Coupled oscillators, entangled oscillators, and Lorentz-covariant harmonic oscillators

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS

J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S458–S467 doi:10.1088/1464-4266/7/12/005

Coupled oscillators, entangled oscillators,and Lorentz-covariant harmonicoscillatorsY S Kim1 and Marilyn E Noz2

1 Department of Physics, University of Maryland, College Park, MD 20742, USA2 Department of Radiology, New York University, New York, NY 10016, USA

E-mail: yskim@physics.umd.edu and noz@nucmed.med.nyu.edu

Received 9 August 2005, accepted for publication 11 October 2005Published 4 November 2005Online at stacks.iop.org/JOptB/7/S458

AbstractOther than scattering problems where perturbation theory is applicable, thereare basically two ways to solve problems in physics. One is to reduce theproblem to harmonic oscillators, and the other is to formulate the problem interms of two-by-two matrices. If two oscillators are coupled, the problemcombines both two-by-two matrices and harmonic oscillators. This methodthen becomes a powerful research tool which can be used in many differentbranches of physics. Indeed, the concept and methodology in one branch ofphysics can be translated into another through the common mathematicalformalism. Coupled oscillators provide clear illustrative examples for someof the current issues in physics, including entanglement and Feynman’s restof the universe. In addition, it is noted that the present form of quantummechanics is largely a physics of harmonic oscillators. Special relativity isthe physics of the Lorentz group which can be represented by the group oftwo-by-two matrices commonly called SL(2, c). Thus the coupledharmonic oscillator can play the role of combining quantum mechanics withspecial relativity. It is therefore possible to relate the current issues ofphysics to the Lorentz-covariant formulation of quantum mechanics.

Keywords: entangled oscillators, space–time entanglement

1. Introduction

Because of its mathematical simplicity, the harmonic oscillatorprovides soluble models in many branches of physics. Itoften gives a clear illustration of abstract ideas. In manycases, the problems are reduced to the problem of two coupledoscillators. Soluble models in quantum field theory, suchas the Lee model [1] and the Bogoliubov transformation insuperconductivity [2], are based on two coupled oscillators.Recently, two coupled oscillators formed the mathematicalbasis for squeezed states in quantum optics [3, 4], especiallytwo-mode squeezed states [5, 6].

More recently, it was noted by Giedke et al thatentanglement realized in two-mode squeezed states can beformulated in terms of symmetric Gaussian states [7]. Froma mathematical point of view, the subject of entanglement hasbeen largely a physics of two-by-two matrices. It is gratifying

to note that harmonic oscillators can also play a role inclarifying the physical basis of entanglement. The symmetricGaussian states can be constructed from two coupled harmonicoscillators. The entanglement issues in two-mode squeezedstates can therefore be added to the physics of coupledharmonic oscillators. Since many physical models are basedon coupled oscillators, entanglement ideas can be exported toall those models.

In this paper, we construct a model of Lorentz-covariantharmonic oscillators based on the coupled oscillator. Sincethe covariance requires coupling of space and time variables,the covariant oscillator formalism allows expansion ofentanglement ideas to the space–time region.

Combining quantum mechanics with special relativityis a fundamental problem in its own right. Why do weneed covariant harmonic oscillators while there is quantumfield theory with Feynman diagrams? Since this is also

1464-4266/05/120458+10$30.00 © 2005 IOP Publishing Ltd Printed in the UK S458

Coupled oscillators, entangled oscillators, and Lorentz-covariant harmonic oscillators

a fundamental problem in its own right, we would like toaddress this issue in the first sections of this paper. Thepoint is that quantum mechanics deals with waves. Thereare running waves and standing waves. The present formof quantum mechanics and its S-matrix formalism are onlyfor running waves, and cannot directly deal with standingwaves satisfying boundary conditions. Of course standingwaves are superpositions of two running waves, but we donot know how to approach this problem when we includeLorentz transformations. The simplest approach is to workwith a soluble model based on harmonic oscillators and two-by-two matrices.

From the mathematical point of view, special relativity isa physics of Lorentz transformations or the Lorentz group. Itis gratifying to note that the six-parameter Lorentz group canbe represented by two-by-two matrices with unit determinant.The elements can be complex numbers. This group is knownas SL(2,C) which forms the universal covering group of theLorentz group. Thus, special relativity is a physics of two-by-two matrices.

The standard approach to two coupled oscillators is toconstruct a two-by-two matrix of two oscillators with differentfrequencies. Thus, it is not surprising to note that themathematics of the coupled oscillator is directly applicableto Lorentz-covariant harmonic oscillators. Therefore, thecovariant oscillators, defined in the space–time region, can beenriched by the physics of entanglement.

As for using coupled harmonic oscillators for combiningquantum mechanics with relativity, we examine in thispaper the earlier attempts made by Dirac and Feynman.We first examine Dirac’s approach which was to constructmathematically appealing models. We then examine howFeynman approached this problem. He observed theexperimental world, told the story of the real world in hisstyle, and then wrote down mathematical formulae as needed.We use coupled oscillators to combine Dirac’s approachand Feynman’s approach to construct the Lorentz-covariantformulation of quantum mechanics.

