On the equivalence principle in quantum theory

Richard Hughes

The Bohr–Einstein debate over the ‘‘weighing of energy’’ and the validity of the time–energy uncertainty relation is reexamined in the context of gravitation theories that do not respect the equivalence principle. Bohr’s use of the equivalence principle is shown to be sufficient, but not necessary, to establish the validity of this uncertainty relation in Einstein’s ‘‘weighing-of-energy’’ gedanken experiment. The uncertainty relation is shown to hold in any energy-conserving theory of gravity, and so a failure of the equivalence principle does not engender a failure of quantum mechanics. The relationship between the gravitational redshift and the equivalence principle is reviewed.

C. JAYARATNAM ELIEZER, P. G. L. Leach

Alon E. Faraggi, Marco Matone

Sebastiano Segreto, Matteo Bruno

We analyze the issue of unitary equivalence within Generalized Uncertainty Principle (GUP) theories in the one-dimensional case. For a deformed Heisenberg algebra, its representation in terms of Hilbert space and conjugate operators is not uniquely determined, raising the question of whether different realizations of the same algebra are equivalent and describe the same physics. After proposing a definition of a quantum GUP theory, we establish conditions for unitary equivalence. Using this framework, we rigorously prove that two commonly used representations are unitarily equivalent, specifying the conditions under which this equivalence holds. We demonstrate this equivalence explicitly by providing a unitary map and showing how both GUP formulations yield the same physical results in two examples: the quantum harmonic oscillator and a free-falling particle. Finally, we discuss a case where equivalence fails, suggesting that a generalization of the Stone-von Neumann theorem may not be possible within the GUP framework under our definition of unitary equivalence.

Cláus Lämmerzahl

The role of the equivalence principle in the context of non-relativistic quantum mechanics and matter wave interferometry, especially atom beam interferometry, will be discussed. A generalised form of the weak equivalence principle which is capable of covering quantum phenomena too, will be proposed. It is shown that this generalised equivalence principle is valid for matter wave interferometry and for the dynamics of expectation values. In addition, the use of this equivalence principle makes it possible to determine the structure of the interaction of quantum systems with gravitational and inertial fields. It is also shown that the path of the mean value of the position operator in the case of gravitational interaction does fulfill this generalised equivalence principle.

Daniel M. Greenberger