aa r X i v : . [ qu a n t - ph ] F e b The Algebraic Way.
B. J. Hiley ∗ .Physics Department, UCL and TPRU, Birkbeck, University ofLondon, Malet Street, London WC1E 7HX.(22 Janurary 2015) Abstract
In this paper we examine in detail the non-commutative symplec-tic algebra underlying quantum dynamics. We show that this algebracontains both the Weyl-von Neumann algebra and the Moyal algebra.The latter contains the Wigner distribution as the kernel of the densitymatrix. The underlying non-commutative geometry can be projectedinto either of two Abelian spaces, so-called ‘shadow phase spaces’. Oneof these is the phase space of Bohmian mechanics, showing that it isa fragment of the basic underlying algebra. The algebraic approachis much richer, giving rise to two fundamental dynamical time de-velopment equations which reduce to the Liouville equation and theHamilton-Jacobi equation in the classical limit. They also include theSchr¨odinger equation and its wave function, showing that these fea-tures are a partial aspect of the more general non-commutative struc-ture. We discuss briefly the properties of this more general mathemat-ical background from which the non-commutative symplectic algebraemerges.
The basic principle of the algebraic approach is to avoid start-ing with a specific Hilbert space scheme and rather to emphasisethat the primary objects of the theory are the fields (or theobservables) considered as purely algebraic quantities , togetherwith their linear combinations, products and limits in the appro-priate topology (Emch [14]). ∗ E-mail address [email protected].
1n order to understand the motivation for “The Algebraic Way” we needto recall the origins of quantum theory. History tells us that the first pio-neering papers to develop a mathematical approach to quantum phenomenawere those of Born, Dirac, Heisenberg and Jordan [7,8,12]. Their attempts toaccommodate the Ritz-Rydberg combination principle, an empirical discov-ery in atomic spectra, into a dynamical theory forced the classical variablesof Hamiltonian dynamics to be replaced by non-commuting analogues.With the emerging law of non-commutative multiplication, the need fora matrix representation of x and p was soon recognised even though thephysical meaning of such a change was unclear. To the physicist, these ma-trix representations opened up a new field of unfamiliar non-commutativealgebras with which they were not very comfortable and since the mathemat-ics itself appeared to have no obvious physical interpretation, the approachwas eventually abandoned in favour of the Schr¨odinger wave mechanics ap-proach.This approach gained greater impetus when particles were found to ex-hibit the wave-like behaviour predicted by de Broglie. These experimentalresults encouraged Schr¨odinger [38] to look for what he called a “Hamil-tonian undulatory mechanics” by modifying the Hamilton-Jacobi offshootof Hamiltonian dynamics. His motivation came from noting that while rayoptics could be explained using equations that were analogous in form toHamilton’s equation of motion for particles, the Hamilton-Jacobi theory con-tained surfaces of constant action, which suggested an analogy with the wavefronts used in the Huygens construction to explain interference phenomenain light.This exploration led Schr¨odinger [38] to a differential equation whichimmediately produced energy levels that conformed with the Ritz-Rydbergdata. The mathematical techniques involved in solving differential equationswere well known to physicists at that time and the faith in this equationwas further reinforced with the introduction of Born’s probability postulate,establishing the relation between wave and particle. Although this relationwas not entirely clear conceptually, it enabled the formalism to be appliedwith outstanding success.Conceptually the wave and algebraic approaches were very different, onebeing based on a very familiar wave phenomenon, the other being based onan unfamiliar non-commutative dynamics with no obvious interpretation.Soon Schr¨odinger himself showed how the two approaches were related and,since the techniques for solving the Schr¨odinger equation were very familiar,this approach became established as the way to understand the physics ofquantum phenomena. 2evertheless many conceptual problems remained, generating many dif-ferent interpretations, some naive others quite bizarre, all based on the as-sumption that the Schr¨odinger equation tells the whole story, not only forunderstanding individual experimental phenomena, but in defining what ul-timately constitutes ‘reality’. However as we will show in this paper that, inspite of its great successes, it is only a part of the whole story. In order tosee this, we need to return to examine the details of the original algebraicapproach in some detail. Before discussing these issues, I would like to briefly highlight the relevantfeatures of the Schr¨odinger approach that we will need in order to motivateour presentation. Of course, we will start with the Schr¨odinger equation,even though it is not clear exactly how it was derived from the Hamilton-Jacobi theory: i ~ ∂ψ∂t = ˆ Hψ with H ( x, p ) → ˆ H ( ˆ X, ˆ P )where the classical Hamiltonian H ( x, p ) is replaced by its operator formˆ H ( ˆ X, ˆ P ).To work with the equation, we must go to a specific representation. Itcustomary to use the Schr¨odinger representation for whichˆ X → x ˆ P → − i ~ ∂∂x ψ → ψ ( x, t )so that we are working in configuration space ( x , x , . . . , x n ).However this is not the only representation. We can use the p − representationwhere ˆ X → i ~ ∂∂p ˆ P → p ψ → ψ ( p, t )so that in this case we are working in momentum space ( p , p , . . . p n ). Againwe have the oscillator representation whereˆ X → ( a † + a ) / √ P → i ( a † − a ) / √ N = a † a. This representation enables us to work more easily with an arbitrary num-ber of particles and is essential for quantum field theory. Of course the3chr¨odinger representation is favoured because we believe that quantumprocesses actually occur in Minkowski space-time.Although there is an abundance of mathematical representations, theStone-von Neumann theorem proves that all irreducible representations areunitarily equivalent. By this we mean that if there are two unitary repre-sentations, π and π , in their respective Hilbert spaces H and H , π : G → U ( H ) and π : G → U ( H )and there exists an operator A : H → H , then these representations areequivalent iff there exists an operator A such that Aπ ( g ) = π ( g ) A ∀ g ∈ G. Having established mathematical equivalence, we are left with the question,“Are the representations also physically equivalent?” This, in turn, leavesanother question “Of what mathematical structures are they representa-tions?”
