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.

Filter by
Sorted by
Tagged with
1
vote
0answers
26 views

According to Hartle-Hawking state, could we build a sum over all possible metrics (including non-compact ones)?

Physicists Stephen W Hawking and James B Hartle 1 proposed that the universe, in its origins, had no boundary conditions both in space and time. To do that, they proposed a sum over all compact ...
1
vote
0answers
39 views

Expression for sum over paths

In an introductory lecture on the path integral formalism, I came across the following. Suppose that $\gamma$'s are paths such that a particle travelling along any of them reaches the position co-...
3
votes
0answers
46 views

Symmetries in quantum field theory and anomalies

Suppose we have a lagrangian quantum field theory, thus a theory where we can write an action in the form \begin{equation} S = \displaystyle \int d^4 x \; \mathcal L \, \left( \partial_{\mu} \phi , \...
1
vote
1answer
52 views

Eliminating residual gauge in BRST quantization of Yang-Mills theory

I would like to know if there is a procedure to completely fix a gauge, which I believe we must do in order to make sense of the path integral? In chapter 74 Sredniki introduces the Lagrangian $$ \...
0
votes
1answer
30 views

D'Alembert Operator on Fermionic Field in Path Integral

I am learning the Faddeev–Popov path integral formlism with Schwartz's QFT textbook. In the section 25.4.2 "BRST invariance", I came across the Lagrangian as: $$\mathcal{L}=-\frac{1}{4} F_{\mu \nu}^{2}...
1
vote
1answer
35 views

Fermionic Harmonic Oscillator Partition Function

I am reading Nakahara Geometry, Topology, and Physics. In the section on fermionic harmonic oscillator, after some math, the partition function is given by $$\begin{aligned} Z(\beta) &=\mathrm{e}^{...
5
votes
0answers
32 views

Examples of path integral where path of extremal action does not contribute the most?

I have learnt that by doing a saddle point approximation in the path integral formulation of quantum mechanics, the classical action (extremal action where $\delta S=0$) is the one that contributes ...
-1
votes
0answers
18 views

Does changing the direction of paths change the value of the path-integral?

When we use path-integral formalism to calculate some quantities, we often specify the starting and the ending points in the phase space, say $A$ and $B$. Let say $\gamma_A^B$ denote a path that ...
5
votes
1answer
70 views

Proving that a Wick rotation is valid for a quantum field theory

While trying to find out if there is a rigorous justification for Wick rotating a QFT, I came across this other question (link below [1]) that mentions the Osterwalder-Schrader Theorem that gives a ...
0
votes
0answers
19 views

How to deal with integral operators in the action, in the path integral of a field theory?

One could imagine adding to the free action of a scalar field theory some non-local operators given as integrals over the base manifold (or over the boundary) of some smooth function of the scalar ...
0
votes
1answer
14 views

Confusion regarding a few sign conventions in appyling faraday's law to inductive circuits

In the mit ocw lecture by Prof. Lewin on EMI, He quotes a few statements from here to couple of seconds of the lecture. I am confused why the sign of $L\frac{dI}{dt}$ changes, when we go around and ...
0
votes
1answer
35 views

Why does $q(t) \to-i\hbar \frac{\delta}{\delta J(t) }$ for the generating functional of a perturbed harmonic oscillator?

When computing a generating functional, $Z[J]$, in terms of the generating functional of Green functions, $Z[0]$, in my lecturer's notes we reach the following terms: $$Z[J]= \mathcal{N} \int Dq \...
1
vote
0answers
39 views

Gauge invariance of the regulator when calculating the chiral (ABJ) anomaly by the Fujikawa method

I am currently studying the calculation of chiral anomaly using fermionic path integral. In all texts I looked at, the authors simply use a regulator of the following form $e^{(\gamma_{\mu}D^{\mu})^...
1
vote
0answers
50 views

Derive Feynman rules for interacting Proca theory

Is there a smooth way to derive the Feynman rule for the interaction term $$\mathcal{L}_{int}=gA_{\mu}A^{\mu}A_{\nu}A^{\nu}?$$ $A_{\mu}$ denotes a massive vector field.
1
vote
1answer
56 views

Harmonic Oscillator: extract the ground state wave function from the propagator

I am currently studying the path integral formulation of quantum mechanics and have done a couple of problems (free particle and simple harmonic oscillator). Now, I am already done calculating the ...
1
vote
0answers
48 views

Fourier Transform in the Path Integral of a Harmonic Oscillator

My question comes directly from Section 7 of Srednicki's QFT textbook. I'm not able to reproduce Equation (7.5): $$\begin{aligned} [\cdots]=\frac{1}{2} \int_{-\infty}^{+\infty} \frac{d E}{2 \pi} \...
0
votes
0answers
43 views

