Typeset by FoilTEX · N-particle theory at the 1927 Solvay conference (see Valentini book). – Typeset by FoilTEX – 8. Quantum Hamilton-Jacobi theory and the quantum potential

– Typeset by FoilTEX – 1

Another Look at Pilot Wave Theory

De Broglie and Bohm’s solution to the QM interpretation problem.Together with reflections on possible connections with quantum Monte Carlo.

Mike Towler

Theory of Condensed Matter GroupCavendish Laboratory

University of Cambridge

www.tcm.phy.cam.ac.uk/∼mdt26 and www.vallico.net/tti/tti.html

[email protected]


Opinion Poll

What is your opinion on pilot-wave theory? (otherwise known as de Broglie-Bohmtheory, Bohmian mechanics, the causal or ontological interpretation of quantummechanics).

A: I have never heard of it.

B: I have heard of it and..

B1: ..don’t know enough details to have an opinion.

B2: ..understand that it has been shown to be incorrect or impossible.

B3: ..believe while internally self-consistent it is pointless metaphysical speculation.

B4: ..believe is interesting and stands equal with one or more other interpretations.

B5: ..believe it is the best available interpretation of QM.

C: Some other viewpoint.


Feynman on the two-slit experiment

•“A phenomenon which is impossible, absolutely impossible, to explain in anyclassical way, and which has in it the heart of quantum mechanics. In reality itcontains the only mystery.”

•“Do not keep saying to yourself, if you can possibly avoid it, ‘But how can it be likethat?’ because you will get ‘down the drain,’ into a blind alley from which nobodyhas yet escaped. Nobody knows how it can be like that.”

•As to the question “How does it really work? What machinery is actually producingthis thing? Nobody knows any machinery. Nobody can give you a deeper explanationof this phenomenon than I have given; that is, a description of it.”

•“Many ideas have been concocted to try to explain the curve for P12 [that is, theinterference pattern] in terms of individual electrons going around in complicatedways through the holes. None of them has succeeded.”

•This experiment “has been designed to contain all of the mystery of quantummechanics, to put you up against the paradoxes and mysteries and peculiarities ofnature one hundred per cent.”


What problems are we trying to address?

•Though QM treats statistical ensembles in a satisfactory way, we are unableto describe individual quantum processes without bringing in unsatisfactoryassumptions, such as the collapse of the wave function.

•We do not really understand the well-known nonlocality that has been brought outby Bell in connection with the EPR experiment.

•What is the mysterious ‘wave-particle duality’ in the properties of matter that isdemonstrated in a quantum interference experiment.

•Also where is boundary between a quantum system and the rest of the world(including apparatus and experimenter)? Where does microscopic definiteness giveway to microscopic indefiniteness? Somewhere between pollen grains, viruses andmacromolecules? No defined boundary between ‘classical domain’ where ‘real state’is valid concept and ‘quantum domain’ where it is not.

•Above all, there is the inability to give a clear notion of what the reality of aquantum system should be.

Why Pilot-waves?

“There is hardly any other interpretation that is consistent, accepts the existence ofan outside reality, and agrees with the predictions of the quantum formalism.”


Basic idea of pilot waves

De Broglie: “a freely moving body follows a trajectory that is orthogonalto the surfaces of an associated guiding wave.”

Prediction of ‘diffraction’ of electrons by small holes =⇒ Nobel prize in 1929.

•System with wave function Ψ(x, t) has definite configuration x(t) at all times,

with velocity x(t) = j(x,t)

|Ψ(x,t)|2i.e. ratio of quantum probability current j to quantum

probability density ρ (note classical formula for current is density times velocity).

•Recall that probability current j = ih2m(Ψ∇Ψ∗ − Ψ∗∇Ψ) = h

mIm(Ψ∗∇Ψ) andsatisfies the quantum-mechanical continuity equation

∂ρ

∂t+∇ · j = 0 or

∂

∂t

∫V

|Ψ|2dV +∫

S

j · dA = 0

which is the conservation law for probability density in QM. i.e. the time derivativeof the change of the probability of the particle being measured in V is equal to therate at which probability flows into V .

•Theory based on this “completely avoids all the quantum paradoxes, all themysticism of Bohr and Heisenberg, and replaces it with sharp mathematics”.


