We study here the properties of some quantum mechanical wave functions, which, in contrast to the regular quantum mechanical wave functions, can be predetermined with certainty (probability 1) by performing dense measurements (or continuous observations). These specific “certain” states are the junction points through which pass all the diverse paths that can proceed between each two such neighboring “sure” points. When we compare the properties of these points to the properties of the well-known universal wave functions of Everett we find (...) a strong similarity between these two apparently uncorrelated entities, and in this way find the same similarity between the Feynman path integrals and Everett's universal wave functions. (shrink)

The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman's path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a (...) simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman's derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other. (shrink)

This paper argues that the path integral formulation of quantum mechanics suggests a form of holism for which the whole (total ensemble of paths) has properties that are not strongly reducible to the properties of the parts (the single trajectories). Feynman’s sum over histories calculates the probability amplitude of a particle moving within a boundary by summing over all the possible trajectories that the particle can undertake. These trajectories and their individual probability amplitudes are thus necessary in (...) calculating the total amplitude. However, not all possible trajectories are differentiable, thus suggesting that they are not physical possibilities, but only mathematical entities. It follows that if the possible differentiable trajectories are taken to be part of the physical system, they are not sufficient to calculate the total probability amplitude. The conclusion is that the total ensemble is weakly non-supervenient upon the physically possible trajectories. (shrink)

Starting from the earlier notions of stationary action principles, these tutorial notes shows how Schwinger's Quantum Action Principle descended from Dirac's formulation, which independently led Feynman to his path-integral formulation of quantum mechanics. Part I brings out in more detail the connection between the two formulations, and applications are discussed. Then, the Keldysh-Schwinger time-cycle method of extracting matrix elements is described. Part II will discuss the variational formulation of quantum electrodynamics and the development of source theory.

The spin of a free electron is stable but its position is not. Recent quantum information research by G. Svetlichny, J. Tolar, and G. Chadzitaskos have shown that the Feynman position path integral can be mathematically defined as a product of incompatible states; that is, as a product of mutually unbiased bases (MUBs). Since the more common use of MUBs is in finite dimensional Hilbert spaces, this raises the question “what happens when spin path integrals are (...) computed over products of MUBs?” Such an assumption makes spin no longer stable. We show that the usual spin-1/2 is obtained in the long-time limit in three orthogonal solutions that we associate with the three elementary particle generations. We give applications to the masses of the elementary leptons. (shrink)

The (Feynman) propagator G(x2,x1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(x_2,x_1)$$\end{document} encodes the entire dynamics of a massive, free scalar field propagating in an arbitrary curved spacetime. The usual procedures for computing the propagator—either as a time ordered correlator or from a partition function defined through a path integral—requires introduction of a field ϕ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (x)$$\end{document} and its action functional A[ϕ(x)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...) \setlength{\oddsidemargin}{-69pt} \begin{document}$$A[\phi (x)]$$\end{document}. An alternative, more geometrical, procedure is to define a propagator in terms of the world-line path integral which only uses curves, xi(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^i(s)$$\end{document}, defined on the manifold. I show how the world-line path integral can be reinterpreted as an ordinary integral by introducing the concept of effective number of quantum paths of a given length. Several manipulations of the world-line path integral becomes algebraically tractable in this approach. In particular I derive an explicit expression for the propagator GQG(x2,x1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_\mathrm{QG}(x_2,x_1)$$\end{document}, which incorporates the quantum structure of spacetime through a zero-point-length, in terms of the standard propagator Gstd(x2,x1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_\mathrm{std}(x_2,x_1)$$\end{document}, in an arbitrary curved spacetime. This approach also helps to clarify the interplay between the path integral amplitude and the path integral measure in determining the form of the propagator. This is illustrated with several explicit examples. (shrink)

The proposal made 50 years ago by Schulman :1558–1569, 1968), Laidlaw and Morette-DeWitt :1375–1378, 1971) and Dowker to decompose the propagator according to the homotopy classes of paths was a major breakthrough: it showed how Feynman functional integrals opened a direct window on quantum properties of topological origin in the configuration space. This paper casts a critical look at the arguments brought by this series of papers and its numerous followers in an attempt to clarify the reason why the (...) emergence of the unitary linear representation of the first homotopy group is not only sufficient but also necessary. (shrink)

The concept of “transaction,” introduced by Cramer in his realistic nonlocal interpretation of quantum mechanics (QM), is herein reformulated in the language of the Feynman graphs' technique.

