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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Textbook on QFT in curved space-time via path integrals

I am looking for an introductory textbook on QFT in curved space-time via the path integral method. I want to understand the following: How to build a generic perturbative QFT in curved space-time ...
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Discontinuity of paths in phase space path integrals

Berezin's famous paper "Feynman path integrals in a phase space" discusses the space of paths on which the phase space path integral is concentrated. In particular, these paths are known to be ...
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How to evaluate this odd path integral?

Is there any analytical solution on this path integral? \begin{align} &P(\theta_{N} t_{N}| \theta_{0} ...
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118 views

Why can't quantum field theory be quaternion instead of complex?

So, the definition of QFT in terms of path integrals is that the partition function is: $$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$ But does it have any meaning if instead of this $U(1)$ ...
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All of Physics! [duplicate]

In several of Neil Turok's talks, he talks about this equation that encompasses all of physics. Here it is: How much of it is true? If it isn't, then is it possible to put all of our knowledge of ...
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Loopwise expansion of effective action $\Gamma[\phi]$

My question is about the loopwise expansion of the effective action $\Gamma(\varphi)$ up to 1-loop contributions. I've understood well the results for both $Z[J]$ and $W[J]$ functionals loopwise ...
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Obtaining quantum Hamiltonian for charged particle from path integral formulation

I was working on Shankar 8.6.4, which is about obtaining in one dimension the Hamiltonian operator of a charged particle from the path integral formulation. First, I get the propagator over a time ...
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What type of non-differentiable continuous paths contribute for the path integral in quantum mechanics

Consider the path integral for a 1D particle subjected to a potential $V(x)$ in imaginary time $$ \int_{x(0)=x_0}^{x(T)=x_T} [dx] \, e^{- \int_0^T d\tau \left[\frac{1}{2}\dot{x}^2 + ...
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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 ...
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70 views

Path Integral Evaluation

I've seen the path integral formulation now in a couple contexts (propagator in quantum mechanics, and coherent state functional integral in many body physics). I'm now struggling with how to actually ...
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52 views

Fastest path of light [duplicate]

Fermat's principle of least time says that light always takes the fastest path to any point. So how can light know which is the fastest path without taking all the paths first?
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162 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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49 views

Kraus operators + path integrals = Lindblad equation?

The other day our professor was talking about Kraus representation of density operator and the derivation of Lindblad equation. He told that this was related to the Feynman path integrals and that we ...
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70 views

Can I Weyl-order the following Hamiltonian?

I am trying to perform a path integral but I am having trouble with the Weyl ordering of my Hamiltonian. The Lagrangian of the system in question is $$L~=~\frac{1}{2}f(q)\dot{q}^2,$$ where $f(q)$ ...
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72 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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64 views

Why can we not choose the stress tensor in a CFT to be identically symmetric?

The stress tensor for a conformal field theory (or any quantum field theory) can be derived from the action $S$ by the functional derivative $$T^{\mu \nu} ~=~ -\frac{2}{\sqrt{|g|}}\frac{\delta ...
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124 views

Can photons travel faster than $c$? (Feynman Lectures)

I apologise for the very non-technical nature of this question. I am new to QED and perhaps am interpreting things in the wrong way, but I'll ask anyway, and hopefully someone can provide a ...
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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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Path integral measure Fourier transformation for case of real field

Let's have $$ Z[J] = \int D \varphi e^{iS[\varphi , J]}, $$ where $\varphi$ denotes real scalar field. Let's make Fourier transform, $$ \varphi (x) = \int e^{iqx}\varphi (q), \quad \varphi^{*} (q) = ...
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Does the path integral measure have dimension?

For example, in the field functional integral: $$\int D\phi \ e^{S[\phi]} $$ Does the $D\phi$ here have dimensions?
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What is the relation between phase space formulation with Wigner quasi-probability distributions and path integral formulation of quantum mechanics?

I am trying to conceptually connect the two formulations of quantum mechanics. The phase space formulation deals with Wigner quasi-probability distributions on the phase space and the path integral ...
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156 views

Path integral in quantum mechanics

I am confused by the derivation in Srednicki QFT's chapter 6 from (6.8) to (6.9). In (6.8), we have $$<q'',t''|q',t'>~=~\int DqDp \exp[i\int_{t'}^{t''}dt(p\dot{q}-H(p,q))],\tag{6.8}$$ and ...
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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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Path integrals and Weyl ordering in Peskin and Schroeder [duplicate]

On pages 280-281 of "An Introduction to Quantum Field Theory" by Peskin and Schroeder, the authors discuss the path integral formulation for a general quantum system and briefly mention Weyl ordering. ...
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78 views

How to calculate the expectation value of position/momentum using path integrals?

