The Particle as a Statistical Ensemble of Events in Stueckelberg-Horwitz-Piron Electrodynamics 3 rd International Electronic and Flipped Conference on Entropy and its Applications Martin Land Hadassah College Jerusalem www.hadassah.ac.il/cs/staff/martin November 2016 Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 1 / 18
Stueckelberg Covariant Mechanics Worldline Theory of Particles and Antiparticles (1941) Dynamical theory of spacetime events Equations of motion for event x µ ( τ ) Evolution of x µ ( τ ) traces worldline Coordinate time t = x 0 may increase or decrease under evolution Single worldline describes pair annihilation and creation Requires new evolution parameter τ Monotonic replacement for t = x 0 Poincar´ e invariant Independent of spacetime coordinates Distinguishes chronological time τ and coordinate time x 0 Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 2 / 18
Covariant Canonical Mechanics Upgrade nonrelativistic classical and quantum mechanics Newtonian time t Evolution parameter τ + + − → Galilean symmetry Poincar´ e symmetry Inherit methods of nonrelativistic classical and quantum mechanics = p µ x µ = ∂ K p µ = − ∂ K ˙ ˙ = 0 ∂ p µ M ∂ x µ 1 2 M p µ p µ K = − → x 0 = p 0 ˙ d x dt = p M − → p 0 x = p ˙ M Free particle permits reparameterization τ − → t Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 3 / 18
Covariant Mechanics with Interactions Two-body Hamiltonian — Horwitz and Piron (1973) p 1 µ p µ + p 2 µ p µ 1 2 K = + V ( x 1 , x 2 ) 2 M 1 2 M 2 Generalize classical central force problems � ( x 1 − x 2 ) 2 − ( t 1 − t 2 ) 2 V ( x 1 , x 2 ) = V ( ρ ) ρ = where Separation of center of mass and relative motion K = P µ P µ 2 M + p µ p µ 2 m + V ( ρ ) = P µ P µ 2 M + K rel where p µ = M 2 p µ 1 − M 1 p µ m = M 1 M 2 P µ = p µ 1 + p µ 2 M = M 1 + M 2 2 M M Relativistic bound states and scattering solutions Selection rules, radiative transitions, perturbation theory, Zeeman and Stark effects, bound state decay 4-vector and scalar potentials required to reproduce well-known phenomenology Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 4 / 18
Stueckelberg-Horwitz-Piron (SHP) Canonical Mechanics Irreducible chronological time τ — determines temporal ordering of events Order of physical occurrence may differ from order of observed coordinate times x 0 as events appear in measuring apparatus Event occurrence x µ ( τ 1 ) at τ 1 is irreversible — unchanged by subsequent ( τ 2 > τ 1 ) event at same spacetime coordinates x µ ( τ 2 ) = x µ ( τ 1 ) Resolves grandfather paradoxes No closed timelike curves — return trip to past coordinate time x 0 takes place while chronological time τ continues to increase In SHP QED, particle propagator G ( x 2 − x 1 , τ 2 − τ 1 ) vanishes unless τ 2 > τ 1 Super-renormalizable QED with no matter loops τ -retarded causality equivalent to Feynman contour for propagators — follows from vacuum expectation value of τ -ordered operator products Covariant Hamiltonian generates evolution of 4D block universe defined at τ to infinitesimally close 4D block universe defined at τ + d τ Standard Maxwell electrodynamics = equilibrium limit Dynamic system → τ -independent and static 4D block universe Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 5 / 18
Stueckelberg-Horwitz-Piron (SHP) Electrodynamics Unified gauge theory — Saad, Horwitz, and Arshansky (1989) Generalized Stueckelberg-Schrodinger equation � 1 p µ − e p µ − e − e � � c a µ � � � h ∂ τ ψ ( x , τ ) = K ψ ( x , τ ) = ψ ( x , τ ) i ¯ c a µ c φ 2 M Invariant under local gauge transformations ie hc Λ ( x , τ ) ψ ( x , τ ) ψ ( x , τ ) → e ¯ Vector potential a µ ( x , τ ) → a µ ( x , τ ) + ∂ µ Λ ( x , τ ) Scalar potential φ ( x , τ ) → φ ( x , τ ) + ∂ τ Λ ( x , τ ) ∂ µ j µ + ∂ τ ρ = 0 Global gauge invariance − → conserved current 2 j µ = − i ψ ∗ ( ∂ µ − ie c a µ ) ψ − ψ ( ∂ µ + ie � c a µ ) ψ ∗ � � � ρ = � ψ ( x , τ ) � � 2 M � Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 6 / 18