The way from the path integral to Feynman diagrams is sketched. The emphasis is put on the decrease of complexity in this process, from infinite-dimensional integrals down to the apparent simplicity of child’s play. On the other hand, also the subsequent increase in complexity when using Feynman diagrams to make realistic physical predictions is described, thus illustrating the dialectic between the simplicity and clarity of Feynman diagrams, and the complexity in their practical applications.

Despite the importance of the variational principles of physics, there have been relatively few attempts to consider them for a realistic framework. In addition to the old teleological question, this paper continues the recent discussion regarding the modal involvement of the principle of least action and its relations with the Humean view of the laws of nature. The reality of possible paths in the principle of least action is examined from the perspectives of the contemporary metaphysics of modality and Leibniz's (...) concept of essences or possibles striving for existence. I elaborate a modal interpretation of the principle of least action that replaces a classical representation of a system's motion along a single history in the actual modality by simultaneous motions along an infinite set of all possible histories in the possible modality. This model is based on an intuition that deep ontological connections exist between the possible paths in the principle of least action and possible quantum histories in the Feynman path integral. I interpret the action as a physical measure of the essence of every possible history. Therefore only one actual history has the highest degree of the essence and minimal action. To address the issue of necessity, I assume that the principle of least action has a general physical necessity and lies between the laws of motion with a limited physical necessity and certain laws with a metaphysical necessity. (shrink)

An appropriate kind of curved Hilbert space is developed in such a manner that it admits operators of $\mathcal{C}$ - and $\mathfrak{D}$ -differentiation, which are the analogues of the familiar covariant and D-differentiation available in a manifold. These tools are then employed to shed light on the space-time structure of Quantum Mechanics, from the points of view of the Feynman ‘path integral’ and of canonical quantisation. (The latter contains, as a special case, quantisation in arbitrary curvilinear coordinates (...) when space is flat.) The influence of curvature is emphasised throughout, with an illustration provided by the Aharonov-Bohm effect. (shrink)

The quasiclassical theory in terms of Feynman path integrals is used to calculate the decay of the Cooperon amplitude caused by transverse gauge field fluctuations in a disordered electron system. It is found that the phase relaxation rate in two dimensions varies linearly with the temperature as in the more common case of electric field fluctuations, but is proportional to the conductance rather than the resistance. A logarithmic correction factor is found in comparison to an earlier qualitative estimate.

Nonrelativistic quantum mechanics can be derived from real Markov diffusion processes by extending the concept of probability measure to the complex domain. This appears as the only natural way of introducing formally classical probabilistic concepts into quantum mechanics. To every quantum state there is a corresponding complex Fokker-Planck equation. The particle drift is conditioned by an auxiliary equation which is obtained through stochastic energy conservation; the logarithmic transform of this equation is the Schrödinger equation. To every quantum mechanical operator there (...) is a stochastic process; the replacement of operators by processes leads to all the well-known results of quantum mechanics, using stochastic calculus instead of formal quantum rules. Comparison is made with the classical stochastic approaches and the Feynman path integral formulation. (shrink)