Basic equations - de Broglie form (1927)

Focussing for simplicity on the nonrelativistic quantum theory of a system of N(spinless) particles with 3-vector positions xi(i = 1, 2, . . . , N) it is now generallyagreed that, with appropriate initial conditions, quantum physics may be accountedfor by the deterministic dynamics defined by two differential equations, theSchrodinger equation

ih∂Ψ∂t

=N∑

i=1

− h2

2mi∇2

iΨ + V Ψ (1)

for a ‘pilot wave’ Ψ(x1,x2, . . . ,xN , t) in configuration space, and the de Broglieguidance equation

midxi

dt= ∇iS (2)

for particles trajectories xi(t), where the phase S(x1,x2, . . . ,xN , t) is locally definedby S = hIm lnΨ so that Ψ = |Ψ| exp[(i/h)S]. The particles are thus ‘pushed along’by the wave along trajectories perpendicular to surfaces of constant phase.


Basic equations - Bohm form (1952)

Bohm’s presentation of 1952 was somewhat different. Take time derivative ofguidance equation, then use TDSE to get Newton’s law for acceleration

∂

∂t∇iS(x, t) = ∇i

∂

∂thIm lnΨ = ∇iIm

[h

Ψ∂

∂tΨ]

=

∇iIm

[i

Ψ

(h2

2mi∇2

iΨ− V Ψ

)]= −∇i

[1Ψ

(−h2

2mi∇2

iΨ + V Ψ

)]= −∇i [V + Q]

mixi = −∇i(V + Q) ,

where

Q ≡ −∑

i

h2

2mi

∇2i |Ψ||Ψ|

is the ‘quantum potential’. Bohm regarded this as the law of motion, with thede Broglie guidance equation added as a constraint on initial momenta. For deBroglie, in contrast, law of motion for velocity had a fundamental status, and forhim represented a unification of the principles of Maupertuis and Fermat.

Usually believed de Broglie did one-particle theory only, and that Bohm generalizedit to N particles (as claimed in Bohm’s book). However, de Broglie gave the fullN -particle theory at the 1927 Solvay conference (see Valentini book).


Quantum Hamilton-Jacobi theory and the quantum potential

Bohm theory usually presented as follows. Substitute amplitude-phase decomposition(polar form) of time-dependent wave function Ψ(x, t) = R(x, t) exp(iS(x, t)/h)into time-dependent Schrodinger equation. Rewrite to get pair of coupled evolutionequations, namely the continuity equation for ρ = R2:

∂R2

∂t+∇ ·

(R2∇S

m

)= 0

and a modified Hamilton-Jacobi equation for S:

∂S

∂t= −(∇S)2

2m− V −Q where Q = − h2

2m

∇2R

R

This equation relates the change in the ‘action’ S to the quantum Lagrangian,which is the excess kinetic energy flow over the total potential energy.

Only difference between Hamilton-Jacobi theory (a standard reformulation ofclassical mechanics) and this ‘quantum version’ is the quantum potential Q. ThusNewtonian mechanics emerges from ‘Bohmian mechanics’ in classical limit, and sizeof quantum potential provides a measure of deviation of pilot-wave theory from itsclassical approximation.


The quantum potential Q

Good points •Q gives measure of deviation of Bohm from classical approximation.Can be used to develop approximation schemes for solutions to Schrodinger equation.

Bad points •Increased complexity (Schrodinger equation simple and linear. ModifiedHamilton-Jacobi equation complicated, highly nonlinear, and requires continuityequation for closure). Gives false impression of lengths to which we must go toconvert orthodox QM into something more rational. Suggests in order to makeSchrodinger equation into theory acounting for quantum phenomena in ‘realisticterms’ we must add complicated quantum potential of grossly nonlocal character.

In fact the quantum potential need not be mentioned in the formulation of pilot-wavetheory - it is merely a reflection of the wave function, which it shares with orthodoxquantum theory. Use de Broglie formulation instead!


Bell on the two-slit experiment

While each trajectory passes through but one of the slits, the wave passes throughboth; the interference profile that therefore develops in the wave generates a similarpattern in the trajectories guided by this wave.

Compare Feynman’s discussion (see earlier) with Bell’s:

“Is it not clear from the smallness of the scintillation on the screen that we haveto do with a particle? And is it not clear, from the diffraction and interferencepatterns, that the motion of the particle is directed by a wave? De Broglie showedin detail how the motion of a particle, passing through just one of two holes inthe screen, could be influenced by waves propagating through both holes. And soinfluenced that the particle does not go where the waves cancel out, but is attractedto where they cooperate. This idea seems to me so natural and simple, to resolvethe wave-particle dilemma in such a clear and ordinary way, that it is a great mysteryto me that it was so generally ignored.”– Typeset by FoilTEX – 11

Trajectories

•Occam’s razor?“Best theory is the one with fewest premises.”

But though Bohm adds a particle equation of motion to QM (extra premise) heeliminates the - to most people uncompelling - postulates about measurement (fewerpremises). Also gives a completely new interpretation of quantum phenomena inwhich e.g. probability plays no fundamental role. Descriptive content identical buttheories probably not equivalent at all. No basis for Occam.