We have the formula: \begin{equation} \langle F \rangle = \frac{\int Dx \times F[\phi] exp\{i/\hbar S[\phi]\}}{\int Dx \times exp\{i/\hbar S[\phi]\}} \end{equation} Now, I am wondering how a change ...
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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) = ...
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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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Length path integral

Let's consider a 2-dimensional Euclidean plane. The length between two points $a$ and $b$ can be defined in the following way: $$ (ab) := \inf_{\gamma} \,\int_0^1 d\tau \,\sqrt{\delta_{ab} ...
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Issues with the Operator to State map using Path Integral

Suppose your QFT has a Hilbert space $\mathcal{H}$, and let $\text{End}(\mathcal{H})$ be the set of operators on $\mathcal{H}$. It is often stated that in QFT there is a map $$\mathcal{F}: ...
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114 views

Perspectives of QFT [closed]

From the answer to this question Computing $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$, I have discovered that there is two perspectives to QFT. I am doing a course which is unfortunately a summary of QFT and ...
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68 views

Feynman Path integrals in space with holes in it [closed]

Feynman Path Integrals are a way of calculating the wave function of quantum mechanics. It usually integrates every possible path through all of space. I wonder if there is any study of Feynman path ...
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267 views

Why isn't the path integral rigorous?

I've recently been reading Path Integrals and Quantum Processes by Mark Swanson; it's an excellent and pedagogical introduction to the Path Integral formulation. He derives the path integral and shows ...
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Writing down many particle Hamiltonian

We are given that \begin{align}\mathrm{tr} e^{-\frac{i}{\hbar}\hat{H}t}&= \int D[a_1,\dots,a_n]\times\\&\qquad\exp\left[\int_0^t dt' \left(\frac{1}{2}\sum_j ...
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Computing $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$

Given Hamiltonian $H=\frac{P^2}{2}+\frac{\omega^2}{2}Q^2$, compute $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$, where $T$ is the time-ordering of the product, $|0\rangle$ is the ground state. Now set ...
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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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How do we arrive on this kernel equation?

In Feynman and Hibbs, we see the following equation: $$K(b,a)~=~\sum_{\text{paths from $a$ to $b$}} \phi [x(t)] \tag{2-14}$$ which is valid always. Now, they write $$\phi[x(t)] ~=~ \text{const} ...
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Tadpole diagrams in $\phi^3$ theory

In "Quantum Field Theory" by Mark Srednicki, Chapter 9 page 67, after he proves that $\langle 0|\phi(x)|0 \rangle$ vanishes (meaning sum of all connected diagrams with a single source is zero), he ...
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How is the functional integral over momentum performed in the case of the real scalar field?

Let's follow Peskin and Schroeder section 9.2, page 282. The Hamiltonian of a free real scalar field is $$H=\int{}d^3x[\frac{1}{2}\pi^2+\frac{1}{2}(\nabla\phi)^2+V(\phi)]$$ so the expression for ...
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Reducing unitary evolution operator of a two-spin system to the evolution operator of one of the spins

Consider a system of two spins $s_1$ and $s_2$, each of which can be in one of two states, represented by 0 or 1. A basis for the Hilbert space of this system would be {|0,0>,|0,1>,|1,0> and |1,1>}, ...
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Given a QFT Hamiltonian, is there a unique Lagrangian?

Consider a QFT in one spatial dimension specified by the following Hamiltonian density: $\mathcal{H} = -i \phi^\dagger \frac{\partial}{\partial x} \phi + V(\phi^\dagger,\phi)$ where $\phi$ is a ...
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“Find the Lagrangian of the theory”

I've heard a few of my professors throw around the term "finding the Lagrangian of a theory". What exactly is this referring to. From what I understand it seems that you determine invariances ...
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Feynman paths of FTL velocity have imaginary momentum?

In this Phys.SE answer it is discussed that Feynman path integrals sums amplitudes for all possible paths, including those that are not time-like. If you take the momentum-space path integrals, I ...
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If path integrals aren't well-defined, how can they have any physical meaning?

I am confused about a particular point about the nature of path integration. According to what I've read, what we really mean when we say functional integration is \begin{equation} ...
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Feynman diagrams with classical apparatus on the perturbative region

on QFT, one usually simplifies the interaction between fields and classical apparatus (sources, detectors, etc.) by assuming the classical devices only interact with the asymptotic on-shell states ...
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Delta functional in path integrals - reference needed

In a few articles dealing with path integral quantization I came across some calculations where apparently identities of the form \begin{equation} \int (\mathcal{D}\Phi)\, ...
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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 ...
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Can someone explain how to do a functional integral through an example?

I want to learn how to do path integrals as used in quantum field theory. I understand the specific type of integral used in qft is called a functional integral. I looked at Wikipedia and checked in ...
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How do I solve this Gaussian path integral?

Suppose $$ Z = \int \mathcal D[\phi^*] \mathcal D[\phi] \exp(\phi^*A\phi + \phi B\phi) $$ where $A$ and $B$ are operators. I know how to solve a Gaussian path integral involving only $\phi^* A \phi$ ...
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limit as $x_1 \to x_0$, propagator for the harmonic oscillator

Consider a non-relativistic particle of mass $m$, moving along the $x$-axis in a potential $V(x) = m\omega^2x^2/2$. use path-integral methods to find the probability to find the particle between ...
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Uses for Action from Lagrangian Mechanics

In my course on Lagrangian/Hamiltonian mechanics I noticed that we dealt with finding the stationary point of the change in action $ \delta S $ and we were never really doing anything with $ S $ ...