We consider a theory in which spacetime is a 4-dimensional manifold V4 embedded in an N-dimensional space VN. The dynamics is given by a first-order action which is a straightforward generalization of the well-known Nambu-Gotto string action. Instead of the latter action we then consider an equivalent action, a generalization of the Howe-Tucker action, which is a functional of the (extrinsic) embedding variables ηa(x) and of the (intrinsic) induced metric gυv (x) on V4. In the quantized theory we can define (...) an effective action by means of the Feynman path integral in which we functionally integrate over the embedding variables. What remains is functionally dependent solely on the induced metric. It is well known that the effective action so obtained contains the Ricci scalar R and its higher orders. But due to our special choice of a quantity, the so-called “matter” density ω(η) in VN entering the original first-order action, it turns out that the effective action contains also the source term. The latter is in general that of a p-dimensional membrane (p-brane). In particular we consider the case of bosonic point particles. Finally we discuss and clarify certain interpretational aspects of quantum mechanics from the viewpoint of our embedding model. (shrink)

We first define a class of processes which we call regular quantum Markov processes. We next prove some basic results concerning such processes. A method is given for constructing quantum Markov processes using transition amplitude kernels. Finally we show that the Feynman path integral formalism can be clarified by approximating it with a quantum stochastic process.

We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. [Formula: see text] as “almost finite”, while the latter allows one to treat uncountable sets like e.g. [Formula: see text] (...) as “almost countable”. This reduction of the uncountable to the finite/countable turns out to have a considerable logical and computational cost: we show that the aforementioned lemmas, and many related theorems, are extremely hard to prove, while the associated sub-covers are extremely hard to compute. Indeed, in terms of the standard scale (based on comprehension axioms), a proof of these lemmas requires at least the full extent of second-order arithmetic, a system originating from Hilbert–Bernays’ Grundlagen der Mathematik. This observation has far-reaching implications for the Grundlagen’s spiritual successor, the program of Reverse Mathematics, and the associated Gödel hierarchy. We also show that the Cousin lemma is essential for the development of the gauge integral, a generalization of the Lebesgue and improper Riemann integrals that also uniquely provides a direct formalization of Feynman’s path integral. (shrink)

Natural modalities are often analysed from an abstract point of view where they are associated with putative laws of nature. However, the way possibilities are represented in physics is more complex. Lagrangian mechanics, for instance, involves two different layers of modalities: kinematical and dynamical possibilities. This paper examines the status of these two layers, both in the classical and quantum case. The quantum case is particularly problematic: we identify four possible interpretive options. The upshot is that a close inspection of (...) the way possibilities are represented in physics could lead to new ways of thinking about natural modalities. (shrink)

The relationship between the canonical operator and the path integral formulation of quantum electrodynamics is analyzed with a particular focus on the implementation of gauge constraints in the two approaches. The removal of gauge volumes in the path integral is shown to match with the presence of zero-norm ghost states associated with gauge transformations in the canonical operator approach. The path integrals for QED in both the Feynman and the temporal gauges are examined and (...) several ways of implementing the gauge constraint integrations are demonstrated. The upshot is to show that both the Feynman and the temporal gauge path integrals are equivalent to the Coulomb gauge path integral, matching the results developed by Kurt Haller using the canonical formalism. In addition, the Faddeev–Popov form for the Feynman gauge and temporal gauge Lagrangian path integrals are derived from the Hamiltonian form of the path integral. (shrink)

Cognitive decisions are best described by quantum mathematics. Do quantum information devices operate in the brain? What would they look like? Fuss and Navarro () describe quantum lattice registers in which quantum superpositioned pathways interact (compute/integrate) as ‘quantum walks’ akin to Feynman's path integral in a lattice (e.g. the ‘Feynman quantum chessboard’). Simultaneous alternate pathways eventually reduce (collapse), selecting one particular pathway in a cognitive decision, or choice. This paper describes how quantum walks in a (...) class='Hi'>Feynman chessboard are conceptually identical to ‘topological qubits’ in brain neuronal microtubules, as described in the Penrose-Hameroff 'Orch OR' theory of consciousness. (shrink)

Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such “crown jewels” of mathematical economics as the existence of general equilibrium and the second welfare theorem. He (...) also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser [51] in his extension of Gödel’s incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him. (shrink)

