The trajectory approach of quantum mechanics

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Warsaw - Oct. 17, 2016

Patrick Peter

Institut d’Astrophysique de Paris GRεCO

The trajectory approach of quantum mechanics

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Homogeneous & Isotropic metric (FLRW):

Last

Scattering surface

ds² = −dt² + a² (t)

! dr²

1 − Kr² + r² "dθ² + sin² θdφ²#

$ ,

dη = dt

a(t) =⇒

ds² = a²(η) "−dη² + γ_ij dxⁱdx^j # S =

% √

−g

&

− R

6ℓ²_P + p '

d⁴x

p = ωρ

n_T = n_S − 1 = 12ω 1 + 3ω T

S ≃ 4 × 10⁻²(n_S − 1

1 Matter component: perfect fluid:

Last

Scattering surface

T_µν = pg_µν + (ρ + p)u_µu_ν

ds² = −dt² + a² (t)

! dr²

1 − Kr² + r² "dθ² + sin² θdφ²#

$ ,

dη = dt

a(t) =⇒

ds² = a²(η) "−dη² + γ_ij dxⁱdx^j #

S =

% √

−g

&

− R

6ℓ²_P + p '

d⁴x p = ωρ

n_T = n_S − 1 = 12ω 1 + 3ω

1

{

^radiationdust

Last

Scattering surface

H² + K

a² = 1

3 (8πG_N ρ + Λ) ω = 1

3 p = ωρ

T_µν = pg_µν + (ρ + p)u_µu_ν

ds² = −dt² + a² (t)

! dr²

1 − Kr² + r² "dθ² + sin² θdφ²#

$ ,

dη = dt

a(t) =⇒

ds² = a²(η) "−dη² + γ_ij dxⁱdx^j#

1

+ cosmological constant = Einstein equation:

Id

τ_a

H ≡ ˙a

a = da a dt

U(1) × U(1) ̸∈ SU(2)

ℑm(ω₁)

P₄(ω) = ω⁴ + #₃ω³ + #₂ω² + #₁ω + #₀ = 0

k ≡ k_z ∈ R & ω ≡ k_t

A_zu^z_s 1 Hubble rate

Spatial curvature

or w ⇤ 1 perfect fluid ?

¨ a

a = 1

3 [ 4⇥G_N (⇤ + 3p)]

L = m² M ²

2 ln 1 + w M ²

⇥

m dx(t)

dt = ~ ⌅m ⇥^⇤r⇥

|⇥(x)|² = ~rS

⇤ ⇥ T [⇤]

|⌅⌃⇧⌅| ⇥

⇤

d³xP (x)|⌅^x⌃⇧⌅^x|

|| |⌅^x⌃||² = T [|⌅⌃⇧⌅|]

Lⁱ_x =

⇥

⇥3/4

e ^(qⁱ ^x)²^/2

1 p = w⇤

d| ⇧ = i ˆH| ⇧dt + ⌃ ⇣ ˆC ⌅ ˆC⇧⌘

dW_t| ⇧

2

⇣Cˆ ⌅ ˆC⇧⌘2

dt| ⇧ E (dW^tdW_t⁰) = dtdt⁰⇥(t t⁰)

⌅ ˆC⇧ ⇤ ⌅ | ˆC| ⇧

5.9 ⇥ 10 ²⁸ m for the Earth

1

w = 0 d| ⇧ = i ˆH| ⇧dt + ⌃ ⇣ ˆC ⌅ ˆC⇧⌘

dW_t| ⇧

2

⇣Cˆ ⌅ ˆC⇧⌘2

dt| ⇧ E (dW^tdW_t⁰) = dtdt⁰⇥(t t⁰)

⌅ ˆC⇧ ⇤ ⌅ | ˆC| ⇧

1

w = ¹₃ dT

T ⌅ ˆv d|⇥⌃ = i ˆH|⇥⌃dt + ⌥ ⇣ ˆC ⇧ ˆC⌃⌘

dW_t|⇥⌃

2

⇣Cˆ ⇧ ˆC⌃⌘2

dt|⇥⌃

E (dW^tdW_t⁰) = dtdt⁰⇥(t t⁰)

⇧ ˆC⌃ ⇤ ⇧⇥| ˆC|⇥⌃

1

Motivations: (quantum) cosmology

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Particular solution: matter & radiation

