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Schrodinger revisited: the role of Dirac’s ‘standard’

ket in the algebraic approach.

M. R. Brown and B. J. Hiley

Theoretical Physics Research Unit

Birkbeck College, University of LondonMalet Street, London WC1E 7HX, England

Abstract

We follow Dirac and write the Schrodinger equation in an algebraicform which is representation-free. From this equation and its Hermi-tian conjugate we obtain respectively the continuity equation, whichinvolves the commutator of the Hamiltonian with the density opera-tor and an equation for the time development of the phase operatorthat involves the anti-commutator of the Hamiltonian with this den-sity operator. We show this latter equation plays two important roles:(i) it expresses the conservation of energy in a system where energyis well defined and (ii) it provides a simple way to evaluate the gaugechanges that occur in the Aharonov-Bohm, the Aharonov-Casher, andBerry phase effects. Both these operator (i.e. purely algebraic) equa-tions also allow us to re-examine the Bohm interpretation, showingthat it is in fact possible to construct Bohm interpretations in repre-sentations other than the x-representation. We discuss the meaning ofthe Bohm interpretation in the light of these new results in terms ofnon-commutative structures and this enables us to clarify its relationto standard quantum mechanics.

1 Introduction

In a previous paper, Monk and Hiley [1] have suggested that instead ofusing the traditional the Hilbert space description of quantum phenomena,one should give primary consideration to the algebraic structure, not onlybecause it has a number of mathematical advantages that have already beenpointed out by Dirac [2] [3] [4], but because it offers the possibility of aradically different interpretation of the quantum formalism. By algebraic

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structures we mean exploiting the rich possibilities contained in, for example,C* and W* (von Neumann) algebras, which play an important role in fieldtheory [5] [6] as well as in equilibrium [7] [8] and non-equilibrium statisticalmechanics [9] [10]. In spite of the potential richness of these methods, theyhave not been used in the general debate on the foundations of quantumtheory mainly because of the abstract nature of the mathematics.

In their paper Monk and Hiley [1] outlined how this mathematics canbe simplified so that the approach becomes much more transparent. Fur-thermore one can develop an interpretation for this formalism provided oneis willing to give up the basic ideas of particles- and/or fields-in-interactionand instead, think in terms of process. Indeed recent work [11] [12] [13][14] has shown how the algebraic approach does have a potential for takingthe discussions of the meaning of the quantum formalisms into new do-mains. It is this background that provides the motivation for the presentpaper. However, we will not assume any detailed prior knowledge of thisabstract algebraic structure. Our purpose here is to show how we can re-write the equations of elementary quantum mechanics in a purely algebraicway, which, as we show, yields some interesting new insights.

Traditionally, the algebraic approach has implied the use of the Heisen-berg picture. Here the operators (or elements of the algebra, in our case)become time dependent and carry the dynamics of the quantum system.One clear advantage of this approach is that all the elements of the algebraare representation-independent and so we are not tied to any one partic-ular representation. A further advantage is that the equations of motionhave a close structural similarity to the classical equations of motion, viz,commutator brackets directly replace Poisson brackets [2] [3].

In contrast to this, the Schrodinger picture has the time development en-tirely tied into the wave function, and as such is a representation dependentobject which appears to exists only in Hilbert space and does not appearin the algebra. Thus the Schrodinger picture does not seem to have thegenerality of the representation-independent Heisenberg picture.

In this paper we will show that this representation dependence of theSchrodinger equation is only apparent and in section 2 we will show that itcan be written in a representation independent form. This means writingthe wave function as a “wave operator” (left ideal in the algebra) so thatthe Schrodinger equation becomes purely algebraic and independent of anyrepresentation in a Hilbert space.

The way to do this has been known for a long time [6] [15]. IndeedMonk and Hiley [1] have already shown in simple terms that the key stepinvolves expressing the wave operator as a left ideal. This means that the

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wave function must be replaced by the density operator, ρ, even in the caseof pure states. Here the density operator plays the role of an idempotentand it is this idempotency that is central role in the whole approach.

Writing the wave function in operator form and using both the Schrodingerequation and its Hermitian conjugate, we obtain the two equations (6) and(7) below. The first, expressed in terms of the commutator of the Hamilto-nian and the density operator, is the Liouville equation and, as is well known,describes the conservation of probability. The second describes the time de-pendence of the phase and is expressed in terms of the anti-commutator ofthe Hamiltonian and the density operator. This second equation, which doesnot appear in the literature as far as we are aware, becomes an equation forthe conservation of energy in systems when energy is well defined.

In section 3, the time dependent phase operator equation (7) is shown tobe gauge invariant and reproduces some well known results of gauge theoryin a very direct and simple way. For example, one can immediately derive theAharonov-Bohm phase for a particle travelling in a vector potential while atrivial extension incorporates the Aharonov-Casher phase. This latter phasearises when a neutral particle with a magnetic moment passes a line-charge.In addition to these examples, the Berry phase and its associated energyfollow almost trivially from the same equation.

In sections 4 and 5 we use these equations to explore in more detailthe Bohm interpretation [BI] [20] [18]. Because the equations (6) and (7)are representation independent, we can construct a BI based on trajectoriesin any representation. To do this, we introduce a generalisation of thecurrent density operator and demonstrate its use in both ordinary spaceand momentum space. In section 5 and 6 we show in detail how one canconstruct a consistent BI in the p-representation. This is contrary to theassertion that this is not possible even in “the simplest case to construct anacceptable causal interpretation” [24] and we discuss the significance of thisstatement in the light of our examples.

