aa r X i v : . [ qu a n t - ph ] F e b Aspects of Algebraic Quantum Theory: a Tributeto Hans Primas
B. J. Hiley ∗ .Physics Department, University College, London, Gower Street,London WC1E 6BT.TPRU, Birkbeck, University of London, Malet Street,London WC1E 7HX. It is a privilege to be invited to contribute to this volume dedicated to HansPrimas whose work on the foundations of quantum theory has had a stronginfluence on my own thinking on the subject. I first came across his ideason algebraic quantum mechanics in a bound manuscript entitled
QuantumMechanical System Theory [1] in David Bohm’s room at Birkbeck College in1977. The manuscript, co-authored with Ulrich M¨uller-Herold, was to proveinvaluable for my thinking about quantum theory.I had been working with David Bohm trying to develop a new way ofthinking about quantum theory based on a process philosophy, in which wewere trying to formulate in terms of an algebraic structure along the linesof the original proposals of Born, Heisenberg and Jordan [2]. The idea ofusing an algebraic structure to describe process has an even longer historygoing back to Hamilton [3], Grassmann [4, 5] and Clifford [6], but for onereason or another it fell into disrepute, in spite of its use by Eddington [7].Fortunately today the notion of process as fundamental is undergoinga revival, particularly with the appearance of category theory especially inthe hands of Abramsky and Coecke [8] and Coecke [9] who use the theoryin the context of quantum mechanics, explaining in greater detail their mo-tivations for using a process approach. In this paper I prefer to motivate ∗ E-mail address [email protected]. C ∗ -algebraic approach,an algebraic structure that I was completely unaware of at the time.My interests in an algebraic approach had already been aroused by Pen-rose’s [10] twistor theory, a generalisation of the Dirac Clifford algebra in-troduced by Dirac to describe the relativistic electron. At the time Penrosewas in the mathematics department at Birkbeck and, together with Bohm,we would meet regularly for seminars that were concerned with the possi-bility of developing quantum space-time structures, a radical idea that wethought necessary in order to unite quantum theory with general relativity.Penrose [10] was also exploring the possibility of developing a descriptionbased on a discrete spin network, thus avoiding the need to assume an apriori given space-time continuum [11]. This idea of a network structurefitted in very nicely with the topic of my PhD, although that was in a verydifferent field.My thesis involved investigating certain aspects of the Ising model used inthe study of cooperative phenomena in solid state physics. The simple modelthat I was exploring involved determining the thermodynamics of a many-particle lattice system with nearest neighbour interactions. It was basedon a method of finite clusters, using an idea first proposed by Domb [12].The evaluation of the partition function, and hence the thermodynamicalproperties, necessitated developing a technique for embedding finite graphsin regular tessellations. What I noticed was that some of these properties,essentially combinatorial in nature, depended only on the dimensionality ofthe embedding space and not on the detailed structure of the tessellation.In other words, simply by counting embeddings, one could determine thedimensionality of the embedding space [13]. It was only later that I becameaware of the fact that the partition function could be obtained much moresimply using an algebraic approach used in knot theory. This approach isdescribed in Kauffman [14] who illustrated the technique on small clusters.The phrase ‘quantum space-time’ was a generic term to refer to anystructure that did not take a continuum of points as fundamental, but ratherthe points were assumed to emerge from a deeper structure. That was, infact, the idea behind the Penrose twistor which is used to describe a complexof light rays whose intersections define the points of space-time. He alsofound that congruences of light rays twisting around each other could beused to define sets of ‘extended points’ which he hoped would avoid some ofthe singularities that plague quantum electrodynamics.But surely finding partition functions of a spin lattice is a long way2rom the problems of developing a quantum space-time? Not so because itturns out that the algebraic techniques lying behind both twistors and thealgebraic evaluation of partition functions are closely related to the seminalwork of Vaughn Jones [15] on von Neumann algebras. In a remarkable paper,he showed the connection between these algebras and the combinatorialproperties of knots which, as we have already remarked, lie at the heart ofthe techniques involved in evaluating the partition function of finite clustersof spin systems. The connection becomes even more suggestive when it isrealised that the Onsager exact solution [16] for the two-dimensional Isingmodel