In section 2, it is noted that quantum mechanics dealswith waves, and there are running waves and standing waves.While it is somewhat straightforward to make running wavesLorentz covariant, there are no established prescriptions forconstructing standing waves consistent with special relativity.We stress in this section why standing waves are differentfrom running waves. In section 3, we discuss the quantummechanics of two coupled oscillators, and study how thesystem could absorb the physical ideas developed in the caseof two-mode squeezed states.

In section 4, we study systematically Dirac’s lifetimeefforts to combine quantum mechanics with relativity. Hewas concerned with space–time asymmetry associated withposition–momentum and time–energy uncertainty relations.We examine carefully what more had to be done to completethe task initiated by Dirac. In section 5, we studyFeynman’s efforts to combine quantum mechanics with specialrelativity. Here also, we carefully examine the shortcomingsin Feynman’s papers on harmonic oscillators. It is shownpossible in section 6 that the works of Feynman and Diraccan be combined to produce a covariant harmonic oscillatorsystem. In section 7, it is shown that the covariant oscillator

formalism shares the same mathematical base as that of twocoupled oscillators, and much of the physical ideas, especiallythe entanglement idea, can be translated into the space–timevariables of the Lorentz-covariant world.

The physics of space–time has its own merit, and is notbound to import ideas developed in other areas of physics. Insection 8, we note that there is a decoherence effect observedfirst by Feynman. It is known widely as Feynman’s partonpicture in which partons appear like incoherent entities. Itis widely believed that partons are Lorentz-boosted quarks.Then, the question is how the Lorentz boost, which is a space–time symmetry operation, can destroy coherence. We addressthis question in this section.

2. Scattering states and bound states

In this section, we would like to address the question of why weneed covariant harmonic oscillators while there is the Lorentz-covariant formulation of quantum field theory which allows usto calculate scattering amplitudes using Feynman diagrams.

When Einstein formulated his special relativity onehundred years ago, he was considering point particles.Einstein’s energy–momentum relation is known to be validalso for particles with space–time extensions. There have beenefforts to understand special relativity for rigid particles withnonzero size, without any tangible result. On the other hand,the emergence of quantum mechanics made the rigid-bodyproblem largely irrelevant. Thanks to wave–particle duality,we talk about wavepackets and standing waves, instead of rigidbodies. The issue becomes whether those waves can be madeLorentz-covariant.

Of course, here, the starting point is the plane wave, whichcan be written as

eip·x = ei( �p·�x−Et). (1)

Since it takes the same form for all Lorentz frames, we do notneed any extra effort to make it Lorentz covariant.

Indeed, the S-matrix derivable from the present formof quantum field theory calls for calculation of all S-matrixquantities in terms of plane waves. Thus, the S-matrix isassociated with perturbation theory or Feynman diagrams.Feynman propagators are written in terms of plane waves onthe mass shell.

We should realize however that the S-matrix formalismis strictly for running waves, starting from a plane wavefrom one end of the universe and ending with another planewave at another end. How about standing waves? Thisquestion is illustrated in figure 1. Of course, standingwaves can be regarded as superpositions of running wavesmoving in opposite directions. However, in order to guaranteelocalization of the standing waves, we need a spectral functionor boundary conditions. The covariance of standing wavesnecessarily involves the covariance of boundary conditions orspectral functions. How much do we know about this problem?This problem has not yet been systematically explored.

On the other hand, some concrete models for covariantbound states were studied in the past by a number ofdistinguished physicists, including Dirac [8], Yukawa [9], andFeynman and his co-authors [10]. We shall return to thisproblem in section 4.

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Y S Kim and M E Noz

Running Waves

StandingWaves

Running Waves

Figure 1. Running waves and standing waves in quantum theory. Ifa particle is allowed to travel from infinity to infinity, it correspondsto a running wave according to the wave picture of quantummechanics. If, on the other hand, it is trapped in a localized region,we have to use standing waves to interpret its location in terms ofprobability distribution.

Finally, let us see what kind of problems we expect if weuse S-matrix methods for bound-state problems. If we use thespherical coordinate system where the scattering centre is atthe origin, the S-matrix consists of both incoming and outgoingwaves. If we make analytic continuations of these waves tobound states with negative total energy, the outgoing wavebecomes localized, but the incoming wave increases to infinityat large distance from the origin, as indicated in figure 2. Thereare no methods of eliminating this unphysical wavefunction.

Indeed, this was the source of the mistake made by Dashenand Frautschi in their once-celebrated calculation of theneutron and proton mass difference using an S-matrix formulacorresponding to the first-order energy shift [11]. They used aperturbation formula derivable from S-matrix considerations,but their formula corresponds to the perturbation formula [12]:

δE = (φgood, δVφbad), (2)

where the good and bad bound-state wavefunctions are like

φgood ∼ e−br , φbad ∼ ebr , (3)

for large values of r , as illustrated in figure 2. We are not awareof any S-matrix method which guarantees the localization ofbound-state wavefunctions.

3. Coupled oscillators and entangled oscillators

The coupled oscillator problem can be formulated as that of aquadratic equation in two variables. The diagonalization of thequadratic form includes a rotation of the coordinate system.However, the diagonalization process requires additionaltransformations involving the scales of the coordinatevariables [13, 14]. Indeed, it was found that the mathematicsof this procedure can be as complicated as the group theory ofLorentz transformations in a six-dimensional space with threespatial and three time coordinates [15].

