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Physica E 42 (2010) 317–322

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/physe

The quark-gluon plasma and the stochastic interpretation of quantummechanics

Mark Davidson

Spectel Research Corp, 807 Rorke Way, Palo Alto CA 94303, USA

a r t i c l e i n f o

Available online 24 June 2009

Keywords:

Turbulence

Quantum

Fluid

Soliton

Quark-gluon plasma

Burgers

77/$ - see front matter & 2009 Elsevier B.V. A

016/j.physe.2009.06.076

ail address: mdavid@spectelresearch.com

a b s t r a c t

Analysis of data from the relativistic heavy-ion collider (RHIC) has established that a quark-gluon

plasma is formed in heavy-ion collisions, and that it is well described by an ideal classical fluid model

with nearly zero viscosity. It is believed that a similar state of matter permeated the entire universe at

about three microseconds after the big bang. The Reynolds number for this primordial fluid is estimated

to be on the order 1019 suggesting that the early universe was extremely turbulent. Could quantum

behavior as we observe today possibly be a stochastic manifestation of this turbulent beginning? This

paper explores this question by examining a particularly simple class of ideal fluids exhibiting

turbulence and chaos in numerical simulations implemented utilizing a Fourier–Galerkin spectral

truncation method. This proto-fluid model ignores pressure completely and is referred to in the

cosmological literature as the Lambda-CDM model or cosmic dust model. It is compatible with a flat

space-time provided a cosmological constant is included in the equations of general relativity, and it can

be transformed into the inviscid Burgers equation. A new class of solitons is presented for this system. It

is shown that these can be combined with the known chaotic solutions, and that they then move

randomly with a mean velocity that does not decay with time, exhibiting a time-reversible stochastic

motion without any friction, similar to quantum particles. The turbulent fluid is statistically a Markov

process, and the soliton’s position can be described by a hidden Markov model. It is also found that if the

initial conditions for the fluid contain a single filament of vorticity, then a complex wave function which

is single valued can be defined.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Recent experiments at the relativistic heavy-ion collider (RHIC)have revealed that our universe probably consisted entirely of anearly ideal classical fluid at 3ms after the big bang. Heavy-ioncollisions at 40 TeV produced a quark-gluon plasma in theseexperiments [1–5]. This result opens new possibilities for thestochastic interpretation of quantum mechanics. For the first time,a good candidate exists for its stochastic origins, namelyturbulence in a cosmological proto-fluid of the very earlyuniverse. Various hydrodynamic models have been used todescribe the fluid discovered at RHIC, most owing their genesisto Landau’s seminal hydrodynamic model [6–8]. Therefore, allthat we observe today appears to have its origin in an essentiallyclassical ideal fluid that filled the entire universe at 3ms after thebig bang, and probably earlier. While many physicists are trying toexplain how such a classical fluid can come to be with hindsightfrom the laws of the standard model, is it not also conceivable thatthe logical connection might flow in the opposite direction? Could

ll rights reserved.

not the laws of physics that we observe today, and especially thequantum laws, somehow have their origin in this violentlyturbulent classical fluid past which the RHIC experiments havenow provided the first glimpse of? The idea that quantummechanics is related to fluid mechanics is originally due toMadelung [9]. This paper takes some exploratory steps to examinethis idea. The lack of an underlying reality to quantum mechanicsis a weakness of modern physics in the opinion of a sizable groupof physicists. The derivation of quantum mechanics fromrelativistic inviscid turbulent fluid dynamics would go a longway to remedying this situation.

The classical correspondence principle is a well-establishedmaxim. What if there was a dual ‘‘Quantum CorrespondencePrinciple’’, so that in some limit the quantum theory emergedfrom a particular and very special classical system? Finding such atheory has so far proven to be extremely difficult. But perhaps thatis a strength rather than a weakness. Today there are too manypossible physical theories that cannot be ruled out. Owing tomathematical breakthroughs, the landscape of string theories hasbecome enormous [10], and the only principle that seemsavailable to distinguish the standard theory of particle physicsfrom a myriad of alternatives is the anthropic principle.