Our examples not only remove one of serious criticisms of this inter-pretation, namely that it does not use the full symplectic symmetry of thequantum formalism, but provides us with new insights into the meaning ofBI and its relation to standard quantum mechanics. In all of this work noappeal is made to any classical formalism whatsoever, showing that the BIis quantum through-and-through.

Perhaps the most important conclusion of this work is to show the BIarises directly from the non-commutative structure of the quantum mechan-ical phase space. Non-commutative geometries are not built on any form ofwell defined continuous manifolds. We are forced to construct “shadow man-

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ifolds” [21] [22]. As we show in section 5, these shadow manifolds have thestructure of a phase space. One is constructed using the x-representationand the other uses the p-representation. These spaces are different but con-verge to the same phase space in the classical limit.

In the final section we will discuss the consequences of this constructionfor the BI. In fact we will show that our approach more clearly illustrates theideas that Bohm and Hiley presented in the final chapter of their book [18].There it was argued that a new way of exploring the meaning of the quantumformalism required a new order, the implicate order, this order having itsorigins in the mathematics of non-commutative geometry [23]. The shadowphase spaces are examples of explicate orders. Finally we briefly discuss howthe BI fits into this general scheme.

2 The algebraic approach to the Schrodinger equa-

tion

We begin by writing the Schrodinger equation in a general representation

i∂ψ(ai, t)

∂t= Hψ(ai, t) (h = 1) (1)

where ai are the eigenvalues of the algebraic element or operator1A. Nowthe idea is to write the wave function as a function2 of the operator A.

Rather than approach this through an abstract formalism expressing thewave operator as a left ideal, we will follow Dirac’s symbolic approach [25]and write the wave function in the form

ψ(ai, t) = 〈ai|ψ(A, t)〉

Notice that in contrast to the usual approach we have written ψ as a functionof the operator A and not its eigenvalue.

Now for any arbitrary function f(A), we can write

f(A)|ψ〉 = f(A)|ψ(A)〉 = |f(A)ψ(A)〉

Dirac argued that since the bar | is movable, we could actually dispense withit altogether and write the wave function in the form ψ(A) 〉. The symbol 〉

1Throughout this paper we will use the convention of representing operators by capitalsand eigenvalues by lower case letters.

2To keep things simple, we consider a function of a single operator A only. A generali-sation to a complete commuting set is possible, but we do not need such a generalisationfor this paper.

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is left in place to remind us that we cannot multiply this wave operator fromthe right. Dirac calls this symbol the standard ket. It is chosen to reflectthe specific representation we are using, so that in this case we must writeit in the form 〉A. We then write the wave operator in the form A(t) 〉A.

We now express this operator in the form

ψ(A, t) 〉A = eiSQ(A,t) 〉A

where SQ = S − i lnR and observe, in passing, that

∂ψ(A, t)

∂t〉A= i

∂SQ(A, t)

∂tψ(A, t)〉

A.

However, we want to work in an arbitrary representation, in this case thea-representation. For convenience of presentation, we will suppress the la-bel A on the standard ket as no ambiguity will arise. We can now writeequation (1) as

− ∂SQ∂t

ψ〉 = Hψ〉. (2)

This equation is the quantum operator equivalent of the Hamilton-Jacobiequation of classical mechanics:

∂Scl∂t

+H = 0

where Scl is the classical action. It is natural, therefore, to call the operatorSQ the quantum action.

We now introduce the standard bra and write 〈ψ† = 〈e−iS†Q . The Her-

mitian conjugate of equation (1) is then

− 〈ψ†∂S†

Q

∂t= 〈ψ†H†. (3)

Defining the density operator3 ρ = ψ〉〈ψ†, post- and pre-multiplying equa-tions (2) and (3) by 〈ψ† and ψ〉 respectively and then adding and subtractingthe resulting equations, we obtain

(−∂SQ∂t

ρ+ ρ∂S†

Q

∂t) = [ρ,H]− (4)

and

(−∂SQ∂t

ρ− ρ∂S†

Q

∂t) = [ρ,H]+ (5)

3In this derivation we use a pure state density operator which is idempotent.

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in both of which (and hereafter) we assume that H is Hermitian.Equation (4) may be written in Hermitian form as

∂ρ

∂t+

1

i[ρ,H]− = 0. (6)

In fact, this equation can be obtained from the Schrodinger equation andits complex conjugate without using the polar form of the wave operator.Equation (6) will immediately be recognised the Liouville equation. The di-agonal elements of this equation determine the time dependent behaviour ofthe probability density of the wave function and therefore it simply expressesthe conservation of probability.

The general form of equation (5) cannot be further simplified. However,its diagonal component reduces to the form

ρR∂S

∂t+

1

2[ρ,H]+ = 0 (7)

where ρR = R2. This equation describes the time development of the phaseof the wave operator and so we will call it (and the general form (5)) thequantum phase equation. In a state in which the energy is well defined

∂S

∂t= −E

Thus, equation (7) becomes an expression for the conservation of energy inthis case.

The Liouville equation (6) is well known and plays a prominent role inquantum statistical mechanics. The quantum phase equation (7) does notusually appear in the literature, although something similar has been usedby George et al [10] in their discussions of irreversible quantum processes.In their case the anti-commutator is simply introduced by defining it to bethe energy super-operator. What we show here is that this operator comesdirectly from the Schrodinger equation and although this extension to superoperator status is possible, this generalisation not necessary for the purposesof this paper.

