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Causality in the Modern Approach to Foundations of Quantum Field Theory

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Causality in the Modern Approach to Foundations of Quantum Field Theory Causality in the modern approach to foundations of quantum field theory Kasia Rejzner 01.06.2015 University of York Causality in the modern approach to foundations of quantum field theory 1 / 40 N Outline of the talk 1 Introduction 2 Preliminaries 3 Spacetime geometry 4 AQFT 5 Entanglement 6 QFT on curved spacetimes Causality in the modern approach to foundations of quantum field theory 2 / 40 N 1 Introduction 2 Preliminaries 3 Spacetime geometry 4 AQFT 5 Entanglement 6 QFT on curved spacetimes Causality in the modern approach to foundations of quantum field theory 3 / 40 N Introduction What is AQFT? QFT, PARTI- COSMOLOGY CLE PHYSICS Renor- KMS states malization QFT on curved spacetime Defor- mations PAQFT QUANTUM STATISTICAL MECHANICS MODEL BUILDING QIT CFT in 2D QFT ON NONCO- MUTATIVE AQFT NCG SPACETIMES ALGEBRAIC QIT CONSTRUC- TIVE QFT Does it contradict the causality? Can the empty space be entangled? How to generalize these notions to curved spactimes? (black holes, early Universe) Introduction Questions for today: What does the entanglement mean in QFT? Causality in the modern approach to foundations of quantum field theory 5 / 40 N Can the empty space be entangled? How to generalize these notions to curved spactimes? (black holes, early Universe) Introduction Questions for today: What does the entanglement mean in QFT? Does it contradict the causality? Causality in the modern approach to foundations of quantum field theory 5 / 40 N How to generalize these notions to curved spactimes? (black holes, early Universe) Introduction Questions for today: What does the entanglement mean in QFT? Does it contradict the causality? Can the empty space be entangled? Causality in the modern approach to foundations of quantum field theory 5 / 40 N Introduction Questions for today: What does the entanglement mean in QFT? Does it contradict the causality? Can the empty space be entangled? How to generalize these notions to curved spactimes? (black holes, early Universe) Causality in the modern approach to foundations of quantum field theory 5 / 40 N 1 Introduction 2 Preliminaries 3 Spacetime geometry 4 AQFT 5 Entanglement 6 QFT on curved spacetimes Causality in the modern approach to foundations of quantum field theory 6 / 40 N Preliminaries Scales in the Universe Causality in the modern approach to foundations of quantum field theory 7 / 40 N Special relativity is a theory proposed in 1905 by Albert Einstein in the paper On the Electrodynamics of Moving Bodies. As the name of the paper suggest, the motivation was to make Electrodynamics compatible with Mechanics. This turned out to be impossible within Newton’s theory. Preliminaries Physics at high velocities When matter moves at high velocities (close to the velocity of light), special relativity starts to play a role. Causality in the modern approach to foundations of quantum field theory 8 / 40 N As the name of the paper suggest, the motivation was to make Electrodynamics compatible with Mechanics. This turned out to be impossible within Newton’s theory. Preliminaries Physics at high velocities When matter moves at high velocities (close to the velocity of light), special relativity starts to play a role. Special relativity is a theory proposed in 1905 by Albert Einstein in the paper On the Electrodynamics of Moving Bodies. Causality in the modern approach to foundations of quantum field theory 8 / 40 N Preliminaries Physics at high velocities When matter moves at high velocities (close to the velocity of light), special relativity starts to play a role. Special relativity is a theory proposed in 1905 by Albert Einstein in the paper On the Electrodynamics of Moving Bodies. As the name of the paper suggest, the motivation was to make Electrodynamics compatible with Mechanics. This turned out to be impossible within Newton’s theory. Causality in the modern approach to foundations of quantum field theory 8 / 40 N The mathematical foundations of this theory were systematically formulated by Born, Heisenberg and Jordan in late 1925 in the famous Dreimännerarbeit. The new mathematical framework proposed in this work, “matrix mechanics”, is known today as operator-algebraic approach to QM. Independently, in 1926, Erwin Schrödinger proposed another approach to QM, the “wave mechanics”, which later became much more popular than its predecessor. Preliminaries What about quantum? At small scales the behavior of matter is governed by Quantum Mechanics. Causality in the modern approach to foundations of quantum field theory 9 / 40 N The new mathematical framework proposed in this work, “matrix mechanics”, is known today as operator-algebraic approach to QM. Independently, in 1926, Erwin Schrödinger proposed another approach to QM, the “wave mechanics”, which later became much more popular than its predecessor. Preliminaries What about quantum? At small scales the behavior of matter is governed by Quantum