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.
1,703 questions
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Path integral in quantum mechanics and expectation values
How to calculate any expectation value from path integral in quantum mechanics? In QM path integral the initial and final points are fixed and points between them are varied. But as far as i ...
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How to implement constraints via delta functions?
I have a question regarding the implementation of constraint equations as delta functions in integrals.
My confusion can best be illustrated with a quick example:
Consider a Gaussian integral of the ...
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Mathematical rigor behind renormalization [closed]
I struggle to understand how theories that are based on renormalization can be considered mathematically rigorous. I understand how renormalization works for non-abelian theories, through loop ...
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Classical mechanics vs quantum mechanics [closed]
In classical mechanics we say that a classical particle obeys the equations of motion, whereas in quantum mechanics a particle can take any path, not just the classical one. But when we quantize a ...
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Proof for scalar field VEV solving classical equations of motion to lowest order [duplicate]
Page 370 (start of section 11.4) of Peskin and Schroeder claims that the VEV of a scalar field in the presence of an external source,
$$\phi_\text{cl} \equiv \langle 0_J|\phi(x)|0_J\rangle,\tag{11.46}$...
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QFT Path Integrals: How to define a precise initial state if vacuum fluctuations can alter it?
I'm trying to solidify my understanding of the path integral formalism in Quantum Field Theory, and I've run into a conceptual paradox regarding the definition of initial states. I would be grateful ...
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How does the Berezin integral for fermions represent a "sum over all configurations"?
In quantum field theory, the path integral for a bosonic field has a very intuitive interpretation as a "sum over all possible field configurations."
To make this concrete, let's consider a ...
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How to calculus the corresponding operators of bosonic beta-gamma ghost systems? [closed]
In Polchinski's book, it states that the corresponding operators of $|1\rangle, |-1\rangle$ are $\delta(\beta),\delta(\gamma)$, and suggests that it can be shown by path integral. I'm a little ...
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Functional derivative of the partition function as a path integral [closed]
I'm having trouble understanding the following functional derivative, which appears in David Tong's notes on Statistical field theory, page 40:
Let $Z[-]$ be the partition function that takes in a ...
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Does the semiclassical path integral formulation of quantum tunnelling admit a non-linear (curved) extremal path in imaginary time? [closed]
In the semiclassical (WKB / path-integral) description of quantum tunnelling, the dominant contribution to the transmission amplitude comes from the extremum of the Euclidean action
$$
S_E = \int \...
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Correlation functions in perturbative QFT
In perturbative quantum field theory, one starts with interaction Lagrangian density
$$\mathcal{L} = \mathcal{L}_{\text{free}}+g\mathcal{L}_{\text{I}}\tag{1}$$
Where $g$ is coupling strength. ...
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How does the path integral for the Universal Wavefunction account for future internal decoherence events?
I'm trying to solidify my understanding of the path integral formalism when applied to the entire universe, and I've run into a conceptual point that I'd like to clarify.
Let's assume a purely ...
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When is Wick Rotation Justified?
Suppose I have a QFT defined by a Lagrangian in Minkowski space and one in Euclidean space related by a Wick Rotation. What sort of objects/properties in general stay the same between either theory; ...
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Compute probability current from stochastic path integral
I would like to compute the probability current associated with a stochastic differential equation, say
$$
\frac{\mathrm{d} X}{\mathrm{d} t}
=
v
+
\sigma \xi(t)
$$
where $v$ is a drift velocity, $\xi$ ...
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Classical Equivalent of time evolution given by on-shell action?
In Quantum Mechanics, the time evolution of an observable in the Heisenberg picture is determined by the Dirac bracket with the Hamiltonian operator
$$ i\hbar\frac{d}{dt}\hat{\mathcal{O}}(t)=[\hat{\...
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Why is Fractional Quantum Hall (FQH) physics independent of the charge of the dynamical gauge field in Tong’s lecture notes?
In Chapter 5 of David Tong's lecture notes on the Quantum Hall Effect, he introduces an action which is a functional of some dynamical gauge field $a_\mu$. He takes the charge of the dynamical gauge ...
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On the convergence of path integral [closed]
Why does the euclidean path integral for a free particle diverge, but if we add mass term it converges?
For example this two-dimensional path integral converges absolutely.
$$\int_{-\infty}^{\infty}\...
