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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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Witten Index of Riemannian Manifold

Consider a system on a Riemannian manifold with the Lagrangian $$L = \frac{1}{2}g_{IJ} \dot{\phi}^I \dot{\phi}^J + \frac{i}{2}g_{IJ}(\overline{\psi}^I D_t \psi^J - D_t \overline{\psi}^I \psi^J) - \...
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What justifies passing the limit to the exponent in the derivation of the path integral?

In the usual derivation of the path integral there is one strange passage. Taking Weinberg's derivation for instance from his QFT book, chapter 9, we have the following equation $$\langle q';t'|q;t\...
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Meaning of “Exactly solvable in the large $N$ limit” for the SYK model

Every presentation on the SYK Model (check any youtube lecture by Douglas Stanford, Juan Maldacena, Subir Sachdev, Alexei Kitaev, etc.) claims that it is exactly solvable in the large $N$ limit, thus ...
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Vanishing partition function [duplicate]

I am currently stuck with the following partition function Let the action be $$S(X, \psi^1, \psi^2) = \frac{1}{2} (\partial h)^2 - \partial^2h\psi^1 \psi^2 ,$$ where $h$ is a real function of the ...
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In 2d CFT, why the $T_{zz}$ component of energy-momentum tensor is holomorphic even at quantum level?

In 2d Conformal Field Theory, the $T_{zz}$ component of energy-momentum tensor is treated as a holomorphic function $T(z)=T_{zz}$ at quantum level such as in OPE involved energy-momentum tensor. I ...
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Uniqueness in the path integral vs canonical quantisation

In quantum mechanics it is well known that if you have a Lagrangian $\mathcal{L}$ and you want to quantise it, there is no unique way of doing this. This is because when you construct the Hamiltonian $...
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Convergence Property of Path-Integral

Let the action be $$S= \int \bigg\{ \frac{1}{2} \big(\frac{dX}{dt}\big)^2 - V(X) \bigg\} d\tau$$ and the corresponding Path-Integral $$Z= \int DX(t) e^{iS}.$$ Since the convergence is not clear we ...
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How to compute thermodynamic magnitudes with the Green's function?

I'm studying the SYK model and there seems two equivalent approaches for solving it. One is the diagrammatic expansion in the large $N$ limit, where we get self-consistent equations (in imaginary time)...
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Real and Imaginary time Green's Functions

In real time, one can calculate the two point function of a given theory using \begin{equation} G(\vec{x},t)=\langle \Omega | \phi(\vec{x},t)\phi^\dagger (0,0)|\Omega\rangle =\int_{\phi(0,0)}^{\phi(\...
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Grassmann-even action

I am currently studying supersymmetric quantum mechanics with the help of the book Mirror Symmetry by Kentaro Hori (and others). On page 155 where they introduce Grassmann variables they say that the ...
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Quantum Fluctuation Contribution in the Path Integral of a Meta-stable Potential

In Wen XiaoGang's QFT of Many-Body Systems Sect 2.4.2, he claimed that in the integral of the quantum fluctuation of the bounce process, the determination of the operator $-\frac{d^2}{d\tau^2}+V''(x_{...
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Path integral formulation of amplitude from initial to final state

In path integral formulation we say that we are summing over all possible ways for the system get from initial to final state. Now if we just write the amplitude and then insert complete set of states,...
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Deriving the path integral for periodic boundary conditions

I'm thinking about path integrals with the Euclidean time formalism, where I have partition function $Z=\operatorname{Tr} e^{-\beta \hat H}$. I'm used to the following derivation of the path integral: ...
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Path integral calculations $e^{i\omega 0^+}$

When computing correlation functions using the path integral formulation, I often need to compute integrals such as $$ \int_{-\infty}^\infty \frac{d\omega}{2\pi} \frac{1}{i\omega -\epsilon} $$ ...
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Euclidean path integral for density matrix with chemical potential

I am generally familiar with how to represent a thermal density matrix $\rho\propto e^{-\beta H}$ with a path integral, namely the partition function is $$\int D\Psi \exp \bigg[-\int_0^\beta dt\int d^...
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Infinite sum: Renormalisation

Trying to do the calculation made in a physics article Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling (page 10 to go from equation 56 to 57), I ...
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Relativistic path integral

