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 ℏ 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 ℏ 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 ℏ 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 ℏ 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 ℏ 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 ℏ 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 ℏ 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 ℏ 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:

▶ Interferometry with clocks Zych et al. 2011, Nature Communications 2, 505 ▶ Loss of visibility in photon interference due to Shapiro time delay Zych et al. 2012, Classical and Quantum Gravity 29, 224010 ▶ 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? ▶ Diósi 2015 ▶ Zeh 2015 ▶ Bonder et al. 2015/16 ▶ Adler & Bassi 2016 ▶ Pang et al. 2016 ▶ Carlesso & Bassi 2016 Here: Relativistic centre of mass? What about external potentials? 2 .time dilation & quantum matter ▶ Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame. ▶ Comparison shows a time difference Δτ.

− ℏ ▶ Superposition yields phase difference ⇒ Δφ = e iHclockΔτ/ ? ▶ 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 − ℏ ▶ Time evolution of clock: | ψ(τ)⟩ = e iHclockτ/ | ψ(0)⟩

4 ▶ Comparison shows a time difference Δτ.

− ℏ ▶ Superposition yields phase difference ⇒ Δφ = e iHclockΔτ/ ? ▶ 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 − ℏ ▶ Time evolution of clock: | ψ(τ)⟩ = e iHclockτ/ | ψ(0)⟩ ▶ Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame.

4 − ℏ ▶ Superposition yields phase difference ⇒ Δφ = e iHclockΔτ/ ? ▶ 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 − ℏ ▶ Time evolution of clock: | ψ(τ)⟩ = e iHclockτ/ | ψ(0)⟩ ▶ Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame. ▶ Comparison shows a time difference Δτ.

4 ▶ 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 − ℏ ▶ Time evolution of clock: | ψ(τ)⟩ = e iHclockτ/ | ψ(0)⟩ ▶ Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame. ▶ Comparison shows a time difference Δτ.

− ℏ ▶ Superposition yields phase difference ⇒ Δφ = e iHclockΔτ/ ?

4 interference of clocks

. U

U U U Δτ − ℏ ▶ Time evolution of clock: | ψ(τ)⟩ = e iHclockτ/ | ψ(0)⟩ ▶ Clocks on different paths evolve with proper time τ. τ is the time coordinate in the co-moving Lorentz frame. ▶ Comparison shows a time difference Δτ.

− ℏ ▶ Superposition yields phase difference ⇒ Δφ = e iHclockΔτ/ ? ▶ Different paths in superposition evolve with proper time, i. e. the time coordinate of different inertial frames. ⇒ Hypothesis!

4 ▶ For many degrees of freedom: recurrence of coherence becomes rare

interference of clocks

Δτ .

.

▶ For a single degree of freedom (“clock”): oscillating visibility of interference pattern

5 interference of clocks

Δτ .

.

▶ For a single degree of freedom (“clock”): oscillating visibility of interference pattern ▶ For many degrees of freedom: recurrence of coherence becomes rare

5 ∙ kinetic energy (“special relativistic time dilation”) ∙ gravitational energy (“gravitational time dilation”)

▶ 2 → 2 Relativistic energy: mc mc + Hinternal ( ) 1 p2 H = H − H − mgx total cm mc2 internal 2m

▶ Coupling of internal Hamiltonian to

simple “derivation”

▶ 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”)

▶ Coupling of internal Hamiltonian to

simple “derivation”

▶ Center of mass dynamics described by one particle Hamiltonian

p2 H = + mgx + terms of order 1/c2 cm 2m

▶ 2 → 2 Relativistic energy: mc mc + Hinternal ( ) 1 p2 H = H − H − mgx total cm mc2 internal 2m

6 simple “derivation”

▶ Center of mass dynamics described by one particle Hamiltonian

p2 H = + mgx + terms of order 1/c2 cm 2m

▶ 2 → 2 Relativistic energy: mc mc + Hinternal ( ) 1 p2 H = H − H − mgx total cm mc2 internal 2m

▶ 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

▶ need canonical commutation relations for Quantum Mechanics

8 ▶ 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 . ∑ ∑ 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 √ 1 H = P2c2 + H2 , K = {R, H} − tP . internal 2c2

▶ Search coordinates such that the generators have this form

9 ▶ 2 (1) { } { } To lowest order in 1/c : φ [R, P, ρμ N, πμ N] ▶ Commutation relations ⇒ internal Hamiltonian

∑N 2 1 π H = Mc2+H(0)+ H(1)[P, {π } , φ(1)] , H(0) = μ +U(0) internal i c2 i μ N i 2m int μ=1 μ

▶ 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)

▶ Introduce a unitary transformation exp(iφ) that maps Galilean to relativistic variables.