•But you can’t measure trajectories?

Bell: “But to admit things not visible to the gross creatures that we are is, inmy opinion, to show a decent humility, and not just a lamentable addiction tometaphysics.”


The quantum equilibrium distribution

P (x, t) = |Ψ(x, t)|2

•Pilot-wave theory requires assumption that we have a statistical ensemble withprobability density P (x, t) = |Ψ(x, t)|2. If this is true at any particular time it willalways be satisfied (‘equivariance’). Think of quantum equilibrium P = |Ψ|2 as

roughly analagous to thermal equilibrium P = exp(−H/kT )Z .

What happens if this assumption is not satisfied?

If the guidance equation is exact:

Any initial distribution not satisfying equivariance would tend towards thatdistribution as a result of essentially random perturbations of the particle’s wavefunction coming from the environment in which a particle is found.

If the guidance equation is only true ‘on average’:

The particle velocity as given by the guidance equation is additionally subject torandom fluctuations. Then, in order to satisfy equivariance, it is necessary to assumea suitable ‘osmotic velocity’ so that in equilibrium the diffusion current is balancedby the osmotic velocity and the mean velocity is given by the guidance equation(similar to diffusion and drift motion in DMC..).


The measurement problem - definition

What is the problem?

If you believe that the wave function is all there is, and you include both the systemand the measuring apparatus then the wave function after the measurement is asuperposition of all possible results and continues to evolve according to the TDSE.QM thus implies that measurements typically fail to have outcomes of the sort thetheory was created to explain.

OR, EQUIVALENTLY

QM has two rules for evolution of wave function:

•A deterministic dynamics given by Schrodinger’s equation for when the system isnot being ‘measured’ or observed.•A random non-local collapse of the wave function to an eigenstate of the ‘measuredobservable’ for when it is.

Objection because standard QM does not provide a coherent account of how thesetwo apparently incompatible rules can be reconciled. Why should interactionsbetween system and apparatus that we happen to call measurements be governedby laws different from those governing all other interactions?


The measurement problem in pilot-wave theory

If you admit the existence of configurations then:

•The description of the after-measurement situation includes, in addition to thewave function, the values of the variables that register the result, thus there is nomeasurement problem. In pilot wave theory pointers always point.

•The wave function of the universe does not collapse, only the wave function of thesubsystem being measured.

•You can derive the collapse of the subsystem wave function mathematically.. Incontrast to the orthodox collapse, the collapse takes place objectively, takes a finiteamount of time, and does not depend on anything like the observer’s knowledge.

•All paradoxes related to measurement (Schrodinger’s Cat, Wigner’s Friend etc. etc.disappear).

•Observers are not required, thus the theory works before life evolved and doesn’tdepend on observation by God. Nor is it necessary to postulate that each term inthe superposition forms a whole new universe.

Most rational beings would consider this to be an improvement.


Stationary states

Ψ(x, t) = Ψ0(x)e−iEt/h

Eigenfunctions of the Hamiltonian H = −h2

2m∇2 + V with V independent of time. In

Ψ = R exp(iS/h) form we have R(x, t) = R0(x) and S(x, t) = S0(x)− Et.

Consequences

•Probability density is independent of the time: |Ψ|2 = R20(x).

•Quantum potential Q = (−h2/2mR0)∇2R0 is time-independent. Therefore so isthe total effective potential ∂(V + Q)/∂t = 0.

•The velocity field is independent of time. If Ψ0(x) is a real function then thevelocity is zero (Recall probability current j = h

2mi(Ψ∗∇Ψ−Ψ∇Ψ∗)). The particle

is at rest where one would classically expect it to move since the quantum force(−∇Q) cancels the classical force (−∇V ).

•The energy of all particles in the ensemble, −∂S/∂t, is a constant of the motionand equal to the energy eigenvalue E.


The Uncertainty Principle

∆x∆p ≥ h2

•Uncertainty emerges in pilot-wave theory simply as a constraint on our knowledgeof quantum systems (as opposed to a constraint on their objective definiteness).

•Put another way, uncertainty principle describes statistical scatter in resultsof specific pairs of (non-commuting) measurements carried out on ensemble ofidentically prepared systems. Does not describe inherent impossibility for a particleto have e.g. a simultaneously well-defined position and momentum.

•Closely related to QEH, which says particle positions are Born-rule distributed.Trivial consequence is that position uncertainty (square root of expectation value ofx2 minus squared expectation value of x) matches that of standard QM.

•Statements about momentum uncertainty more subtle. In principle, momentumnot indefinite (if know exact x and Ψ can infer precise velocity from guidanceformula). But in general, QEH prevents you from knowing x and hence velocity.

