Classical dynamics, arrow of time, and genesis of the Heisenberg - PowerPoint PPT Presentation

Classical dynamics, arrow of time, and genesis of the Heisenberg commutation relations Detlev Buchholz & Klaus Fredenhagen Mathematics of interacting QFT models University of York, July 1st 2019 1 / 14

Background Perturbative AQFT led to a new constructive scheme for quantum physics Ingredients: classical systems, orbits in configuration space, Lagrangeans operations (perturbations of system), labelled by functionals on orbits (fixed by potentials, durations in time) arrow (direction) of time; entering into the microworld by order (succession) of operations Result: dynamical C*-algebra for given Lagrangean; commutation relations etc arise from its intrinsic structure. New look at quantum physics; no a priori "quantization rules" This talk: application of scheme to classical mechanics 2 / 14

Classical mechanics Notation: N particles in R s , equal masses, distinguishable positions x = ( x 1 , . . . , x N ) ∈ R sN continuous orbits x : R → R sN , family denoted by C , loops x 0 : R → R sN with compact support, form vector space C 0 ⊂ C , velocities ˙ x : R → R sN Perturbations: (given by potentials, time dependencies) are described by space F of functionals F : C → R � F [ x ] . = dt F ( t , x ( t )) � t , x �→ F ( t , x ) = f 0 ( t ) x + g k ( t ) V k ( x ) k where f 0 ∈ C 0 , g k ∈ D ( R s ) , V k continuous, bounded Support of F : union of supports of underlying test functions x �→ F x 0 [ x ] . Shifts: F �→ F x 0 , x 0 ∈ C 0 given by = F [ x + x 0 ] 3 / 14

Classical mechanics Lagrangeans: t �→ L ( x ( t )) . x ( t ) 2 − V ( x ( t )) , = ( 1 / 2 ) ˙ x ∈ C ; � dt L ( x ( t )); relative action for loops x 0 ∈ C 0 : action � � L ( x ( t ) + x 0 ( t )) − L ( x ( t )) � δ L ( x 0 )[ x ] = dt χ ( t ) with χ ↾ supp x 0 = 1 ( note: element of F , linear term x appears) Stationary points of action: Euler-Lagrange equation ¨ x ( t ) + ∂ V ( x ( t )) = 0 “inverses” of K = − d 2 Propagators: i.e. K ∆ • = ∆ • K = 1 dt 2 mean ∆ D . advanced ∆ A , retarded ∆ R , = ( 1 / 2 )(∆ A + ∆ R ) ∆ . commutator function: = ∆ R − ∆ A , K ∆ = ∆ K = 0 4 / 14

Dynamical algebra Step 1: given a Lagrangean L , construct a dynamical group G L Definition: G L is the free group generated by symbols S L ( F ) , F ∈ F , modulo the relations S L ( F ) = S L ( F x 0 + δ L ( x 0 )) (i) for all F ∈ F , x 0 ∈ C 0 (ii) S L ( F 1 ) S L ( F 2 ) = S L ( F 1 + F 2 ) whenever F 1 has support in the future of F 2 Remarks: the elements of G L describe the effects of perturbations on the underlying system without stipulating their concrete action F = 0 implies S L ( δ L ( x 0 )) = S ( 0 ) = 1 (Euler-Lagrange equation) (i) (ii) constant functionals F h : F → h ∈ R have arbitrary support in time; thus S ( F ) S ( F h ) = S ( F + F h ) = S ( F h ) S ( F ) (form central subgroup) 5 / 14

Dynamical algebra Step 2: proceed from G L to *-algebra A L sums: � cS , c ∈ C , S ∈ G L span A L adjoints: ( � cS ) ∗ . = � cS − 1 ( S unitary operators) products: fixed by distributive law fixing scale: S ( F h ) = e i h 1, h ∈ R (amounts to atomic units) norm: algebra has faithful states and thus a (maximal) C*-norm Definition: Given L , the corresponding dynamical algebra A L is the C*-algebra determined by the dynamical group G L . No quantization conditions, functional integrals etc ; only classical concepts used (“common language”, cf. Bohr’s doctrine) 6 / 14

