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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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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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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 lenght. ...
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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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Faddeev-Popov determinant and topology of the wordline

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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144 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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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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118 views

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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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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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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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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Computational content of Feynman integral

I am looking for references with rigorous (not just numerical) studies deriving convergence rates of discretized path integrals to their "true" values in some interesting special cases (since the ...
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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?
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Path integral in interacting quantum field theory

From my understanding we do not yet know how to make much out of interacting QFT other than scattering amplitude at asymptotic infinity. (Correct me if I misunderstand.) But path integral, in ...
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Interaction term in free energy for Gaussian Fixed Point

In general in statistical field theory, the free energy $F_0$ as a function of our order parameter $\phi$ can be written as $$F_0[\phi]=F_0[\phi^-]+F_0[\phi^+]+F_I[\phi^-,\phi^+]$$ where the last ...
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Path Integral Notation [closed]

In my Statistical Field Theory lectures, I was told that $$Z=\int \mathcal{D}\phi\ e^{-F[\phi]}=\int\prod_{k<\Lambda}d\phi_k\ e^{-F[\phi_k]}$$ I want to clarify that I understand the ...
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Is there a multiscale analysis for field theories?

Consider a (zero dimensional) Gaussian field theory described by the dynamical action $$S = \int_t \tilde{\phi}(t) \left[\partial_t \phi(t) + M(t) \phi(t)\right] - \gamma \tilde{\phi}(t)^2\, .$$ $\...
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Path integral formulation of the density matrix ρ

In Feynman's Statistical Mechanics - A Set of Lectures, upon the introduction of the path integral, a series of approximations are made in order to calculate integrals. I am unsure how exactly to get ...
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How to calculate path integral of polymer system?

We have the functional integral $(1)$ of viriable $\delta p_j$. In our case this intergral follows from RPA (random phase approximation) method assuming $\ p_j(r)=p_j^*+\delta p_j(r)$. The task is to ...
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Why Parity Anomaly in Odd Dimensions?

In section 13.6 of Nakahara, the parity anomaly is in odd dimensional spacetime. From the paper Fermionic Path Integral And Topological Phases by Witten, the problem appears as one cannot define the ...
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Transition amplitudes

We have a forced simple harmonic oscillator Lagrangian $$L = \frac{\dot{\phi}^2}{2} - \frac{m^2{\phi}^2}{2} + f(t)\phi \, .$$ The external force goes to $0$ as $t \to \pm \infty$. I'm trying to ...
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Vacuum Energy Calculation using Path Integral

I am currently reading Zee's book on quantum field theory, and I am in the chapter where he is introducing Grassmann integrals. He re-introduces the path integral evaluated for the vacuum, i.e. no ...
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Determinant of d'Alembert Operator $\mathop\Box-m^{2}$

In quantum field theory, the partition function of a free scalar is $$\mathcal{Z}=\int\mathcal{D}\phi\exp i\int d^{n}x\frac{1}{2}\left[(\partial_{\mu}\phi)(\partial^{\mu}\phi)-m^{2}\phi^{2}\right]$$ $...
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Is there a quantum analogue of Mean Value Theorem theorem?

Background I was thinking of Mean Value Theorem in the context of classical mechanics I have $2$ points $A$ and $B$ and my particle goes from $A$ to $B$ then I know the velocity of the particle at a ...
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Is the path integral amplitude a wavefunction?

The probability amplitude for a particle to travel from $\mathbf{x}_i $ to $\mathbf{x}_f$ in a time $t$ is given by the path integral $$ \langle \mathbf{x}_f | e^{-iHt} |\mathbf{x}_i \rangle = \int \...
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Are there limitations to the type of paths needed in the path integral formulation of quantum mechanics?

In some places it is stated that one needs to include all paths in the path integral approach to quantum mechanics. But in the implementations I have seen one has been content with paths that goes in ...
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Clarification of Path Integral formulation

I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N \exp(i\...
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Two-point Green for Free Dirac Fields

I am trying to compute the $2$-point Green function $\tau_2(x,y)$ for free Dirac fields. The corresponding formula for $\tau_2(x,y)$ is given by $$\tau_2(x,y) = -\frac{\delta^2}{\delta\eta_x \delta \...
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What is the relationship between velocity-dependent potentials and non-Abelian gauge fields?

My (limited) understanding of non-Abelian gauge fields is that they arise from the construction of a theory using a non-Abelian Lie group (as a generalization of the Abelian group underlying E&M) ...
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The background field method of deriving a 2PI effective action. Calzetta and Hu book

I am going through "Nonequilibrium Quantum Field Theory" by Calzetta and Hu right now and it seems that I cannot fully understand the derivations in chapter 6.5. There, they consider the derivation of ...
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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 ...
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If the path integral formulation includes future events, why doesn't that imply retrocausality?

I know that such events would cancel out in the math, but if an extreme event were to happen in the future (say a black hole forming or something on that par), would a particle in the present react to ...
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Is the target space metric a dynamic field in the Polyakov action?

In Quantum Fields and String, A Course For Mathematicians in the lecture on string theory (volume II), the Polyakov action is described: $$S(\xi, g, G) = \kappa\int_\Sigma d\mu_g \text{Tr}_g\,\xi^\...
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Functional derivative for the same function expressed before and after Wick rotation

This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf On page 17, they ...
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Multi-instanton contribution to path integral

Briefly, I would like to have a reference to a clear detailed exposition of the computation of the multi-instanton contribution to the path integral while computing the energy levels splitting of the ...
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Derivation of propagator for free particle

On Sakurai page 127 he gives the formula $$ (1)~~~~~\langle x_n,t_n|x_{n-1},t_{n-1}\rangle = \left[\frac{1}{w(\Delta t)}\right] \exp\left[\frac{im(x_n-x_{n-1})^2}{2\hbar\Delta t} \right]$$ Noting ...
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Feynman Path Integrals and Bohr-Sommerfeld Quantisation Condition [duplicate]

I'm currently learning about Feynman Path Integrals, and I came across the following paragraph: "Periodic classical orbits will carry a complex phase which will in general average to zero over many ...