Integrate conservation equation:

for the conservation equation,

H ² + K = 8 G_N

3 ⇥ +

3

⇥

a² (21)

for the constraint, and finally

H = ⇤ 4 G_N

3 (⇥ + 3p) +

3

⌅

a². (22)

There exists a special solution, which happens to be realized in our Universe, at least so seem to say the data, namely that for which the spatial curvature K vanishes. It defines a density, called the critical density ⇥_c given by

⇥_c ⇤ 3H²

8 G_N =⌃ ⇥ ⇤ ⇥

⇥_c , (23)

in terms of which one can express all densities in a dimensionless way. For each fluid component but the cosmological constant, one can set ⇥_a = 8 G_N ⇥_a/(3H²) = ⇥_a/⇥_c; we also introduce an equivalent curvature "density" as ⇥_K = K/(a²H²) and finally

⇥ = /(3H²), and then the Friedmann constraint simply reads:

⌥

a

⇥_a + ⇥ + ⇥_K = 1, (24)

so the Friedmann equation is understandable as an energy budget: all possible contri- butions basically sum up to 100%! Numerically, the Hubble constant today is measured to be of the order of H₀ = 100hkm · s ¹ · Mpc ¹, where h = 0.704 ± 0.025. Similarly, the relative densities are also measured in units of the critical density, estimated as

⇥_c ⌅ 1.9 ⇥ 10 ²⁹g · cm ³; they frequently are found expressed as ⇥⁰_i = ⇥⁰_i h² to account for the indeterminacy of the Hubble expansion rate as well as on the density parameter itself, the subscript “0” meaning the present-day value.

Special solution: matter and radiation

With a varying equation of state w(t) and a scale factor a(t), which is a monotonic function of time, it is always possible to parameterize all functions of time as functions of a, and in particular w. On can then formally integrate the conservation equation as

⇥[a(t)] = ⇥_ini exp

⇧

3 [1 + w(a)]dlna

⌃

w⇧cst= ⇥_ini a a_ini

⇥ _3(1+w)

, (25)

which gives an exact solution for the constant equation of state situation. This is pre- cisely the case when matter (w = w_m = 0) or radiation (w = w_r = ¹₃ ) dominates over everything else. Eq. (25) then shows that matter scales as ⇥_m ⌥ a ³, as expected from mass conservation in an expanding volume, while radiation gets an extra power, scaling for the conservation equation,

H ² + K = 8 G_N

3 ⇥ +

3

⇥

a² (21)

for the constraint, and finally

H = ⇤ 4 G_N

3 (⇥ + 3p) +

3

⌅

a². (22)

There exists a special solution, which happens to be realized in our Universe, at least so seem to say the data, namely that for which the spatial curvature K vanishes. It defines a density, called the critical density ⇥_c given by

⇥_c ⇤ 3H²

8 G_N =⌃ ⇥ ⇤ ⇥

⇥_c , (23)

in terms of which one can express all densities in a dimensionless way. For each fluid component but the cosmological constant, one can set ⇥_a = 8 G_N ⇥_a/(3H²) = ⇥_a/⇥_c; we also introduce an equivalent curvature "density" as ⇥_K = K/(a²H²) and finally

⇥ = /(3H²), and then the Friedmann constraint simply reads:

⌥

a

⇥_a + ⇥ + ⇥_K = 1, (24)

so the Friedmann equation is understandable as an energy budget: all possible contri- butions basically sum up to 100%! Numerically, the Hubble constant today is measured to be of the order of H₀ = 100hkm · s ¹ · Mpc ¹, where h = 0.704 ± 0.025. Similarly, the relative densities are also measured in units of the critical density, estimated as

⇥_c ⌅ 1.9 ⇥ 10 ²⁹g · cm ³; they frequently are found expressed as ⇥⁰_i = ⇥⁰_i h² to account for the indeterminacy of the Hubble expansion rate as well as on the density parameter itself, the subscript “0” meaning the present-day value.

