## LAWRENCE P. HORWITZ, PROFESSOR

972-3-6408724 (Tel-Aviv University) or 972-3-5318956 (Bar-Ilan University)

972-3-6407932 or 972-3-6429306 (Tel-Aviv University) or 972-3-5353298 (Bar-Ilan University)

### Topics of research:

1. Relativistic quantum mechanics and quantum field theory
2. Theory of classical and quantum unstable systems and chaos
3. Quantum theory on hypercomplex Hilbert modules
4. Complex projective spaces in quantum dynamics [CP(N)]
5. Relativistic statistical mechanics and thermodynamics

### List of recent publications

• L.P. Horwitz and W.C. Schieve. Horseshoes in a relativistic Hamiltonian system in 1+1 dimensions. Phys. Rev A 46, 743 (1992).
• L.P. Horwitz and W.C. Schieve . Horseshoes in perturbations of a relativistic Hamiltonian system in 1+1 Dimensions. Proceedings of the Conference on Differential Geometric Methods in Physics, New York June 1991.
• L.P. Horwitz and A. Razon. Tensor product in quaternion quantum mechanics. Proceedings of the II International Wigner Symposium, Goslar, July 16-20, 1991, World Scientific (1993).
• L.P. Horwitz. On the definition and evolution of states in relativistic classical and quantum mechanics. Foundations of Physics 22, 421 (1992) (invited, Sir Karl Popper issues).
• L.P. Horwitz. Dynamical group of the Relativistic quantum mechanical Kepler problem. Jour. Math. Phys. 34, 645 (1993)}.
• L.P. Horwitz and N. Shnerb . On the group theory of the polarization states of a gauge theory. Proc. of the Symposium, Symmetries in Science VI, from the Rotation Group to Quantum Algebras, Aug. 2-7, 1992, Bregenz, Austria, Plenum Press, N.Y. (1992)}.
• L.P. Horwitz and N. Shnerb. On the group theory of the polarization states of a massless field. Jour. Math. Phys. 35,1658 (1994).
• L.P. Horwitz and C. Piron. The unstable system and irreversible motion in quantum theory. Helv. Phys. Acta. 66, 693 (1993).
• A. Razon and L.P. Horwitz. Uniqueness of the scalar product in the tensor product of quaternion Hilbert modules. Jour. Math. Phys. 33, 3098 (1992).
• I. Antoniou, J. Levitan and L.P. Horwitz. Time dependence and intrinsic irreversibilities of the Pietenpol model. Jour. Phys. A: Math. and Gen. 26, 6033 (1993).
• I. Antoniou, J. Levitan and L.P. Horwitz. Estimate for regeneration up to the golden rule time. Jour. Phys. A: Math and Gen. 26, 3243 (1993).
• L.P. Horwitz. Some spectral properties of anti-self adjoint operators on a quaternionic Hilbert space. Jour. Math. Phys. 34, 3405 (1993).
• R. Faibish and L.P. Horwitz. Non-compact dynamical groups for the solution of bound state problems in relativistic quantum theory: I The relativistic harmonic oscillator in 1+1 dimensions. Jour. Group Theory in Phys. 2, 41 (1994).
• L.P. Horwitz and R. Faibish. Dynamical groups of the relativistic Kepler problem and the harmonic oscillator. Vol. II, p.450, Anales de Fisica, Monografias, Vols. I, II, Editors M.A. del Olmo, M. Santander and J.M. Guilarte, CIEMAT-RSEF, Madrid (1993).
• L. Burakovsky and L.P. Horwitz. Equilibrium relativistic mass distribution. Physica A 201, 666 (1993).
• E. Eisenberg and L.P. Horwitz. Time, irreversibility and the unstable system in quantum and classical physics. To be published (invited review) in Advances in Chemical Physics.
• N. Shnerb and L.P. Horwitz. Canonical quantization of 4-and 5-dimensional U(1) gauge fields. Phys. Rev. A. 48, 4068 (1993).
• S. Tasaki, E. Eisenberg and L.P. Horwitz. Measurement theory in the Lax-Phillips formalism. Foundations of Physics (invited paper in honor of C. Piron) 24, 1179 (1004).
• M.C. Land, R.I. Arshansky and L.P. Horwitz. Selection rules for dipole radiation from a relativistic bound state. Foundations of Physics 24, 563 (1994) (invited paper, in honor of F. Rohrlich).
• M.C. Land, N. Shnerb and L.P. Horwitz. On Feynman's Approach to the Foundations of Gauge Theories. Jour. Math. Phys. 36, 3263 (1995).