In summary then, equations (6) and (7) are simply the algebraic equiv-alents of the Schrodinger equation when it is written in a way that does notdepend on a specific representation. It is now easy to confirm that these twoequations, when expressed in a particular representation, are simply the realand imaginary parts of the Schrodinger equation under a polar decomposi-tion of the wave function written in that particular representation.

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3 The quantum phase equation

3.1 Gauge invariance

We will first examine equation (7) in some detail. Let us begin by lookingat the form of this equation in the x-representation when we choose theHamiltonian H = p2/2m+ V (x). Here equation (7) becomes

∂S

∂t+

1

2m

(

∂S

∂x

)2

+ V (x)− 1

2mR

(

∂2R

∂x2

)

= 0 (8)

This equation is the real part of the Schrodinger equation in the x-representation.Before examining this equation in detail we must ensure that it is gauge

invariant. To show that this is the case, let us first introduce the gaugetransformation V ′(x) = V (x)+V0. This must be accompanied by the phasetransformation ψ′(x, t) = φ(x)exp[−i(E + V0)t]. Since we are considering awell defined energy state, the transformed equation (5) will read

((−∂SQ∂t

+ V0)ρ− ρ(∂S†

Q

∂t+ V0)) = [ρ,H]+ + [ρ, V0]+

because ρ′ = ρ, H ′ = H+V0 and S′Q = SQ+V0. Then, since [ρ, V0]+ = 2ρV0,

we immediately recover equation (5), so establishing its gauge invariance andthat of equation (7).

Gauge invariance in this case involves the phase change S′ = S+S0 thensince

V ′(x, t) = V (x) + V0(t)

our equation gives∂S0∂t

= V0(t)

so that

S0 =

∫ t

t0V0(t

′)dt′

It will immediately be recognised that this is the expression for the scalarpart of the Aharonov-Bohm effect [26].

We can also obtain the magnetic phase shift from equation (7) by startingfrom the Hamiltonian

H =1

2m(P− eA)2 (c = 1)

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On expanding this Hamiltonian, we find

H = 12mP

2 − e2m (P.A+A.P) + e2

2mA2 (9)

= Hfree particle +Hinteraction +Hfree field

The corresponding phase will then consist of three terms

S = Sfree particle + Sinteraction + Sfree field

so that the diagonal form

ρR

(

∂Sint∂t

)

+1

2[Hint, ρ]+ = 0

gives

ρR∂Sint∂t

= eA.jx

in the x-representation. If we then write jx = ρRv, we find

Sint = e

∫ t

t0A.vdt = e

∫ x

x0

A.dx (10)

We immediately recognise that this is the expression for the Aharonov-Bohmphase for the vector potential4.

The phase for the Aharonov-Casher effect [28], which involves a neutralparticle with a magnetic moment passing a line of electric charges, alsofollows trivially once the Hamiltonian

H =1

2m(P−E× µ)2 − µE2/m

is assumed. The additional phase change also follows trivially from the sameprocedure used for the vector potential.

It should also be noted that the Berry phase [29] [30] emerges directlyfrom equation (7). In this case the behaviour of the quantum system dependson some additional cyclic parameter B(t). The phase now becomes a functionof this parameter. Thus equation (7) becomes

ρR

(

∂S

∂t+ B∂S

∂B

)

+1

2[ρ,H]+ = 0 (11)

4 This effect was derived in the x-representation from the ‘guidance’ condition byPhilippidis et al [32]. A more recent discussion using this approach rather than themethod we use can be found in Sjoqvist and Carlsen [27]

8

giving an extra phase factor B ∂S∂B . Thus the contribution to the phase from

this extra degree of freedom is

SBerry phase =

∫ t

t0B∂S∂Bdt

To evaluate this term, we need to consider specific problems which meansgoing to a specific Hamiltonian in a specific representation. This represen-tation is generally the x-representation. Berry [29] considered the case ofthe precession of nuclear spin in a magnetic field in his original paper, andshowed that

∂S

∂B = ℑ〈B(t), t|∇B|B(t), t〉

so that

SBerry phase = ℑ∫ B

B0

〈B(t), t|∇B|B(t), t〉dB

which is exactly the result obtained by Berry [29].

3.2 The x- and p-representations

Having seen how the additional phase changes arise for these simple gaugefields, we now return to examine the details of equation (8). To do thislet us consider the case of the harmonic oscillator with Hamiltonian H =p2/2m+Kx2/2. In this case equation (8) reads

∂Sx∂t

+1

2m

(

∂Sx∂x

)2

+Kx2

2− 1

2mRx

(

∂2Rx

∂x2

)

= 0 (12)

where we have inserted the suffix x to emphasise that this is equation (7)expressed in the x-representation.

Now let us write down the corresponding equation in the p-representation.This takes the form

∂Sp∂t

+p2

2m+K

2

(

∂Sp∂p

)2

− K

2Rp

(

∂2Rp

∂p2

)

= 0 (13)

It should be noted that although the functional forms of these two equa-tions are clearly different, they nevertheless have the same energy content.This can be very easily checked for the ground state of the harmonic oscilla-tor. One can quickly show that both equations give the ground state energyto be ω/2, the zero-point energy.

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In spite of the differences in functional form, there are structural sim-ilarities between these two equations. These arise essentially because wehave chosen a symmetric Hamiltonian. For example, instead of the p thatappears in what looks like a kinetic energy term in equation (13), we have(∂Sx/∂x) in equation (12), and instead of x in the potential energy term inequation (13), we have (∂Sp/∂p). The last term in each equation has thesame general form except with the roles of x and p interchanged.