Special solution: matter and radiation

With a varying equation of state w(t) and a scale factor a(t), which is a monotonic function of time, it is always possible to parameterize all functions of time as functions of a, and in particular w. On can then formally integrate the conservation equation as

⇥[a(t)] = ⇥_ini exp

⇧

3 [1 + w(a)]dlna

⌃

w⇧cst= ⇥_ini a a_ini

⇥ _3(1+w)

, (25)

which gives an exact solution for the constant equation of state situation. This is pre- cisely the case when matter (w = w_m = 0) or radiation (w = w_r = ¹₃ ) dominates over everything else. Eq. (25) then shows that matter scales as ⇥_m ⌥ a ³, as expected from mass conservation in an expanding volume, while radiation gets an extra power, scaling

as _r ⇤ a ⁴, due to the redshift of its wavelength. Now consider an initial condition con- sisting of given relative amounts of matter and radiation. When the Universe begins its evolution, with a small value of the scale factor, radiation dominates and the total density is _tot ⇥ ^r until it gets caught up by the dustlike matter. This remarkably accurate picture for the Universe density evolution is illustrated in figure 4.

γ O

Visibility function, 10 3 e –τ

dt dz 5

4

3

2

1

0

0 500 1000

Redshift W₀ h² = 1

W₀ h² = 0.1 W₀ h² = 0.01

Equality a

a

Today r

a^–3

Nucleosynthesis

Decoupling a^–4

FIGURE 4. Top – Evolution of densities: the Universe begins dominated by radiation, whose density decreases faster than that of matter, so the latter ultimately dominates. Not shown is the final phase of domination by a cosmological constant which, as its name indicates, behaves as a constant. The point at which radiation and matter contribute equally is, not surprisingly, called equality. Bottom – On the same scale, matter density is depicted together with a typical light ray, whose mean free path is initially much shorter than the Hubble scale, as e.g. during nucleosynthesis; as the matter density gets smaller and smaller, the mean free path eventually becomes larger than the Hubble scale after what is therefore denoted decoupling. The Universe becomes transparent to this radiation we now observe as the microwave background.

The meaning of the equation of state is clarified when one considers a perturbation propagating in the fluid. As is well known in fluid dynamics and as we shall also discuss later, the sound velocity c_s is given² by c²_s = dp/d = p^⌅/ ^⌅. It can be shown (and the reader is encouraged to do so!), that the relation

w^⌅ = 3H (1 + w) c²_s w⇥

(26) holds, so that a constant equation of state means w = c²_s .

With the solution for the density as a function of the scale factor and the equation of state given, it is an easy matter to solve the Friedman equation. For a vanishing spatial

2 In fact, it should be partial derivative for constant entropy.

Phenomenologically valid description for almost 14 Gyrs!!!

as _r ⇤ a ⁴, due to the redshift of its wavelength. Now consider an initial condition con- sisting of given relative amounts of matter and radiation. When the Universe begins its evolution, with a small value of the scale factor, radiation dominates and the total density is _tot ⇥ ^r until it gets caught up by the dustlike matter. This remarkably accurate picture for the Universe density evolution is illustrated in figure 4.

γ O

Visibility function, 10 3 e –τ

dt dz 5

4

3

2

1

0

0 500 1000

Redshift W₀h² = 1

W₀h² = 0.1 W₀h² = 0.01

Equality a

a

Today r

a^–3

Nucleosynthesis

Decoupling a^–4

FIGURE 4. Top – Evolution of densities: the Universe begins dominated by radiation, whose density decreases faster than that of matter, so the latter ultimately dominates. Not shown is the final phase of domination by a cosmological constant which, as its name indicates, behaves as a constant. The point at which radiation and matter contribute equally is, not surprisingly, called equality. Bottom – On the same scale, matter density is depicted together with a typical light ray, whose mean free path is initially much shorter than the Hubble scale, as e.g. during nucleosynthesis; as the matter density gets smaller and smaller, the mean free path eventually becomes larger than the Hubble scale after what is therefore denoted decoupling. The Universe becomes transparent to this radiation we now observe as the microwave background.

The meaning of the equation of state is clarified when one considers a perturbation propagating in the fluid. As is well known in fluid dynamics and as we shall also discuss later, the sound velocity c_s is given² by c²_s = dp/d = p^⌅/ ^⌅. It can be shown (and the reader is encouraged to do so!), that the relation

w^⌅ = 3H (1 + w) c²_s w⇥

(26) holds, so that a constant equation of state means w = c²_s.

