We introduce the notion of an extended moment in time, the duron. This is a region of temporal am... more We introduce the notion of an extended moment in time, the duron. This is a region of temporal ambiguity which arises naturally in the nature of process which we take to be basic. We introduce an algebra of process and show how it is related to, but different from, the monoidal category introduced by Abramsky and Coecke. By considering the limit as the duration of the moment approaches the infinitesimal, we obtain a pair of dynamical equations, one expressed in terms of a commutator and the other which is expressed in terms of an anti-commutator. These two coupled real equations are equivalent to the Schrödinger equation and its dual. We then construct a bi-algebra, which allows us to make contact with the thermal quantum field theory introduced by Umezawa. This allows us to link quantum mechanics with thermodynamics. This approach leads to two types of time, one is Schrödinger time, the other is an irreversible time that can be associated with a movement between inequivalent vacuum states. Finally we discuss the relation between our process algebra and the thermodynamic origin of time.
We begin by giving correct expressions for the short-time action; following the work of one of us... more We begin by giving correct expressions for the short-time action; following the work of one of us and Makri-Miller. We use these estimates to derive a correct expression modulo ∆t 2 for the quantum propagator and we show that the quantum potential is negligible modulo ∆t 2 for a point source. We finally prove that this implies that the quantum motion is classical for very short times.
Mainstream cognitive neuroscience typically ignores the role of quantum physical effects in the n... more Mainstream cognitive neuroscience typically ignores the role of quantum physical effects in the neural processes underlying cognition and consciousness. However, many unsolved problems remain, suggesting the need to consider new approaches. We propose that quantum theory, especially through an ontological interpretation due to Bohm and Hiley, provides a fruitful framework for addressing the neural correlates of cognition and consciousness. In particular, the ontological interpretation suggests that a novel type of “active information”, connected with a novel type of “quantum potential energy”, plays a key role in quantum physical processes. After introducing the ontological interpretation we illustrate its value for cognitive neuroscience bydiscussing it in the light of a proposal by Beck and Eccles about how quantum tunneling could play a role in controlling the frequency of synaptic exocytosis. In this proposal, quantum tunneling would enable the “self” to control its brain withou...
In this paper we start from a basic notion of process, which we structure into two groupoids, one... more In this paper we start from a basic notion of process, which we structure into two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford algebras respectively. We show how the orthogonal Clifford algebra, which include the Schrödinger, Pauli and Dirac formalisms, describe the classical light-cone structure of space-time, as well as providing a basis for the description of quantum phenomena. By constructing an orthogonal Clifford bundle with a Dirac connection, we make contact with quantum mechanics through the Bohm formalism which emerges quite naturally from the connection, showing that it is a structural feature of the mathematics. We then generalise the approach to include the symplectic Clifford algebra, which leads us to a non-commutative geometry with projections onto shadow manifolds. These shadow manifolds are none other than examples of the phase space constructed by Bohm. We also argue that this provides us with a mathematical structure that fits the implicateexplicate order proposed by Bohm.
A method for measuring the weak value of spin for atoms is proposed using a variant of the origin... more A method for measuring the weak value of spin for atoms is proposed using a variant of the original Stern–Gerlach apparatus. A full simulation of an experiment for observing the real part of the weak value using the impulsive approximation has been carried out. Our predictions show a displacement of the beam of helium atoms in the metastable 23S1 state, Δw, that is within the resolution of conventional microchannel plate detectors indicating that this type of experiment is feasible. Our analysis also determines the experimental parameters that will give an accurate determination of the weak value of spin. Preliminary experimental results are shown for helium, neon and argon in the 23S1 and 3P2 metastable states, respectively.
The claim of Kocsis et al. to have experimentally determined “photon trajectories” calls for a re... more The claim of Kocsis et al. to have experimentally determined “photon trajectories” calls for a re-examination of the meaning of “quantum trajectories”. We will review the arguments that have been assumed to have established that a trajectory has no meaning in the context of quantum mechanics. We show that the conclusion that the Bohm trajectories should be called “surreal” because they are at “variance with the actual observed track” of a particle is wrong as it is based on a false argument. We also present the results of a numerical investigation of a double Stern-Gerlach experiment which shows clearly the role of the spin within the Bohm formalism and discuss situations where the appearance of the quantum potential is open to direct experimental exploration.
