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.

Filter by
Sorted by
Tagged with
6
votes
1answer
74 views

A relationship between the proof of a renormalizability and gauge fixing conditions?

I already know that QCD is renormalizable in several gauges, including the $\xi$ gauge and the background field gauge. That is, the divergence of the quantum effective action is limited by symmetry, ...
4
votes
1answer
96 views

Propagator in Path Integrals

I am reading Section I.2 in Zee's QFT in a Nutshell. The amplitude for a particle to start at position $I$ and end at $F$ is (eq. (I.2.6)): $$ \langle q_f|e^{-iHT}|q_I\rangle=\int Dq(t)\ e^{i\int_0^T ...
0
votes
1answer
73 views

How to prove that the fields' contribution to the action in Feynman's Path Integral are quantized?

The Lagrangian density in the path integral contains spinor, vector and number fields. However, their combinations in the action are scalars such as $\bar \psi \psi$, just numbers. Let's say I have an ...
3
votes
0answers
67 views

Do the Ward identities contain contact terms in Euclidean QFT?

In derivations of the Ward identities, I have never seen the signature of spacetime explicitly specified, so I'd always assumed they hold regardless of signature. However, the argument below seems to ...
2
votes
1answer
65 views

Specific commutator calculation using the path integral

Consider the path integral quantisation of a scalar field $\phi$ on flat spacetime. Let the Lagrangian be $\mathcal{L}$. I would like to prove the following equal time commutation relation: \begin{...
2
votes
1answer
126 views

Fields and Path integrals in the axiomatic Wightman setting

I have recently looked at the Wightman approach to axiomatically define a continuum QFT, using these notes [1] in particular. I am confused about where distributions appear in both the classical and ...
5
votes
1answer
143 views

Math behind photon stopwatch path integral in Feynman's QED book

I want to connect the math and the exact concepts hidden behind the simplified picture of photon propagation provided in Feynman's QED book. Background Feynman's photon stopwatches In his book QED - ...
0
votes
1answer
85 views

Derivation of Feynman rules from generating functional

I have to factorise the Sudakov form factor in six dimensional $\phi^3$-theory, but first I want to determine the Feynman rules using path integrals. The Lagrangian of the theory reads $$ \mathcal{L} =...
2
votes
1answer
61 views

How to come up with Feynman rules: Proof of the multiplicity factor from functional derivative?

Consider $(\phi^*\phi)^2$ theory of complex scalar field. The goal is to come up with Feynman rules from functional derivatives, and the emphasis is on how does the symmetry factors or the ...
4
votes
1answer
120 views

Interpretation of the fermionic path integral

The bosonic path integral computes transition amplitudes. E.g. for a scalar field $\phi$, the amplitude between state $|\phi_1\rangle$ on Cauchy surface $\Sigma_1$ and $|\phi_2\rangle$ on $\Sigma_2$ ...
2
votes
1answer
72 views

QFT generating functional and Green function and propagator

I am confused about why does the generating functional gives the propagator by differentiation, and why that propagator is the Green function. I understand how to take the functional derivative like ...
0
votes
0answers
44 views

Free Energy vs. Partition Function in QFT

The partition function of QFT is defined as $$Z=\int\mathcal{D}\varphi e^{iS[\varphi]}.$$ Now, it is a general fact that this formal path integral can be computed perturbatively as (sketchy) $$Z=\sum_{...
5
votes
2answers
463 views

What the role of classical equation of motion in quantum field theory?

I've learnt quantum field theory for a semester but I still can't understand the role of classical equation of motion in QFT. I have looked up for several books. They all discuss classical field ...
0
votes
0answers
34 views

Chiral anomaly of Weyl fermion is half of Dirac

How can one mathematically see that the anomaly for a Weyl fermion is half of Dirac in the Fujikawa path integral method? Edit I do understand that a Dirac fermion is two Weyl fermions. What I wish to ...
4
votes
0answers
37 views

Where are the multi-instantons in Supersymmetric QM?

Instantons can be used to find non perturbative corrections to ground state energies. However, the way in which they are used seems to me to be very different between the two common toy models of the ...
2
votes
1answer
73 views

Inserting a position operator in the path integral in QFT

With the usual path integral description, we have the formula $$\langle q''t''|q't'\rangle =\int\mathcal{D}q \exp{(iS)}$$ where $S=\int_{t'}^{t''}L(q,\dot{q})$ is the action evaluated for $t\in (t',t''...
0
votes
0answers
31 views

What is the relation between the partition function from Stat. Mech. And the Path Integral? [duplicate]

Beside the fact that they look identical when you take imaginary time in the path integral formulation. I understand we doing statistics and we are just integrating over all states with a relative ...
2
votes
1answer
112 views

Expectation values in path integral formalism

In quantum field theory, it is often assumed that the expectation value $\langle A\rangle$ of an operator $A$ can be written in the path integral formalism in the following way: $$ \langle A\rangle = \...
2
votes
1answer
76 views

Conditions on the covariance operator in Gaussian Path Integrals

In field theory, one typically encounters integrals of the form: $$ \mathcal{Z}[J] = \int \mathcal{D}[\phi] \exp \left( - \frac{1}{2} \int d^Dx d^Dx' \ \phi(x)A(x,x')\phi(x')+ \int d^Dx \phi(x) J(x)\...
1
vote
1answer
99 views

Why do we use two ways to write the kinetic term in a Lagrangian?

