Zeno dynamics and constraints

Abstract

Abstract. We investigate some examples of quantum Zeno dynamics, when a system undergoes very frequent (projective) measurements that ascertain whether it is within a given spatial region. In agreement with previously obtained results, the evolution is found to be unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. By using a new approach to this problem, this result is found to be valid in an arbitrary N -dimensional compact domain. We then propose some preliminary ideas concerning the algebra of observables in the projected region and finally look at the case of a projection onto a lower dimensional space: in such a situation the Zeno ansatz turns out to be a procedure to impose constraints.

The Geometry of the Quantum Zeno Effect

“…Beppe has been for a long time an outstanding advocate of such a geometric understanding. On many different occasions, we have discussed with him the basic notions of the quantum Zeno effect [2][3][4]. A geometrical picture of such a fundamental quantum mechanical effect was proposed by Beppe at the conference for the 75th birthday of George Sudarshan [5], who fathered the quantum Zeno effect (QZE) in a seminal article written in collaboration with Baidyanath Misra [6].…”

Zeno Product Formula Revisited

“…The effect was analyzed mathematically by Friedman [12] but became popular only after the authors of [21] invented the above stated name. Recent interest is motivated mainly by the fact that now the effect is within experimental reach; an up-todate bibliography can be found, e.g., in [10] or [22]. The physical interpretation of this formula can be given in the context of particle decay, cf.…”

Sufficient conditions for the anti-Zeno effect

“…For the first time it was formulated explicitly in this context by Beskow and Nilsson [3] and soon after a mathematical analysis [5,14] revealed sufficient conditions under which it exists; it became truly popular after the authors of [20] coined its present name. Recently the effect attracted a new wave of mathematical [8,9,19,22] and physical [11,10,12,13,16,18] interest; in the mentioned papers one can find a more complete bibliography.Although the opposite situation, in which a frequent measurement can on the contrary speed up the decay, or ideally to lead to an immediate disappearance of the unstable system, was also noticed early [6], it attracted attention only recently -see, e.g., [1,2,17,21] and also [22] and references therein. As in the case of the Zeno effect, the problem can be tackled from two points of view.…”

“…For the first time it was formulated explicitly in this context by Beskow and Nilsson [3] and soon after a mathematical analysis [5,14] revealed sufficient conditions under which it exists; it became truly popular after the authors of [20] coined its present name. Recently the effect attracted a new wave of mathematical [8,9,19,22] and physical [11,10,12,13,16,18] interest; in the mentioned papers one can find a more complete bibliography.…”

“…One might expect that the sufficient condition is given by (11) again but this guess is wrong. To find the answer, we denote by {χ j } an orthonormal basis in P H; it allows to write the second term in (5) as…”