Here, in our approximation of polaron theory, we examine the importance of introducing theT product, which turn out to be a very convenient theoretical approach for the calculation of thermodynamical averages.We focus attention on the investigation of the so-called linear polaron Hamiltonian and present in detail the calculation of the correlation function, spectral function, and Green function for such a linear system.It is shown that the linear polaron Hamiltonian provides an exactly solvable model of our system, and the result obtained (...) with this approach holds true for an arbitrary coupling constant which describes the strength of interaction between the electron and the lattice vibrations. Then, with the help of a variational technique, we show the possibility of reducing the real polaron Hamiltonian to a socalled trial or approximate linear model Hamiltonian.We also consider the exact calculation of free energy with a special technique that reduces calculations with the help of the T product, which, in our opinion, works much better and is easier than other analogous considerations, for example, the path-integral or Feynman-integral method.(1,2)Here we furthermore recall our own work,(4) where it was shown that the results of Refs. 7 and 8 concerning the impedance calculation in the polaron model may be obtained directly without the use of the path-integral method.The study of the polaron system's thermodynamics is carried out by us in the framework of the functional method. A calculation of the free energy and the momentum distribution function is proposed.Note also that the polaron systems with strong coupling(9) proved to be useful in different quantum field models in connection with the construction of dynamical models of composite particles. A rigorous solution of the special strong-coupling polaron problem, describing the interaction of a nonrelativistic particle with a quantum field, was given by Bogolubov.(3) The works of Tavkhelidze, Fedyanin, Khrustalev, and others(10–13) are dedicated to the further development and generalization of the Bogolubov method.Notice, too, that the electron-photon interaction effects play an important part in many problems of modern solid state theory (see, e.g., Refs. 7 and 14–19).The present paper summarizes a set of lectures delivered as a special course in the physics department of Moscow State University. (shrink)

A mapping of a finite directed graph onto a curve in space-time is considered. The mapping induces the dynamics of a free particle moving along the curve. The distinction between the Lagrangian and the Hamiltonian formulation of particle mechanics is expressed in terms of the distinction between referring to a particle in space and time and referring to the points in space which the particle occupies, respectively. These elements are combined to yield an interpretation of Feynman's path (...) class='Hi'>integral formulation of quantum mechanics. Describing a bound state of a system as a particle is discussed. (shrink)

In the traditional form of canonical quantization, certain field components (not having “conjugate” momenta) must be regarded as noncanonical. This long-known distinction enters modern gauge theories, when they are canonically quantized as by Kugo and Ojima. We avoid that peculiarity by not using any conjugate “momenta” at all. In our formulation, canonical quantization can be related to Feynman's path integral.

Twenty extremal principles of the natural sciences are reformulated to the general ontological scheme. The hypothesis is substantiated that the unique role of the principle of least action is based on its probabilistic interpretation. It is shown how most of the variational principles can be reduced to the principle of maximal probability, which is based on a realistic interpretation of Feynman’s path integral method.

Decoherence is the name for the complex of phenomena leading to appearance of classical features of quantum systems. In the present paper decoherence in continuous measurements is analyzed with the help of restricted path integrals (RPI) and (equivalently in simple cases) complex Hamiltonians. A continuous measurement results in a readout giving information in the classical form on the evolution of the measured quantum system. The quantum features of the system reveal themselves in the variation of possible measurement readouts. For (...) example, the monitoring energy of a multi-level system may visualize a transition between levels as a process evolving in time but with an unavoidable quantum noise. Decoherence of a continuously measured system is completely determined by the measurement readout, i.e., by the information recorded in its environment. It is shown that the ideology of RPI makes the Feynman version of quantum mechanics closed, contrary to the conventional operator form of quantum mechanics which needs quantum theory of measurement as a necessary additional part. (shrink)

