Asymptotic Equivalence of KMS States in Rindler spacetime Maximilian Kähler Institute of Theoretical Physics Leipzig University LQP workshop, 29-30 May, 2015
Introduction - The Unruh effect "An accelerated observer perceives an ambient inertial vacuum as a state of thermal equilibrium." [Fulling-Davies-Unruh 1973-1976] Modern formulation in mathematical physics: The Minkowski vacuum restricted to the Rindler spacetime is a KMS state with real parameter 1 β Unruh = 2 π g . Can 1 / β Unruh be interpreted as a local temperature? 1 natural units c = � = k B = 1 2 / 17
Outline 1 KMS Condition and Previous Results 2 Main Result 3 Proof of Main Result (Sketch) 3 / 17
References D. Buchholz, C. Solveen Unruh Effect and the Concept of Temperature Class. Quantum Grav. 30(8):085011, Mar 2013 arXiv:1212.2409 D. Buchholz, R. Verch Macroscopic aspects of the Unruh Effect arXiv:1412.5892 , Dec 2014 M. Kähler On Quasi-equivalence of Quasi-free KMS States restricted to an unbounded Subregion of the Rindler Spacetime http://lips.informatik.uni-leipzig.de/pub/2015 Diploma Thesis, Jan 2015 4 / 17
KMS states Let ( A , α t ) be a C*-algebraic dynamical system. Definition A state ω on A is called a β -KMS state for β > 0, if for all A , B ∈ A there exists a bounded continuous function F A , B : S β := R × i [ 0 , β ] − → C , holomorphic in the interior of S β , such that for all t ∈ R F A , B ( t ) = ω ( A α t ( B )) , F A , B ( t + i β ) = ω ( α t ( B ) A ) . 5 / 17
Recent doubts on the thermal interpretation of β Unruh Buchholz-Solveen 03/2013: Two distinct definitions of temperature in classical thermodynamics: A) empirical temperature scale based on zeroth law, B) absolute temperature scale based on second law ("Carnot Parameter"). Observation One-to-one correspondence of these definitions does only hold in inertial situations. Review of temperature definitions in an algebraic framework, KMS-parameter corresponds to the second law definition, ⇒ KMS-parameter loses interpretation of inverse temperature in non-inertial situations. 6 / 17
Recent doubts on local thermal interpretation of β Unruh Example: comparison of KMS states in Minkowski and Rindler space Buchholz and Solveen exhibit a empirical temperature observable θ y for every y ∈ M Minkowski Spacetime M Rindler Spacetime R � � β ( θ y ) = C 1 1 1 ω M ω R C β ( θ y ) = β 2 − β x ( y ) 2 ( 2 π ) 2 spatially homogeneous spatial temperature gradient 7 / 17
Far away all KMS states look the same Buchholz-Verch 12/2014: Let O ⊂ R be a causally complete bounded subset, Let x ( s ) = ( 0 , s , 0 , 0 ) ∈ R 4 , s > 0 , be a family of 4-vectors, Consider the translates O + x ( s ) and the corresponding local field algebra A ( O + x ( s )) of a massless scalar field. 8 / 17
Far away all KMS states look the same Buchholz-Verch 12/2014: Let O ⊂ R be a causally complete bounded subset, Let x ( s ) = ( 0 , s , 0 , 0 ) ∈ R 4 , s > 0 , be a family of 4-vectors, Consider the translates O + x ( s ) and the corresponding local field algebra A ( O + x ( s )) of a massless scalar field. Then for all β 1 , β 2 > 0 � � lim � ω β 1 − ω β 2 � | A ( O + x ( s )) � = 0 . � � � s →∞ spatial inhomogeneity is independent of chosen observable 8 / 17
