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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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 $$\int_{\varphi(t_a, \vec{x})}^{\varphi(t_b, \vec{x})} \mathcal{D}\varphi(x)e^{iS[\varphi(x)]}$$ ...
6
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238 views

Deriving Feynman rules from Renormalized Lagrangian

In the context of Renormalized Pertubation Theory Peskin Schröder says: The Lagrangian $$ \mathcal{L}=\frac{1}{2} (\partial_\mu\phi_r)^2-\frac{1}{2}m^2\phi_r^2-\frac{\lambda}{4!}\phi_r^4 + \frac{1}{2} ...
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193 views

Relationship between the Black-Scholes model and path integrals

This question was inspired by some interesting comments by Rod Vance on this answer. Could you (Rod), or someone else, expand on these comments and give a brief summary of the connection between the ...
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222 views

What is the point of path integral for boson and fermion?

I am a beginner to study QFT and confused about path integral for boson or fermion. I have read about the path integral for single particle, and finished some problems. But I cannot understand the ...
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598 views

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\...
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274 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}...
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75 views

${\phi}^4$ description of Ising ferromagnet

Suppose the coupling between two spins is $C_{i,j}<0$, then the classical partition function is given by $$Z=\sum_{\{s_i\}}e^{\sum_{i,j}s_iK_{ij}s_j+h\sum_{i}s_i}$$ where $K_{ij}=-\beta C_{ij}$ and ...
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45 views

Where can I find a video on Liouville Non-Critical String Path Integral?

Would anybody know of a video lecture course in which Polyakov's non-critical string path integral $$Z = \int D [\psi(\xi)]\, \, \exp \left\{ \frac{D-26}{48 \pi}\int_{\xi}\left(\tfrac{1}{2}(\...
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105 views

How to calculate gravity path integrals about an AdS background?

Suppose I have some Lagrangian of some higher derivative gravity coupled to a may be matter fields. Now I want to fluctuate it to quadratic order about an AdS background and calculate the 1-loop ...
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251 views

Why do we use functional integration in QFT?

Recently I learned functional integral's formalism in quantum field theory. I have realized that I don't understand why exactly do we introduce it. We have the expression for $S$-matrix, then we may ...
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103 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 ...
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46 views

Linear Response And path integral

I'm following Wen's book on Quantum field theory, and I'm struggling with section 2.2.1 on linear response and response functions. Specifically I'm unable to reproduce equation 2.2.7 in which the ...
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88 views

Why are coherent states necessary for defining the fermionic path integral?

I am following the discussion of fermionic path integrals and Grassmann variables in QFT for the Gifted Amateur (ch. 28). It defines a coherent state for fermions $\rvert \eta \rangle$ as \begin{...
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68 views

How should the path integral change under a dilation?

Let's say I have a two-point function of a scalar field in flat space: $$ \langle \phi(x)\phi(y)\rangle = \int \mathcal D \phi \, \phi(x)\phi(y)\,e^{iS[\phi]} $$ Then I dilate things: $$ \langle \...
3
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106 views

Fermionic path integral on the disk - Recovering the vacuum state

I'm trying to get a better feel for the operator to state map in quantum field theory. There is a general claim for 2d theories that doing the path integral on a disk with no operator insertions gives ...
3
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135 views

Polology in Functional Integration

Completeness of Hilbert space (on-shell states) is a very powerful concept in canonical quantization, for example, to study the nonperturbative characteristics of the S-matrix, like polology (pole and ...
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64 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 ...
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290 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_{...
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167 views

Normal ordering and path integrals

What is the manifestation of normal ordering for creation/annihilation operators in the path-integral formalism?
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206 views

Path integral measure and symmetry

For a generic field theory the path integral measure is defined as, \begin{equation} \mathcal{D}\Phi = \prod_i d\Phi(x_i), \end{equation} where $\Phi$ is a generic field (i.e. it may be scalar, ...
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311 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 ...
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83 views

Wilsonian Renormalisation — Peskin & Schroder Sect. 12.1

I'm working my way through Peskin & Schroeder, but some of the details of the calculations done in their introduction to the renormalisation group are slipping past me. For concreteness, the ...
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50 views

Stratonovich integral in quantum field theory

I'm reading a paper on Wick renormalization and there are a couple of things that are not that clear to me. The paper ends with the following sentence: In Euclidean quantum field theory, the ...
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57 views

Determinants in path integrals in gauge theories and geometry

I know that in the formalism of path integral it is easy to show how determinants, corresponding to gauge fixing condition and FP ghosts, appear. But there is strict explanation of these determinants ...
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34 views

Topological susceptibility of electroweak theta-term

Suppose EW theory generating functional: $$ Z[\text{sources}] = \int D(A,\psi,\bar{\psi}, H,H^{\dagger})\text{exp}\bigg[i\int d^{4}x\bigg(-\frac{1}{4g_{EW}^2}F_{EW}^2 + \bar{\psi}(D - m)\psi + DH^{\...
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79 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_{-...
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63 views

Help me calculate the Euclidean action of a gravitating system!

I recently read Gibbons and Hawking's paper Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752. I am interested in repeating their calculations. It is fairly ...
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183 views

Klein-Gordon propagator integral in the light-like case

In Kerson Huang's Quantum Field Theory From Operators to Path Integrals (Amazon link), pages 28 and 29, he calculates the propagator in the following case: time-like, space-like and light-like. First ...
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64 views

What's the importance of background field gauge?

