# What is Path integral: Definition and 180 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. ### A Lagrangian in the Path Integral

Using free scalar field for simplicity. Hi all, I have a question which is pretty simple, we have the path integral in QFT in the presence of a source term: $$Z[J] = \int \mathcal{D}\phi \, e^{i \int d^4x \left( \frac{1}{2} \phi(x) A \phi(x) + J(x) \phi(x) \right)}$$ So far so good. Now...
2. ### I What do the authors of the paper mean here exactly by path integral?

Hi, I am new here so apologies if i am not using the right subforum. I don't have a physics background so i am not very technical but i do have a little bit of understanding. I was reading this paper by hawking/hertog and came across something that ended up confusing me. Here is it: "Pre-big...

5. ### Is the Result of the Path Integral Positive Due to Negative dx?

In the book it is mentioned that, in path c, the line integral would be: $$\int \vec{F}\cdot \vec{dr} = A \int_{1}^{0}xy dx = A\int_1^0 x dx = -\dfrac{A}{2}$$. but I think that dx is negative in that case, the result would be positive, right?
6. ### A Grassmann number: a more rigorous treatment?

Most textbooks on fermionic path integral only briefly introduce Grassmann numbers. However, I want a more systematic treatment to feel comfortable about this approach. For illustration, I have several examples here. Example 1: Consider a system with only one state, how to calculate ##\langle...
7. ### I Can a Path Integral Formulation for Photons Start from a Massless Premise?

I am aware that one usually starts from the Maxwell equations and then derives the masslessness of a photon. But can one do it the other way round? The action of photon would be ##S = \hbar \int \nu (1 - \dot{x}^2) \mbox{d}t##, where ##\nu## is the frequency acting as a Lagrange multiplier...
8. ### A Orthogonality of variations in Faddev-Popov method for path integral

Hi there, I've been stuck on this issue for two days. I'm hoping someone knowledgeable can explain. I'm working through the construction of the quantum path integral for the free electrodynamic theory. I've been following a text by Fujikawa ("Path Integrals and Quantum Anomalies") and also...
9. ### I Feynman path integral and events beyond the speed of light

In Richard Feynman's book "The Strange Theory of Light and Matter", in chapter 2, he explains how to calculate the probability that light from some source will be reflected by a mirror and be detected at some location. He explains how you sum up all of the probability amplitudes (represented...
10. ### I Understanding how to derive the Feynman rules out of the path integral

I am studying interacting scalar fields (from Osborn) using the path integral approach. We define the functional integral \begin{equation*} Z[J] := \int d[\phi] e^{iS[\phi] + i\int d^d x J(x) \phi(x)} \tag{1} \end{equation*} The idea is to differentiate ##Z[J]## with respect to ##J## and end...

13. ### A Derive the Principle of Least Action from the Path Integral?

Several weeks ago I had considered the question as to how one can start from the Schroedinger Equation, and after several transformations, derive F=ma as a limiting case. At some point in my investigations of this derivation, it occurred to me that this is simply too much work. While in...
14. ### 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...
15. ### B What Is the Occurrence Rate of Path Integral in a Photon Double-Slit Experiment?

Hi forgive my very ignorant question. How frequently does 'Path Integral', curved shot and normal shot happen out of say 100 shots with a photon?
16. ### A Does Feynman's path integral ignore alternative histories?

Summary: Does Richard Feynman's multiple histories ignore alternative histories? Did Richard Feynman's multiple histories (https://en.wikipedia.org/wiki/Multiple_histories) ignore the existence of other alternarive histories or paths? I ask this referring to this comment from this page...
17. ### I About the Feynman Path Integral and Principle of Least Time

I don't entirely get why we usually say that only the shortest path contributes in the path integral. If you calculate the volume of nth fresnel zones which is the locus where the path length is between n-1 and n wavelengths from the shortest path in 3 dimensions, they are the same I believe. So...
18. ### A Path Integral Approach To Derive The KG Propagator

I'm having trouble understanding a specific line in my lecturers notes about the path integral approach to deriving the Klein Gordon propagator. I've attached the notes as an image to this post. In particular my main issue comes with (6.9). I can see that at some point he integrates over x to...
19. ### I Counterpart of Feynman integral in de Broglie-Bohm theory

I am looking for good references / clarifications on the subject. First of all, my question is concerned only with mathematical formulation of something that sort of plays the role of the Feynman path integral of the "standard" QFT. It is not concerned with the physical or philosophical...
20. ### A Question about equivalence of Path Integral and Schrodinger

I've seen a proof that the path integral formulation of quantum mechanics is equivalent to solving Schrodinger's equation. However, it appears to me that the proof actually depended on the Hamiltonian having a particular form. I'm wondering how general is the equivalence. Let me sketch a...
21. ### I QFT Path Integral: Confusions on Left & Right Most Terms & Vacuum State

Hello! I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N exp(i\delta t \int d^3x L[\Phi_j,\partial_t \Phi_j])$$ What happens when we have the left and right...
22. ### 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?
23. ### Path Integral Setup for Given Initial and Final Points

Homework Statement The path integral from (0,0,0) to (1,1,1) of $$<x^2,2yz,y^2>$$. I am a little confused about the setup.Homework Equations $$\int_{a}^{b} v.dl$$The Attempt at a Solution Here is how I set it up. $$\int_{0}^{1}x^2 dx + \int_{0}^{1}2yz dy + \int_{0}^{1}y^2 dz$$ Since the...
24. ### I Is the Action Always a Minimum in the Principle of Least Action?

Hello, When we applying the principle of least action, we require ##\delta S=0##, which corresponding to the action S being an extremum. I am just wondering why do we say that the action is a minimum instead of a maximum for a physical path? Can I use the path integral to explain this problem...

