What is Path integrals: Definition and 60 Discussions

The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these are coordinate space or Feynman path integrals), than the Hamiltonian. Possible downsides of the approach include that unitarity (this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the S-matrix is obscure in the formulation. The path-integral approach has been proved to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by deriving either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s, which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.The basic idea of the path integral formulation can be traced back to Norbert Wiener, who introduced the Wiener integral for solving problems in diffusion and Brownian motion. This idea was extended to the use of the Lagrangian in quantum mechanics by Paul Dirac in his 1933 article. The complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier in his doctoral work under the supervision of John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian (rather than a Hamiltonian) as a starting point.

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  1. qft-El

    A Heisenberg picture and Path integrals (Zee QFT)

    Reading the introduction to path integrals given in the latest edition of Zee's "Quantum field theory in a nutshell", I have found a remark which I don't really understand. The author is evaluating the free particle propagator ##K(q_f, t; q_i, 0)## $$\langle q_f\lvert e^{-iHt}\lvert q_i...
  2. Sophrosyne

    I Question about Feynman’s path integrals

    In Feynman’s path integrals, there is: ∫dq″Π0(t″,t′;q″,q′)=1 What is the funny pi looking symbol?
  3. U

    Engineering Path integrals in scalar fields when the path is not provided

    I cannot seem to start answering the question as a result of the path not being provided. How do I solve this when the path is not provided? See picture below
  4. C

    I Quantization of Quasiperiodic Orbits in the Bohr-Sommerfeld Model

    Recall that in the semi-classical Bohr-Sommerfeld quantization scheme from the early days of quantum mechanics, bound orbits were quantized according to the value of the action integral around a single loop of a closed path. Clearly this only makes sense if the orbits in question permit closed...
  5. M

    Path Integrals in Wolfram Mathematica

    Hello everyone ! I am new to this site so I 'd better say hello to you all ! I am finishing my BR in physics and part of this ending is to deliver a thesis . Long story short I must compute path-integrals in SU(2) and SU(3) pure yang-mills fields . Problem is that i was never very good with...
  6. I

    I Phase space path integrals

    I have heard of phase space path integrals, but couldn't find anything in Wikipedia about it, so I am wondering, what does it compute ? In particular, are the endpoints points of definite position and momentum? If so, how does one convert them to quantum states ? Also, how is it related to...
  7. Yellotherephysics

    A Functional Determinant of a system of differential operators?

    So in particular, how could the determinant of some general "operator" like $$ \begin{pmatrix} f(x) & \frac{d}{dx} \\ \frac{d}{dx} & g(x) \end{pmatrix} $$ with appropriate boundary conditions (especially fixed BC), be computed? And assuming that it diverges, would it be valid in a stationary...
  8. W

    I Propagator of a Scalar Field via Path Integrals

    I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from: $$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$ To...
  9. Sophrosyne

    B Speed of light with quantum path integrals

    Richard Feynman formulated quantum path integrals to show that a single photon can theoretically travel infinitely many different paths from one point to another. The shortest path, minimizing the Lagrangian, is the one most often traveled. But certainly other paths can be taken. Using single...
  10. saadhusayn

    Perturbation expansion with path integrals

    I expanded the exponential with the derivative to get: ## Z = \Bigg(1 + \frac{1}{2} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} + \frac{1}{4} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{k}} A^{-1}_{kl}...
  11. J

    A Can the Huygens-Fresnel principle be translated into path integrals?

    Wave optics, including diffraction, seems to be apt for path integral language. In fact, Feynman's double slit language is purely "diffraction". Also, the PDE for the wave equation results in a solution via Green's function, and the Green function is where "the path integral lives". I have...
  12. C

    I Feynman, Hibbs Transition Amplitudes and Energy

    I’m currently self-studying from Feynman & Hibbs Quantum Mechanics and Path Integrals, but having trouble with a statement in the chapter on time-dependent perturbations. Background: They define $$V_{mn}(t_c) = \int_{-\infty}^\infty \phi_m^*(x_c)V(x_c,t_c)\phi_n(x_c)\,dx_c,$$ where V(x,t) is...
  13. H

    A How to evaluate path integrals numerically?

