scholarly journals Path Integral Implementation of Relational Quantum Mechanics

2021 ◽  
Author(s):  
Jianhao M. Yang
Keyword(s):  
Quantum Mechanics ◽  
Quantum System ◽  
Path Integral ◽  
Measurement Theory ◽  
Entanglement Entropy ◽  
Quantum Probability ◽  
Probability Amplitude ◽  
Path Integral Representation ◽  
Relational Quantum Mechanics ◽  
Influence Functional

Abstract Relational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In the recent works (J. M. Yang, Sci. Rep. 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born's Rule, Schr\"{o}dinger Equations, and measurement theory. This paper gives a concrete implementation of the relational probability amplitude by extending the path integral formulation. The implementation not only clarifies the physical meaning of the relational probability amplitude, but also gives several important applications. For instance, the double slit experiment can be elegantly explained. A path integral representation of the reduced density matrix of the observed system can be derived. Such representation is shown valuable to describe the interaction history of the measured system and a series of measuring systems. More interestingly, it allows us to develop a method to calculate entanglement entropy based on path integral and influence functional. Criteria of entanglement is proposed based on the properties of influence functional, which may be used to determine entanglement due to interaction between a quantum system and a classical field.

scholarly journals Path integral implementation of relational quantum mechanics

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Jianhao M. Yang
Keyword(s):  
Quantum Mechanics ◽  
Reference Frame ◽  
Path Integral ◽  
Measurement Theory ◽  
Entanglement Entropy ◽  
Quantum Probability ◽  
Frame Theory ◽  
Probability Amplitude ◽  
Relational Quantum Mechanics ◽  
Measured System

AbstractRelational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In a recent paper (Yang in Sci Rep 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born’s Rule, Schrödinger Equations, and measurement theory. This paper further extends the reformulation effort in three aspects. First, it gives a clearer explanation of the key concepts behind the framework to calculate measurement probability. Second, we provide a concrete implementation of the relational probability amplitude by extending the path integral formulation. The implementation not only clarifies the physical meaning of the relational probability amplitude, but also allows us to elegantly explain the double slit experiment, to describe the interaction history between the measured system and a series of measuring systems, and to calculate entanglement entropy based on path integral and influence functional. In return, the implementation brings back new insight to path integral itself by completing the explanation on why measurement probability can be calculated as modulus square of probability amplitude. Lastly, we clarify the connection between our reformulation and the quantum reference frame theory. A complete relational formulation of quantum mechanics needs to combine the present works with the quantum reference frame theory.

scholarly journals p-Adic Path Integrals for Quadratic Actions

1997 ◽  
Vol 12 (20) ◽  
pp. 1455-1463 ◽  
Author(s):  
G. S. Djordjević ◽  
B. Dragovich
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Path Integrals ◽  
Probability Amplitude ◽  
Exact Form ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
One Dimensional ◽  
Ordinary Quantum

The Feynman path integral in p-adic quantum mechanics is considered. The probability amplitude [Formula: see text] for one-dimensional systems with quadratic actions is calculated in an exact form, which is the same as that in ordinary quantum mechanics.

Euclidean path integrals and quantum mechanics (QM)

2021 ◽  
pp. 18-41
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Correlation Functions ◽  
Path Integrals ◽  
Continuous Phase ◽  
Functional Integral ◽  
Smooth Transition ◽  
Path Integral Representation ◽  
The Matrix ◽  
Functional Measure

Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.

scholarly journals SUPERSYMMETRY, HOMOLOGY WITH TWISTED COEFFICIENTS AND n-DIMENSIONAL KNOTS

2006 ◽  
Vol 21 (19n20) ◽  
pp. 4185-4196 ◽  
Author(s):  
EIJI OGASA
Keyword(s):  
Integral Representation ◽  
Natural Number ◽  
Quantum System ◽  
Path Integral ◽  
Topological Invariant ◽  
Supersymmetric Theory ◽  
Path Integral Representation ◽  
Alexander Polynomials ◽  
Usual Manner ◽  
Infinitesimal Transformations