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Significance of index theorem for quantum anomalies
I've recently been studying anomalies in quantum field theory, and have heard mention of index theorems coming in use for them. However, I cannot understand, at a deep level, how this is the case.
For ...
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What kind of procedure is the Wick rotation to Euclidean formulation?
I'm learning QFT via the path integral formalism. I've been struggling understanding the Wick rotation to Euclidean formulation, towards which I feel very uncomfortable. In particular I cannot find a ...
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Defining quantum field theory via path integrals
For simplicity, consider a quantum field theory with a single quantum field $\phi$. It is well known that if we know all correlation functions of the field $\phi$, that is we know all functions
\begin{...
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On the absolute convergence of Euclidean path integral
I am interested in knowing under which conditions Euclidean path integral absolutely converges.
I define an exponentially decaying function as
$$ f(x) = e^{-kg(x)},k>0$$
Where $kg(x)\geq C|x|^p+D$ ...
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Why can we use a non-symmetric propagator if the currents symmetrize the integral?
Imagine a simple QFT with action principle
\begin{equation}
S[\phi,J]=\int d^4x \Big\{-\frac{1}{2}\phi\Box \phi+J\phi\Big\}.
\end{equation}
As usual, we solve the field equations $\Box \phi=J$ to get
\...
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Euler-Lagrange Equations under constraint in Keldysh formalism
I have 2 questions regarding the minimization procedure of an effective action in the Keldysh formalism. In the Keldysh path integral one deals (after Keldysh rotation) with a partition function of ...
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How to find the correct propagator prefactor in the dilute instanton gas in 1D quantum mechanics?
In Altland & Simons (2nd ed., pp. 117-124), there is a discussion on path integrals and instantons where I cannot understand where the factor $e^{-\omega\tau}$ comes from. The calculation goes the ...
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Regularization of determinants in QFT
I have a question regarding regularization in quantum field theory. Hagen Kleinert talks about analytic regularization in his book "Path Integrals". In chapter 2.15 he calculates the ...
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Why Does the Path Integral Seemingly Lack Interference for One Particular Time Step?
Consider the path integral equation:
$$\langle x(t_f)|\hat{U}(t_f,t_i)|x(t_i)\rangle=\lim_{\delta t\to 0}\int\prod_{n=1}^N\mathrm{d}x(t_i+n\delta t)\,e^{iS_n/\hbar} \langle x(t_f)|x(t_i+n\delta t)\...
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Question on Quanutm Physics regarding the wave model and paths taken [duplicate]
I was wondering if I am understanding Quantum probabilities are particles correct, so hypothetically if we have a photon and dont know what path it is taking then it will act as a wave, right, and as ...
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Why "virtual photons" have no defined propagation speed? [closed]
All of our physics QED and QFT in order to comply with conservation laws and the speed of causality c, are based on a so called mathematical constract, fitting parameter, the "virtual photon&...
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What are the challenges in extending the path integral from particle paths to field configurations?
As far as I have been able to see, the Feynman path integral can be given a precise mathematical meaning via analytic continuation of Wiener Gaussian integrals. First you define the Wiener measure on ...
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Gaussian integration of path integrals
I have a question regarding the Gaussian integration of path integrals with quadratic action.
Lets say the path integral has the standard form
$$ \int D[\Phi^\dagger,\Phi]e^{iS}$$
with a quadratic ...
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Path integral for a run-and-tumble particle
I am familiar with the path integral formalism for stochastic differential equations of the form (in 1d for simplicity)
\begin{equation}
\dot{x}(t) = f(x(t)) + \sqrt{2 D} \ \xi(t).
\end{equation}
It ...
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Definition of Euclidean generating functional [duplicate]
I'm following a class on QFT. I'm having a hard time understanding the rotation to Euclidean of the generating functional $W[J]$ of some scalar theory $L(\phi, x)$.
$$
W[J] := \mathcal{N} \int [D\phi] ...
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What does Weinberg mean by “integrating out the auxiliary field $h$” in the BRST formalism for QED?
In The Quantum Theory of Fields, Volume II, Weinberg defines the BRST variation $\delta_\theta$ as follows:
\begin{align}
\delta_\theta \psi & = i t_\alpha \theta \omega_\alpha \psi \tag{15.7.7} \\...
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Classical (stationary phase) trajectory in quantum tunneling
When we talks about quantum mechanical tunneling in the formalism of path integral, we normally say that there's no classical (stationary-phase) path connecting the two minima of the potential so we ...