•Thus momentum measurements do not simply reveal the value of the pre-measurement velocity (times mass). See next slide.


Momentum

Particle in a 1-D box

Ψ ∝ eikx + e−ikx

•k chosen so that wave function vanishes at walls.

•Standard QM says momentum is hk or −hk.

•These are stationary states so pilot-wave theorysays particle at rest (no, really!). Assuming QEH,position uncertainty is correct.

•’Momentum’ doesn’t have meaning in pilot-wave theory. But I can measure it!!How? Take away walls, let particle move freely, then detect position x. Divide bytime, multiply by mass =⇒ ‘momentum’.

•In pilot-wave theory, wave function forms two packets which fly off in the twodirections. Particle carried away by one of them (50/50). It accelerates until it hasvelocity ±hk/m at point x. Thus measurement agreeing with QM prediction butnot revealing pre-experiment value of zero! =⇒ ‘Naive realism about operators’.

•Bohm momentum uncertainty correct. Says how initial spread of positions relatesto spread of momentum values that could be measured if momentum measurementwere performed (related to Fourier transform of Ψ(x), roughly speaking).


‘Naive realism about operators’

•Unlike position, concepts of ‘momentum’, ‘energy’, ‘angular momentum’ only havemeaning in Newtonian mechanics (as conserved quantities). Newtonian mechanicsis wrong. Heisenberg insisted momentum etc. should have a meaning in QM. Hisidea: defining physical quantity means specifying how to measure it. This is OK ifresults don’t depend on experimental arrangement, but with random results can’tsay if meaningful as any definition will always produce some result. Heisenbergchose definitions which, in Newtonian world, would measure Newtonian value (??).But correct experimental arrangement for measuring momentum (if it exists) innon-Newtonian world should differ from Newtonian?

•In pilot-wave theory can define actual momentum as mv (but this is not a conservedquantity) or < Ψ|(−ih)∇|Ψ > (which sometimes is) but not helpful as these neednot agree with outcome of ‘momentum measurement’.

•Kochen-Specker theorem showed (1967) that relationships among spin componentsof spin 1 particle are incompatible with the idea that measurements of theseobservables merely reveal their preexisting values rather than, as we are urged tobelieve in QM, creating them. Supposed to be definitive evidence against ‘hiddenvariables’ ! (Pilot-wave theory −→ bin.) However, ‘naive realism about operators’has seduced us into insisting, despite evident impossibility, that the spin operatorscorrespond to actual properties of the particles. But they don’t!


SpinPilot waves also make sense for particleswith spin, i.e., for particles whose wavefunctions are spinor-valued. Whendirected toward Stern-Gerlach magnets,they emerge moving in a discrete set ofdirections e.g. 2 directions for a spin-12particle, having 2 spin components. Spinproperty of wave function not of particles.

•Magnets designed/oriented so that incident wave packet will, due to Schrodingerevolution, separate into distinct packets - corresponding to spin components of wavefunction - moving in the discrete set of directions. Particle, depending on initialposition, ends up in one of these packets. Probability distribution convenientlyexpressed in terms of quantum-mechanical spin operators (for a spin-12 particle givenby the Pauli spin matrices).

•Clear from contextuality that particles don’t carry spin. Imagine spin-12 particlecoming towards SG device that ‘measures its spin in the z-direction’. Assume initialposition such that it emerges from ‘spin-up’ port. Now mentally rotate SG apparatus180◦ so B-field points in opposite direction and repeat. According to standard QM,this is still a device that will ‘measure z-component of spin’, but Bohm says particlenow comes out of ‘spin down’ port (i.e. in same direction as before).


John Bell on measurement

“A final moral concerns terminology. Why did such serious people take so seriouslyaxioms which now seem so arbitrary? I suspect that they were misled by thepernicious misuse of the word ‘measurement’ in contemporary theory. This wordvery strongly suggests the ascertaining of some preexisting property of some thing,any instrument involved playing a purely passive role. Quantum experiments are justnot like that, as we learned especially from Bohr. The results have to be regardedas the joint product of ‘system’ and ‘apparatus’, the complete experimental set-up.But the misuse of the word ‘measurement’ makes it easy to forget this and then toexpect that the ‘results of measurements’ should obey some simple logic in whichthe apparatus is not mentioned. The resulting difficulties soon show that any suchlogic is not ordinary logic. It is my impression that the whole vast subject of‘Quantum Logic’ has arisen in this way from the misuse of a word. I am convincedthat the word ‘measurement’ has now been so abused that the field would besignificantly advanced by banning its use altogether, in favour for example of theword ‘experiment’.”


Non-locality

Definition: a direct influence of one object on another, distant object, contrary to ourexpectation that an object is influenced directly only by its immediate surroundings.

