CUQM-101HEPHY-PUB 774/03UWThPh-2003-XXmath-ph/0311032 November 2003

The energy of a system of relativistic massless bosons

bound by oscillator pair potentials

Richard L. Hall, Wolfgang Lucha, and Franz F. Schöberl

Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montréal, Québec, Canada H3G 1M8 Institut für Hochenergiephysik, Österreichische Akademie der Wissenschaften, Nikolsdorfergasse 18, A-1050 Wien, Austria Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria

, ,

Abstract

We study the lowest energy of a semirelativistic system of identical massless bosons with Hamiltonian

We prove

where and with an error less than for all The average of these bounds determines

PACS: 03.65.Ge, 03.65.Pm, 11.10.St One of the advantages of studying the semirelativistic “spinless-Salpeter” Hamiltonian [1,2] is that it captures some aspects of a full relativistic treatment and at the same time allows us to express the many-body problem in a tractable form. The principal source of mathematical difficulty is the kinetic-energy operator , which is defined in momentum space as a multiplicative operator [3], and becomes, via the Fourier transform, a non-local operator in configuration space. We have earlier found energy bounds [4] for systems of bosons in the case In the nonrelativistic limit the kinetic energy has the Schrödinger asymptotic form Since the Schrödinger many-body harmonic-oscillator problem is exactly soluble [5-7], we were able to derive energy bounds that are asymptotically exact as The bounds were weakest in the ultrarelativistic limit It is the purpose of this paper to present accurate bounds for this limiting case The Hamiltonian for the system we study is given by

We shall prove that the lowest energy of this system satisfies the inequalities

where the coefficients and are given by

The energy of the -body system is therefore determined by the average of the bounds in Eq. (2) for all couplings and all with an error less than In order to establish the energy bounds we must consider two fundamental symmetries: translational invariance, and boson permutation symmetry. The Hamiltonian includes the kinetic energy of the centre of mass. Therefore we choose a set of relative coordinates so that the kinetic energy of the centre of mass can be eliminated. The most convenient relative coordinates for our purposes are Jacobi coordinates defined in terms of the (column) vector of individual-particle coordinates by an orthogonal matrix Thus we write Since is orthogonal, the conjugate momenta are given in terms of the individual momenta by the expression The first of the Jacobi coordinates, is proportional to the centre-of-mass variable, so that the elements of the first row are all equal to The other two coordinates which we shall need to refer to specially are and which are given explicitly in terms of the ‘other set’ by

The expression of the boson permutation-symmetry constraint in terms of Jacobi coordinates can be a source of complication. But we do not need to face this difficulty here: we simply exploit the ‘reducing power’ of the necessary boson symmetry to relate the -body problem to a scaled two-body problem. Let us assume that is a normalized boson wave function of the relative coordinates By Lemma (1) established in Ref. [4], we know that an operator acting on with leading term may be replaced by zero, even when the term is inside the kinetic-energy square root. This will be important later. We shall also use another important relation [4, Eq. (2.5)], namely

An arbitrary boson wave function is not necessarily symmetric in the but Eq. (5) is generally true, and is very useful. A special boson wave function which certainly is symmetric in the is the Gaussian function which also has another unique [8,9] and useful property, namely it factors into single-variable Gaussians as follows:

That has the correct boson symmetry follows immediately from the following identity valid for Jacobi relative coordinates

In momentum space the Gaussian transforms to a Gaussian by the three-dimensional Fourier transform as follows:

The lower energy bound is found by the following argument. We suppose that is the exact ground-state wave function for the -body system corresponding to energy and, using the necessary boson symmetry, we write

By employing (4) and (5) in succession, and noting that the lemma allows us to remove the operator from the square root, we arrive at the relation

Thus the -body energy is bounded below by the lowest energy of the one-body Hamiltonian

But is the Hamiltonian for the Schrödinger problem of a single particle moving in a linear potential Thus we find

is the first zero of Airy’s function, and is also exactly the bottom of the spectrum of in three dimensions. This establishes the lower energy bound.

The upper bound is found by means of the Gaussian ‘trial’ function discussed above. We have

That is to say

By minimizing with respect to the variational parameter we obtain

This result establishes the upper bound. It is perhaps tempting to try to improve the upper bound by the use of a more flexible trial function. However, it is not trivially easy to accomplish this, and to keep the calculation and result simple, since we must use a translation-invariant boson function.

It is interesting that there is a relationship between the problem discussed in this paper and the corresponding Schrödinger problem with a linear potential and To be more precise, if we consider the nonrelativistic problem with Hamiltonian given by

then we have shown [10, Eq. (4.16)] that the lowest energy of is bounded by the inequalities

where the coefficients and are exactly as given in Eq. (3) above. The two problems are brought more nearly into ‘coincidence’ if we set The remaining differences, involving and , can then be understood if one notes that the factor must be associated with the ‘potential terms’ in each case, and, similarly, the and couplings must be made to ‘correspond’ by scaling, as the Fourier transformation, which relates the two systems, is applied. With the aid of such arguments, the ‘almost equivalence’ of the problems could eventually be used formally to extract our main result. However, we present these considerations here merely as a confirmation of our results; we prefer to develop more widely applicable direct approaches for the semirelativistic many-body problem itself, valid for all

Acknowledgements

Partial financial support of this work under Grant No. GP3438 from the Natural Sciences and Engineering Research Council of Canada, and the hospitality of the Institute for High Energy Physics of the Austrian Academy of Sciences in Vienna, is gratefully acknowledged by one of us [RLH].

References [1]E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 (1951). [2]E. E. Salpeter, Phys. Rev. 87, 328 (1952). [3]E. H. Lieb and M. Loss, Analysis (American Mathematical Society, New York, 1996). The definition of the Salpeter kinetic-energy operator is given on p. 168. [4]R. L. Hall, W. Lucha, and F. F. Schöberl, J. Math. Phys. 43, 1237 (2002); Erratum ibid. 44, 2724 (2003). [5]W. M. Houston, Phys. Rev. 47, 942 (1935). [6]H. R. Post, Proc. Phys. Soc. London 66, 942 (1953). [7]R. L. Hall, Phys. Rev. A 51, 3499 (1995). [8]R. L. Hall, Can. J. Phys. 50, 305 (1972). [9]R. L. Hall, Aequ. Math. 8, 281 (1972). [10]R. L. Hall, J. Math. Phys. 29, 990 (1988).