Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal... more Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.
In [6] it was proposed to define "god" as a region of a universe that is subject to cir... more In [6] it was proposed to define "god" as a region of a universe that is subject to circular causality. While we do not adapt the exact definition of "god" that is being introduced in that paper, we do accept the concept that "god" has something to do with causal anomalies: either circular causality or else two competing causal structures. We will show that the presence of a causal anomaly (whatever it happens to be) might allow us to define trinity in non-contradictory way. That is, we will show how the members of trinity can be separate entities and, yet, have the same identity.
The Clifford spectrum is an elegant way to define the joint spectrum of several Hermitian operato... more The Clifford spectrum is an elegant way to define the joint spectrum of several Hermitian operators. While it has been know that for examples as small as three 2-by-2 matrices the Clifford spectrum can be a two-dimensional manifold, few concrete examples have been investigated. Our main goal is to generate examples of the Clifford spectrum of three or four matrices where, with the assistance of a computer algebra package, we can calculate the Clifford spectrum.
Mensky has suggested to account for "continuous measurement" by attaching to a path integral a we... more Mensky has suggested to account for "continuous measurement" by attaching to a path integral a weight function centered around the classical path that the integral assigns a probability amplitude to. We show that in fact this weight function doesn't have to be viewed as an additional ingredient put in by hand. It can be derived instead from the conventional path integral if the infinitesimal term iǫ in the propagator is made finite; the "classical trajectory" is proportional to the current.
Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal... more Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.
The goal of this paper is to re-express QFT in terms of two "classical" fields living i... more The goal of this paper is to re-express QFT in terms of two "classical" fields living in ordinary space with single extra dimension. The role of the first classical field is to set up an injection from the set of values of extra dimension into the set of functions, and then said injection will be used in order to convert the second field into a coarse grained functional, thereby approximating QFT state. It turns out that this work also has a side-benefit of modeling ensemble of states in terms of one single state which, in turn, is interpretted in the above way. It is important to clarify that by "classical" we mean functions over ordinary space rather than configuration, Fock or function space. The "classical" theory that we propose is still non-local.
The goal of this article is to come up with interpretation of quantum phenomena that is both loca... more The goal of this article is to come up with interpretation of quantum phenomena that is both local and deterministic. This is done by the means of envoking two different metrics, g_o and g_s. These two metrics give very different "speeds of light": c_o and c_s, respectively. The g_o and c_o are, respectively, "ordinary" metric and speed of light that we are used to. On the other hand, c_s is superluminal. In this paper I propose a model in which newly introduced signals, which are subject to g_s, are responsible for key quantum phenomena.
Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal... more Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.
The purpose of this paper is to come up with a framework that "converts" existing conce... more The purpose of this paper is to come up with a framework that "converts" existing concepts from configuration space to ordinary one. This is done by modeling our universe as a big "computer" that simulates configuration space. If that "computer" exists in ordinary space and is ran by "classical" laws, our theory becomes "classical" by default. We have first applied this concept to a version of quantum field theory in which elementary particles have size (that is, a theory that does not yet exists). After that, we have also done the same with Pilot Wave model of discrete jumps, due to D\"urr et el.
The goal of this paper is to propose an approach to the formulation of dynamics for causal sets a... more The goal of this paper is to propose an approach to the formulation of dynamics for causal sets and coupled matter fields. We start from the continuum version of the action for a Klein-Gordon field coupled to gravity, and rewrite it first using quantities that have a direct correspondent in the case of a causal set, namely volumes, causal relations, and timelike lengths, as variables to describe the geometry. In this step, the local Lagrangian density L(f;x) for a set of fields f is recast into a quasilocal expression L_0(f;p,q) that depends on pairs of causally related points p ≺ q and is a function of the values of f in the Alexandrov set defined by those points, and whose limit as p and q approach a common point x is L(f;x). We then describe how to discretize L_0(f;p,q), and use it to define a discrete action.