This paper discusses some of the technical problems related to a Weylian geometrical interpretation of the Schrödinger and Klein-Gordon equations proposed by E. Santamato. Solutions to these technical problems are proposed. A general prescription for finding out the interdependence between a particle's effective mass and Weyl's scalar curvature is presented which leads to the fundamental equation of geometric quantum mechanics, $$m(R)\frac{{dm(R)}}{{dR}} = \frac{{\hbar ^2 }}{{c^2 }}$$ The Dirac equation is rigorously derived within this formulation, and further problems to be solved (...) are proposed in the conclusion. The main one is based on obtaining the relationship between Feynman's path integral quantization method, among others, and the methods of geometric quantum mechanics. The solution of this problem will be a crucial test for this theory that attempts to “geometrize” quantum mechanics rather than the conventional approach in the past of quantizing geometry. A numerical prediction of this theory yields a 3×10−35 eV correction to the ground-state energy of the hydrogen atom. (shrink)

Although the path-integral formalism is known to be equivalent to conventional quantum mechanics, it is not generally obvious how to implement path-based calculations for multi-qubit entangled states. Whether one takes the formal view of entangled states as entities in a high-dimensional Hilbert space, or the intuitive view of these states as a connection between distant spatial configurations, it may not even be obvious that a path-based calculation can be achieved using only paths in ordinary space and (...) time. Previous work has shown how to do this for certain special states; this paper extends those results to all pure two-qubit states, where each qubit can be measured in an arbitrary basis. Certain three-qubit states are also developed, and path integrals again reproduce the usual correlations. These results should allow for a substantial amount of conventional quantum analysis to be translated over into a path-integral perspective, simplifying certain calculations, and more generally informing research in quantum foundations. (shrink)

The action for a massive particle in special relativity can be expressed as the invariant proper length between the end points. In principle, one should be able to construct the quantum theory for such a system by the path integral approach using this action. On the other hand, it is well known that the dynamics of a free, relativistic, spinless massive particle is best described by a scalar field which is equivalent to an infinite number of harmonic oscillators. (...) We clarify the connection between these two—apparently dissimilar—approaches by obtaining the Green function for the system of oscillators from that of the relativistic particle. This is achieved through defining the path integral for a relativistic particle rigorously by two separate approaches. This analysis also shows a connection between square root Lagrangians and the system of harmonic oscillators which is likely to be of value in more general context. (shrink)

A comparison is made between the Bohm trajectory and Feynman path approaches to the long-standing problem of determining the average lime taken for a particle described by the Schrödinger wave function ψ to tunnel through a potential barrier. The former approach follows simply and uniquely from the basic postulates of Bohm's causal interpretation of quantum mechanics; the latter is intimately related to the most frequently cited approaches based on conventional interpretations. Emphasis is given to the fact that fundamentally (...) different transmission (T)-reflection (R) decompositions, particlelike and wavelike respectively, are central to the two methods: ¦ψ¦2=[¦ψ¦2]T+[¦ψ¦2]R (Bohm trajectory approach); ψ=ψT+ψR (Feynman path approach). (shrink)

Barut's classical model of the spinning particle having external dynamical variables x and p and internal dynamical variables $\bar z$ and z is taken into account. The path integrations over holomorphic spinors $\bar z$ and z are discussed. This quantization gives the kernel of the relativistic particles with higher spin as well as the Dirac electron. The Green's function of the spin-n/2 particle is obtained.

Williams (WS) and Down (DS) syndromes are neurodevelopmental disorders with distinct genetic origins and different spatial memory profiles. In real-world spatial memory tasks, where spatial information derived from all sensory modalities is available, individuals with DS demonstrate low-resolution spatial learning capacities consistent with their mental age, whereas individuals with WS are severely impaired. However, because WS is associated with severe visuo-constructive processing deficits, it is unclear whether their impairment is due to abnormal visual processing or whether it reflects an inability (...) to build a cognitive map. Here, we tested whether blindfolded individuals with WS or DS, and typically developing (TD) children with similar mental ages, could use path integration to perform an egocentric homing task and return to a starting point. We then evaluated whether they could take shortcuts and navigate along never-traveled trajectories between four objects while blindfolded, thus demonstrating the ability to build a cognitive map. In the homing task, 96% of TD children, 84% of participants with DS and 44% of participants with WS were able to use path integration to return to their starting point consistently. In the cognitive mapping task, 64% of TD children and 74% of participants with DS were able to take shortcuts and use never-traveled trajectories, the hallmark of cognitive mapping ability. In contrast, only one of eighteen participants with WS demonstrated the ability to build a cognitive map. These findings are consistent with the view that hippocampus-dependent spatial learning is severely impacted in WS, whereas it is relatively preserved in DS. (shrink)

