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Quantum Mechanics for Non-Inertial Observers

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Quantum Mechanics for Non-Inertial Observers quantum mechanics for non-inertial observers . André Großardt Università degli Studi di Trieste, Italy FPQP2016, Bengaluru, 8 Dec 2016 quantum systems in gravitational fields G Newton + QM Newtonian Gravity ! tested! QFT on curved General Relativity spacetime . nonrel. class. Quantum ~ mechanics Mechanics Special Relativity rel. QFT 1/c Only external gravitational fields! 1 quantum systems in gravitational fields G Newton + QM Newtonian Gravity ! tested! QFT on curved General Relativity spacetime . nonrel. class. Quantum ~ mechanics Mechanics Special Relativity rel. QFT 1/c Only external gravitational fields! 1 quantum systems in gravitational fields G Newton + QM Newtonian Gravity ! tested! QFT on curved General Relativity spacetime . nonrel. class. Quantum ~ mechanics Mechanics Special Relativity rel. QFT 1/c Only external gravitational fields! 1 quantum systems in gravitational fields G Newton + QM Newtonian Gravity ! tested! QFT on curved General Relativity spacetime . nonrel. class. Quantum ~ mechanics Mechanics Special Relativity rel. QFT 1/c Only external gravitational fields! 1 quantum systems in gravitational fields G Newton + QM Newtonian Gravity ! tested! QFT on curved General Relativity spacetime . nonrel. class. Quantum ~ mechanics Mechanics Special Relativity rel. QFT 1/c Only external gravitational fields! 1 quantum systems in gravitational fields G Newton + QM Newtonian Gravity ! tested! QFT on curved General Relativity spacetime . nonrel. class. Quantum ~ mechanics Mechanics Special Relativity rel. QFT 1/c Only external gravitational fields! 1 quantum systems in gravitational fields G Newton + QM Newtonian Gravity ! tested! QFT on curved General Relativity spacetime . nonrel. class. Quantum ~ mechanics Mechanics Special Relativity rel. QFT 1/c Only external gravitational fields! 1 quantum systems in gravitational fields G Newton + QM Newtonian Gravity ! tested! QFT on curved General Relativity spacetime . nonrel. class. Quantum ~ mechanics Mechanics Special Relativity rel. QFT 1/c Only external gravitational fields! 1 time dilation induces decoherence … or does it? Effects of gravitational time dilation on quantum systems: I Interferometry with clocks Zych et al. 2011, Nature Communications 2, 505 I Loss of visibility in photon interference due to Shapiro time delay Zych et al. 2012, Classical and Quantum Gravity 29, 224010 I Decoherence due to gravitational time dilation Pikovski et al. 2015, Nature Physics 11, 668 Discussion: Agrees with known principles? Is it decoherence? Is it relevant? I Diósi 2015 I Zeh 2015 I Bonder et al. 2015/16 I Adler & Bassi 2016 I Pang et al. 2016 I Carlesso & Bassi 2016 Here: Relativistic centre of mass? What about external potentials? 2 .time dilation & quantum matter I Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame. I Comparison shows a time difference Δτ. − ~ I Superposition yields phase difference ) Δφ = e iHclockΔτ= ? I Different paths in superposition evolve with proper time, i. e. the time coordinate of different inertial frames. ) Hypothesis! interference of clocks . U U U U Δτ − ~ I Time evolution of clock: j ψ(τ)i = e iHclockτ= j ψ(0)i 4 I Comparison shows a time difference Δτ. − ~ I Superposition yields phase difference ) Δφ = e iHclockΔτ= ? I Different paths in superposition evolve with proper time, i. e. the time coordinate of different inertial frames. ) Hypothesis! interference of clocks . U U U U Δτ − ~ I Time evolution of clock: j ψ(τ)i = e iHclockτ= j ψ(0)i I Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame. 4 − ~ I Superposition yields phase difference ) Δφ = e iHclockΔτ= ? I Different paths in superposition evolve with proper time, i. e. the time coordinate of different inertial frames. ) Hypothesis! interference of clocks . U U U U Δτ − ~ I Time evolution of clock: j ψ(τ)i = e iHclockτ= j ψ(0)i I Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame. I Comparison shows a time difference Δτ. 4 I Different paths in superposition evolve with proper time, i. e. the time coordinate of different inertial frames. ) Hypothesis! interference of clocks . U U U U Δτ − ~ I Time evolution of clock: j ψ(τ)i = e iHclockτ= j ψ(0)i I Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame. I Comparison shows a time difference Δτ. − ~ I Superposition yields phase difference ) Δφ = e iHclockΔτ= ? 4 interference of clocks . U U U U Δτ − ~ I Time evolution of clock: j ψ(τ)i = e iHclockτ= j ψ(0)i I Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame. I Comparison shows a time difference Δτ. − ~ I Superposition yields phase difference ) Δφ = e iHclockΔτ= ? I Different paths in superposition evolve with proper time, i. e. the time coordinate of different inertial frames. ) Hypothesis! 