It is generally believed that quantum phenomena “demand a fundamentalmodification of the basic physical concepts and laws” [34]. In other wordswe require a totally different description from that used in classical physics.However there are some obvious similarities in the form of the dynamicalequations of motion. In classical mechanics, Hamilton’s equations of motioncan be written in the form˙ x i = { x i , H } ; ˙ p i = { p i , H } and generally ˙ f ( x i , p i ) = { f ( x i , p i ) , H } (1)where H is the Hamiltonian and { ., . } are the Poisson brackets defined by { f, g } = X i (cid:20) ∂f∂x i ∂g∂p i − ∂f∂p i ∂g∂x i (cid:21) , giving the special case { x i , p j } = δ ij .On the other hand, in quantum mechanics, Heisenberg’s equations ofmotion appear in the form i ~ d ˆ X i dt = [ ˆ X i , ˆ H ]; i ~ d ˆ P i dt = [ ˆ P i , ˆ H ] and generally i ~ d ˆ Fdt = [ ˆ
F , ˆ H ] . (2)4ere ˆ H is the classical Hamiltonian where x and p are replaced by operatorsˆ X and ˆ P and [ ., . ] are the commutator brackets defined by[ ˆ F , ˆ G ] = ˆ F ˆ G − ˆ G ˆ F giving in the special case [ ˆ X i , ˆ P j ] = i ~ δ ij .The similarity in the form of the two sets of equations is quite remark-able, in spite of the differences in the nature of the elements involved. Theclassical equations of motion are ordinary functions on a continuous phasespace while, in the quantum case, they are operators acting on vectors in anabstract Hilbert space. However there is one other striking similarity. Theyare both invariant under the Heisenberg group.The Heisenberg equations of motion directly use elements of the Heisen-berg (Lie) algebra defined by the canonical commutation relations[ ˆ X i , ˆ X j ] = [ ˆ P i , ˆ P j ] = 0 , [ ˆ X i , ˆ P j ] = δ ij ˆ T , [ ˆ X i , ˆ T ] = [ ˆ P i , ˆ T ] = 0 . Here we have written ˆ T = i ~ ˆ I for convenience, so that the elements ( ˆ X i , ˆ P i , ˆ T )generate the Heisenberg group, H n .On the other hand, the classical dynamical variables are representations of the Heisenberg algebra in which commutators are replaced by Poissonbrackets. Thus the Heisenberg group is not only significant in the quantumdomain but also operates in the classical domain. In fact it plays a vital rolein radar theory [1], which is in no way a quantum phenomenon.There is a further invariance which is more directly seen in the classicalmechanics in the dynamical equations of motion (1). They are are invari-ant under transformations of the symplectic group Sp (2 n ) (i.e. canonicaltransformations) for a 2 n -dimensional phase space. These transformationsleave invariant the antisymmetric bilinear form ω ( x, p ; x ′ , p ′ ) = xp ′ − x ′ p .Although one can prove this directly, it can also be thought of as arisingfrom the group of automorphisms of the underlying Heisenberg group.If we write two elements of the Heisenberg group in the formˆ U = n X i =1 x i ˆ X i + p i ˆ P i + t ˆ T , ˆ U ′ = n X i =1 x ′ i ˆ X i + p ′ i ˆ P i + t ′ ˆ T , we find [ ˆ
U , ˆ U ′ ] = ω ( x, p ; x ′ , p ′ ) ˆ T , (3)where ω is again an antisymmetric bilinear form. The appearance of ω ( x, p ; x ′ , p ′ )in equation (3) implies that the Heisenberg group and, hence the Heisenberg5quations of motion, are invariant under the group of symplectic transforma-tions. In other words the group of automorphisms of the Heisenberg groupis the symplectic group.This means that the mathematical structure underlying both classicaland quantum dynamical behaviour arises from symplectic geometry. It turnsout that, in the quantum case, the symplectic geometry is non-commutative,while in the classical case, it is commutative. Although these structures areclearly related mathematically, we still have a puzzle as to why there is notrace of an underlying phase space in the quantum algebra and, even if wewere to find one, then how would it accommodate the Heisenberg uncertaintyprinciple? To find the role of phase space in quantum mechanics, we must put aside anyobjections based on the uncertainty principle and follow some early work ofvon Neumann [36]. Let us consider, not only translations in x -space, butalso translations in p -space. As is well known, space translations can bedescribed using the Taylor expansion so that f ( x + a ) = exp (cid:20) α ∂∂x (cid:21) f ( x ) . α ∈ R In the case of a translation in momentum space, we may similarly write g ( p + β ) = exp (cid:20) β ∂∂p (cid:21) g ( p ) . β ∈ R By recalling the Schr¨odinger representation, we can write these exponentialsin operator form, namelyˆ U ( α ) = exp( iα ˆ P ) and ˆ V ( β ) = exp( iβ ˆ X ) . We immediately see that these translations do not commute becauseˆ U ( α ) ˆ V ( β ) = e iαβ ˆ V ( β ) ˆ U ( α ) . The operators ˆ U ( α ) and ˆ V ( β ) generate the Weyl-von Neumann algebra.6 .1 Relation between the Weyl-von Neumann Algebra andHilbert Space To make the link with the Hilbert space formalism, von Neumann introducedthe algebraic element ˆ S ( α, β ) = e − iαβ/ ˆ U ( α ) ˆ V ( β ) . (4)Then for a system described by | ψ i in the usual Hilbert space, we can formthe expectation value S ψ ( α, β ) = h ψ | ˆ S ( α, β ) | ψ i . von Neumann then shows that any linear operator ˆ A can be symbolicallywritten as ˆ A = Z Z a ( α, β ) ˆ S ( α, β ) dαdβ. This leads to a quantum expectation value of the operator ˆ A , via h ψ | ˆ A | ψ i = Z Z a ( α, β ) S ψ ( α, β ) dαdβ where the kernel a ( α, β ) is defined by a ( α, β ) = Z h α + γ | ˆ A | α − γ i e − iβγ dγ. In this way we can completely reproduce the expectation values of quantummechanics in terms of functions of real variables ( α, β ). I refer the reader tovon Neumann for the details.In the Weyl-von Neumann approach, then, the operators of the quantumformalism are replaced by differential functions on the α, β -space. Howevervon Neumann made no attempt to explain the physical meaning of thespace spanned by the parameters α and β . Nevertheless one fact emerges:the multiplication of two of these functions, say, a ( α, β ) and b ( α, β ), must benon-commutative in order to reproduce the results of quantum mechanics.Suppose ˆ A ↔ a ( α, β ) and ˆ B ↔ b ( α, β ), then if ˆ A. ˆ B ↔ a ( α, β ) ⋆ b ( α, β ),von Neumann shows that a ( α, β ) ⋆ b ( α, β ) = Z Z e [ i ( αβ ′ − α ′ β ) / a ( α − α ′ , β − β ′ ) b ( α ′ , β ′ ) dα ′ dβ ′ . (5)7ot only is this star product non-commutative, it is also non-local. Thusnon-locality appears as a basic feature of the α, β plane, so if we want toreplace the operators of the quantum formalism by continuous functions,then the resulting structure must be non-local. Moyal [35] arrived at exactly the same mathematical structure as von Neu-mann but by starting from a very different approach. He was trying tounderstand the nature of the statistics that is needed in quantum mechan-ics, so he asked the question “How can we generalise the statistics of randomvariables if these variables are non-commutative?”With a pair of commutative random variables
X, Y , one defines the ex-pectation values by introducing the characteristic function e i ( Xt + Y s ) . Thenthe expectation value of some function f X,Y ( x, y ) is φ X,Y ( t, s ) = E h e i ( Xt + Y s ) i = Z Z e i ( xt + ys ) f X,Y ( x, y ) dxdy. Moyal proposed that, in the non-commutative case, the characteristic func-tion be replaced by h ψ | e i ( α ˆ P + β ˆ X ) | ψ i so that we can form the function F ψ ( x, p ) = 14 π Z Z h ψ | e i ( α ˆ P + β ˆ X ) | ψ i e i ( αp + βx ) dαdβ. (6)He then proposed that the average of any quantum operator ˆ A can be foundusing h ψ | ˆ A | ψ i = Z Z a ( x, p ) F ψ ( x, p ) dxdp. (7)Note that Moyal has now introduced the two parameters x and p throughthe Fourier transform (6) and since we are dealing with a single particle, ithas been assumed that these parameters are the position and momentum ofa single particle. If that were where the case, then from the form of equation(7), we could regard F ψ ( x, p ) as a probability distribution for the particlehaving coordinates ( x, p ) and we can then regard equation (7) as giving thequantum expectation value for the operator ˆ A by averaging a ( x, p ) over aphase space.There are two difficulties in making such an assumption. This product, although first defined by von Neumann, is now known as the Moyalstar product.