    Since we only know Gaussian integration, could one get Green's function numerically with interacting action. Usual perturbation theory is tedious and limited, could one get high accurate result with PC beyond perturbation?
  14. rocdoc

    I Decomposing a Certain Exponential Integral

    There is nothing wrong with the well known $$e^{i\theta}=\cos\theta+i\sin\theta$$ for real ## \theta## but what about $$\int_{-\infty}^\infty~e^{i\theta(p)}\mathrm{d}p=\int_{-\infty}^\infty~\cos\theta(p)\mathrm{d}p+i\int_{-\infty}^\infty~\sin\theta(p)\mathrm{d}p$$ I have been trying to use...
  15. rocdoc

    I Path Integrals in Quantum Theory

    I have found a general result for certain exponential integrals that may be of interest to those involved with using path integrals. I am not certain that I am applying it correctly but it appears to work, and I can reproduce results quoted in various textbooks , using it. This may however be...
  16. Seanra

    I The speed of light in a medium and path integrals

    So I've heard from multiple sources that one explanation for why light slows down whilst traveling through mediums other than a vacuum is that the light "takes every possible path at the same time" through the medium. Below I've drawn my two possible interpretations of what that means. Can...
  17. H

    Propagator in the derivation of path integrals

    Why isn't (5.298) the following instead? ##K(x, t_1; x', t_0) = \delta(x-x')\,e^{-\frac{i}{\hbar}H(t_1-t_0)}## My reasoning: Since \Psi(x, t_1) = e^{-\frac{i}{\hbar}H(t_1-t_0)}\,\Psi(x, t_0)\\ = e^{-\frac{i}{\hbar}H(t_1-t_0)}\,\int\delta(x-x')\Psi(x', t_0)\,dx' The exponential operator with...
  18. C

    Propagator in 2D Euclidean space

    Homework Statement Consider the following scalar theory formulated in two-dimensional Euclidean space-time; S=∫d2x ½(∂μφ∂μφ + m2φ2) , a) Determine the equations of motion for the field φ. b) Compute the propagator; G(x,y) = ∫d2k/(2π)2 eik(x-y)G(k). Homework Equations Euler-Lagrange equations...
  19. H

    Confused about perturbation theory with path integrals

    Homework Statement Hi, I am just trying to wrap my head around using path integrals and there are a few things that are confusing me. Specifically, I have seen examples in which you can use it to calculate the ground state shift in energy levels of a harmonic oscillator but I don't see how you...
  20. F

    Feynman Path integrals in space with holes?

    Feynman Path Integrals are a way of calculating the wave function of quantum mechanics. It usually integrates every possible path through all of space. I wonder if there is any study of Feynman path integrals through a space with holes in it - with regions of space excluded from the integration...
  21. K

    Path Integrals in QFT: Beyond Peskin's Reference

    Can anyone suggest me a good reference for path integrals (QFT), apart from peskin.
  22. pellman

    Path integrals as usually presented - what does it tell us?

    Typical introductions to path integrals start with asking for the value of \langle x_1,t_1 | x_2,t_2 \rangle. This is usually interpreted as the probability amplitude of observing a particle at x_2 at time time t_2 given that it is located at x_1 at t_1. But is this so? I am having trouble...
  23. M

    The Classical Path, QM Path Integrals and Paths in Curved Spacetime

    "The" Classical Path, QM Path Integrals and Paths in Curved Spacetime Hey Guys! I've got an exciting question! It's been burning on my mind for years, but I think I can formulate it now. It's not so much a specific question, but rather a physical story which perhaps this thread can uncover...
  24. W

    Path integrals important for undergrads to know?

