 4.2 the limit exists if and only if the limit exists, in which case both limits are equal. Notice also that if then and for every family and . Furthermore, for every family with uniformly bounded trace norms and for every sequence convergent to one has .

Lemma 4.5. Let and be as in Definition 4.2. Then Moreover, if the operators , are self-adjoint then

Proof. Implication (4.2) follows from Definition 4.2 and the estimate
Now assume that , which is equivalent to the condition where . The operators have the spectral representations where are the projectors onto orthogonal one dimensional subspaces of eigenvectors of , corresponding to eigenvalues . Since , for every there exists such that . Thus the operators satisfy the condition which, in view of implication (4.2) proved and condition (4.5), yields . In particular, where Observe that , hence condition (4.9) gives Since , conditions (4.8) and (4.11) yield which proves implication (4.3).

The following lemma follows from Lemma 4.5.

Lemma 4.6. Fix . Let and be families of self-adjoint operators such that , and let be a family of operators with uniformly bounded trace norms . Then

In the sequel denotes a family of nonnegative definite self-adjoint operators , and for every 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 defined for fixed and every , .

Remark 4.7. For every the operator is invertible and . Furthermore, if then is invertible and .

The next theorem is a version of a theorem studied in [11, 12] (see Remark 4.3).

Theorem 4.8. If and the reals , , are uniformly bounded then If and the reals , , are uniformly bounded then If, additionally, for some and every then

Proof. By Theorem 3.1 and the assumption one has Since , relation (4.20) yields (4.16), in view of Remark 4.4.
Now turn to the proof of relation (4.17). According to Remark 4.7, The explicit form of given by Theorem 3.4 shows that commutes with , and since both operators are self-adjoint, is also self-adjoint. Thus conditions (4.16), (4.21), and Lemma 4.5 yield (4.17).
The proof of relations (4.18), (4.19) runs parallel to that of (4.16), (4.17). Notice that in this case the expression from estimate (4.21) is replaced by (see Remark 4.7).

The following theorem for (with the reservation of Remark 4.3) was obtained in [11, 12]. The author of [12] gave also arguments that can be used to check the assumptions of this theorem.

Theorem 4.9 (Asymptotic formulae I). If for every and then, for every , ,
If for every and then, for every , ,

Proof. First equivalence (4.23) will be proved. Observe that hence Since the operators are trace class, . Thus, by assumption (4.22) and the self-adjointness of the operators , one obtains
The rest of the proof of (4.23) is by induction with respect to .
() () By Theorem 3.1 for one has Assumption (4.22) gives , hence, by (4.29), Remark 4.4, and the assumption , one obtains Thus, in view of equivalence (4.16) from Theorem 4.8 and Lemma 4.6, one has Furthermore, assumption (4.22) implies that the trace norms of the operators on the r.h.s of (4.31) are uniformly bounded. Therefore, according to Remark 4.4, The explicit form of , given by Theorem 3.4 implies that and commute, which proves the self-adjointness of the operator on the l.h.s of (4.32). Thus conditions (4.32), (4.27) for , (4.28), and Lemma 4.5 yield relation (4.23) for .
() Assuming validity of equivalence (4.23) for , where , , its validity will be proved for .
By Theorem 3.1 for one has