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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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Equivalence Picard-Lefschetz path integrals and “Feynman's” path integrals

I have just seen the Picard lefschetz method applied to path integrals in order to make these more convergent. I understand how we could modify the contour of integration for a real integral but what ...
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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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103 views

Why is Dirac's remark that inspired Feynman's path integrals approach to quantum mechanics regarded as mysterious? [on hold]

In his book Dirac remarked a relation between the transformation function connecting two different representations and the classical action, stated as: $\langle q_t | q_T \rangle$ corresponds to $exp [...
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38 views

Path integrals and fourier series

I am currently reading the Feynman and Hibbs about Quantum mechanics and path integrals and I found something pretty confusing ( for me ) at page 72. At this page, they are replacing an integration on ...
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35 views

WKB connection formulae from the path integral

The semiclassical, or WKB, approximation is one that is far more natural in the path integral formalism than it is when derived from the Schrodinger equation directly. Furthermore, the connection ...
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48 views

When is the Path Integral exact?

Feynman's path integral is exactly solvable for quadratic Hamiltonians and the Coulomb potential. I recently heard without proof that there is a larger class of systems for which it is also exactly ...
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67 views

Euler-Maclaurin formula for path integral

Is there a corresponding Euler-Maclaurin formula for path integral when we divide the path integral into discrete lattice? What is the error correction when we divide the space into lattice of length ...
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48 views

Difference between different approximations in QM and other definition of the integral

I am currently studying the path integral formalism by myself and I am a bit lost within all the different way to solve the integrals we have. I have one big question: It sounds maybe a bit strange ...
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Feynman $i\epsilon$-prescription for fermion propagator via path integrals

In Section 9.4 of S. Weinberg's book "The quantum theory of fields" it is shown how to get the Feynman $i\epsilon$-prescription in the propagator of a free scalar field using path integrals and ...
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92 views

Summation of an exponential operator on quantum amplitude

For a quantum Dirac field interacting with a classical EM field, one can (through the Quantum Dynamical Principle) write the vacuum transition amplitude as $$\langle0_+|0_-\rangle=\exp\left[ie_0\int ...
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41 views

Path integral: uncoupling via staging variables

I am studying the transformation to staging variables for the calculation of path integrals in quantum mechanics, following the scheme presented in the book of Mark Tuckerman "Statistical Mechanics: ...
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32 views

Fourier transform of variable in path integral

In Sredinicki's QFT given below, he changed the integration variables in eq(174). This step confuses me. I only know some basics about path integral. In my opinion, when he used fourier transform of ...
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Paths of least action and loops in time

In the book Quantum Field Theory for the Gifted Amateur link: https://books.google.ca/books?hl=en&lr=&id=nIk6AwAAQBAJ&oi=fnd&pg=PP1&ots=JZjwG_qDt5&sig=...
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Why can we treat the path integral as an integral over a infinite dimensional vectorspace?

The path integral is usually introduced by integrating over all piecewise linear paths in discrete time and then taking the time step $\varepsilon$ to zero, i.e. $$\int Dx e^{iS[x]} \sim \int dx_1 ...
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47 views

Dependence of BRST Quantization on the Choice of Gauge-Fixing Function

There is a point which confuses me about BRST procedure. One shows that, if we define physical states as the ones that are annihilated by BRST charge $Q$, the scattering amplitudes don't depend on ...
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Two Questions about Path Integral from “Gauge Fields and Strings” by Polyakov

My questions are about worldline path integrals from the book Gauge Fields and Strings of Polyakov. On page 153, chapter 9, he says Let us begin with the following path integral \begin{align} &...
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A Question about Path Integral Measure

I want to do the following path integral. $$\mathcal{Z}=\int\mathcal{D}x e^{iS[\dot{x}]}$$ The action only denpends on $\dot{x}$. For some reason, I want to replace the integral measure $\mathcal{D}...
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Different Schwinger-Dyson Equations

In the literature on QFT there are a lot of different equations that are all called "Schwinger-Dyson equation" so I wanted to know how are they related and if they have proper names. The first ...
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How does the Weyl anomaly imply $\langle T^{\mu}_{\mu} \rangle \neq 0$

I want to consider the case of euclidean field theory in 2 dimensions with the action $$S[\phi]=\int \! d^2\!x \sqrt{\det(g)}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$$ which leads to a partition ...
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Calculation of path integral in QFT

I am studing QFT using the text book of Srednicki's. And I am stuck on one of calculations of the integrals in his book. Consider a harmonic oscillator with hamiltonian: We can write the following ...
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76 views

Are powers of the harmonic oscillator semiclassically exact?