With the solution for the density as a function of the scale factor and the equation of state given, it is an easy matter to solve the Friedman equation. For a vanishing spatial

2 In fact, it should be partial derivative for constant entropy.

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Numerical simulation for

large scale structure formation

Comparison with observational

Numerical simulations: C. Pichon @ IAP

data

10⁵

10⁴

1000

100

10

1

10⁴ 1000

0.001 0.01 0.1 1 10

100 10 1

Wavelength (h^–1 Mpc)

Wave number k (h / Mpc) Power spectrum today P(k) (h–1 Mpc)3

CMB

SDSS Galaxies Cluster abundance Gravitational lensing Lyman – α forest

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A central problem (though not often formulated thus…) : the singularity

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Singularity problem Quantum effect?

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w ⌘

^ab

@

_a

@

_b

⌃ (x

^↵

) = (r)e

^{i(!t kz)}

⌘ x

^?

e

^{i (⇠}^a⁾

!

²

(⌘, k) )

s/m

³

n/m

³

⌧ 1 2

3 + H

¹

˙

1 + w + a ' ˙

H (

8

gaussian signal

almost scale invariant

w⌘ab @ a @ b ⌃(x↵ )=(r)ei(!tkz) ⌘x? ei (⇠a ) !2 (⌘,k) ) s/m3 n/m3 ⌧1 2 3+H1 ˙ 1+w+ a˙' H ( 8

excluded

isocurvature _{. 1%}

16

compatible with INFLATION E (dW^tdW_t⁰) = dtdt⁰ (t t⁰)

h ˆCi ⌘ h | ˆC| i

f_NL^local 2 [20, 50]

f_NL^local = ±5 + 3

2₃p

✏

⇢ + p < 0

⇢_ekp / a ³(^1+w^ekp) a ⁶

n_s & 1

f_NL^ortho = 25 ± 39

n_s = 0.9639 ± 0.0047 15

. 1%

dt = a(⌘)d⌘

⌘

H ⌘ 1 a

da

dt ⌘ ˙a a H ⌘ 1

a

da

d⌘ ⌘ a⁰ a

⇢ + P  0

10 ²h ¹ Mpc  k_phys¹  10³h ¹ Mpc

¨ a

a = 1

3 V '˙ ² T

T / GNE_inf² ⇠

✓ E_inf M_Pl

◆2

⌦_K = 0.000 ± 0.005

16

r < 0.11

17

E (dW^tdW_t⁰) = dtdt⁰ (t t⁰)

f_NL^local 2 [20, 50]

2₃p

✏

⇢ + p < 0

⇢_ekp / a ³(^1+w^ekp) a ⁶

n_s & 1

f_NL^ortho = 25 ± 22

n_s = 0.9639 ± 0.0047 15

f_NL^local 2 [20, 50]

2₃p

✏

⇢ + p < 0

⇢_ekp / a ³(^1+w^ekp) a ⁶

n_s & 1

f_NL^equil = 9.5 ± 44

n_s = 0.9639 ± 0.0047 15

f_NL^local 2 [20, 50]

2₃p

✏

⇢ + p < 0

⇢_ekp / a ³(^1+w^ekp) a ⁶

n_s & 1

f_NL^local = 0.71 ± 5.1

n_s = 0.9639 ± 0.0047 15

quantum vacuum fluctuations of a single scalar d.o.f

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Quantum mechanics

Physical system = Hilbert space of configurations State vectors

Observables = self-adjoint operators

Measurement = eigenvalue A|aⁿ⇥ = aⁿ|aⁿ⇥

⇥ [a(t), ⌅(t)]

h_ij dxⁱdx^j = a²(t)

 dr²

1 kr² + r² d⇥² + sin² ⇥d⇧² (x) = ⌅(t) ˆ

⇤⇥ = i ⇥

N = 0 ˆ

⇤ⁱ⇥ = i ⇥

Nⁱ = 0 {h^ij(x), (x)}

⇥ [h_ij (x), (x)]

1

Evolution = Schrödinger equation (time translation invariance) i~ d

dt|⇧(t)⇥ = ˆH|⇧(t)⇥

A|aⁿ⇥ = aⁿ|aⁿ⇥

⇥ [a(t), ⌅(t)]

h_ijdxⁱdx^j = a²(t)

 dr²

1 kr² + r² d⇥² + sin² ⇥d⌃² (x) = ⌅(t) ˆ

⇤⇥ = i ⇥

N = 0 ˆ

⇤ⁱ⇥ = i ⇥

Nⁱ = 0 {h^ij(x), (x)}

1

Hamiltonian Born rule Prob[a_n; t] = |⇥aⁿ|⇧(t)⇤|²

i~ d

dt |⇧(t)⇤ = ˆH |⇧(t)⇤

A|aⁿ⇤ = aⁿ|aⁿ⇤

⇥ [a(t), ⌅(t)]

h_ij dxⁱdx^j = a²(t)

 dr²

1 kr² + r² d⇥² + sin² ⇥d⌃² (x) = ⌅(t) ˆ

⇤⇥ = i ⇥

N = 0 ˆ

⇤ⁱ⇥ = i ⇥

Nⁱ = 0 1

Collapse of the wavefunction:

Prob[a

_n

; t] = |⇥a

ⁿ

|⇧(t)⇤|

²

i ~ d

dt |⇧(t)⇤ = ˆ H |⇧(t)⇤

A |a

ⁿ

⇤ = a

ⁿ

|a

ⁿ

⇤

⇥ [a(t), ⌅(t)]

h

_ij

dx

ⁱ

dx

^j

= a

²

(t)

 dr

²

1 kr

²

+ r

²

d⇥

²

+ sin

²

⇥d⌃

²

(x) = ⌅(t) ˆ

⇤⇥ = i ⇥

N = 0 ˆ

⇤

ⁱ

⇥ = i ⇥

N

ⁱ

= 0 1

before measurement,

Prob[a_n; t] = |⇥aⁿ|⇧(t)⇤|² i~ d

dt |⇧(t)⇤ = ˆH |⇧(t)⇤

A|aⁿ⇤ = aⁿ|aⁿ⇤

⇥ [a(t), ⌅(t)]

h_ijdxⁱdx^j = a²(t)

 dr²

1 kr² + r² d⇥² + sin² ⇥d⌃² (x) = ⌅(t) ˆ

⇤⇥ = i ⇥

N = 0 ˆ

⇤ⁱ⇥ = i ⇥

Nⁱ = 0 1

after

Schrödinger equation = linear (superposition principle) / unitary evolution

Wavepacket reduction = non linear / stochastic

}

^Mutuallyincompatible + External observer

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Usually in a lab:

repeat the experiment

Ensemble average over

experiments

Quantum average

Ergodicity Here one has a single

experiment (a single universe)

Spatial

average over directions in

the sky

Quantum average

Predictions for a quantum theory

Predictions of the theory: Calculated by quantum average

| ˆ O | ⇥

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The measurement problem in quantum mechanics

Definite outcome: we don’t measure superpositions

Preferred basis: no unique definition of measured observables

collapse of the wave function

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Stern-Gerlach

x dT

T ⌅ ˆv d|⇥⌃ = i ˆH|⇥⌃dt + ⌥ ⇣ ˆC ⇧ ˆC⌃⌘

dW_t|⇥⌃

2

⇣Cˆ ⇧ ˆC⌃⌘2

dt|⇥⌃

E (dW^tdW_t⁰) = dtdt⁰⇥(t t⁰)

⇧ ˆC⌃ ⇤ ⇧⇥| ˆC|⇥⌃

1

x dT

T ⌅ ˆv d|⇥⌃ = i ˆH|⇥⌃dt + ⌥ ⇣ ˆC ⇧ ˆC⌃⌘

dW_t|⇥⌃

2

⇣Cˆ ⇧ ˆC⌃⌘2

dt|⇥⌃ E (dW^tdW_t⁰) = dtdt⁰⇥(t t⁰)

