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. ShayanJ

    Path integral propagator as Schr's eq Green's function

    Its usually said that the propagator ## K(\mathbf x'',t;\mathbf x',t_0) ## that appears as an integral kernel in integrals in the path integral formulation of QM, is actually the Green's function for the Schrodinger equation and satisfies the equation below: ## \left[ -\frac{\hbar^2}{2m}...
  2. D

    Looking for online videos on QFT using path integral method

    Hi. I am just starting to study QFT using the path integral method and for which the main textbook is by Srednicki. Does anyone know of any good online videos which would be suitable Thanks
  3. B

    Spherical coordinates path integral and stokes theorem

    Homework Statement Homework Equations The path integral equation, Stokes Theorem, the curl The Attempt at a Solution [/B] sorry to put it in like this but it seemed easier than typing it all out. I have a couple of questions regarding this problem that I hope can be answered. First...
  4. C

    Definition of the Path Integral

    Why is it that, in the definition of the path integral, we have the product of neighboring integrals of the form : ∫Φdx1...dxn when the whole idea is based on adding the contribution of neighboring paths. I need some help understanding why it is of the form ∫Φdx1...dxn and not the form...
  5. nikosbak

    Compute 3-Point Function QFT Homework with Fermions

    Homework Statement I'm working on path integrals for fermions and I came across an exercise that ask to compute the three point functions , one of that is the: $$<0|J^{\mu}(x_1)J^{\nu}(x_2)J^{\rho}(x_3)|0> $$ where $$J^{\mu}$$ is the current $$J^{\mu}=\bar{\psi}\gamma^{\mu}\psi$$. ***Can you...
  6. A

    How can I derive the path integral using the Baker-Campbell-Hausdorff formula?

    On the attached picture I have started trying to derive the path integral but I don't know how I get further. Can anyone help me? Also I have used that: exp(-iHΔt/ħ) = exp(-i(T+V)Δt/ħ) = exp(-iTΔt/ħ)exp(-iVΔt/ħ) But my book says that this identity is only correct to first order, i.e. there is...
  7. A

    Greens functions from path integral

    Let me post this question again in a slightly modified form. On the attached picture the path integral for the partion function: Z = Tr(exp(-βH)) Now according to what it says on the picture it should be easy from this to get the Green's function in the path integral formalism. The Green's...
  8. A

    Greens function path integral representation

    In my book the path integral representation of the green's function is given as that on the attached picture. But how do you go from the usual trace formula for the Green's function 2.6 to this equation?
  9. A

    Construction of the path integral

    I am reading about the construction of the path integral in which states that the propagator is given by: <qf l exp(-iHt/ħ l qi > = ∫Dq exp(i/ħ∫dt L(q,q')) and Dq is the integration measure given by limN→∞Πn=1N-1 dqn. Can someone help me understand this integration measure a bit better? I...
  10. Matta Tanning

    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...
  11. T

    Understanding the Role of Multiple Integrals in the Path Integral Formula

    Here's the 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. What is that supposed to mean? Ain't we supposed to sum up all the paths but why do they do the multiple integral thing? Also regarding page 4 of...
  12. F

    Can the Feynman Path Integral Account for Two Non-Interacting Particles?

    Is there such a thing as a Feynman Path Integral for two non-interacting particles? I find myself wondering how the wave function of a single particle is changed in the presence of a second particle. The Feynman path integral takes into account every possible path that a particle can take. So...
  13. G

    Continuing to Euclidean Space Justified in Path Integral?

    It seems to me that in a path integral, since you are integrating over all field configurations, that going into Euclidean space is not valid because some field configurations will give poles in the integrand of your action, and when the integrand has poles you can't make the rotations required...
  14. P

    Mathematical state of Path Integral?

    So I've just recently started learning path integral methods in QFT and string theory, and I've heard from numerous sources that the path integral (specifically fermionic path integrals, perhaps?) are objects which are not at all on solid mathematical ground. The feeling I get is that perhaps...
  15. D

    Feynman Diagram for phi^4 theory (path integral)

    Homework Statement Hey guys! So basically in the question I'm given the action S=\int d^{d}x \left[ \frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi\eta^{\mu\nu} - \frac{m^{2}}{2}\phi^{2} -\frac{\lambda}{4!}\phi^{4}\right]. I have use the feynman rules to calculate the tree level diagram with...
  16. M

    Unitarity and locality on patgh integrals

    my question is this: you know than in feynman path integra, you integrate eiS/hbar along all the fields. you also know that S is real and that it is the integral of local functions (fields and derivatives of fields). you also know that path integral generates an unitary and local...
  17. Xiaomin Chu

    Path integral and discontinuous paths

    Somebody asked about this but that thread was closed very soon. In physics, discontinuous paths breaks locality so they must be 0; but mathematically, they causes some problems. Discontinuous functions must not be differentiable, so it's impossible to calculate the action over that path. However...
  18. Quarlep

    Understanding the Path Integral Formulation in Feynman's Theory

    What's the main logic of path integral formulation ? (Feyman path integral formulation) I mean what's the reason to think this way ? Thanks
  19. nuclearhead

    Where does the 'i' come from in QFT path integral?