34. ### Deriving an influence functional is deriving me mad (Help)

Homework Statement I am attempting to derive Caldeira-Leggett's influence functional found in their paper "Path Integral Approach To Quantum Brownian Motion". If you find my following statements confusing, then pages 16-18 of http://web.science.uu.nl/itf/Teaching/2006/MxWakker.pdf show the...
35. ### Flux-flux correlation function under Feynman's path integral

This is a chemically inspired problem, but the path is fully quantum mechanics and a bunch of integrals. How does one calculate fully quantum mechanical rate ($\kappa$) in the golden-rule approximation for two linear potential energy surfaces? Attempt: Miller (83) proposes...
36. ### Quantum Path integral formulation

Hello! Can someone suggest me a good reading about path integral formulation of quantum mechanics? I took 2 undergrad courses on QM, so I would like something focusing on path integral (maybe some problems too). I don't necessary want a book, even some online pdf that contains some good...
37. ### I Difference between statistical and dynamical properties

Hi All, What are the main differences between statistical and dynamics properties in physics? Could you please explain the difference for problems in both classical and quantum mechanics. For instance, path integral molecular dynamics is supposed to give statistical properties of a quantum...
38. ### A On the equivalence of operator vs path integral in QFT

I have read many textbooks and googled google times for a clear explanation, but I could not find one. How does raising and lowering -annihilation/ creation-(is that energy or particle number?) translate to transition probabilities of path integral.
39. ### A Path integral formula for a state with non-trivial time dependency

Consider a system with a time-dependent Hamiltonian. We know that the evolution of the state of this system, is given by ## \displaystyle |\Psi(t_1)\rangle=T \exp\left( -i \int_{t_0}^{t_1} dt H(t) \right) |\Psi(t_0)\rangle ##. Do you think you can prove that the path integral formula for the...
40. ### I Path integral implies superluminal motion?

In the path integral formalism, where we treat a photon as if it takes every possible path, aren't the possible paths limited by the speed of light? If we were to perform the double slit experiment, and shield the detector after a specified time frame to limit the time for a photon to make the...
41. ### A Path integral formula for vacuum to vacuum amplitude

I'm reading the path integral chapter of Schwartz's "Quantum Field theory and the Standard model". Something seems wrong! He starts by putting complete sets of states(field eigenstates) in between the vacuum to vacuum amplitude: ## \displaystyle \langle 0;t_f|0;t_i \rangle=\int D\Phi_1(x)\dots...
42. ### A Ground state wave function from Euclidean path integral

From the path integral approach, we know that ## \displaystyle \langle x,t|x_i,0\rangle \propto \int_{\xi(0)=x_i}^{\xi(t_f)=x} D\xi(t) \ e^{iS[\xi]}##. Now, using ## |x,t\rangle=e^{-iHt}|x,0\rangle ##, ## |y\rangle\equiv |y,0\rangle ## and ## \sum_b |\phi_b\rangle\langle \phi_b|=1 ## where ## \{...
43. ### I Vanishing Hamiltonian for quantum path integral

The path integral in quantum mechanics involves a factor ##e^{iS_{N}/\hbar}##, where ##S_{N}\equiv \sum\limits_{n=1}^{N+1}[p_{n}(x_{n}-x_{n-1})-\epsilon H(p_{n},x_{n},t_{n})].## In the limit ##N \rightarrow \infty##, ##S_{N}## becomes the usual action ##S## for a given path.When the...
44. ### I Path integral and causally disconnected parts of universe....

One more question: it appears that portions of the universe are expanding away from us faster than the speed of light. Given this, particles in two 'parts' of the universe that are no longer causally connected should not be able to influence each other (due to speed of light constraint). So...
45. ### A Path integral computes time-ordered products

In general, ##\displaystyle{\langle q_{f}|e^{-iHt/\hbar}|q_{i}\rangle=\int\mathcal{D}q(t)\ e^{iS[q]/\hbar}}## and ##\displaystyle{\langle q_{f}|\hat{Q}(t)|q_{i}\rangle=\int\mathcal{D}q(t)\ e^{iS[q]/\hbar}}\ q(t).##How can one switch from the above expressions to the following...
46. ### A Utility of path integral formulation of quantum mechanics

How does the path integral formulation of quantum mechanics as given by ##\langle q_{f}|e^{-iHt/\hbar}|q_{i}\rangle = \int \mathcal{D}q(t)\ e^{iS[q]/\hbar}## make manifest aspects of quantum mechanics such as symmetries?
47. ### I Field eigenstates and path integral formulation

Recently I have reviewed by reference books to get a better understanding of the fundamentals of QFT and there is one thing I still do not understand. In the QFT derivation of the path integral formula, it seems that every book and online resource makes the assumption that for the field operator...

Trying to accomplish a monte carlo simulation on the condensed state of 4He, yet I am in my sophomore year and know only a bit of quantum statistical physics. Is there any documentations recommended for beginners to the algorithm applied to 4He? I've found some but they are not friendly to...
49. ### Comparing Path Integral Conventions in QED

Hello. I am doing some reading on QED and am getting a bit confused on the different conventions used. In Matthew Schwartz's book we have the Lagrangian given as: LQED = -¼ FμνFμν + iψ*γμ(∂μ + ieAμ)ψ - V(ψ*ψ) And the path integral factor is exp(iS). In another text, however, I see the QED...
50. ### Why Do Path Integrals Involve Multiple Integrations Instead of Simple Summation?

Source: http://web.mit.edu/dvp/www/Work/8.06/dvp-8.06-paper.pdf Regarding page 5 of 14, I don't understand the multiple integrals thing. Ain't we supposed to sum up all the paths like in this equation (4) but why do they instead multiply and integrate over integration and so on?Also for...