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. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:An Integrative Habit of Mind: John Henry Newman on the Path to Wisdom by Frederick D. AquinoDavid FleischackerAn Integrative Habit of Mind: John Henry Newman on the Path to Wisdom. By Frederick D. Aquino. DeKalb, Ill.: Northern Illinois University Press, 2012. Pp. x + 129. $29.00 (cloth). ISBN: 978-0-87580-452-1.Frederick Aquino has spent a number of years digesting Newman’s thought and interfacing it with a number of (...) facets of modern epistemology that cover both the subjective and social conditions of knowledge. In this volume he focuses on qualities that lead to an integrative habit of mind specific to a pluralistic world filled with seemingly incommensurable horizons. Aquino [End Page 481] thinks that this integrative habit is the key to how we can live together in such a pluralistic context.The introduction identifies the meaning of an integrative habit of mind. It requires informed judgment and a growing understanding of the unity of things. To be clear, Aquino does not want to present the outlines of a perfected mind, one characterized by a particular set of judgments and beliefs. Rather, his point is that the human mind is always developing, and he is seeking what promotes that ongoing development in a pluralistic context. He gleans and develops these characteristics from Newman, whose lifelong work as an educator and pastor provides a fertile field of thought on this topic. Aquino dwells on three texts written by Newman: the University Sermons, The Idea of a University, and An Essay in Aid of a Grammar of Assent. These three texts especially are rich in describing the interior and social conditions of the development of the human intellect.The first chapter examines a wide umbrella of elements that fall under the heading of the personal and social conditions required of an integrative habit of mind. It starts with a need to attend to the particular way that a person has developed within a tradition because from that tradition the mind pushes forth to broaden its horizon within a fruitful matrix of interlocutors. Aquino describes this path as a sacramentally embodied integration of the mind. His formulation of the sacramental highlights how this knowledge of the world is a pointer to reality, but not the same as reality. Knowledge here is a kind of allegory.The second chapter explores how one comes properly to exercise the mind in real-world situations. Aquino notes that it is not always possible or reasonable reflectively to articulate the subjective and social conditions in which one reaches beliefs and judgments. So, how does one navigate this in life? The answer is a proper fit. All human beings have some innate capabilities or faculties, such as reason, memory, and the senses. Moving from an unreflective to a reflective use of these—or to use Newman’s language, from an uncultivated to a cultivated illative sense—depends upon a number of factors (e.g., who one has become, what one is doing, and the world in which one lives). One individual may be too young to shift to this reflective mode; another may be in the midst of a battle or asleep. The union of these personal and social conditions into a whole that determines whether one shifts into a reflective mode or not is the discovery of a proper fit. Again, recognizing this proper fit is the act of an integrative habit of mind.The third and final chapter introduces the last key facet of the integrative habit that leads to wisdom and a right use of the illative sense, namely, a “connected view.” Such a view brings together into relation a multitude of data and ideas. A connected view is one of the primary concerns in Newman’s Idea of a University, and it is essential to a university education. This is why Aquino spends a good portion of the chapter exploring practical ways within [End Page 482] the university that one can awaken and sustain this quest for a connected view.Throughout the book, the method used by the author strikes one as a manifestation of Hans-Georg Gadamer’s fusion of horizons, in this case the author’s horizon... (shrink)

It is shown in the present paper that the transformation relating a parallel transported vector in a Weyl space to the original one is the product of a multiplicative gauge transformation and a proper orthochronous Lorentz transformation. Such a Lorentz transformation admits a spinor representation, which is obtained and used to deduce the transportation properties of a Weyl spinor, which are then expressed in terms of a composite gauge group defined as the product of a multiplicative gauge group and the (...) spinor group. These properties render a spinor amenable to its treatment as a particle coupled to a multidimensional gauge field in the framework of the Kaluza–Klein formulation extended to multidimensional gauge fields. In this framework, a fiber bundle is constructed with a horizontal, base space and a vertical, gauge space, which is a Lie group manifold, termed its structure group. For the present, the base is the Minkowski spacetime and the vertical space is the composite gauge group mentioned above. The fiber bundle is equipped with a Riemannian structure, which is used to obtain the classical description of motion of a spinor. In its classical picture, a Weyl spinor is found to behave as a spinning charged particle in translational motion. The corresponding quantum description is deduced from the Klein–Gordon equation in the Riemann spaces obtained by the methods of path-integration. This equation in the present fiber bundle reduces to the equation for a Weyl spinor, which is close to but differs somewhat from the squared Dirac equation. (shrink)

