INTRIQ 11-12 November 2014

Analogies between quantum and fluid mechanics

Is quantum mechanics, deep down, a fluid-dynamical theory ?

Louis Vervoort

Post-doc @ Département de Physique (R. MacKenzie) Université de Montréal and CIRST, UQAM (Y. Gingras)

[email protected]

Overview

1. Intro: recent experiments by Couder et al. (fluids mimicking QM)

2. The Schrödinger Eq. as a fluid-mechanical Eq. (Madelung 1927)

3. Can Bell’s theorem exclude that local ‘fluid-dynamical’ theories

complete QM ?

4. Conclusions, experiments & link with QIT

This is not a crusade against QM ! Introduction: quantum-like features in fluid-dynamical systems

 Experiments Yves COUDER et al. (Paris) (Nature, PRL 2005 – 2014)

 Quantum-like behavior of oil droplets on an oil-film

 https://www.youtube.com/watch?v=nmC0ygr08tE Introduction: quantum-like features in fluid-dynamical systems

Huygens’ principle Introduction: quantum-like features in fluid-dynamical systems

 Quantum-like behaviour through a ‘pilot-wave’ = surface wave on the oil film

created by external vibration + back-reaction of the droplet

 « First macroscopic particle + pilot wave » (recall de Broglie) Introduction: quantum-like features in fluid-dynamical systems Introduction: quantum-like features in fluid-dynamical systems Introduction: quantum-like features in fluid-dynamical systems

Quantization of angular momentum, Couder et al., PNAS 2010

Laboratory frame

Rotating frame

Average radius is quantized ! Introduction: quantum-like features in fluid-dynamical systems Degeneracy ‘co-rotating’ and ‘contra-rotating’ states is lifted, as in Zeeman Splitting (B // ) II. The Schrödinger equation as a hydrodynamic equation

Erwin Madelung, Zeitschrift für Physik, 1927

The Navier-Stokes equation reads: u (u )u = P . 2u f . (3a) M t

Here M = dM / dV is the mass density of an elementary fluid element with mass dM and volume dV; u the fluid element’s velocity (a vector field); P the thermodynamic pressure (related to

and the temperature T); P the pressure gradient; f the body force per unit of volume (

dF = ) acting on the fluid element (typically: gravity); and the dynamic viscosity (in units dV N.s/m2 = kg/(m.s)) depending on the thermodynamic state (the temperature). II. The Schrödinger equation as a hydrodynamic equation

The Navier-Stokes equation is only valid under a list of certain conditions – conditions which are however satisfied by most ‘normal’ fluids, hence its capital importance for fluid mechanics. The conditions of validity of Eq. (3a) are: (C1) the fluid is incompressible, (C2) the fluid is in thermodynamic equilibrium or nearly so; (C3) the fluid’s stress-strain tensor is linear and isotropic; (C4) the thermodynamic pressure in the fluid is equal to the mean of the normal stresses (Stokes’ assumption); (C5) the fluid is ‘Newtonian’ (this is implied by (C2-C4)); (C6) the viscosity is spatially constant in the fluid (the temperature differences in the fluid are small enough). II. The Schrödinger equation as a hydrodynamic equation

The second fundamental equation of fluid mechanics we will use is the mass conservation or ‘continuity’ equation:

M ( u) = 0. (3b) t M This equation expresses, in differential form, that the rate of increase of mass within a fixed volume must equal the rate of inflow through the boundaries.

To bring about the analogy between Eq. (3a-b) and the Schrödinger equation, we have to further suppose that our fluid consists of ‘particles’ or entities of constant mass ‘m’; so that a mass M of the fluid can be understood as composed of N masses m. This allows to introduce, in the usual manner, a probability density for the ‘fluid particles’ as follows: dM d(N.m) dN = = = m. ≡ .m . (4) M dV dV dV

Particle of mass m Fluid element (‘singularity’ following the fluid element, cf. Bohm & Vigier 1954) The Schrödinger equation as a hydrodynamic equation

Let us now re-write the equations (3a-b), making further simplifying hypotheses. The second term on the left in Eq. (3a) can be rewritten as follows, using standard vector differential calculus: u2 (u )u = [u ( u)] , (5) 2 where u ≡ is the vorticity which underlies vortex formation in the fluid. Let us assume (assumption C8) that the fluid is irrotational, i.e. its vorticity = is zero everywhere. Then we can suppose that the velocity derives from a scalar function (a field) S: 1 u ≡ S . (6) m

