The arrow of time and quantum physics: difficulties and resolutions - PowerPoint PPT Presentation

The arrow of time and quantum physics: difficulties and resolutions Detlev Buchholz Quantum physics meets mathematics Syposium on the occasion of Klaus Fredenhagen ’s 70th birthday Universität Hamburg, December 8th 2017 1 / 24

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Arrow of time Only parts of Minkowski space (forward lightcones) are accessible 9 / 24

Arrow of time Physical time evolution (inertial observer) acts as a semi-group 10 / 24

Quantum physics Basic conepts Observables: A unital algebra of bounded operators in some cone Arrow of time: time evolution (inertial observer) acts by morphisms α t ( A ) ⊂ A , t ∈ R + States: expectation functionals in A ∗ . Preceeding structure suffices to characterize ground states ω (invariance, analyticity, mixing) Facts Let ω be a ground state on ( A , α ) with GNS representation ( π, H , Ω) . There is a continuous unitary representation U of R with positive 1 generator s.t. Ad U ( t ) ◦ π = π ◦ α t , t ∈ R + , and U ( t )Ω = Ω , t ∈ R . ′′ = B ( H ) (massive theories) There are the alternatives: (i) π ( A ) 2 ′′ type III 1 (presence of massless particles) (ii) π ( A ) 11 / 24

Quantum physics Interpretation Let ω be a ground state on A with GNS representation ( π, H , Ω) . The unitary representation U (fixed by theory) allows to extend 1 the state ω to the past, from the data taken in any given future directed lightcone. (Justification of treatment of time as R ). In massive theories these date uniquely determine this extension. 2 In presence of massless particles the extension is not unique , leading to conceptual problems. 12 / 24

Quantum physics Incomplete information about the past (outgoing radiation) 13 / 24

Quantum physics Fiat lux! Implications: Standard theoretical concepts of quantum physics become operationally irrelevant pure states? : incomplete information! superposition principle? : no lifts to rays in a Hilbert space! transition probabilities? : no minimal projections! Are there other theoretical concepts describing the same physics? Proposal (DB, Erling Størmer): funnels of algebras: provide locally complete information generic states: can be superimposed primitive observables: replace minimal projections 14 / 24

Funnels Observations and operations are made in (fuzzy) spacetime regions Algebra of observables generated by A 1 ⊂ A 2 ⊂ · · · ⊂ A n · · · factors of type I ∞ ≃ B ( H ) � A n + 1 infinite dimensional (hence type I ∞ ), n ∈ N A ′ n A = � n A n proper sequential type I ∞ funnel (Takesaki) Examples: relativistic QFTs (split property), lattice theories, . . . 15 / 24

Generic states States ω : A → C , GNS–representation ( π, H , Ω) locally normal, i.e. weakly continuous on unit balls of A n , n ∈ N , faithful, i.e. ω ( A ∗ A ) = 0 for A ∈ A implies A = 0 � A n + 1 , n ∈ N generic, i.e. representing vector Ω cyclic for A ′ n Remark: Generic vector states “ G δ dense” in H 1 (Dixmier, Marechal) Definition Let ω be generic. Its orbit under non-mixing operations is given by ω A . = { ω A = ω ◦ Ad A : A ∈ A , ω A ( 1 ) = 1 } , where Ad A ( B ) = A ∗ B A , B ∈ A . 16 / 24

Generic states Physical interpretation: Generic states ω describe a “global background” in which physical operations are performed (“state of the world”). Given such a state, these operations produce the corresponding orbit ω A . Examples: vacuum states in relativistic QFT thermal equilibrium states in relativistic and non-relativistic QFT Hadamard states in curved spacetimes 17 / 24

Superpositions Fix a generic state ω with orbit ω A . Norm distance of states � ω A − ω B � . = sup | ω A ( C ) − ω B ( C ) | , ω A , ω B ∈ ω A . C ∈ A 1 Proposition There exists a canonical lift from ω A to rays in A which is bijective: ω A = ω B iff B = t A for t ∈ T 1 locally continuous: if � ω A m − ω A � → 0 for (bounded) A m , A ∈ A n , 2 then t m A m → A in the strong operator topology locally complete: if � ω A l − ω A m � → 0 for (bounded) A l , A m ∈ A n , 3 there is A ∈ A n such that t m A m → A and � ω A m − ω A � → 0 . 18 / 24