Spin-statistics transmutation is the phenomenon occurring when a “dressing” transformation introduced for physical reasons (e.g. gauge invariance) modifies the “bare” spin and statistics of particles or fields. Historically, it first appeared in Quantum Mechanics and in semiclassical approximation to Quantum Field Theory. After a brief historical introduction, we sketch how to describe such phenomenon in Quantum Field Theory beyond the semiclassical approximation, using a path-integral formulation of euclidean correlation functions, exemplifying with anyons, dyons and skyrmions.

Traditional interpretations of quantum theory in terms of wave function collapse are particularly unappealing when considering the universe as a whole, where there is no clean separation between classical observer and quantum system and where the description is inherently relativistic. As an alternative, the consistent histories approach provides an attractive “no collapse” interpretation of quantum physics. Consistent histories can also be linked to path-integral formulations that may be readily generalized to the relativistic case. A previous paper described how, (...) in such a relativistic spacetime path formalism, the quantum history of the universe could be considered to be an eigenstate of the measurements made within it. However, two important topics were not addressed in detail there: a model of measurement processes in the context of quantum histories in spacetime and a justification for why the probabilities for each possible cosmological eigenstate should follow Born’s rule. The present paper addresses these topics by showing how Zurek’s concepts of einselection and envariance can be applied in the context of relativistic spacetime and quantum histories. The result is a model of systems and subsystems within the universe and their interaction with each other and their environment. (shrink)

Several authors have made claims about the compatibility between the Free Energy Principle and theories of autopoiesis and enaction. Many see these theories as natural partners or as making similar statements about the nature of biological and cognitive systems. We critically examine these claims and identify a series of misreadings and misinterpretations of key enactive concepts. In particular, we notice a tendency to disregard the operational definition of autopoiesis and the distinction between a system’s structure and its organization. Other misreadings (...) concern the conflation of processes of self-distinction in operationally closed systems and Markov blankets. Deeper theoretical tensions underlie some of these misinterpretations. FEP assumes systems that reach a non-equilibrium steady state and are enveloped by a Markov blanket. We argue that these assumptions contradict the historicity of sense-making that is explicit in the enactive approach. Enactive concepts such as adaptivity and agency are defined in terms of the modulation of parameters and constraints of the agent-environment coupling, which entail the possibility of changes in variable and parameter sets, constraints, and in the dynamical laws affecting the system. This allows enaction to address the path-dependent diversity of human bodies and minds. We argue that these ideas are incompatible with the time invariance of non-equilibrium steady states assumed by the FEP. In addition, the enactive perspective foregrounds the enabling and constitutive roles played by the world in sense-making, agency, development. We argue that this view of transactional and constitutive relations between organisms and environments is a challenge to the FEP. Once we move beyond superficial similarities, identify misreadings, and examine the theoretical commitments of the two approaches, we reach the conclusion that far from being easily integrated, the FEP, as it stands formulated today, is in tension with the theories of autopoiesis and enaction. (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)

Koopman-von Neumann in the 30’s gave an operatorial formulation of Classical Mechanics. It was shown later on that this formulation could also be written in a path-integral form. We will label this functional approach as CPI (for classical path-integral) to distinguish it from the quantum mechanical one, which we will indicate with QPI. In the CPI two Grassmannian partners of time make their natural appearance and in this manner time becomes something like a three (...) dimensional supermanifold. Next we introduce a metric in this supermanifold and show that a particular choice of the supermetric reproduces the CPI while a different one gives the QPI. (shrink)

Shahriar S. Afshar claimed that his 2007 modified version of the double-slit experiment violates complementarity. He makes two modifications to the standard double-slit experiment. First, he adds a wire grid that is placed in between the slits and the screen at locations of interference minima. The second modification is to place a converging lens just after the wire grid. The idea is that the wire grid implies the existence of interference minima, while the lens can simultaneously obtain which-way information. More (...) recently, Cramer argued that the experiment bolstered the Transactional Interpretation of Quantum mechanics. His argument scrutinizes Bohr’s complementarity in favor of TIQM. We analyze this experiment by simulation using the path integral formulation of quantum mechanics and find that it agrees with the wave-particle duality relation given by Englert, Greenberg and Yasin. We conclude that the use of Afshar’s experiment to provide a testbed for quantum mechanical interpretations is limited. (shrink)