I got totally messed up with Problem 2-6 from Feynman's Quantum Mechanics and Path Integrals. The body of the problem is to find the kernel of a relativistic particle between the points a to b, that ...
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Feynman's derivation of Schrödinger equation. Potential spatial dependence

I am working on the book "Quantum Mechanics and Path Integrals" from Feynman and Hibbs. When finding the correspondence with Schrödinger equation he takes $$\eqalign{&\psi(x,t+\epsilon) = {}\...
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Euclidean Feynman Rules derivation using Wick Theorem

When studying perturbation theory and Feynman rules, the standard derivation seems to start from the S-Matrix element in the interaction picture, and expands it into some series, after which Wick's ...
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Diagrammatic expansion of an operator insertion in path integral for Trace Anomaly calculation

Starting with a scale invariant classical field theory, we can prove that the energy-momentum tensor will be traceless. \begin{equation} \Theta^\mu_{\ \mu }=0 \end{equation} In the context of the ...
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Defining a metric on the space of paths

Imagine the following path integral $$\int_{x(0)=x_i}^{x(T)=x_f} \mathcal{D}x \, e^{\frac{i}{\hbar}S[x]}.$$ This integral is defined over the space of all paths that satisfy the boundary conditions ...
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Density matrix expression by path integral

I came across an expression which I don't understand for the density matrix $\rho$ given by the path integral method (Fradkin, p.760) - $$ \left< \phi(x) \left| \rho\right| \phi\left(x'\right) \...
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Feynman's derivation of the Schroedinger equation by expanding path integrals to first order in $\epsilon$

As discussed in the answer to How can one derive Schrödinger equation?, one should be able to "derive" the Schrodinger equation from the path integral formulation of quantum mechanics. However, ...
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What actually led Feynman to the Path Integral? [duplicate]

It is commonly known that Feynman's path integral was inspired by Dirac's observation that the kernel is proportional to $\exp{\dfrac{i}{\hbar} S}$. It was Feynman, however, who had the idea of ...
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How to derive Eq. (6.21) in Srednicki?

I'm reviewing Srednicki's chapter on path integrals and am having trouble understanding how he arrives at formula 6.21: $$\left<0|0\right>_{f,h}= \int \mathcal{D}q \,\mathcal{D}p \, \exp \left[...
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Peskin equation on the treatment of chiral anomaly

In page 666 (it couldn't be other way - bad joke), chapter 19, the Eq. (19.73) claims (see properties of the $\phi_n(x)$ functions in this post: Change of variables in path integral measure): $$ \...
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Change of variables in path integral measure

In fermion's path integral we have a measure that you can write, in terms of the Grassmann variables $\psi, \bar{\psi}$ as $$ D\bar{\psi}D\psi, \quad \psi(x) = \sum_n a_n\phi_n(x), \quad \bar{\psi}(x)...
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What is the role of $\hbar$ in quantum mechanics? [duplicate]

Planck's constant $\hbar$ appears in the Schrodinger equation: $$i\hbar \frac{d|\psi\rangle}{dt}\ = \hat{H}|\psi\rangle$$ which implies for stationary states, $$|\psi(x,t)\rangle=e^{-iE_n/\hbar}|\...
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Calculation of current from path integral

I would like to calculate $\langle\bar{\psi}\psi\rangle$ in free theory. I start from the following generating functional: $$Z[J]=\int\mathcal{D}[\bar{\psi},\,\psi]\exp\left(i\int d^dx\,[\bar{\psi}(i\...
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Equivalence Picard-Lefschetz path integrals and “Feynman's” path integrals

I have just seen the Picard lefschetz method applied to path integrals in order to make these more convergent. I understand how we could modify the contour of integration for a real integral but what ...
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Peskin & Schroeder eq. 9.26 and functional methods

I have been reading chapter 9 in Peskin & Schroeder's QFT book and has been stuck in transition from equation 9.26 to 9.27. Equation 9.26 reads: $$\frac{1}{V^2} \Sigma_{m,l} \exp{[-i(k_m.x_1+k_l ....
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Path integrals and fourier series

I am currently reading the Feynman and Hibbs about Quantum mechanics and path integrals and I found something pretty confusing ( for me ) at page 72. At this page, they are replacing an integration on ...
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WKB connection formulae from the path integral

The semiclassical, or WKB, approximation is one that is far more natural in the path integral formalism than it is when derived from the Schrodinger equation directly. Furthermore, the connection ...
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When is the Path Integral exact?

