Questions tagged [path-integral]

Path integral formulation (Due to Feynman) is a major formulation of Quantum Mechanics along with Matrix mechanics (Due to Heisenberg and Pauli), Wave Mechanics (Due to Schrodinger), and Variational Mechanics (Due to Dirac). DO NOT USE THIS TAG for line/contour integrals.

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69 views

In the Feynman Path Integral, why must “contributing paths” be continuous? Or is this a false notion?

I have been studying the Feynman path integral and its various derivations, and I've run into a bit of a problem. The standard Feynman path integral appears as follows: $$ \int \mathcal{D}[x(t)]\exp\...
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1answer
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Question about an OPE for the free massless scalar CFT

In page 78 of David Tong's notes on CFT https://www.damtp.cam.ac.uk/user/tong/string/four.pdf, he finds that the propagator for a theory of free massless scalars is $$\langle X(\sigma)X(\sigma')\...
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Wilson loop as path integral of parallel transport action

I am trying to get that the path integral of the parallel transport action is the Wilson loop. Here is the setting: Let $w$ be a complex vector dimension $N$, and $A_{\mu}$ a fixed Yang-Mills ...
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Functional integral in spontaneous symmetry breaking

So, functional integral is defined to be (with $\lvert\Omega\rangle$ is the vacuum state): $$\frac{\langle\Omega\rvert X \lvert\Omega\rangle}{\langle\Omega\vert\Omega\rangle} = \frac{\int \mathcal{D} ...
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Measure in the Fourier Representation of the Coherent States Path Integral

The problem I have arises in the context of condensed matter physics. I am largely following chapter 4 about functional integration in the book by Altland and Simon. Consider the coherent states path ...
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How to unify the cumulant expansion and Feynman diagram expansion?

In most QFT books, the perturbation theory is given by "Taylor expansion". When evaluating 2-points, the numerator gives all the diagrams, i.e. $$\int D[\phi]e^{iS[\phi]}\phi_1\phi_2=\int D[\phi]\...
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Scalar field propagator in curved space from path integral

Consider a scalar masless field (in 2d for concreteness) in a curved space with standard action $$S=\frac{1}{4\pi}\int d^2x \sqrt{g}g^{ab}\nabla_a\phi\nabla_b \phi$$ There is an elegant way to derive ...
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1answer
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How to verify this statement in optical theorem?

There is a statement in optical theorem (From Peskin & Schroeder): It is easily checked (in QED, for example) that each diagram contributing to an S-matrix element $\mathcal{M}$ is purely real ...
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Decoupling of ghost fields in axial-gauge QCD

After quantizing QCD using the Faddeev-Popov "prescription", we end up with the original QCD Lagrangian plus the gauge-fixing term, \begin{equation} -\frac{1}{2\alpha}(n\cdot A)^2, \end{equation} and ...
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What's the path of least action for fermions off-shell?

The Lagrangian of fermions is first order both in space-derivatives and time-derivatives. In the variation of the action one usually fixes both the initial point and end point. I have the following ...
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1answer
520 views

Srednicki Eqs. (6.22) and (9.6). How to get rid of $i\epsilon$ in the interaction term?

I'm studying qft from Srednicki's book. If one writes down the full $i\epsilon$ terms, passing from Eq. (6.21) (non-perturbative definition) to Eq. (6.22) (perturbative definition) yields $$\left<0|...
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Getting Feynman propagator using path integral

In QM using Feynman path integral(FPI) we derive the propagator of free particle which comes out to $$(f(t))e^{iS_{cl}/\hbar}$$ But in QFT the Feynman propagator is derived using the differential ...
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Does this path integral give the minimum proper time-squared between to points?

Taking the usual definition of the propagator for a massless scalar field and taking the reciprocal: $$f(x,y)\propto\left(\int \phi(x)\phi(y) e^{i\int \sqrt{-g}g^{\mu\nu}(z)\partial_\mu \phi(z)\...
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Supersymmetric localisation of 2D super YM on $S^2$

I wanna to understand how to calculate partition function for pure abelian Yang-Mills theory. To do this, I need follow some usual step's (I follow Benini, Localization in supersymmetric field ...
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Is there a deep reason why action comes from a local lagrangian?

In both classical and quantum physics Lagrangians play a very important role. In classical physics, paths that extremize the action $S$ are the solutions of the Euler-Lagrange equations, and the ...
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1answer
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Is the Feynman's path integral a density?

The Feynman-Kac path integral formula is used to solve parabolic equations related to stochastic processes. Considering the probabilistic expression, the solution is indeed not a density. However, ...
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1answer
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How to connect Green function to propagator?