involves a Clifford algebra, an algebra that is one example of a vonNeumann algebra. Note also that these algebras are the very algebras thatPenrose [10] used to construct his twistors. However all these ideas werethen yet to unfold in the future. In those early days, Bohm [17, 18] was developing his notions of “structure-process” which emphasised the relationships, order and structure of a net-work of elementary processes. Not relations that could be embedded in theCartesian order of points, but a new order from which the classical Cartesianorder could be abstracted in some suitable limit. This structure, we believed,would provide a more natural way of accounting for quantum phenomena.The basic ideas of ‘structure’ had already been introduced by Edding-ton [19] when he raised the question “What sort of thing is it that I know”?For him the answer was structure , structure that could be captured by math-ematics. For example, the concept of space is not an empty ‘container’, buta relationship of the ensemble of movements that is experienced as we probeour surroundings, using light signals or other suitable physical processes.For Eddington, the structure of these experiences could be captured by agroup, which in the relativistic case would be the Lorentz group, giving riseto Minkowski space-time. Of course in the presence of a gravitational field,this group must be replaced by a larger group, the group of general co-ordinate transformations but for Penrose the conformal group was generalenough to be explored initially. When we come to quantum phenomena,Weyl [20] pointed out that we must turn our attention, not to the group,but to the group algebra. It is the group algebras that form the backgroundto non-commutative geometry. 3 .3 The Role of Clifford Algebras
However, again, I go too fast because initially Bohm and I thought thata natural mathematical expression of this structure would be provided bycombinatorial topology alone [21]. Although this provided some interestinginsights, it misses a vital ingredient, namely, the activity or movement thatwas necessary to describe process. But then I noticed that Penrose’s spinnetwork had Clifford algebras at its heart, the algebra that Onsager used tosolve the two-dimensional Ising model. Could it be that the combinatorialaspects could be captured by an algebra itself, so that we could use algebrasto describe a dynamic structure-process?To my surprise I found that Clifford [6] was led to his algebra, not bythinking of a quantum system, but by considering the dynamical activity ofclassical mechanical systems. He noticed that Hamilton’s quaternion alge-bra, a way of describing rotations in space through action, could be gener-alised to capture the Lorentz group and even leads to the conformal groupwhich is used in twistor theory. Algebraic elements could be understoodin terms of how movements could be combined to form new movements.Clifford introduced terms like ‘versors’, ‘rotators’, and ‘motors’ emphasisingactivity. Unfortunately these ideas seemed to add nothing new to physicsthat was not already described more simply by the vector calculus, so thealgebraic approach was ignored. However that changed when Dirac, facedwith the negative energies appearing in the relativistic generalisation of theSchr¨odinger equation, rediscovered the Clifford algebra. It provided a de-scription of spin, relativity and the twistor in one algebraic hierarchy.Unfortunately the appearance of the Dirac Clifford algebra did not leadto a reconsideration of Clifford’s ideas. Rather the algebra was seen as ageneralisation of the quantum operator algebra that was already used inthe standard Hilbert space formalism taught to undergraduates. In thatapproach the wave function played a key role and gave rise to the so called‘wave-particle duality’, a notion that I find very unhelpful, being a totallyconfused idea. Somehow this wave function is used to describe the so called‘state of the system’ which was, in turn, assumed to evolve in the Cartesianorder of space and time. While this approach was a predictive success, it hasmany, as yet, unsolved interpretational problems, such as the measurementproblem, schizophrenic cats and the like. All of these could be handled asa set of rules for getting ‘correct’ results, but one is left with the uneasyfeeling that something is not quite right because the nature of the physicalprocesses themselves remains very unclear.This view was shared by Hans Primas who proposed the question “Why4 Hilbert space model?” He then explained that this Hilbert space was but aparticular representation of a more general quantum mechanics. The algebraemphasises a non-commutative structure, a structure that has its origins inthe early work of Born, Heisenberg and Jordan [2]. For Primas [22]Algebraic quantum mechanics starts with an abstract B ∗ -algebra, A , of observables. From this algebraic realisation of quantummechanics, we can get the corresponding