8. As is well known, F ψ ( x, p ) is the Wigner function and can becomenegative. The assumption that F ψ ( x, p ) is a probability density thenopens up a debate as to the validity of the whole approach. Howeverwe will show that F ψ ( x, p ) is not a probability distribution, but thekernel of a density matrix which is not necessarily positive definite oreven real. Thus it is the interpretation of F ψ ( x, p ) being a probabil-ity distribution that is not valid, not the method in which it arises,so we can follow Feynman [17] and use equation (7) as a valid wayto evaluate the quantum expectation values without worrying aboutthe appearance of negative values of F ψ ( x, p ). We need to rememberthat we are dealing with a non-commutative structure and not simplyaveraging over classical coordinates.2. As is not so well known, the parameters ( x, p ) are not the positionand momentum of a localised particle, but the mean values of a cell inphase space associated with the particle. Thus in this approach, theparticle cannot be considered as a point-like object. Rather it is a non-local distribution of energy, the quantum blob [21, 23]. This region,which we associate with the particle, explains the non-local nature ofthe ⋆ product. We will now show the function F ψ ( x, p ) is, in fact, the one particle Wignerfunction, the many-body generalisation of which was first introduced byWigner [41] to discuss the thermodynamic properties of quantum systems.First consider the operator ˆ S ( α, β ) defined in equation (4) written in aslightly modified formˆ S ′ ( α, β ) := e iαβ ˆ U ( α ) ˆ V ( β ) = e iα ˆ P / e iβ ˆ X e iα ˆ P / . It is not difficult to show that h ψ | ˆ S ′ ( α, β ) | ψ i = Z ψ ∗ ( x − α/ e iβx ψ ( x + α/ dx. By taking the Fourier transform, we find F ψ ( x, p ) = 12 π Z ψ ∗ ( x − α/ e − iαp ψ ( x + α/ dα, (8)which we recognise as the Wigner function. Thus we see the Wigner func-tion is intimately connected with the Weyl-von Neumann-Moyal algebraicapproach. We will show this later in section 4.3. .4 Non-Commutative Phase Space In order to confirm that we are dealing with a non-commutative phase space,we will follow Moyal [35], who showed that the star-product (5) can bewritten in a more convenient way, a ( x, p ) ⋆ b ( x, p ) = a ( x, p ) exp[ i ~ ( ←− ∂ x −→ ∂ p − −→ ∂ x ←− ∂ p ) / b ( x, p ) . (9)It is not difficult to show that this expression when applied to x and p gives x ⋆ p − p ⋆ x = i ~ . Thus we see that although we are dealing with functions of ordinary real( x, p ) variables, the usual commutative inner product must be replaced bya non-commutative product.Once we have a non-commutative product we must distinguish betweenleft and right multiplication. However we find it easier to take this intoaccount by introducing two types of bracket, namely, { a, b } MB = a ⋆ b − b ⋆ ai ~ and { a, b } BB = a ⋆ b + b ⋆ a . The first is the Moyal bracket, while the second is the Baker bracket (orthe Jordan product). Using the expression for the product (9), it is easy toshow { a, b } MB = a ( x, p ) sin[ ~ ( ←− ∂ x −→ ∂ p − −→ ∂ x ←− ∂ p ) / b ( x, p )and { a, b } BB = a ( x, p ) cos[ ~ ( ←− ∂ x −→ ∂ p − −→ ∂ x ←− ∂ p ) / b ( x, p ) . The importance of these brackets is that they become classical objects inthe limit O ( ~ ). The Moyal bracket becomes the Poisson bracket { a, b } MB = { a, b } P B + O ( ~ ) = [ ∂ x a∂ p b − ∂ p a∂ x b ] + O ( ~ )while the Baker bracket to the same approximation reduces to the simpleproduct { a, b } BB = ab + O ( ~ ) . Thus we see that the non-local ⋆ -product now becomes the local inner prod-uct used in classical mechanics. Thus in one single formalism we have a wayof dealing with both quantum and classical mechanics . In the earlier sections we have used the parameter p without giving it a physicalmeaning. If we want to interpret it as a momentum, we must replace it by p/ ~ . These results form the basis of deformation quantisation [31]. Non-Commutative Dynamics: the Phase SpaceApproach
As we have seen, an important lesson when dealing with a non-commutativealgebra is to carefully distinguish between left and right multiplication . Wehave been able to avoid this distinction by going to the Schr¨odinger repre-sentation which gives a simpler algorithm that only uses left multiplication.To exploit the full implications of the non-commutative structure we haveto go deeper.To define the dynamics in such a mathematical structure, we have toconsider the following two equations H ( x, p ) ⋆ F ψ ( x, p, t ) = i (2 π ) − Z e − iτp ψ ∗ ( x − τ / , t ) −→ ∂ t ψ ( x + τ / , t ) dτ (10)and F ψ ( x, p, t ) ⋆ H ( x, p ) = − i (2 π ) − Z e − iτp ψ ∗ ( x − τ / , t ) ←− ∂ t ψ ( x + τ / , t ) dτ. (11)Subtracting these two equations gives us one time development equationexpressed in terms of the Moyal bracket: ∂ t F ψ = ( H ⋆ F ψ − F ψ ⋆ H ) / i = { H, F ψ } MB . (12)While by adding the two equations, we get another time development equa-tion expressed in terms of the Baker bracket [2]:2 { H, F ψ } BB = i (2 π ) − i Z e − iτp [ ψ ∗ ( x − τ / , t ) ←→ ∂ t ψ ( x + τ / , t )] dτ (13)where ψ ∗ ←→ ∂ t ψψ ∗ ψ = [ ψ ∗ −→ ∂ t ψ − ψ ∗ ←− ∂ t ψ ] ψ ∗ ψ . (14)It should be noted that we need both equations to get a complete descriptionof quantum mechanics. For a more detailed discussion see Zachos [42].We have already seen that equation (12) leads to the classical Liouvilleequation in the classical limit. To see what equation (13) gives in the classical More formally the mathematical structure of quantum mechanics is a bimodule. ψ = Re iS into equation (13), expand out and then take thelimit to O ( ~ ). We find { H, F ψ } BB = H.F ψ + O ( ~ ) = − ∂ t S ) F ψ + O ( ~ )which then gives the classical Hamilton-Jacobi equation, ∂S∂t + H = 0 . A related approach to the classical limit will be found in Schleich [37].This is a very interesting result when we recall that Schr¨odinger actu-ally started from the classical Hamilton-Jacobi equation in order to find an“Hamiltonian undulatory mechanics”. One of the reasons why he was forcedto guess his equation was because he not did fully appreciate the significanceof non-commutativity.