Main Result Local quasi-equivalence of quasi-free KMS states Theorem Let A be the Weyl algebra of the real massive Klein-Gordon field on the Rindler spacetime. Let A ( B ) be the local subalgebra corresponding to the causal completion of the unbounded region ( 0 , y 1 , ξ ) ∈ R 1 , 3 | y 1 > X 0 , � ξ � < R � � B := , with R > 0 , X 0 > 0 . For β > 0 denote by ω β the unique quasi-free β -KMS states on A with non-degenerate β -KMS one-particle structure. ⇒ Then for all 0 < β 1 < β 2 ≤ ∞ the states ω β 1 | A ( B ) and ω β 2 | A ( B ) are quasi-equivalent. 9 / 17
Main Result Local quasi-equivalence of quasi-free KMS states Theorem Let A be the Weyl algebra of the real massive Klein-Gordon field on the Rindler spacetime. Let A ( B ) be the local subalgebra corresponding to the causal completion of the unbounded region ( 0 , y 1 , ξ ) ∈ R 1 , 3 | y 1 > X 0 , � ξ � < R � � B := , with R > 0 , X 0 > 0 . For β > 0 denote by ω β the unique quasi-free β -KMS states on A with non-degenerate β -KMS one-particle structure. ⇒ Then for all 0 < β 1 < β 2 ≤ ∞ the states ω β 1 | A ( B ) and ω β 2 | A ( B ) are quasi-equivalent. 9 / 17
Restricted Spacetime Region in Rindler spacetime R := { ( y 0 , y 1 , y 2 , y 3 ) ∈ R 1 , 3 | | y 0 | < y 1 } 10 / 17
Restricted Spacetime Region in Rindler spacetime R := { ( y 0 , y 1 , y 2 , y 3 ) ∈ R 1 , 3 | | y 0 | < y 1 } 10 / 17
Restricted Spacetime Region in Rindler spacetime ( 0 , y 1 , ξ ) ∈ R 1 , 3 | y 1 > X 0 , � ξ � < R � � B := 10 / 17
Main Result Local quasi-equivalence of quasi-free KMS states Theorem Let A be the Weyl algebra of the real massive Klein-Gordon field on the Rindler spacetime. Let A ( B ) be the local subalgebra corresponding to the causal completion of the unbounded region ( 0 , y 1 , ξ ) ∈ R 1 , 3 | y 1 > X 0 , � ξ � < R � � B := , with R > 0 , X 0 > 0 . For β > 0 denote by ω β the unique quasi-free β -KMS states on A with non-degenerate β -KMS one-particle structure. ⇒ Then for all 0 < β 1 < β 2 ≤ ∞ the states ω β 1 | A ( B ) and ω β 2 | A ( B ) are quasi-equivalent. 11 / 17
Proof Description of Quasi-free States Classical Klein-Gordon field can be described by a real symplectic space ( S , σ ) . Let µ : S × S → R be a real inner product on ( S , σ ) , such that | σ (Φ 1 , Φ 2 ) | ≤ 2 µ (Φ 1 , Φ 1 ) 1 / 2 · µ (Φ 2 , Φ 2 ) 1 / 2 . Then � − 1 � ω µ ( W (Φ)) := exp 2 µ (Φ , Φ) defines a state on A . ω µ is called a quasi-free state. 12 / 17
Sufficient Criterion for Quasi-equivalence Theorem (Araki-Yamagami 1982; Verch 1992) Let ω 1 , ω 2 be quasi-free states on the Weyl-Algebra A , uniquely characterised by real inner products µ 1 , µ 2 : S × S → R . Consider the complexification S C := S ⊕ iS and the sesquilinear extensions µ C 1 , µ C 2 to S C . Then ω 1 , ω 2 are quasi-equivalent if the following two conditions hold i) µ C 1 , µ C 2 induce equivalent norms on S C ii) The operator T : S C → S C defined through µ C 1 (Φ 1 , Φ 2 ) − µ C 2 (Φ 1 , Φ 2 ) = µ C 1 (Φ 1 , T Φ 2 ) , for all Φ 1 , Φ 2 ∈ S C , is of trace class in ( S C , µ C 1 ) . 13 / 17