In this paper, we start with a simple problem of twoidentical oscillators. Then the Hamiltonian takes the form

H = 1

2

{1

mp2

1 +1

mp2

2 + Ax21 + Ax2

2 + 2Cx1x2

}. (4)

If we choose coordinate variables

y1 = 1√2(x1 + x2),

y2 = 1√2(x1 − x2),

(5)

φ(r)

φ ∼ ebrbad

φ ∼ e--brgood

r

Figure 2. Good and bad wavefunctions contained in the S-matrix.Bound-state wavefunctions satisfy the localization condition and aregood wavefunctions. Analytic continuations of plane waves do notsatisfy the localization boundary condition, and become badwavefunctions at the bound-state energy.

the Hamiltonian can be written as

H = 1

2m{p2

1 + p22} +

K

2{e−2ηy2

1 + e2ηy22}, (6)

where

K =√

A2 − C2,

exp(2η) =√

A − C

A + C.

(7)

The classical eigenfrequencies are ω± = ωe±2η with ω =√K/m.

If y1 and y2 are measured in units of (mK)1/4, the ground-state wavefunction of this oscillator system is

ψη(x1, x2) = 1√π

exp

{−1

2(e−2ηy2

1 + e2ηy22)

}. (8)

The wavefunction is separable in the y1 and y2 variables.However, for the variables x1 and x2, the story is quite different,and can be extended to the issue of entanglement.

There are three ways to excite this ground-state oscillatorsystem. One way is to multiply Hermite polynomials for theusual quantum excitations. The second way is to constructcoherent states for each of the y variables. Yet another way isto construct thermal excitations. This requires density matricesand Wigner functions [14].

The key question is how the quantum mechanics in theworld of the x1 variable is affected by the x2 variable. If we usetwo separate measurement processes for these two variables,these two oscillators are entangled.

Let us write the wavefunction of equation (8) in terms ofx1 and x2; then

ψη(x1, x2) = 1√π

exp

{−1

4[e−2η(x1 + x2)

2 + e2η(x1 − x2)2]

}.

(9)When the system is decoupled with η = 0, this wavefunctionbecomes

ψ0(x1, x2) = 1√π

exp

{−1

2(x2

1 + x22 )

}. (10)

The system becomes separable and becomes disentangled.

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Coupled oscillators, entangled oscillators, and Lorentz-covariant harmonic oscillators

As was discussed in the literature for several differentpurposes [3, 16, 17], this wavefunction can be expanded as

ψη(x1, x2) = 1

cosh η

∑k

(tanh η)kφk(x1)φk(x2), (11)

where φk(x) is the normalized harmonic oscillator wavefunc-tion for the kth excited state. This expansion serves as themathematical basis for squeezed states of light in quantum op-tics [3], among other applications.

The expansion given in equation (11) clearly demonstratesthat the coupled oscillators are entangled oscillators. Thisexpression is identical to equation (1) of the recent paper byGiedke et al [7]. This means that the coupled oscillators canabsorb most of the current entanglement issues, and serve asa reservoir of entanglement ideas for other physical systemsmodelled after the coupled oscillators. We are particularlyinterested in expanding these ideas to relativistic space andtime through the covariant oscillator formalism.

In section 4, we shall see that the mathematics of thecoupled oscillator can serve as the basis for the covariantharmonic oscillator formalism where the x1 and x2 variablesare replaced by the longitudinal and time-like variables,respectively. This mathematical identity will lead to theconcept of space–time entanglement in special relativity, aswe shall see in section 7.

4. Dirac’s harmonic oscillators and light-conecoordinate system

Dirac is known to us through the Dirac equation forspin-1/2 particles. However, his main interest was infoundational problems. First, Dirac was never satisfied withthe probabilistic formulation of quantum mechanics. This isstill one of the hotly debated subjects in physics. Second, ifwe tentatively accept the present form of quantum mechanics,Dirac was insisting that it has to be consistent with specialrelativity. He wrote several important papers on this subject.Let us look at some of his papers.

During World War II, Dirac was looking into thepossibility of constructing representations of the Lorentz groupusing harmonic oscillator wavefunctions [8]. The Lorentzgroup is the language of special relativity, and the presentform of quantum mechanics starts with harmonic oscillators.Presumably, therefore, he was interested in making quantummechanics Lorentz covariant by constructing representationsof the Lorentz group using harmonic oscillators.

In his 1945 paper [8], Dirac considered the Gaussian form

exp{− 12 (x

2 + y2 + z2 + t2)}. (12)

This Gaussian form is in the (x, y, z, t) coordinate variables.Thus, if we consider Lorentz boost along the z direction, wecan drop the x and y variables, and write the above equationas

exp{− 12 (z

2 + t2)}. (13)

This is a strange expression for those who believe inLorentz invariance. The expression (z2 + t2) is not invariantunder Lorentz boost. Therefore Dirac’s Gaussian form ofequation (13) is not a Lorentz-invariant expression.

