The Unruh effect in general boundary QFT Daniele Colosi ENES, Unidad Morelia, UNAM Morelia, 13 June 2018 Contents The Unruh effect GBF and quantum field theory Schrödinger-Feynman quantization Holomorphic quantization GBF in Minkowski and Rindler spacetimes Unruh effect in the GBF Global Unruh effect Local Unruh effect Conclusions Importance: É Relation between the Minkowski vacuum and the notion of particle in Rindler space (naturally associated with an accelerated observer): particle content of a field theory is observer dependent É Relation with the Hawking effect and cosmological horizons É Possible experimental detection Moreover: É Perfect arena for the GBF Unruh effect The Unruh effect states that linearly uniformly accelerated observers perceive the Minkowski vacuum state (i.e. the no-particle state of inertial observers) as a mixed particle state described by a density matrix at a temperature T = , a being the constant acceleration of the observer. 2πkB Moreover: É Perfect arena for the GBF Unruh effect The Unruh effect states that linearly uniformly accelerated observers perceive the Minkowski vacuum state (i.e. the no-particle state of inertial observers) as a mixed particle state described by a density matrix at a temperature T = , a being the constant acceleration of the observer. 2πkB Importance: É Relation between the Minkowski vacuum and the notion of particle in Rindler space (naturally associated with an accelerated observer): particle content of a field theory is observer dependent É Relation with the Hawking effect and cosmological horizons É Possible experimental detection Unruh effect The Unruh effect states that linearly uniformly accelerated observers perceive the Minkowski vacuum state (i.e. the no-particle state of inertial observers) as a mixed particle state described by a density matrix at a temperature T = , a being the constant acceleration of the observer. 2πkB Importance: É Relation between the Minkowski vacuum and the notion of particle in Rindler space (naturally associated with an accelerated observer): particle content of a field theory is observer dependent É Relation with the Hawking effect and cosmological horizons É Possible experimental detection Moreover: É Perfect arena for the GBF ,2 d Rindler R η = const. ρ = const. 2d Minkowski t x η = const. ρ = const. 2d Minkowski,2 d Rindler t R x 2d Minkowski,2 d Rindler t R x η = const. ρ = const. É Particle interpretation: the vacuum state in Minkowski corresponds to an entangled state between the modes of the field defined in the left and right Rindler wedges. É Crispino et al., The Unruh effect and its applications, Rev. Mod. Phys. 80 (2008), 787–838 «the Unruh effect is the equivalence between the Minkowski vacuum and a thermal bath of Rindler particles» É Operational interpretation: A uniformly accelerated Unruh-DeWitt detector responds as if submersed in a thermal bath when interacting with a quantum field in the Minkowski vacuum state. É Operational interpretation: A uniformly accelerated Unruh-DeWitt detector responds as if submersed in a thermal bath when interacting with a quantum field in the Minkowski vacuum state. É Particle interpretation: the vacuum state in Minkowski corresponds to an entangled state between the modes of the field defined in the left and right Rindler wedges. É Crispino et al., The Unruh effect and its applications, Rev. Mod. Phys. 80 (2008), 787–838 «the Unruh effect is the equivalence between the Minkowski vacuum and a thermal bath of Rindler particles» Outline The Unruh effect GBF and quantum field theory Schrödinger-Feynman quantization Holomorphic quantization GBF in Minkowski and Rindler spacetimes Unruh effect in the GBF Global Unruh effect Local Unruh effect Conclusions 2 quantization schemes have been studied, that transform a classical field theory into a general boundary quantum field theory: É Schrödinger-Feynman quantization É holomorphic quantization GBF and QFT Standard QFT can be formulated within the GBF GBF and QFT Standard QFT can be formulated within the GBF 2 quantization schemes have been studied, that transform a classical field theory into a general boundary quantum field theory: É Schrödinger-Feynman quantization É holomorphic quantization Schrödinger-Feynman quantization É Schrödinger representation + Feynman path integral quantization The state space for a hypersurface is the space of functions on Σ Σ field configurationsH K on . Σ Σ É Inner product, Z 2 1 = ' 1(') 2('). h j i K D Σ É Amplitude for a region M, M , 2 H@ Z Z iSM (φ) ρM ( ) = ' (') φe . K M D KM ,φ M =' D @ j@ É A classical observable F in M is modeled as a function on KM . The F quantization of F is the linear map ρ : M defined as M H@ ! C Z Z F iSM (φ) ρM ( ) = ' (') φF(φ)e . K M D KM ,φ M =' D @ j@ Holomorphic quantization Linear field theory: L is the vector space of solutions near the É Σ hypersurface Σ. L carries a non-degenerate symplectic structure and a complex É Σ !