Hilbert-space model H .... as the universal representation ( π, H ) of the B ∗ -algebra A .Thus Hilbert space is a mere representation, but a representation of what?Could algebraic structure itself provide a description of structure-processand in doing so, clarify the nature of quantum processes? We now come to the point where algebra meets logic. Primas highlightedthe close relationship between the von Neumann algebras and orthomodularlattices of the type used in the analysis of formal logic. In fact the set ofprojections in a von Neumann algebra forms a complete orthomodular latticeso that investigating the properties of this lattice gives a different insightinto the algebraic structure.Projection operators are idempotents, E = E and because their eigen-values are 0 and 1, they can be used to define the truth or falsity of aset of propositions. We thus have an alternative method of analysing theSchr¨odinger formalism in terms of a non-Boolean logic, a generalisation ofthe Boolean logic of classical physics.The generalised non-Boolean logic contains a new notion of incompatiblepropositions , tied intimately to the appearance of non-commuting operators.This difference led Finkelstein [23] to conclude that the appearance of quan-tum processes causes a fracture in physical logic. Indeed Finkelstein showedthat in this non-Boolean logic, the distributivity law of classical logic wasviolated.This raises the important question as to whether this change in logic hasto do with the fact that we can only obtain incomplete knowledge of a quan-tum system or whether this fact stems from a profound change in the basic reality underlying quantum phenomena. Bohr offered an epistemological in-terpretation in which he proposed that the incompatibility of propositions5rises from our inability, in principle , to obtain complete knowledge of thesystem. For Bohr, quantum phenomena confirmed that there was a newprinciple of epistemology, namely the principle of complementarity to whichall knowledge must conform. If this was a fundamental principle then, nomatter what underlies appearance, it would be impossible, even in principle,to construct intuitive pictures of this underlying reality, pictures of the typeused in the classical world.However quantum phenomena occur without the need for anyone to in-terpret them or have knowledge of them. There is an actual process un-folding, independent of any observer and this fact demands an underlyingontology. As Primas [25] insistsAccordingly, practically all high-level theories adopt some kindof scientific realism i.e. the view that biological, chemical andphysical objects have existence independent of some mind per-ceiving them.The key question is then, “What form is this ontology going to take”? Is itgoing to be ‘veiled reality’ as suggested by d’Espagnat [24] or do we followPrimas [25] and insist that “the most fundamental theory has to be phrasedin an individual and ontic interpretation ”. Our hope was that the notionof structure-process would provide the intuitive basis of such a fundamentaltheory.Any generalised theory must be based on non-commutative algebras thatlie at the heart of quantum processes. Since geometry forms the basis ofclassical physics, its generalisation, non-commutative geometry, must be theway forward to explore the nature of the underlying ontology.Such a possibility had already been anticipated by Murray and von Neu-mann [26], who presented a very detailed, but intimidating mathematicaldiscussion of what are now called von Neumann algebras, algebras thatwould play a fundamental role in non-commutative geometry [27]. Fortu-nately for the purposes of this paper we will not require this detailed knowl-edge as we can illustrate the essential ideas using the orthogonal Cliffordalgebra, a specific von Neumann algebra but one with which physicists andchemists are very familiar through the use of the Pauli σ -matrices and theDirac γ -matrices.What the physicist or the chemist may not realise, however, is that aClifford algebra over a complex field is a particular example of a type II von Neumann algebra with a Jones index of 4 cos ( π/
4) [28]. From thecomments above, it should be clear that the Clifford algebra will play animportant role in our discussion of a non-commutative geometry, a point of6iew shared by Finkelstein [29] when he writes, “I am strongly tempted bythe example of Clifford”.
As we have already remarked, the conventional view among physicists isto regard the Clifford algebra merely as a formal mathematical device, butour introductory remarks suggest that it is more than that, describing anunderlying structure-process. However to proceed down that route meanswe must give up, as a fundamental form, the classical notion of a particleevolving along a well defined trajectory in an a priori given space-time.Instead we should adopt a thoroughgoing process philosophy along the linessuggested by Eddington [19], Finkelstein [30] and Bohm [31].
