Gauge invariant quantum gravitational decoherence Teodora Oniga Department of Physics, University of Aberdeen BritGrav 15, Birmingham, 21 April 2015
Outline Open quantum systems have so far been successfully applied in the description of quantum vacuum effects on matter (mainly EM interactions). Zero-point quantum gravity is expected to have analogous effects. The theory of open quantum systems can be used as an approach to quantum gravity phenomenology. Coupling between low-energy gravitons and general matter master equation for open quantum gravitational systems with bosonic matter.
Open quantum systems Consider a total system made up of a system S of interest surrounded by and interacting with an environment treated as a reservoir R. Only the reduced dynamics of the system S are considered (the reservoir degrees of freedom are traced out). This theory has provided descriptions of quantum optics, quantum Brownian motion and quantum information. Has also been used to study decoherence phenomena.
Open quantum gravitational systems In addition to quantum superposition, particles in a box open to spacetime fluctuations develop into a statistical mixture of states through entanglement with environmental gravitons.
Gravitational decoherence The existence of gravitational fields will induce decoherence in quantum systems. There have been growing recent interest in quantum gravity phenomenology including decoherence. [Refs later.] I will consider low-energy quantum gravity effects where a model independent quantization of linearized general relativity is expected to be valid. A gauge invariant approach is adopted.
Dirac quantization of linearized gravity The interaction of a general bosonic matter field with a weak gravitational field is considered: Dirac s constraint quantization used to ensure gauge invariance. We use the ADM formalism for canonical gravity, where the matter interaction Hamiltonian density is given by:
Dirac quantization of linearized gravity In linearized gravity: From the reservoir Hamiltonian, given by the GR Hamiltonian in linearized gravity, we obtain the constraint functions: where is the metric perturbation and is the canonical conjugate momentum. These constraints are not put to zero so that gauge transformation can be generated after quantization.
Gravitational gauge transformation In linearized GR, gauge transformations are induced by infinitesimal coordinate transformations i.e. diffeomorphisms. Under a gauge transformation, the constraints act as generators: The gauge transformations for both gravity and matter are then generated by: where
Gravitational gauge transformation Under quantization, Poisson brackets become commutators, thus we obtain the following commutation relations: For linearized gravity in Heisenberg picture: Up to this point, no gauge has been fixed.
Gravitational gauge transformation We can reduce the degrees of freedom by applying the Lorenz gauge condition as operator equation: with Then we can expand: This leads to the Hamiltonian density in the following form, upon applying the TT gauge:
Density matrix description A density matrix description is for decoherence A mixed state (statistical ensemble) density matrix is given by: For two interacting systems A and B, the partial trace can be taken to obtain the reduced density matrix of system A: The off-diagonal terms of the density matrix represent coherence between different states.
The influence functional Starting from the Liouville von Neumann equation for an open quantum system: with and from the hamiltonian density obtained from the previous calculation: by tracing out the environmental degrees of freedom, we obtain:
The influence functional With a weak-coupling assumption, after a number of steps, we obtain a phase influence functional of the form: with:
Master equation To study decoherence, we take a leading approximation to the influence functional method using the master equation. In this case, by applying the Markov assumption, we obtain the following closed form of the master equation:
Applications This theory can be applied to a generic bosonic field such as cold atoms, BECs and starlight. Further inclusion of fermionic matter is possible. The particle number conservation from the master equation can be readily applied in a relatively simple study of decoherence for a single particle in a box. It can be massive/massless scalar particle in relativistic or non-relativistic motion.
References T. Oniga, C. Wang, Open quantum gravitational systems, in preparation (2015) C. Wang, R. Bingham, J. Mendonca. Quantum gravitational decoherence of matter waves, Classical and Quantum Gravity 23, L59 (2006) C. Anastopoulos, B. Hu. A master equation for gravitational decoherence: probing the textures of spacetime, Classical and Quantum Gravity 30, 165007 (2013) H. Breuer, F. Petruccione. The theory of open quantum systems, Oxford University Press (2002)
Acknowledgements