10 ▶ Commutation relations ⇒ internal Hamiltonian

∑N 2 1 π H = Mc2+H(0)+ H(1)[P, {π } , φ(1)] , H(0) = μ +U(0) internal i c2 i μ N i 2m int μ=1 μ

▶ 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)

▶ Introduce a unitary transformation exp(iφ) that maps Galilean to relativistic variables. ▶ 2 (1) { } { } To lowest order in 1/c : φ [R, P, ρμ N, πμ N]

10 ▶ 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)

▶ Introduce a unitary transformation exp(iφ) that maps Galilean to relativistic variables. ▶ 2 (1) { } { } To lowest order in 1/c : φ [R, P, ρμ N, πμ N] ▶ Commutation relations ⇒ internal Hamiltonian

∑N 2 1 π H = Mc2+H(0)+ H(1)[P, {π } , 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)

▶ Introduce a unitary transformation exp(iφ) that maps Galilean to relativistic variables. ▶ 2 (1) { } { } To lowest order in 1/c : φ [R, P, ρμ N, πμ N] ▶ Commutation relations ⇒ internal Hamiltonian

∑N 2 1 π H = Mc2+H(0)+ H(1)[P, {π } , φ(1)] , H(0) = μ +U(0) internal i c2 i μ N i 2m int μ=1 μ

▶ 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 .

10 .non-inertial observers ▶ Shell observer hovering at constant altitude: ≈ Rindler coordinates (homogeneously accelerated observer) ( ) gx 2 ds2 = − 1 + c2dt2 + dx2 + dy2 + dz2 c2 On Earth: g ≈ -10 m/s2

co-moving observer

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γ

▶ Co-moving observer frame: Fermi normal coordinates

12 co-moving observer

.

γ

▶ Co-moving observer frame: Fermi normal coordinates ▶ Shell observer hovering at constant altitude: ≈ Rindler coordinates (homogeneously accelerated observer) ( ) gx 2 ds2 = − 1 + c2dt2 + dx2 + dy2 + dz2 c2 On Earth: g ≈ -10 m/s2

12 time evolution in the rindler frame

Time-like Killing vectors:

t Minkowski observer Rindler: (t’,x’) Local Lorentz: (t,x) t4'

t ' 3 ∂t Killing vector on t ' 2 = ∂t t 0 hypersurface: t1' ( ) ′ g x ∂t = 1 + c2 ∂t x' Rindler observer spacetime region 2 x' = -c /g of interest

∂t'

∂t'

13 time evolution in the rindler frame (ii)

Hamiltonian for the Rindler observer:

H ? H g { H } Rindler. Mink. + 2.c2 x, Mink. . ( ) ′ g x ∂t . 1 +. c2 ∂t

( ) 1 g HRindler = HMinkowski + M g X + {X, P2} + H(0) g X cm cm c2 4 M i .

14 .supporting potential ▶ What about a particle at rest in the lab frame?

Classically:

0 = −P˙ ∂H = (∂X ) H0 ∂U = M + i g X + ext c2 ∂X

1 ⇒ U = −M g X − H(0) g X ext c2 i cancels gravity to all orders

particle on a table

▶ So far: free falling particle seen by accelerated observer

g -g . y g

16 particle on a table

▶ So far: free falling particle seen by accelerated observer ▶ What about a particle at rest in the lab frame?

Classically:

− ˙ g -g 0 = P ∂H . = g (∂X ) H0 ∂U y i ext = M + g X + c2 ∂X

1 ⇒ U = −M g X − H(0) g X ext c2 i cancels gravity to all orders

16 If we want to satisfy this by a potential Uext(X, t): ( ) 1 ⟨P2⟩ − 2⟨P2⟩ U = −M g X − ⟨H(0)⟩ + x g X ext c2 i 2M .

▶ in general time dependent, except for stationary states ▶ must be tuned to internal and( external energy) ▶ g (0) − ⟨ (0)⟩ resulting coupling term: c2 X Hi Hi