Dirac: Uncertaintywithout Excitations

Heisenberg: Uncertaintywith Excitations

t

z

Figure 3. Space–time picture of quantum mechanics. There arequantum excitations along the space-like longitudinal direction, butthere are no excitations along the time-like direction. Thetime–energy relation is a c-number uncertainty relation.

On the other hand, this expression is consistent with hisearlier papers on the time–energy uncertainty relation [18].In those papers, Dirac observed that there is a time–energyuncertainty relation, while there are no excitations alongthe time axis. He called this the ‘c-number time–energyuncertainty’ relation. When one of us (YSK) was talking withDirac in 1978, he clearly mentioned this phrase again. He saidfurther that this space–time asymmetry is one of the stumblingblocks in combining quantum mechanics with relativity. Thissituation is illustrated in figure 3.

In 1949, the Reviews of Modern Physics published aspecial issue to celebrate Einstein’s 70th birthday. Thisissue contains Dirac paper entitled ‘Forms of RelativisticDynamics’ [19]. In this paper, he introduced his light-conecoordinate system, in which a Lorentz boost becomes a squeezetransformation.

When the system is boosted along the z direction, thetransformation takes the form(

z′t ′

)=

(cosh η sinh ηsinh η cosh η

)(zt

). (14)

The light-cone variables are defined as [19]

u = (z + t)/√

2, v = (z − t)/√

2, (15)

and the boost transformation of equation (14) takes the form

u ′ = eηu, v′ = e−ηv. (16)

The u variable becomes expanded while the v variable becomescontracted, as is illustrated in figure 4. Their product

uv = 12 (z + t)(z − t) = 1

2 (z2 − t2) (17)

remains invariant. In Dirac’s picture, the Lorentz boost is asqueeze transformation.

If we combine figures 3 and 4, then we end up withfigure 5. This transformation changes the Gaussian form ofequation (13) into

ψη(z, t) =(

1

π

)1/2

exp

{−1

2

(e−2ηu2 + e2ηv2

)}. (18)

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Y S Kim and M E Noz

A=4u ′v ′

t

z

u

v

A=4uv=2(t2–z2)

Figure 4. Lorentz boost in the light-cone coordinate system.

Let us go back to section 3 on the coupled oscillators. Theabove expression is the same as equation (8). The x1 variablenow became the longitudinal variable z, and the x2 variablebecame the time-like variable t .

We can use the coupled harmonic oscillator as the startingpoint of relativistic quantum mechanics. This allows us totranslate the quantum mechanics of two coupled oscillatorsdefined over the space of x1 and x2 into quantum mechanicsdefined over the space–time region of z and t .

This form becomes (13) when η becomes zero.The transition from equations (13) to (18) is a squeezetransformation. It is now possible to combine what Diracobserved into a covariant formulation of the harmonicoscillator system. First, we can combine his c-number time–energy uncertainty relation described in figure 3 and his light-cone coordinate system of figure 4 into a picture of covariantspace–time localization given in figure 5.

The wavefunction of equation (13) is distributed within acircular region in the uv plane, and thus in the zt plane. On theother hand, the wavefunction of equation (18) is distributedin an elliptic region with the light-cone axes as the majorand minor axes respectively. If η becomes very large, thewavefunction becomes concentrated along one of the light-cone axes. Indeed, the form given in equation (18) is a Lorentz-squeezed wavefunction. This squeeze mechanism is illustratedin figure 5.

There are two homework problems which Dirac left us tosolve. First, in defining the t variable for the Gaussian form ofequation (13), Dirac did not specify the physics of this variable.If it is going to be the calendar time, this form vanishes in theremote past and remote future. We are not dealing with thiskind of object in physics. What is then the physics of thistime-like t variable?

The Schrodinger quantum mechanics of the hydrogenatom deals with localized probability distribution. Indeed, thelocalization condition leads to the discrete energy spectrum.Here, the uncertainty relation is stated in terms of the spatialseparation between the proton and the electron. If we believein Lorentz covariance, there must also be a time separationbetween the two constituent particles, and an uncertaintyrelation applicable to this separation variable. Dirac did notsay in his papers of 1927 and 1945, but Dirac’s ‘t’ variable

Figure 5. Effect of the Lorentz boost on the space–timewavefunction. The circular space–time distribution in the rest framebecomes Lorentz squeezed to become an elliptic distribution.

is applicable to this time-separation variable. This time-separation variable will be discussed in detail in section 5 forthe case of relativistic extended particles.

Second, as for the time–energy uncertainty relation,Dirac’s concern was how the c-number time–energyuncertainty relation without excitations can be combined withuncertainties in the position space with excitations. How canthis space–time asymmetry be consistent with the space–timesymmetry of special relativity?

Both of these questions can be answered in terms of thespace–time symmetry of bound states in the Lorentz-covariantregime [17]. In his 1939 paper [20], Wigner worked outinternal space–time symmetries of relativistic particles. Heapproached the problem by constructing the maximal subgroupof the Lorentz group whose transformations leave the givenfour-momentum invariant. As a consequence, the internalsymmetry of a massive particle is like the three-dimensionalrotation group which does not require transformation into time-like space.