I know that there has already been many questions related to this question, such as in Differentiating Propagator, Green's function, Correlation function, etc. However, that question mainly ...
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Can the real-time Green's function be written in the form of path integral on the real axis? [closed]

In every textbook, the path integral of the Green's function is written in imaginary-time. I wonder whether we could write real-time green function in the path integral form.
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Fourier transform of matrix element of evolution operator

I learn ''Path integrals in quantum mechanics'' by Jean Zinn-Justin now. There is a chapter about calculating the path integral for particle on a ring (rigid rotator). So, after some calculations we ...
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Normalizing Propagators (Path Integrals)

In the context of quantum mechanics via path integrals the normalization of the propagator as $$\left | \int d x K(x,t;x_0,t_0) \right |^2 ~=~ 1$$ is incorrect. But why? It gives the correct pre-...
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Integrating the gauge covariant derivative by parts

I was watching a set of lectures on effective field theory and the lecturer said that you can always integrate the covariant derivative by parts due to gauge symmetry. For example, if I understand ...
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1answer
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Fixing time in Feynman phase space path integral

The phase space version of Feynman's path integral expression for the free particle propagator involves a (formal) sum over paths in phase space with fixed $q$ endpoints and (as far as I'm aware) ...
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1answer
63 views

Self-intersecting paths in Feynman path integral

The self-intersecting paths are not included in the Feynman path integral (FPI) approach because by definition a curve is defined as $\gamma:\mathbb{R}\rightarrow \mathbb{R}^3$ and since we take the ...
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Relation between Feynman path integral and time ordering operator

Is there any relationship between Feynman path integral approach and time ordering operator because both of the approaches obliterate the idea of noncommutativity of operators since in FPI the ...
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1answer
173 views

Which vacuum do I use for the path-integral?

In Weinberg, vol. 1, Section 9.2, Weinberg defines the in and out vacua as states with no particles (9.2.4): $$a_{\rm in}|{\rm VAC,in}\rangle=0$$ $$a_{\rm out}|{\rm VAC,out}\rangle=0$$ He does this ...
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Coherent state basis of (relativistic) particle Fock space

For a neutral scalar bosonic particle of mass $m$, I consider a Fock space with an orthonormal basis of momenta eigenstates \begin{equation}\label{Fock-p-states} \left|p_1p_2\cdots p_n\right\rangle=\...
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1answer
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Path integral measure in Coleman's “Aspects of symmetry”

Suppose we have a integral measure $[\mathrm dx]$, and suppose we make the change of variables $$x=y+\sum_n c_nx_n, \tag{2.6}$$ where $x_n$ is a certain orthonormal basis. Why we should have $$[\...
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Does finite temperature QFT partition function capture both quantum and statistical fluctuations?

I've been working on understanding finite temperature field theory and am stuck with the following concepts: For our thermal system in equilibrium we have the usual partition function \begin{equation} ...
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1answer
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The quantization of the electromagnetic field in Peskin and Schroeder (Eq.9.52)

I'm working on the quantization of the electromagnetic field in Peskin (page 294). However, I'm confused about the Eq.(9.52). Peskin says Eq.(9.51) and Eq.(9.52) are equivalent, but why? Is Eq.(9.52) ...
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Main idea behind this paper on Closed-time-path functional formalism

I tried to understand following paper: Closed-time-path functional formalism in curved spacetime: Application to cosmological back-reaction problems but I can't understand what is going on because I ...
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Detailed calculation of the scattering amplitude between two particles in $\phi^4$-perturbation theory using the path integral formalism

I'm looking for detailed example/calculation of the scattering amplitude between two particles (mesons) in $\phi^4$-perturbation theory using the path integral formalism. Do anyone of you guys have a ...
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Deriving the path integral from the Trotter product formula

Can the path integral be derived in the following way? $$ \left< \psi \right| \hat{U} \left| \psi \right>=\left< \psi \right| e^{-i t (\hat{T}+\hat{V})/\hbar} \left| \psi \right> $$ ...
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Simulation of Feynman path integral in literature

Can someone provide me a numerical simulation of Feynman path integral? Where the contribution of each path is added individually so that I can understand how much do the paths outside the light cone ...
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Why is it hard to extend the Feynman Checkerboard to more than 1+1 dimensions?