What EPR shows us:

•Result of experiment on one side immediately predicts result on other (for parallelanalyzers).•If do not believe one side can have causal influence on the other, then requirethat results on both sides are determined in advance. But this has implications fornon-parallel settings which conflict with quantum mechanics (Bell again).

Thus Bell’s analysis showed that any account of quantum phenomena needs to benon-local, not just any ‘hidden variables’ account i.e. non-locality is implied by thepredictions of standard quantum theory itself. Thus, if nature is governed by thesepredictions (which it is, according to experiments) then nature is non-local.


Non-locality and pilot waves

•Non-locality of pilot-wave theory derives solely from non-locality built into structureof standard QM, as provided by wave function on configuration space, an abstractionwhich combines or binds distant particles into a single irreducible reality.

•Velocity from guiding equation of any particle of a many-particle system willtypically depend upon the positions of the other, possibly distant, particles wheneverwave function of the system is entangled, i.e., not a product of single-particle wavefunctions.

•All usual results (such as EPR) follow from pilot-wave mathematical analysis usingthe concept of the ‘conditional wave function’ (see textbooks).

•Only problem therefore acceptance of non-locality without necessarily knowing‘mechanism’. Concept perfectly rational since leads to neither internal logicalcontradiction nor disagreement with any facts. Can still isolate systems sufficientlyto study them, and non-local effects not significant at the large-scale level.

•Does not violate special relativity. Consequence of QEH that non-local effects inpilot-wave theory don’t yield observable consequences that are also controllable - wecan’t use them to send instantaneous messages (with ‘non-equilibrium matter’ youcan! - see Valentini).


Implicate order and metaphysics

See Bohm’s book ‘Wholeness and the Implicate Order ’ (1983).

•Local notion of order by means of a Cartesian grid (extended where necessary tocurvilinear) is inadequate in the quantum domain, so Bohm suggests new notion oforder, somewhat analagous to using a hologram to make an image, to replace theold, which is analagous to using a lens.

•Each region of the hologram makes possible an image of the whole object. Thehologram does not look like the object, but gives rise to an image only when suitablyilluminated. The whole object is ‘enfolded’ in each part of the hologram rather thanbeing in point-to-point correspondence. Order in the hologram = implicate. Orderin the object and image = explicate.

•Order of the whole universe is enfolded into each region. Suggestion that implicateorder will have the kind of general necessity that is suitable for expressing the basiclaws of physics, while the explicate order (what you see) is important only as aparticular case.

•Relativity and QM incompatible because of using wrong type of order. Obviouslythis explains non-locality! Oh yes.

‘Undivided Wholeness in Flowing Movement’


Bohm huge in nutter community

Notions of implicate order have led to existence of huge ‘Bohm industry’ involvingEastern philosophy, consciousness, life, wholeness etc. This is actually ratherinteresting..

For Bohm, life is a continuous flowing process of enfoldment and unfoldmentinvolving relatively autonomous entities. DNA ‘directs’ the environment to form aliving thing. Life can be said to be implicate in ensembles of atoms that ultimatelyform life.


Connections with QMC, or, why did I get into this?

•Vague notion that sampling configurations in QMC is like sampling trajectories (orlike Bohm’s stochastic version of pilot-wave theory, where random fluctuations shovean arbitrary quantum system towards ‘quantum equilibrium’ P (R, t) = |Ψ(R, t)|2).

•Quantum potential looked a bit like the local kinetic energy in QMC (also likeLaplacian of the charge density that Bader has shown characterizes many featuresof molecular structure and reactivity).

•Equations in stochastic processes of random walk practically identical in stochasticpilot wave theory and in DMC.

•Guidance equation in stochastic pilot-wave theory very similar (‘osmotic term’serves to direct particles to regions where |Ψ|2 is large - a bit like drift velocity inimportance-sampled DMC).

•[from Bohm’s book] Relation between wave function at one time and its formlater is determined by propagator or Green’s function G through Ψ(x, t) =

∫G(x−

x′, t − t′)Ψ(x′, t′) dx′. Simple picture of movement is that waves from the wholespace enfold into each region and waves from each region unfold back into thewhole space. Green’s function ultimately derived from Cartesian order by solvingdifferential equations. But if we question that this order holds fundamentally wecan adopt view that Green’s function is more basic than the differential equation.


Diffusion Monte CarloConsider imaginary time behaviour of time-dependent Schrodinger equation(

H − ET

)Ψ(R, t) = −∂Ψ(R, t)

i∂t

For eigenstate, general solution is clearly

φ(R, t) = φ(R, 0)e−i(H−ET )t

Then expand arbitrary Ψ(R, t) in eigenfunctions of H

Ψ(R, t) =∞∑

n=0

cnφn(R)e−i(En−ET )t

Substitute it → τ (imaginary time). Oscillatory behaviour becomes exponential.