The Schrödinger equation as a hydrodynamic equation

Using Eq. (4-6), it is straightforward to transform the Navier-Stokes equation (3a) into: 1 1 f S ( S)2 = P . (7) t 2m

f Fluid dynamics usually assumes that the body forces f are conservative (C9), so that (or

) derives from a potential U, as is the case of gravity. If we finally assume (C10) that the flow is barotropic, i.e. that is a function of P only, then one easily proves (Kundu and Cohen 2008 If wep. 118now): define:

1 1 dP PP = = .≡ Q , ( 8 ) (9) then we readily obtain from (7):

1 S ( S)2 (U Q) 0 (10a) t 2m .

Now, with the same assumption and definition as introduced in (6), our second fundamental equation, the continuity equation (3b) becomes: S ( ) = 0. (10b) t m

IfII. we Thenow define:Schrödinger equation as a hydrodynamic equation 1 dP P = ≡ Q , (9) then we readily obtain from (7):

1 S ( S)2 (U Q) 0 (10a) t 2m (A1) .

Now, with the same assumption and definition as introduced in (6), our second fundamental equation, the continuity equation (3b) becomes: S ( ) = 0. (A2)(10b) t m

II. The Schrödinger equation as a hydrodynamic equation

It is then straightforward to relate the Eqs. (10a-b) to the 1-particle (or N-particle) Schrödinger equation: 2 i = 2 U . (11) t 2m Indeed, any solution to (11) can be written as:

By inserting ( 12 ) into ( 11 ) and = separating R.exp(iS / real) = and imaginary.exp(iS /  parts,), we readily obtain two(12 ) equatiwhereons R and for S are and real S, functions,equivalent and to where the Schrödinger = R2 is the equation probability for density. The firstof the equation particle is preciselywith mass (10 m,b ),according our (transformed) to the Born continuity interpretation equation of theof hydrodynamics. wave function. SThe is thesecond phase equation of the is:wave function. S ( S)2 U Q = 0, (13) If we plug (12) into (11)t 2wem obtain (A1) and (A2), if we define if we define:

2 2 2 2 2R Q ≡ = . (14) 2m 2 2 2mR

Finally, if we take the gradient of Eq. (13) we precisely obtain the (transformed) Navier- Stokes equation (10a). Alternatively, we could integrate Eq. (10a) and obtain Eq. (13) (if we put the integration constant equal to zero). II. The SchrödingerBy inserting equation (12) as into a ( 11hydrodynamic) and separating real equation and imaginary parts, we readily obtain two In conclusion, there is a formalequati analogyons for between and S, equivalent the Schrödinger to the Schrödinger equation equation and basic for equations. The first equation is of fluidIn dynamics conclusion,; at there least is aif preciselyformal we make analogy (10 certainb), ourbetween (transformed) assumptions the Schrödinger continuity for theequation equationfluid, and namely of basic hydrodynamics. equations C1- C10 a Thebove second. equation is: Clearly,of even fluid dynamicsif assumptions; at least ifC1 we –make C10 certain are ‘natural’ assumptions and for describe the fluid, namelya wide C1 variety- C10 a boveof fluids,. the Clearly, even if assumptions C1 – C10 are ‘natural’ and describeS ( S )a2 wide variety of fluids, the analogy is not perfect. We would have a perfect analogy if we wereU ableQ = 0,to derive the precise (13) analogy is not perfect. We would have a perfect analogy ift we were2m able to derive the precise form ofform the ofquantum the quantum potential potentialif we Q define: Qgiven given inin (14)) by by using using fluid fluid-mechanical-mechanical arguments arguments only; i.e. only ; i.e. 2 dPdP 2 2 2 2R if we wereif we able were toable show to show that that in in certain certain fluids fluids Q = =Q≡ Qin in(14 ()14. ). = = Q . (14) 2m 2 2 2mR

Finally, if we take the gradient of Eq. (13) we precisely obtain the (transformed) Navier- Stokes equation (10a). Alternatively, we could integrate Eq. (10a) and obtain Eq. (13) (if we Newton’s put the integration constant equal to zero). m.u = U Q Q = Quantum potential, due to 2nd law: inner stresses in the Madelung fluid

Wilhelm, Phys. Rev. 1970 III. Link between experiments of Couder et al. & Schrödinger Eq.