The purpose of this paper is to review relativistic quantum theories with an invariant evolution parameter. Parametrized relativistic quantum theories (PRQT) have appeared under such names as constraint Hamiltonian dynamics, four-space formalism, indefinite mass, micrononcausal quantum theory, parametrized path integral formalism, relativistic dynamics, Schwinger proper time method, stochastic interpretation of quantum mechanics and stochastic quantization. The review focuses on the fundamental concepts underlying the theories. Similarities as well as differences are highlighted, and an extensive bibliography is provided.

Heidegger holds that our age is dominated by the ambition of reason to possess the world. And he sees in Leibniz the man who formulated the theorem of our modern age: nothing happens without a reason. He calls this attitude `calculating thought' and opposes to it a kind of thought aimed at preserving the essence of things, which he calls `meditating thought'. Cristin's book ascribes great importance to this polarity of thinking for the future of contemporary philosophy, and thus compares (...) the basic ideas of the two thinkers. Leibniz announces the conquest of reason; Heidegger denounces the dangers of reason. Their diversity becomes manifest in the difference between the idea of reason and the image of the path. But is Leibniz's thought really only `calculating'? And do we not perhaps also encounter the traces of reason along Heidegger's path? With these questions in mind we may begin to redefine the relation between the two thinkers and between two different conceptions of reason and philosophy. The hypothesis is advanced that Heidegger's harsh judgment of Leibniz may be mitigated, but it also becomes clear that Heidegger's rewriting of the code of reason is an integral part of our age, in which many signs point to new loci of rationality. With his original interpretation, aware of the risks he is taking, Renato Cristin offers a new guide to the understanding of reason: he shows forth Leibniz as one who defends the thought of being in the unity of monadology, and Heidegger as a thinker who preserves the sign of reason in his meditating thought. (shrink)

We interpret the concept of determinism for a classical system as the requirement that the solution to the Cauchy problem for the equations of motion governing this system be unique. This requirement is generally believed to hold for all autonomous classical systems. Our analysis of classical electrodynamics in a world with one temporal and one spatial dimension provides counterexamples of this belief. Given the initial conditions of a particular type, the Cauchy problem may have an infinite set of solutions. Therefore, (...) random behavior of closed classical systems is indeed possible. With this finding, we give a qualitative explanation of how classical strings can split. We propose a modified path integral formulation of classical mechanics to include indeterministic systems. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Notes and Fragments TO THE UGHTHOUSE AND THE FEMINIST PATH TO POSTMODERNITY by Bill Martin Postmodernity is in part the existence of an unprecedented space for feminism. Already in this formulation, however, we encounter two major terms that require explication. I will argue in this essay that Virginia Woolf's To the Lighthouse provides the basis for productively understanding postmodernism and feminism in relation to each other.1 The (...) question of placing feminism temporally has of course been raised before; I will simply mention two occasions. In Sexual/Textual Politics, Toril Moi argues for a kind of natural affinity of feminism with literary modernism's implicit critique of notions of the unified self.2 Here the comparison of Woolf with Joyce is appropriate. Moi moves from this perceived affinity toward the notion of écriturefeminine. Practitioners of the latter have been much more interested in Joyce than Woolf. Joyce, especially in Finnegans Wake, thoroughly shatters the self, reaching the highest expression of modernism. Woolf does something different. In To tL· Lighthouse, she not only decenters the subject, she also reconstitutes the subject. If Woolf had only reinstated the subject, as a matter of simply returning to the previous paradigm, then obviously her writing would have to be considered anachronistic. Though the affinity posited by Moi is real and useful (if made problematic by what seem to be real affinities that also exist between modernism and misogyny ), the force of another affinity can also be felt: in her deconstruction /reconstitution, Woolf achieves a writing that is both postmodern and feminist. The both/and relation here is itself a cultural creation, 307 308PHrLOSOPHY and Literature however, which points to the elective affinity involved in joining a particular politics with a particular artistic practice. The question is whether a political argument that does not avoid questions of artistic integrity can be marshaled to the task of showing that the affinity of feminism and postmodernism is more viable in the contemporary context than the affinity posited by Moi.3 Elaine Showalter, in her formulation of "women's time," raises the possibility ofanother kind of temporalized feminism. In "Feminist Criticism in the Wilderness," Showalter pursues the idea that there might be a time and space which women occupy without their location being known or knowable to men, and perhaps without women fully knowing it themselves.4 In short, this "time" is conditional and predicated upon a possible future in which people will remember and be more fully aware of their spatial/temporal context. I would propose that this same conditional and uncertain sense of time applies to postmodernism. For the very terms "postmodernism" and "postmodernity" connote a transitional period in which it is not certain how elements which are for the time being in flux will ultimately setde down and congeal. The "aftereffect of the "post" in postmodernity will be truly understood only in the longer run, if even then. Showalter takes it that there is a sense in which "women's time," the "Age of Feminism," is occurring right now, but in a concealed way. Again, something like this may be said of postmodernism. In either case there is the sense of creating artifacts for a future which may or may not exist. There is also the sense of permanent postponement or deferral which is integral to the very notion of "possibility." Add to these references the by now familiar postmodern and feminist criticisms of grand narratives that crush difference, and I think we are in possession of a set of basic issues common to both feminism and postmodernism: the nature of the self and its possible recreation, the question of periodization and our "time," problems of difference and questions of doing something different (in the latter case feminism is much more explicit about doing something different that is also emancipatory, though some forms ofpostmodernism stress this as well). Given these common concerns I simply want to set the new affinity as follows: postmodernism is a social and cultural atmosphere in which feminism and other emancipatory trends might flourish. In this context, To tL· Lighthouse is a trailblazing feminist-postmodern artifact. The project here is not to generate or force a postmodern reading (any more... (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)

