Projection operators are central to the algebraic formulation of quantum theory because both wavefunction and hermitian operators(observables) have spectral decomposition in terms of the spectral projections. Projection operators are hermitian operators which are idempotents also. We call them quantum idempotents. They are also important for the conceptual understanding of quantum theory because projection operators also represent observation process on quantum system. In this paper we explore the algebra of quantum idempotents and show that they generate Iterant algebra (defined in the paper), Lie algebra, Grassmann algebra and Clifford algebra which is very interesting because these later algebras were introduced for the geometry of spaces and hence are called geometric algebras. Thus the projection operator representation gives a new meaning to these geometric algebras in that they are also underlying algebras of quantum processes and also they bring geometry closer to the quantum theory. It should be noted that projection operators not only make lattices of quantum logic but they also span projective geometry. We will give iterant representations of framed braid group algebras, parafermion algebras and the $su(3)$ algebra of quarks. These representations are very striking because iterant algebra encodes the spatial and temporal aspects of recursive processes. In that regard our representation of these algebras for physics opens up entirely new perspectives of looking at fermions,spins and parafermions(anyons).Projection operators are central to the algebraic formulation of quantum theory because both wavefunction and hermitian operators(observables) have spectral decomposition in terms of the spectral projections. Projection operators are hermitian operators which are idempotents also. We call them quantum idempotents. They are also important for the conceptual understanding of quantum theory because projection operators also represent observation process on quantum system. In this paper we explore the algebra of quantum idempotents and show that they generate Iterant algebra (defined in the paper), Lie algebra, Grassmann algebra and Clifford algebra which is very interesting because these later algebras were introduced for the geometry of spaces and hence are called geometric algebras. Thus the projection operator representation gives a new meaning to these geometric algebras in that they are also underlying algebras of quantum processes and also they bring geometry closer to the quantum theory. It should be noted that projection operators not only make lattices of quantum logic but they also span projective geometry. We will give iterant representations of framed braid group algebras, parafermion algebras and the $su(3)$ algebra of quarks. These representations are very striking because iterant algebra encodes the spatial and temporal aspects of recursive processes. In that regard our representation of these algebras for physics opens up entirely new perspectives of looking at fermions,spins and parafermions(anyons).

arXiv:1705.06600May 17, 20172 citations

Algebraic Approach to Quantum Theory

Kasia Rejzner

Mathematical physics studiesJan 1, 20161 citation

Some basic concepts of algebraic quantum theory

J. E. Roberts, Gert Roepstorff

Communications in Mathematical PhysicsDec 1, 196951 citations

Operator Algebras in Quantum Computation

Mathys Rennela

In this master thesis, I discuss how the theory of operator algebras, also called operator theory, can be applied in quantum computer science.

arXiv:1510.06649Oct 22, 20158 citations

Operational Quantum Theory I

Heinrich Saller

Springer eBooksJan 1, 200619 citations

Quantum measurements as Type-I approximations to Type-III von Neumann algebras

Blind Review

In an algebraic understanding of holographic gravity, a major question is: how can a boundary theory with Type-III von Neumann algebras of local observables (where density operators and entropy are ill-defined) be equivalent to a bulk theory with Type-I von Neumann algebras of local observables (where density operators and entropy are well-defined)? Replica wormholes and problems like the factorization problem suggest that instead of full equivalence, there is an approximation. When approximate equality breaks down, corrections to bulk theory predictions are required, just as in replica wormholes. Quantum measurements are then described as operations that can only be described as identity operations in the bulk theory but are non-trivial in the boundary theory. The breakdown of approximate equality between the boundary theory and the bulk theory generates illusions of wavefunction collapse, when there actually is no wavefunction collapse in the boundary theory with Type-III von Neumann algebras.In an algebraic understanding of holographic gravity, a major question is: how can a boundary theory with Type-III von Neumann algebras of local observables (where density operators and entropy are ill-defined) be equivalent to a bulk theory with Type-I von Neumann algebras of local observables (where density operators and entropy are well-defined)? Replica wormholes and problems like the factorization problem suggest that instead of full equivalence, there is an approximation. When approximate equality breaks down, corrections to bulk theory predictions are required, just as in replica wormholes. Quantum measurements are then described as operations that can only be described as identity operations in the bulk theory but are non-trivial in the boundary theory. The breakdown of approximate equality between the boundary theory and the bulk theory generates illusions of wavefunction collapse, when there actually is no wavefunction collapse in the boundary theory with Type-III von Neumann algebras.

Aug 17, 20230 citations

su(3)

algebra of quarks. These representations are very striking because iterant algebra encodes the spatial and temporal aspects of recursive processes. In that regard our representation of these algebras for physics opens up entirely new perspectives of looking at fermions,spins and parafermions(anyons).