# What is Path integral formulation: Definition and 39 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. ### I What is the significance of the T - V Lagrangian of a system?

Let E be a fixed immutable quantity. E can be freely exchanged between T and V, as long as $$T + V = E$$ 1. What does the quantity $$\int_x T - V$$ signify? What is the importance of this quantity? -------------------- Let E now be the budget of a factory. E can either be spent on T or V in...
2. ### 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...
3. ### Proof involving exponential of anticommuting operators

For ##N=1##, I have managed to prove this, but for ##N>1##, I am struggling with how to show this. Something that I managed to prove is that $$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$ which generalizes what I initially...
4. ### 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...
5. ### 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...
6. ### 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...
7. ### 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...
8. ### 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...
9. ### I A Closer Look at the Randomness of Quantum Measurements in QED

In all Quantum Physics experiments, the sequence of measurement results is inherently random. Consider just the position observable. In the Schrodinger picture of non-relativistic QM, in each measurement-event, nature steps in and randomly selects one of the observable's eigenvalues/vectors to...
10. ### I The propagator and the Lagrangian

I note the following: $$\begin{split} \langle\vec{x}_n|e^{-i \frac{\mathcal{H}_n}{\hbar} (t_n-t_0)}|\vec{x}_{0}\rangle &=\delta(\vec{x}_n-\vec{x}_0)e^{-i \frac{\mathcal{H}_n}{\hbar} (t_n-t_0)} \end{split}$$I divide the time interval as follows...
11. ### I Checking My Understanding: Lagrangian & Path Integral Formulation

I note the following: \begin{split} \langle \vec{x}| \hat{U}(t-t_0) | \vec{x}_0 \rangle&=\langle \vec{x}| e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} | \vec{x}_0 \rangle \\ &=e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} \delta(\vec{x}-\vec{x}_0)...
12. ### A Help to understand the derivation of the solution of this equation

Please, help here people. Im reading this article Wave Optics in Gravitational Lensing (T. T. Nakamura, 1999) . In the article start work with $$(\nabla ^2 +\omega)\tilde\phi = 4\omega^2U\tilde\phi$$ where $$\tilde\phi = F(\vec r)\tilde\phi_{0}(r)$$. Using...
13. ### 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...
14. ### 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...
15. ### A Taylor series expansion of functional

I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field, L=½(∂φ)^2 - m^2 φ^2 in the equation, S[φ]=∫ d4x L[φ] ∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2) Particularly, it is in the Taylor series...
16. ### 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?
17. ### I Classical/QM justification of principle of least action

Hi. Is the principle of least (better: stationary) action only an axiom in classical mechanics, or can it be derived from a more profound (classical) principle? As far as I know, it can be derived from the path integral formulation of QM. Is this a more profound justification for the principle...
18. ### 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...
19. ### 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...
20. ### Invariance of integration measure under shifts in field

I've been trying to teach myself the path integral formulation of quantum field theory and there's a point that's really bugging me: why is the integration measure ##\mathcal{D}\phi(x)## invariant under shifts in the field of the form \phi(x)\rightarrow\tilde{\phi}(x)=\phi(x)+\int...
21. ### Relation between phase space and path integral formulation?

I am trying to conceptually connect the two formulations of quantum mechanics. The phase space formulation deals with quasi-probability distributions on the phase space and the path integral formulation usually deals with a sum-over-paths in the configuration space. I see how they both lead...
22. ### How does uncertainty apply to an electron's magnetic moment measurement?

The electron paths in the path integral formulation of qm are in ordinary space-time. The end-points of the paths in the steady states must be in ordinary space-time as well. The electron's potential is exact or nearly exact either by measurement or by calculation. The electron's locations...
23. ### Two-point correlation function in path integral formulation

Suppose that I have already calculated the two-point correlation function for a Lagrangian with no interations using the path integral formulation. \langle \Omega | T[\phi(x)\phi(y)] | \Omega \rangle = \frac{ \int \mathcal{D}\phi \phi(x)\phi(y) \exp[iS_0] }{ \int \mathcal{D}\phi \exp[iS_0] }...
24. ### Not seeing the action of a free particle in the Path Integral Formulation

In the very first example of Feynman and Hibb's Path Integral book, they discuss a free particle with \mathcal{L} = \frac{m}{2} \dot{x}(t)^2 In calculating it's classical action, they perform a simple integral over some interval of time t_a \rightarrow t_b. S_{cl} = \frac{m}{2}...
25. ### General Question About Path Integral Formulation of QM

Hello all, I will be learning about the path integral formulation, among other topics, in an advanced QM class during this upcoming semester, so I read ahead a little. I understand that, essentially, the propagator between two points in spacetime is the normalized sum of exp(i*2pi*S/h) over...
26. ### Path Integral Formulation: Allowable Paths?