The physical concept of locality is first analyzed in the special relativistic quantum regime, and compared with that of microcausality and the local commutativity of quantum fields. Its extrapolation to quantum general relativity on quantum bundles over curved spacetime is then described. It is shown that the resulting formulation of quantum-geometric locality based on the concept of local quantum frame incorporating a fundamental length embodies the key geometric and topological aspects of this concept. Taken in conjunction with the strong (...) equivalence principle and the path-integral formulation of quantum propagation, quantum-geometric locality leads in a natural manner to the formulation of quantum-geometric propagation in curved spacetime. Its extrapolation to geometric quantum gravity formulated over quantum spacetime is described and analyzed. (shrink)

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 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)

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 search for a quantum theory of the gravitational field is one of the great open problems in theoretical physics. This book presents a self-contained discussion of the concepts, methods and applications that can be expected in such a theory. The two main approaches to its construction - the direct quantisation of Einstein's general theory of relativity and string theory - are covered. Whereas the first attempts to construct a viable theory for the gravitational field alone, string theory assumes that (...) a quantum theory of gravity will be achieved only through a unification of all the interactions. However, both employ the general method of quantisation of constrained systems, which is described together with illustrative examples relevant for quantum gravity. There is a detailed presentation of the main approaches employed in quantum general relativity: path-integral quantisation, the background-field method and canonical quantum gravity in the metric, connection and loop formulations. The discussion of string theory centres around its quantum-gravitational aspects and the comparison with quantum general relativity. Physical applications discussed at length include the quantisation of black holes, quantum cosmology, the indications of a discrete structure of spacetime, and the origin of irreversibility. This book will be of interest to researchers and students working in relativity and gravitation, cosmology, quantum field theory and related topics. It will also be of interest to mathematicians and philosophers of science. (shrink)

