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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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Why can the time-ordered exponentials be brought to the right?

Having worked through almost all calculations in section 4.2 of Peskin & Schroeder's An Introduction to QFT, I still don't get why we can get to Eq. (4.31) \begin{equation} <\Omega|\mathcal{T}\...
Drarp's user avatar
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10 votes
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Does a good path integral exist in Loop Quantum Gravity?

The Hamiltonian operator of Loop quantum gravity is a totally constraint system $$H = \int_\Sigma d^3x\ (N\mathcal{H}+N^a V_a+G)$$ Here, $\Sigma$ is a 3-dimensional hypersurface; a slice of ...
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9 votes
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Different features of Gravity and Yang-Mills

I am reading a famous paper by S.Hawking - "Quantum gravity and path integrals" https://doi.org/10.1103/PhysRevD.18.1747. On the third page left column there is a statement, after the ...
spiridon_the_sun_rotator's user avatar
9 votes
0 answers
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Lattice QFT of the Jones Polynomial

Start with a gauge theory with Chern-Simons action $$ S[A] = \frac{k}{4 \pi} \text{Tr} \intop_{M} \left( A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right) $$ and a Wilson loop observable in the ...
Prof. Legolasov's user avatar
9 votes
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296 views

Path Integral on Feynman Hibbs: Interaction of EM field and matter, how can we get to equation (9.68) from (9.67)?

On Feynman Hibbs "Quantum Mechanics and Path Integrals", the equation (9.67) describes the transition amplitude of the matter (for example an atom) to go from the state $M$ to the state $M$ ...
M. M. R.'s user avatar
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8 votes
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379 views

Definition of gravity path integral?

In a non-abelian gauge theory there is a "fundamental" gauge field $A_\mu^a$ with gauge index $a$ often called connection. Although $ A_\mu^a$ is not gauge invariant, gauge invariant quantities can be ...
user47224's user avatar
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8 votes
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Is it possible to do a path integral between two boundaries analytically on a quantum lattice?

I have been trying to perform some path integral between two boundaries for a massless scalar field $$\int_{\varphi(t_a, \vec{x})}^{\varphi(t_b, \vec{x})} \mathcal{D}\varphi(x)e^{iS[\varphi(x)]}$$ ...
Slereah's user avatar
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8 votes
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Gaussian Integrals : Functional determinant expressed as a trace

Be $A_{ij}$ a symmetric matrix. Then I can easily write $$ \int \exp\left(-\frac{1}{2}\sum_{i,j}x_i A_{ij} x_j+\sum_{i} B_i x_i\right)\; d^nx= \sqrt{(2\pi)^n}\exp\left\{-\frac{1}{2}\mathrm{Tr}\log A\...
Juan Sebastian Totero's user avatar
7 votes
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267 views

References on the relation between path integrals and algebraic QFT

I am wondering on whether there are any references discussing how path integrals relate to the algebraic approach for quantum field theory. More specifically, in the algebraic approach, states are ...
7 votes
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Violation of Bell-like inequalities with spatial Boltzmann path ensemble: Ising model?

Quantum mechanics is equivalent with Feynman path ensemble, which after Wick rotation becomes Boltzmann path ensemble, which can be normalized into stochastic process as maximal entropy random walk (...
Jarek Duda's user avatar
7 votes
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465 views

2D Chern Simons action by integrating out fermions

In Qi, Hughes, and Zhang's paper (https://arxiv.org/abs/0802.3537), they show how the Chern number appears as a coefficient of response function. Given the Hamiltonian (49) of a (2+1) or (4+1)D ...
Yu-An Chen's user avatar
7 votes
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424 views

What is the link between path integral source terms and the propagator?

Consider a diffusion process defined by $$\frac{\partial \phi}{\partial t} = D \frac{\partial ^2 \phi}{\partial x^2}$$ where $\phi$ is the probability density of a particle's position and $D$ is the ...
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6 votes
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232 views

Operator insertions vs boundary conditions in AdS/CFT

This question is motivated by AdS/CFT, but really it's just about AdS quantum gravity. Consider quantum gravity in asymptotically AdS spacetime. For simplicity, assume there are no matter fields: the ...
nodumbquestions's user avatar
6 votes
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206 views

Saddle point contributions to the gravitational path integral

In his lectures on black holes and quantum information, Tom Hartman states that the gravitational path integral can be approximated as $$ Z(\beta) \approx \sum_{g_\text{cl}} e^{-I_E[g_\text{cl}, \phi]}...
einsteinfanboy98's user avatar
6 votes
0 answers
165 views

Relationship between product integrals and functional determinants

This is in reference to the answer posted to this question. The person who answered the question claims that the functional determinant of any operator $O$ is given by a product integral $$\det O = \...
Dr. user44690's user avatar
6 votes
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180 views

SUSY sigma model in QM, bosonic sector?

