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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Thermodynamic free energy of interacting system
This question concerns an interacting system's thermodynamic free energy $\Omega$. Generally speaking, The action $S$ for an interacting system has the following form:
\begin{equation}
S(\phi,\psi) = ...
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Can the standard Quantum-Mechanical Path Integral also be evaluated on a lattice?
I have been trying to learn about lattice path integrals. Unfortunately, majority of the literature on this topic is in regard to Lattice Quantum Field Theory and Lattice Quantum Chromodynamics. That ...
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Clarification about Weinberg effective action and effective potential
I'm going over Weinberg Quantum effective action section in his second book, I understood his derivation and reasoning up to the equation:
$$iW[J]=\int_{\substack{Connected\\trees}}\left[\mathcal{D}\...
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How to integrate a Gaussian path integral of free particle using zeta function regularization?
I am attempting to integrate this path integral in Euclidean variable $\tau $ (but this need not be the same as the $X^0$ field):
$$Z=\int _{X(0)=x}^{X(i)=x'}DX\exp \left(-\int _0^i d\tau \left[\frac{...
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Why must the propagator exponent be imaginary?
In response to asmaier's question, qmechanic showed why the propagator must be $\exp(cS)$. That made perfect sense. But can it also be shown that $c$ is imaginary? I believe it follows from ...
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How is the free particle propagator derived? [closed]
The free particle propagator is a well known function,
for example, see Wikipedia.
However, I cannot find a source that explains how to derive
the free particle propagator.
Please explain how the free ...
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Momentum Space Propagator from Path Integral Formulation of “Polyakov-style” action for a massive relativistic point particle
I have the derived the following expression for the propagator of a “Polyakov-style” action for a massive relativistic point particle:
\begin{equation*}
G(X_2 - X_1) = \mathcal{N}'\int_{0}^{\infty}...
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Multiplicative property of the functional determinant
If we consider two differential operators $\mathcal{D}_1$ and $\mathcal{D}_2$, we can compose them to create the differential operator $\mathcal{D}_1 \mathcal{D}_2$. Then we could consider an action (...
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Free particle probability to go from $a$ to $b$ [duplicate]
Feynman and Hibbs write that the probability
for a particle to go from $a$ to $b$ is
\begin{equation*}
P(b,a)=|K(b,a)|^2
\end{equation*}
The kernel for a free particle is given as
\begin{equation*}
K(...
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Ambiguities of the Hubbard-Stratonovich decoupling
Consider an interacting fermionic theory on some lattice $\Lambda$ with action
$$
\mathcal{S}[\bar{\psi},\psi] = \int_{0}^{\beta} \mathrm{d}\tau \sum_{i\in \Lambda} \sum_{\sigma = \left\{ \uparrow,\...
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How do we interpret the second-order differential operator in the QFT path integral?
For the free scalar field theory, the path integral has a differential operator term in the exponent,
$$
Z[J] = \int \mathcal{D}\phi \, \exp\left( i \left[ -\frac{1}{2} \int d^d x \, \phi(x) A \phi(x) ...
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Fermionic path integral with boundary
Given a path integral:
$$K(\eta,\xi) = \int\limits_{\psi(0)=\eta}^{\psi(1)=\xi} e^{\int_0^1\dot{\psi}(t)\psi(t) dt} D\psi\tag{1}$$
where $\psi(t)$ are a real Grassmann fields.
I get two answers ...
user84158
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Are non-trivial topologies in the gravitational path integral related to large gauge transformations in Yang-Mills?
While the gravitational path integral is not a well-understood concept mathematically, a number of works (particularly in recent research connected to AdS/CFT) emphasize the importance of integrating ...
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Is the path integral emergent?
I have recently read a couple of papers on lattice QCD and found that there is a well-established connection between Boltzmann distribution and the path integral in QFT (disclaimer: I am not a huge ...
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What do the authors of the paper mean here exactly by path integral?
First of all, please forgive me if i am asking a dumb question. I don't have a physics background. I was reading this paper by Hawking & Hertog on populating string theory landscape and came ...
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Justification for the derivative expansion in the Exact Renormalization Group
In the Exact Renormalization Group formalism, specifically the formalism of Wetterich, one writes down an evolution equation for the effective average action $\Gamma_k[\varphi]$, see f.ex
$$
\...
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In formalizing QFT, are mathematical issues of canonical quantization approach and that of path integral approach related?