To summarise then, in a process philosophy, we must give up the commonsense idea that the world consists of material objects with definite size,shape and properties. But this notion has already been called into questionin special relativity where we are forced to adopt a description based on thenotion of a point event. There is no consistent description of an extendedrigid object; a particle must be treated as a complex structure of eventsthat can be regarded as forming a ‘world tube’. The tube itself cannothave a sharp boundary but must be identified with a pattern of events,distinguishable, but not separate from, a complex of interrelated backgroundevents. In this approach the ‘particle’ is a semi-stable, quasi-local featurethat can preserve its form in time. However under suitable conditions itcan undergo, not only quantitative changes, but also qualitative changes, inits basic elements, a phenomenon that is well-known in high energy particlephysics.In passing, note that Primas [25] also has a similar structural notion ofa ‘particle’. He stresses that the so-called ‘fundamental’ entities, such aselectrons, protons, or quarks, must not be taken as the building blocks ofreality. They are merely what he calls patterns of reality . For Primas thesepatterns emerge operationally from the empirical domain, a point to whichI will return later.A limitation of the notion of an elementary ‘rock-like’ particle becomeseven more apparent in the quantum domain. To bring the difficulty outclearly, consider the following example inspired by Weyl. Suppose we retain7he classical notion of a particle with specific properties. To keep thingssimple, consider a quantum world in which we have a collection of objectswith two distinct shapes, either spheres or cubes, and two distinct colours,either red or blue. Our task is to separate these objects into four distinctgroups – red spheres, blue spheres, red cubes and blue cubes. In a classicalworld there is no problem, but in this quantum world, shape and colour areobservables, represented by non-commuting operators, their ‘values’ beingrepresented by their corresponding eigenvalues. This means that to sep-arate colours and shapes, we must have two different types of observinginstruments. In our case we call these instruments ‘spectacles’.Suppose we require to collect together an ensemble of red spheres. Firstwe put on the ‘shape-distinguishing’ spectacles and collect together spheres,discarding all the cubes. Then we put on the ‘colour-distinguishing’ specta-cles and collect together the red spheres, discarding all the rest. We are done;we have a collection of red spheres. So what is the problem? Just recheckthat the objects in the ensemble are still spheres. We use the first pair ofglasses again and find that half the objects are now cubes! No permanent either/or in this world. No permanent and/and either!Clearly quantum phenomena do not have their existence defined in termsof classical objects with well defined properties! Finkelstein has alreadystressed this feature and argues that “to speak about the wave function ofthe system is a syntactic error” [29]. We do not simply ‘find’ the state ofa system. We have to ‘probe’ the system with another physical process,the ‘observing instrument’. In other words our instruments are part of theunderlying structure-process and therefore change the system itself, or betterstill, change the process that is the system. How, then, do we encompassthese radically new ideas without losing features of the standard formalismthat have been used with outstanding success?Let us begin by following Eddington [19] who suggests that the elementsof existence , the individuals, in a process world, should be described byidempotents, E = E . The eigenvalues, λ e , of an idempotent are 1 or 0,existence or non-existence. In symbols E = E, with λ e = 1 or 0 . If all idempotents commute, as in classical physics, existence is always welldefined. We have a Boolean logic. In quantum theory we have a difference,idempotents do not always commute[ E a , E b ] = 0 . E a or E b , never E a and E b . Existence, non-existence and in between? This is the consequence of a non-Boolean logic.