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M. Davidson / Physica E 42 (2010) 317–322318

In reviewing the turbulence literature, this author was drawnto recent research on the inviscid Burgers equation, and inparticular to some very interesting numerical results of simula-tions to this equation using Galerkin truncation [11–13].Other authors have noted a mathematical similarity betweenthe forced inviscid Burgers equation and quantum field theory[14,15]. Similarities between vortex turbulence theory andquantum mechanics have been noted previously as well [16],based on an incompressible fluid model related to the vortexsponge model of the 19th and early 20th century [17]. The Burgersequation has also been used in a spontaneous wave functioncollapse model that is quite relevant to this paper [18].

We start from an ideal relativistic fluid. A more detailedtreatment can be found in [19]. An approximate estimate wasmade for the Reynolds number of the quark-gluon plasma in theearly universe [19]

ReU�2:26� 1019: ð1Þ

This large value guarantees that the fluid will be extremelyturbulent. We work with an ideal fluid (zero viscosity) for thesimple reason that quantum mechanics is time reversible, andviscosity would lead inevitably to irreversibility in the statisticaldynamics of any emergent theory.

2. Relativistic ideal fluid mechanics and the inviscid Burgersequation

The relativistic equation for an ideal fluid in curved space-timeis

rbTab ¼ 0; ð2Þ

where rb is the covariant derivative and where

Tab ¼ pgab þ ðpþ rÞUaUb: ð3Þ

U is the proper 4-velocity of the fluid, c ¼ 1, p the pressure, andr the mass density.

We assume that the fluid is time-like. Space-like velocities,perhaps as a second component, might be entertained asa concession to Bell non-locality arguments, although Bell’stheorem actually doesn’t apply to a classical fluid modelof quantum mechanics as explained later in this paper and inRef. [19]. Also, the RHIC collision data exhibit rapid thermalizationthat poses a fundamental causality problem for the interpretationof the experiments. The early universe seems to have thermalizedtoo rapidly as well, as evidenced by the horizon problemin cosmology. Could these phenomena be related? If they are,then the usual theory of inflation is not likely to be theexplanation, since no inflation has been observed at RHIC.This equation is completed by an independent equation of massconservation

rmrUm ¼ 0; ð4Þ

and usually by a barotropic equation of state. The hydrodynamicequations that are used to simulate the RHIC experiments are much more complex than these [20,21]. We are looking for atractable and a simple model to begin looking for the emergenceof quantum-like behavior. The Burgers theory ignores pressurealtogether, and we shall do the same. Ignoring pressure,the energy momentum tensor and equation of motion become

Tab ¼ rUaUb; and UaraUb ¼ 0; ð5Þ

where we have used the product law for covariant derivatives.The equation involves only the velocity field. It was shown

in Ref. [19] that Eq. (5) is consistent with a flat space-timemetric provided that a cosmological constant is allowed ingeneral relativity. The fluid equations in Minkowski spacebecome

Ua@aUn ¼ 0: ð6Þ

This equation has an infinite number of invariants as is shownin Ref. [19]. In cosmology, this form for the energy momentumtensor is called the cosmic dust model [22]. This type of universeis generally associated with cosmic inflation and is calledthe lambda-CDM model [23].

As was shown in Ref. [19], one can define a proper timevariable in place of the usual time variable for the fluid, andin terms of this time, the equations become the inviscidBurgers equation

@

@tuðx; tÞ þ ðuðx; tÞrÞuðx; tÞ ¼ 0; ð7Þ

3. Background on the Burgers equation

The Burgers equation was first proposed as a model for zeropressure gas dynamics [24]. A review is given in Ref. [25].The Galerkin-truncated version of this equation has been shownto have a number of interesting properties in numericalexperiments [11–13,26] including ergodic chaotic behaviormimicking turbulence.

In Ref. [12] it was shown that there are three conservedquantities for this system, and that for many randomly selectedstarting conditions the equations are chaotic and ergodic andresult in equipartition of energy, but also that for certainnon-typical starting conditions, the system does not resultin equipartition, but rather has a tilted energy spectrum.