The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact....
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Wick rotation vs. Feynman $i\varepsilon$-prescription

The generating functional $Z[J]$ of some scalar field theory is \begin{equation} Z[J(t,\vec{x})]=\int \mathcal{D}\phi e^{i\int (\mathcal{L}+J\phi)d^4x} \end{equation} This integral is not well ...
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Path integral and Out-of-time-ordered (OTOC) correlator

A simple observation that any insertions within the path integral are classical variables (Not operators) and hence, objects inside the path integral "commute" (is symmetric under exchange). Hence, ...
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A Naive Question about Delta Function and Wick Rotation

A delta function can be written as $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dp\,e^{ipx}.$$ I have a very poor understanding of the Wick rotation technique used in quantum field theory. ...
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How does the functional measure transform under a field redefinition?

My question is: how does the path integral functional measure transform under the following field redefinitions (where $c$ is an arbitrary constant and $\phi$ is a scalar field): \begin{align} \phi(x)&...
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Grassmann's variables under integration

If $\eta$ is a Grassmann variable, due to invariance under translations we get that, $$\int d\eta\ \eta = 1 \tag1$$ Nevertheless, for being Grassmann's, $\eta$ satisfies $\eta^2 = 0$. ...
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Path Integral in Electric-Magnetic Duality

The action of electromagnetic field is $$S=\int\left(-\frac{1}{2e^{2}}F\wedge\ast F+\frac{\theta}{8\pi^{2}}F\wedge F\right),$$ where $F=dA$ is the curvature $2$-form, and $A$ is the connection $1$-...
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119 views

Complex Gaussian integral with different source terms

Do the source terms multiplying a complex field and its conjugate need to be conjugates for the Gaussian identity to hold? E.g. is $$\int D({\phi,\psi,b}) e^{-b^\dagger A b +f(\phi, \phi^\dagger,\...
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Distinct choice of partition in the Path Integral

Practically all books in Quantum Mechanics and Quantum Field Theory define the non-relativistic path integral by taking one interval $[a,b]$ and breaking it up into $N$ subintervals of equal length. ...
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51 views

Fermionic ghost path integral results in $\delta$ function?

This is related to a statement in pg 20 of hep-th/9408074 formula (2.39). Suppose $$\mathcal{L}\sim\frac{i}{\lambda^{\prime}}\bar{\eta}^xg_{ij}U_x{}^i\psi^j+\cdots \tag{2.35}$$where $\bar{\eta}$ to ...
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Calcluating the photon propagator with gauge fixing parameter

I'm trying to calculate the photon propagator via the functional integral, with lagrangian (plus source) $L = -\frac{1}{4}F^{ab}F_{ab} - \frac{\lambda}{2}\left(\partial^aA_a\right)^2 + J^aA_a $ ...
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Smoothness of sum over histories?

Considering the sum over histories approach to quantum mechanics. This considers all histories consistent with certain starting configurations and ending configurations. How "smooth" do these ...
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About Witten's path integral formulation of Jones polynomial

In his landmark paper Quantum field theory and the Jones polynomial, Witten proposed that the Jones polynomial can be obtained by the expectation value of the Wilson loop operators over links in the ...
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172 views

Faddeev-Popov determinant and topology of the worldline

I am studying the path integral quantization of relativistic particles, using the BRST quantization method. I have to compute the integral \begin{equation} Z\sim \int Dx \det(\partial_\tau)e^{-\int_0^...
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187 views

Non-local field redefinition and effects on path-integral measure

Consider the partition function $$ Z[0] = \int \left[\mathcal{D}A_\mu\right]\left[\mathcal{D}\pi\right] e^{-i \int d^4x \left(-\frac{1}{2}(\partial\pi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{a}{M^2}...
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Inverse of a matrix in a Path Integral