The suggestion is that the idempotent will provide a means of focusing onthe sub-process that is the individual. The individual is a process that iscontinually changing into itself, E.E = E . While probing the individual, theprocess may change the quality of the idempotent, it nevertheless remainsan idempotent, enabling us to track the individual as a sub-process withinthe whole structure-process. In an algebra, an idempotent can be used todefine a set of elements within a minimal left ideal of the total algebra.These elements carry all the information contained in the ‘wave’ functionbut now have the advantage of being an integral part of the whole algebra.In a semi-simple algebra, we can always form an element of such anideal by writing Ψ L ( A ) = ψ L ( A ) E . Mathematically we are constructinga left module or left vector space, but we need not be familiar with thismathematical structure to see how it works. Consider a spin-half systemwhich requires the observables to be expressed in terms of the Pauli spinmatrices. As is well known the spin ‘wave’ function is a column two-matrix,the spinor, Ψ = (cid:18) ψ ψ (cid:19) . From the algebraic point of view, the Pauli spin matrices define the Cliffordalgebra C , ( σ ) generated by the three Pauli spin matrices σ i . An element ofa minimal left ideal can be written in the form Ψ L ( σ ) = ψ L ( σ ) E where E issome idempotent. It is conventional to choose E = (1 + σ ) /
2, which breaksthe rotational symmetry and defines a preferred z -axis while ψ L ( σ ) ∈ A .If we then polar decompose the algebraic spinor, we can write Ψ L ( σ ) = RU where U = U † and R is a positive definite matrix. It is then easy toshow that the spinor can be written in the formΨ L ( σ ) = g + g σ + g σ + g σ ; g i ∈ R . Here we have written the elements of the algebra in terms of Pauli matrices, σ ij = σ i σ j , a rotor. To make contact with the usual spinor, we have the9dentities g = ( ψ ∗ + ψ ) / g = i ( ψ ∗ − ψ ) / g = ( ψ ∗ + ψ ) / g = i ( ψ ∗ − ψ ) / . (1)Let us emphasise again that we have chosen a specific idempotent, namely, E = (1 + σ ) / z -axis. This is usually doneby introducing a homogeneous magnetic field, so the choice of idempotent isdefined operationally , just as Primas’ patterns are defined operationally. Inother words we are changing the process that is the system under investiga-tion. In Wheeler’s words [32], we are participating in the process to inducea change in the process that constitutes the system.This is exactly what we need to account for our toy model of a quantumworld using ‘shapes’ and ‘colours’. The change that we find when checkingthe content of the final ensemble arises from the participatory nature ofour ‘instrument’. Looking through the ‘quantum spectacles’ is not a passiveprocess, it is an action , which must not be thought of as a mere ‘disturbance’.It is an inescapable change in the structure-process that is the system. Moredetails of this idea will be found in Hiley and Frescura [33] and in Hiley andCallaghan [34].This example explains very succinctly how the Pauli algebraic spinorappears and is used in the description of the algebra. It is easily generalisedto the Dirac spinor and indeed the twistor, which is a semi-spinor of theconformal Clifford. These Clifford algebras form a hierarchy or tower ofalgebras, C , → C , → C , → C , of the type considered by Jones [15].It is interesting to note that the Schr¨odinger ‘wave’ function can also beconsidered as an element of a minimal left ideal in the Clifford algebra C , ,with the quaternions appearing in C , .In addition to elements of the left ideal, we also have dual elements,Ψ R ( A ) = Eψ R ( A ), chosen from an appropriate minimal right ideal. Thisenables us to give a complete specification of the structure-process of anindividual system by writing ρ c ( A ) = Ψ L ( A ) e Ψ L ( A )where ρ c ( A ) is an element that characterises the system. It is the algebraicanalogue of the density matrix.If we define e Ψ L ( A ) = Ψ R ( A ) = E e ψ L ( A ) then, by suitable choice of thetilde operation, we find ρ c = ρ c , a signature of what is known in the standardapproach as a pure state . It should be noted that the corresponding dual10lement introduced by Primas and M¨uller-Herold [1] was called a normalisedpositive linear functional . Using this additional mathematical structure, wehave the possibility of a generalisation to mixed states, but in this paper,we confine our attention to pure states for simplicity.As well as rotational symmetries, we must also consider translation