Mechanics. The mathematical foundations of this theory were systematically formulated by Born, Heisenberg and Jordan in late 1925 in the famous Dreimännerarbeit. Causality in the modern approach to foundations of quantum field theory 9 / 40 N Independently, in 1926, Erwin Schrödinger proposed another approach to QM, the “wave mechanics”, which later became much more popular than its predecessor. Preliminaries What about quantum? At small scales the behavior of matter is governed by Quantum Mechanics. The mathematical foundations of this theory were systematically formulated by Born, Heisenberg and Jordan in late 1925 in the famous Dreimännerarbeit. The new mathematical framework proposed in this work, “matrix mechanics”, is known today as operator-algebraic approach to QM. Causality in the modern approach to foundations of quantum field theory 9 / 40 N Preliminaries What about quantum? At small scales the behavior of matter is governed by Quantum Mechanics. The mathematical foundations of this theory were systematically formulated by Born, Heisenberg and Jordan in late 1925 in the famous Dreimännerarbeit. The new mathematical framework proposed in this work, “matrix mechanics”, is known today as operator-algebraic approach to QM. Independently, in 1926, Erwin Schrödinger proposed another approach to QM, the “wave mechanics”, which later became much more popular than its predecessor. Causality in the modern approach to foundations of quantum field theory 9 / 40 N Experiments with particle collisions (for example at CERN) can be understood with the use of the scattering theory, In QFT spacetime is fixed, it has no dynamics. Preliminaries Matter at small scales and high ve- locities Universe at small scales (particle physics) is described by quantum field theory (QFT), Causality in the modern approach to foundations of quantum field theory 10 / 40 N In QFT spacetime is fixed, it has no dynamics. Preliminaries Matter at small scales and high ve- locities Universe at small scales (particle physics) is described by quantum field theory (QFT), Experiments with particle collisions (for example at CERN) can be understood with the use of the scattering theory, Causality in the modern approach to foundations of quantum field theory 10 / 40 N Preliminaries Matter at small scales and high ve- locities Universe at small scales (particle physics) is described by quantum field theory (QFT), Experiments with particle collisions (for example at CERN) can be understood with the use of the scattering theory, In QFT spacetime is fixed, it has no dynamics. Causality in the modern approach to foundations of quantum field theory 10 / 40 N 1 Introduction 2 Preliminaries 3 Spacetime geometry 4 AQFT 5 Entanglement 6 QFT on curved spacetimes Causality in the modern approach to foundations of quantum field theory 11 / 40 N Each event (anything that happens) is represented by a point in this diagram. Wether we move or stand still, we can describe our position in space and time by drawing a curve in the spacetime diagram. Spacetime geometry Space and time t What is spacetime? For simplicity assume that the space is one dimensional. We can draw a diagram, where time is flowing along the vertical axis and horizontal axis represents the direction in space. x Causality in the modern approach to foundations of quantum field theory 12 / 40 N Wether we move or stand still, we can describe our position in space and time by drawing a curve in the spacetime diagram. Spacetime geometry Space and time t What is spacetime? For simplicity assume that the space is one dimensional. We can draw a diagram, where time is flowing along the vertical axis and horizontal axis represents the direction in space. Each event (anything that happens) is represented by a point in this diagram. x Causality in the modern approach to foundations of quantum field theory 12 / 40 N Spacetime geometry Space and time t What is spacetime? For simplicity assume that the space is one dimensional. We can draw a diagram, where time is flowing along the vertical axis and horizontal axis represents the direction in space. Each event (anything that happens) is represented by a point in this diagram. Wether we move or stand still, we can describe our position in space and time by x drawing a curve in the spacetime diagram. Causality in the modern approach to foundations of quantum field theory 12 / 40 N On the spacetime diagram, we can draw at each point two lines (a cone) representing jx − x0j = jt − t0j, which limits the region of spacetime accessible from that point. This object is called the lightcone with apex (t0; x0). Spacetime geometry Space and time t The main principle of special relativity says that nothing can move faster than light, so dx cannot be higher than c, the speed of dt light. From now on we choose units in which c = 1. (t0; x0) x Causality in the modern approach to foundations of quantum field theory 13 / 40 N Spacetime geometry Space and time t The main principle of special relativity says that nothing can move faster than light, so dx cannot be higher than c, the speed of dt light.
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