Since equation (12) is the real part of the Schrodinger equation, we canidentify5

pr = ℜ [ψ∗(x)Pψ(x)] /|ψ(x)|2 =

(

∂Sx∂x

)

(14)

Substituting this into equation (12) we find

∂Sx∂t

+p2r2m

+K

2x2 − 1

2mRx

(

∂2Rx

∂x2

)

= 0 (15)

In the Bohm interpretation, pr was identified with the “beable” momentum.With this identification equation (14) makes it now quite clear why thebeable momentum is a function of x, in contrast to the classical momentumwhich is always an independent variable.

If pr is a momentum, then clearly equation (15) looks like an equationfor the total energy of the quantum system. If we make the assumptionthat this is an expression for the conservation of energy then, in quantumtheory, we must have an additional quality of energy represented by the lastterm on the RHS. This term is, of course, the quantum potential energy. Ashas been shown elsewhere this new quality of energy offers an explanationof quantum processes like interference, barrier penetration and quantumnon-separability, all of which are quantum phenomena [18].

Notice that equation (15) allows the possibility of approaching the clas-sical limit smoothly. In this limit Sx → Scl, pr → pcl = (∂Scl/∂x) and thequantum potential energy becomes negligible so that equation (15) becomesthe classical Hamilton-Jacobi equation.

Before continuing, we wish to stress a point that has not been oftenfully appreciated, namely, that equation (12) is a quantum equation andconverting it to equation (15) requires no appeal whatsoever to classicalphysics. It is true that in the traditional approach to this equation, theBI has made use of the relation p = (∂Sx/∂x) by appealing to classicalcanonical theory. This is a backward step and is totally unnecessary.

5Holland [31] has called expressions of this type ‘local expectation values’.

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It is because equations (7) and (12) are part of the quantum formalismthat we were able to derive quantum effects such as the Aharonov-Bohm,Aharonov-Casher and Berry phases from equation (7). In passing it shouldalso be noted that both the x- and p-representations of this equation (i.e.,(12) and (13)) contain a term which we have called the quantum poten-tial. This potential is modified by the presence of the gauge effects as wasfirst shown by Philippidis, Bohm and Kaye [32]. The quantum potentialis central to ensuring energy is conserved and, furthermore, it encapsulatesquantum non-separability or quantum non-locality[33]. The quantum po-tential plays a key role in our approach and must be distinguished fromBohmian mechanics advocated by Durr et al.[16].

If we now turn to the p-representation, i.e., equation (13), we can writeit in the form

∂Sp∂t

+p2

2m+K

2x2r −

K

2Rp

(

∂2Rp

∂p2

)

= 0 (16)

by introducing

xr = ℜ [ψ∗(p)Xψ(p)] /|ψ(p)|2 = −(

∂Sp∂p

)

(17)

Here xr is the position “beable”, which now supplements the momentum p.Again in the classical limit, we have Sp → Scl, xr → xcl = −(∂Scl/∂p) andthe last term on the RHS of (16) becomes negligible. It should be noted thatin this limit equations (15) and (16) reduce to the same equation giving riseto a unique phase space, which is identical to the classical phase space.

All the above equations are part of standard quantum mechanics. Al-though we have drawn attention to the significance of equations (15) and(16) to the BI, we have yet to discuss the interpretation in any detail. To dothis we first need to find a way to calculate “trajectories”. In the traditionalapproach to BI this is done by regarding p = (∂Sx/∂x) as a “guidance”condition and then using x = p/m from which one can calculate a set oftrajectories. These trajectories are then identical to the streamlines of theprobability current in the co-ordinate representation. However this is notthe general way to do it as can be seen by considering the p-representation.The analogous expression in this representation is x = −(∂Sp/∂p) and thisclearly cannot be regarded as a “guidance” condition. Something is notquite right here. In order to find out what is involved it is necessary toexplore the Liouville equation (6) in more detail.

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4 Probability currents

In this section we will focus our attention on the Liouville equation (6). Theusual way of showing that this equation gives rise to probability currents isto develop the equation in the x-representation by forming P (x) = 〈x|ρ|x〉so that the equation can be written in the form

∂P

∂t+∇.j = 0 (18)

where j is the usual probability current

j =1

2mi[ψ∗(∇ψ)− (∇ψ∗)ψ] (19)

However what we need to do is find an expression for the current that isnot representation specific. To do this we first consider the classical Liouvilleequation

∂ρ

∂t+ {ρ,H} = 0

where {} is the Poisson bracket. It is easy to verify that this equation canbe written in the form

∂ρ

∂t+ {Jx, p} − {Jp, x} = 0 (20)

withJx = ρ∇pH and Jp = −ρ∇xH (21)

If we now follow Dirac’s suggestion and replace Poisson brackets withcommutators, we find

i∂ρ

∂t+ [JX,P]− [JP,X] = 0 (22)

HereJX = ∇P (ρH) and JP = −∇X(ρH) (23)

where the derivatives are on operators. These are defined through [35]

f = ρxnpm then∂f

∂x= xn−1pmρ+ xn−2pmρx+ ...+ pmρxn−1.

and∂f

∂p= pm−1ρxn + pm−2ρxnp+ ...+ ρxnpm−1.

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In the simple case of a free particle we have

JX = ρP + Pρ and JP = 0.

To see how this connects to the conventional results let us evaluate equa-tion (22) in the x-representation. Here we find

i∂〈x|ρ|x〉

∂t+ 〈x|[JX,P]|x〉 − 〈x|[JP,X]|x〉 = 0 (24)

If H = p2

2m + V (x) then the first commutator gives 〈x|[JX,P]|x〉 = ∇xjxand the second commutator vanishes. Thus equation (24) becomes

∂P (x)

∂t+∇xjx = 0 (25)

which is iust equation (18) and

jx = 〈x|∇p(ρp2

2m)|x〉 (26)

which is gives an expression for the current that is identical to the usualexpression given by equation (19). Furthermore it is unique since it is inde-pendent of the form of the potential used in the Hamiltonian.