I have just started reading Schwartz's book on QFT and I see from the first few chapters that he writes the kinetic part of the Lagrangian in a way I find strange. As an example, for the massless ...
1
vote
0answers
60 views

Quantum to classical mapping

I'm having troubles understanding precisely how the mapping from a quantum system to a classical one works. Let's say that I have a quantum system in $d$ dimensions with Hamiltonian $H$ at temperature ...
2
votes
1answer
73 views

Propagator of harmonic oscillator at specific times

It is well known that the propagator (kernel) of a simple harmonic oscillator is given by $$ U\left(x_{b},T;x_{a},0\right)=\sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega T}}\exp\left\{ \frac{im\omega}{2\...
0
votes
1answer
85 views

What does a loop integral do to the Feynman path integral?

So the Feynman path integral tells us to sum over all the possible paths a particle can take and to weight each path by the exponential of i times the action along the path. What does a loop-integral ...
0
votes
0answers
14 views

Time ordered product of position operators in non-relativistic path-integral

In many books the path integral formulation is described starting by unidimensional position states in non-relativistic formulation, taken in the Heisenbeg picture: so we have $|r,t\rangle\doteq e^{\...
3
votes
1answer
108 views

How does the $+i\varepsilon$ prescription in the propagator comes from analytic continuation of the Euclidean 2-point function?

Let $S_0[\phi]$ be the action for a real Klein-Gordon field $$S_0[\phi]=\dfrac{1}{2}\int d^Dx \phi(x)(\Box-m^2)\phi(x)\tag{1}.$$ If we try to construct the generating functional $Z_0[j]$ we find that ...
1
vote
0answers
53 views

Coherent state path integral for Dirac fermions

I’m trying to derive the fermionic path integral for the Dirac theory using the coherent state path integral, but I’m not able to get around the presence of a $\gamma_0$ making it look different from ...
3
votes
0answers
43 views

Path Integral for Fokker-Planck equation

As per Wio, the special case of the Fokker-Planck equation (in SDE form) \begin{equation*} dX = f(x)dt + \sqrt{2D} dW_t \end{equation*} has the path integral representation in the Ito scheme as \...
1
vote
0answers
60 views

Physical meaning of correlation functions, why does obtaining all of them mean solving a given QFT?

I vaguely understand that a 2-point function would tell us the propagation from an exited point to another, but I have no idea what, for any natural number n, $$\langle\phi(x_1)...\phi(x_n)\rangle=\...
3
votes
0answers
93 views

How to compute the beta-functions given the euclidean action for two scalar fields?

Given the Euclidean action for two scalar fields in $d$ dimensions: \begin{equation} S_E = \int d^dx\frac{1}{2}((\nabla\phi_1)^2 + \nabla\phi_2)^2) + \frac{\lambda}{4!}(\phi^4_1 + \phi^4_2) + \frac{2\...
2
votes
1answer
70 views

Inconsistency of numbers of $d p$ and $d q$ in path integrals over phase space

I am new to QFT. In books like Fradkin's QFT an integrated approach, and Stefan's Gauge field theories 2nd Ed., they derive the path integral from first writing down the integral over the phase space, ...
0
votes
0answers
35 views

Understanding of transition amplitude

In QFT, suppose we have state at initial position and time $x_i$ and $t_i$ respectively and by corresponding state is $|x_i;t_i\rangle$. Then, according to most textbooks, transition amplitude when ...
7
votes
1answer
109 views

Does Feynman's path integral include complex trajectories?

The WKB approximation provides the correct exponential decay of eigenstates inside classically forbidden regions if one allows classical momenta to be imaginary. The typical example is a double well ...
4
votes
1answer
157 views

Transitioning from Path integral in QM to QFT

I've been recently introduce in the Path Integral formulation of Quantum Mechanics. The typical place to start is where Feynman himself, I believe, started. That is, the one dimensional case where a ...
-1
votes
1answer
57 views

Why in $e^+ e^-\rightarrow \mu^+ \mu^-$ there is only two $S_F$ propagator instead of four from path integral?

This was a homework to calculate the $e^+ e^-\rightarrow \mu^+ \mu^-$ following Peskin & Schroeder chapter 5.1. However, I got confused with the path integral aspect of the calculation.(Which was ...
2
votes
0answers
64 views

How to justify $\bar \psi(x) \psi(x) \bar\psi(y)\psi(y)\Rightarrow -\bar \psi(x) \psi(y) \bar\psi(y)\psi(x)$ in path integral?