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Algebraic proof of Dyson-Schwinger equation in Zinn-Justin
On p.100 of Quantum Field Theory and Critical Phenomena by Justin Zinn-Justin, as part of the algebraic proof of the Dyson-Schwinger equation in Euclidean spacetime, Eq. (5.34) gives the following ...
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Imposing boundary conditions in QFT
I have a question regarding the implementation of twisted boundary condtions in a path integral.
In this paper: https://doi.org/10.1103/PhysRevX.11.041004
the authors claim in appendix G that for a ...
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Can we derive Klein-Gordon, Dirac equation from Feynman path ensembles like Schrödinger?
While there is this standard derivation of Schrödinger equation from Feynman path ensembles, can we also derive/imagine Klein-Gordon, Dirac equations through path ensembles?
The main difficulty seems ...
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Why are there no ghost terms arising in the Lagrangian of an Abelian gauge theory?
As the title writes, I am trying to understand from a mathematical pov, why is it that abelian gauge theories have no ghost terms?
One can argue that because the generators commute, we do not have ...
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Expectation value of non-time-ordered product of field operators from the path integral
In the path integral formulation of QFT, we have the general formula relating the vacuum expectation value of a time-ordered product of fields to the path integral:
$$\langle 0 | T\{ \phi(x_1)...\phi(...
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How to perform Wilsonian RG for the $O(N)$ nonlinear sigma model?
I am interested in deriving the $\beta$ function and anomalous dimension $\gamma$ of the $O(N)$ nonlinear $\sigma$ model, in particular defined by the action $S = \frac{1}{2g} \int d^2 x (\partial_{\...
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How to draw these Feynman diagrams from double Taylor expansion? [duplicate]
M. Srednicki in his book "Quantum field theory" on page 60 computes the path integral for interacting theory (in $\varphi^3$ theory) as follows: $$Z_1 (J)\propto \sum_{V=0}^{\infty}\frac{1}{...
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Why do the path integrals of different inequivalent canonical quantisations (due to the choice of different Cauchy foliations) agree?
Suppose we have a field theory on Minkowski spacetime. This theory can be quantised in several inequivalent ways:
We can foliate the spacetime into Cauchy hypersurfaces, and write the action in co ...
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Partition function of free boson on torus
In Section 10 of the "Conformal Field Theory" by Francesco et al. The
Hamiltonian is defined as $H=(2\pi/L)(L_0+\bar{L_0}-c/24)$ so that partition function should be:
$$Z(\tau)=Tr\exp(2\pi i ...
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Formal proof that a path integral over node-crossing paths vanishes
In statistical physics of fermionic systems, one is often interested in evaluating path integrals of the form
$$
\rho_F(\mathbf{R}, \mathbf{R}; \beta) = \frac{1}{N!} \sum_{\mathcal{P}} (-1)^{\mathcal{...
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Zeta-function regularization of an uncountably infinite product of constant factors
I want to calculate a functional determinant coming from a Gaussian path integral with operator Matrix $M$. The determinant is given by the product over the eigenvalues according to
$$\text{det}(M) = \...
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Initial condition of the canonical density matrix in path integral formulation
The imaginary-time path-integral expression for the many-body density matrix of $N$ bosons is
$$
\rho\left(R,R_{0};\beta\right)=\left\langle R\right|e^{-\beta\hat{H}}\left|R_{0}\right\rangle \propto\...
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Beginner friendly materials for functional method in QFT
When I ask questions on this site regarding Feynman diagram, I see a lot of answers using functional method in QFT (e.g. this post and this post).
However, they seems quite confusing to me because I'...
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Experiment demonstrating light actually explores all the paths in a Veritasium YouTube video [duplicate]
This is the link to the Veritasium YouTube video where Derek and his friend show that light indeed explores all the paths. A laser beam is made to fall on a point say $P$ on a reflecting surface at an ...
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Infrared zeromodes and the physical status of infinite-wavelength photons in quantum field theory
In quantum field theory, photons are described as quantized excitations of the electromagnetic field, with energy given by $E = ℏω = hc/λ$.
In the infrared (IR) limit, where the wavelength $λ$ tends ...
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Gaussian integral (Hubbard-Stratonovich) with complex field
I am trying to perform a Hubbard-Stratonovich transformation to decouple a quartic interaction, which technically is just a Gaussian integral over an extra field complex field
\begin{equation}
e^{V \...
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