The Feynman Checkerboard Wikipedia article states: "There has been no consensus on an optimal extension of the Chessboard model to a fully four-dimensional space-time." Why is it hard to extend ...
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1answer
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General Gaussian integral in Peskin and Schroeder [closed]

I'm working on the Eq. (9.24) in Peskin & Schroeder. I tried to derive it but I have difficulties. I canʻt follow this step: $$ \left(\prod_{k} \int d \xi_{k}\right) \exp \left[-\xi_{i} B_{i j} \...
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1answer
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Path integral formulation of an Abelian Field Theory, unclear identity

TL;DR: How exactly does one come to this identity $$\int\mathcal{D}G(A^\alpha)\delta(G(A^\alpha)) = \int\mathcal{D}\alpha(x)\delta(G(A^\alpha)) \mathrm{det}\left(\frac{\delta G(A^\alpha)}{\delta\...
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Path integral for free fermion on torus

If one will consider free fermion on torus,one will face with different spin structures. There are four spin structures, usually labeled ±±. The ++ spin structure has a single positive chirality zero-...
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The discrete Fourier series in Peskin and Schroeder (page 285)

I'm working on the discrete Fourier series in Peskin (page 285),but I have two questions. Question 1: I tried to derive Eq.(9.21): Consider $$ f(x)=\int \frac{d^{4} k}{(2 \pi)^{4}} e^{-i k \cdot x} ...
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Boson or fermion density matrix in path-integral Monte Carlo

In path-integral Monte Carlo literature the thermal density matrix of $N$-particle boson or fermion system is written as symmetrized or antisymmetrized sum $$ \rho_\mathrm{B,F}(R,R',\beta)=\frac1{N!}\...
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Symmetries in QFT preserving only combination of action and measure

Could we list examples of symmetries that preserve only the combination of the measure $\mathcal{D}\phi$ together with $e^{-S}$ but not each on their own? (That is, symmetries which have no classical ...
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1answer
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Proof for getting delta function on $t \to t_0 $ from the equation of the propagator for the free particle in 1 dimension

From Sakurai's quantum mechanics equation 2.5.16 give propagator for a free particle in 1 dimension. Equation 2.5.16 is $$K (x^",t;x',t_0)=\sqrt {m\over {2\pi i\hbar (t-t_0)}} \exp \Biggl [{im (x^"...
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Do Feynman path integrals satisfy Bell locality assumption?

There are generally two basic ways to solve physics models: Directional, e.g. Euler-Lagrange equation in CM, Schrödinger equation in QM. We evolve some initial conditions in some direction, can ...
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Clarification regarding animation on path integrals

The wikipedia page on path integrals contains the following animation. It's a pretty animation. Sadly the wiki page only says the following on the animation: The diagram shows the contribution of ...
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Derivative interaction: $\mathcal{H}_\mathrm{int}\neq - \mathcal{L}_\mathrm{int}$. Question about Feynman Rules

As we known, if there is time derivative interaction in $\mathcal L_\mathrm{int}$, then $\mathcal{H}_\mathrm{int}\neq -\mathcal{L}_\mathrm{int}$. For example, Scalar QED, $$ \begin{aligned} \mathcal{...
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1answer
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The Functional Methods in Peskin and Schroeder (page 280)

I'm working on the Functional Methods in Peskin (page 280) However, I canʻt obtain Eq.(b) and Eq.(c) from Eq.(a) Consider Eq.(a) \begin{align} \left\langle q_{k+1}|f(q)| q_{k}\right\rangle&= ...
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Why there's no missing determinant in Gaussian integration with Grassmann variables?

This integral appears in Ashok p. 82-83. We have the integral $$ I = \int \prod_{i,j}d\theta^*_id\theta_j e^{-(\theta_i^*M_{ij}\theta_j + c_i^*\theta_i + \theta_i^*c_i)}, $$ and if the inverse of $...
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SUSY sigma model in QM, bosonic sector?

The bosonic sigma model in ordinary QM (i.e. a 'free' particle trapped on a curved manifold $\mathcal{M}$), has a Hamiltonian which is just the negative Laplacian on $\mathcal{M}$. For any $\mathcal{...
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Integral over a total functional derivative is identically vanishing

In following an extension course on quantum field theory, a problem popped up that my TAs couldn't quite explain to my satisfaction. I suspect the answer is really simple, so I hope somebody with a ...
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Path integrals for brownian motion in a harmonic potential

The problem is as follows: Use the path-integral formulation of stochastic dynamics for a particle in a harmonic potential $U(r)= \frac{1}{2}kr^2$ to show that $$P(x,t|x_0,t_0)=(\frac{\beta k}{2\...
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1answer
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Relation between worldsheets and worldlines

If we want to find the propagator of say a zero-spin particle, the formal inverse can be computed by introducing $$-i\int_{0}^{\infty}e^{i(p^2 - m^2)s}\mathrm{d}s$$ We can rewrite this in path ...

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