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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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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Integrating out massive degrees of freedom of Super Yang-Mills Action

From this paper, I want to integrate out the massive degrees of freedom. The total action $S$ is given by $$S = S_{Y} + S_{A} + S_{Fermi} + S_{ghost} $$ where the terms are given in equations (2.9),(...
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3answers
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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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44 views

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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83 views

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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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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110 views

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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62 views

Can the real-time Green's function be written in the form of path integral on the real axis?

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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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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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
60 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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43 views

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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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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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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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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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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55 views

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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185 views

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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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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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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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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23 views

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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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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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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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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Boundary conditions of fermionic coherent states path integral

Given the algebra of a fermionic oscillator $$ \{\hat{a},\hat{a}^\dagger \}=1\,, \qquad \hat{a}^2=(\hat{a}^\dagger)^2=0, $$ with coherent states $ \hat{a}|\xi\rangle=\xi|\xi\rangle $, let's ...
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231 views

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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Boundary conditions of fermionic path integrals

When considering path integrals with grassmann variables, as stated on page 159 of Quantization of gauge systems - Henneaux, Teitelboim we only have one boundary condition, since the equations of ...
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Path Integral in QM with a position-dependent kinetic energy

I'm studying p. 160 in Ryder's book of QFT and there is an example where the standard path integral equation is not valid $$\langle q_ft_f|q_it_i\rangle = N \int Dq \exp \left( \frac{i}{\hbar}S \...
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59 views

Propagator in Path Integral Quantum Mechanism as Green Function of Schrodinger Equation

I'm studying in Ryder's book of QFT. I'm dealing with QM in the path integral approach and he is trying to prove that the propagator $K(x_f t_f;x_i t_i)$ is the Green function of the Schrodinger (S.) ...
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Propagation amplitude from ground state to ground state in Zee 's book of 'QFT in a Nut Shell'

Iam studying path integral approach to QFT through zee's book of 'QFT in a Nut Shell'. In page number 12 equation 6 states that $\langle q_F \lvert e^{-iHt} \lvert q _I \rangle = \int Dq(t) e^{({...
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Propagator of a quantum system between two points in spacetime

How is $U(x,t;x',t')= \left< \Psi(x,t)\right. \left| \Psi(x',t') \right> $ the propagator for the quantum system between two points in spacetime? I would think because $\left< x\right.\...
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Polchinski Weyl Anomaly from perturbing the flat background. Eq (3.4.22)

In deriving the Weyl anomaly for the bosonic string using a perturbation around a flat background, Polchinksi uses Eq. (3.4.22), i.e. $$ \ln \frac{ Z[\delta+h] }{Z[\delta]} \approx\, \frac{1}{8\pi^2}\...
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Why is there an difference between the exponent of the determinant of these two path integral?

When I read about Altland and Simons “Condensed matter field theory”, I came across with the path integral (3.28). $$\langle {q_f}|e^{-iHt/\hbar} |q_i\rangle = \det(\frac{i}{2\pi \hbar} \frac{\...
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Functional Integral in Statistical Mechanics

In this work, the author state that many problems in statistical mechanics center on the analysis of functional integrals of the form: \begin{equation} Z(\varphi') = \int d\mu(\varphi) e^{-V(\varphi+\...
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Wick's rotation for the harmonic oscillator. Explicit computation

I'm stuck in this computation; it shouldn' be difficult but it's always better to check with a lot of details these things. Consider the propagator for the harmonic oscillator Ashok p.55 bottom of the ...
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Why is it justified to discard off-shell momenta contributions in the exponent of the expression for a path integral amplitude?

Let us consider a free field theory with one field $\phi$. The Lagrangian density is $L(\phi, \partial_{\mu} \phi)$ and the corresponding Hamiltonian density is $H(\phi,\pi,\partial_{\mu \neq 0}\phi)$....
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I don't understand the Feynman path integral!

Let's say that in 2D space, in a vacuum, there is a point A and a point B. A particle x travels from point A to B. I've heard people say, that in this case, probabilistically, the particle will ...
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Missing factor in Dimensionally reduced Yang Mills Ghost Field

I'm trying to calculate the ghost field in the background field gauge for the dimensionally reduced Yang Mills action in this paper. I am using the expression from Srednicki's book, chapter 78. The ...
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1answer
172 views

What is the physical meaning of $W[J]=\frac{\hbar}{i}\ln Z[J]$?

The quantity $Z[J]$ (which is the generating functional for all Green functions) physically represents the probability amplitude for a system to remain in the vacuum state. Can we find a similar ...
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Question about the expression for vacuum persistence amplitude $Z[J]$ as a ratio of two Feynman kernels

The vacuum persistence amplitude $Z[J]=\langle0,+\infty|0,-\infty\rangle$ (See eqs. 11.83 and 11.86, Field Quantization by W. Greiner, J. Reinhardt) can be shown to equal to the ratio of two Feynman ...
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How can I find out the 'Klein-Gordon' equation by using Path Integral techniq? [closed]

I get the following relation from eqn. (6.52) in Sakurai(2nd edition) $$\Psi +\Delta t \frac{\partial \Psi}{\partial t}+\Delta t^2\frac{1}{2!}\frac{\partial^2\Psi}{\partial t^2}+...=\lim_{\Delta t\to ...
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Violation of Bell-like inequalities with spatial Boltzmann path ensemble: Ising model?

Quantum mechanics is equivalent with Feynman path ensemble, which after Wick rotation becomes Boltzmann path ensemble, which can be normalized into stochastic process as maximal entropy random walk (...

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