Derivation of Heisenberg relations Consider non-interacting Lagrangean x ( t ) 2 t �→ L 0 ( x ( t )) = ( 1 / 2 ) ˙ � Simplest (linear) perturbations � f 0 , x � . = dt f 0 ( t ) x ( t ) x �→ F f 0 [ x ] . = � f 0 , x � + ( 1 / 2 ) � f 0 , ∆ D f 0 � , f 0 ∈ C 0 W ( f 0 ) . = S L 0 ( F f 0 ) , f 0 ∈ C 0 . Definition: Theorem (1) W ( K x 0 ) = 1 , x 0 ∈ C 0 (2) W ( f 0 ) W ( g 0 ) = e − ( i / 2 ) � f 0 , ∆ g 0 � W ( f 0 + g 0 ) , f 0 , g 0 ∈ C 0 Interpretation: Weyl operators W ( x 0 ) . = e i � x 0 , Q � , x 0 ∈ C 0 (1) generators solutions of Heisenberg eq.: t �→ Q ( t ) = Q + t ˙ Q (2) [ Q k , ˙ [ Q k , Q l ] = [ ˙ Q k , ˙ Q l ] = i δ kl 1, Q l ] = 0 Physics: operators of position Q and momentum P . = ˙ Q 7 / 14

Derivation of Heisenberg relations Proofs: recall: K = − d 2 (1) (dynamics) x 0 ∈ C 0 dt 2 x �→ F K x 0 [ x ] = � K x 0 , x � + ( 1 / 2 ) � K x 0 , ∆ D K x 0 � = � ˙ x 0 , ˙ x � + ( 1 / 2 ) � ˙ x 0 , ˙ x 0 � = δ L 0 ( x 0 )[ x ] S L 0 ( F K x 0 ) = S L 0 ( δ L 0 ( x 0 )) = S L 0 ( 0 ) = 1; hence S L 0 ( F f 0 + K x 0 ) = S L 0 ( F f 0 ) similarly (2) (Weyl relations) Given f 0 , g 0 , let f 0 + K x 0 be later than g 0 . Then S L 0 ( F f 0 ) S L 0 ( F g 0 ) = S L 0 ( F f 0 + K x 0 ) S L 0 ( F g 0 ) = S L 0 ( F f 0 + K x 0 + F g 0 ) . Linearity of F f [ x ] with regard to x implies S L 0 ( F f 0 + K x 0 + F g 0 ) = S L 0 ( F f 0 + K x 0 + g 0 + F h f 0 + K x 0 , g 0 ) = e i h f 0 + K x 0 , g 0 S L 0 ( F f 0 + K x 0 + g 0 ) = e i h f 0 + K x 0 , g 0 S L 0 ( F f 0 + g 0 ) where � h f 0 + K x 0 , g 0 = · · · = − ( 1 / 2 ) � f 0 + K x 0 , ∆ g 0 � = − ( 1 / 2 ) � f 0 , ∆ g 0 � . 8 / 14

Derivation of Heisenberg relations Proofs: recall: K = − d 2 (1) (dynamics) x 0 ∈ C 0 dt 2 x �→ F K x 0 [ x ] = � K x 0 , x � + ( 1 / 2 ) � K x 0 , ∆ D K x 0 � = � ˙ x 0 , ˙ x � + ( 1 / 2 ) � ˙ x 0 , ˙ x 0 � = δ L 0 ( x 0 )[ x ] S L 0 ( F K x 0 ) = S L 0 ( δ L 0 ( x 0 )) = S L 0 ( 0 ) = 1; hence S L 0 ( F f 0 + K x 0 ) = S L 0 ( F f 0 ) similarly (2) (Weyl relations) Given f 0 , g 0 , let f 0 + K x 0 be later than g 0 . Then S L 0 ( F f 0 ) S L 0 ( F g 0 ) = S L 0 ( F f 0 + K x 0 ) S L 0 ( F g 0 ) = S L 0 ( F f 0 + K x 0 + F g 0 ) . Linearity of F f [ x ] with regard to x implies S L 0 ( F f 0 + K x 0 + F g 0 ) = S L 0 ( F f 0 + K x 0 + g 0 + F h f 0 + K x 0 , g 0 ) = e i h f 0 + K x 0 , g 0 S L 0 ( F f 0 + K x 0 + g 0 ) = e i h f 0 + K x 0 , g 0 S L 0 ( F f 0 + g 0 ) where � h f 0 + K x 0 , g 0 = · · · = − ( 1 / 2 ) � f 0 + K x 0 , ∆ g 0 � = − ( 1 / 2 ) � f 0 , ∆ g 0 � . 8 / 14