    So I've been thinking about the axioms of quantum field theory. In particular the expression for the particle amplitudes: G(x1,x2,...,xn) = ∫Φ(x1)Φ(x2)...Φ(xn)ei S[Φ]/ħ D[Φ] / ∫Φ(xn)ei S[Φ]/ħ D[Φ] But I've been struggling to explain the existence of the 'i'. It seems like this is a...
  20. G

    Extracting ground state in path integral

    Suppose you have the transition amplitude in the presence of a source <q''t''|q't'>_{f} To extract the ground state, we change the Hamiltonian to H-i\epsilon , because we can write: $$|q't'>=e^{iHt'} |n><n|q> \rightarrow e^{iE_0t'} |0><0|q>=<0|q>e^{iHt'} |0>=<0|q> |0 t'> $$ where only...
  21. B

    Path Integral From Heisenberg Uncertainty?

    Two questions about the path integral: a) Intuitively, how does one (as Landau does) start quantum mechanics from Heisenberg's uncertainty principle, which states there is no concept of the path of a particle, derive Schrodinger equation i \hbar \tfrac{\partial \psi}{\partial t} = H \psi the...
  22. S

    Action corresponding to photon emission

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  23. Spinnor

    Lattice QCD, path integral, single "path", what goes on at a point?

    Say we try and calculate the ground state energy of the bound state of a quark antiquark meson via lattice QCD. Say I look at one space time lattice point of one path. Do the fermi fields "live" on the lattice points? Do the boson fields "live" on the legs between the space time lattice points...
  24. J

    Use of Trotter Theorem in Path Integral Molecular Dynamics

    I am unable to prove step 8.3 in this proof of the path integral formulation of molecular dynamics https://files.nyu.edu/mt33/public/jpc_feat/node11.html Any help would be much appreciated.
  25. B

    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] }...
  26. Ravi Mohan

    What happens to infinitesimal time in path integral

    Hi, I am studying path integral formulation from Ballentine. Till equation 4.50, I follow quiet well. G(x,t;x_0,t_0) = \lim_{N \to \infty}\int\ldots\int\left(\frac{m}{2\pi i\hbar\Delta t}\right)^{\frac{N+1}{2}}\exp{\sum_{j=0}^{N}\left(\frac{im(x_{j+1}-x_j)^2}{2\hbar\Delta...
  27. Matterwave

    Path integral Quantum Mechanics

    Hi guys, I have a few questions regarding Feynman's formulation of quantum mechanics. Given the propagator K(x',t';x,t), when is this propagator equal to: $$K(x',t';x,t)=Ae^{\frac{i}{\hbar}S_{cl}(x',t';x,t)}$$ Where S_cl is the classical action evaluated along the classical path of motion...
  28. G

    Srednicki's normalization of path integral

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  29. C

    A challenging vector field path integral

    Homework Statement Evaluate ∫F dot ds Homework Equations F = < 1 - y/ (x^2 + y^2) , 1 + x/(x^2 + y^2) , e^z > C is the curve z = x^2 + y^2 -4 and x + y + z = 100 The Attempt at a Solution I don't think Stokes theorem applies since the vector field is undefined at the origin, so I'm...
  30. marcus

    A purely geometric path integral for gravity

    This paper seems to me especially interesting: http://arxiv.org/abs/1308.2946 Purely geometric path integral for spin foams Atousa Shirazi, Jonathan Engle (Submitted on 13 Aug 2013) Spin-foams are a proposal for defining the dynamics of loop quantum gravity via path integral. In order for...
  31. E

    Path Integral to determine Work Done

    When doing a path integral to determine the work done by a particle : http://latex.codecogs.com/gif.latex?\int&space;\textbf{w}\cdot&space;d\mathbf{s} Where F is some vector. Now, I can't remember what ds is. I vaguely seem to remember that it is some unit vector parallel to and in the...
  32. L

    Path integral and gaussian integral

    I am trying to calculate the functional for real scalar field: W[J] = \int \mathcal{D} \phi \: exp \left[{ \int \frac{d^4 p}{(2 \pi)^4}[ \frac{1}{2} \tilde{\phi}(-p) i (p^2 - m^2 +i \epsilon) \tilde{\phi}(p)} +\tilde{J}(-p) \tilde{\phi}(p)] \right] Using this gaussian formula...
  33. A

    Feynman’s path integral and an electron in a Penning trap

    Last night the BBC repeated Brian Cox, A Night With the Stars (). At some point a calculation is done using a simplified version of Feynman’s path integral, where the mean time is estimated for a diamond to be found outside a small box: t>\frac{x Δx m}{h} The box was not expected to be...
  34. S

    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}...
  35. G

    Jacobian in path integral equal to one?