• E. Eisenberg and L.P. Horwitz. Instrinsic mechanism for entropy change in classical and quantum evolution. Phys. Rev. A 52, 70 (1995).
• L.C. Biedenharn and L.P. Horwitz. On the equivalence of the Skyrme-Witten model and large Nc quark model. Foundations of Physics 24, 401 (1994) (invited paper in honor of F. Rohrlich) .
• L.P. Horwitz. A soluble model for scattering and decay in quaternionic quantum mechanics I: Decay. Jour. Math. Phys. 35, 2743 (1994).
• L.P. Horwitz. A soluble model for scattering and decay in quaternionic quantum mechanics II: Scattering. Jour. Math. Phys. 35, 2761 (1994).
• N. Shnerb and L.P. Horwitz. Gauge and group properties of massless fields in any dimension. Jour. Phys. A: Math. and Gen. 27, 3565 (1994).
• L. Burakovsky and L.P. Horwitz. Covariant thermodynamics and a realistic'' Friedman model. Accepted Foundations of Physics.
• N. Shnerb and L.P. Horwitz. Gauge invariant quantization and induced braid statistics in $2+1$ dimensional $U(1)$ theories. Phys. Lett. B 339, 90 (1994).
• L.P. Horwitz. The unstable system in relativistic quantum mechanics. Found. Phys. 25, 39 (1995).
• L. Burakovsky and L.P. Horwitz. Distribution for indistinguishable events. Found. Phys. 25 , 785 (1995).
• L. Burakovsky and L.P. Horwitz. Relativistic mass distribution of event-anti-event systems and a realistic'' equation of state for hot hadronic matter. Accepted Found. of Phys.
• L. Burakovsky and L.P. Horwitz. Manifestly covariant relativistic thermodynamics and avoidance of gravitational singularities. Submitted.
• L. Burakovsky and L.P. Horwitz. $5D$ Inflationary Cosmology. Accepted Gen. Rel. and Gravitation.
• L.C. Biedenharn and L.P. Horwitz (IAS). Quarks in the Skyrme-'t Hooft-Witten model. Zeits. f. Phys. C 65, 551 (1995).
• L.P. Horwitz and E. Eisenberg. Lax-Phillips theory and the unstable system in quantum mechanics. Proceedings of conference on the Quantum-Classical Correspondence, Drexel University 8 Sept. 1994.
• M. Land and L.P. Horwitz. Off-shell quantum electrodynamics. Submitted.
• M.C. Land and L.P. Horwitz. Relativistic Zeeman effect. J. Phys. A:Math and Gen 28, 3289 (1995).
• J. Frastai and L.P. Horwitz. Off-shell fields and Pauli-Villars regularization. Accepted, Foundations of Physics.
• L. Burakovsky and L.P. Horwitz. Generalized Boltzmann equation in a manifestly covariant relativistic statistical mechanics. Accepted, Foundations of Physics.
• Y. Ashkenazy, L.P. Horwitz, J. Levitan and M. Lewkowitch. Chaotic-like behavior of a quantum system without classical limit. Phys. Rev. Lett. 75, 1070 (1995).
• Y. Ashkenazy, R. Berkovits, L.P. Horwitz and J. Levitan. Spectral analysis of a quantum system without classical limit. Submitted.
• L. Burakovsky, L.P. Horwitz and W.C. Schieve. Towards a realistic equation of state of strongly interacting matter. Accepted, Found. of Phys. Lett.
• L. Burakovsky, L.P. Horwitz and W.C. Schieve. On relativistic Bose-Einstein condensation. Submitted.
• B. Sarel and L.P. Horwitz. A chiral spin theory in the framework of an invariant evolution parameter formalism. Submitted.
• L. Burakovsky and L.P. Horwitz. Galilean limit of equilibrium mass distribution. J. Phys. A:Math. and Gen. 27, 2623 (1994).
• L. Burakovsky and L.P. Horwitz. Galilean limit of equilibrium mass distribution for indistinguishable events. J. Phys. A:Math and Gen. 27, 4725 (1994).
• L.P. Horwitz and P. Leifer. The semiclassical limit from the geometry of the projective state space. Submitted.
• O. Pelc and L.P. Horwitz. Generalization of the Coleman-Mandula Theorem to higher dimensions. Submitted.
• O. Pelc and L.P. Horwitz. Construction of a complete set of states in relativistic scattering theory. Submitted.
• L.P. Horwitz. Hypercomplex quantum mechanics. To be published in Foundations of Physics, issues in honor fo Max Jammer (invited).
• Y. Ashkenazy, L.P. Horwitz and J. Levitan. Complexity, tunneling and geometrical symmetry. Submitted.

