A. Smirnov

Moscow State University

e-mail: AlSmirnov@nes.ru

UDC 512.643

Key words: BMV-conjecture, positive-semideﬁnite matrices, quaternions, octonions. Abstract

A.S. Smirnov. BMV-conjecture, its noncommutative and nonasscociative cases. This paper investigates generalizations of BMV-conjecture for quaternionic and octonionic matrices. For quaternions the correctness of the formulation is shown as well as its equivalence to the original conjecture for complex matrices. General properties of octonions and hermitian matrices over them are examined for BMV-conjecture formulation over octonions.

1

Introduction

A conjecture with an intriguing name, which is more than a quarter century old, was stated by Bessis, Moussa and Villani [1] in 1975 in attempt to simplify the calculation of partition functions of quantum mechanical systems. It concerns with a positivity property of traces of matrices. If this property holds, it will permit to calculate explicit error bounds in a sequence of Pade approximants. BMV-conjecture is easy to state. Let A, B ∈ Mn (C) be hermitian matrices. And let B be a positivesemideﬁnite matrix. Then the following function, deﬁned as F (λ) = tr[eA−λB ] is the Laplace transform of a positive measure supported on [0, ∞).

This fact can be easily veriﬁed for quantum mechanics deﬁned by the Schrodinger equation for bosons without magnetic ﬁelds because in that case the partition function can be represented by Wiener integral. This representation cannot be applied for fermions, and that is of importance for condensed matter physics.

Also, the BMV-conjecture appears in other branches of matrix analysis as well and there are number of papers devoted to it. For 2 × 2 matrices there is an easy proof of its validity. For matrices of the larger size almost nothing is known up to this day. Some recent works [11], [12] showed the validity of the conjecture in some ‘average’ or ‘typical’ sense. In 2004 Lieb and Seiringer in [2] have reformulated the BMV-conjecture in pure algebraic form: Conjecture 1. For any two positive-semideﬁnite matrices A, B ∈ Mn (C) all coeﬃcients of polynomial f (t) = tr[(A + Bt)m ] are nonnegative.

The coeﬃcient of tk in polynomial f (t) for a ﬁxed m is the trace of matrix Sm,k (A, B ), where Sm,k (A, B ) is a sum of all words of the length m in A and B , in which B appears k times. For instance, S4,2 (A, B ) = A2 B 2 + AB 2 A + ABAB + BABA + B 2 A2 + BA2 B .

1

Statement 1.1. Let A, B ∈ Mn (C) be hermitian positive-semideﬁnite matrices. Then 1.

2.

3.

4.

An is hermitian positive-semideﬁnite matrix.

tr(A) 0.

Re(tr[AB ]) 0.

For any C ∈ Mn (C) matrix C ∗ AC is positive-semideﬁnite.

From the stated above properties of hermitian positive-semideﬁnite matrices over complex numbers the validity of BMV-conjecture follows for all m 5. In their work Hillar and Johnson [3] veriﬁed ﬁrst nontrivial case m = 6, k = 3 for 3 × 3 positive-semideﬁnite matrices and constructed an example of such matrices A, B that tr(ABAB 2 A)

0. Hagele [4] established, that BMV-conjecture holds for the case m = 7. Moreover, Hillar in his paper [5] proved the following fact: Theorem 1.2. If conjecture 1 is true for some m = m0 , then it is true for all m

m0 .

Proof . [5, corollary 1.8].

Therefore, BMV-conjecture turns to be valid for the case m

have proved the following theorem

Theorem 1.3. Conjecture 1 holds for all m

7. Recently, Klep and Schweighofer [6]

13.

Also in Burgdorf’s work [7] it was shown, that for 0

k

4 and m − 4

k

m for arbitrary m

tr(Sm,k (A, B )) 0. Therefore coeﬃcient of tk in f (t) for such k is nonnegative. In this paper the noncommutative case (conjecture 1 over quaternionic matrices) and nonassociative case (conjecture 1 over octonionic matrices) of conjecture 1 are examined.

2

Quaternionic case

2.1

Matrices over quaternions

The linear space of quaternions is a four-dimensional vector...