⇧ ˆC⌃ ⇤ ⇧⇥| ˆC|⇥⌃

1

Statistical mixture

|⇥ⁱⁿ = 1

⌦2 (|⇤ + |⌅ ) ⇥ |SGⁱⁿ n|⇤ ⇥ |SGⁱⁿ o

⌥ n

|⌅ ⇥ |SGⁱⁿ o n (|⇤ + |⌅ ) ⇥ |SGⁱⁿ o

n|⇤ ⇥ |SG o

⌥ n

|⌅ ⇥ |SG⇥

o

⇥_x(⌃) =

✓

4m

◆ ¹₄

dT

T ⇧ ˆv d|⇥ = i ˆH|⇥ dt + ⌦ ⇣ ˆC Cˆ ⌘

dW_t|⇥

2

⇣Cˆ Cˆ ⌘2

dt|⇥

1

|⇥ⁱⁿ = 1

⌦2 (|⇤ + |⌅ ) ⇥ |SGⁱⁿ n|⇤ ⇥ |SGⁱⁿ o

⌥ n

|⌅ ⇥ |SGⁱⁿ o n (|⇤ + |⌅ ) ⇥ |SGⁱⁿ o

n|⇤ ⇥ |SG o

⌥ n

|⌅ ⇥ |SG⇥

o

⇥_x(⌃) =

✓

4m

◆ ¹₄

dT

T ⇧ ˆv d|⇥ = i ˆH|⇥ dt + ⌦ ⇣ ˆC Cˆ ⌘

dW_t|⇥

2

⇣Cˆ Cˆ ⌘2

dt|⇥

1

|⇥ⁱⁿ = 1

⌦2 (|⇤ + |⌅ ) ⇥ |SGⁱⁿ n|⇤ ⇥ |SGⁱⁿ o

⌥ n

|⌅ ⇥ |SGⁱⁿ o n (|⇤ + |⌅ ) ⇥ |SGⁱⁿ o

n|⇤ ⇥ |SG o

⌥ n

|⌅ ⇥ |SG_⇥ o

⇥_x(⌃) =

✓

4m

◆ ¹₄

dT

T ⇧ ˆv d|⇥ = i ˆH|⇥ dt + ⌦ ⇣ ˆC Cˆ ⌘

dW_t|⇥

2

⇣Cˆ Cˆ ⌘2

dt|⇥ 1

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Stern-Gerlach

What about situations in which one has only one realization?

What about the Universe itself?

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What do we do with the wave function of the Universe???

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Possible extensions and a criterion: the Born rule

Superselection rules Modal interpretation

Many worlds / many minds Hidden variables

Modified Schrödinger dynamics

}

Born rule not put by hand!

A. Bassi, G.C. Ghirardi / Physics Reports 379 (2003) 257–426 277

Fig. 1.

4. Possible ways out of the macro-objecti!cation problem

Various ways to overcome the measurement problem have been considered in the literature: in this section we brie!y describe and discuss them. It is useful to arrange the various proposals in a hierarchical tree-like structure [14], taking into account the fundamental points on which they di"er:

in the #gure below we present a diagram which may help in following the argument. Subsequently we will comment on the various options.

4.1. Listing the possible ways out

A #rst distinction among the alternatives which have been considered in the literature derives from taking into account the role which they assign to the statevector | ⟩ of a system (Fig. 1). This leads to the Incompleteness versus Formal Completeness option:

Incompleteness: this approach rests on the assertion that the speci#cation of the state | ⟩ of the system is insu$cient: further parameters, besides the wavefunction, must be considered, allowing us to assign de#nite properties to physical systems.

Formal Completeness: it is assumed that the assignment of the statevector represents the most accurate possible speci#cation of the state of a physical system.

When the assumption of Formal Completeness is made, two fundamentally di"erent positions can be taken about the status of an ensemble—a pure case in the standard scheme—all individuals of

A. Bassi & G.C. Ghirardi, Phys. Rep. 379, 257 (2003) U T 6= 0

> 1 m_' = p

2 ⌘ m_C = p

2e⌘

= m²_' m²_C

bckd =

'(r)e^in✓

0

⌃ / e^t/⌧

9 !² < 0 1

Consistent histories

+ TESTABLE!

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Schrödinger

Polar form of the wave function Hamilton-Jacobi

quantum potential Hidden Variable Theories

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1927 Solvay meeting and von Neuman mistake ... ‘In 1952, I saw the impossible done’ (J. Bell)

Louis de Broglie (duke) David Bohm (communist)

Ontological formulation (dBB)

(17)

⇧ x(t)

⇥ ⇥

^tot

crit

|

^f

⌥ = exp

⇤

_t_f

t_in

H (⇥ ) d⇥ ˆ

⇥

|

ⁱⁿ

⌥

= 1

2 ( |⇤ ⌥ |SG ⌥ + |⌅ ⌥ |SG

⇥

⌥) (1)

|

ⁱⁿ

⌥ = 1

2 ( |⇤ ⌥ + |⌅ ⌥) |SG

ⁱⁿ

⌥

⌅ |⇤ ⌥ |SG

ⁱⁿ

⌥ ⇧

⌃ ⌅

|⌅ ⌥ |SG

ⁱⁿ

⌥ ⇧

⌅ ( |⇤ ⌥ + |⌅ ⌥) |SG

ⁱⁿ

⌥ ⇧

1

Ontological formulation (dBB) Trajectories satisfy (de Broglie) m dx

dt = ⌥m r

| (x, t)|² = rS

⌃ x(t)