Equation of state of a path integral

How does one take the equation of state of a path integral? In "discrete" statistical physics, one has this partition function: $$ Z=\sum_{i}\exp(-\beta E[i]) $$ And the equation of state is the ...
0
votes
0answers
43 views

Path integral as a partition function (math)

I am reading the following Wikipedia page, but I am skeptical about what I am reading (it sounds too good to be true). Specifically, I am looking at the passage which states: The number of ...
1
vote
1answer
30 views

$R_\xi$ gauges and the EM-field

$R_\xi$-gauges are said to be a generalization of the Lorenz gauge. I dont quite get why we add the term $$ \mathcal L_{GF} = - \frac{(\partial_\mu A ^\mu)^2}{2\xi} $$ to the Lagrangian. If i ...
2
votes
1answer
48 views

In string theory path integral, what happens if I fix worldsheet metric?

In string theory worldsheet path integral, integral is done over all possible topologies, metric and coordinates. And I was wondering if there is something in string theory similar to quantum field ...
1
vote
1answer
49 views

Why is the Nambu-Goto path-integral ill-defined?

I have found a lot of places saying that the Nambu-Goto action is ill-defined, that the squareroot exponential is a complicated thing to make sense of in a path-integral and so on. Then people go on ...
3
votes
2answers
67 views

Wick Rotation & Scalar Field Value & Mapping

Wick Rotation helps to solve the problem of the convergence of the path integral, by changing the integral contour in the complex plane. But my question is: In the scalar field path integral, the ...
4
votes
0answers
29 views

Questions about the large-instanton problem

The Problem. The issue that I'm talking about is the large-instanton problem of asymptotically-free non-abelian gauge theories. You can read about it in: [1.] Section 15 of 't Hooft's 1976 paper on ...
3
votes
1answer
60 views

I am stuck in the derivation of Schwinger-Dyson equation for 1-point Function in Schwartz's QFT book

This is from chapter 14.7.1 in Schwartz's QFT book. I am trying to derive contact terms starting from field redefinition $\phi\rightarrow\phi(x)+\epsilon(x)$. For the 1-point function we have from ...
4
votes
3answers
133 views

Gauge invariance of Faddeev-Popov determinant in bosonic string theory

I am, once again, going through an introduction to (bosonic) string theory, following the lecture notes by David Tong on the subject, and once again I am stumbling on technicalities around the ...
2
votes
1answer
93 views

Does the quadratic mass term $\phi^2$ belong to the free Lagrangian or is it an interaction term?

$$L = -\frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{m^2}{2}\phi^2.$$ Why is the $\phi^2$ term in the scalar Lagrangian not considered a self-interaction?
1
vote
0answers
54 views

Completeness relation in QFT

I am studying QFT using Peskin and shroeder, At page 284,I am stucked at a point he stated. $$\int D\phi| \phi \rangle\langle\phi|=1$$ If space is $2$-dimensional, Integral over $D\phi $ is a ...
4
votes
2answers
127 views

Why does a square root term make the quantisation of action difficult?

When going over my lecturer's notes on String Theory and trying to understand a particle as a theory of gravity in 1D, it is mentioned that the action $(1)$ is regularisation invariant, $$S=-m\...
3
votes
2answers
64 views

I want to understand a trick in the derivation of the Schwinger-Dyson equations

In the book of Ashok Das, Field theory-path integral approach, he begin the demonstration of the Schwinger-Dyson equation using the fact that the $\delta Z[J]=0$, so \begin{equation} \delta Z[J]=\int ...
6
votes
1answer
88 views

Choice of folliation in path integral

Assume we have a scalar field theory for a field $\phi$. Can we think of the Hilbert space as being spanned by states of the form $|\varphi\rangle$ for configurations $\varphi\in C^\infty(\mathbb{R}^3)...
3
votes
0answers
75 views

Functional determinant in scalar QED

I'm trying to integrate out the scalars from the path integral in scalar QED, but I encountered an integral I don't know how to do. The model is $S = \int_{\mathbb{R}^4}d^4x \left( -\frac{(F_{\mu\nu})...
8
votes
4answers
196 views

Why can we shift the field $\phi$, so that $\langle \Omega | \phi(x) | \Omega \rangle = 0$?