A method for measuring the real part of the weak value of spin for non-zero rest mass atoms is pr... more A method for measuring the real part of the weak value of spin for non-zero rest mass atoms is presented using a variant on the original Stern-Gerlach apparatus. The experiment utilises helium in the metastable 23S1 state. A full simulation for observing the real part of the weak value using the impulsive approximation has been carried out and it predicts a displacement of the beam, Δw, that is within the resolution of our detector. It also indicates how this shift might be increased. The full analysis also indicated that there is a limit, L, to the applicability of the weak value approximation and has been evaluated for our apparatus. This experiment has the possibility to be expanded to utilise other nobal gas species such as neon and argon in the 3P2 metastable state, but we shall restrict this paper to metastable helium only.
There has been a recent revival of interest in the notion of a `trajectory' of a quantum part... more There has been a recent revival of interest in the notion of a `trajectory' of a quantum particle. In this paper we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are able to conclude that a Bohm `trajectory' is the average of an ensemble of actual individual stochastic Feynman paths. This implies that they can be interpreted as the mean momentum flow of a set of individual quantum processes and not the path of an individual particle. This enables us to give a clearer account of the experimental two-slit results of Kocsis {\em et al.}}
The experimental results of Kocsis et al., Mahler et al. and the proposed experiments of Morley e... more The experimental results of Kocsis et al., Mahler et al. and the proposed experiments of Morley et al. show that it is possible to construct "trajectories" in interference regions in a two-slit interferometer. These results call for a theoretical re-appraisal of the notion of a "quantum trajectory" first introduced by Dirac and in the present paper we reexamine this notion from the Bohm perspective based on Hamiltonian flows. In particular, we examine the short-time propagator and the role that the quantum potential plays in determining the form of these trajectories. These trajectories differ from those produced in a typical particle tracker and the key to this difference lies in the active suppression of the quantum potential necessary to produce Mott-type trajectories. We show, using a rigorous mathematical argument, how the active suppression of this potential arises. Finally we discuss in detail how this suppression also accounts for the quantum Zeno effect.
It is argued that in order to address the mind/matter relationship, we will have to radically cha... more It is argued that in order to address the mind/matter relationship, we will have to radically change the conceptual structure normally assumed in physics. Rather than fields and/or particles-in-interaction described in the traditional Cartesian order based a local evolution in spacetime, we need to introduce a more general notion of process described by a non-commutative algebra. This will have radical implications for both for physical processes and for geometry. By showing how the Bohm interpretation of quantum mechanics can be understood within a noncommutative structure, we can give a much clearer meaning to the implicate order introduced by Bohm. It is through this implicate order that mind and matter can be seen as different aspects of the same general process.
In this paper an account of Bohm's own attitude to his early papers on an alternative interpretat... more In this paper an account of Bohm's own attitude to his early papers on an alternative interpretation to quantum theory is presented. He has made it clear that it was never his intention to return to a deterministic mechanical theory, rather it was an attempt to show that an alternative was possible and, in itself, may provide clues for further developments. From the earliest he felt that a more radical approach was needed, an approach that depended on his notion of structure process. In such an approach the particle must not be regarded as some a priori given immutable entity, but must take its dynamical properties from the environment in which it finds itself so that the notion of wholeness, so important to both Bohr and Bohm, could be a fundamental part of the description. It will be argued that the essential feature of his later ideas of the implicate order were already present in his early work in "Causality and Chance" written shortly after his original proposals.
We reexamine the claim made by Englert, Scully, Süssman and Walther that in certain 'Welcher Weg'... more We reexamine the claim made by Englert, Scully, Süssman and Walther that in certain 'Welcher Weg' (Which Way) interference experiments, the Bohm trajectories behave in such a bizarre and unacceptable way that they must be considered as unreliable and even 'surreal'. We show that this claim cannot be correct and is based on an incorrect use of the Bohm approach.
