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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3answers
595 views

Why does spin appear in quantum systems but not classical systems?

It is often claimed that spin is a purely quantum property with no classical analogue. However (as was very recently pointed out to me), there is a classical analogue to spin whose action is given (in ...
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1answer
309 views

What exactly is a Feynman propagator?

Let $p,q$ be two points. On pg 671 of "Road to Reality", Penrose says that integrating the amplitudes of all paths between $p$ and $q$ would be infinite. Hence, we need the concept of a Feynman ...
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Understanding the statement “orbifold theories are QFTs with finite gauge group”

I'd like to understand the equivalence of orbifold theories in string theory and (2D worldsheet) QFTs with finite gauge group, using the path integral. Suppose my action is $$S= \frac{1}{2\pi \alpha'...
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47 views

Where to find a Path Integral treatment of 1D Quantum Heisenberg model or Quantum Spin Chain?

All: Where to find a Path Integral treatment of 1D Quantum Heisenberg model or Quantum Spin Chain? I would like to find a detailed calculation of path amplitude in such situation. I did some google ...
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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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1answer
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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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54 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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1answer
270 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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2answers
290 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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0answers
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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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1answer
118 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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57 views

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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1answer
242 views

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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2answers
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Derive Schwinger-Dyson equations in Srednicki

In eq. (22.20) on p. 135 in Srednicki he defines the functional integral $$Z(J) = \int\mathcal{D}\phi\,\exp\Big[\mathrm{i}\big(S+\int\mathrm{d^4}y \,J_a\phi_a\big)\Big], \tag{A}$$ where $S$ and $...
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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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1answer
79 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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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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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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1answer
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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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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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1answer
46 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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1answer
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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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1answer
106 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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2answers
259 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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2answers
122 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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710 views

One-loop Correction to Effective Action

This might be a stupid question. In Bailin and Love's "Cosmology in gauge field theory and string theory", the authors are describing how to calculate the effective potential at a finite temperature ...
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1answer
120 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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1answer
126 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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1answer
301 views

Gaussian integral in momentum space

My question is related to p. 353 of Altland and Simon (section 6.7) which concerns about the following field integral where $\beta = 1/T$ and $V_n$ is defined in the following way: It seems to be a ...
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31 views

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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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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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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1answer
79 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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1answer
89 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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2answers
654 views

How to directly evaluate path integral for harmonic oscillator by brute force method?

It is easy to evaluate the green's function using path integral approach by evaluating classical action and using functional calculus method. Is it possible to evaluate path integral for harmonic ...
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1answer
102 views

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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548 views

Vacuum to vacuum transition amplitude [duplicate]

I have two questions about Vacuum to vacuum transition amplitude. Can any particle stay in $|0\rangle$? I was studying this topic from Srednicki's QFT book. He writes in eq.$(6.22)$ $$\langle0|0 \...
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1answer
811 views

SUSY QM and Atiyah-Singer index theorem

Consider maps $t\mapsto x^i(t)$ from circle to some Riemannian (spin) manifold and Lagrangian $$ \mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k \...
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3answers
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How does light know which path is fastest?

We know from Fermat's principle of least time that light follows the fastest path. But how does light know which path is the fastest?
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1answer
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In the semiclassical approximation, should I expand the generating functional around saddles of the sourced or the unsourced action?

Consider a Euclidean path integral say in a real scalar field theory. $$ \int d[\phi]\exp(-I[\phi]) $$ In the semiclassical approximation, we consider stationary points of the action and expand ...
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1answer
55 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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1answer
51 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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1answer
298 views

Deriving the Lagrangian form of the Feynman path integral through Gaussian integration

The hamiltonian form of the path integral for the time evolution of a single particle in one dimension (in non-relativistic quantum mechanics) is: $$\langle x|\hat U(t_2,t_1)|x'\rangle=\int \mathcal ...
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1answer
276 views

Is Feynman gauge reduce always physical gauge?

Is Feynman gauge reduce always physical gauge? I heard in QCD, Feynman gauge does not always give correct physics. The lecture says, "Feynman gauge gives physical gauge, if the theory contains ...
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1answer
265 views

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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2answers
101 views

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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1answer
166 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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1answer
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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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