# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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{... 1answer 82 views ### 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&= ... 0answers 40 views ### 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)}, $$and if the inverse of ... 0answers 123 views ### SUSY sigma model in QM, bosonic sector? The bosonic sigma model in ordinary QM (i.e. a 'free' particle trapped on a curved manifold \mathcal{M}), has a Hamiltonian which is just the negative Laplacian on \mathcal{M}. For any \mathcal{... 0answers 59 views ### Integral over a total functional derivative is identically vanishing In following an extension course on quantum field theory, a problem popped up that my TAs couldn't quite explain to my satisfaction. I suspect the answer is really simple, so I hope somebody with a ... 0answers 40 views ### Path integrals for brownian motion in a harmonic potential The problem is as follows: Use the path-integral formulation of stochastic dynamics for a particle in a harmonic potential U(r)= \frac{1}{2}kr^2 to show that$$P(x,t|x_0,t_0)=(\frac{\beta k}{2\...
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 ...