Defining Inner Products Inner product for quasi-free β -KMS states: µ β (Φ 1 , Φ 2 ) = �� � � A 1 / 2 β � � � � A 1 / 2 β � � 1 f 1 , A 1 / 2 coth p 1 , A − 1 / 2 coth f 2 + p 2 , 2 2 2 L 2 ( R 3 ) L 2 ( R 3 ) 0 ( R 3 ) × C ∞ 0 ( R 3 ) , j = 1 , 2. for Φ j = ( f j , p j ) ∈ S := C ∞ Involves the partial differential operator A x 1 + e 2 x 1 ( m 2 − ∂ 2 A = − ∂ 2 x 2 − ∂ 2 x 3 ) , positive and essentially self-adjoint on C ∞ 0 ( R 3 ) ⊂ L 2 ( R 3 ) . Norm equivalence can be easily asserted. 14 / 17
Proving trace class property Kontorovich-Lebedev transform: special integral transform U provides explicit spectral representation of operator A Integral operator: U used to rewrite T 1 / 2 as integral operator on weighted L 2 spaces I := U − 1 T 1 / 2 U : L 2 ( M , d ν β 1 ) → L 2 ( M , d ν β 1 ) , � K ( m , m ′ ) φ ( m ′ ) d ν β 1 ( m ′ ) ( I φ )( m ) = M Hilbert-Schmidt Theorem: T 1 / 2 is Hilbert-Schmidt class ⇔ K ∈ L 2 ( M × M , d ν β 1 ⊗ d ν β 1 ) 15 / 17
Summary Three levels of content: A) Conceptual level: Interpretation of Unruh effect requires careful application of thermodynamic concepts, 1 / β need not be a meaningful temperature scale. B) Abstract quasi-equivalence result: first result to establish local quasi-equivalence on unbounded subregion "Accelerated" KMS states coincide at large distance. C) Specific functional analytic techniques: Explicit spectral calculations, Analysis of integral operators. 16 / 17
References D. Buchholz, C. Solveen Unruh Effect and the Concept of Temperature Class. Quantum Grav. 30(8):085011, Mar 2013 arXiv:1212.2409 D. Buchholz, R. Verch Macroscopic aspects of the Unruh Effect arXiv:1412.5892 , Dec 2014 M. Kähler On Quasi-equivalence of Quasi-free KMS States restricted to an unbounded Subregion of the Rindler Spacetime http://lips.informatik.uni-leipzig.de/pub/2015 Diploma Thesis, Jan 2015 17 / 17
Construction of the operator T Use the Riesz-lemma to define the operator T : S → S by µ β 1 (Φ 1 , Φ 2 ) − µ β 2 (Φ 1 , Φ 2 ) = µ β 1 (Φ 1 , T Φ 2 ) for all Φ 1 , Φ 2 ∈ S . As a matrix acting on f - and p -components of S � � s β 1 ,β 2 ( A 1 / 2 ) 0 T = , s β 1 ,β 2 ( A 1 / 2 ) 0 with � 1 − coth ( β 2 τ/ 2 ) � s β 1 ,β 2 ( τ ) := coth ( β 1 τ/ 2 ) 18 / 17
Intuition for the operator T 1 − coth ( β 2 τ/ 2 ) � � s β 1 ,β 2 ( τ ) := coth ( β 1 τ/ 2 ) 19 / 17
Previous result on local quasi-equivalence Theorem (Verch, CMP 160) Let ω 1 and ω 2 be two quasi-free Hadamard states on the Weyl algebra A of the Klein-Gordon field in some globally hyperbolic spacetime ( M , g ) , and let π 1 and π 2 be their associated GNS representations. Then π 1 | A ( O ) and π 2 | A ( O ) are quasi-equivalent for every open subset O ⊂ M with compact closure. 20 / 17
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