⇥ ⇤ ^tot

crit

| ^f = exp

⇤ _t_f

t_in

H (⇥ ) d⇥ˆ

⇥

| ⁱⁿ

= 1

⌦2 (|⌅ ⇥ |SG⇥ + |⇧ ⇥ |SG⇤ ) (1)

| ⁱⁿ = 1

⌦2 (|⌅ + |⇧ ) ⇥ |SGⁱⁿ

⌅|⌅ ⇥ |SGⁱⁿ ⇧ ⌅

|⇧ ⇥ |SGⁱⁿ ⇧ 1

(18)

⇧ x(t)

⇥ ⇥

^tot

crit

|

^f

⌥ = exp

⇤

_t_f

t_in

H (⇥ ) d⇥ ˆ

⇥

|

ⁱⁿ

⌥

= 1

2 ( |⇤ ⌥ |SG ⌥ + |⌅ ⌥ |SG

⇥

⌥) (1)

|

ⁱⁿ

⌥ = 1

2 ( |⇤ ⌥ + |⌅ ⌥) |SG

ⁱⁿ

⌥

⌅ |⇤ ⌥ |SG

ⁱⁿ

⌥ ⇧

⌃ ⌅

|⌅ ⌥ |SG

ⁱⁿ

⌥ ⇧

⌅ ( |⇤ ⌥ + |⌅ ⌥) |SG

ⁱⁿ

⌥ ⇧

1

Ontological formulation (BdB)

Trajectories satisfy (Bohm) ^Q ⇥ 1

2m

r²| |

| | m d²x

dt² = r(V + Q) m dx

dt = ⌅m r

| (x, t)|² = rS

⇤ x(t)

⇥ ⇥ ^tot

crit

| ^f ⇧ = exp

⇤ _t_f

t_in

H (⇥ ) d⇥ˆ

⇥

| ⁱⁿ⇧

1

Q ⇥ 1 2m

r

²

| |

| | m d

²

x

dt

²

= r(V + Q) m dx

dt = ⌅m r

| (x, t)|

²

= rS

⇤ x(t)

⇥ ⇥

^tot

crit

|

^f

⇧ = exp

⇤

_t_f

t_in

H (⇥ ) d⇥ ˆ

⇥

|

ⁱⁿ

⇧

1

(19)

⇧ x(t)

⇥ ⇥

^tot

crit

|

^f

⌥ = exp

⇤

_t_f

t_in

H (⇥ ) d⇥ ˆ

⇥

|

ⁱⁿ

⌥

= 1

2 ( |⇤ ⌥ |SG ⌥ + |⌅ ⌥ |SG

⇥

⌥) (1)

|

ⁱⁿ

⌥ = 1

2 ( |⇤ ⌥ + |⌅ ⌥) |SG

ⁱⁿ

⌥

⌅ |⇤ ⌥ |SG

ⁱⁿ

⌥ ⇧

⌃ ⌅

|⌅ ⌥ |SG

ⁱⁿ

⌥ ⇧

⌅ ( |⇤ ⌥ + |⌅ ⌥) |SG

ⁱⁿ

⌥ ⇧

1

Ontological formulation (dBB) Trajectories satisfy (de Broglie) m dx

dt = ⌥m r

| (x, t)|² = rS

⌃ x(t)

⇥ ⇤ ^tot

crit

| ^f = exp

⇤ _t_f

t_in

H (⇥ ) d⇥ˆ

⇥

| ⁱⁿ

= 1

⌦2 (|⌅ ⇥ |SG⇥ + |⇧ ⇥ |SG⇤ ) (1)

| ⁱⁿ = 1

⌦2 (|⌅ + |⇧ ) ⇥ |SGⁱⁿ

⌅|⌅ ⇥ |SGⁱⁿ ⇧ ⌅

|⇧ ⇥ |SGⁱⁿ ⇧ 1

Properties:

classical limit well defined state dependent

intrinsic reality

no need for external classical domain/observer!

strictly equivalent to Copenhagen QM

probability distribution (attractor)

non local …

⇤ x(t)

m d

²

x(t)

dt

²

= ⌃ (V + Q)