It was shown in Ref. [19] that there are soliton solutions tothis system of equations, and that they are related to the deviationfrom equipartition, and to extremal values of the Hamiltonian.These soliton solutions are similar to the delta solitons proposedand analyzed by Sarrico [27] which were inspired by the seminalworks of Maslov et al. [28,29].

4. The inviscid Burgers equation in one dimension

The continuum version of the equation is

@u

@tþ

1

2

@

@xðu2Þ ¼ 0; ð8Þ

with periodic boundary conditions u(x+2p, t) ¼ u(x, t). PLf ¼ fLdenotes the Fourier projection operator with cutoff for wave-numbers beyond L

uk ¼1

2p

Z 2p

0uðxÞe�ikx dx; ð9Þ

where k is integer, and u(x) is 2p periodic and real valued so thatu�k ¼ (uk)�. The truncated equation is [11–13,26]

@uL

@tþ

1

2

@

@xPLðu

2LÞ ¼ 0; ð10Þ

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M. Davidson / Physica E 42 (2010) 317–322 319

the energy density is [12]

E ¼1

2

Xjkjo¼L

jukj2 ¼ju0j

2

2þXLk¼1

jukj2; ð11Þ

and the Hamiltonian is [12]

H ¼1

12p

Z 2p

0PLðu

3LÞdx: ð12Þ

It can be shown that

d

dtu0 ¼

dE

dt¼

dH

dt¼ 0: ð13Þ

The equations are invariant under Galilean transformations,parity, and time reversal.

Fig. 1. A simple soliton.

Fig. 2. A time series of a single moving soliton viewed in the frame where the

mean fluid velocity is zero. It maintains its shape as it moves to the left with

soliton velocity of �1 rad/s.

5. Extremal condition and solitons

As found in Ref. [19], extremal solutions (E extremal holding H

fixed) give solitons. We use the Lagrange multiplier method. Tosimplify, we work in the Galilean frame of reference where u0 ¼ 0.There is no loss of generality in doing this due to Galileaninvariance. The condition becomes

@

@uk½H þ lE� ¼ 0; ka0: ð14Þ

We obtain the relation [19]

@uk

@t� ikluk ¼ 0; ð15Þ

which is equivalent to

@uðx; tÞ

@t� l

@uðx; tÞ

@x¼ 0; ð16Þ

and this has solutions of the form

uðx; tÞ ¼ uðxþ ltÞ: ð17Þ

And therefore the extremal solutions are traveling waves (withthe direction determined by the sign of l) we obtain a value for l[19]

l ¼ �3H

2E: ð18Þ

Since the equations are invariant under Galilean transforma-tions, we can slow down the traveling wave to zero velocity bymoving along with it, and then the solution becomes just a staticfunction of x. This static solution must satisfy [19]

@uk

@t¼ 0 ¼

�ik

2

Xjk0 j;jk�k0 jo¼Ljkjo¼L

uk�k0 uk0 : ð19Þ

Fig. 1 shows a soliton numerically calculated taken from Ref.[19]. The soliton’s width varies roughly inversely as the cuttoff L

6. Time series of a single soliton with noise

In Fig. 2, we see a time series for a pure soliton. It moveswithout changing shape. In Fig. 3, we see the same soliton, butwith additional random starting velocity noise field of the form

uðkÞ ¼sffiffiffi2p Xk

k ¼ 1 to L; ^uðkÞ ¼ uðkÞ�

;

ð20Þ

where Xk is a normally distributed complex random variablewhose real and imaginary parts have variance 1. This noise isequipartitioned on the average, and this mimics the equilibriumchaos of the typical starting conditions as found in Ref. [12]. Thenoise is simply added to the pure soliton solution for the initialconditions. The result shows that the soliton moves in a randomand chaotic velocity field. Despite the chaotic background, thesoliton maintains its identity and average velocity as timeprogresses without noticeable deterioration or slowing down

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Fig. 3. A time series of a single moving soliton with random noise initial starting

condition. The soliton still maintains its identity over long times.