In this paper we discuss the relevance of the algebraic approach to quantum phenomena first intro... more In this paper we discuss the relevance of the algebraic approach to quantum phenomena first introduced by von Neumann before he confessed to Birkoff that he no longer believed in Hilbert space. This approach is more general and allows us to see the structure of quantum processes in terms of non-commutative probability theory, a non-Boolean structure of the implicate order which contains Boolean substructures which accommodates the explicate classical world. We move away from mechanical 'waves' and 'particles' and take as basic what Bohm called a structure process. This enables us to learn new lessons that can have a wider application in the way we think of structures in language and thought itself.
In this paper we discuss some of the background to the notion of active information introduced by... more In this paper we discuss some of the background to the notion of active information introduced by Bohm and Hiley to account for quantum processes. To appreciate the full significance of this new notion, we show why it is essential to distinguish our approach from the approach that goes under the name 'Bohmian mechanics'. We then show for the first time how the quantum potential emerges from the Heisenberg picture thus providing a new perspective to the whole approach. This enables us to clarify the role of the energy associated with the quantum potential and the status of active information. We conclude with some remarks on the relation between active information and Shannon information.
In this paper we show the deep connection between the Wigner-Moyal approach and the Bohm approach... more In this paper we show the deep connection between the Wigner-Moyal approach and the Bohm approach to quantum mechanics. We point out that the key equations used in the Bohm approach were al-ready contained in Moyal's classic 1949 paper. Furthermore we argue that these two approaches can be seen as different but related aspects of standard quantum formalism when the algebraic approach, rather than the Hilbert space approach is taken as primary. This leads us to consider the non-commutative Moyal *algebra in which two key equa-tions obtained are shown to be the analogue of algebraic equations de-rived by Brown and Hiley using the algebraic approach. The different phase spaces exploited in each case appear as necessary consequences of the non-commutative algebra since they are but shadow manifolds already implicit in the algebra defining the quantum formalism.
In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's starting point, the c... more In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's starting point, the characteristic function $M(\tau,\theta)=\int \psi^{*}(x)e^{i(\tau {\hat p}+\theta{\hat x})}\psi(x)dx$, is essentially the primitive idempotent used by von Neumann in his classic paper "Die Eindeutigkeit der Schr\"odingerschen Operatoren". This paper provides the original proof of the Stone-von Neumann equation. Thus the mathematical structure Moyal develops is simply a re-expression of what is at the heart of quantum mechanics and reproduces exactly the results of the quantum formalism. The "distribution function" $F(X,P,t)$ is simply the quantum mechanical density matrix expressed in an $( X,P)$-representation, where $X$ and $P$ are the mean co-ordinates of a cell structure in phase space. The whole approach therefore clearly has little to do with classical statistical theories but is a consequence of a non-commutative nature of the theory.
We analyse the track of an {\alpha}-particle passing through a cloud chamber using the Bohm theor... more We analyse the track of an {\alpha}-particle passing through a cloud chamber using the Bohm theory and show that the resulting classical track has its origins in the quantum Zeno effect. By assuming the ionised gas molecules reveal the positions of the {\alpha}-particle on its trajectory and using these positions in a short time propagator technique developed by de Gosson, we show it is the failure of the quantum potential to develop quickly enough that leads to the appearance of the classical trajectory. Bohm and Hiley have already argued that it is this failure of the quantum potential to develop appropriately that prevents an Auger electron from undergoing a transition if continuously watched. This allows us to conclude that, in general, it is the suppression of the quantum potential that accounts for the quantum Zeno effect.
In this paper we approach the question of the existence of a (x, p) phase space in a new way. Rat... more In this paper we approach the question of the existence of a (x, p) phase space in a new way. Rather than abandoning all hope of constructing such a phase-space for quantum phenomena, we take aspects from both the Wigner-Moyal and Bohm approaches and show that although there is no unique phase space, we can form `shadow' phase spaces. We then argue that this is a consequence of the non-commutative geometry defined by the operator algebra.
The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerou... more The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerous than is usually believed. We show that the Schrödinger equation for a nonrelativistic spinless particle is a classical equation which is equivalent to Hamilton's equations. Our discussion is quite general, and incorporates time-dependent systems. This gives us the opportunity of discussing the group of Hamiltonian canonical transformations which is a non-linear variant of the usual symplectic group.