The normalization in the path integral approach to quantum field theory, in contrast with statistical field theory, can contain physical information. The main claim of this paper is that the inner product on the space of field configurations, one of the fundamental pieces of data required to be added to quantize a classical field theory, determines the normalization of the path integral. In fact, dimensional analysis shows that the introduction of this structure necessarily introduces a scale (...) that is left unfixed by the classical theory. We study the dependence of the theory on this scale. This allows us to explore mechanisms that can be used to fix the normalization based on cutting and gluing different integrals. “Self-normalizing” path integrals, those independent of the scale, play an important role in this process. Furthermore, we show that the scale dependence encodes other important physical data: we use it to give a conceptually clear derivation of the chiral anomaly. Several explicit examples, including the scalar and compact bosons in different geometries, supplement our discussion. (shrink)

The essence of the path integral method in quantum physics can be expressed in terms of two relations between unitary propagators, describing perturbations of the underlying system. They inherit the causal structure of the theory and its invariance properties under variations of the action. These relations determine a dynamical algebra of bounded operators which encodes all properties of the corresponding quantum theory. This novel approach is applied to non-relativistic particles, where quantum mechanics emerges from it. The method works (...) also in interacting quantum field theories and sheds new light on the foundations of quantum physics. (shrink)

We summarize the essential ingredients, which enabled us to derive the path-integral for a system of harmonically interacting spin-polarized identical particles in a parabolic confining potential, including both the statistics (Bose–Einstein or Fermi–Dirac) and the harmonic interaction between the particles. This quadratic model, giving rise to repetitive Gaussian integrals, allows to derive an analytical expression for the generating function of the partition function. The calculation of this generating function circumvents the constraints on the summation over the cycles of (...) the permutation group. Moreover, it allows one to calculate the canonical partition function recursively for the system with harmonic two-body interactions. Also, static one-point and two-point correlation functions can be obtained using the same technique, which make the model a powerful trial system for further variational treatments of realistic interactions. (shrink)

From the very beginning, coherent state path integrals have always relied on a coherent state resolution of unity for their construction. By choosing an inadmissible fiducial vector, a set of “coherent states” spans the same space but loses its resolution of unity, and for that reason has been called a set of weak coherent states. Despite having no resolution of unity, it is nevertheless shown how the propagator in such a basis may admit a phase-space path integral (...) representation in essentially the same form as if it had a resolution of unity. Our examples are toy models of similar situations that arise in current studies of quantum gravity. (shrink)

In this paper, I explore the feasibility of a realistic interpretation of the quantum mechanical path integral - that is, an interpretation according to which the particle actually follows the paths that contribute to the integral. I argue that an interpretation of this sort requires spacetime to have a branching structure similar to the structures of the branching spacetimes proposed by previous authors. I point out one possible way to construct branching spacetimes of the required sort, and (...) I ask whether the resulting interpretation of quantum mechanics is empirically testable. (shrink)

The central limit theorem has been found to apply to random vectors in complex Hilbert space. This amounts to sufficient reason to study the complex–valued Gaussian, looking for relevance to quantum mechanics. Here we show that the Gaussian, with all terms fully complex, acting as a propagator, leads to Schrödinger’s non-relativistic equation including scalar and vector potentials, assuming only that the norm is conserved. No physical laws need to be postulated a priori. It thereby presents as a process of irregular (...) motion analogous to the real random walk but executed under the rules of the complex number system. There is a standard view that Schrödinger’s equation is deterministic, whereas wavefunction “collapse” is probabilistic —we have now a demonstrated linkage to the central limit theorem, indicating a stochastic picture at the foundation of Schrödinger’s equation itself. It may be an example of Wheeler’s “It from bit” with “No underlying law”. Reasons for the primary role of \ are open to discussion. The present derivation is compared with recent reconstructions of the quantum formalism, which have the aim of rationalizing its obscurities. (shrink)

In the transdisciplinary field of ‘mindfulness,’ originally a Buddhist concept (Pāli sati; Sanskrit smṛti; Chinese nian 念; Tibetan dran pa), the two tendencies represented by Buddhist traditional a...

