Product of Representations and Further Physical Applications

Franco Strocchi

Francisco Marcelo Fernández

The Schrödinger equation with a time-dependent quadratic Hamiltonian is investigated. The time-evolution operator is written as a product of exponential operators determined by the Heisenberg equations of motion. This product operator is shown to be global in the occupation number representation when the Hamiltonian is Hermitian. The success of some physical applications of the product-form representation is explained.

Alan D. Rendall

It is shown that if a representation of a *-algebra on a vector space V is an irreducible *-representation with respect to some inner product on V, then under appropriate technical conditions this property determines the inner product uniquely up to a constant factor. Ashtekar(1991) has suggested using the condition that a given representation of the algebra of quantum observables is a *-representation to fix the inner product on the space of physical states. This idea is of particular interest for the quantization of gravity where an obvious prescription for defining an inner product is lacking. The results of this paper show rigorously that Ashtekar's criterion does suffice to determine the inner product in very general circumstances. Two versions of the result are proved: a simpler one which only applies to representations by bounded operators and a more general one which allows for unbounded operators. Some concrete examples are worked out in order to illustrate the meaning and range of applicability of the general theorems.

László Lovász

R. Aldrovandi

Markus Bachmayr, Michael Götte, Pfeffer, Max

The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to generalize to other classes of operators. Imposing block sparsity yields a scheme for conserving the particle number that is commonly used in applications in physics. Operations on such block structures, their rank truncation, and implications for numerical algorithms are discussed. Explicit and rank-reduced matrix product operator representations of one- and two-particle operators are constructed that operate only on the non-zero blocks of matrix product states.