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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11
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3answers
145 views

Does a path integral necessarily mean there is a quantum mechanical description?

Given a path integral for a system $$Z(\phi) = \int [D\phi] e^{-S[\phi]},$$ where I am working in the Euclidean signature, necessarily mean that the system described is quantum mechanical? In the ...
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1answer
49 views

Are there other possible options to represent the amplitude in the path integral formalism?

The path intgral formalism of quantum mechanics states that the amplitude to go from $\left(x_i,t_i\right)$ to $\left(x_f,t_f\right)$ is $$K\left(x_f,t_f,x_i,t_i\right) = \int \mathcal{D}x\quad e^{i\...
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0answers
52 views

Path integral for free fermion: meaning of boundary terms

In article Deriving Projective Hyperspace from Harmonic there is discussion of JWKB (page 7) path integral for free fermion. Here I briefly rederive the statement: If one start with Lagrangian: $$ L = ...
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26 views

How to change the integration measure between vectors and complex numbers?

When using $CP(1)$ representation, we will write the unit vector $\boldsymbol{n}$ as the the spinor form, i.e. complex number: $$\boldsymbol{n}=\left(\begin{array}{cc}z_{1}^{\dagger} & z_{2}^{\...
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1answer
51 views

Meaning of capital pi symbol in sum over histories integral

This question is primarily mathematical in nature. I have been reading Quantum Field Theory for the Gifted Amateur and I am reading about Feynman’s path integral approach. The definition of the “sum ...
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1answer
29 views

Conjugate momentum in the vacuum functional for the fermionic oscillator

The vacuum functional for the fermionic oscillator is given by $$ Z[0] = N\int\mathcal{D}\overline{\psi}\mathcal{D}\psi \exp\left(i\int_0^Tdt\left(i\overline{\psi}\psi-w\overline{\psi}\psi \right)\...
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0answers
17 views

Applying the heat-kernel method?

I was wondering if the strategy I am going to try is correct? Am I missing anything important? I am currently trying to apply the stationary phase approximation to a path integral. I need to find the ...
7
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1answer
180 views

One-loop effective action for scalar field on the curved background in large potential

I hope to compute a functional integral $Z=\int \mathcal{D}\phi\,\, e^{-S[\phi]}$ with an action $$S[\phi]=\int d^2x \sqrt{g}\Big((\nabla \phi)^2+\frac{1}{\lambda}M^2(x) \phi^2\Big)$$ The scalar field ...
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1answer
73 views

Partition function in quantum field theory

Why does the partition function include current term in free scalar field $$Z[J] = \int \mathcal{D}\phi \, e^{i \left(S[\phi] + \int d^4x \,J(x) \phi(x) \right)}~$$
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2answers
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Why the classical configuration always static when applying saddle point (semi-classical) approximation?

For an Green function/partition function: $$\int D[\phi]e^{\frac{i S[\phi]}{\hbar}}$$ We can make saddle point approximation and gives classical configuration: $$\delta \mathcal{S}=0\Longrightarrow \...
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2answers
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Harmonic oscillator partition function via Matsubara formalism

I am trying to understand the solution to a problem in Altland & Simons, chapter 4, p. 183. As a demonstration of the finite temperature path integral, the problem asks to calculate the partition ...
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1answer
58 views

Generating Functional for Complex Scalar Theory

The generating functional for a free complex scalar field theory is given by: $$W[J,J^*]=\int D\phi D\phi^* \exp (i \int_{}^{} d⁴ x [(\partial_{\mu}\phi)^*(\partial^{\mu}\phi) -m^2\phi^*\phi + J^*\...
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1answer
65 views

Confusions on expectation value for $\hbar$ going to zero

In Matthew D. Schwartz's QFT book, Chapter 28, the author claims when $\hbar \rightarrow 0$, the following equality (eq 28.4) holds: So how can I see the second "$=$" holds? It seems the ...
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27 views

For the QFT path integral action for a pair of sources and sinks $W_{ij}(J_1,J_2)$, is $W_{12} = W_{21}$ ever NOT exactly upheld?

The path integral's action $W_{ij}$ as a function of a pair of sources and sinks $J = J_1 + J_2$ can be written $$W_{ij} = -\frac{1}{2} \int \frac{d^4k}{(2 \pi)^4}J_j^*(k)\frac{1}{k^2-m^2+i\epsilon}...
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Chern-Simons Path integral restricting to small gauge transformations

How does one compute the Chern-Simons path integral in 2+1 dimensions considering only small gauge transformations? Is this even a well-defined theory?
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1answer
20 views

Stationary Phase approximation with multiple coordinates?

The stationary phase approximation can be used to find an approximate value for the path integral \begin{equation}\int Dx e^{-S[x]} \approx e^{-S[\bar{x}]} \left(\det{\frac{\hat{A}}{2 \pi}}\right)^{-1/...
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0answers
57 views

Measure of Feynman path integral

Feynman path integral for non-relativistic case is defined as: $$\int\mathcal{D}[x(t)]e^{iS/\hbar}$$ where $$\int \mathcal{D[x(t)]}=\lim_{N\rightarrow\infty}\Pi_{i=0}^{i=N}\bigg(\int_{-\infty}^{\infty}...
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1answer
67 views

Derivation of transition amplitude probability between two adjacent space points for general time independent hamiltonian

I am studying Srednicki book of Quantum Field theory. In chapter 6 regarding the path integral there was derived equation of transition probability for hamiltonian of type: $$H(\hat{P},\hat{Q})= \frac{...
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What is the entropy and/or equation of state of a partition function such as $Z=\int D\phi \exp (i S[\phi]/\hbar)$?

At this link https://en.wikipedia.org/wiki/Partition_function_(mathematics), it is claimed that the following partition function: $$ Z=\int D\phi \exp (-\beta H[\phi]) \tag{1} $$ is a consequence of ...
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1answer
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Question about Faddeev-Popov gauge-fixing in Schwartz textbook

I am trying to understand equation (25.91) from Schwartz's Quantum Field Theory textbook. The goal is to gauge-fix the path integral for Quantum chromodynamics using the Faddeev-Popov trick. Briefly, ...
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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 ...
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43 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-...
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55 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 , \...
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1answer
59 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 $$ \...
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1answer
34 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}...
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1answer
47 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}^{...
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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 ...
5
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1answer
77 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 ...
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20 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 ...
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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 ...
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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 \...
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49 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})^...
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56 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.
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1answer
62 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 ...
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0answers
52 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} \...
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45 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 ...
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46 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 variables $...
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1answer
36 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 ...
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1answer
49 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 ...
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1answer
53 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 ...
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2answers
78 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
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0answers
33 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 ...
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1answer
63 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
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3answers
146 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 ...
3
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2answers
131 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?
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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 ...
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2answers
131 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
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2answers
68 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 ...
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1answer
91 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)...
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0answers
78 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})...

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