Fig. 4. The top graph shows the peak position of the soliton with noise after

subtracting out the mean velocity motion, and the bottom graph shows the Fourier

power spectrum of this trajectory.

M. Davidson / Physica E 42 (2010) 317–322320

seemingly forever, and it moves with approximately constantvelocity slightly different from its unperturbed velocity. Thevelocity of the soliton is modified slightly by the presence ofnoise, even on the average. The peak position of the soliton showsa random diffusive motion in Fig. 4.

These results are rather amazing. We have here a system thatprovides random diffusion of localized solitons without anyviscosity. The system is time reversible and the diffusion persistsforever. It is similar to quantum mechanics in this regard, yet thebasic model is extremely simple.

7. Properties of solitons

The solitons suffer considerable damage on collision, and thisdamage does not seem to be decreasing with increasing L [19].They attract one another when placed in proximity and allcharacteristic curves asymptote to the leading edge of a soliton.

We may add an external body force to the fluid. In this case,both characteristic curves and solitons obey Newton’s equation,provided that the force is weak [19], but there would be anadditional high spatial frequency force correction, stochastic innature, due to the truncation or momentum cutoff. For strongexternal forces, the soliton would be distorted, and the subse-quent motion would be complicated. Simulations confirm thesequalitative behaviors [19].

It was noted in Ref. [11] that there are invariant subspaces forthis system. Solitons can be constructed in higher number ofdimensions too [19].

8. Possible relationship to string theory

Moffat’s topological invariants [30] provide a mechanism forthe endurance of closed tubes of vorticity. If these tubes arenarrow in cross section, then they look like strings. There is aninteresting connection between string theory and magnetohy-drodynamics with narrow magnetic flux tubes, and also withvortex flux tubes in superfluid turbulence theory [31–33]. Theobserved stochastic vortex tangle in superfluid turbulence He IIsuggests a similar phenomenon should occur in the truncatedinviscid Burgers system. In fact, if the inviscid ideal fluid (in 3D ofcourse) started off with just a single filamentary vortex, i.e. asingle string-like tube of vorticity, then as time progressed thissingle flux tube could become extraordinarily entangled, fillingthe whole fluid volume with a tangle of vorticity, but every singlepiece of vortex tube would still have the same flux as the startingparent vortex tube. This is due to the Kelvin circulation theorem.Thus, the Kelvin circulation around any loop would be quantizedin essentially the same manner as in a superfluid. This is apotential candidate for a mechanism underlying quantum en-tanglement in a theory such as is being proposed here.

9. The fluid as a Markov process

The truncated inviscid Burgers equation is first order in timeand by Picard’s existence theorem it has a unique solution for anyinitial conditions. Therefore, the state of the entire fluid at oneinstant in time determines the fluid for all other times, both pastand future. Consequently any statistical treatment of the fluid willbe a Markov process in both the forward and backward timedirections. This is similar to Nelson’s stochastic mechanics [34,35]which is based on a dynamical assumption that is time reversibleand stochastic. Since the inviscid Burgers equation is timereversible and inherently stochastic, it is remarkably similar toNelson’s theory. One of the problems with Nelson’s theory is thatthe main examples of Brownian motion in physics are due tohighly dissipative systems that are not time reversible. In ourpresent work, we have a system that achieves a stochasticbehavior and Markov behavior without dissipation. Stochasticmechanics assumes, on the microscopic level, that the trajectorieslook locally like Wiener processes. This is clearly not true for thefluid model, except in some coarse time approximation.

The motion of soliton peaks or of characteristic curves, on theother hand, can be derived from the fluid, but they are notnecessarily Markov processes except in some approximation. Inthe presence of ergodic noise, the soliton’s peak becomes

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M. Davidson / Physica E 42 (2010) 317–322 321

effectively a random variable that depends on the fluid. Thus, thesoliton’s peak trajectory will be describable by a hidden Markovmodel [36]. Stochastic mechanics is known to be highly non-unique [37,38], and therefore it is conceivable that it could beequivalent to such a hidden Markov model in some approxima-tion.

