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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1answer
270 views

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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2answers
190 views

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
507 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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2answers
77 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 ...
3
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2answers
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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0answers
41 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 ...
4
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2answers
114 views

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 ...
3
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1answer
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, ...
3
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1answer
91 views

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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2answers
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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1answer
119 views

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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1answer
2k views

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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2answers
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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
176 views

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
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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0answers
46 views

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 ...
2
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1answer
167 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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1answer
127 views

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=\...
3
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1answer
195 views

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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42 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} ...
0
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1answer
55 views

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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29 views

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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2answers
347 views

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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0answers
35 views

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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2answers
53 views

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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3answers
969 views

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
53 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} \...
2
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1answer
69 views

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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2answers
81 views

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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1answer
70 views

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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2answers
137 views

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
94 views

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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2answers
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 ...
3
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1answer
51 views

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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2answers
2k views

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
69 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&= ...
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34 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 $...
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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{...
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47 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 ...
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0answers
36 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\...
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1answer
138 views

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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98 views

Physicist path integral and cylinder set measures

Path integral via discretization So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is just what I understood!). Let a quantum system ...
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1answer
38 views

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 ...
3
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1answer
190 views

Why can the time-ordered exponentials be brought to the right?

Having worked through almost all calculations in section 4.2 of Peskin & Schroeder's An Introduction to QFT, I still don't get why we can get to Eq. (4.31) \begin{equation} <\Omega|\mathcal{T}\...
4
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1answer
382 views

Boundary conditions in holomorphic/coherent state path integral

Consider the holomorphic representation of the path integral (for a single degree of freedom): $$ U(a^{*}, a, t'', t') = \int e^{\alpha^{*}(t'') \alpha(t'')} \exp\left\{\intop_{t'}^{t''} dt \left( -a^...
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0answers
327 views

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 (...
6
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1answer
715 views

What is the constraint on the Gauge Potential in the Covariant Gauges?

One of the most common gauges in QED computations are the $R_{\xi}$ gauges obtained by adding a term \begin{equation} -\frac{(\partial_\mu A^{\mu})^2}{2\xi} \end{equation} to the Lagrangian. ...
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11 views

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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1answer
251 views

Calculating $S$-matrix in string theory

To calculate string $S$-matrix, we mainly use Faddeev-Popov gauge fixing method, as in chapter 6 of Polchinsky's book 《string theory》. But in section 6.2, 'tree level amplitude', I didn't find ...

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