In QFT, many mathematical issues arise. Setting aside renormalization, these deal with rigorous constructions of objects underlying QFT:
i) In the canonical quantization approach, the main issue comes ...
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Connection between a saddle point approximation and plain perturbation theory
I am currently studying functional integration in the context of classical and quantum equilibrium thermodynamics. However one thing puzzles me:
In the book "Phase Transitions and Renormalization ...
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Hubbard-Stratonovich for 4-component attractive interaction
Suppose we are dealing with a hamiltonian similar to a BCS one, but instead of the 2-component interaction we are dealing with something of the form: $-g\psi^\dagger_1\psi^\dagger_2\psi^\dagger_3\psi^\...
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Probability amplitudes in Richard Feynman’s QED [duplicate]
So i’ve been reading Richard Feyman’s book, QED, and in it, he simplifies the idea of how physicists calculate the probability of a photon hitting a certain detector. He lets the magnitude of a vector ...
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Calculating the energy using path integrals vs hamiltonian
I'm reading A. Zee's "Quantum Field Theory in a Nutshell" section I.4 in which he used path integrals to calculate the energy of a real scalar field and 2 sources depending only on the ...
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Derivation of Fokker-Planck equation from Langevin equation
I have been trying for a long while to wrap my head around this step in the derivation of the Fokker-Planck equation in Appendix 8 of Nigel Goldenfeld's "Lectures On Phase Transitions And The ...
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Understanding Classical Contributions to the Quantum Field Theory Path Integral
I'm a data scientist with no physics background who sometimes reads about physics in my spare time, so please take it easy on me, I know these are really obvious questions to physicists but feel free ...
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How to understand confinement of 2+1d $U(1)$ gauge theory from instanton effects?
In Section 6.3.2 of XG Wen's book Theory of Quantum Many-body System, the confinement of $2 + 1 d$ compact $U \left( 1 \right)$ gauge theory was explained by the instanton effect. I want to know how ...
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Thirring Model effective potential
I have the lagrangian for the Thirring model coupled with vector interaction $V^\mu$ in the large-$N$ approximation:
\begin{equation}
\mathscr{L}=\bar{\psi}\left(i \gamma^\mu\partial_\mu-m - \gamma_\...
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Functional derivative of the generating functional with respect to the source term
To be specific, let us use the notations in W. Metzner et al., Rev. Mod. Phys. 84,299 (2012).
The generating functional $G[\eta, \bar\eta]$ is given by [Eq. 4]
\begin{align}
G[\eta, \bar\eta] = - \ln \...
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Normalisation and Dirac's Formulation of the Path Integral
Zee's Quantum Field Theory in a Nutshell (Dirac's Formulation in Chapter 1.2) contains the following passage (in attached image). Can someone please explain where the normalization is used in getting ...
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Can anyons exist on a torus without any additional conditions?
While learning recently some more "advanced" stuff about path integral formalism I was introduced to the topological conditions that specify the process of construction of the propagator, i....
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External/Background Fields Meaning
(I'll work in the Euclidean for convenience)
In the path integral formulation of QFT given a field $\phi$, or a set of them if you want to, we have that the partition function is given by:
$$Z[J] = \...
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Is Vacuum Expectation Value equivalent to the sum of Tadpole Diagram in the QFT?
I am recently learning Spontaneously Symmetry Breaking in the QFT. For example let us just focus on the potential $V[\Phi]=a\Phi^2 + b\Phi^4$ where $a<0,b>0$ and denote the vacuum expectation ...
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Why does the path integral formulation not provide an ontological basis for quantum theory?
I have looked at several books on the foundations of quantum theory and found that the path integral formulation is hardly ever discussed in detail. I find this surprising because this formulation of ...
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On what space of maps is Polyakov path integral actually defined?
This is a question more concerned about mathematical detail involving the Polyakov path integral.
In section $3.2$ of Polchinski's 1st String Theory book it is stated the following about Polyakov path ...
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General validity of the Linked Cluster Theorem in QFT for arbitrary correlation functions
In QFT, either at zero or finite temperature, the Linked Cluster Theorem (LCT) ensures that all disconnected diagrams appearing in the numerator of the interacting Green's function exactly cancel with ...