In Feynman’s Path Integral formulation of QM, one starts by considering all possible paths between two fixed space-time events. Question: Must the wave-length associated with each allowable path divide evenly into the spatial length of the path?
27. ### Path integral formulation

Hey guys, can anyone suggest good learning materials (books, lectures, pdfs...) for the path integral formulation of QM? I don't need anything too advanced, just a thorough intro. Are Feynman's books any good? EDIT: Oh yeah, some quantum thermodynamics too in the mix would be cool.
28. ### Path integral formulation of non-relativistic quantum mechanics

I am looking for a textbook that introduces and discusses the path integral formulation of non-relativistic quantum mechanics? Would you have some suggestions for me? Thanks.
29. ### What is the derivation for the path integral formulation of quantum mechanics?

I'm not quite satisfied by the derivation I've found in Sakurai (Modern Quantum Mechanics) and was trying to 'derive' it myself. I'd like some help to seal the deal. I've described below what I've done. Please tell me where to go from there. I know the solution to the Schrodinger equation can...
30. ### Path integral formulation - uses/related topics

Hi, I'm writing a paper on the PI formulation and i wondered if anyone has any other ideas as to what its uses are and what other topics it is used in. I came across the CDT (causal dynamical triangulation) theory and this uses a non perturbativ PI approach so i will talk about that in the...
31. ### Correlation Functions in Path Integral Formulation of QFT

Hi, I was going through section 9.2 of Peskin and Schroeder, and came across equation 9.16 which reads \int\mathcal{D}\phi(x) = \int \mathcal{D}\phi_{1}({{\bf{x}}}) = \int \mathcal{D}\phi_{2}({{\bf{x}}}\)int_{\phi(x_{1}^{0},{\bf{x}})\\\phi(x_{1}^{0},{\bf{x}})}\mathcal{D}\phi(x) What does the...
32. ### Does Multi Path Integral Formulation Violate Special Relativity?

does the multi Path integral formulation violate special relativity ! do we get speeds faster than c.
33. ### Pre-requisites for path integral formulation?

Pre-requisites for path integral formulation? Does anybody have any idea of the pre-requisites to learn Feynmann's path integral formulation? (properly) Right about now, I'm still learning about Lagrangian and Hamiltonian mechanics which focuses on the principle of least action. Right now, the...
34. ### Feynman's Path integral formulation

Does Feynman's path integral formulation violate relativity , we get path's that are faster than c.
35. ### Path Integral formulation of Loop Cosmology (a first)

Today (17 March) we got our first news of a Path Integral formulation of LQC. Adam Henderson is a PhD student in Ashtekar's group at Penn State. He gave an internationally distributed seminar talk on his research. http://relativity.phys.lsu.edu/ilqgs/henderson031709.pdf...
36. ### Understanding Non-Relativistic Path Integral Formulation

I'm doing a project for my quantu class on the non-relativistic path integral formulation. I took out "quantum mechanics and path integrals" feynmann, but he doesn't seem to like explaining explicitly how certain results are obtained... so my two main questions are should the weight...
37. ### Path integral formulation of wave-optics

A few years back,I stumbled upon a nice idea which I am reporting below:- Just as classical mechanics is the h \rightarrow 0 limit of quantum mechanics(rather action >> \hbar,from path integral formulation),so should it be possible to argue from a path integral approach, that ray optics is...
38. ### Path integral formulation of Bohmian mechanics

If Bohmain mechanics is true then the path integral: \int{d[\phi]}e^{(i/\hbar)\int_{a}^{b}Ldt where the Lagrangian is: L=(1/2)m(dx/dt)^{2}-V(x)+(\hbar^{2}/2m)\nabla^{2}\rho should be equal to its semiclassical expansion...(as in both cases are trajectories) my question is how would one...
39. ### Equivalence between path integral formulation and matrix formulation

Does anyone know where to find the "direct" (not by prove they are both equal to Schrodinger formualtion )proof?