5D Notations and Conventions Formal designations in analogy with x 0 = ct ∂ 5 = 1 x 5 = c 5 τ and ∂ τ c 5 Five explicitly τ -dependent gauge fields a µ ( x , τ ) and a 5 ( x , τ ) = 1 φ ( x , τ ) c 5 Index conventions λ , µ , ν = 0, 1, 2, 3 α , β , γ = 0, 1, 2, 3, 5 and g αβ = diag ( − 1, 1, 1, 1, ± 1 ) Gauge transformations a α ( x , τ ) → a α ( x , τ ) + ∂ α Λ ( x , τ ) Conserved current ∂ α j α = 0 Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 7 / 18
Classical Lagrangian Mechanics Lagrangian x µ p µ − K = L = 1 x µ + e x µ ˙ x α a α L = ˙ 2 M ˙ c ˙ Lorentz force d ∂ L − ∂ L d x µ + e = e � � c a µ ( x , τ ) x α ∂ µ a α ( x , τ ) = 0 − → M ˙ c ˙ d τ ∂ ˙ x µ ∂ x µ d τ � ˙ x ν ∂ ν + ∂ τ ) a µ � = e x µ = e c f µ x α ∂ µ a α − ( ˙ x α M ¨ α ( x , τ ) ˙ c where x 5 = c 5 ˙ f µ α = ∂ µ a α − ∂ α a µ τ = c 5 ˙ Particles and fields may exchange mass d ec 5 f 5 µ ˙ d τ ( − 1 x 2 ) = g 55 2 M ˙ x µ c Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 8 / 18
Kinetic Term for Field Standard considerations Velocity-potential → current-potential integral a α ( x , τ ) = 1 � � X α ( τ ) δ 4 � � X α a α → ˙ d 4 x ˙ d 4 x j α ( x , τ ) a α ( x , τ ) x − X ( τ ) c X α ( τ ) δ 4 � � j α ( x , τ ) = c ˙ x − X ( τ ) Kinetic action term for field Not imposed by physical foundations L kinetic = 1 4 c f αβ ( x , τ ) f αβ ( x , τ ) Most obvious candidate Lorentz and gauge invariant Contains only first order derivatives Produces Maxwell-like field equations Admits wave equation and Green’s function e � � t − | x | �� a 0 ( x , τ ) = Coulomb scattering → wrong dynamics 4 π | x | δ τ − c Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 9 / 18
Kinetic Term for Field f αβ f αβ → f αβ f αβ + ( λ /2 ) 2 � ∂ τ f αβ � � � Higher-order derivative term ∂ τ f αβ Electromagnetic action � e � ds 1 �� � � d 4 xd τ c 2 j α ( x , τ ) a α ( x , τ ) − f αβ ( x , τ ) Φ ( τ − s ) f αβ ( x , s ) S em = λ 4 c Field interaction kernel � λ � λκ � � 2 � � 2 d κ δ ′′ ( τ ) = � e − i κτ Φ ( τ ) = δ ( τ ) − 1 + 2 2 π 2 Inverse function � ds e i κτ d κ � 1 + ( λκ /2 ) 2 = e − 2 | τ | / λ λ ϕ ( τ − s ) Φ ( s ) = δ ( τ ) → ϕ ( τ ) = λ 2 π Field equations ds Φ ( τ − s ) f αβ ( x , s ) = e � ∂ β f αβ c j α ( x , τ ) Φ ( x , τ ) = ∂ β Variation wrt a α ∂ β f αβ ( x , τ ) = e ds ϕ ( τ − s ) j α ( x , s ) = e � c j α Invert with ϕ ϕ ( x , τ ) c Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 10 / 18
Ensemble of Events Shift integral in current � � ds ϕ ( τ − s ) j α ( x , s ) = ds e − 2 | s | / λ j α ( x , τ − s ) j α ϕ ( x , τ ) = ϕ ( x , τ ) = weighted superposition of instantaneous currents j α ( x , τ − s ) j α Originate at events X µ ( τ − s ) displaced from X µ ( τ ) by s on worldline Regard j α ϕ as current produced by ensemble of events in neighborhood of X µ ( τ ) Independent random events with constant average rate = 1/ λ events per second Poisson distribution of events Average time between events = λ Probability e − s / λ / λ at τ that next event will occur following time interval s > 0 Extend displacement to positive and negative values: e − 2 | s | / λ Construct ensemble of events ϕ ( s ) X µ ( τ − s ) along worldline Weight ϕ ( s ) = probability of event delayed from τ by interval at least | s | Green’s function selects from ensemble unique event at lightlike separation Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 11 / 18
Field Equations 5D pre-Maxwell equations ∂ β f αβ ( x , τ ) = e c j α ǫ αβγδǫ ∂ α f βγ = 0 ϕ ( x , τ ) 4D component form ∂ ν f µν − 1 ∂ τ f 5 µ = e ∂ µ f 5 µ = e ϕ = c 5 c j µ c j 5 c e ρ ϕ ϕ c 5 ∂ ν f 5 µ − ∂ µ f 5 ν + 1 ∂ µ f νρ + ∂ ν f ρµ + ∂ ρ f µν = 0 ∂ τ f µν = 0 c 5 Analog of 3-vector Maxwell equations ∇ × B − 1 c ∂ t E = e ∇ · E = e c J 0 c J ∇ × E + 1 ∇ · B = 0 c ∂ t B = 0 Martin Land — ECEA 2016 Particle as a Statistical Ensemble November 2016 12 / 18
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