Superpostions Physical interpretation: superposition of states in ω A is a meaningful operation, 1 ω A , ω B ↔ T A , T B → T ( c A A + c B B ) ↔ ω ( c A A + c B B ) relative phase between c A , c B ∈ C matters ω A maximal set reached by localized non-mixing operations 3 Mixtures: Conv ω A . � � m pm ω A m : ω A m ∈ ω A , pm > 0 , � � = m pm = 1 Proposition Let ω A ∈ ω A s.t. ω A = � M m = 1 pm ω A m ; then ω A 1 = · · · = ω A M = ω A . ω A extreme points of Conv ω A ; analogue of pure states. 19 / 24

Transition probabilities Definition Let ω A , ω B ∈ ω A . Transition probability given by: ω A · ω B . = | ω ( A ∗ B ) | 2 (Defintion meaningful in view of the bijective relations ω A ↔ T A , ω B ↔ T B ) Remark: comparison with Uhlmann transition probability U · ω B = sup Ω A , Ω B |� Ω A , Ω B �| 2 . ω A · ω B ≤ ω A Proposition Let ω A , ω B ∈ ω A . 0 ≤ ω A · ω B ≤ 1 (notion of orthogonality) , 1 ω A · ω B = ω B · ω A 2 ω A · ω B ≤ 1 − 1 4 � ω A − ω B � 2 ; equality holds iff ω is pure (usual sense) 3 ω A , ω B �→ ω A · ω B is locally continuous 4 there are complete families of orthogonal states { ω A m ∈ ω A } m ∈ N , 5 i.e. � m ω B · ω A m = 1 for any ω B ∈ ω A . 20 / 24

Primitive observables Question: How can one relate these transition probabilities to observations? Recall: ω A ∈ ω A , non-mixing operations B ∈ A , ω A �→ ( 1 /ω A ( B ∗ B )) ω A ◦ Ad B . Restrict operations B to unitary operators U (observable); result ω A �→ ω A ◦ Ad U = ω UA , ω A ∈ ω A . Examples: effects of temporary perturbation of dynamics Transition probability (fidelity of operation): ω A · ( ω A ◦ Ad U ) = ω A · ω UA = | ω A ( U ) | 2 . Can be observed by measurements of U in state ω A . 21 / 24

Primitive observables Definition Primitive observables are fixed by unitaries U ∈ A . For given ω A ∈ ω A ω A �→ ω UA describes the effect of the corresponding operation ω A · ω UA = | ω A ( U ) | 2 is the fidelity of this operation Example: U = E + t ( 1 − E ) with E projection, t ∈ T . Fidelity ω A · ω UA = ω A ( E ) 2 + ω A ( 1 − E ) 2 + 2 Re ( t ) ω A ( E ) ω A ( 1 − E ) Standard expectation values of observables can be recovered: Proposition Given projection E ∈ A , (finite number of) states ω A ∈ ω A , and ε > 0 . There exists a unitary U ∈ A √ | ω A · ω UA − ω A ( E ) 2 | < ε , i.e. “usual probatilities ≈ fidelities” 1 ω UA ( 1 − E ) < ε (compare von Neumann projection postulate) 2 22 / 24

Primitive observables Question: Is ω A · ω B operationally defined for any ω A , ω B ∈ ω A ? (This requires that there are unitaries U ∈ A such that � ω B − ω UA � < ε .) Theorem (Connes, Haagerup, Størmer) Let ω be of type III λ and let 0 ≤ λ < 1 . There are ω A , ω B ∈ ω A s.t. inf U � ω B − ω UA � > ε ( λ ) . 1 λ = 1 . Then inf U � ω B − ω UA � = 0 for any ω A , ω B ∈ ω A . 2 Concept of transition probabilities (operationally) meaningful for pure states ω on A generic states ω on A of type III 1 . These are exactly the two cases of interest in quantum field theory! 23 / 24

Conclusions Features of time: arrow of time is a fundamental fact (can be encoded in theory) statements about the past require some theory (are ambiguous) conflicts with quantum physics (modification of concepts needed) New look at quantum physics: fixed algebra replaced by funnel of algebras generic states and their excitations replace concept of pure states superpositions defined, based on bijective lifts to funnel transition probabilities can be defined primitive (unitary) observables determine transition probabilities meaningful framework for states in QFT (type I ∞ and III 1 ) 24 / 24