Σ structure J : L L compatible with the symplectic structure: Σ Σ ! Σ 2 J = id and ! (J ( ),J ( )) = ! ( , ). Σ − Σ Σ Σ · Σ · Σ · · J and combine to a real inner product g , 2 ,J and to É Σ !Σ Σ( ) = !Σ( Σ ) a complex inner product , g , 2i · · , which· makes· L Σ = Σ( ) + !Σ( ) Σ into a complex Hilbert space.{· ·} · · · · The Hilbert space associated with is the space of holomorphic É Σ Σ functions on L withH the inner product Σ Z 1 , exp g , d , 0 Σ = (φ) 0(φ) Σ(φ φ) µ(φ) h i L −2 Σ where is a (fictitious) translation-invariant measure on L . µ Σ Holomorphic quantization (II) É The amplitude map ρM : M associated with the spacetime H@ ! C region M for a state M is given by 2 H@ Z 1 ρM ( ) = (φ)exp g@ M (φ,φ) dµM˜ (φ). L −4 Σ É The observable map associated to a classical observable F in a region M is Z F 1 ρM ( ) = (φ)F(φ)exp g@ M (φ,φ) dµM˜ (φ). L −4 Σ Klein-Gordon theory in Minkowski É Action of a real massive Klein-Gordon field on1 + 1-dimensional Minkowski spacetime 1 Z 2 µν 2 2 S[φ] = d x η @ φ∂ φ m φ . 2 µ ν − É The GBF is defined in a region M bounded by the disjoint union of two spacelike hypersurfaces represented by two equal time hyperplanes. É It is convenient to expand the field in the basis of the boost modes Z 1 1 p(x,t) = dq exp im(x sinhq t coshq) ipq 23=2π − − −∞ Klein-Gordon theory in Minkowski All the relevant structures can be defined and the Hilbert space constructed. É The complex structure results to be @ J t Σi = Æ @ 2 − t É The vacuum state is the standard Minkowski vacuum state É Amplitude and observable maps are implementable in terms of !( , ),g( , ) and , · · · · {· ·} Klein-Gordon theory in Rindler space 2 2 2 2 É Rindler space is defined by ds = ρ dη dρ , where − t = ρ sinhη, x = ρ coshη It corresponds to the right wedge of Minkowski space, 2 := x : x 0, x > 0 . R f 2 M ≤ g É We consider the region R bounded by the disjoint union of two equal-Rindler-time hyperplanes.⊂ R É The field is expanded in the basis of the Fulling modes sinh p 1=2 R ( ( π)) ipη φ (ρ,η) = Kip(mρ)e− , p > 0, p π Kip is the modified Bessel function of the second kind (Macdonald function). Klein-Gordon theory in Rindler space All the relevant structures can be defined and the Hilbert space constructed. É The complex structure results to be @η J R Σi = q @ 2 − η É Amplitude and observable maps are implementable in terms of !( , ),g( , ) and , · · · · {· ·} The relevance of this boundary condition manifests at the level of the algebraic structures, e.g. ( ) R R ! R φ,φ0 ! R φ ,φ 0 Σ0 ( ) = Σ0 ( ) Z ε cosh(pπ) h i + limi dp φ(p)φ(p)0 φ(p)φ0(p) , ε 0 sinh p ! 0 ( π) − R R where Σ0 hyperplane t = 0, Σ is the semi-hyperplane η = 0, Σ = Σ0 . 0 0 \R the two quantum theories, in Minkowski and in Rindler spaces, are inequivalent) Boundary condition In order for the quantum theory to be well defined the following condition must be imposed R φ (ρ = 0,η) = 0 Boundary condition In order for the quantum theory to be well defined the following condition must be imposed R φ (ρ = 0,η) = 0 The relevance of this boundary condition manifests at the level of the algebraic structures, e.g. ( ) R R ! R φ,φ0 ! R φ ,φ 0 Σ0 ( ) = Σ0 ( ) Z ε cosh(pπ) h i + limi dp φ(p)φ(p)0 φ(p)φ0(p) , ε 0 sinh p ! 0 ( π) − R R where Σ0 hyperplane t = 0, Σ is the semi-hyperplane η = 0, Σ = Σ0 . 0 0 \R the two quantum theories, in Minkowski and in Rindler spaces, are inequivalent) Outline The Unruh effect GBF and quantum field theory Schrödinger-Feynman quantization Holomorphic quantization GBF in Minkowski and Rindler spacetimes Unruh effect in the GBF Global Unruh effect Local Unruh effect Conclusions Unruh effect Two notions: 1. Global Unruh effect: Comparison of particle states in Minkowski and Rindler spaces, e.g. É Crispino et al.: «The Unruh effect is defined in this review as the fact that the usual vacuum state for QFT in Minkowski spacetime restricted to the right Rindler wegde is a thermal state.» É Jacobson, ”Introductory lectures on black hole thermodynamics”: «The essence of the Unruh effect is the fact that the density matrix describing the Minkowski vacuum, traced over the states in the regionz < 0, is precisely a Gibbs state for the boost Hamiltonian at a temperatureT = 1=2π.» 2.