1995 | SERGIO DOPlicher, KLAUS FREDENHAGEN, JOHN E. ROBERTS
This paper proposes a model of Quantum Spacetime based on uncertainty relations derived from Heisenberg's principle and Einstein's theory of classical gravity. The authors introduce a non-commutative algebra to describe spacetime, where the commutative algebra of functions on spacetime is replaced by a non-commutative algebra. This model replaces local interactions with nonlocal effective interactions in Minkowski space. The quantum nature of spacetime leads to uncertainty relations between the coordinates of spacetime events, which are essential for the model.
The authors derive algebraic relations that imply these uncertainty relations, which have the form of commutators and antisymmetric tensors. These relations lead to a quantum spacetime structure that, in the classical limit, reduces to Minkowski space times a two-component space homeomorphic to the tangent bundle of the 2-sphere. The model is connected to Connes' theory of the standard model, suggesting potential applications in particle physics.
The paper discusses the definition of free fields and interactions over Quantum Spacetime, and outlines the first steps towards quantum field theory on quantum spacetime. It shows that the commutator of free fields at spacelike distances decreases like a Gaussian, and provides a formal recipe for defining interaction Hamiltonians and the perturbative expansion of the S-matrix.
The authors also explore the implications of the quantum structure of spacetime on the classical limit, where the Planck length approaches zero. In this limit, the quantum spacetime reduces to Minkowski space times a two-component space. The paper also discusses the role of the manifold Σ, which is related to the quantum structure of spacetime and plays a key role in the uncertainty relations.
The paper concludes by discussing the implications of the quantum structure of spacetime for quantum field theory and the potential for new physical phenomena. It also notes that the model is a semiclassical approximation to a theory where gravity and quantum physics are truly unified. The authors suggest that further research is needed to fully understand the implications of the quantum structure of spacetime and its potential applications in physics.This paper proposes a model of Quantum Spacetime based on uncertainty relations derived from Heisenberg's principle and Einstein's theory of classical gravity. The authors introduce a non-commutative algebra to describe spacetime, where the commutative algebra of functions on spacetime is replaced by a non-commutative algebra. This model replaces local interactions with nonlocal effective interactions in Minkowski space. The quantum nature of spacetime leads to uncertainty relations between the coordinates of spacetime events, which are essential for the model.
The authors derive algebraic relations that imply these uncertainty relations, which have the form of commutators and antisymmetric tensors. These relations lead to a quantum spacetime structure that, in the classical limit, reduces to Minkowski space times a two-component space homeomorphic to the tangent bundle of the 2-sphere. The model is connected to Connes' theory of the standard model, suggesting potential applications in particle physics.
The paper discusses the definition of free fields and interactions over Quantum Spacetime, and outlines the first steps towards quantum field theory on quantum spacetime. It shows that the commutator of free fields at spacelike distances decreases like a Gaussian, and provides a formal recipe for defining interaction Hamiltonians and the perturbative expansion of the S-matrix.
The authors also explore the implications of the quantum structure of spacetime on the classical limit, where the Planck length approaches zero. In this limit, the quantum spacetime reduces to Minkowski space times a two-component space. The paper also discusses the role of the manifold Σ, which is related to the quantum structure of spacetime and plays a key role in the uncertainty relations.
The paper concludes by discussing the implications of the quantum structure of spacetime for quantum field theory and the potential for new physical phenomena. It also notes that the model is a semiclassical approximation to a theory where gravity and quantum physics are truly unified. The authors suggest that further research is needed to fully understand the implications of the quantum structure of spacetime and its potential applications in physics.