Ψ(R, τ) =∞∑

n=0

cnφn(R)e−(En−ET )τ

Get time independence by choosing ET to be ground state eigenvalue E0.As τ →∞, Ψ → ground state φ0.

Ψ(R, τ) = c0φ0 +∞∑

n=1

cnφn(R)e−(En−ET )τ


Diffusion Monte Carlo

•How do we propagate solution in imaginary time?

Ψ(R, τ + δτ) =∫

G(R,R′, δτ)Ψ(R′, τ) dR′

•How do we find Green’s function G(R,R′, δτ)? Consider Schrodinger equation intwo parts:

−12∇2

RΨ = −∂Ψ∂τ

diffusion equation

V Ψ = −∂Ψ∂τ

rate equation

•Green’s function for diffusion equation known: 3N dimensional Gaussian withvariance δτ in each dimension.

G(R,R′, δτ) = (2πδτ)−3N2 exp

(−|R−R′|2

2δτ

)•Extra branching factor from rate equation.

× exp[−δτ

(V (R) + V (R′)− 2ET

2

)]


Diffusion Monte Carlo

•So use distribution (ensemble) of Brownian particles (‘walkers’) to representΨ(R, τ). The Green’s function G(R,R′, δτ) is then interpreted as the probabilityof a walker moving from point R′ to R in a time δτ . Branching factor determinespopulation of walkers: In regions of high V , walkers will be killed off; in low Vregions, walkers will multiply.

•Propagate distribution defined by Green’s function in imaginary time. At longtimes, excited states will decay away. Can then continue propagation and accumulateaverages of observables.

V(x)

Ψinit(x)

Ψ0(x)

t

τ {

x


Importance sampling in diffusion Monte Carlo

•Problem 1: Ψ is not a probability distribution → sign problem.

•Problem 2: Rate equation is poorly behaved.

•Solution: Work with distribution f(R, τ) = Ψ(R, τ)ΦT (R) and deal with signproblem via fixed-node approximation.

−12∇

2Rf(R, τ) +∇R · [F(R)f(R, τ)]

− (EL(R)− ET) f(R, τ) = −∂f(R,τ)∂τ

F(R) ≡ ∇RΦTΦT

drift vector

f(R, τ + δτ) =∫

G(R′,R, δτ)f(R, τ) dR′


Importance sampling in DMC

In short time approximation G consists of diffusion, drift and branching processes

G(R′,R, δτ) = (2πδτ)−3N2 exp

[−(

R′ −R− δτF(R)2δτ

)2]

× exp[−δτ

(EL(R) + EL(R′)

2− ET

)]•Drift term enhances density of configurations where ΦT is large.

•Branching term contains the local energy which fluctuates less than the potentialenergy

•(Ignore additional details for comparison with pilot-waves)


QM computations with trajectories

Computing the wavefunction from trajectories: particle and wave pictures inquantum mechanics and their relation

P. Holland (2004)

“The notion that the concept of a continuous material orbit is incompatible witha full wave theory of microphysical systems was central to the genesis of wavemechanics. Early attempts to justify this assertion using Heisenberg’s relationswere subsequently shown to be flawed, and indeed no credible proof forbidding thetreatment of quantum processes in terms of precisely defined spacetime trajectorieshas ever been offered. The idea nevertheless entered the folklore of the subjectand even now is invoked to highlight alleged paradoxical implications of quantummechanics (e.g. Schrodinger’s cat). So great was the philosophical bias that notonly was the material orbit ruled out as an aid to comprehension, but the possibilitiesof using the trajectory as a computational tool, or even as the basis of an alternativerepresentation of the quantum theory - the twin subjects of this paper - wereforegone.”

Notice the excellent self-confident tone of this introductory paragraph!

Has pilot-wave theory finally ‘matured’?


QM computations with trajectories - path integral approach

Feynman recalls the Pocono conference (1948):

“Bohr was also at the meeting. After I had tried many times to explain what I wasdoing and didn’t succeed. I talked about trajectories, then I would swing back - Iwas being forced back all the time. I said that in quantum mechanics one coulddescribe the amplitude of each particle in such and such a way. Bohr got up andsaid ’Already in 1925, 1926, we knew that the classical idea of a trajectory or a pathis not legitimate in quantum mechanics: one could not talk about the trajectoryof an electron in the atom, because it was something not observable.’ In otherwords, he was telling me about the uncertainty principle. It became clear to methat there was no communication between what I was trying to say and what theywere thinking. Bohr though that I didn’t know the uncertainty principle, and wasactually not doing QM right either. He didn’t understand at all what I was saying.I got a terrible feeling of resignation. ”

In path integral approach, trajectory is integral to computation of wave function:propagator linking two spacetime points calculated by linearly superposing elementaryamplitudes eiS/h associated with all paths connecting the points. Wave functionat point then found from Huygen’s principle by superposing contributions comingfrom all other points (weighted by initial wave function). Implies propagation of aparticle somehow involves simultaneous traversal of multiple equally likely paths.