Since the Moyal algebra gives the correct quantum expectation values ofquantum operators by averaging over a symplectic phase space and sincethe Bohm approach gives the same expectation values using what seems tobe a different phase space defined in terms of ( x, p = ∇ S ), there surely mustbe a relation between these two approaches. To bring out this relationship,let us follow Moyal and treat F ψ ( x, p ) as a quasi-probability distribution.We can then define the conditional expectation value of the momentum.A value of this momentum can be obtained from the general relationgiven by Moyal [35], namely ρ ( x ) p n = Z p n F ψ ( x, p ) dp = (cid:18) ~ i (cid:19) n [( ∂ x − ∂ x ) ψ ( x ) ψ ( x )] x = x = x . (15)For n = 1 we find, by writing ψ = Re iS , that p ( x ) = 12 i [ ψ ∗ ∇ ψ − ( ∇ ψ ∗ ) ψ ] = ∇ S ( x ) . This is identical to the Bohm momentum defined by the relation p = ∇ S ,the so called “guidance relation”. However in the approach we are exploring12ere, there are no waves of any form and the notion of guiding wave is mean-ingless. Everything that emerges is a consequence of the non-commutativesymplectic geometry.This connection between the Bohm momentum and the conditional ex-pectation value of the momentum can be made even stronger. Moyal showsthat by starting from the Heisenberg equations of motion, the transport ofthe momentum p ( x, t ) is given by ∂ t ( ρp k ) + X i ∂ x i ( ρp k ∂ x i H ) + ρ∂ x k H = 0 . Then after some work and again writing ψ = Re iS , Moyal finds ∂∂x k (cid:20) ∂S∂t + H − ∇ ρ mρ (cid:21) = 0 . If we choose H = p / m + V where p = ( ∇ S ) − ~ (cid:18) ∇ RR (cid:19) + ~ ∇ ρρ . Then ∂S∂t + H − ∇ ρ mρ = ∂S∂t + 12 m ( ∇ S ) + V − m ∇ RR = 0 . (16)Here the RHS of equation (16) is the quantum Hamilton-Jacobi equation,the real part of the Schr¨odinger equation that plays a key role in the Bohmapproach [5]. But since the Moyal algebra contains the Bohm approach,and in fact is exactly the von Neumann algebra (i.e. an algebra upon whichquantum mechanics is based) then clearly the Moyal and the Bohm approachare simply different aspects of precisely the same mathematical structure.Full details of the above derivations can be found in the appendix of theoriginal Moyal paper [35]. Further details of the relation between the Moyaland the Bohm approach can be found in Hiley [23]. What the previous subsection 6.1 shows is that if we take the variable x as one axis of the phase space, we can take p to be the other axis of thephase space. Thus we have constructed a phase space out of the variables13 x, p ). In this phase space, the time development equation is the quantumHamilton-Jacobi equation ∂ t S ( x, t ) + ( ∇ x S ( x, t )) / m + Q x ( x, t ) + V ( x, t ) = 0 . (17)Here the quantum potential, Q x ( x, t ), is given by Q x ( x, t ) = − m (cid:18) ∇ R ( x, t ) R ( x, t ) (cid:19) . Thus we can construct trajectories in this ( x, p ) space.However notice that the distribution F ψ ( x, p ) is symmetric in x and p so that we can also find the conditional expectation value of the position, x ( p, t ), in terms of the momentum p . We will again follow Moyal and definethis value x as ρ ( p ) x = Z xF φ ( x, p ) dx = Z xψ ∗ ( x ′ ) ψ ( x ′′ ) δ [ x − ( x ′ + x ′′ ) / e ip ( x ′ − x ′′ ) dxdx ′ dx ′′ , which in the p -representation takes the simpler form ρ ( p ) x = Z xF φ ( x, p ) dx = 12 i [( ∂ p − ∂ p ) φ ∗ ( p ) φ ( p )] p = p = p . Writing φ ( p ) = R ( p ) e iS ( p ) , we find the conditional expectation value of theposition, x ( p ), given the value of p is x ( p ) = −∇ p S ( p ) . Again in analogy with the previous case, we have another quantum Hamilton-Jacobi equation, only this time in p -space. Thus ∂ t S ( p, t ) + p / m + Q p ( p, t ) + V ( −∇ p S ( p, t ) , t ) = 0 , (18)where Q p ( p, t ) = − mR p (cid:18) ∂ R p ∂p (cid:19) (19)is the quantum potential in a second phase space constructed in terms ofthe coordinates ( x = −∇ p S, p ). An example of how this works for the caseof a particle in a potential V ( x ) = Ax will be found in Brown and Hiley [9]where more details of the whole approach are given.14hus we find that there are, at least, two shadow phase spaces we canaccess. Each gives a different phase space picture of the same overall alge-braic structure, a feature that has already been recognised in the Wignerapproach by Leibfried et al. [30] who call these spaces shadow phase spaces,a term Hiley [25] has also used.These shadow spaces are an example of what Bohm calls ‘explicate or-ders’ in his general notion of the implicate order [4]. In this case the algebraicstructure defines the implicate order, while the two shadow phase spaces area pair of explicate orders. One should note that both equations (12) and (13)do not contain quantum potentials explicitly. They only appear explicitlyin equations (17) and (18), namely at the level of conditional expectationvalues. One should also note that in the classical limit p → p and x → x , sothat, in this limit, both quantum potentials vanish and we have one uniquephase space. We can get more insight into this whole approach by returning to the op-erator approach and exploiting the one-to-one relation ˆ A ↔ a ( x, p ). Thismeans we should be able to form the operator equivalent of the two equa-tions (12) and (13). In order to motivate this, let us return to consider howthe Schr¨odinger equation emerges from the Heisenberg equation for the timedevelopment of the density operator ˆ ρ , i d ˆ ρdt = [ ˆ H, ˆ ρ ] . (20)Let us follow Dirac [13] and write ˆ ρ = ˆ ψ. ˆ φ . Notice that both ˆ ψ and ˆ φ are operators , not vectors in a Hilbert space. Substituting this expression intoequation (20), we get i d ˆ ψdt ˆ φ + iψ d ˆ φdt = ( ˆ H ˆ ψ ) ˆ φ − ˆ ψ ( ˆ φ ˆ H ) . i d ˆ ψdt = ˆ H ˆ ψ (21)and − i d ˆ φdt = ˆ φ ˆ H. (22)We say ‘Schr¨odinger-like’ because ˆ ψ and ˆ φ are elements of the operatoralgebra. Notice the order of the operators in these two equations; in equation(21) the operators act from the left, while in equation (22) the operatorsact from the right. In fact these equations are left and right translationSchr¨odinger equations, the analogues of equations (10) and (11) proposedin the von Neumann-Moyal algebra.Recall that to obtain equation (12), we subtracted equations (10) and(11), so we see that the Heisenberg equation of motion can be formed by sub-tracting equations (21) and (22). There is a clear analogy with the bra andket vectors, but here ˆ ψ and ˆ φ are taken to be elements of the non-commutingalgebra, not elements of an external abstract Hilbert space. ˆ ψ and ˆ φ are, infact, elements of a specific left and right ideal respectively that exists withinthe non-commuting symplectic algebra itself. The implications of this forany possible physical interpretation have been discussed in Hiley [24] andHiley and Callaghan [26].Thus in our approach all the elements we use appear in the algebra it-self and there is no essential need to introduce an exterior Hilbert space,although this alternative is available if required for ease of calculation. Thisthen shows clearly that the Schr¨odinger equation is, as Bohr [6] claimed,merely an algorithm for calculating the outcome of given experimental sit-uations. But unlike Bohr, we are giving attention to the algebra, in thiscase the non-commutative symplectic group algebra. It is this algebra thatprovides a complete mathematical description of the quantum dynamics.We will now bring out this algebraic structure more clearly by adoptinga change of notation, in which ‘operators’ simply become elements of thealgebra because they ‘operate’ on themselves. Thus we will drop the ‘hats’and write ˆ ψ → Ψ L and ˆ φ → Φ R . Here Ψ L is an element of a suitableleft ideal and Φ R an element of some suitable right ideal defined by thephysics of the problem we are considering. These elements contain all theinformation about the state of the system. Mathematically they are centralfeatures of the symplectic Clifford algebra [11]. Similar features appear in16he orthogonal Clifford algebra used to describe the spin and relativisticproperties of quantum phenomena [29]. A detailed discussion of how onechooses these ideals will be found in that paper. Let us replace the density operator ˆ ρ of a pure state by ρ = Ψ L Ψ R , whereΨ L is a left ideal in the algebra and Ψ R is the right ideal. Then the left andright equations of motion are i d Ψ L dt = H Ψ L and − i d Ψ R dt = Ψ R H. Next we form i ( −→ ∂ t Ψ L )Ψ R = ( −→ H Ψ L )Ψ R and − i Ψ L (Ψ R ←− ∂ t ) = Ψ L (Ψ R ←− H ) . Now we can subtract and add these two equations as before and obtain thetwo algebraic equations i h ( −→ ∂ t Ψ L )Ψ R + Ψ L (Ψ R ←− ∂ t ) i = ( −→ H Ψ L )Ψ R − Ψ L (Ψ R ←− H ) (23) i h ( −→ ∂ t Ψ L )Ψ R − Ψ L (Ψ R ←− ∂ t ) i = ( −→ H Ψ L )Ψ R + Ψ L (Ψ R ←− H ) . (24)Since we are writing ρ = Ψ L Ψ R , equation (23) can be written in the form i∂ t ρ = [ H, ρ ] − . (25)This is, in fact, just the quantum Liouville equation. Equation (24) can bewritten in the form i Ψ R ←→ ∂ t Ψ L = [ H, ρ ] + (26)where we have used definition (14). This equation is simply the expressionfor the conservation of energy. Thus equations (25) and (26) then are thealgebraic equivalents of (12) and (13) and give a complete algebraic descrip-tion of a single quantum system. In the previous sub-section we showed equations (25) and (26) to be thedefining equations for the time development of a single quantum system interms of the non-commutative symplectic structure. Notice once again thatthere is no explicit quantum potential in these equations.17o see how these equations are related to the usual Hilbert space ap-proach, we first introduce a projection operator Π a = | a ih a | and apply it toeach equation in turn. We obtain i ∂P ( a ) ∂t + h [ ρ, H ] − i a = 0 , P ( a ) ∂S∂t + h [ ρ, H ] + i a = 0 . Here P ( a ) is the probability of finding the system in the quantum state ψ ( a )which we have written in polar form ψ ( a ) = R ( a ) e iS ( a ) .In order to get a feel for this approach, it is useful to consider particularexamples. Therefore let us consider the harmonic oscillator, H = p m + Kx for its simplicity and for the fact that it is symmetric in x and p . We willchoose two specific projection operators, Π x = | x ih x | and Π p = | p ih p | .We will begin by projecting into the x -representation using Π x = | x ih x | to obtain ∂P ( x ) ∂t + ∇ x . (cid:18) P ( x ) ∇ x S x m (cid:19) = 0 (27) ∂S x ∂t + 12 m (cid:18) ∂S x ∂x (cid:19) − mR x (cid:18) ∂ R x ∂x (cid:19) + Kx . (28)Thus we see that equation (27) is the Liouville equation which is the expres-sion for the conservation of probability in the x -representation. Equation(28) is the quantum Hamilton-Jacobi equation in the x -representation thatappears in Bohmian mechanics.Let us now project into the p -representation by choosing the projectionoperator Π p = | p ih p | to obtain ∂P p ∂t + ∇ p . (cid:18) P p ∇ p S p m (cid:19) = 0 (29) ∂S p ∂t + p m − K R p (cid:18) ∂ R∂p (cid:19) + K (cid:18) ∂S p ∂p (cid:19) . (30)Notice the appearance again of a quantum potential Q p = − K R p (cid:16) ∂ R∂p (cid:17) .Thus we see the quantum potential becomes manifest only as a result ofthe projections. Notice that when the quantum potential is negligible, werecover the classical behaviour, equations (28) and (30) being related by acanonical transformation. Although we have illustrated these projections for18he harmonic oscillator, it follows trivially that they work for any generalHamiltonian.Thus projections from the non-commutative algebraic time developmentequations (25) and (26) produce exactly the same results as obtained fromthe two von Neumann-Moyal equations (12) and (13). Both lead to the samepair of shadow phase spaces. Both produce the same quantum Hamilton-Jacobi equations, namely, equations (17) and (18). Let us now return to our original motivation, namely, that the primarymathematical structures necessary to describe quantum phenomena are non-commutative geometric algebras. In this paper we have concentrated onthe non-commutative symplectic geometry, restricting ourselves to specificexamples to motivate the general method. In a series of papers Hiley andCallaghan [26, 27, 29] have shown how the orthogonal Clifford algebras canbe used to describe the spin and relativistic properties of quantum systems.These two algebraic approaches are very similar in their mathematicalstructure, so there is clearly a more general structure of which these algebrasare specific examples. Indeed they are both simple examples of von Neu-mann algebras and general methods for handling these non-commutativealgebras now exist [15].We will be particularly interested in their relevance to non-commutativeprobability theory, and in particular, the appearance of conditional expec-tation values in these structures, which has non-commutative integrationtheory at its heart [39]. We have seen the need to consider left and rightdifferentiation, so that the inverse of differentiation, namely, integration hasto take this two-sidedness into account. Equation (7) has been interpretedas providing the expectation value of a ( x, p ) taken over F ψ ( x, p ), treating itas if it were a classical probability density. When it was subsequently dis-covered that F ψ ( x, p ) can become negative, alarm bells may have soundedas has been discussed in [3, 16, 17] and more recently in [32]. Yet in spite ofthese difficulties, the expectation values h ψ | ˆ A | ψ i calculated by these meth-ods always turn out to be positive.The explanation of these results lies in non-commutative measure theory,particularly in the papers of Umegaki [40] and Jones [33]. What Umegakishows is that a positive definite conditional expectation value always existsin a sub-algebra N of a type II factor von Neumann algebra M, which is the19ype of algebra we are discussing in this paper. In particular the conditionalexpectation E N : M → N is defined by the relation tr ( E N y ) = tr ( xy ) for x ∈ M and y ∈ N . The map E N is normal and has the following properties: E N ( axb ) = aE N ( x ) b for x ∈ M, a, b ∈ ME N ( x ∗ ) = E N ( x ) ∗ ∀ x ∈ ME N ( x ∗ ) E N ( x ) ≦ E N ( x ∗ x ) and E N ( x ∗ x ) = 0 ⇒ x = 0 . Since the von Neumann-Moyal algebra we are discussing here is a typeII von Neumann algebra, a trace exists and it remains to evaluate this tracefor the two possible projections from the ( x, p ) algebra to the two Abeliansub-algebras, one spanned by x and the other by p . Our case is trivial sincewe are considering the special case of a single particle.One of the projections we have introduced is E P : ( x, p ) → ( x ) which wasdefined by equation (15). A careful examination of the origins of F ψ ( x, p )shows that it is actually the kernel of the density matrix itself. This resulthas already been pointed out in Hiley [25] but we will outline the argumentbriefly again here.Let us start with the density operator ˆ ρ ψ = | ψ ih ψ | and form ρ ψ ( x , x ) = ψ ∗ ( x ) ψ ( x ) which is the kernel of the density matrix [20]. Now let us go tothe momentum representation and write ψ ( x ) = 1 p (2 π ) Z φ ( p ) e ipx dp. Then the density kernel can be written as ρ ψ ( x , x ) = 12 π Z Z φ ∗ ( p ) e − ix p φ ( p ) e ix p dp dp . Now let us change co-ordinates to X = ( x + x ) / η = x − x and P = ( p + p ) / π = p − p , so that the density kernel can be written in the form ρ ψ ( X, η ) = 12 π Z Z φ ∗ ( P − π/ e iXπ φ ( P + π/ e iηP dP dπ. Take the Fourier transform ρ ψ ( X, η ) = Z F ψ ( X, P ) e iηP dP F ψ ( X, P ) = 12 π Z φ ∗ ( P − π/ e iXπ φ ( P + π/ dπ. (31)Recalling that φ ∗ ( P − π/