The bosonic sigma model in ordinary QM (i.e. a 'free' particle trapped on a curved manifold $\mathcal{M}$), has a Hamiltonian which is just the negative Laplacian on $\mathcal{M}$. For any $\mathcal{...
octonion's user avatar
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6 votes
0 answers
427 views

Formulating BCS theory in the functional integral with a real order parameter

Often in BCS theory, people take the order parameter $\Delta$ to be real. I tried to construct a BCS theory with a real order parameter from the start and ran into some trouble. I'd be interested to ...
Mason's user avatar
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610 views

Relationship between the statistical mechanics partition function and the path integral correlation function

In the path integral formulation I have $Z[J]$, the generating functional of correlation functions, and $W[J]=\frac{i}{\hbar} \ln{Z[J]}$, the generating functional of connected correlation functions. ...
PsycoPulcino's user avatar
6 votes
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648 views

Deriving the Path Integral Representation of the Fokker-Planck Equation

Suppose $$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$$ is a 1D nonlinear stochastic differential equation ($dW_t$ is typically assumed to be Brownian). According to wikipedia the distribution of $X_t$ at ...
6 votes
0 answers
795 views

What is Hawking Hartle vacuum state and why does the following Euclidean path integral gives the wave functional of it?

I am studying the wave function of black hole via the paper by Sergey Solodukhkin, Entanglement entropy of black holes,arXiv:hep-th: 1104.3712. In the paper, equation (53) is as follows: $$\Psi[\psi_{-...
xjtan's user avatar
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5 votes
1 answer
53 views

Onsager-Machlup functional and the Boltzmann distribution

I've been looking into path integral representations of stochastic processes lately and came across the Onsager-Machlup functional description of the Langevin equation. In the overdamped case, where ...
aQuarkyName's user avatar
5 votes
0 answers
216 views

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 ...
user341440's user avatar
5 votes
1 answer
117 views

What is the meaning of the function of $x$ and $p$ that you get when you cut open the phase space path integral?

When we "cut" an ordinary path integral, we obtain a state in the position representation. That is, if we fix some initial position $x_i$, then the path integral $$\int_{x_i}^{x_f}Dx e^{-S}$$...
nodumbquestions's user avatar
5 votes
2 answers
335 views

Gaussian Grassmann integral with complex/bosonic source term

I'm interested in solving the following multi-dimensional integral $$ \int d \theta d \bar{\theta} e^{-\bar{\theta}M \theta +\Lambda \theta + \bar{\theta} J } $$ where $\theta$ is a $N$-dimensional ...
Jsf73's user avatar
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5 votes
0 answers
294 views

Schwinger-Keldysh contour and $i\epsilon$ prescription

In Tom Hartman's notes on path integrals, he describes the Schwinger-Keldysh (or "in-in") formalism for calculating vacuum correlators in QFT. He explains that Lorentzian time-ordered vacuum ...
nodumbquestions's user avatar
5 votes
0 answers
169 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 , \...
PPIP's user avatar
  • 141
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0 answers
109 views

New symmetries upon quantization

In standard field theory texts, a “classical symmetry” is defined to be a transformation $\phi\to\phi’$ such that the corresponding action is left invariant. The symmetry is said to survive ...
Bob Knighton's user avatar
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5 votes
0 answers
217 views

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 just what I understood!). Let a quantum system ...
Gold's user avatar
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5 votes
0 answers
913 views

Normal ordering in path integral of QFT

In QFT, we use normal ordering to eliminate infinity from hamiltonian. In path integral formulation of QFT though, since what we integrate over is "classical field configuration", instead of operators,...
Brion Brion's user avatar
5 votes
0 answers
166 views

Path integral and gauge redundancy for slave particle

In the slave boson, we have $c^\dagger = b f^\dagger$ where $b$ is boson and $f$ is fermion. There is also a local constraint $b^\dagger b+f^\dagger f=1$ to retrict the Hilbert space and a $U(1)$ ...
Goodfish's user avatar
  • 301
5 votes
2 answers
376 views

What's the path of least action for fermions off-shell?

The Lagrangian of fermions is first order both in space-derivatives and time-derivatives. In the variation of the action one usually fixes both the initial point and end point. I have the following ...
WIMP's user avatar
  • 2,645
5 votes
1 answer
288 views

Weinberg's spontaneous broken symmetries

Steven Weinberg in his second volume of QFT's book (in section about spontaneously broken symmetries, in subsection about Goldstone bosons) writes following: if we have linear transformation of ...
Name YYY's user avatar
  • 8,841
5 votes
0 answers
1k views

Getting Slavnov-Taylor identity

Let's have generating functional in path integral form for gauge $SU(n)$ theory with interaction: $$ \tag 1 Z[J] = \int DB D\bar{\Psi}D\Psi D\bar{c}Dc e^{iS}. $$ Here $$ S = S_{YM}(B, \partial B) + S_{...
Andrew McAddams's user avatar
5 votes
0 answers
153 views

Path Integral on Einstein Cartan Manifold

In condensed matter, crystal with disclination and dislocation has both curvature and torsion. I am looking for a reference in which path integral quantization of Dirac equation on manifold with ...
5 votes
0 answers
527 views

What is the relationship between consistent histories and path integrals?