Quantum hydrodynamics

•Can single trajectories linking two spacetime points be made the basis of analternative computational scheme in QM? i.e. can they have a ‘use’ instead of beingmerely interpretative? Yes - see recent chemical physics literature. Uses analogybetween QM and hydrodynamics (similarity in form between Schrodinger equationin position space, suitable expressed as two real equations - pilot waves! - and theequations of fluid mechanics).

•Probability density is proportional to fluid density, and the phase of the wavefunction is a velocity potential. Novel feature of quantum fluid is the appearance ofquantum stresses (usually represented through the quantum potential).

•Can thereby achieve same end as path integral method - computation of wavefunction given initial value - using quite different and conceptually simpler trajectorymethod in which two spacetime points are connected by at most a single continuousorbit. Two steps in Feynman’s approach (propagator then Huygens) thus condensedinto one. Wave function as a whole generated from its initial form by single-valuedcontinuum of trajectories. Therefore not dealing with just interpretation but ratheran alternative mathematical representation or picture of QM.

•Ψ encodes the temporal history of a space point, while v encodes its ‘spatial’history. Given the wave function, the time-dependence of either state function canbe computed and implies the other. Wave-particle duality.


Time-dependent quantum Monte Carlo

Sadly, MDT beaten to it: I.P. Christov (University of Sofia, 2007) has invented ‘timedependent quantum Monte Carlo’ (TDQMC) with moving configurations guided byde Broglie-Bohm pilot waves.

Christov claims several advantages over DMC:

•Distribution of walkers in configuration space corresponds to quantum probabilitydensity; method thus not sensitive to sign of many-body wave function.=⇒ SOLVED FERMION SIGN PROBLEM!!

•Guide functions/nodes evolve together with particle configurations. No initialguess for many-body wave function required. =⇒ NO MORE SUCKING UP TOOWNERS OF CASTEP/CRYSTAL FOR INTERFACES TO QMC PROGRAM!!

•Can do time-dependent studies of quantum objects and their interactions withexternal electromagnetic fields, while preserving fully correlated quantum dynamics.=⇒ TIME-DEPENDENCE (HARD IN DMC)!!.

•Uses first-order de Broglie pilot waves (linear Schrodinger equation and velocities).Easier numerically than what chemical physics chaps do (Bohm-style second-orderHamilton-Jacobi and quantum potential). =⇒ FASTER AND SCALES BETTER!!.

•Useful for promotion of world peace, love and harmony and is the moon on a stick.=⇒ YEAH!!


TDQMC algorithm•Synthesis between QMC and quantum hydrodynamics. Each electron representedby ensemble of M configs, each of which moves according to quantum trajectoryimplied by guidance equation,

drik =

(1m∇S + α

h

2m

∇|Ψ|2

|Ψ|2

)dt + η

√αh

mdt,

corresponding to conventional pilot-wave mechanics (α = 0) or stochastic version(α = 1) where equilibrium between last two terms means mean velocity still given by∇S/m. Guide function (‘orbital’) for kth config from ith electron ensemble evolvesaccording to time-dependent Schrodinger equation i.e. into a steady-state (for Vindependent of time) where RHS of guidance equation approaches zero.

•Complex time used in evolution; if wfn real during GS calc, then RHS zero at alltimes rather than converging to zero. Imaginary part causes guide waves to relaxto GS. Real part gives time-dependent phase which guides walkers to stationarypositions. Amplitude α of random part decreasing function of time.

•No spin variables in Schrodinger eqn. Electron stats determined by symmetryproperties of many-electron wfn. Nodes from dependence of Ψ on individualtime-dependent orbitals (through Slater dets). Trajectories can never cross nodesand nodes are time-dependent. No integrals calculation - all quantities of interestexpressed entirely in terms of walker configurations.– Typeset by FoilTEX – 36

TDQMC - relaxation to ground state

Time evolution of the coordinates of twoarbitrary walkers for 1D helium (a) forDMC and (b) for TDQMC.

•TDQMC trajectories tend to steady-state positions close to the end of the interval(distributed according to |ΨGS|2 over the ensemble), whereas the DMC trajectoriesexperience random jumps at all times. Different behaviour due to essentially differentmechanisms for guiding of configs.