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Path Integral Formalism and Two-Point Function
I'm given an action,
$$A[\vec{S}] = \frac{\Theta}{2} \int_{-\infty}^{\infty} dt \left(\frac{d\vec{S}}{dt}\right)^2, \tag{1}$$
with $$\vec{S}^2 = 1.\tag{2}$$ I'm asked to calculate the two point ...
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Why can a real field have an imaginary saddlepoint?
Suppose we have a functional integral of the form
$$
\int D\psi \exp\left( -S \right)
$$
over a real field $\psi$ with an action
$$
S=\int_r L(r,\psi(r))=\int_r \frac{1}{2}\psi(r)^2+i\psi(r) j(r).
$$
...
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Propagator from free scalar path integral
Let $\langle{0|}T\phi(x)\phi(y)|0\rangle$ be the vacuum expectation value of the 2 point correlator for the free scalar field. Page six in these notes say that we can calculate this correlator by ...
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QED photon path (direction of photon emission)
In QED we look at all possible path a photon could go from S to P, and I understand the most significant contributions to the final arrow are the few near straight paths connecting S and P while other ...
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Inverse square law of a photon in QED
So in Feynman's QED book strange theory of light and matter, he mentioned as a photon travels, it spreads a little, thus the "arrow" shrinks inversely with distance, and that is the inverse ...
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Semiclassic limit of a QFT in Zinn-Justin
I am reading the Zinn-Justin book "Quantum Field Theory and Critical Phenomena" and i have come across a perplexing point.
Given the partition functional, in Euclidean QFT:
$$Z[J, \hbar] = \...
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Generating Functional for Massless Spin 2 Particle
I'm trying to derive the generating functional for a massless, spin 2 field. However, I am getting a left over term that needs to go away. I'm working in de Donder gauge so that $\partial_\mu h^{\mu\...
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Is the Euclidean generating functional $Z_{E}[J]$ identified with original Minkowskian generating functional $Z[J]$?
In quantum field theory, it is common to perform wick rotation $t\rightarrow -i\tau$ and get Euclidean generating functional $Z_{E}[J]$. When I first studied QFT, I just saw this a magic trick to ...
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The underlying cause of ill-defined loop-integrals in Quantum Field Theory
One of the main causes which leads to ill-defined loop integrals in Quantum Field Theory is that the variables of a Field Theory, $\varphi(x)$ for instance, are Quantum Fields which are governed by ...
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What's the correspondence between Feynman diagrams and field configurations?
If I understand correctly, a Feynman diagram represents a finite set of "interactions", such as the exchange of a photon between two electrons. You can think of it as a graph in which the ...
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Transition amplitude between field configurations from the path integral
In the path integral formulation of QFT, we should in principle be able to calculate the transition amplitude from a classical field configuration $\phi_{in}(x,t=0)$ to $\phi_{out}(x,t=T)$ using the ...
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Functional measure variable change
On Peskin & Schroeder's QFT, page 285, the book introduces the functional quantization of scalar fields.
We can replace the variables $\phi(x)$ defined on a continuum of points by variables $\phi(...
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Peskin and Schroeder path integral discretization
I'm reading section 9.2 of Peskin and Schroeder, specifically where they begin the explicit computation of the two point correlation function for a free scalar field using the path integral (p.285). ...
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Can higher loops cancel one-loop divergences?
Let's say that after dimensional regularization we have a one-loop effective action of the form
$$\Gamma^{\text{div}}_1=\int\frac{\mu^\epsilon}{\epsilon}\Big[ c_1\phi^2(\partial_\mu A^\mu) +c_2 \phi ...
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Calculating a Gaussian-like path integrals with Grassmann variables and real variables
I want to compute the following path integral
$$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \prod_{i=1}^{n}d\overline{\theta}_id\theta \: \exp{\left(-\overline{\theta}_i \partial_j w_i(x)\theta_j -\...
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Proving that the path integral formulation of scalar QED theory is independent of the choice of the gauge-fixing parameter $\xi$
I am considering the following scalar QED lagrangian:
$$L = −\frac{1}{4}F_{\mu\nu}^2 + |D_{\mu\varphi}|^2 − m^2|\varphi|^2− \frac{1}{2\xi}(\partial_\mu A^\mu)^2.$$
Where I want to show that the ...
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Taking imaginary time to show damping relation in path integral formalism
I'm having trouble with a step in the reasoning in pg.9 of Bailin & Love - Introduction to gauge field theory:
To find the ground-state to ground-state amplitude we have a term:(given a basis of ...
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