2) = 1 √ π Z ψ ∗ ( x ) e − i ( P − π/ x dx φ ∗ ( P + π/
2) = 1 √ π Z ψ ∗ ( x ) e − i ( P + π/ x dx . Using these in equation (31), we find F ψ ( X, P ) = 12 π Z ψ ∗ ( X − η/ e − iηP ψ ( X + η/ dη which is just the expression we used in equation (8) with η = α .Notice in this construction that the resulting Wigner function is a func-tion in the ( X, P ) phase space. This phase space has been constructed froma pair of points in ( x , x ) configuration space and the coordinates ( X, P )are the mean position and mean momentum of a cell in an ( x, p ) phasespace. Thus the Wigner function F ψ ( X, P ) is a density matrix over a cellconstructed in the underlying ( x, p ) classical phase space. We have kept ourarguments deliberately simple to arrive at this result. A rigorous geometricapproach that produces this result and its generalisation can be found inCari˜nena et al [10].The first point to notice is that the Wigner function is a complex densitymatrix, not a probability density. This shows why it is incorrect to regard F ψ ( x, p ) as a probability distribution of particle positions and momenta.Thus the worries about negative and complex “probabilities” are totallyunfounded [32].The second point to notice is that the Wigner approach, when appliedto a single particle, is non-local depending on a region rather than a singlepoint. This means we must represent the particle by a region in phase space,namely, the “quantum blob” introduced by de Gosson [21]. However thisnon-locality should not be surprising because as we have already pointed out,the ⋆ -product is non-local. The fact that non-locality is an essential featureof the description should again not be surprising. Indeed the phase spacemust be non-local otherwise we would be in violation of the uncertaintyprinciple. That the ⋆ -product must be a non-local product has already beenpointed out by Gracia-Bondia and V´arilly [18, 19]. Indeed further detailsof the mathematical structure lying behind some of the results discussed inthis paper will be found in these papers.21 Conclusion
The aim of this paper has been to show that the algebraic structure ofthe quantum operators defined by von Neumann [36] and later developedby Moyal [35] gives a more general mathematical structure in which theusual Schr¨odinger representation with its wave function provide but a par-tial mathematical account of quantum phenomena. Elsewhere [28] we haveshown that the information contained in the wave function can be encodedin the algebra in terms of certain ideals already contained in the algebraitself. Hence there is no fundamental need to postulate an external Hilbertspace, and this is in accord with the principle outlined in the above quota-tion taken from Emch [14], namely, that the primary objects of the quantumformalism should be purely algebraic quantities.The geometries underlying these structures are non-commutative in gen-eral and by concentrating on a non-commutative symplectic geometry, wehave shown that the quantum dynamics can be described either by the ele-ments of an abstract algebra or by functions on a generalised phase space.The multiplication rule for combining these functions is necessarily the non-commutative ⋆ -product introduced by von Neumann [36] and Moyal [35].Moyal’s contribution was to show how the algebra generalised classicalstatistics to a non-commutative statistics that emerges from a more generalnon-commutative probability theory [15]. By recognising this generalisation,we have shown that the Wigner function emerges from a representation ofthe kernel of the density matrix. We argue that it is therefore incorrect toregard this kernel as a probability density. Furthermore this fact explainswhy the negative values of the Wigner function present no difficulty.Within this theory we can introduce conditional expectation values fromwhich Bohmian mechanics emerges under the assumption that space-time isbasic. But one has an x ↔ p symmetry in the algebra so that it is possibleit define an alternative “mechanics” taking the momentum space as basic.Thus the Bohm approach does contain the ( x, p ) symmetry that Heisenbergclaimed it lacked [22]. Moreover this symmetry produces shadow phasespaces as used in [30]. In Bohm’s implicate order, these are what he callsexplicate orders. We have also shown how these shadow manifolds mergeinto a single commutative phase space in the classical limit.We noted that the ⋆ -product is a non-local product, as does [18]. Fur-thermore we have shown that the kernel of the density matrix describes acell-like structure, rather than a point particle in phase space. Again thissuggests that the quantum particle is represented by a region of the under-lying non-commutative symplectic space, so that the quantum formalism22s basically non-local in a radically new way even for the single particle,locality arising only at the classical limit.
10 Acknowledgements
I would like to thank Robert Callaghan, Maurice de Gosson, Glen Dennisand David Robson for their invaluable and enthusiastic discussions.
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