Feynman's path integral is exactly solvable for quadratic Hamiltonians and the Coulomb potential. I recently heard without proof that there is a larger class of systems for which it is also exactly ...
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Euler-Maclaurin formula for path integral

Is there a corresponding Euler-Maclaurin formula for path integral when we divide the path integral into discrete lattice? What is the error correction when we divide the space into lattice of length ...
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Difference between different approximations in QM and other definition of the integral

I am currently studying the path integral formalism by myself and I am a bit lost within all the different way to solve the integrals we have. I have one big question: It sounds maybe a bit strange ...
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Feynman $i\epsilon$-prescription for fermion propagator via path integrals

In Section 9.4 of S. Weinberg's book "The quantum theory of fields" it is shown how to get the Feynman $i\epsilon$-prescription in the propagator of a free scalar field using path integrals and ...
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Summation of an exponential operator on quantum amplitude

For a quantum Dirac field interacting with a classical EM field, one can (through the Quantum Dynamical Principle) write the vacuum transition amplitude as $$\langle0_+|0_-\rangle=\exp\left[ie_0\int ...
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Path integral: uncoupling via staging variables

I am studying the transformation to staging variables for the calculation of path integrals in quantum mechanics, following the scheme presented in the book of Mark Tuckerman "Statistical Mechanics: ...
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Fourier transform of variable in path integral

In Sredinicki's QFT given below, he changed the integration variables in eq(174). This step confuses me. I only know some basics about path integral. In my opinion, when he used fourier transform of ...
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Paths of least action and loops in time

In the book Quantum Field Theory for the Gifted Amateur link: https://books.google.ca/books?hl=en&lr=&id=nIk6AwAAQBAJ&oi=fnd&pg=PP1&ots=JZjwG_qDt5&sig=...
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Why can we treat the path integral as an integral over a infinite dimensional vectorspace?

The path integral is usually introduced by integrating over all piecewise linear paths in discrete time and then taking the time step $\varepsilon$ to zero, i.e. $$\int Dx e^{iS[x]} \sim \int dx_1 ...
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Dependence of BRST Quantization on the Choice of Gauge-Fixing Function

There is a point which confuses me about BRST procedure. One shows that, if we define physical states as the ones that are annihilated by BRST charge $Q$, the scattering amplitudes don't depend on ...
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Two Questions about Path Integral from “Gauge Fields and Strings” by Polyakov

My questions are about worldline path integrals from the book Gauge Fields and Strings of Polyakov. On page 153, chapter 9, he says Let us begin with the following path integral \begin{align} &...
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A Question about Path Integral Measure

I want to do the following path integral. $$\mathcal{Z}=\int\mathcal{D}x e^{iS[\dot{x}]}$$ The action only denpends on $\dot{x}$. For some reason, I want to replace the integral measure $\mathcal{D}...
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Different Schwinger-Dyson Equations

In the literature on QFT there are a lot of different equations that are all called "Schwinger-Dyson equation" so I wanted to know how are they related and if they have proper names. The first ...
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How does the Weyl anomaly imply $\langle T^{\mu}_{\mu} \rangle \neq 0$

I want to consider the case of euclidean field theory in 2 dimensions with the action $$S[\phi]=\int \! d^2\!x \sqrt{\det(g)}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$$ which leads to a partition ...
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Calculation of path integral in QFT

I am studing QFT using the text book of Srednicki's. And I am stuck on one of calculations of the integrals in his book. Consider a harmonic oscillator with hamiltonian: We can write the following ...
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Are powers of the harmonic oscillator semiclassically exact?

The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact....
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Wick rotation vs. Feynman $i\varepsilon$-prescription

The generating functional $Z[J]$ of some scalar field theory is \begin{equation} Z[J(t,\vec{x})]=\int \mathcal{D}\phi e^{i\int (\mathcal{L}+J\phi)d^4x} \end{equation} This integral is not well ...