We introduce the notion of an extended moment in time, the duron. This is a region of temporal am... more We introduce the notion of an extended moment in time, the duron. This is a region of temporal ambiguity which arises naturally in the nature of process which we take to be basic. We introduce an algebra of process and show how it is related to, but different from, the monoidal category introduced by Abramsky and Coecke. By considering the limit as the duration of the moment approaches the infinitesimal, we obtain a pair of dynamical equations, one expressed in terms of a commutator and the other which is expressed in terms of an anti-commutator. These two coupled real equations are equivalent to the Schrödinger equation and its dual. We then construct a bi-algebra, which allows us to make contact with the thermal quantum field theory introduced by Umezawa. This allows us to link quantum mechanics with thermodynamics. This approach leads to two types of time, one is Schrödinger time, the other is an irreversible time that can be associated with a movement between inequivalent vacuum states. Finally we discuss the relation between our process algebra and the thermodynamic origin of time.
We begin by giving correct expressions for the short-time action; following the work of one of us... more We begin by giving correct expressions for the short-time action; following the work of one of us and Makri-Miller. We use these estimates to derive a correct expression modulo ∆t 2 for the quantum propagator and we show that the quantum potential is negligible modulo ∆t 2 for a point source. We finally prove that this implies that the quantum motion is classical for very short times.
Mainstream cognitive neuroscience typically ignores the role of quantum physical effects in the n... more Mainstream cognitive neuroscience typically ignores the role of quantum physical effects in the neural processes underlying cognition and consciousness. However, many unsolved problems remain, suggesting the need to consider new approaches. We propose that quantum theory, especially through an ontological interpretation due to Bohm and Hiley, provides a fruitful framework for addressing the neural correlates of cognition and consciousness. In particular, the ontological interpretation suggests that a novel type of “active information”, connected with a novel type of “quantum potential energy”, plays a key role in quantum physical processes. After introducing the ontological interpretation we illustrate its value for cognitive neuroscience bydiscussing it in the light of a proposal by Beck and Eccles about how quantum tunneling could play a role in controlling the frequency of synaptic exocytosis. In this proposal, quantum tunneling would enable the “self” to control its brain withou...
In this paper we start from a basic notion of process, which we structure into two groupoids, one... more In this paper we start from a basic notion of process, which we structure into two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford algebras respectively. We show how the orthogonal Clifford algebra, which include the Schrödinger, Pauli and Dirac formalisms, describe the classical light-cone structure of space-time, as well as providing a basis for the description of quantum phenomena. By constructing an orthogonal Clifford bundle with a Dirac connection, we make contact with quantum mechanics through the Bohm formalism which emerges quite naturally from the connection, showing that it is a structural feature of the mathematics. We then generalise the approach to include the symplectic Clifford algebra, which leads us to a non-commutative geometry with projections onto shadow manifolds. These shadow manifolds are none other than examples of the phase space constructed by Bohm. We also argue that this provides us with a mathematical structure that fits the implicateexplicate order proposed by Bohm.
A method for measuring the weak value of spin for atoms is proposed using a variant of the origin... more A method for measuring the weak value of spin for atoms is proposed using a variant of the original Stern–Gerlach apparatus. A full simulation of an experiment for observing the real part of the weak value using the impulsive approximation has been carried out. Our predictions show a displacement of the beam of helium atoms in the metastable 23S1 state, Δw, that is within the resolution of conventional microchannel plate detectors indicating that this type of experiment is feasible. Our analysis also determines the experimental parameters that will give an accurate determination of the weak value of spin. Preliminary experimental results are shown for helium, neon and argon in the 23S1 and 3P2 metastable states, respectively.
The claim of Kocsis et al. to have experimentally determined “photon trajectories” calls for a re... more The claim of Kocsis et al. to have experimentally determined “photon trajectories” calls for a re-examination of the meaning of “quantum trajectories”. We will review the arguments that have been assumed to have established that a trajectory has no meaning in the context of quantum mechanics. We show that the conclusion that the Bohm trajectories should be called “surreal” because they are at “variance with the actual observed track” of a particle is wrong as it is based on a false argument. We also present the results of a numerical investigation of a double Stern-Gerlach experiment which shows clearly the role of the spin within the Bohm formalism and discuss situations where the appearance of the quantum potential is open to direct experimental exploration.