    I'm going through a whole undergrad quantum book (Townsend) by myself. It has a chapter on path integral QM. He said in the intro that it can be skipped, but I was wondering if knowledge of this subject is immediately helpful when starting graduate level quantum. I start grad school in the...
  25. P

    Path Integrals- Multivariable Calculus

    Path Integrals-- Multivariable Calculus Hi all-- really stuck here, help would be greatly appreciated. :) 1. Evaluate ∫Fds (over c), where F(x, y, z) = (y, 2x, y) and the path c is de fined by the equation c(t) = (t, t^2, t^3); on [0, 1]: 2. Homework Equations L = sqrt(f'(t)^2 +...
  26. W

    Path Integrals Harmonic Oscillator

    Hi, I am reading through the book "Quantum Mechanics and Path Integrals" by Feynman and Hibbs and am having a bit of trouble with problem 3-12. The question is (all Planck constants are the reduced Planck constant and all integrals are from -infinity to infinity): The wavefunction for a...
  27. R

    How do you compute the circulation of this fluid (path integrals)

    Homework Statement A fluid as velocity field F(x, y, z) = (xy, yz, xz). Let C denote the unit circle in the xy-plane. Compute the circulation, and interpret your answer.Homework Equations The Attempt at a Solution Since the unit circle is a closed loop, I assumed that ∫ F * dr = 0 (the ∫ symbol...
  28. R

    Path integrals and parameterization

    Homework Statement Evaluate ∫ F ds over the curve C for: a) F = (x, -y) and r(t) = (cos t, sin t), 0 ≤ t ≤ 2∏ b) F = (yz, xz, xy) where the curve C consists of straight-line segments joining (1, 0, 0) to (0, 1, 0) to (0, 0, 1) Homework Equations The Attempt at a Solution a) I first found the...
  29. D

    2D Systems and 4D Minkowski Space: Exploring Path Integrals

    The combination of special relativity and quantum mechanics in a single framework makes our understanding of such systems to be true only in 4D, Minkowski space...I have noticed that recent published work concerning 2D systems and I am not sure about this reduction of 4D to only 2D, does it mean...
  30. E

    Dirac Notation in building Path Integrals

    Alright, so I was wondering if anyone could help me figure out from one step to the next... So we have defined |qt>=exp(iHt/\hbar)|q> and we divide some interval up into pieces of duration τ Then we consider <q_{j+1}t_{j+1}|q_{j}t_{j}> =<q_{j+1}|e-iHτ/\hbar|q_{j}>...
  31. A

    Brownian Motion and Path Integrals

    I am reading "Quantum Mechanics and Path Integrals" (by Feynman and Hibbs) and working out some of the problems... as a hobby of sorts. I have run into a problem in section 12-6 Brownian Motion. On page 339 (of the emended edition), the authors demonstrate, by example, a method for...
  32. E

    Feynman's book: Quantum Mechanics and Path Integrals

    Hello, I tried to read Feynman's book: Quantum Mechanics and Path Integrals but it is so difficult. Is it a really important book if you want to learn Quantum Mechanics? If so what should I do in preparation to read it? Thanks
  33. Demystifier

    Path integrals and foundations of quantum mechanics

    It is frequently stated that path integral formulation of quantum mechanics is equivalent to the more traditional canonical quantization. However, I don't think it is really true. I claim that, unlike canonical quantization, path integral quantization is not self-sufficient. That's because...
  34. H

    What is the functional iW[J] for a system with 2 sources or 3 sources?

    Hello! I have a hard time getting to know what this exponential W(J) really is about. What is it even called? Zee writes: Z(J) = Z(J=0) * e^(i W(J)), and I suppose that this is the term that should be evaluated by summing over all possible pair of sources? What is W(J) for a system with 2...
  35. H

    Path Integrals in finite dimensions

    Hello fellow physicists! Last meeting with my supervisor I had just recovered from disease so all I have left are some equations for the math behind path integrals that don't make to much sense.. I was wondering if, maybe someone can help and clarify what he was trying to get at. It would be...
  36. S

    Exploring the Connection: Quantum & Classical Path Integrals

    What is the relationship between the quantum mechanical path integral and classical mechanics path integral?
  37. M

    Path integrals and supposed sum over paths

    If I understand correctly--a big if--the path integration method, at least when applied to plain old QM, is described as (1) every possible path the particle could take is assigned an amplitude, (2) sum up (integrate over) these amplitudes for all possible paths. The problem I have with this...
  38. H