Good morning! I can't make sense of an inverse of a matrix appearing in a calculation for a Wiener Path Integral. In discretized form: $$\int \prod_{i=1}^N \frac{dx_i}{\sqrt{\pi \epsilon}} e^{-\frac{1}...
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Follow up on understanding path integral measures

A while ago I asked the following question: Understanding Measure in Path integrals and got to the conclusion that path integral measures are infinite products of $d\phi(x_i)$ for some scalar field $\...
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Expectation values of states in QFT via the path integral

The path integral in QFT is usually computed only for the vacuum state, $$\langle 0 | T\{ A \} | 0 \rangle = \int \mathcal{D}\phi(x) A e^{iS[\phi]}$$ Doing it for different states is a bit trickier,...
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Is this actually the rigorous definition of the path integral in Quantum Mechanics?

Let a quantum system with a single degree of freedom be given. We want to define the path integral so that we get the representation for the propagator as $$\langle q' |e^{-iHT}|q\rangle=\int_{x(a)=...
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How do I derive Pauli's exclusion principle with path integrals?

I am trying to prove Pauli's exclusion principle using path integrals. My starting point is the configuration space $\mathcal{C}$ for two indistinguishable particles in 3D: $$ \mathcal{C} = \{ \{x_1,...
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Deriving Ward identity directly from a given formula for the conserved current only using the equal-time canonical commutation relation

I have a very technical question on deriving a Ward identity directly from a given explicit form of the "conserved current". Let me emphasize that I do not start with an apriori knowledge on the ...
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82 views

Primary field in CFT and path integral

I should feel ashamed to ask such a naive question, but anyway let me start with the $\phi^4$ theory in the Minkowski spacetime, which has a Lagrangian of the form $$\frac{1}{2}(\partial\phi)^2-\frac{...
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Is one allowed to split path integrals in the Feynman-Vernon Influence theory

In QFT the propagator $J(t,t_0,x_f,x_i) = \langle x_f | U(t,t_0) | x_i \rangle$ fulfills the property $$ J(t,t_0,x_f,x_i) = \int_{-\infty}^{\infty}dx' J(t,t',x_f,x')J(t',t_0,x',x_i) $$ and can be ...
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Paths contributing to the path integral measure in Gross' book

My question regards a comment D. Gross makes in his unpublished lecture notes about quantum field theory (the one with no chapter 1). In chapter 8 (path integrals) pag. 136, he reaches at the ...
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130 views

Understanding Measure in Path integrals

I know this is still a current topic of study, so my question is less about mathematical underpinnings, and more about the transition from the "sum-over-all-possible-intermediate-points" measure $$\...
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Calculating the generating function for a linear interaction in scalar QFT

So i'm asked to calculate the generating function $Z(J)$ for a Lagrangian density $$L = -\left( \partial \phi \right) ^2 + m^2\phi^2 + f\left(x\right) \phi$$ for a fixed function $ f(x) $ , and ...
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How does extending a Chern-Simons theory to the bulk fix potential singularities?

According to ref.1 (§A.3), the naive definition of Chern-Simons $$ S[A]=k\int_M \mathrm{CS}[A]\tag{A.17} $$ is ill-defined, because $A$ may have "Dirac string singularities". The solution is to extend ...
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108 views

Path integrals vs operator

I have a statement that the path integrals formalism is eqivalent to operator formalism in quantum mechanics. Is it a correct statement? I understand that each of these two formalisms has its ...
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53 views

Calculation of a 4-point function by path integrals

In Srednicki's book in chapter 8 a four-point function is computed as a sum of products of propagators: $$<0|T\phi(x_1)\phi(x_2) \phi(x_3)\phi(x_4)|0> = \frac{1}{i^2}[\Delta(x_1 -x_2)\Delta(...
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Should we start from Euclidean QFT if we are to be rigorous? [closed]

Path integral is only rigorous in Euclidean QFT. This suggests that one should start from Eucliden QFT and transport back the results back into Minkowski time. Is this how I should think of QFT?