If we extend Wigner’s concept to relativistic bound states,the space–time asymmetry which Dirac observed in 1927is quite consistent with Einstein’s Lorentz covariance [21].Indeed, Dirac’s time variable can be treated separately.Furthermore, it is possible to construct a representations ofWigner’s little group for massive particles [17]. As for the timeseparation which can be linearly mixed with space-separationvariables when the system is Lorentz boosted, it has its role ininternal space–time symmetry.

Dirac’s interest in harmonic oscillators did not stop withhis 1945 paper on the representations of the Lorentz group.In his 1963 paper [5], he constructed a representation ofthe O(3, 2) de Sitter group using two coupled harmonicoscillators. This paper contains not only the mathematicsof combining special relativity with the quantum mechanicsof quarks inside hadrons, but also forms the foundations oftwo-mode squeezed states which are so essential to modernquantum optics [3, 6]. Dirac did not know this when hewas writing this 1963 paper. Furthermore, the O(3, 2) deSitter group contains the Lorentz group O(3, 1) as a subgroup.Thus, Dirac’s oscillator representation of the de Sitter groupessentially contains all the mathematical ingredient of what weare studying in this paper.

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Coupled oscillators, entangled oscillators, and Lorentz-covariant harmonic oscillators

Harmonic

Feynman Diagrams

Oscillators

Feynman Diagrams

Figure 6. Feynman’s roadmap for combining quantum mechanicswith special relativity. Feynman diagrams work for running waves,and they provide a satisfactory resolution for scattering states inEinstein’s world. For standing waves trapped inside an extendedhadron, Feynman suggested harmonic oscillators as the first step.

5. Feynman’s oscillators

In his invited talk at the 1970 spring meeting of the AmericanPhysical Society held in Washington, DC (USA), Feynmanwas discussing hadronic mass spectra and a possible covariantformulation of harmonic oscillators. He noted that the massspectra are consistent with degeneracy of three-dimensionalharmonic oscillators. Furthermore, Feynman stressed thatFeynman diagrams are not necessarily suitable for relativisticbound states and that we should try harmonic oscillators.Feynman’s point was that, while plane-wave approximations interms of Feynman diagrams work well for relativistic scatteringproblems, they are not applicable to bound-state problems. Wecan summarize what Feynman said in figures 2 and 6.

In their 1971 paper [10], Feynman et al started theirharmonic oscillator formalism by defining coordinate variablesfor the quarks confined within a hadron. Let us use thesimplest hadron consisting of two quarks bound together withan attractive force, and consider their space–time positions xa

and xb, and use the variables

X = (xa + xb)/2, x = (xa − xb)/2√

2. (19)

The four-vector X specifies where the hadron is located inspace and time, while the variable x measures the space–timeseparation between the quarks. According to Einstein, thisspace–time separation contains a time-like component whichactively participates as in equation (14), if the hadron is boostedalong the z direction. This boost can be conveniently describedby the light-cone variables defined in equation (15).

What do Feynman et al say about this oscillatorwavefunction? In their classic 1971 paper [10], they start withthe following Lorentz-invariant differential equation.

1

2

{x2µ − ∂2

∂x2µ

}ψ(x) = λψ(x). (20)

This partial differential equation has many differentsolutions depending on the choice of separable variablesand boundary conditions. Feynman et al insist on Lorentz-invariant solutions which are not normalizable. On theother hand, if we insist on normalization, the ground-statewavefunction takes the form of equation (13). It is thenpossible to construct a representation of the Poincare groupfrom the solutions of the above differential equation [17]. Ifthe system is boosted, the wavefunction becomes as given inequation (18).

Although this paper contained the above-mentionedoriginal idea of Feynman, it contains some serious

mathematical flaws. Feynman et al start with a Lorentz-invariant differential equation for the harmonic oscillator forthe quarks bound together inside a hadron. For the two-quarksystem, they write the wavefunction of the form

exp{− 12 (z

2 − t2)}, (21)

where z and t are the longitudinal and time-like separationsbetween the quarks. This form is invariant under the boost,but is not normalizable in the t variable. We do not know whatphysical interpretation to give to this the above expression.

On the other hand, Dirac’s Gaussian form given inequation (13) also satisfies Feynman’s Lorentz-invariantdifferential equation. This Gaussian function is normalizable,but is not invariant under the boost. However, the word‘invariant’ is quite different from the word ‘covariant’. Theabove form can be covariant under Lorentz transformations.We shall return to this problem in section 6.

Feynman et al studied in detail the degeneracy of the three-dimensional harmonic oscillators, and compared their resultswith the observed experimental data. Their work is completeand thorough, and is consistent with the O(3)-like symmetrydictated by Wigner’s little group for massive particles [17, 20].Yet, Feynman et al make an apology that the symmetry is notO(3, 1). This unnecessary apology causes a confusion notonly to the readers but also to the authors themselves, andmakes the paper difficult to read.

6. Can harmonic oscillators be made covariant?

The simplest solution to the differential equation ofequation (20) takes the form of equation (13). If weallow excitations along the longitudinal coordinate and forbidexcitations along the time coordinate, the wavefunction takesthe form

ψn0 (z, t) = Cn Hn(z) exp{− 1

2 (z2 + t2)}, (22)

where Hn is the Hermite polynomial of the nth order, and Cn

is the normalization constant.If the system is boosted along the z direction, the z and t

variables in the above wavefunction should be replaced by z′and t ′ respectively with

z′ = (cosh η)z − (sinh η)t, t ′ = (cosh η)t − (sinh η)z.(23)

The Lorentz-boosted wavefunction takes the form

ψnη (z, t) = Hn(z

′) exp{− 12 (z

′2 + t ′2)}. (24)

It is interesting that these wavefunctions satisfy theorthogonality condition [22].