This is the second paper in a series on the dynamics of matter fields in the causal set approach ... more This is the second paper in a series on the dynamics of matter fields in the causal set approach to quantum gravity. We start with the usual expression for the Lagrangian of a charged scalar field coupled to a SU(n) Yang-Mills field, in which the gauge field is represented by a connection form, and show how to write it in terms of holonomies between pairs of points, causal relations, and volumes or timelike distances, all of which have a natural correspondence in the causal set context. In the second part of the paper we present an alternative model, in which the gauge field appears as the result of a procedure inspired by the Kaluza-Klein reduction in continuum field theory, and the dynamics can be derived simply using the gravitational Lagrangian of the theory.
A while ago a proposal have been made regarding Klein Gordon and Maxwell Lagrangians for causal s... more A while ago a proposal have been made regarding Klein Gordon and Maxwell Lagrangians for causal set theory. These Lagrangian densities are based on the statistical analysis of the behavior of field on a sample of points taken throughout some "small" region of spacetime. However, in order for that sample to be statistically reliable, a lower bound on the size of that region needs to be imposed. This results in "unwanted contributions" from higher order derivatives to the Lagrangian density, as well as non-trivial curvature effects on the latter. It turns out that both gravitational and non-gravitational effects end up being highly non-linear. In the previous papers we were focused on leading order terms, which allowed us to neglect these nonlinearities. We would now like to go to the next order and investigate them. In the current paper we will exclusively focus on the effects of higher order derivatives in the flat-space toy model. The gravitational effects will ...
The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that inco... more The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that incorporates particle creation and annihilation without sacrificing determinism. This has been previously attempted in an article by the same author titled "Incorporating particle creation and annihilation in Pilot Wave model", in a much less satisfactory way. In this paper I would like to "clean up" some of the things. In particular, I would like to get rid of a very unnatural concept of "visibility" of particles, which makes the model much simpler. On the other hand, I would like to add a mechanism for decoherence, which was absent in the previous version.
Mensky has suggested to account for "continuous measurement" by attaching to a path int... more Mensky has suggested to account for "continuous measurement" by attaching to a path integral a weight function centered around the classical path that the integral assigns a probability amplitude to. We show that in fact this weight function doesn't have to be viewed as an additional ingredient put in by hand. It can be derived instead from the conventional path integral if the infinitesimal term i\epsilon\ in the propagator is made finite; the "classical trajectory" is proportional to the current.
The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that inco... more The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that incorporates particle creation and annihilation without sacrificing causality. In some sense, this work echoes the work of Nikoli\'c (I call "visibility" what he calls "effectiveness"), but in my work I choose position and visibility as beables, as opposed to field beables that are used in his.
We present a framework for the dynamics of causal sets and coupled matter fields, which is a simp... more We present a framework for the dynamics of causal sets and coupled matter fields, which is a simplification and generalization of an approach we recently proposed. Given a set of fields including the gravitational one, the main step in implementing our proposal consists in writing their continuum-based action using as variables for the spacetime geometry the causal order and volume element. One then discretizes the resulting expression, with a procedure designed to maintain covariance. After a discussion of the general framework, we treat in detail the case of scalar fields, Yang-Mills gauge fields and the gravitational field.
The purpose of this paper is to propose a "classical" model of "quantum" fiel... more The purpose of this paper is to propose a "classical" model of "quantum" fields which is local. Yet it admittedly violates relativity as we know it and, instead, it fits within a bimetric model with one metric corresponding to speed of light and another metric to superlumianl signals whose speed is still finite albeit very large. The key obstacle to such model is the notion of functional in the context of QFT which is inherently non-local. The goal of this paper is to stop viewing functionals as fundamental and instead model their emergence from the deeper processes that are based on functions over R^4 alone. The latter are claimed to be local in the above bimetric sense.
The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that inco... more The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that incorporates particle creation and annihilation without sacrificing causality. In some sense, this work echoes the work of Nikolić in [4] and [5] (I call "visibility" what he calls "effectiveness"), but in my work I choose position and visibility as beables, as opposed to field beables that are used in his.
The purpose of this paper is to come up with one of the many possible schemes of "adding" gravity... more The purpose of this paper is to come up with one of the many possible schemes of "adding" gravity to Pilot Wave models.
Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal... more Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.