A method for measuring the real part of the weak value of spin for non-zero rest mass atoms is pr... more A method for measuring the real part of the weak value of spin for non-zero rest mass atoms is presented using a variant on the original Stern-Gerlach apparatus. The experiment utilises helium in the metastable 23S1 state. A full simulation for observing the real part of the weak value using the impulsive approximation has been carried out and it predicts a displacement of the beam, Δw, that is within the resolution of our detector. It also indicates how this shift might be increased. The full analysis also indicated that there is a limit, L, to the applicability of the weak value approximation and has been evaluated for our apparatus. This experiment has the possibility to be expanded to utilise other nobal gas species such as neon and argon in the 3P2 metastable state, but we shall restrict this paper to metastable helium only.
There has been a recent revival of interest in the notion of a `trajectory' of a quantum part... more There has been a recent revival of interest in the notion of a `trajectory' of a quantum particle. In this paper we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are able to conclude that a Bohm `trajectory' is the average of an ensemble of actual individual stochastic Feynman paths. This implies that they can be interpreted as the mean momentum flow of a set of individual quantum processes and not the path of an individual particle. This enables us to give a clearer account of the experimental two-slit results of Kocsis {\em et al.}}
The experimental results of Kocsis et al., Mahler et al. and the proposed experiments of Morley e... more The experimental results of Kocsis et al., Mahler et al. and the proposed experiments of Morley et al. show that it is possible to construct "trajectories" in interference regions in a two-slit interferometer. These results call for a theoretical re-appraisal of the notion of a "quantum trajectory" first introduced by Dirac and in the present paper we reexamine this notion from the Bohm perspective based on Hamiltonian flows. In particular, we examine the short-time propagator and the role that the quantum potential plays in determining the form of these trajectories. These trajectories differ from those produced in a typical particle tracker and the key to this difference lies in the active suppression of the quantum potential necessary to produce Mott-type trajectories. We show, using a rigorous mathematical argument, how the active suppression of this potential arises. Finally we discuss in detail how this suppression also accounts for the quantum Zeno effect.
It is argued that in order to address the mind/matter relationship, we will have to radically cha... more It is argued that in order to address the mind/matter relationship, we will have to radically change the conceptual structure normally assumed in physics. Rather than fields and/or particles-in-interaction described in the traditional Cartesian order based a local evolution in spacetime, we need to introduce a more general notion of process described by a non-commutative algebra. This will have radical implications for both for physical processes and for geometry. By showing how the Bohm interpretation of quantum mechanics can be understood within a noncommutative structure, we can give a much clearer meaning to the implicate order introduced by Bohm. It is through this implicate order that mind and matter can be seen as different aspects of the same general process.
In this paper an account of Bohm's own attitude to his early papers on an alternative interpretat... more In this paper an account of Bohm's own attitude to his early papers on an alternative interpretation to quantum theory is presented. He has made it clear that it was never his intention to return to a deterministic mechanical theory, rather it was an attempt to show that an alternative was possible and, in itself, may provide clues for further developments. From the earliest he felt that a more radical approach was needed, an approach that depended on his notion of structure process. In such an approach the particle must not be regarded as some a priori given immutable entity, but must take its dynamical properties from the environment in which it finds itself so that the notion of wholeness, so important to both Bohr and Bohm, could be a fundamental part of the description. It will be argued that the essential feature of his later ideas of the implicate order were already present in his early work in "Causality and Chance" written shortly after his original proposals.
We reexamine the claim made by Englert, Scully, Süssman and Walther that in certain 'Welcher Weg'... more We reexamine the claim made by Englert, Scully, Süssman and Walther that in certain 'Welcher Weg' (Which Way) interference experiments, the Bohm trajectories behave in such a bizarre and unacceptable way that they must be considered as unreliable and even 'surreal'. We show that this claim cannot be correct and is based on an incorrect use of the Bohm approach.