The path integral is not typically utilized for analyzing entanglement experiments, in part because there is no standard toolbox for converting an arbitrary experiment into a form allowing a simple sum-over-history calculation. After completing the last portion of this toolbox (a technique for implementing multi-particle measurements in an entangled basis), some interesting 4- and 6-particle experiments are analyzed with this alternate technique. While the joint probabilities of measurement outcomes are always equivalent to conventional quantum mechanics, differences in the (...) calculations motivate a number of foundational insights, concerning nonlocality, retrocausality, and the objectivity of entanglement itself. (shrink)

For time-independent fields the Aharonov-Bohm effect has been obtained by idealizing the coordinate space as multiply-connected and using representations of its fundamental homotopy group to provide information on what is physically identified as the magnetic flux. With a time-dependent field, multiple-connectedness introduces the same degree of ambiguity; by taking into account electromagnetic fields induced by the time dependence, full physical behavior is again recovered once a representation is selected. The selection depends on a single arbitrary time (hence the so-called holonomies (...) are not unique), although no physical effects depend on the value of that particular time. These features can also be phrased in terms of the selection of self-adjoint extensions, thereby involving yet another question that has come up in this context, namely, boundary conditions for the wave function. (shrink)

Hamilton-Jacobi theory is applied to find appropriate canonical transformations for the calculation of the phase-space path integrals of the relativistic particle equations. Hence, canonical transformations and Hamilton-Jacobi theory are also introduced into relativistic quantum mechanics. Moreover, from the classical physics viewpoint, it is very interesting to find and to solve the Hamilton-Jacobi equations for the relativistic particle equations.

In an accompanying paper Gomes, we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe M, existing on timeless configuration space \ of some field over M. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. Locality in full can only emerge dynamically, and is not postulated. This new understanding of locality is based solely on (...) the properties of extremal paths in configuration space. I do not demand locality from the start, as it is usually done, but showed conditions under which certain systems exhibit it spontaneously. In this way we recover semi-classical local behavior when regions dynamically decouple from each other, a notion more appropriate for extension into quantum mechanics. The dynamics of a sub-region O within the closed manifold M is independent of its complement, \, if the projection of extremal curves on \ onto the space of extremal curves intrinsic to O is a surjective map. This roughly corresponds to \, where \ is a linear projection. This criterion for locality can be made approximate—an impossible feat had it been already postulated—and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are mutually independent according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is the ultimate aim of a definition of locality. (shrink)

We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be therefore held responsible for the emergence at larger scales of various symmetries. For illustration (...) we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed. (shrink)

The unified treatment of the Dirac monopole, the Schwinger monopole, and the Aharonov-Bohm problem by Barut and Wilson is revisited via a path integral approach. The Kustaanheimo-Stiefel transformation of space and time is utilized to calculate the path integral for a charged particle in the singular vector potential. In the process of dimensional reduction, a topological charge quantization rule is derived, which contains Dirac's quantization condition as a special case. “Everything that is made beautiful and fair (...) and lovely is made for the eye of one who sees.” Jelaluddin Rumi,Mathnawi [I, 2383]. (shrink)

Recent studies of the contribution made by the hippocampus to spatial behavior suggest that it plays a role in integrating and double integrating distance and direction information using cues generated by self-movement. This and other evidence that the hippocampus plays a central role in spatial behavior seems inconsistent with proposals that it is primarily involved in episodic memory.