Recently I've read that background field gauge is very convenient for gauge theories, because it fixes the connection between normalization constants of gauge field and gauge coupling constant one. I ...
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174 views

Anomalies in QFT books

Why in most QFT books when author discusses of non-invariance of measure of path integral (massless fermions interact with gauge fields) $$ \int D\bar{\Psi} D\Psi \to |\Psi \to U\Psi , \quad \bar{\Psi}...
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99 views

Proximity effect and integrating out the quasiparticle degrees of freedom

I am reading at the moment the paper http://arxiv.org/abs/1401.5203 and try to reproduce the results. One result is the proximity correction $S_{\Sigma}$ to the system $$ e^{-S_{\Sigma}} =\frac{\int\...
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148 views

Feynman rules of a theory in non-standard form

I am currently studying lecture notes by Akhmedov on interacting scalar field theory in de Sitter space. In these notes, he considers a scalar field theory of the form $$\mathcal{L}=\partial_\mu\...
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64 views

When can I use semiclassical approximation?

I know that I can use semiclassical approximation for path integral approach (in quantum mechanics) $\int d[q]e^{iA}$ when action $A >>1 $. But how shall I use such condition? For example, ...
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136 views

Is time ordering defined for a single operator depending of two time variables?

The time ordering for the purpose of quantum mechanics is e.g. given by $${\mathcal T} \left[A(x) B(y)\right] := \begin{matrix} A(x) B(y) & \textrm{ if } & x_0 > y_0 \\ \pm B(y)A(x) & \...
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33 views

The central limit theorem from a path integral?

On https://en.wikipedia.org/wiki/Path_integral_formulation it is noted that the central limit theorem can be interpreted as the first historical evaluation of a statistical path integral. Is this true?...
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54 views

Chern Simons Theory over S^3 as integral - what is domain of integration?

I found these nice lecture notes Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories so I am hoping to understand some parts of the Chern Simons theory better. ...
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71 views

Making mathematical sense on a Feynman's path integral equation

Usually we find this relation in the context of Feynman's path integral (see, for example, Maggiore's book on QFT, pg 223): $ \int_{q_i}^{q_f}[dq] = \int_{-\infty}^{\infty}d\bar{q}\int_{q_i}^{\bar q}[...
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59 views

How does satisfying the Euler-Lagrange equation put a Classical Path on-shell?

I am thinking of what the Euler-Lagrange equation, $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0 $$ specifically represents in satisfying the ...
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91 views

Unsolved Potentials in Path Integral

I just started learning on path integral on my own. It seems that the path integral method is not always able to be solved, depending on the potential. On the other hand, these potentials are ...
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78 views

Transition Amplitude in Field Theory

I am currently reading the "Quantum Field Theory" by Lewis Ryder. In chapter 5 he is talking about path integrals and says that the transition amplitude $ \langle q_f t_f \vert q_i t_i\rangle $ is $$ \...
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69 views

Theta-parameter vacuum energy

Suppose we have $\theta$-field in QCD (in a special case of constant $\theta$ it reduces to ordinary $\theta$ parameter): $$ \tag 1 Z_{\theta} = \int D[\psi_{QCD}]e^{iS}, $$ where $$ \tag 2 S = \int d^...
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121 views

How do instantons look in real time/spacetime?

Instantons, as I understand it, are mathematical constructions in Eucledean spacetime. Does it imply that instantons do not exist in real spacetime or instanton tunneling effects are does not have ...
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102 views

Relation of field creation operators to path integral?

Applying two field creation operators to a vacuum I get: $$\hat{\psi}^\dagger(x)\hat{\psi}^\dagger(y)|0\rangle = (\hat{\phi}(x)\hat{\phi}(y) - s^{-1}(x-y)) |0\rangle$$ where the quantum field ...
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How to arrive on the diffraction pattern for the double slit experiment using path integrals for the Gaussian slit case?

I wish to take the path integral route to derive the diffraction pattern for the double slit experiment using the Gaussian slits as the nature of the slits. The kernel looks like: \begin{equation} K((...
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144 views

Vacuum to vacuum transition amplitude confusion

I keep seeing this for the vacuum to vacuum transition amplitude: $\langle 0,\infty|0,-\infty\rangle_J\,.$ What is $|0,-\infty\rangle_J?$ Am I to take the Hamiltonian with the source present, ie. $H(t)...
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268 views

Epstein-Glaser causal perturbation theory

Why does causal perturbation theory in the sense of Epstein Glaser fall under algebraic QFT rather than heuristic QFT in renormalization?
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91 views

If you are only interested in deriving Feynman diagrams can you skip path integrals and just compute greens functions?

I've been reading about the path integral approach to quantum field theory and I noticed that at the end you are just computing greens functions that you could have started computing in the beginning. ...
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156 views

Deriving effective model without integrating out degrees of freedom in path integral formalism?

In path integral formalism of quantum field theory (particle physics or condensed matter), one can in principle integrate out part of the degrees of freedom so as to attain an effective model ...
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67 views

Anomaly and Weyl spinors

I try to better understand anomalies in QFT and I've got a question concerning derivation of axial anomaly in Terning's lectures (page 12) Consider a theory of Weyl fermions coupled to a gauge field $...
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51 views

Integration measure

Consider the field being decomposed into a orthogonal and completed basis: $\Phi(x) = \sum_n c_n \phi_n(x)$ (or $\Phi(x) = \int dk c_k \phi_k (x)$, if continuous) The notation: $\phi_n(x) = <x|\...