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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32 views

Constraints in path integral and the Lagrange multiplier

I was reading some references on the slave-particle approach to the Kondo problem and Anderson model. It is known that the slave-particle is introduced in the large Hubbard $U$ limit of the system so ...
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Calculating kernel [closed]

A particle with mass m in one dimension is kept in a constant external force field F.The lagrangian of the system is L=mv^2/2 + Fx.Calculate the kernal if classical action is {[m(xf-xi)^2/2(tf-ti)]+[F(...
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33 views

Generalization of Gaussian integral for tensors

How do you generalize the formula for matrices (or operators) $$\int d^d x \, \exp \Big\{ - \frac{1}{2} x^i A_{ij} x^j \Big\} = \sqrt{\frac{(2 \pi)^d}{\det A}} = \sqrt{\det (2 \pi A^{-1})}$$ for ...
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Calculation of 3-point function given a generating funcional $Z[J]$

With: $$\ln Z[J]= \int dt \frac{J^2(t)}{2} f(t) + C \int dt \frac{J^3(t)}{3!}$$ I am asked to calculate the 3-point funcion. Attempted solution: The 3-point funcion is given by $\frac{ \delta^3 }{\...
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130 views

Calculating the numerical factor from Feynman diagram

I kind of understood the symmetry factor quite well. However, I just do not understand how one can relate the Feynman diagram to the term (especially the numerical factor in front of it) in the ...
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Proper path integral of a field theory

I have been trying to find out the sweet middle ground of describing path integration of field theories, in between the physicist way and the mathematician way, but it seems hard to find something ...
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74 views

Why any expectation value can be computed by this path integral, and not just the time-ordered ones?

This is quite a basic question about the path integral. In Polchinki's String Theory book, Chapter 2, he says: Expectation values are defined by the path integral $$\langle \mathscr{F}[X]\...
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202 views

Why can't quantum field theory be complex instead of imaginary?

In the following question 1, the author claims that a QFT is defined as: $$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$ Then uses this definition to explore the possibility of formulating a QFT ...
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Solving the Line Integral in Ampère's Law mathematical correctly

Imagen we have a infinite long, cylindrical conductor with radius $\varrho_0$ and $\textbf{j}=\begin{cases}j_0 \textbf{e_z} &r\leq \varrho_0\\ 0 &r>\varrho_0\end{cases}$ We have Ampère's ...
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Using the Martin-Siggia-Rose (MSR) formalism for oscillator with general non-harmonicity

I am wondering if using the Martin-Siggia-Rose (MSR) formalism can be convenient/treatable for calculating correlation functions [or their spectral densities] of a linear [underdamped] oscillator with ...
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91 views

Wick rotation: still trouble in getting how it works

I'm preparing my second exam in QFT and I still have trouble in getting the Wick rotation and its analytic continuation. I know that this topic have been discussed a lot in previous threads, but I ...
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Does the concept of the photon as a particle exist in QFT in the path integral formalism? Does the concept of a particle exist?

In the second quantization approach to quantum field theory, how I understand it, the field is decomposed in components of definite momentum which are treated as non-interacting harmonic oscillators, ...
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53 views

Diagrammatic Representation of non-Gaussian perturbation expansions

I have no experience in graph theory and am a little confused with how Hugh Osborn represents a perturbation expansion with diagrams on page 15 of these notes. We have a perturbation expansion My ...
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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 what I understood!). Let a quantum system with ...
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Relation between functional measures and states in AQFT

Let $(M,g)$ be a globally hyperbolic spacetime and $\phi$ a KG field. In AQFT we consider the algebra of observables $\mathfrak{A}$ generated by $\phi(f)$ where $f\in C^\infty_0(M)$ is a test function....
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25 views

Fermionic thermal density matrix

Usually to describe the density matrix of a system at finite temperature, we use the Euclidean path integral $$\rho[\psi_1,\psi_2] = \int _{\psi_1}^{\psi_2}\mathcal D \psi e^{-S_{E}[\psi]}, $$ where $...
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Quantum corrections to metric on non-linear sigma model target space

I am trying to make sense of what physicists mean when they talk of quantum corrections to the metric on the target spaces of nonlinear sigma models, for example [GHL99]. First some quick notation. ...
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81 views

Book reference: Quantum field theory from path integrals

a) What are some good references to learn quantum field theory from the approach of path integrals? Like books which start from path integral formulation of quantum mechanics and then do calculations ...
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30 views