∫ψn

0 (z, t)ψmη (z, t) dz dt =

(√1 − β2

)nδnm , (25)

where β = tanh η. This orthogonality relation is illustratedin figure 7. The physical interpretation of this in terms ofLorentz contractions is given in our book [17], but seems torequire further investigation.

It is indeed possible to construct the representation ofWigner’s O(3)-like little group for massive particles using

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Y S Kim and M E Noz

Figure 7. Orthogonality relations for covariant oscillatorwavefunctions. The orthogonality relations remain invariant underLorentz boosts, but their inner products have interesting contractionproperties.

these oscillator solutions [17]. This allows us to use thisoscillator system for wavefunctions in the Lorentz-covariantworld.

However, presently, we are interested in space–timelocalizations of the wavefunction dictated by the Gaussianfactor or the ground-state wavefunction. In the light-cone coordinate system, the Lorentz-boosted wavefunctionbecomes

ψη(z, t) =(

1

π

)1/2

exp

{−1

2

(e−2ηu2 + e2ηv2

)}, (26)

as given in equation (18). This wavefunction can be written as

ψη(z, t) =(

1

π

)1/2

exp{−1

4

[e−2η(z + t)2 + e2η(z − t)2

]}.

(27)Let us go back to equation (9) for the coupled oscillators. Ifwe replace x1 and x2 by z and t respectively, we arrive at theabove expression for covariant harmonic oscillators.

We of course talk about two different physical systems.For the case of coupled oscillators, there are two one-dimensional oscillators. In the case of covariant harmonicoscillators, there is one oscillator with two variables. TheLorentz boost corresponds to coupling of two oscillators. Withthese points in mind, we can translate the physics of coupledoscillators into the physics of covariant harmonic oscillators.

7. Entangled space and time

Let us now compare the space–time wavefunction ofequation (27) with the wavefunction equation (9) for thecoupled oscillators. We can obtain the latter by replacingx1 and x2 in the coupled-oscillator wavefunction by z and trespectively.

ψη(z, t) = 1

cosh η

∑k

(tanh η)kφk(z)φk(t). (28)

This expansion is identical to that for the coupled oscillators ifz and t are replaced by x1 and x2 respectively.

Thus the space variable z and the time variable t areentangled in the same manner as given in [7]. However,there is a very important difference. The z variable is welldefined in the present form of quantum mechanics, but thetime-separation variable t is not. First of all, it is differentfrom the calendar time. It exists because the simultaneity inspecial relativity is not invariant in special relativity. This pointhas not yet been systematically examined.

All we can say at this point is that the Lorentzentanglement requires one variable we can measure, and theother variable we do not pretend to measure. In his bookon statistical mechanics [23], Feynman makes the followingstatement about the density matrix. When we solve a quantum-mechanical problem, what we really do is divide the universeinto two parts—the system in which we are interested and therest of the universe. We then usually act as if the system inwhich we are interested comprised the entire universe. Tomotivate the use of density matrices, let us see what happenswhen we include the part of the universe outside the system.

Does this time-separation variable exist when the hadronis at rest? Yes, according to Einstein. In the present form ofquantum mechanics, we pretend not to know anything aboutthis variable. Indeed, this variable belongs to Feynman’s restof the universe.

We can use the coupled harmonic oscillators to illustratewhat Feynman says in his book. Here we can use x1 and x2

for the variable we observe and the variable in the rest of theuniverse. By using the rest of the universe, Feynman doesnot rule out the possibility of other creatures measuring the x2

variable in their part of the universe.Using the wavefunction ψη(z, t) of equation (9), we can

construct the pure-state density matrix

ρ(z, t; z′, t ′) = ψη(z, t)ψη(z′, t ′), (29)

which satisfies the condition ρ2 = ρ:

ρ(z, t; z′, t ′) =∫ρ(z, t; z′′, t ′′)ρ(z′′, t ′′; z′, t ′) dz′′ dt ′′.

(30)If we are not able to make observations on t , we should takethe trace of the ρ matrix with respect to the t variable. Thenthe resulting density matrix is

ρ(z, z′) =∫ρ(z, t; x ′

1, t) dt. (31)

The above density matrix can also be calculated from theexpansion of the wavefunction given in equation (11). If weperform the integral of equation (31), the result is

ρ(z, z′) =(

1

cosh(η)

)2 ∑k

(tanh η)2kφk(z)φ∗k (z

′). (32)

The trace of this density matrix is 1. It is also straightforwardto compute the integral for Tr(ρ2). The calculation leads to

Tr(ρ2) =(

1

cosh(η)

)4 ∑k

(tanh η)4k . (33)

The sum of this series is 1/ cosh(2η) which is less than one.