    Consider: \int d\phi e^{iS[\phi]}=\int d\phi' J e^{iS'[\phi']} where J is the Jacobian. If the transformation of variables to phi' is a symmetry of the action [i.e., S'=S], then this becomes: \int d\phi e^{iS[\phi]}=\int d\phi' J e^{iS[\phi']} But doesn't this imply that the Jacobian has...
  36. C

    Using Path Integral to calculate propagator

    Hi, It's great to find this forum. I'm teaching myself QM using Shankar, it's a great book, I've covered 8 chapters so far. My question is about the notion of using Path Integral method to calculate the propagator. The recipe given by Shankar says the propagator is U(x,t;x')=A\int...
  37. R

    Interpreting path integral averages as measure integrals

    Hi all, Sorry if this is in the wrong place. I'm trying to understand probability theory a bit more rigorously and so am coming up against things like lebesgue integration and measure theory etc and have a couple of points I haven't quite got my head around. So starting from the basics...
  38. L

    Feynman Path Integral - All Possible Paths - Relativity Conflict?

    Hello I have not familiarised myself with the mathematics etc I merely have the conceptual idea that, for example, with the two slit experiement and electron is permitted to take "all possible paths" from the electron gun to the detector screen. The thought occurred (and as will all thoughts...
  39. C

    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...
  40. marcus

    Solving the Quantum Zeno problem with path integral QG

    Schroeren's new paper "Decoherent histories of spin networks" made me aware of some work by two people at Cambridge DAMTP and London Imperial's Blackett Lab on the Quantum Zeno (QZ) effect in the path integral e.g. decoherent histories (DH) context. Schroeren's paper...
  41. E

    Can the Path Integral be Calculated from the Propagator?

    Homework Statement Itzykson-Zuber ch. 9-1-1: If H=\frac{P^2}{2m}-QF(t) then \frac{\delta}{i\delta F(t)}\langle f\mid i\rangle_F=\langle f\mid Q(t)\mid i \rangle_F Ok, I understand that. But then it states: if H=\frac{P^2}{2m}+V(Q) then \int\mathcal{D}(q)\exp\left\{i\int...
  42. LarryS

    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?
  43. C

    Path integral over a function with image in a circunference

    Homework Statement Let \vec{F}: ℝ^{2}->ℝ^{2} be a continuous vector field in which, for every (x, y), \vec{F}(x, y) is parallel to x\vec{i}+y\vec{j}. Evaluate \int_{γ}\vec{F}\cdot d\vec{r} where γ:[a, b]->ℝ^{2} is a curve of class C^{1}, and it's imagem is contained in the circunference...
  44. U

    Why Does the Path Integral in Quantum Mechanics Include an Imaginary Unit 'i'?

    I've been studying the path integral approach to QM on my own, and trying to draw some analogies between the partition function of QM \begin{equation}Z_{QM}=\int D\varphi e^{\frac{i}{\hbar}S[\phi]}\end{equation} and that of statistical mechanics...
  45. K

    Path Integral of Triangle: Parameterization & Solution Explanation

    Homework Statement The problem asks: find the integral of gamma F.ds where F(x,y,z) = (e^z, e^y, x+y). gamma being a triangle with vertices: (1,0,0) (0,1,0) (0,0,1) going in a counterclockwise direction Homework Equations The Attempt at a Solution So I'm not even sure...
  46. S

    How Does Path Integral Formalism Derive the Quantum Propagator?

    Path Integral Formalism Reading through Shankar atm, up to page 232/233. Reference to pages if interested. http://books.google.co.nz/books?id=2zypV5EbKuIC&printsec=frontcover&source=gbs_vpt_reviews#v=onepage&q=232&f=false(sorry I am too noob at latex to type all the formulas out..) It's...
  47. K

    Path integral and partition function

    I have some confusions identifying the following objects: (1)Some transition amplitude involving time evolution(Peskin page 281, eqn 9.14): \langle\phi_b(\mathbf x)|e^{-iHT}|\phi_a(\mathbf x)\rangle=\int{\cal D\phi \;exp[i\int d^4x\cal L]} (2)Partition function(after wick rotation)...
  48. T

    Path Integral example, analysis of integral limits for K-G theory

    In general I find in books that the path integral approach is an equivalent alternative of the hamiltonian approach for QFT (and for QT in general, but my concern is with QFT). There I usually find that this method is usually developed in a formal way and used to derive Feynman rules, gauge...
  49. Z

    Heat death, quantum uncertainty and Feynman's Path Integral

    Hi all, as a complete noob, I must first ask that people understand that I have only a layman's understanding of cosmology. However, after watching a few of Brian Cox's lectures on entropy and the heat death of the universe, I had a rather interesting thought (although as I am not a...
  50. E

    Path Integral Basics (Why dimension increases in the integrals?)

    Alright, I have a kind of dumb question: Why do I distinguish between dq and dqi when considering the propagation from qi to q to qf? For example, if we want the wave function at some qf and tf given qi and ti, we may write: ψ(qf,tf)=∫K(qftf;qiti)ψ(qi,ti)dqi Why do we distinguish between dqi...
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