4 I For many degrees of freedom: recurrence of coherence becomes rare interference of clocks Δτ . I For a single degree of freedom (“clock”): oscillating visibility of interference pattern 5 interference of clocks Δτ . I For a single degree of freedom (“clock”): oscillating visibility of interference pattern I For many degrees of freedom: recurrence of coherence becomes rare 5 ∙ kinetic energy (“special relativistic time dilation”) ∙ gravitational energy (“gravitational time dilation”) I 2 ! 2 Relativistic energy: mc mc + Hinternal ( ) 1 p2 H = H − H − mgx total cm mc2 internal 2m I Coupling of internal Hamiltonian to simple “derivation” I Center of mass dynamics described by one particle Hamiltonian p2 H = + mgx + terms of order 1=c2 cm 2m 6 ∙ kinetic energy (“special relativistic time dilation”) ∙ gravitational energy (“gravitational time dilation”) I Coupling of internal Hamiltonian to simple “derivation” I Center of mass dynamics described by one particle Hamiltonian p2 H = + mgx + terms of order 1=c2 cm 2m I 2 ! 2 Relativistic energy: mc mc + Hinternal ( ) 1 p2 H = H − H − mgx total cm mc2 internal 2m 6 simple “derivation” I Center of mass dynamics described by one particle Hamiltonian p2 H = + mgx + terms of order 1=c2 cm 2m I 2 ! 2 Relativistic energy: mc mc + Hinternal ( ) 1 p2 H = H − H − mgx total cm mc2 internal 2m I Coupling of internal Hamiltonian to ∙ kinetic energy (“special relativistic time dilation”) ∙ gravitational energy (“gravitational time dilation”) 6 .centre of mass coordinates relativistic centre of mass Galilean c.m. coordinates weighted with rest mass not at rest in zero total momentum frame no uniform motion for interacting particles Relativistic: centre of… …energy coordinates weighted with energy (“relativistic mass”) non-covariant, non-canonical …inertia previous def. in frame with zero total momentum, translated to other frames by Lorentz transformations covariant, non-canonical …spin mean of previous definitions non-covariant, canonical I need canonical commutation relations for Quantum Mechanics 8 I Search coordinates such that the generators have this form perturbative definition of centre of mass coordinates Krajcik & Foldy 1974, PRD 10, 1777 Galilei group generators P P P H J H [ i; j] = [ i; ]= [ i; ] = 0 ; J J J J P P [ i; j] = i εijk k ; [ i; j] = i εijk k ; J K K K H P [ i; j] = i εijk k ; [ i; ] = i i ; K P K K [ i; j] = i δijM ; [ i; j] = 0 : P P mμ With the centre of mass coordinates R = M rμ, P = pμ these assume the single particle form: P = P ; J = R × P P2 H = + H ; K = MR − tP : 2M internal 9 perturbative definition of centre of mass coordinates Krajcik & Foldy 1974, PRD 10, 1777 Poincaré group generators P P P H J H [ i; j] = [ i; ]= [ i; ] = 0 ; J J J J P P [ i; j] = i εijk k ; [ i; j] = i εijk k ; J K K K H P [ i; j] = i εijk k ; [ i; ] = i i ; K P H 2 K K − J 2 [ i; j] = i δij =c ; [ i; j] = i εijk k=c : With the centre of mass coordinates R = ???, P = ??? these assume the single particle form: P = P ; J = R × P q 1 H = P2c2 + H2 ; K = fR; Hg − tP : internal 2c2 I Search coordinates such that the generators have this form 9 I 2 (1) f g f g To lowest order in 1=c : φ [R; P; ρμ N; πμ N] I Commutation relations ) internal Hamiltonian XN 2 1 π H = Mc2+H(0)+ H(1)[P; fπ g ; φ(1)] ; H(0) = μ +U(0) internal i c2 i μ N i 2m int μ=1 μ I Expanding Poincaré generator H) centre-of-mass Hamiltonian ! (0) P2 1 P4 P2H HMinkowski = Mc2 + + H(0) + − + H(1) − i cm 2M i c2 8M3 i 2M2 . perturbative definition of centre of mass coordinates (ii) I Introduce a unitary transformation exp(iφ) that maps Galilean to relativistic variables. 10 I Commutation relations ) internal Hamiltonian XN 2 1 π H = Mc2+H(0)+ H(1)[P; fπ g ; φ(1)] ; H(0) = μ +U(0) internal i c2 i μ N i 2m int μ=1 μ I Expanding Poincaré generator H) centre-of-mass Hamiltonian ! (0) P2 1 P4 P2H HMinkowski = Mc2 + + H(0) + − + H(1) − i cm 2M i c2 8M3 i 2M2 . perturbative definition of centre of mass coordinates (ii) I Introduce a unitary transformation exp(iφ) that maps Galilean to relativistic variables. I 2 (1) f g f g To lowest order in 1=c : φ [R; P; ρμ N; πμ N] 10 I Expanding Poincaré generator H) centre-of-mass Hamiltonian ! (0) P2 1 P4 P2H HMinkowski = Mc2 + + H(0) + − + H(1) − i cm 2M i c2 8M3 i 2M2 . perturbative definition of centre of mass coordinates (ii) I Introduce a unitary transformation exp(iφ) that maps Galilean to relativistic variables. I 2 (1) f g f g To lowest order in 1=c : φ [R; P; ρμ N; πμ N] I Commutation relations ) internal Hamiltonian XN 2 1 π H = Mc2+H(0)+ H(1)[P; fπ g ; φ(1)] ; H(0) = μ +U(0) internal i c2 i μ N i 2m int μ=1 μ 10 perturbative definition of centre of mass coordinates (ii) I Introduce a unitary transformation exp(iφ) that maps Galilean to relativistic variables.
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