In the p-representation we find

∂P (p)

∂t+∇p.jp = 0 (27)

wherejp = −〈p|∇x(ρV (X)|p〉 (28)

Thus we can now calculate probability currents in the p-representation.Unfortunately equation (28) does not give us a model independent expres-sion for the probability current because the specific form of the currentdepends on the form of V (x). On reflection this is not surprising becausethe rate of change of momentum must depend upon the externally appliedpotential. We will examine the consequences of these results for the Bohminterpretation in the next section.

5 Re-examination of the Bohm approach

Let us now reappraise the Bohm approach in the light of the new resultspresented above.

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It has been assumed that it is not possible to construct a BI using anyrepresentation other than the x-representation. This belief arises from anearly correspondence between Epstein [34] and Bohm [24]. Epstein suggestedthat it should be possible to develop an alternative causal interpretation bystarting in the momentum representation. Bohm replied agreeing that a newcausal interpretation could possibly arise from such a procedure providedthe canonical transformation on the particle variables were simultaneouslyaccompanied by a corresponding linear transformation on the wave function.But, he concluded that this did not seem to lead, even in the simplest ofcases, to an acceptable causal interpretation. He does not explain why hecame to this conclusion but this position has remained the accepted wisdom.

The general results with the harmonic oscillator presented above showthat, at least as far as the mathematics is concerned, it does seem possible todevelop a causal interpretation in the p-representation based upon equation(16), (17) and (28). We will illustrate how this can be done using specificexamples in section 6, but here we will simply discuss the general principlesinvolved.

We have already pointed out in section 4 that the so called “guidance”condition is assumed to play a pivotal role in what is known as “Bohmianmechanics” [16] does not generalise to the p-representation. However whatdoes generalise is a method based on probability currents. Thus in anyq-representation we use

dq/dt = jq/P (q)

which can be integrated immediately to find a set of streamlines in a generalq-space.

In the x-representation we have

dx/dt = jx/P (x)

which, when integrated, give the streamlines that are assumed to correspondto the particle trajectories of the BI. It could be argued that this assumptionis not very different from the implicit assumption used in pragmatic quantummechanics where the probability current is assumed to describe the flux ofparticles emerging from, say, a scattering process. Here the flux at a detectoris interpreted as the rate of arrival of the scattered particles.

The additional assumption made in BI is that particles exist with si-multaneously well defined positions and momenta and each particle followsone of the one-parameter curves. Such an assumption is clearly excluded instandard quantum mechanics, but this leaves us with the difficulty of un-derstanding how to incorporate the Born probability postulate and its rolein the continuity equation (18) except in some abstract sense.

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If we do follow BI in the x-representation, then the position of the particleis clearly defined and the momentum, pr, associated with the particle mustbe provided through the relation

pr = ℜ [ψ∗(x)Pψ(x)] /|ψ(x)|2 =

(

∂Sx∂x

)

= mjx/P (x) = mdx/dt

Here the “beable” momentum pr is wholly quantum in origin showing thatBI has its origins entirely within quantum mechanics.

Now in the p-representation we use

dp/dt = jp/P (p)

to give a set of one-parameter curves in momentum space. In this approachthe momentum of the particle has a clear meaning, while the position beablexr is given by equation (17), namely

xr = ℜ [ψ∗(p)Xψ(p)] /|ψ(p)|2 = −(

∂Sp∂p

)

This means that the derivative in the current jp = −〈p|∇x(ρV (X)|p〉 givenby equation (28) must be evaluated at x = xr. Thus we again have aspecification of the particle with a given momentum at a given “beable”position xr.

Thus the BI based on equations (15) and (16) leads to two distinct phasespaces, one constructed on each representation. Each phase space contains aset of trajectories, one derived from jx and the other from jp. Although thesephase spaces are actually different, they carry structures that are consistentwith the content of the Schrodinger equation. This is in contrast to theclassical limit where there is a unique phase space. However we have alreadynoted that equation (15) and (16) reduce to a single equation in the classicallimit so the existence of (at least) two phase spaces is a consequence of thequantum formalism.

Now the existence of at least two phase spaces may come as a surpriseto those who see BI as a return to classical or quasi-classical notions. Whatwe have shown here is that BI enables us to construct what we may call“shadow phase spaces”, a construct that is a direct consequence of the non-commutative nature of the quantum algebra. Giving ontological meaningto the non-commutative algebra implies a very radical departure from theway we think about quantum processes. This was the central theme ofBohm’s work on the implicate order [23]. The work presented in this paperfits directly into this conceptual structure, a point that will be discussed atlength elsewhere.

15

Our present purpose is to clarify the structure of the mathematics lyingbehind the BI. To this end note that choosing a representation is equivalentto choosing an operator which is to be diagonal. Thus in the phase spacedescribed by equation (15) the position eigenvalues are used for the x co-ordinates and we then construct the momentum co-ordinate through thecondition pr = (∂Sx/∂x) to provide the “beable” momentum.

On the other hand, equation (16) describes a phase space constructedusing the momentum eigenvalues together with the “beable” position xrdefined by xr = −(∂Sp/∂p). In this way we see exactly how it is possible toconstruct two different phase spaces, one for each representation.