classical stationarity condition

⟨ ⟩ d i ∂A Ehrenfest theorem: ⟨A⟩ = − ⟨[A, H]⟩+ dt ℏ ∂t

d2 Require no class. acceleration: ⟨X⟩ = 0 dt2

17 ▶ in general time dependent, except for stationary states ▶ must be tuned to internal and( external energy) ▶ g (0) − ⟨ (0)⟩ resulting coupling term: c2 X Hi Hi

classical stationarity condition

⟨ ⟩ d i ∂A Ehrenfest theorem: ⟨A⟩ = − ⟨[A, H]⟩+ dt ℏ ∂t

d2 Require no class. acceleration: ⟨X⟩ = 0 dt2

If we want to satisfy this by a potential Uext(X, t): ( ) 1 ⟨P2⟩ − 2⟨P2⟩ U = −M g X − ⟨H(0)⟩ + x g X ext c2 i 2M .

17 ▶ must be tuned to internal and( external energy) ▶ g (0) − ⟨ (0)⟩ resulting coupling term: c2 X Hi Hi

classical stationarity condition

⟨ ⟩ d i ∂A Ehrenfest theorem: ⟨A⟩ = − ⟨[A, H]⟩+ dt ℏ ∂t

d2 Require no class. acceleration: ⟨X⟩ = 0 dt2

If we want to satisfy this by a potential Uext(X, t): ( ) 1 ⟨P2⟩ − 2⟨P2⟩ U = −M g X − ⟨H(0)⟩ + x g X ext c2 i 2M .

▶ in general time dependent, except for stationary states

17 ( ) ▶ g (0) − ⟨ (0)⟩ resulting coupling term: c2 X Hi Hi

classical stationarity condition

⟨ ⟩ d i ∂A Ehrenfest theorem: ⟨A⟩ = − ⟨[A, H]⟩+ dt ℏ ∂t

d2 Require no class. acceleration: ⟨X⟩ = 0 dt2

If we want to satisfy this by a potential Uext(X, t): ( ) 1 ⟨P2⟩ − 2⟨P2⟩ U = −M g X − ⟨H(0)⟩ + x g X ext c2 i 2M .

▶ in general time dependent, except for stationary states ▶ must be tuned to internal and external energy

17 classical stationarity condition

⟨ ⟩ d i ∂A Ehrenfest theorem: ⟨A⟩ = − ⟨[A, H]⟩+ dt ℏ ∂t

d2 Require no class. acceleration: ⟨X⟩ = 0 dt2

If we want to satisfy this by a potential Uext(X, t): ( ) 1 ⟨P2⟩ − 2⟨P2⟩ U = −M g X − ⟨H(0)⟩ + x g X ext c2 i 2M .

▶ in general time dependent, except for stationary states ▶ must be tuned to internal and( external energy) ▶ g (0) − ⟨ (0)⟩ resulting coupling term: c2 X Hi Hi

17 relativistic interaction

Consider a Klein-Gordon field, minimally coupled to an

electromagnetic field, i. e. iℏ∂t → iℏ∂t − e Φ.

To order 1/c2: Schrödinger equation ( ) p2 p4 iℏψ˙ = − + V ψ 2m 8m3c2

with e2Φ2 eΦ p2 V = −e Φ + + 2mc2 4m2c2 .

▶ relativistic correction of the interaction depends on momentum

18 supporting quantum interaction

d2 ⟨ ⟩ If we want to satisfy the condition dt2 X = 0 by a quantum interaction:

1 g U (X, P) = −M g X − H(0) g X − {X, P2} + f(P) ext c2 i 4Mc2 .

This exactly cancels all gravitational terms in the Hamiltonian:

Rindler, ext. pot. Minkowski, free Hcm = Hcm + f(P) .

19 .conclusion summary

Minkowski observer (a) Rindler observer (b) ↑ g y x = y= x

(c) (d) ↑ g ↑ g y x = y= x

21 . summary

Minkowski observer (a) Rindler observer (b) ↑ g y x = y= x

(c) (d) ↑ g ↑ g y x = y= x

21 . summary

Minkowski observer (a) Rindler observer (b) ↑ g y x = y= x

(c) (d) ↑ g ↑ g y x = y= x

21 . summary

Minkowski observer (a) Rindler observer (b) ↑ g y x = y= x

(c) (d) ↑ g ↑ g y x = y= x

21 . summary

Minkowski observer (a) Rindler observer (b) ↑ g y x = y= x

(c) (d) ↑ g ↑ g y x = y= x

21 . this is joint work with Marko Toroš & Angelo Bassi

Thank You!

Questions? .

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