    Path integrals (Feynman and Hibbs) clarification

    Hi all, I have been systematically working through the wonderful book Quantum mechanics and path integrals by Feynman and Hibbs and have come to realize that it has a shockingly large number of typos. I have been trying to derive eqn 5-13 on Pg 103 starting from eqn 3-42 Pg. 57 by using...
  39. R

    The harmonic oscillator in terms of path integrals without dirac notation

    Hi, I'm desperately searching for some literature which discusses the harmonic oscillator, preferably simple, in terms of the path integral formulation. I am unfamiliar with dirac notation and want something as simple as possible which gives general results of the harmonic oscillator in terms of...
  40. R

    Definitoon of trajectory in quantum physics, especially in path integrals

    What is the correct way to use the term trajectory in physics when discussin path integral forumaltion calculations. Here is the sentence i am trying to complete and am unsure if i may use the term trajectory: So the wavefunction offers a much more simplistic and perhaps more beautiful way to...
  41. LarryS

    Imaginary Time and Path Integrals?

    Steven Hawking writes in A Brief History of Time that time itself must sometimes have an imaginary component in order for Feynman's Sum-Over-Histories approach to work. Why, in a nutshell, is this so? Thanks in advance.
  42. maverick280857

    Srednicki Path Integrals eq 6.22

    Hi everyone, I was reading through the section on path integrals in Srednicki's QFT book. I came across equation 6.22 \langle 0|0\rangle_{f,h} = \int\mathcal{D}p\mathcal{D}q\exp{\left[i\int_{-\infty}^{\infty}dt\left(p\dot{q}-H_{0}(p,q)-H_{1}(p,q)+fq+hp\right)\right]} =...
  43. R

    Transition from QFT to Classical Physics using Path Integrals?

    I have a question in Srednicki's book regarding path integrals, but first I'll set it up so that no familiarity of the book is required to answer the question. The vacuum to vacuum transition amplitude for the photon field in the presence of a source is given by: <0|0>_J=\int \mathcal D A...
  44. R

    Path Integrals in QED: Solve by Substituting Classical Solution?

    When calculating a path integral, if the Lagrangian is quadratic in the field, then you can perform the path integral by just substituting in the classical solution for the field. So if you have free-field Lagrangians for electrons and photons, and add the standard QED interaction term - which...
  45. malawi_glenn

    Understanding the Breakdown of Path Integrals in Peskin

    Hello I am trying to follow how one can define a correlation function of two quantum fields using Path integrals. I have stumbled on equation 9.16 in Peskin, where they states that the functional integral can be split into: \int D \phi(x) = \int D\phi _1 (\vec{x}) \int D \phi _2 (\vec{x} )...
  46. T

    How is Euclideanization of Path Integrals Justified in Quantum Field Theory?

    I can't find any good references on Euclideanizing path integrals (from Minkowski to Euclidean metric). I understand how this is done in perturbative 1-loop calculations, where the pole structure of the Feynman propagators are used to perform the so-called Wick Rotation. This seems to be a...
  47. R

    Some Confusion with Path Integrals

    Hey folks, I'm reading Zee 'Introduction to QFT' and have a quick question on some terminology. On page 10 he describes how: \langle q_f|e^{-iHT}|q_i \rangle "is the amplitude for a particle to go from some initial state to some final state". He then derives the path integral...
  48. pellman

    Path integrals and supposed sum over paths

    If I understand correctly--a big if--the path integration method, at least when applied to plain old QM, is described as (1) every possible path the particle could take is assigned an amplitude, (2) sum up (integrate over) these amplitudes for all possible paths. The problem I have with this...
  49. R

    QED Path Integrals: Weinberg QFT Book Explained

    Does anyone understand how weinberg in his QFT book does QED via path integrals: specifically, how does he integrate out the adjoint dirac spinor?
  50. P

    What are the fundamental mathematics behind path integrals in QM?

    What is the 'fundamental' maths behind path integrals in QM?