10. Quantization

The single valuedness of the quantum mechanical wavefunction is not an automatic property of stochastic models orhydrodynamic models [39,40]. Suppose the cosmological proto-fluid started out with one infinitesimally thin string-like vortextube or filament. This would be an initial condition for the entireuniverse. Then, over time, assuming the fluid were turbulent, thissingle vortex tube would become immensely tangled andconfused, but its flux would be maintained [19]. In developing amodel based on this idea we might allow vortex crossing andrecombination effects [41] to occur which would allow closedloops of thin vortex tubes to separate from the main string and gotheir own way inside the fluid while inheriting the flux of theparent vortex. These topological solitons might persist andinteract. Assuming that an arbitrary closed curve C does notexactly intersect any vortex string we have thatIC

udl ¼ Nk; ð21Þ

for N integer and k a constant,where k is the universal flux of all the vortex strings and N

depends on how many vortex strings are linked by the curve C andon their orientation.

In this situation, for any smooth and simply connected domainO that does not contain any sections of any vortex strings we canwrite

u ¼ DS; x 2 O: ð22Þ

This is the weak form of the Helmholtz theorem. By takingoverlapping domains we can analytically continue S to the wholespace. But S will not be single valued because if it were then thecirculation would vanish for all closed curves. The multiple valuesof S can be {S+lk, l integer}. Because S is not single valued, it is notconvenient mathematically to describe the fluid in terms of it.However, note that e7iS2p/k is single valued. Therefore, there is areal advantage to transforming the fluid equations into complexones to deal with single-valued functions and still have thesimplicity of a fluid described by a potential function. Let us nowconsider a conserved quantity moving in the fluid. It might bemass, charge, or probability density of a diffusion. The fluidequations are then equivalent to the following Schrodinger-typeequation by a Madelung transformation for any conservedquantity r [19]

�1

2

k2p

� �2

Dþ Vk�eRþiS2p=k ¼ i

k2p

� � @@t

eRþiS2p=k where r ¼ e2R;

�

ð23Þ

Vk ¼1

2

k2p

� �2DeR

eR; ð24Þ

defining a ‘‘wave function’’ by

C ¼ eRþiS2p=k: ð25Þ

We have a Schrodinger-like equation except that the potentialgiven by Eq. (24) is nonlinear. The single valuedness of this wave

function is thus a consequence of the starting conditions for thefluid having a single infinitely thin vortex tube.

The extra terms Vk result in a nonlinear Schrodinger equation.They need to be cancelled somehow if we are to have linearsuperposition. A possible suppression mechanism based on fluidshock reduction is given in Ref. [19].

Shocks may play still another role. They allow discontinuousjumps in the state of the fluid mimicking quantum jumps. Thus,fluid shocks might be a mechanical explanation for spontaneouswave function collapse [42,43]. In fact, there is just such aspontaneous wave function collapse model which is based on theBurgers equation [18]. So the historical and popular concept of aquantum jump actually might have a mechanical explanation inthis picture. Photon emission or absorption could be accompaniedby a fluid shock too; this might help to resolve the profoundlyparadoxical aspects of photon emission (or absorption) as burstsof radiation coincident with fluid shocks.

11. The Bell theorem

The Bell theorem [44] does not apply to a deterministic theorysince all physical variables in such a theory are determined by theinitial conditions. The ‘‘free will axiom’’ (which is incompatiblewith determinism) is required for any conclusions to be maderegarding non-locality and quantum mechanics [45]. Althoughdeterministic, the turbulent fluid model is essentially incalculablefor any significant time interval. No experiment has ever beenproposed that would test the ‘‘free will axiom’’. It seems to be anunverifiable assumption, and therefore the Bell theorem cannot beused to rule out a turbulent fluid model for quantum mechanics.In fact, all theorems based on contrafactual logic are similarlyrendered irrelevant [46].

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