Consider an interaction term of the form $$(e\int dx^4\bar \psi(x) \psi(x))(e\int dy^4 \bar\psi(y)\psi(y))$$ where the generating function was $$\bar\eta(x)\psi(x)+\bar\psi(x)\eta(x)\Rightarrow \int ...
2
votes
0answers
70 views

Complete the square for the generating functional of the Dirac field

Quote Peskin page 302 the Dirac generating function was $$Z[\bar \eta ,\eta ]=\int D\bar\psi D\psi\exp[i\int dx^4 (\bar\psi (i\gamma^\mu\partial_\mu -m )\psi+\bar\eta \psi+\bar\psi \eta)]$$ could be ...
0
votes
0answers
34 views

Non-Abelian Chern-Simons path integral on a torus

Is it possible to exactly evaluate the Chern-Simons path integral with a non-compact gauge group (say $SU(2)$) on a torus? I am asking this because 3d gravity is an $SL(2,\mathbb{R})$ Chern-Simons ...
3
votes
1answer
156 views

How do I show that the $n$-point correlator $\left\langle\phi(x_1)\phi(x_2)...\phi(x_n)\right\rangle$ is equal to this expression?

Given the Euclidean action \begin{equation} S_E(\phi) = \int d^d x \frac{1}{2}\big(\nabla\phi\cdot\nabla\phi + m^2\phi^2\big)\end{equation} and the partition function \begin{equation}\mathcal{Z} = \...
2
votes
0answers
42 views

What is the meaning of integrating by parts of the generating functional?

Let $ \varphi^{i}(x) \equiv \varphi^{A} $ be boson fields and let $$ S_{0}=S_{0}(\varphi)=\int d x L\left(\varphi^{i}(x), \partial_{\mu} \varphi^{i}(x)\right) $$ be a classical action. Consider the ...
3
votes
1answer
110 views

In QFT, why are the vacuum partition function and the zero-temperature imaginary-time partition function the same?

When doing thermal field theory, one can start with the definition of the (thermal) partition function $Z = Tr[e^{-\beta H}]$, and inserting a number of completness-relations, we can arrive at (I am ...
1
vote
0answers
19 views

Why is the variation of powers of reduced density matrices of an interval given by integrals of the Stress tensor around that interval

I have a questions which seems trivial but I don't know how to handle it properly.. In Cardy and Calabreses derivation of the entanglement entropy of a line (https://arxiv.org/abs/hep-th/0405152) in ...
2
votes
0answers
70 views

Calculating the path integral for cubic interactions (perturbatively)

I'm trying to apply the Coleman-Weinberg mechanism to the weakly interacting, $g \ll 1$, $\mathbb{Z}_2$-symmetric $\phi^6$-theory in $d = 3 - \epsilon$ dimensions (in Euclidean signature) \begin{...
0
votes
0answers
42 views

Questions of fermionic coherent states (page 166 of Atland and Simons)

This is quite a basic question but I just can not find the solution. The question is how do you show equation (2) below. Let me explain the details. From page 166 of condensed matter book by Atland ...
4
votes
0answers
72 views

Defining the functional integral measure from the generating functional

In standard QFT we define the generating functional from the functional integral as $${\cal Z}[j]=\int\mathfrak{D}\phi e^{-S[\phi]+i\int d^Dx j(x)\phi(x)}\tag{1}.$$ On the other hand, intuitively ...
1
vote
1answer
58 views

Did $\int dx^4 \partial ^\mu \phi(x) \partial_\mu \phi(x)= -\int dx^4 \phi(x) \partial^2 \phi(x)$ in the path integral formalism?

In the canonical quantization, one assumed the condition that the field $\phi(x)$ vanished at both space and time infinity. However, in the path integral formalism, thought the field $\phi(x)$ was ...
2
votes
1answer
97 views

Berezin integral of a Grassmann field

Consider a time dependent Grassmann field i.e. $\theta(t)$. Now, consider the following Berezin integral $$\int [\mathcal{D}\theta] ~\prod_{t}\dot{\theta}\tag{1}$$ where $\dot{\theta}$ time derivative ...
2
votes
0answers
72 views

Why are vertex operators integrated over the worldsheet?

In chapter $3$ of Polchinski after discussing why vertex operators are used for preparing states in S-matrix. We are given the vertex operator for closed string tachyon is $$V_0=2g_c\int d^2\sigma\...
1
vote
0answers
73 views

How to justify $\int D\phi\exp[-\frac{1}{2} \int d^4 x'\int d^4 x\phi(x')M(x',x)\phi(x)]$

It's related to a homework and exercise. The homework was more complicated, but I needed this to figure out the convention that was used. Consider the integral $$\int D\phi\exp[-\frac{1}{2} \int d^4 x'...
0
votes
0answers
62 views

Can we find the functional for a quantum field in QFT

I've heard that QFT is really described by a functional which dictates the probability that a field will be in a certain configuration $\Phi[\phi]$. My question is, can we find an equation that will ...
2
votes
1answer
66 views

A footnote in Altland Simons

On page 212 footnote 18 says: Remember that, in a theory with complex or Grassmann fields, only contractions $\sim \langle \bar{\psi}\psi\rangle_0$ exist, i.e., there is a total of $n!$ distinct ...

1
2 3 4 5
24