⇤t

⁰

; ⇤ (x, t

₀

) = | (x, t

⁰

)|

²

Q ⇥ 0

⇤

ds

²

= N

²

(⌅)d⌅ a

²

(⌅)

_ij

dx

ⁱ

dx

^j

p = p

₀

⇤ ˙⌃ + ⇥ ˙s

N (1 + ⇧)

⌅

¹⁺

(⌃, ⇥, s) =

T = p

_s

e

^s/s⁰

p

_⇥ ^{(1+ )}

s

₀

⇤

₀

H = p

²_a

4a Ka + p

_T

a

³

⇥

N a

³

2

Q ⇥ 1 2m

r²| |

| | m d²x

dt² = r(V + Q) m dx

dt = ⌅m r

| (x, t)|² = rS

⇤ x(t)

⇥ ⇥ ^tot

crit

| ^f ⇧ = exp

⇤ _t_f

t_in

H (⇥ ) d⇥ˆ

⇥

| ⁱⁿ⇧

1

( )

(20)

The two-slit experiment:

Surrealistic trajectories?

Non straight in vacuum...

X

… a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics.

R. P. Feynman (1961)

(21)

Warsaw - Oct. 17, 2016January 23rd 2014 21

Diffraction by a potential simple understanding of tunnelling ...

(22)

Aside: a nice hydrodynamical analogy Faraday waves… ⁽¹⁸³¹⁾

A₀ sin (2⇡f t) 70 Hz

0.19 mm

⇢ = | |²

⇢_ini 6= | |² v_ini / rS

⇢_ini 6= | |² v_ini 6/ rS S =

Z

dtL

S = 1

16⇡G_N

Z

M

d⁴xp

g ⁴R + S^matter

Prob(v_k) = | ^k|² / e ^2⌦^k^v^k² 2

A₀ sin (2⇡f t)

70 Hz

0.19 mm

⇢ = | |²

⇢_ini 6= | |² v_ini / rS

⇢_ini 6= | |² v_ini 6/ rS

S = Z

dtL

S = 1

16⇡G_N Z

M

d⁴xp

g ⁴R + S^matter

Prob(v_k) = | ^k|² / e ^2⌦^k^v^k² 2

0.19 mm

⇢ = | |²

⇢_ini 6= | |² v_ini / rS

⇢_ini 6= | |² v_ini 6/ rS S =

Z

dtL S = 1

16⇡G_N Z

M

d⁴xp

g ⁴R + S^matter Prob(v_k) = | ^k|² / e ^2⌦^k^v^k²

2

forced standing surface waves

Just above Faraday wave threshold

⇧₂₂ Silicone Oil

0.19 mm

⇢ = | |²

⇢_ini 6= | |² v_ini / rS

⇢_ini 6= | |² v_ini 6/ rS S =

Z

dtL

2

Y. Couder et al. ^(>2006) J. Walker ⁽¹⁹⁷⁸⁾

http://math.mit.edu/~bush/?page_id=252

(23)

(24)

Typical values for the

experiment

(25)

Bouncing droplet…

(26)

Bouncing droplet…

(27)

or bouncing droplets…

(28)

+ subharmonic modulation (larger forcing amplitude) => instability => motion!!!

droplets become walkers…

(29)

one image per bounce => suppress vertical motion => horizontal mode only

(30)

apparent randomness of the motion…

(31)

(32)

integrate over time…

(33)

longer times…

and reconstruct the

standing wave pattern!

(34)

Probability Distribution Function

(35)

Comparison with actual Faraday wave pattern

(36)

forms quantised bound states

(37)

Warsaw - Oct. 17, 2016 Morgan Freeman's "Through the Wormhole" / Science Channel  

Season II, episode 6 - How Does The Universe Work?

(38)

self-interfering classical particle!

experimental setup

(39)

actual snapshots

(40)

a couple of trajectories…

(41)

more trajectories!

apparently random again

(42)

Warsaw - Oct. 17, 2016 Morgan Freeman's "Through the Wormhole" / Science Channel  

Season II, episode 6 - How Does The Universe Work?

(43)

One slit + fit

Y. Couder and E. Fort, Phys. Rev. Lett. 97, 154101 (2006)

statistical determinacy

(44)

Tao slit + fit

… a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics.

R. P. Feynman (1961)

Y. Couder and E. Fort, Phys. Rev. Lett. 97, 154101 (2006)