In [6] it was proposed to define "god" as a region of a universe that is subject to cir... more In [6] it was proposed to define "god" as a region of a universe that is subject to circular causality. While we do not adapt the exact definition of "god" that is being introduced in that paper, we do accept the concept that "god" has something to do with causal anomalies: either circular causality or else two competing causal structures. We will show that the presence of a causal anomaly (whatever it happens to be) might allow us to define trinity in non-contradictory way. That is, we will show how the members of trinity can be separate entities and, yet, have the same identity.
The Clifford spectrum is an elegant way to define the joint spectrum of several Hermitian operato... more The Clifford spectrum is an elegant way to define the joint spectrum of several Hermitian operators. While it has been know that for examples as small as three 2-by-2 matrices the Clifford spectrum can be a two-dimensional manifold, few concrete examples have been investigated. Our main goal is to generate examples of the Clifford spectrum of three or four matrices where, with the assistance of a computer algebra package, we can calculate the Clifford spectrum.
Mensky has suggested to account for "continuous measurement" by attaching to a path integral a we... more Mensky has suggested to account for "continuous measurement" by attaching to a path integral a weight function centered around the classical path that the integral assigns a probability amplitude to. We show that in fact this weight function doesn't have to be viewed as an additional ingredient put in by hand. It can be derived instead from the conventional path integral if the infinitesimal term iǫ in the propagator is made finite; the "classical trajectory" is proportional to the current.
Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal... more Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.
The goal of this paper is to re-express QFT in terms of two "classical" fields living i... more The goal of this paper is to re-express QFT in terms of two "classical" fields living in ordinary space with single extra dimension. The role of the first classical field is to set up an injection from the set of values of extra dimension into the set of functions, and then said injection will be used in order to convert the second field into a coarse grained functional, thereby approximating QFT state. It turns out that this work also has a side-benefit of modeling ensemble of states in terms of one single state which, in turn, is interpretted in the above way. It is important to clarify that by "classical" we mean functions over ordinary space rather than configuration, Fock or function space. The "classical" theory that we propose is still non-local.
The goal of this article is to come up with interpretation of quantum phenomena that is both loca... more The goal of this article is to come up with interpretation of quantum phenomena that is both local and deterministic. This is done by the means of envoking two different metrics, g_o and g_s. These two metrics give very different "speeds of light": c_o and c_s, respectively. The g_o and c_o are, respectively, "ordinary" metric and speed of light that we are used to. On the other hand, c_s is superluminal. In this paper I propose a model in which newly introduced signals, which are subject to g_s, are responsible for key quantum phenomena.
Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal... more Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.
The purpose of this paper is to come up with a framework that "converts" existing conce... more The purpose of this paper is to come up with a framework that "converts" existing concepts from configuration space to ordinary one. This is done by modeling our universe as a big "computer" that simulates configuration space. If that "computer" exists in ordinary space and is ran by "classical" laws, our theory becomes "classical" by default. We have first applied this concept to a version of quantum field theory in which elementary particles have size (that is, a theory that does not yet exists). After that, we have also done the same with Pilot Wave model of discrete jumps, due to D\"urr et el.
The goal of this paper is to propose an approach to the formulation of dynamics for causal sets a... more The goal of this paper is to propose an approach to the formulation of dynamics for causal sets and coupled matter fields. We start from the continuum version of the action for a Klein-Gordon field coupled to gravity, and rewrite it first using quantities that have a direct correspondent in the case of a causal set, namely volumes, causal relations, and timelike lengths, as variables to describe the geometry. In this step, the local Lagrangian density L(f;x) for a set of fields f is recast into a quasilocal expression L_0(f;p,q) that depends on pairs of causally related points p ≺ q and is a function of the values of f in the Alexandrov set defined by those points, and whose limit as p and q approach a common point x is L(f;x). We then describe how to discretize L_0(f;p,q), and use it to define a discrete action.