As can for example be learned from chapter I.2 of Anthony Zee's Quantum field theory in a nutshell, path integrals can be used to to calculate the amplitude for a system to transition from one state ...
Dilaton's user avatar
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5 votes
0 answers
571 views

When can the source term of a partition function be put in?

More specifically, in quantum field theory books, we usually have this: \begin{equation} Z = \int D(\bar{\psi}, \psi) e^{-S + \int_0^\beta d\tau \sum_l [\bar{\eta}_l (\tau) \psi_l (\tau) + \bar{\psi}...
Jordan's user avatar
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4 votes
1 answer
66 views

How does path integral quantization ensure unitarity?

Unitarity can be verified post hoc by examining the optical theorem. In the context of path integral quantization where formal derivation starting from canonical quantization is unavailable, is it ...
Bababeluma's user avatar
4 votes
0 answers
168 views

Quantum field theory with $\ln(\phi)$ potential

Consider a quantum field theory with the following action $$S[\phi] = \int d^dx \left(-\frac{1}{2}\phi\Box\phi -\frac{m^2}{2}\phi^2 +m^d\ln\frac{\phi}{m^{(d-2)/2}}\right)\\=m^d \int d^dx \left(-\frac{...
Dr. user44690's user avatar
4 votes
0 answers
60 views

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) = ...
Mass's user avatar
  • 2,020
4 votes
0 answers
128 views

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 ...
Aleksei Malyshev's user avatar
4 votes
0 answers
104 views

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 $$ \...
Martin Johnsrud's user avatar
4 votes
0 answers
108 views

Path integral representation of the vacuum wavefunctional for theories with massless particles

The vacuum wavefunctional of a quantum field theory is often generally expressed in terms of a path integral as \begin{equation} \langle\Phi(\vec{x})|\Omega\rangle = \int_{\varphi(\tau=0, \vec{x}) = \...
Bruno De Souza Leão's user avatar
4 votes
0 answers
82 views

How to define volume of Weyl transformation and diffeomorphism groups?

I have a trouble in defining Weyl transformation and diffeomorphism group volumes in the formal expression of string partition function on some manifold $M$: \begin{eqnarray} Z_M=\frac{\int Dg [DX]_g}{...
Yuan Yao's user avatar
  • 813
4 votes
0 answers
134 views

Physical VS gauge Symmetry

When we work in path integral formalism in some field theory (or even in QM), we usually look on the action, find its symmetries and than treat them as a gauge symmetry - we sum over all possible ...
ziv's user avatar
  • 1,706
4 votes
0 answers
123 views

Correlators on the Euclidean section of a black hole

In the standard construction of the Euclidean section of a Schwarzschild black hole, we start with the exterior metric in Schwarzschild coordinates: $$\tag{1} ds^2 = -(1-r_s/r)dt^2 + (1-r_s/r)^{-1}dr^...
nodumbquestions's user avatar
4 votes
0 answers
116 views

Question on the implication of Lebesgue integration in Einstein-Hilbert action

Suppose I have two Riemannian manifolds in two dimensions, a 2-sphere $\mathbb{S}^2$ and a disk $\mathbb{D}$. I would like to know whether there is a redundancy in Hawking's path integral approach to ...
Jeanbaptiste Roux's user avatar
4 votes
1 answer
136 views

Issue with path integrals for the partition function

I was going through Kapusta and Gale "Finite temperature Field theory" In chap 2, Eq. 2.24, they need to do the path integral $$Z = Lim_{N-> \infty} \left (\prod_{i=1}^{N} \int_{-\infty}^{...
Angela's user avatar
  • 993
4 votes
0 answers
182 views

Path integral on many-body quantum mechanics

Suppose $\mathscr{H}$ is a Hilbert space describing a one-particle quantum system and $\mathcal{F}(\mathscr{H})$ is its associated Fock space, which is used to describe a many-body quantum system. Let ...
MathMath's user avatar
  • 1,123
4 votes
0 answers
194 views

What is meant by GR being nonrenormalisable, given that "scale" is the very thing being quantised?

TL;DR Usually, RG flow involves scaling the metric $g_{\mu\nu}\to\lambda^2 g_{\mu\nu}$ and seeing how couplings change. But if we're quantising $g_{\mu\nu}$ itself, I don't see how this process can be ...
nodumbquestions's user avatar
4 votes
0 answers
183 views

Feynman Rules from Generating Functional

For the following Lagrangian: $$\mathcal{L}= \overline \psi \left(i \gamma^{\mu}D_{\mu} - m \right)\psi -\frac{1}{2}\left(F_{\mu\nu}\right)^2,$$ I'm trying to find the Feynman rules. I know that the ...
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