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Coupled oscillators, entangled oscillators, and Lorentz-covariant harmonic oscillators

This is of course due to the fact that we are averaging overthe x2 variable which we do not measure. The standard way tomeasure this ignorance is to calculate the entropy defined as

S = − Tr(ρ ln(ρ)), (34)

where S is measured in units of Boltzmann’s constant. Ifwe use the density matrix given in equation (32), the entropybecomes

S = 2{cosh2 η ln(cosh η)− sinh2 η ln(sinh η)

}. (35)

This expression can be translated into a more familiar form ifwe use the notation

tanh η = exp

(− hω

kT

), (36)

where ω is the unit of energy spacing, and k and T areBoltzmann’s constant and absolute temperature respectively.The ratio hω/kT is a dimensionless variable. In terms of thisvariable, the entropy takes the form [24]

S =(

hω

kT

)1

exp(hω/kT )− 1−ln[1−exp(−hω/kT )]. (37)

This familiar expression is for the entropy of an oscillator statein thermal equilibrium. Thus, for this oscillator system, wecan relate our ignorance of the time-separation variable to thetemperature. It is interesting to note that the boost parameter orcoupling strength measured byη can be related to a temperaturevariable.

Let us summarize. At this time, the only theoretical toolavailable to this time-separation variable is through the space–time entanglement, which generates entropy coming from therest of the universe. If the time-separation variable is notmeasured the entropy is one of the variables to be taken intoaccount in the Lorentz-covariant system.

In spite of our ignorance about this time-separationvariable from the theoretical point of view, its existence hasbeen proved beyond any doubt in high-energy laboratories.We shall see in section 8 that it plays a role in producinga decoherence effect observed universally in high-energylaboratories.

8. Feynman’s decoherence

In a hydrogen atom or a hadron consisting of two quarks, thereis a spatial separation between two constituent elements. Inthe case of the hydrogen atom we call it the Bohr radius. Ifthe atom or hadron is at rest, the time-separation variable doesnot play any visible role in quantum mechanics. However,if the system is boosted to the Lorentz frame which moveswith a speed close to that of light, this time-separation variablebecomes as important as the space separation of the Bohrradius. Thus, the time separation plays a visible role in high-energy physics which studies fast-moving bound states. Letus study this problem in more detail.

It is a widely accepted view that hadrons are quantumbound states of quarks having a localized probabilitydistribution. As in all bound-state cases, this localizationcondition is responsible for the existence of discrete mass

spectra. The most convincing evidence for this bound-statepicture is the hadronic mass spectra [10, 17]. However, thispicture of bound states is applicable only to observers in theLorentz frame in which the hadron is at rest. How would thehadrons appear to observers in other Lorentz frames?

In 1969, Feynman observed that a fast-moving hadron canbe regarded as a collection of many ‘partons’ whose propertiesappear to be quite different from those of the quarks [25]. Forexample, the number of quarks inside a static proton is three,while the number of partons in a rapidly moving proton appearsto be infinite. The question then is how the proton looking likea bound state of quarks to one observer can appear different toan observer in a different Lorentz frame. Feynman made thefollowing systematic observations.

(a) The picture is valid only for hadrons moving with velocityclose to that of light.

(b) The interaction time between the quarks becomes dilated,and partons behave as free independent particles.

(c) The momentum distribution of partons becomes widespreadas the hadron moves fast.

(d) The number of partons seems to be infinite or much largerthan that of quarks.

Because the hadron is believed to be a bound state of twoor three quarks, each of the above phenomena appears as aparadox, particularly (b) and (c) together. How can a freeparticle have a widespread momentum distribution?

In order to resolve this paradox, let us constructthe momentum–energy wavefunction corresponding toequation (18). If the quarks have the four-momenta pa

and pb, we can construct two independent four-momentumvariables [10]

P = pa + pb, q = √2(pa − pb). (38)

The four-momentum P is the total four-momentum and is thusthe hadronic four-momentum. q measures the four-momentumseparation between the quarks. Their light-cone variables are

qu = (q0 − qz)/√

2, qv = (q0 + qz)/√

2. (39)

The resulting momentum–energy wavefunction is

φη(qz, q0) =(

1

π

)1/2

exp

{−1

2[e−2ηq2

u + e2ηq2v ]

}. (40)

Because we are using here the harmonic oscillator,the mathematical form of the above momentum–energywavefunction is identical to that of the space–timewavefunction of equation (18). The Lorentz squeeze propertiesof these wavefunctions are also the same. This aspectof the squeeze has been exhaustively discussed in theliterature [17, 26, 27], and they are illustrated again infigure 8 of the present paper. The hadronic structure functioncalculated from this formalism is in a reasonable agreementwith the experimental data [28].

When the hadron is at rest with η = 0, both wavefunctionsbehave like those for the static bound state of quarks. As ηincreases, the wavefunctions become continuously squeezeduntil they become concentrated along their respective positivelight-cone axes. Let us look at the z-axis projection of the

S465

Y S Kim and M E Noz

Ene

rgy

dist

ribut

ion

β=0.8β=0

z

t

z

BOOST

SPACE-TIME

DEFORMATION

Weaker spring constant

Quarks become (almost) free

Tim

e di

latio

n

TIM

E-E

NE

RG

Y U

NC

ER

TA

INT

Y

t

( (

β=0.8β=0

qz

qo

qz

BOOST

MOMENTUM-ENERGY

DEFORMATION

Parton momentum distribution

becomes wider

qo

( (

(

(

QUARKS PARTONS

Figure 8. Lorentz-squeezed space–time and momentum–energywavefunctions. As the hadron’s speed approaches that of light, bothwavefunctions become concentrated along their respective positivelight-cone axes. These light-cone concentrations lead to Feynman’sparton picture.

space–time wavefunction. Indeed, the width of the quarkdistribution increases as the hadronic speed approaches thatof the speed of light. The position of each quark appearswidespread to the observer in the laboratory frame, and thequarks appear like free particles.

