aa r X i v : . [ m a t h - ph ] J un CUQM-121, HEPHY-PUB 840/07
Ultrarelativistic N -boson systems Richard L. Hall
Department of Mathematics and Statistics, Concordia University, 1455 deMaisonneuve Boulevard West, Montr´eal, Qu´ebec, Canada H3G 1M8
Wolfgang Lucha
Institute for High Energy Physics, Austrian Academy of Sciences,Nikolsdorfergasse 18, A-1050 Vienna, AustriaE-mail: [email protected] , [email protected] Abstract.
General analytic energy bounds are derived for N -boson systemsgoverned by ultrarelativistic Hamiltonians of the form H = N X i =1 k p i k + N X i 55% for all N ≥ . PACS numbers: 03.65.Ge, 03.65.Pm Keywords : Semirelativistic Hamiltonians, Salpeter Hamiltonians, boson systems ltrarelativistic N -boson systems 1. Introduction We consider first the semirelativistic N -body Hamiltonian H given by H = N X i =1 p k p i k + m + N X i 2. The general lower bound for m = 0 . It was shown in Ref. [1] that the non-negativity of the expectation h δ ( m, N ) i issufficient to establish the validity of the conjecture (4), where δ ( m, N ) = N X i =1 p k p i k + m − N − N X i 2, isan immediate consequence of the following: Theorem 1 h δ (0 , N ) i = 0. Proof of Theorem 1 Without loss of generality we adopt in momentum space a coordinate origin such that P Ni =1 p i := p = . We define the mean lengths h|| p ||i := k and h|| p − p ||i := d. (7)We wish to make a correspondence between mean lengths such as k and d and thesides of triangles that can be constructed with these lengths. We consider the triangleformed by the three vectors { p , p , p − p } . We suppose that the correspondingangles in this triangle are { φ , θ , θ } (the same notation is used for other similartriples). We now consider projections of one side on a unit vector along an adjacentside and define the mean angles φ and θ by the relations hk p k cos( φ ) i := hk p ki cos( φ )and hk p − p k cos( θ ) i := hk p − p ki cos( θ ) . Thus, on the average, this triangle is isosceles with one angle φ and the other twoangles θ. Since p = 0 , we have h p · p i = 0 . Hence || p || + N X i =2 || p |||| p i || cos( φ i ) = 0 . Thus, by dividing by || p || and using boson symmetry, we find h ( || p || + ( N − || p || cos( φ )) i = h|| p || (1 + ( N − 1) cos( φ )) i = 0 . We therefore conclude that k (1 + ( N − 1) cos( φ )) = 0 , that is to saycos( φ ) = − N − . We now consider again the triangle formed by the three vectors { p , p , p − p } . Wehave immediately from the dot product p · ( p − p ) k p kk p − p k cos( θ ) = k p k ( k p k − k p k cos( φ )) . By dividing by k p k and taking means we obtain d cos( θ ) = k (1 − cos( φ )) . But θ = ( π/ − φ/ 2) and cos( φ ) = − / ( N − . Hence we conclude kd = (cid:18) N − N (cid:19) . This equality establishes Theorem 1. ltrarelativistic N -boson systems 3. The linear potential We apply the new bound to the case of the linear potential V ( r ) = r. The weaker N/ H to obtain a scale-optimizedvariational upper bound E ≤ E g = (Φ , H Φ) . As we showed in Ref. [1], for the linearpotential V ( r ) = r in three spatial dimensions, the conjecture (now proven) impliesthat the N -body bounds are given for N ≥ N (cid:18) ( N − N (cid:19) e = E Lc ≤ E ≤ E Ug = 4 N (cid:18) ( N − N π (cid:19) , (8)where e ≈ . h = k p k + r . From (8) we see that the ratio R = E g /E c = 4 / ( π e ) ≈ . . The energy of the ultrarelativistic many-body system with linear pair potentials istherefore determined by these inequalities with error less than 0.55% for all N ≥ . Earlier we were able to obtain such close bounds for all N only for the harmonicoscillator [3]. 4. Conclusion We have enlarged the number of semirelativistic problems that satisfy the lower-boundconjecture h H i ≥ h H c i to include all problems with m = 0 and N ≥ . An extension ofthe geometric reasoning used in Ref. [1] from pyramids to more general simplices wouldperhaps have provided an alternative proof. However, the more algebraic approachadopted here, relying in the end on mean angles in a triangle, seemed to provide amore independent and robust approach. Acknowledgement One of us (RLH) gratefully acknowledges both partial financial support of his researchunder Grant No. GP3438 from the Natural Sciences and Engineering Research Councilof Canada and hospitality of the Institute for High Energy Physics of the AustrianAcademy of Sciences in Vienna. [1] R. L. Hall and W. Lucha, J. Phys. A: Math. Theor. , 6183 (2007), arXiv:0704.3580 [math-ph].[2] S. Boukraa and J. -L. Basdevant, J. Math. Phys. , 1060 (1989).[3] R. L. Hall, W. Lucha, and F. F. Sch¨oberl, J. Math. Phys.45