The fact that we can find a BI in the p-representation removes the crit-icism that BI does not use the full symplectic symmetry of the quantumformalism. But in removing this asymmetry might, at first sight, destroythe claim that the Bohm interpretation provides a unique ontological inter-pretation. This would only be true if we were insisting that the ontologydemands a unique phase space. However as we have already remarked thequantum algebra is non-commutative and a unique phase space is not pos-sible. It was for this reason that Bohm and one of us (BJH) began toexplore the possibility of giving ontological significance to the algebra itself.This involves thinking in terms of process rather than particles- or fields-in-interaction and this leads, in turn, to introducing the implicate ordermentioned above. This is a very different order from the one assumed bymost physicists, which is essentially what we call the Cartesian order.

Once again we contrast our approach with Bohmian mechanics intro-duced by Durr et al [16]. Their approach requires the x-representation to betaken as basic and the guidance relation to be taken as the defining equa-tion of the approach. In view of the results presented here, we see we couldhave started from the p-representation. But here the relation p = (∂S/∂x)cannot play the role of a guidance condition. Hence making the guidancecondition as the defining equation in the x-representation is arbitrary andcontrary to what Bohm himself had in mind [37] [38] [39].

In regard to the lack of x-p symmetry in the traditional approach toBI, Bohm and Hiley [18] found it necessary to discuss why x was the onlyintrinsic property of the particle, all others depended upon the context. Thiswas certainly felt by one of us (BJH) to be a somewhat arbitrary impositionthat did not seem to be a natural consequence of the symplectic invariance ofthe formalism itself. Had we started with the p-representation we would havefound p to be the intrinsic property, while x depended upon some context.Thus the restoration of symmetry explains why particular variables becomeintrinsic and others not.

16

In the examples we give in this paper, we only consider the two operatorsX and P . If we regard the change from the x-representation to the p-representation as a rotation of π/2 in phase space, we could think aboutexploring rotations through other angles. Such transformation exist and areknown as fractional Fourier transformations which correspond to rotationsthrough any angle α in phase space [40] [41]. These allow us to expressequations (6) and (7) any arbitrary representation. This generalisation hasbeen investigated and will be reported elsewhere [42].

All of this shows that the x variable is not special as far as the mathe-matics goes. The real question is why it is necessary to construct differentphase spaces in the first place, but before we go into this question we wantto present some examples where we can compare in more detail the resultsobtained from both x- and p-representations.

6 Specific examples: comparisons of x- and p-representations

6.1 The free particle described by a Gaussian wave packet

We will start with the simplest case of a particle moving with constantmomentum p in, say, the z-direction and described by a Gaussian wavepacket

φ(p, t) =1

[2π(∆p)2]1

4

exp[

−p2(∆x)2]

exp

[

− ip2

2mt

]

In this representation the current jp = 0 = (dp/dt) so that the streamlinesare of constant momentum.

Equation (7) gives∂S

∂t+

p2

2m= 0

which shows that the quantum potential is zero as is to be expected fromthe form of the wave function φ(p, t)

In the x-representation, the wave packet spreads in the x-direction, hav-ing the wave function

ψ(x, t) =1

(2π(∆x)2)1

4

exp

[

− x2

4D(t)

]

exp

[

ix2t

8m(∆x)2D(t)

]

exp

[

− i

2arctan

(

t

2m(∆x)2

)]

where D(t) = (∆x)2 +(

t2

4m2(∆x)2

)

. The corresponding current is

jx =xt

4m(∆x)2D(t)

17

and equation (7) yields

∂S

∂t+

1

2m

(

∂S

∂x

)2

+1

4mD(t)− x

16m[D(t)]2= 0

where the last two terms constitute the quantum potential.This result can be easily understood since we are starting with the parti-

cle confined in a region ∆x and as time progresses, the wave packet spreadsout as expected. The current jx = P (x)(dx/dt) and the trajectories calcu-lated from this current fan out in a way that exactly accounts for the spreadof the wave packet. Any measurement of the momentum of the particle willalways give the value p since all the quantum potential energy is convertedinto the kinetic energy of the particle during the act of measurement. This isjust an example of the participatory nature of measurement in the standardinterpretation of quantum mechanics [36].

6.2 The quadratic potential

Here we will simply collect the results derived earlier in the paper for easeof comparison.

In the x-representation, where we write ψ(x, t) = Rxexp[iSx], the energyequation becomes

∂Sx∂t

+1

2m

(

∂Sx∂x

)2

+K

2x2 − 1

2mRx

(

∂2Rx

∂x2

)

= 0 (29)

While in the p-representation, where we now write ψ(p, t) = Rpexp[iSp], theconservation of energy equation is

∂Sp∂t

+p2

2m+K

2

(

∂Sp∂p

)2

− K

2Rp

(

∂2Rp

∂p2

)

= 0 (30)

Now we turn to the probability currents and find

jx =1

2mi

[

ψ∗(x)

(

∂ψ(x)

∂x

)

−(

∂ψ∗(x)

∂x

)

ψ(x)

]

(31)

jp =K

2i

[

ψ∗(p)

(

∂ψ(p)

∂p

)

−(

∂ψ∗(p)

∂p

)

ψ(p)

]

(32)

This is to be expected from the symmetry of the Hamiltonian.