The momentum–energy wavefunction is just like thespace–time wavefunction. The longitudinal momentumdistribution becomes widespread as the hadronic speedapproaches the velocity of light. This is in contradictionwith our expectation from nonrelativistic quantum mechanicsthat the width of the momentum distribution is inverselyproportional to that of the position wavefunction. Ourexpectation is that if the quarks are free, they must have theirsharply defined momenta, not a widespread distribution.

However, according to our Lorentz-squeezed space–time and momentum–energy wavefunctions, the space–timewidth and the momentum–energy width increase in the samedirection as the hadron is boosted. This is of course an effectof Lorentz covariance. This indeed is the resolution of thequark–parton puzzle [17, 26, 27].

Another puzzling problem in the parton picture is thatpartons appear as incoherent particles, while quarks arecoherent when the hadron is at rest. Does this mean that thecoherence is destroyed by the Lorentz boost? The answer isNO, and here is the resolution to this puzzle.

Figure 9. Quarks interact among themselves and with an externalsignal. The interaction time of the quarks among themselvesbecome dilated, as the major axis of this ellipse indicates. On theother hand, the external signal, since it is moving in the directionopposite to the direction of the hadron, travels along the negativelight-cone axis. To the external signal, if it moves with velocity oflight, the hadron appears very thin, and the quark’s interaction timewith the external signal becomes very small.

When the hadron is boosted, the hadronic matter becomessqueezed and becomes concentrated in the elliptic region alongthe positive light-cone axis. The length of the major axisbecomes expanded by eη, and the minor axis is contractedby eη.

This means that the interaction time of the quarksamong themselves become dilated. Because the wavefunctionbecomes widespread, the distance between one end of theharmonic oscillator well and the other end increases. Thiseffect, first noted by Feynman [25], is universally observed inhigh-energy hadronic experiments. The period of oscillationincreases like eη.

On the other hand, the external signal, since it is movingin the direction opposite to the direction of the hadron, travelsalong the negative light-cone axis, as illustrated in figure 9.

If the hadron contracts along the negative light-cone axis,the interaction time decreases by e−η. The ratio of theinteraction time to the oscillator period becomes e−2η. Theenergy of each proton coming out of the Fermilab acceleratoris 900 GeV. This leads the ratio to 10−6. This is indeed a smallnumber. The external signal is not able to sense the interactionof the quarks among themselves inside the hadron.

Indeed, Feynman’s parton picture is one concrete physicalexample where the decoherence effect is observed. As for theentropy, the time-separation variable belongs to the rest of theuniverse. Because we are not able to observe this variable, theentropy increases as the hadron is boosted to exhibit the partoneffect. The decoherence is thus accompanied by an entropyincrease.

Let us go back to the coupled-oscillator system. Thelight-cone variables in equation (18) correspond to thenormal coordinates in the coupled-oscillator system given inequation (5). According to Feynman’s parton picture, thedecoherence mechanism is determined by the ratio of widthsof the wavefunction along the two normal coordinates.

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Coupled oscillators, entangled oscillators, and Lorentz-covariant harmonic oscillators

This decoherence mechanism observed in Feynman’sparton picture is quite different from other decoherencesdiscussed in the literature. It is widely understood that theword decoherence is the loss of coherence within a system.On the other hand, Feynman’s decoherence discussed in thissection comes from the way an external signal interacts withthe internal constituents.

9. Concluding remarks

In this paper, we noted first that two-mode squeezed states canplay a major role in clarifying some of the entanglement ideas.Since the mathematical language of two-mode states is that oftwo coupled oscillators, the oscillator system can be a reservoirof physical ideas associated with entanglements. Then, otherphysical models derivable from the coupled oscillators cancarry the physics of entanglement.

We have shown in this paper that the covariant harmonicoscillator system with one space and one time variable sharesthe same mathematical framework as the coupled harmonicoscillator. Thus, the oscillator system gives a concrete exampleof space–time entanglement.

Thanks to its Lorentz covariance, the covariant oscillatorsystem can explain the quark model and parton model as twolimiting cases of the same covariant entity. It can explainthe peculiarities observed in Feynman’s parton picture. Themost controversial aspect in the parton model is that, whilethe quarks interact coherently with external signals, partonsbehave like free particles interacting without coherence withexternal signals. This phenomenon was observed first byFeynman. Thus, it is quite appropriate to call this Feynman’sdecoherence. In this paper, we have provided a resolutionto this parton puzzle. It requires a space–time picture ofentanglement.

Acknowledgments

We would like to thank G S Agarwal, H Hammer, andA Vourdas for helpful discussion on the precise definition ofthe word ‘entanglement’ applicable to coupled systems.

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