 Bohm and Vigier (1954) and Wilhelm (1970) have elaborated Madelung’s theory.  Certain theoretical results led Bohm and Vigier (1954) to hypothesize that the particles of mass m of Madelung’s interpretation are singularities of some nature that follow the ‘Madelung fluid’ (the ether, the physical vacuum, a field,…) while undergoing stochastic fluctuations – fluctuations about the mean values (x,t) and û(x,t) described by the Madelung equations.

Particle of mass m: Fluid element ‘singularity’ following the fluid element, cf. Bohm & Vigier 1954 III. Link between experiments of Couder et al. & Schrödinger Eq.

 This interpretation of the formulas allows to make a direct link with the

Couder experiments:

 The ‘singularity’ of the Madelung theory the point of impact of

the droplets on the fluid film (the droplets follow this point).

 Under this assumption it is possible to use the Madelung equations ( =

Schrödinger Eq.) to describe the trajectories of the individual droplets in

the Couder experiments ! (cf. Sbitnev 2014) IV. Can we push the analogy fluid / quantum mechanics farther ?

 It remains an open question whether the whole of QM (not only the Schrödinger Eq.) can be derived from (the formal analogy of) fluid mechanics.

 Such attempts have been discredited by ‘no-go’ theorems as Bell’s: NO LOCAL THEORY CAN COMPLETE QM

[local = based on sub-luminal, Lorentz-invariant interactions] IV. Is the fluid-mechanical interpretation prohibited by Bell’s theorem ? Bell’s theorem (stochastic version)

a Source b 1 2

Hypothesis of stochastic hidden-variables:

P( i | ) : determines the P of i

M(a,b) = < 1. 2 >a,b = 1 2 1. 2.P( 1, 2|a,b).

XBI = M(a,b) + M(a’,b) + M(a,b’) – M(a’,b’) ≤ 2 IV. Is the fluid-mechanical interpretation prohibited by Bell’s theorem ?

 Variable X is correlated with variable Y IFF P(X | Y) ≠ P(X)

(X,Y are n-vectors)

X Y “correlation graph” IV. Is the fluid-mechanical interpretation prohibited by Bell’s theorem ?

M1. Bell’s hidden-variable model

P( 1 a, ) P( ) P( 2 b, )

LOC:

MI:

M1 P( , a,b, ).P( a,b) P( a, ).P( b, ).P( ) P ( 1, 2 a,b) = 1 1 = 1 2 IV. Is the fluid-mechanical interpretation prohibited by Bell’s theorem ?

M1. Bell’s model a b

P( 1 a, ) P( )

max X BI = 2√2 = 2.83 for certain angles IV. Is the fluid-mechanical interpretation prohibited by Bell’s theorem ?

M1. Bell’s model a b

P( 1 a, ) P( )

LOC XBI ≤ 2 MI

 In sum: the BI allows to experimentally discriminate QM from local HV theories à la Bell (M1). IV. Is the fluid-mechanical interpretation prohibited by Bell’s theorem ?

M1: a b

M2: a b

M3: a b IV. Is the fluid-mechanical interpretation prohibited by Bell’s theorem ?

 In a fluid-mechanical model for the Bell experiment, the Bell particles and analyzers will interact with a fluid-like medium, just as in the experiments of Couder et al. the droplets interact with a fluid.

Alice Bob a b

Background ‘fluid’ / field

Simplest a b model (M2): General Fluid- or ‘Background-based’ models for the Bell experiment

M2. Naïve Fluid Model a b

Now 1 and 2 represent P( 1 a, 1) stochastic properties (field intensities,…) of the fluid / P( 1 a) medium in the neighborhood of the analyzers

Alice Bob a b

Background ‘fluid’ / field

P( |a,b) ≠ P |a’,b’) so MI is violated ! General Fluid- or ‘Background-based’ models for the Bell experiment

M2. Naïve Fluid Model a b

Now 1 represent stochastic properties (field intensities,…) of the fluid / medium in the neighborhood of the analyzers General Fluid- or ‘Background-based’ models for the Bell experiment