(Altland-Simons) Question about a seemingly additional term in the functional field integral

The following is the part of the book from Atland-Simons. My question is about the additional $-\overline{\psi}^{n+1}\psi_n$ in $(4,27)$ of the book. I understand that the term $\overline{\psi}^{n}\...
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57 views

Wick rotation convergence. Functions in the integrand

Performing a Wick rotation over an integral is not equivalent to just a change of variable $t \to \mathrm{i}t = \tau$, after that we rotate the complex plane so that $$\mathrm{i} \int_{-\infty}^{\...
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106 views

Path integral and least action principle

I'm reading Sakurai's book. And there is a part, where it says: let's consider the path that satisfies $$\delta S(N,1) = 0,$$ where the change in $S$ is due to a slight deformation of the path with ...
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82 views

Where the derivative corrections come from in Wilson renormalization?

I known that in the Wilson renormalization process fast modes are integrated out in order to define an effective action for the low modes field. Considering phi to the fourth theory it's easy to see ...
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98 views

Gaussian path integral is equivalent to saddle-point?

If we have a path integral involving many fields, $$Z = \int \mathcal D \phi_1 \cdots \mathcal D \phi_n \exp(-S[\phi_1,\ldots, \phi_n]),$$ and $\phi_n$ occurs only quadratically-- i.e. the $\...
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91 views

Quantum corrections in path integral

I am working the following exercise: Calculate the generating functional $$Z[j]=\int \mathcal{D}\Phi \exp\left(\frac{i}{\hbar}S[\Phi,j]\right),\quad S[\Phi,j]=\int d^4x(\mathcal{L}(\Phi)+j\Phi),$$ $...
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63 views

Why is the Jacobian factor for fermionic variables different from that for bosonic ones?

In Srednicki's textbook Quantum Field Theory, Section 77 discusses anomalies and the path integral for fermions. The path integral over the Dirac field is defined to be \begin{equation} Z(A) \equiv \...
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67 views

Harmonic oscillator path integral: regularizing the functional determinant

From Polchinski's Vol. 1 Appendix A, we can reduce the Euclidean path integral for the 1D harmonic oscillator to computing $(\det\frac{\Delta}{2\pi})^{-1/2}$ where $$\Delta = -\partial_u^2 + \omega^2.$...
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Feynman's path integral and a complete basis

Can we consider the Feynman's probability amplitude as a sum in a complete orthonormal system? And if so, is there a mathematical framework, e.g. an inner-product, that gives the condition? Since all ...
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Must the mean field, in the context of the background field method, satisfy the classical equations of motion?

When deriving the effective action $\Gamma$ in the background field method, one splits the field $\phi = \phi_b + \phi_f$ into a background (or mean field) $\phi_b$ and fluctuations $\phi_f$, then ...
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52 views

Normalization of the integration measure of the Feynman's formula to combine denominators

In Mark Srednicki "Quantum field theory", section 14 -Loop corrections to the propagator-, it is presented the Feynman's formula to combine denominators: $\frac{1}{A_1 ... A_n} = \int dF_n (x_1 A_1 + ....
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63 views

What is the definition of functions of Grassmann numbers?

I understand there are some relevant questions, but none of them solves my issue. From Atland and Simons (Condensed Matter Field Theory), the definition of functions of Grassmann numbers are defined ...
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How can tempered distributions be paths?

I'm reading the Appendix A of Glimm and Jaffe book "Quantum Physics: a functional integral point of view", and there is something that I'm missing In section A.4 the authors talk in a very general ...
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384 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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45 views

Discretization of path integral and linear interpolations

Consider the evaluation by discretization of the path integral $$\int e^{iS[x(t)]}\mathfrak{D}x(t),\quad S[x(t)]=\int_{t}^{t'}\left[\frac{m}{2}\dot{x}(\tau)^2-V(x(\tau))\right]d\tau.$$ One ...
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43 views

Localization Principle (SUSY)

Mirror Symmetry p.200/201 Last section p.200/first p.201 It says, that the localization principle would not work if one would not impose periodic boundary conditions for the fermion integration, ...
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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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126 views

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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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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55 views

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 studied the decay of a Meta-stable state via path integral method. The real-time potential is A state at $x=-a$ decays to $x=\infty$. The ...
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38 views

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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65 views

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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92 views

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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26 views

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 ...