18

The fact that these currents are different should not be too surprising asthey arise in different spaces. Indeed we can bring this out more clearly bywriting

ψ(x) = Rxexp[iSx] and ψ(p) = Rp = Rpexp[iSp]

jx =1

mR2

x

(

∂Sx∂x

)

and jp = KR2p

(

∂Sp∂p

)

dx

dt=

1

m

(

∂Sx∂x

)

= pr/m anddp

dt= K

(

∂Sp∂p

)

= −(

∂V

∂x

)

x=xr

Thus we see that the currents provide the mathematical means of con-structing trajectories in the x-space and p-space respectively. It is a fea-ture of the linear potential and of the quantum harmonic oscillator thatdpdt = −

(

∂V∂x

)

x=xr

, though this is not generally true.

6.3 The linear potential

Here the potential is V (x) = ax. In this case the currents are

Jx =1

2m(ρP + Pρ), and Jp = −aρ.

Thus in the x-representation the probability current is

jx = 〈x|Jx|x〉 =P

m(∂Sx/∂x) = P (dx/dt)

Here Sx is the phase of the wave function.In the p-representation we find

jp = 〈p|Jp|p〉 = −Pa = P (dp/dt)

This is identical to the result obtained from the classical mechanicsthrough the equation

dp

dt= −∂V

∂x= −a.

We can check since this will mean that the quantum potential in thep-representation is zero. Indeed equation (7) gives

∂Sp∂t

+p2

2m− a

∂Sp∂p

= 0 (33)

19

If we now use xr = −(∂Sp/∂p), we find the corresponding energy equa-tion is

∂Sp∂t

+p2

2m+ axr = 0 (34)

This has the same form as the classical Hamilton-Jacobi equation with xr =x. In this case there is no corresponding quantum potential, which is notsurprising as the trajectories in the quantum case are identical to those inthe classical case.

To see if everything is consistent, we need now to compare this withthe results calculated from the x-representation. Here the correspondingSchrodinger equation is

d2ψ

dx2+A3xψ = 0 (35)

where A = −(2mah2 )

1

3 . This equation has an Airy function as a solution:

ψ(x) = CAi(Ax) (36)

which, being real, implies that the probability current is zero:

jx =P

m(∂Sx/∂x) = P (dx/dt) = 0 (37)

Using this result in equation (7) shows that in the x-representation thequantum potential is the negative of the classical potential and is not zeroas in the p-representation.

In passing, we observe that the above solution may be split into incidentand reflected components using the relation:

Ai(Ax) + exp(−2

3iπ)Ai(Axexp(−2

3iπ)) + exp(

2

3iπ)Ai(Axexp(

2

3iπ)) = 0

(38)Taking the incident and reflected wave function separately, one obtains non-zero probability density currents and a non-zero quantum potential. Theresulting trajectories are classical at infinity but are non-classical near theorigin where reflection takes place with an instantaneous change of sign invelocity. This is in sharp distinction to the classical trajectory which turnssmoothly at the origin. It is important to note that the trajectories ofthe incident and reflected waves respectively do not embody the effects ofinterference. It is this interference which produces a stationary trajectoryfor the combined solution.

Finally, we may examine the asymptotic solution when x→ ∞

ψ(x) = Bx−1/4exp[−(2/3)iA3/2x3/2] (39)

20

where B = (2√π)A−1/4exp[iπ/2]. We find that the probability current is

jx =P

m(dS/dx) = −(P/2m)A3/2x1/2 = P (dx/dt)

so that we have x = −at2/2m which is the equation of the classical trajec-tory. Again we can check the value of the quantum potential calculated inthe x-representation using

Q(x) = − 1

2mRx

∂2Rx

∂x2

Because of the nature of the asymptotic solution, this is zero to the orderthat the asymptotic solution (39) satisfies the Schrodinger equation.

We conclude from this example that, whilst their respective contents areconsistent with the Schrodinger equation, the Bohm trajectories obtained indifferent representations may differ.

6.4 The cubic potential

The quantum phase equation (7) in the x-representation using pr = (∂Sx/∂x)gives

∂Sx∂x

+p2r2m

+Ax3 − 1

2mRx

∂2Rx

∂x2= 0

while in the p-representation we

∂Sp∂p

+p2

2m+Axr

3+3A

Rp

∂2Rp

∂p2

(

∂Sp∂p

)

+3A

Rp

(

∂Rp

∂p

)

(

∂2Sp∂p2

)

+A

(

∂3Sp∂p3

)

= 0

where we have used xr = −(∂Sp/∂p). This clearly gives a far more com-plicated quantum potential. Nevertheless the content is still consistentwith Schrodinger’s equation. Both equations reduce to the same classi-cal Hamilton-Jacobi equation when the quantum potential terms reduce tozero.

The respective currents are

jx =1

2mi

[

ψ∗(x)

(

∂ψ(x)

∂x

)

−(

∂ψ∗(x)

∂x

)

ψ(x)

]

and

jp =A

2i

[

ψ(p)∂2ψ∗(p)

∂p2+ ψ∗(p)

∂2ψ(p)

∂p2− ∂ψ(p)

∂p

∂ψ∗(p)

∂p

]

21

which gives

jp = −R2p

(

∂V

∂x

)2

x=xr

+A

[

2Rp

(

∂2Rp

∂p2

)

−(

∂Rp

∂p

)2]

as opposed to the simple expression for jx

jx =1

mR2

x

(

∂Sx∂x

)

.

This clearly shows the limitation of using the condition pr = (∂Sx/∂x) asthe guidance condition. It should also by now be quite clear that the Bohmtrajectories in a particular representation are obtained from the probabil-ity current for that particular representation and not from any additionalguidance condition.

