The Leggett-Garg inequality and Page-Wootters mechanism

Abstract

Violation of the Leggett-Garg inequality (LGI) implies quantum phenomena. In this light we establish that the Moreva et al. [21] experiment demonstrating the Page-Wootter's mechanism [3] falls in the quantum domain. An observer outside a 2-photons world does not detect any change in the 2−photons state,i.e. there is no time parameter for the outside observer. But an observer attached to one of the photons sees the other photon evolving and this means there is an "internal" time. The LGI is violated for the clock photon whose state evolves with the internal time as measured by the system photon. Conditional probabilities in this 2-photons system are computed for both sharp and unsharp measurements. The conditional probability increases for entangled states as obtained by Page and Wootters for both ideal and also unsharp measurements. We discuss how the conditional probabilities can be used to distinguish between massless and massive gravitons. This is important in the context of gravitational waves.1.Introduction-Field theories describing gauge particles are gauge theories and are invariant with respect to some local symmetry group transformations. In Yang-Mills theories these are the non-abelian gauge transformations corresponding to SU (N ) groups. For gravity these are space-time diffeomorphisms. The invariance corresponds to local Lorentz invariance. In gauge theories different solutions arising from same initial conditions become related by the local symmetry. So the general solution of the field equations is non-unique as it contains arbitrary time-dependent functions. So a subset of initial conditions needs to be chosen. This subset is defined by what are called the lagrangian constraints. In the hamiltonian formalism this implies conditions on the allowed initial positions and momenta. Time evolution must conserve these conditions. This can lead to further constraints. A first class constraint is a dynamical quantity in a constrained hamiltonian system whose Poisson bracket with all other constraints vanishes on the constraint surface in phase space. Any constraint which is not first class is a second class constraint. Local symmetry transformations are generated by the first class constraints. So gauge theories are systems with first class constraints.An early attempt to quantise gravity was through the Wheeler-DeWitt (WD) equation [1]. This results from the canonical quantisation of Einstein gravity using Dirac's constrained formalism [2]. An embodiment of the quantum version of the hamiltonian constraint using metric variables gives the WD equation. But this equation is time-independent. All observables are constant and the resulting universe is unchanging and boring.Page and Wootters [3] proposed that the static universe (as observed by an external observer) actually evolves with respect to time as seen by some internal observer within the universe. This is because of quantum correlations (entanglement ) between different con-stituents within the universe. Consider a direct product of two different sta...

Quantum time: Experimental multitime correlations

“…We show that one obtains the correct quantum correlations and we use this to test the Leggett-Garg inequality [14] (which, in this framework, was suggested in [15]). One of the main criticisms of the PW mechanism was that it seemed unable to provide the correct two-time correlations [7,8,16].…”

“…Since we can access arbitrary two-time correlation measurements, we can show that our experiment has a quantum behavior through a violation of the LeggettGarg inequality [14,15]. This inequality is constructed using an observable Q(t) of a two-level physical system, the photon's polarization in our case [18], with |H , |V corresponding to Q = +1, −1 respectively.…”

Three-flavoured neutrino oscillations and the Leggett–Garg inequality

“…A wide range of various quantum system violates the upper bound of LGI and that let the LGI to use it to probe the quantum mechanics (QM) in the macroscopic regime [12][13][14][15][16][17][18][19][20][21][22][23][24]. A detailed review about LGI can be found in [25].…”