Hilbert space structure of covariant loop quantum gravity
Open accessAbstract
We investigate Hilbert space in the Lorentz covariant approach to loop quantum gravity. We restrict ourselves to the space where all area operators are simultaneously diagonalizable, assuming that it exists. In this sector, quantum states are realized by a generalization of spin network states based on Lorentz Wilson lines projected on irreducible representations of an SO(3) subgroup. The problem of infinite dimensionality of the unitary Lorentz representations is absent due to this projection. Nevertheless, the projection preserves the Lorentz covariance of the Wilson lines so that the symmetry is not broken. Under certain conditions, the states can be thought of as functions on a homogeneous space. We define the inner product as an integral over this space. With respect to this inner product, the spin networks form an orthonormal basis in the investigated sector. We argue that it is the only relevant part of a larger state space arising in the approach. The problem of the noncommutativity of the Lorentz connection is solved by restriction to the simple representations. The resulting structure shows similarities with the spin foam approach.
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- Authors
- Alexandrov, S.
- Affiliations
- École Normale Supérieure - PSL (FR)
- Corresponding author
- Alexandrov, Sergei
- ORCID
- Alexandrov, Sergei
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2002-07-22
- Volume
- 66
- Issue
- 2
- Pages
- 024028
- ISSN
- 0556-2821 (ISSN-L)
- 1089-4918
- OpenAlex
- W2134752045
- Cited by
- 42
- References
- 36
- Citations by year
- 2021: 1 · 2018: 1 · 2017: 1 · 2016: 3 · 2015: 3 · 2014: 1 · 2013: 6 · 2012: 3
- FWCI
- 1.66
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- loop quantum gravity, Lorentz covariance, Hilbert space structure, spin network states, Wilson lines, irreducible representations, homogeneous spaces, orthonormal bases, area operator, spin foams
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