7 Conclusions

7.1 Representation independent algebraic forms

In this paper we have shown how it is possible to write the content of theSchrodinger equation in algebraic form without the need to appeal to anyspecific representation. The resulting two equations are respectively theLiouville equation, equation (6), and an equation that describes the timedevelopment of the phase, equation (7), which we have called the quantumphase equation. Furthermore, we show that this equation is gauge invariantand from it we can calculate the Aharonov-Bohm, the Aharonov-Casher andthe Berry phases in a simple and straight forward way.

We have also shown that it is possible to write the probability currentsin operator form which are, once again, independent of any specific rep-resentation. This allows us to define probability currents in any arbitraryrepresentation. All of these results follow from the quantum formalism with-out the need to appeal to any classical formalism.

7.2 The x and p representations: the quantum potential and

the streamlines of probability current

In section 4, we expressed equations (6) and (7) in the x-representation(equations (25) and (15) respectively) and showed that they are identicalto the two defining equations of the traditional Bohm interpretation [18].The quantum potential emerges from equation (12), which in turn comes

22

directly from equation (7), showing that it cannot be “dismissed as artificialand obscuring the essential meaning of the Bohm approach” [17] withoutmissing some of the essential novel features of quantum processes.

In particular, observed characteristic quantum phase or gauge effectscome directly from equations (7) and (15). As Philippidis, Bohm and Kaye[32] have shown many years ago, the presence of the AB effect alters thequantum potential, which in turn accounts for the fringe shifts. Further-more, it is the presence of the quantum potential that offers an explanationof Einstein-Podolsky-Rosen-type correlations [33], as well as quantum stateteleportation [19].

In section 4, we also showed that we can construct a BI in the p-representation. Comparing representations shows very clearly that the Bohmtrajectories are simply the streamlines of the probability currents of the stan-dard theory. The only assumption added to the standard quantum theory inthe Bohm-Hiley [18] version of BI is that particles have simultaneously welldefined positions and momenta and actually follow these streamlines. Fur-ther, we claim that this position is implicit in pragmatic quantum mechanicsin which the probability currents are assumed to be related to particle fluxes.

7.3 Shadow phase spaces

The central point that emerges from our approach is that we can constructtwo different phase spaces. The x-phase space is based on equations (14)and (15), while the p-phase space is built using equations (16) and (17). Thereason why we must resort to constructing different phase spaces is not toodifficult to see once it is realised that we are dealing with a non-commutativestructure6.

For a commutative algebra, the Gel’fand construction allows us to startfrom the algebra and re-construct the underlying manifold [44]. Here thepoints, the topology and the metric structure of the manifold are all carriedby the algebra. No such construction is possible for a non-commutativealgebra. Thus in our case there is no underlying phase space with pointsthat can be specified by the pair of observables (x, p). This is just what theuncertainty principle is telling us. This is the physicist’s way of explanationwhy there can be no single, unique, underlying continuous phase space.

Any attempt to produce a single phase space, such as is done in theWigner-Moyal approach, must necessarily contain unacceptable features [45].In this case, the probability distribution can be negative in certain situations.

6 A simple example of this kind of structure will be found in Hiley [43], and Hiley andMonk[12].

23

For these reasons we must follow what is usually done in non-commutativegeometry and construct shadow manifolds.

In this context equation (12) provides an explanation as to why the en-ergy can be conserved when we attribute to the particle at position x, thebeable momentum pr = ∂Sx/∂x. Since it is a constructed momentum andnot a measured momentum, the kinetic energy will not have the value nec-essary to conserve the total energy. Thus we need another term to “carry”this difference. Since equation (15) is the expression for the conservation ofenergy, the last term on the RHS of equation (15) is the place to “store”this energy difference. This shows that the quantum potential energy is aninternal energy, and clearly does not have an external source.

7.4 Implications for the Bohm Interpretation of quantum

mechanics

Finally, we will briefly comment on the implications of the above analysis onthe BI. The traditional BI assumes the x-representation is special, but thereasons for this were never made clear. It was generally assumed that allphysics must take place in an a priori given space-time arena, a point of viewthat we have called the Cartesian order. Hence the attempt to use the guid-ance condition as a defining equation for Bohmian mechanics [16]. Howeverour mathematical analysis above shows that this condition is a contingentfeature, which is dependent on the asymmetry of the Hamiltonian.

On the other hand, if we take the quantum formalism as primary then wemust place our emphasis on the non-commutative structure of the algebraof formalism. If we do this then attempts to focus on a single phase space,which is equivalent to giving primary relevance to space-time, will fail. Thisin turn calls into question the way we think about quantum processes. In-deed Bohm has already argued that we must abandon the Cartesian orderand replace it by a radically new approach to quantum phenomena which hecalled the implicate order [23]. Here the ontology is provided by the conceptof process which is to be described by the non-commutative algebra. Thisis not a process in space-time, but a process from which space-time is to beabstracted. Abstraction here means to ‘make manifest’ and the order thatis made manifest is called the explicate order.

The key point about this view is that there may be more than oneexplicate order and that these explicate orders cannot be made manifesttogether at the same time. This can be regarded as a direct consequence ofthe participatory nature of the quantum process. Thus the implicate ordercontains an ontological complementarity, which is a necessary consequence

24

of the non-commutative structure. In this picture the BI discussed above issaid to contain two explicate orders, one depending on the x-representationand the other on the p-representation. These are the shadow phase spaces.Both are equally valid descriptions of the outward appearance of a quantumprocess within a given context.

Since our classical world is dominated by appearances in space-time, wewould expect the most relevant explicate order to be that based on the x-representation, with the context being provided by the classical world. Thisis the world in which we place our apparatus and where our measurementstake place. But clearly we need to explore these ideas further as a numberof questions remain unanswered. We will leave this discussion for anotherpaper.

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