sym-metries, which implies turning our attention to the Heisenberg algebra. Herethere is a technical problem because this algebra is nilpotent and thereforedoes not contain any idempotents. However Sch¨onberg [35], and later Hi-ley [36], showed that it was possible to extend this algebra by adding setsof idempotents to form a symplectic Clifford algebra [37]. This then enablesus to employ similar techniques to those used in the orthogonal Clifford al-gebra. One is then able to find time development equations that correspondto the Heisenberg equations of motion.The characteristic element ρ c ( A ) can now be subjected to both left andright translations to determine two fundamental time development equa-tions, i [( ∂ t Φ L ) e Φ L + Φ L ( ∂ t e Φ L )] = i∂ t ρ c = ( −→ H Φ L ) e Φ L − Φ L ( e Φ L ←− H ) (2)and i [( ∂ t Φ L ) e Φ L − Φ L ( ∂ t e Φ L )] = ( −→ H Φ L ) e Φ L + Φ L ( e Φ L ←− H ) . (3)We now have the possibility of two forms of Hamiltonian −→ H = −→ H ( −→ D , V, m )and ←− H = ←− H ( ←− D , V, m ) emphasising the distinction between left and righttranslations. We will not derive these equations here as they have beenderived in Hiley and Callaghan [34]; nevertheless we will use them in thenext section. We merely note that equation (2) is the quantum Liouvilleequation expressing the conservation of probability, while equation (3) isthe quantum Hamilton-Jacobi [QHJ] equation expressing the conservationof energy. A detailed discussion of these equations will be found in Hiley [38].
We must now return to discuss the relation between the non-Boolean struc-ture and its Boolean substructures. Primas [25] offers a formal way tounderstand the relationship between these two logics in terms of a specificphysical process. We will explain his position in the following way.We have argued that there is no such thing as a direct, faithful obser-vation in a quantum process. However as Bohr has pointed out, the results11f any observation must be unambiguously described in terms of a Booleanstructure. This is the only way we can unambiguously communicate theresults of an experiment. How then do we understand the Boolean aspectsof a fundamentally non-Boolean process?Primas suggests that the results of an experiment can be understood asa pattern that is formed by detaching ourselves, and our instruments, fromproperties that we consider to be non-essential. He calls the total process,the factual domain F α , which he distinguishes from the empirical domain E α defined operationally as the result of the α -th pattern recognition technique.The factual domain is non-Boolean and a-local, while the empirical domainis a Boolean and local structure. The link between theory and experiment isthen regarded as a mapping F α → E α which is not required to be one-to-one.Bohm [31] has made, in essence, a similar proposal to understand therelation between Boolean and non-Boolean aspects of physical processes, butin terms of a more general language. Structure-process is defined in terms ofan algebra in which the individual elements of the algebra, like words, taketheir implicit meaning from the way in which the algebra as a whole is used.For example the symbols in the Pauli Clifford algebra take their meaningfrom the rotational symmetries we experience as we rotate in space.In such a structure, all the spin components cannot be made explicit bythe same action. The spin in the z -direction can be made explicit, whilethe other components remain implicit. More generally, as is well known, anensemble of properties corresponding to mutually commuting observablescan be made explicit together. This subset of elements forms a Booleansubstructure within the more general non-Boolean structure. Bohm calledthese substructures explicate orders, while the total non-Boolean structurewas called the implicate order.I have used examples from gestalt psychology as a metaphor to illustratethe notions of the implicate and explicate order. The young lady/old ladygestalt illustrates succinctly what is involved. Our perception constructsor ‘explicates’ a Boolean pattern, say the young lady, by ignoring someof the details in the drawing. When none of the details are ignored, wehave a non-Boolean structure. However metaphors are limited and a deeperanalysis based on equation (3) shows that a projection actually creates theexplicate order. It creates a Boolean substructure within the non-Booleantotality.To see how the projection comes in, let us write the equations (2) and(3) in a more familiar notation i∂ρ = ( H | φ i ) h φ | − | φ i ( φ | H ) (4)12nd i [( ∂ t | φ i ) h φ | − | φ i ( ∂ t h φ | )] = ( H | φ i ) h φ | + | φ i ( φ | H ) . (5)Now introduce the projection operator P a = | a ih a | and take the trace sothat equation (4) becomes ∂P ( a ) ∂t + h [ ρ c , H ] − i a = 0 (6)while equation (5) becomes2 P ( a ) ∂S a ∂t + h [ ρ c , H ] + i a = 0 . (7)To bring out what this means, let us choose an harmonic oscillator Hamil-tonian ˆ H = ˆ p / m + K ˆ x / P x = | x ih x | so that equation (6) becomes ∂P x ∂t + ∇ x . (cid:18) P x ∇ x S x m (cid:19) = 0 . This is just the equation for the conservation of probability in position space.Using the same procedure on equation (7) finally gives us ∂S x ∂t + 12 m (cid:18) ∂S x ∂x (cid:19) − mR x (cid:18) ∂ R x ∂x (cid:19) + Kx Q = − mR x (cid:18) ∂ R x ∂x (cid:19) . (8)Notice that this potential does not appear in the algebraic equation (3) whichwe are regarding as a description of the implicate order. It only appears inthe projected space. This space is a Boolean phase space constructed with( x, p B ( x )) where p B ( x ) is the Bohm or local momentum. It is in this phasespace that trajectories have been constructed by Philippidis, Dewdney andHiley [39]. Thus we have constructed a Boolean explicate order.13e could choose another projection operator P p = | p ih p | so that the twoequations (2) and (3) now become ∂P p ∂t + ∇ p . (cid:18) P p ∇ p S p m (cid:19) = 0and ∂S p ∂t + p m + K (cid:18) ∂S p ∂p (cid:19) − K R p (cid:18) ∂ R p ∂p (cid:19) = 0 . This enables us to project out another Boolean phase space based, this time,on ( x B ( p ) , p ) where x B ( p ) = − (cid:16) ∂S p ∂p (cid:17) . Thus using the momentum repre-sentation we have constructed another explicate order and thereby revealed x, p symmetry – a symmetry that Heisenberg [40] claimed was not presentin the Bohm approach.Bohm chose the x -representation as a preferred representation simplybecause he saw a problem in representing the Coulomb potential in the p -representation. However for other potentials there is no difficulty. IndeedBrown and Hiley [41] showed how the approach worked in the particularcase of a cubic potential.Another criticism that is often made of the Bohm approach is that it doesnot work for the relativistic Dirac particle. However Hiley and Callaghan [42]have shown that we can obtain Lorentz invariant analogues of equations (2)and (3) which can then be put into the form of a relativistic QHJ equation.To do this we need to use the orthogonal Clifford algebra C , . The expres-sion for the quantum potential is more complicated but can be shown toreduce to the expression (8) in the non-relativistic limit [42].These examples show what is involved in what Primas calls pattern recog-nition . It is not a ‘passive’ recognition, it actually involves an active con-struction of the Boolean pattern. But in doing so new features can beintroduced as Primas points out. In the case of the Boolean phase spaceconsidered above, it is the appearance of the quantum potential which canbe considered as the appearance of a force.This is not unlike the nature of the gravitational force which only appearswhen we project the curved space-time geodesic to a flat Minkowski space-time. However there is a significant difference in that the curvature of space-time is universal, whereas the quantum potential is, in a sense, ‘private’,being shared by a group of entangled particles. We could have a situationarising where the quantum potential of one group of entangled particles canbe very different from the quantum potential of another entangled group14f the groups are non-interacting but nevertheless share the same region ofspace-time. The groups do not experience a common quantum potential, itis not universal since they only experience the quantum potential of theirown group. In this paper I have given a limited view of a new way of looking at quantumphenomena that Hans Primas was one of the first to draw to our attention.The disadvantage for pioneers of a new vision is that they do not haveaccess to the later developments, particularly the technical advances, in thiscase the progress in non-commutative mathematics that has been slowlygathering pace since 1977. However without the initial ‘struggle’ to clearthe way, others would not have followed. I will always be grateful to Hansfor his early work and our subsequent discussions which, although at timesheated, always provided new insights.
Acknowledgment
I should like to thank Glen Dennis for his suggestions and helpful comments.
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