These lecture notes present a concise and introductory, yet as far as possible coherent, view of the main formalizations of quantum mechanics and of quantum field theories, their interrelations and their theoretical foundations. The "standard" formulation of quantum mechanics (involving the Hilbert space of pure states, self-adjoint operators as physical observables, and the probabilistic interpretation given by the Born rule) on one hand, and the path integral and functional integral representations of probabilities amplitudes on the other, (...) are the standard tools used in most applications of quantum theory in physics and chemistry. Yet, other mathematical representations of quantum mechanics sometimes allow better comprehension and justification of quantum theory. This text focuses on two of such representations: the algebraic formulation of quantum mechanics and the "quantum logic" approach. Last but not least, some emphasis will also be put on understanding the relation between quantum physics and special relativity through their common roots - causality, locality and reversibility, as well as on the relation between quantum theory, information theory, correlations and measurements, and quantum gravity. Quantum mechanics is probably the most successful physical theory ever proposed and despite huge experimental and technical progresses in over almost a century, it has never been seriously challenged by experiments. In addition, quantum information science ha s become an important and very active field in recent decades, further enriching the many facets of quantum physics. Yet, there is a strong revival of the discussions about the principles of quantum mechanics and its seemingly paradoxical aspects: sometimes the theory is portrayed as the unchallenged and dominant paradigm of modern physical sciences and technologies while sometimes it is considered a still mysterious and poorly understood theory, waiting for a revolution. This volume, addressing graduate students and seasoned researchers alike, aims to contribute to the reconciliation of these two facets of quantum mechanics. (shrink)

We study the coupled dynamics of two populations of random replicators by means of statistical mechanics methods, and focus on the effects of relative population size, strategy correlations and heterogeneities in the respective cooperation pressures. To this end we generalise existing path-integral approaches to replicator systems with random asymmetric couplings. This technique allows one to formulate an effective dynamical theory, which is exact in the thermodynamic limit and which can be solved for persistent order parameters in a fixed-point (...) regime regardless of the symmetry of the interactions. The onset of instability can be determined self-consistently. We calculate quantities such as the diversity of the respective populations and their fitnesses in the stationary state, and compare results with data from a numerical integration of the replicator equations. -/- . (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.

The various physical aspects of the general relativistic principles of covariance and strong equivalence are discussed, and their mathematical formulations are analyzed. All these aspects are shown to be present in classical general relativity, although no contemporary formulation of canonical or covariant quantum gravity has succeeded to incorporate them all. This has, in part, motivated the recent introduction of a geometro-stochastic framework for quantum general relativity, in which the classical frame bundles that underlie the formulation of parallel transport (...) in classical general relativity are replaced by quantum frame bundles. It is shown that quantum frames can take over the role played by complete sets of observables in conventional quantum theory, so that they can mediate the natural transference of the general covariance and the strong equivalence principles from the classical to the quantum general relativistic regime. This results in a geometrostochastic mode of quantum propagation in general relativistic quantum bundles, which is mathematically implemented by path integration methods based on parallel transport along horizontal lifts of geodesics for the vacuum expectation values of a quantum gravitational field in a quantum spacetime supermanifold. The covariance features of this field are embedded in a quantum gravitational supergroup, which incorporates Poincaré as well as diffeomorphism invariance, and resolves the issue of time in quantum gravity. (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)

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 Standard Model of particle physics is usually formulated in terms of fields, it can be equivalently formulated in terms of particles and strings. In this picture particles and open strings are always coupled. This offers an intuitive and graphical explanation for the otherwise mysterious gauge symmetry. In addition, the particle–string formulation avoids introducing redundant path integral configurations that are present in the field formulation. For its explanatory power and economy, the particle–string ontology may be (...) preferred over the field ontology for the Standard Model. (shrink)

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)

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 (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)

Using a framework of Dirac algebra, the Clifford algebra appropriate for Minkowski space-time, the formulation of classical electromagnetism including both electric and magnetic charge is explored. Employing the two-potential approach of Cabibbo and Ferrari, a Lagrangian is obtained that is dyality invariant and from which it is possible to derive by Hamilton's principle both the symmetrized Maxwell's equations and the equations of motion for both electrically and magnetically charged particles. This latter result is achieved by defining the variation of (...) the action associated with the cross terms of the interaction Lagrangian in terms of a surface integral. The surface integral has an equivalent path-integral form, showing that the contribution of the cross terms is local in nature. The form of these cross terms derives in a natural way from a Dirac algebraic formulation, and, in fact, the use of the geometric product of Dirac algebra is an essential aspect of this derivation. No kinematic restrictions are associated with the derivation, and no relationship between magnetic and electric charge evolves from the (classical) formulation. However, it is indicated that in bound states quantum mechanical considerations will lead to a version of Dirac's quantization condition. A discussion of parity violation of the generalized electromagnetic theory is given, and a new approach to the incorporation of this violation into the formalism is suggested. Possibilities for extensions are mentioned. (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.

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)

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)

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)

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)

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.

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)

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)

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...

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 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 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)

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)

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.

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)