In this paper we discuss the relevance of the algebraic approach to quantum phenomena first intro... more In this paper we discuss the relevance of the algebraic approach to quantum phenomena first introduced by von Neumann before he confessed to Birkoff that he no longer believed in Hilbert space. This approach is more general and allows us to see the structure of quantum processes in terms of non-commutative probability theory, a non-Boolean structure of the implicate order which contains Boolean substructures which accommodates the explicate classical world. We move away from mechanical 'waves' and 'particles' and take as basic what Bohm called a structure process. This enables us to learn new lessons that can have a wider application in the way we think of structures in language and thought itself.
In this paper we discuss some of the background to the notion of active information introduced by... more In this paper we discuss some of the background to the notion of active information introduced by Bohm and Hiley to account for quantum processes. To appreciate the full significance of this new notion, we show why it is essential to distinguish our approach from the approach that goes under the name 'Bohmian mechanics'. We then show for the first time how the quantum potential emerges from the Heisenberg picture thus providing a new perspective to the whole approach. This enables us to clarify the role of the energy associated with the quantum potential and the status of active information. We conclude with some remarks on the relation between active information and Shannon information.
In this paper we show the deep connection between the Wigner-Moyal approach and the Bohm approach... more In this paper we show the deep connection between the Wigner-Moyal approach and the Bohm approach to quantum mechanics. We point out that the key equations used in the Bohm approach were al-ready contained in Moyal's classic 1949 paper. Furthermore we argue that these two approaches can be seen as different but related aspects of standard quantum formalism when the algebraic approach, rather than the Hilbert space approach is taken as primary. This leads us to consider the non-commutative Moyal *algebra in which two key equa-tions obtained are shown to be the analogue of algebraic equations de-rived by Brown and Hiley using the algebraic approach. The different phase spaces exploited in each case appear as necessary consequences of the non-commutative algebra since they are but shadow manifolds already implicit in the algebra defining the quantum formalism.
In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's starting point, the c... more In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's starting point, the characteristic function $M(\tau,\theta)=\int \psi^{*}(x)e^{i(\tau {\hat p}+\theta{\hat x})}\psi(x)dx$, is essentially the primitive idempotent used by von Neumann in his classic paper "Die Eindeutigkeit der Schr\"odingerschen Operatoren". This paper provides the original proof of the Stone-von Neumann equation. Thus the mathematical structure Moyal develops is simply a re-expression of what is at the heart of quantum mechanics and reproduces exactly the results of the quantum formalism. The "distribution function" $F(X,P,t)$ is simply the quantum mechanical density matrix expressed in an $( X,P)$-representation, where $X$ and $P$ are the mean co-ordinates of a cell structure in phase space. The whole approach therefore clearly has little to do with classical statistical theories but is a consequence of a non-commutative nature of the theory.
We analyse the track of an {\alpha}-particle passing through a cloud chamber using the Bohm theor... more We analyse the track of an {\alpha}-particle passing through a cloud chamber using the Bohm theory and show that the resulting classical track has its origins in the quantum Zeno effect. By assuming the ionised gas molecules reveal the positions of the {\alpha}-particle on its trajectory and using these positions in a short time propagator technique developed by de Gosson, we show it is the failure of the quantum potential to develop quickly enough that leads to the appearance of the classical trajectory. Bohm and Hiley have already argued that it is this failure of the quantum potential to develop appropriately that prevents an Auger electron from undergoing a transition if continuously watched. This allows us to conclude that, in general, it is the suppression of the quantum potential that accounts for the quantum Zeno effect.
In this paper we approach the question of the existence of a (x, p) phase space in a new way. Rat... more In this paper we approach the question of the existence of a (x, p) phase space in a new way. Rather than abandoning all hope of constructing such a phase-space for quantum phenomena, we take aspects from both the Wigner-Moyal and Bohm approaches and show that although there is no unique phase space, we can form `shadow' phase spaces. We then argue that this is a consequence of the non-commutative geometry defined by the operator algebra.
The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerou... more The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerous than is usually believed. We show that the Schrödinger equation for a nonrelativistic spinless particle is a classical equation which is equivalent to Hamilton's equations. Our discussion is quite general, and incorporates time-dependent systems. This gives us the opportunity of discussing the group of Hamiltonian canonical transformations which is a non-linear variant of the usual symplectic group.
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Papers by Basil Hiley