Problem Introduction In different derivations of the LSZ reduction formula the author makes a shift of the field $\phi(x)$ $$ \phi'(x) = \phi(x) - \langle \Omega | \phi(x) | \Omega \rangle, $$ and ...
7
votes
2answers
92 views

Saddle point approximation and finite action configurations forming a set of zero measure

In Coleman's "Aspects of Symmetry", chapter 7, section 3.2, he makes a claim that configurations of finite action form a set of zero measure and are therefore unimportant. Further, he goes on to prove ...
1
vote
0answers
42 views

How to perform a contour integral for propagators (Peskin section 2.4)

I am currently studying Quantum field theory and I have a problem understunding the way some integrals are performed. I am talking about the ones involving Green functions. I can't understand the ...
2
votes
1answer
41 views

On the prefactor in the path integral formulation

The propagator $K$ from ($x_a,t_a$) to ($x_b,t_b$), as defined by Gottfried, can be written as $$ K(b,a) = F(t_b-t_a)\exp\left(\frac{i}{\hbar}S_{c}(b,a)\right) $$ where $S_c$ is the classical action ...
7
votes
2answers
261 views

Path integrals vs. Diagrammatics

The question is about the approximation techniques available in the path integral formulation and their equivalents in the context of the traditional Feynman-Dyson expansion (aka diagrammatic ...
2
votes
0answers
49 views

Propagator in massive QED/Schwinger model

I'm trying to integrate out the fermions from the path integral in the massive QED/Schwinger model $S = \int_{\mathbb{R}^d}d^{dx} \left( - \frac{(F_{\mu\nu})^2}{4} + \bar{\psi} \left( i\gamma^\mu D_\...
2
votes
2answers
69 views

In the Feynman Path Integral, why must “contributing paths” be continuous? Or is this a false notion?

I have been studying the Feynman path integral and its various derivations, and I've run into a bit of a problem. The standard Feynman path integral appears as follows: $$ \int \mathcal{D}[x(t)]\exp\...
0
votes
2answers
54 views

Momentum-dependent correlator in free scalar field theory

In Klein-Gordon theory we have common representation of n-point correlation function as path integral: $$ \langle 0|T[(\hat{\phi}\left(x_{1}\right) \hat{\phi}\left(x_{2}\right))\dots\hat{\phi}\left(...
2
votes
1answer
52 views

Question about an OPE for the free massless scalar CFT

In page 78 of David Tong's notes on CFT https://www.damtp.cam.ac.uk/user/tong/string/four.pdf, he finds that the propagator for a theory of free massless scalars is $$\langle X(\sigma)X(\sigma')\...
2
votes
1answer
94 views

Wilson loop as path integral of parallel transport action

I am trying to get that the path integral of the parallel transport action is the Wilson loop. Here is the setting: Let $w$ be a complex vector dimension $N$, and $A_{\mu}$ a fixed Yang-Mills ...
10
votes
3answers
247 views

When is Schwartz's method for “integrating out” a field valid?

In Schwartz's QFT book, heavy fields are often "integrated out" by simply solving their equations of motion formally (i.e. allowing things like $\Box^{-1}$) and plugging them back into the Lagrangian. ...
3
votes
1answer
113 views

Path integral for spin?

I'm searching for the path integral formulation for a spin particle and haven't found any precise description yet. Is there a systematic way to construct a non-relativistic path integral formulation ...
0
votes
1answer
42 views

Measure in the Fourier Representation of the Coherent States Path Integral

The problem I have arises in the context of condensed matter physics. I am largely following chapter 4 about functional integration in the book by Altland and Simon. Consider the coherent states path ...
4
votes
2answers
78 views

How to unify the cumulant expansion and Feynman diagram expansion?

In most QFT books, the perturbation theory is given by "Taylor expansion". When evaluating 2-points, the numerator gives all the diagrams, i.e. $$\int D[\phi]e^{iS[\phi]}\phi_1\phi_2=\int D[\phi]\...
3
votes
0answers
52 views

Scalar field propagator in curved space from path integral

Consider a scalar masless field (in 2d for concreteness) in a curved space with standard action $$S=\frac{1}{4\pi}\int d^2x \sqrt{g}g^{ab}\nabla_a\phi\nabla_b \phi$$ There is an elegant way to derive ...
2
votes
0answers
53 views

Integrating out massive degrees of freedom of Super Yang-Mills Action in Matrix Theory

From this paper, I want to integrate out the massive degrees of freedom. The total action $S$ is given by $$S = S_{Y} + S_{A} + S_{Fermi} + S_{ghost} $$ where the terms are given in equations (2.9),(...
2
votes
3answers
79 views

Decoupling of ghost fields in axial-gauge QCD

After quantizing QCD using the Faddeev-Popov "prescription", we end up with the original QCD Lagrangian plus the gauge-fixing term, \begin{equation} -\frac{1}{2\alpha}(n\cdot A)^2, \end{equation} and ...
0
votes
0answers
44 views

Supersymmetric localisation of 2D super YM on $S^2$

I wanna to understand how to calculate partition function for pure abelian Yang-Mills theory. To do this, I need follow some usual step's (I follow Benini, Localization in supersymmetric field ...
3
votes
2answers
95 views

Getting Feynman propagator using path integral

In QM using Feynman path integral(FPI) we derive the propagator of free particle which comes out to $$(f(t))e^{iS_{cl}/\hbar}$$ But in QFT the Feynman propagator is derived using the differential ...

1
2 3 4 5
19