This is the second paper in a series on the dynamics of matter fields in the causal set approach ... more This is the second paper in a series on the dynamics of matter fields in the causal set approach to quantum gravity. We start with the usual expression for the Lagrangian of a charged scalar field coupled to a SU(n) Yang-Mills field, in which the gauge field is represented by a connection form, and show how to write it in terms of holonomies between pairs of points, causal relations, and volumes or timelike distances, all of which have a natural correspondence in the causal set context. In the second part of the paper we present an alternative model, in which the gauge field appears as the result of a procedure inspired by the Kaluza-Klein reduction in continuum field theory, and the dynamics can be derived simply using the gravitational Lagrangian of the theory.
A while ago a proposal have been made regarding Klein Gordon and Maxwell Lagrangians for causal s... more A while ago a proposal have been made regarding Klein Gordon and Maxwell Lagrangians for causal set theory. These Lagrangian densities are based on the statistical analysis of the behavior of field on a sample of points taken throughout some "small" region of spacetime. However, in order for that sample to be statistically reliable, a lower bound on the size of that region needs to be imposed. This results in "unwanted contributions" from higher order derivatives to the Lagrangian density, as well as non-trivial curvature effects on the latter. It turns out that both gravitational and non-gravitational effects end up being highly non-linear. In the previous papers we were focused on leading order terms, which allowed us to neglect these nonlinearities. We would now like to go to the next order and investigate them. In the current paper we will exclusively focus on the effects of higher order derivatives in the flat-space toy model. The gravitational effects will ...
The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that inco... more The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that incorporates particle creation and annihilation without sacrificing determinism. This has been previously attempted in an article by the same author titled "Incorporating particle creation and annihilation in Pilot Wave model", in a much less satisfactory way. In this paper I would like to "clean up" some of the things. In particular, I would like to get rid of a very unnatural concept of "visibility" of particles, which makes the model much simpler. On the other hand, I would like to add a mechanism for decoherence, which was absent in the previous version.
Mensky has suggested to account for "continuous measurement" by attaching to a path int... more Mensky has suggested to account for "continuous measurement" by attaching to a path integral a weight function centered around the classical path that the integral assigns a probability amplitude to. We show that in fact this weight function doesn't have to be viewed as an additional ingredient put in by hand. It can be derived instead from the conventional path integral if the infinitesimal term i\epsilon\ in the propagator is made finite; the "classical trajectory" is proportional to the current.
The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that inco... more The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that incorporates particle creation and annihilation without sacrificing causality. In some sense, this work echoes the work of Nikoli\'c (I call "visibility" what he calls "effectiveness"), but in my work I choose position and visibility as beables, as opposed to field beables that are used in his.
We present a framework for the dynamics of causal sets and coupled matter fields, which is a simp... more We present a framework for the dynamics of causal sets and coupled matter fields, which is a simplification and generalization of an approach we recently proposed. Given a set of fields including the gravitational one, the main step in implementing our proposal consists in writing their continuum-based action using as variables for the spacetime geometry the causal order and volume element. One then discretizes the resulting expression, with a procedure designed to maintain covariance. After a discussion of the general framework, we treat in detail the case of scalar fields, Yang-Mills gauge fields and the gravitational field.
The purpose of this paper is to propose a "classical" model of "quantum" fiel... more The purpose of this paper is to propose a "classical" model of "quantum" fields which is local. Yet it admittedly violates relativity as we know it and, instead, it fits within a bimetric model with one metric corresponding to speed of light and another metric to superlumianl signals whose speed is still finite albeit very large. The key obstacle to such model is the notion of functional in the context of QFT which is inherently non-local. The goal of this paper is to stop viewing functionals as fundamental and instead model their emergence from the deeper processes that are based on functions over R^4 alone. The latter are claimed to be local in the above bimetric sense.
The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that inco... more The purpose of this paper is to come up with a Pilot Wave model of quantum field theory that incorporates particle creation and annihilation without sacrificing causality. In some sense, this work echoes the work of Nikolić in [4] and [5] (I call "visibility" what he calls "effectiveness"), but in my work I choose position and visibility as beables, as opposed to field beables that are used in his.
The purpose of this paper is to come up with one of the many possible schemes of "adding" gravity... more The purpose of this paper is to come up with one of the many possible schemes of "adding" gravity to Pilot Wave models.
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