[Paper Review] Semi-Classical Field Theory as Decoherence Free Subspaces

Jaime Varela|arXiv (Cornell University)|Apr 14, 2014

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper proposes that semi-classical field theory emerges as an approximate decoherence-free subspace (DFS) within a finite-dimensional quantum gravity Hilbert space, where unitary evolution of matter fields is preserved despite gravitational decoherence. The key contribution is a framework linking field theory on curved spacetime to quantum computation, showing that apparent non-unitarity arises from environmental decoherence, while the DFS structure ensures effective unitarity for physical observables.

ABSTRACT

We formulate semi-classical field theory as an approximate decoherence-free-subspace of a finite-dimensional quantum-gravity hilbert space. A complementarity construction can be realized as a unitary transformation which changes the decoherence-free-subspace. This can be translated to signify that field theory on a global slice, in certain space-times, is the simultaneous examination of two different superselected sectors of a field theory. We posit that a correct course graining procedure of quantum gravity should be WKB states propagating in a curved background in which particles exiting a horizon have imaginary components to their phases. The field theory appears non-unitary, but it is due to the existence of approximate decoherence free sub-spaces. Furthermore, the importance of operator spaces in the course-graining procedure is discussed. We also briefly touch on Firewalls.

Motivation & Objective

• To reconcile the infinite-dimensional Fock spaces of quantum field theory with the finite-dimensional Hilbert space of quantum gravity.
• To explain how semi-classical field theory can emerge as an effective, approximately unitary dynamics within a larger quantum-gravitational system.
• To formalize the idea that field theory on a global spacelike slice describes two complementary superselection sectors of a single quantum gravity theory.
• To establish a connection between quantum computing simulations and the emergence of effective field theory in curved spacetime.
• To clarify the role of operator algebras and coarse-graining in defining physical observables in quantum gravity.

Proposed method

• Model the total Hilbert space as a tensor product of an environmental sector (gravitational degrees of freedom) and a system sector (matter fields), with the system Hilbert space further decomposed into a decoherence-free subspace (DFS) and its orthogonal complement.
• Define a coarse-graining map S that maps the full quantum gravity dynamics to a semi-classical field theory on a fixed curved background, with the DFS corresponding to the Fock space of the effective field theory.
• Use the condition that density matrices in the DFS evolve unitarily up to O(ε) corrections under system Hamiltonian evolution, even when the full system-environment dynamics is non-unitary.
• Apply the theory of decoherence-free subspaces via the requirement that error generators (Fα) have degenerate eigenvalues on the DFS, ensuring robustness against environmental interactions.
• Utilize the operator algebra O(H) of the system to enable Taylor expansions in creation/annihilation operators, enabling the emergence of bosonic field-like behavior from finite-dimensional spin-like systems.
• Introduce a time-constant measure αj to quantify non-unitarity, defining DFS as states with max|αj| below a threshold, with exact DFS corresponding to lim αj → 0.

Experimental results

Research questions

• RQ1 How can semi-classical field theory, which has an infinite-dimensional Fock space, emerge from a finite-dimensional quantum gravity Hilbert space?
• RQ2 What is the precise dynamical mechanism by which unitary field theory evolution is preserved in the presence of gravitational decoherence?
• RQ3 How does complementarity in black hole physics relate to the existence of multiple, unitarily inequivalent but physically equivalent DFSs in quantum gravity?
• RQ4 What role do operator algebras and their algebraic structures (e.g., Lie algebras) play in defining the emergence of field-theoretic degrees of freedom from a finite-dimensional quantum gravity framework?
• RQ5 How can the coarse-graining procedure in quantum gravity be formulated such that it preserves effective unitarity in the emergent field theory?

Key findings

• Semi-classical field theory arises as an approximate decoherence-free subspace (DFS) within a finite-dimensional quantum gravity Hilbert space, ensuring effective unitary evolution for matter fields.
• The correspondence Hs → HFock holds when the operator algebra of the system Hilbert space allows Taylor expansions in terms of creation and annihilation operators, enabling field-theoretic behavior.
• Complementarity in black hole physics is interpreted as a unitary transformation between different DFSs, corresponding to different superselection sectors of the same underlying quantum gravity theory.
• The breakdown of unitarity in effective field theory is not fundamental but arises from environmental decoherence, with the DFS structure protecting physical observables from decoherence-induced errors.
• The existence of approximate DFSs is determined by the degeneracy of error generators (Fα) acting on the system, with exact DFSs corresponding to singlet states under a semi-simple Lie algebra of error generators.
• A time-constant measure αj quantifies non-unitary decay; DFSs are defined as states with max|αj| below a threshold, with exact DFSs having lim αj → 0.