•Once GS is achieved in TDQMC, can switch to real time and turn on externalfields to study time-dependent dynamics. Since the algorithm for calculation ofGS guarantees that P (R, t) = |Ψ(R, t)|2, it follows from the continuity equationthat this will hold at all times during the evolution. (e.g. Christov calculatestime-dependent dipole moment of 1D helium and compares with exact results).


Requirements on a solution to the fermion sign problem

AIM: stable algorithm that can sample the exact wave function without resorting tothe fixed-node approximation.

Several solutions per year in literature, but all either :

•have a serious error,

•are known not to work (‘exact’ FN has been discovered at least 3 times),

•work only in special cases (e.g. 1D, certain lattice models, harmonic oscillator),

•cannot scale at large N or low temperature,

•have uncontrolled approximations,

•have not been analyzed enough to know whether they work,

Numerical evidence is nice but not enough, since :

•there are approximation methods that are very accurate in certain problems,

•cancellation of errors is always possible for some limited set of problems.

Scaling must be demonstrated.

TDQMC nice idea, but needs to go through similar analysis


Good things about pilot waves

•Minimum benefit of pilot wave theory, independent of debate about whether it isthe preferable formulation, is disproof of claim that QM implies particles can’t existbefore being measured. Since based on familiar concepts of realism, pilot-waves givesimple, intuitive, non-mystical, and natural answers to what in standard theory areperplexing philosophical questions.

•For supporters, pilot-waves give better formulation of QM since defined moreprecisely than Copenhagen, which is based on theorems not expressed in precisemathematical terms but in words. Pilot-waves subsume quantum concepts ofmeasurement, complementarity, decoherence, and entanglement into mathematicallyprecise guidance conditions on position variables.

•Although implies existence of EPR type of nonlocality, it avoids need for positingother types of alleged nonlocality, such as wave collapse; and also avoids need forsplitting universes as in many-worlds.

•Fact that we can derive time-dependent wave function from trajectories showswe are not dealing just with another interpretation or ‘superfluous ideologicalsuperstructure’, but rather with an alternative mathematical representation orpicture of QM, equivalent in status to, say, Feynman’s path-integral method.

•Can treat e.g. dwell/tunneling times, escape times/escape positions, scatteringtheory, quantum chaos. Difficult or impossible to analyze in pure wave model.


Problems for pilot waves

•Relativity.

“Those paradoxes are simply disposed of by the 1952 theory of Bohm, leaving as thequestion, the question of Lorentz invariance. So one of my missions in life is to getpeople to see that if they want to talk about the problems of quantum mechanics- the real problems of quantum mechanics - they must be talking about Lorentzinvariance.” [Bell again]

Some progress has been made.

•To a first approximation, everybody hates Bohm (except gurus, nutters and theDalai Lama) which is a pity because he was a tremendously nice man.

•If pilot waves are the truth, then the truth is somewhat boring and quantumphysicists might have to stop pretending that their subject was mystical (paralleluniverses! Oh yes.) and only understandable by people with very large brains (likethem), which is a good pose.


Molecular dynamics

•Experimentalists have been able to ‘diffract’ 60 C-atom fullerene molecules passingthrough small holes, so such objects definitely fall on the quantum side of theclassical-quantum boundary.

•Our group in Cambridge is a leading centre for doing molecular dynamics (e.g.with their CASTEP code). So if trajectories don’t exist, and particles don’t haveobjective reality, what exactly do they imagine their molecular dynamics movies aredepicting?


The Last Word

“This isn’t right. It isn’t even wrong.” (W. Pauli)

Guess who he was speaking to at the time?

”..We consider it juvenile deviationism .. we don’t waste our time ... [by] actuallyread[ing] the paper.”

”..[Bohm] is a public nuisance..”

”..[Bohm] is a Trotskyite and a traitor.”


Opinion Poll Again

What is your opinion on pilot-wave theory? (otherwise known as de Broglie-Bohmtheory, Bohmian mechanics, the causal or ontological interpretation of quantummechanics).

A: I have never heard of it [NO LONGER ALLOWED].

B: I have heard of it and..

B1: ..don’t know enough details to have an opinion.

B2: ..understand that it has been shown to be incorrect or impossible.

B3: ..believe while internally self-consistent it is pointless metaphysical speculation.

B4: ..believe is interesting and stands equal with one or more other interpretations.

B5: ..believe it is the best available interpretation of QM.

C: None of the above


Documents

Typeset by FoilTEX · N-particle theory at the 1927 Solvay conference (see Valentini book). – Typeset by FoilTEX – 8. Quantum Hamilton-Jacobi theory and the quantum potential