### Recent Abstracts

1. L. P. Horwitz. The unstable system in relativistic quantum mechanics Found. Phys. 25, 39 (1995)

The recently developed quantum theory utilizing the ideas and results of Lax and Phillips for the description of scattering and resonances, or unstable systems, is reviewed. The framework for the construction of the Lax-Phillips theory is given by a functional space which is the direct integral over time of the usual quantum mechanical Hilbert spaces, defined at each $t$. It has been shown that quantum scattering theory can be formulated in this way. The theory of Lax and Phillips, however, also obtains a simple relation between the poles of the $S$-matrix and the spectrum of the generator of the semigroup corresponding to the reduced motion of the resonant state. It is shown in this work that in order to obtain such a relation in the quantum mechanical case, the evolution operator must act as an integral operator on the time variable. The structure required appears naturally in the Liouville space formulation of the evolution of the state of the system. The resulting $S$ matrix is a function, as an integral operator, of $t-t'$ (i.e., homogeneous), and the semigroup is contractive. A physical interpretation of this structure may be introduced, from which we obtain a quantitative description of the expected age of a created system, and the expected time of decay of an unstable system. The superselection rule which distinguishes between the unstable system and its decay products is realized in this way. It is also shown that from this point of view, one has a natural mechanism for the dynamical mixing of the quantum mechanical states as observed by means of time translation invariant operators. In particular, this provides a model for certain types of irreversible processes, as well as for the measurement process for closed as well as open systems.

2. L.P. Horwitz. A soluble model for scattering and decay in quaternionic quantum mechanics I: Decay. Jour. Math. Phys. 35, 2743-2760 (1994)

The Lee-Friedrichs model has been very useful in the study of decay-scattering systems in the framework of complex quantum mechanics. Since it is exactly soluble, the analytic structure of the amplitudes can be explicitly studied. It is shown in this paper that a similar model, which is also exactly soluble, can be constructed in quaternionic quantum mechanics. The problem of the decay of an unstable system is treated here. The use of the Laplace transform, involving quaternion-valued analytic functions of a variable with values in a complex subalgebra of the quaternion algebra, makes the analytic properties of the solution apparent; some analysis is given of the dominating structure in the analytic continuation to the lower half plane. A study of the corresponding scattering system will be given in a succeeding paper.

3. J. Frastai and L.P. Horwitz. Off-shell fields and Pauli-Villars regularization. Accepted, Foundations of Physics

We analyze the correspondence between a five dimensional U(1) gauge invariant theory and four dimensional scalar QED, where the fifth dimension $(\tau)$ is an invariant parameter of evolution of the manifestly covariant one particle sector as well as for the full Fock space. The correspondence is represented by the limit in which the width of the photon mass distribution $\Delta s$ tends to zero and large $\tau$ correlations occur. In the limiting procedure, calculation of a two point diagram shows that the Pauli-Villars regularization is intrinsically related to the these long range correlations.

### Visiting Appointments:

• University of Geneva, University of Texas at Austin, Institute for Advanced Study (Princeton, N.J.). Each for visits during several years.
• Extended one-time visits: ETH, Zurich; Institute des Hautes Etudes Scientifique, Bures/Yvette, France; Syracuse University.

### Patents

• About 25 patents assigned to the IBM Corporation on pattern recognition, computer design and superconducting elements.

### Prizes

• Samuel F.B. Morse Medal 6.52

### Participation in recent conference organizing committees

• Conference in honor of Max Jammer, Bar Ilan University, 1995.