In this paper, we study and construct a set of Witten indexes for K, where K is any n-dimensional knot in Sn+2 and n is any natural number. We form a supersymmetric quantum system for K by, first, constructing a set of functional spaces (spaces of fermionic (resp. bosonic) states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Our Witten indexes are topological invariant and they are nonzero in general. These indexes are zero if K is equivalent to a trivial knot. Besides, our Witten indexes restrict to the Alexander polynomials of n-knots, and one of the Alexander polynomials of K is nontrivial if any of the Witten indexes is nonzero. Our indexes are related to homology with twisted coefficients. Roughly speaking, these indexes posseses path-integral representation in the usual manner of supersymmetric theory.

scholarly journals PATH INTEGRAL AND PSEUDOCLASSICAL ACTION FOR SPINNING PARTICLE IN EXTERNAL ELECTROMAGNETIC AND TORSION FIELDS

2000 ◽  
Vol 15 (24) ◽  
pp. 3861-3876 ◽  
Author(s):  
BODO GEYER ◽  
DMITRY GITMAN ◽  
ILYA L. SHAPIRO
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Integral Representation ◽  
Path Integral ◽  
Classical Limit ◽  
Equations Of Motion ◽  
Nonrelativistic Limit ◽  
Spinning Particle ◽  
Path Integral Representation ◽  
The Given

Starting from the Dirac equation in external electromagnetic and torsion fields we derive a path integral representation for the corresponding propagator. An effective action, which appears in the representation, is interpreted as a pseudoclassical action for a spinning particle. It is just a generalization of Berezin–Marinov action to the background under consideration. Pseudoclassical equations of motion in the nonrelativistic limit reproduce exactly the classical limit of the Pauli quantum mechanics in the same case. Quantization of the action appears to be nontrivial due to an ordering problem, which needs to be solved to construct operators of first-class constraints, and to select the physical sector. Finally the quantization reproduces the Dirac equation in the given background and, thus, justifies the interpretation of the action.

scholarly journals From the black hole conundrum to the structure of quantum gravity

2021 ◽  
pp. 2130007
Author(s):  
Yasunori Nomura
Keyword(s):  
Black Hole ◽  
Quantum Mechanics ◽  
Quantum Gravity ◽  
Quantum System ◽  
Path Integral ◽  
Recent Progress ◽  
Black Hole Physics ◽  
Interior Volume

We portray the structure of quantum gravity emerging from recent progress in understanding the quantum mechanics of an evaporating black hole. Quantum gravity admits two different descriptions, based on Euclidean gravitational path integral and a unitarily evolving holographic quantum system, which appear to present vastly different pictures under the existence of a black hole. Nevertheless, these two descriptions are physically equivalent. Various issues of black hole physics — including the existence of the interior, unitarity of the evolution, the puzzle of too large interior volume, and the ensemble nature seen in certain calculations — are addressed very differently in the two descriptions, still leading to the same physical conclusions. The perspective of quantum gravity developed here is expected to have broader implications beyond black hole physics, especially for the cosmology of the eternally inflating multiverse.

NON-ADIABATIC ELIMINATION OF VARIABLES IN STOCHASTIC PROCESSES BY MEANS OF PATH INTEGRAL AND INFLUENCE FUNCTIONAL METHODS

1995 ◽  
Vol 09 (06) ◽  
pp. 679-694 ◽  
Author(s):  
HORACIO S. WIO ◽  
C. BUDDE ◽  
C. BRIOZZO ◽  
P. COLET
Keyword(s):  
Stochastic Processes ◽  
Path Integral ◽  
Langevin Equations ◽  
Adiabatic Elimination ◽  
Path Integral Representation ◽  
Functional Methods ◽  
Influence Functional ◽  
Fokker Planck Equations ◽  
Elimination Of Variables ◽  
White Noises

We present a novel scheme for the non-adiabatic elimination of variables in stochastic processes, based on a path integral representation of the probability density and the use of an influence functional. We analyze in particular the case of multivariate Fokker-Planck equations, or equivalently a set of coupled Langevin equations driven by white noises, and discuss some examples where exact or approximate results are obtained.

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