M3. Second Fluid Model

M1: a b

M2: a b

M3: a b General Fluid- or ‘Background-based’ models for the Bell experiment

M3. Second Fluid Model a b

P( 1 , 1,a), P( 1, 2 ), P( 1 a)

LOC MI is violated General Fluid- or ‘Background-based’ models for the Bell experiment

M3. Second Fluid a b Model

Normalization conditions: General Fluid- or ‘Background-based’ models for the Bell experiment

M3. Second Fluid Model a b General Fluid- or ‘Background-based’ models for the Bell experiment

M3. Second Fluid a b Model General Fluid- or ‘Background-based’ models for the Bell experiment

M3. Second Fluid a b Model

 In conclusion, there are fluid or background-based models conceivable of

type M3 (i. e. satisfying (9-10)) for which:

M 3 X BI (a,a’,b,b’) > 2, for some (a,a’,b,b’). □

 It is also possible to prove that, for some choices of the probabilities:

□ = General Fluid- or ‘Background-based’ models for the Bell experiment

M3. Second Fluid a b Model

P( 1 a) = ( 1 a) , P( 2 b) = ( 2 b) (22a) P( , ,a) = with A( , ,a) sgn(a ).sgn(a ) (22b) 1 1 1 , A( , 1 ,a) 1 1 P( , ,b) = with B( , ,b) sgn(b ).sgn(b ) (22c) 2 2 2,B( , 2 ,b) 2 2 1 ( , ) = 1 2 for sgn( ) sgn( ) 1 2 8.( ) 1 2 1 2 1 = 1 2 for sgn( ) sgn( ) . (22d) 8. 1 2 1 2 … lead exactly to the quantum correlations. Vervoort, arXiv (2014), submitted

 In conclusion, no-go theorems are inoperative for fluid- models; they cannot exclude fluid-type / background-based models to exist. (Thee Kochen-Specker theorem is also based on MI.) Examples of M3 (background-based models)

M3. Second Fluid a Model b

 Spin-lattices appear to illustrate most of the characteristics of M3.  They are simple examples of ‘background-based’ models.

1 2 ● ● ● ● ● Alice 3 4 5 Bob

6 7 8 ● ● ● ● ● a b

Fig. 1. 10 spins on a lattice L. Vervoort, Found. Phys. (2013) Ising Hamiltionian:

H( ) = – i,j Jij. i. j – i hi. i. (all i = ±1) Examples of M3 (background-based models)

 The system is local (cf. graph), which can be explicitly calculated: P( , , , ,a,b) 1 2 1 2 = P( 1 , 1, 2 ,a).P( 2 , 1, 2 ,b)

= P( 1 , 1,a).P( 2 , 2 ,b) ( , 1, 2) Examples of M3 (background-based models)

 Locality is satisfied.

 However spin-lattices violate MI.

P( 1, 2 , a,b) ≠ P( 1, 2 , ) Examples of M3 (background-based models)

 If one of the premises of the BI is violated, the BI is possibly violated.  This appears to depend, for a fixed graph, on the parameter values

{hi, Jij} (Jij only ≠ 0 for nearest neighb.)

 For a 10-spin lattice, and for wide ranges of the parameter values {hi,

Jij} XBI > 2. max  X BI ≈ 2.9 [≈ 2√2 = 2.83 ] Conclusion I

 We saw that recent experiments on fluid-dynamical systems (Couder et al.) can mimic several quantum properties, including double-slit interference, quantization of angular momentum, Zeeman splitting, etc.  These experiments put an ancient result by Madelung (1927) in new light.  The Schrödinger Eq. (complex SODE) can quite naturally be interpreted as a hydrodynamic equation (a set of 2 hydrodynamic equations)

 Can Madelung’s project – to interpret the whole of QM as a fluid- mechanical theory – be brought to a good end ?  This research has been restrained by ‘no-go’ theorems: “Any Theory That Explains / Completes QM Must Be Non-Local” (and non-local is impossible).  However that appears to be wrong: a local fluid-like theory can violate the BI in a local manner and reproduce the quantum correlation.  Therefore we should look into Madelung’s (Einstein’s, de Broglie’s, Bohm’s, ‘tHooft’s,…) project again.