Interacting theories Change of Lagrangean (potentials V as before) t �→ L ( x ( t )) = L 0 ( x ( t )) − V ( x ( t )) temporary perturbation ( χ smooth characteristic function) t �→ L χ ( x ( t )) . = L 0 ( x ( t )) − χ ( t ) V ( x ( t )) � �� � V χ ( t , x ( t )) Definition: (cf. relative scattering matrices) S L χ ( F ) . = S L 0 ( − V χ ) − 1 S L 0 ( F − V χ ) ∈ A L 0 , F ∈ F Properties: (elementary computation) S L χ ( F x 0 + δ L χ ( x 0 )) = S L χ ( F ) (i) S L χ ( F 1 ) S L χ ( F 2 ) = S L χ ( F 1 + F 2 ) if F 1 is later than F 2 (ii) Conclusion: defining relations for dynamical algebra A L χ ≃ A L 0 9 / 14

Interacting theories Goal: limit χ → 1 (global dynamics) Note: Let I ⊂ R and χ ↾ I = 1, then δ L χ ( x 0 ) = δ L ( x 0 ) if x 0 ∈ C 0 ( I ) Definition: A L χ ( I ) algebra generated by S L χ ( F ) , F ∈ F ( I ) . Observation: A L χ ( I ) ≃ A L ( I ) and algebras A L χ ( I ) for different χ are related by inner automorphisms of A L 0 Detailed analysis: for increasing intervals I n and functions χ n there exist injective homomorphisms β n : A L ( I n ) → A L 0 ( I n + 1 ) such that γ . = lim n β n point-wise in norm on A L = � I A L ( I ) . Theorem Let L 0 , L be Lagrangeans. There exist monomorphisms γ : A L → A L 0 such that γ ( A L ( I )) ⊂ A L 0 ( � I ) for any I and bounded � I ⊃ I . 10 / 14

Representations Consider Schrödinger representation of Q , P on H S with dynamics L 0 Claim: operators S ( F ) are represented by time ordered exponentials. Problem: For F ∈ F , determine T ( F ) . R ∞ = T e i −∞ dt F ( Q + t P ) • bounded functionals F b : Dyson expansion T ( F b ) = 1 + � k i k � ∞ � t k − 1 −∞ dt 1 . . . −∞ dt k F b ( Q + t 1 P 1 ) · · · F b ( Q + t k P 1 ) • linear (unbounded) L : solution of linear differential equation dt f 0 ( t )( Q + t P ) e − ( i / 2 ) � f 0 , ∆ D f 9 � = W ( f 0 ) e − ( i / 2 ) � f 0 , ∆ D f 9 � R T ( L f 0 ) = e i • combination: T ( F b + L f 0 ) . = T ( F − ∆ A f 0 ) T ( L f 0 ) b Ansatz based on results of structural analysis; it has all required properties 11 / 14

Representations Proof: E.g. “dynamical relation” for bounded functionals F b � �� � T ( F x 0 b + δ L 0 ( x 0 )) = T ( F x 0 F x 0 + F K x 0 ) = T ( + F h + L K x 0 ) b b ) e i h T ( L K x 0 ) = T ( F x 0 − ∆ A K x 0 = T ( F x 0 − ∆ A K x 0 = T ( F b ) � ) T ( F K x 0 ) b b � �� � � �� � 1 F b Definition: Representation ( π S , H S ) of A L 0 fixed by putting π S ( S L 0 ( F )) . = T ( F ) , F ∈ F . Other algebras A L are represented by ( π, H S ) , where π . = π s ◦ γ Theorem (i) The representations ( π, H S ) of A L are “regular” and irreducible (ii) This holds also true for π ↾ A L ( I ) for any finite interval I 12 / 14