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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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1 vote
1 answer
319 views

$S$-matrix elements in path integral approach

How to calculate $S$-matrix elements of quantum electrodynamics using path integral formalism?
1 vote
2 answers
48 views

Infrared regularizing the harmonic oscillator path integral

This is from Laine and Vuorinen’s Basics of Thermal Field Theory. I do not understand why the fact that the integral over $x(\tau)$ implies the following regularization scheme. That is, I don’t ...
4 votes
1 answer
66 views

How does path integral quantization ensure unitarity?

Unitarity can be verified post hoc by examining the optical theorem. In the context of path integral quantization where formal derivation starting from canonical quantization is unavailable, is it ...
1 vote
1 answer
215 views

Reference request for semiclassical approximations for Schwinger-Keldysh path integrals

Can some one provide some resources for understanding semi-classical approximations for Schwinger-Keldysh path integrals. Is there any discussion about instanton (and multi-instanton) (for even single ...
5 votes
2 answers
335 views

Gaussian Grassmann integral with complex/bosonic source term

I'm interested in solving the following multi-dimensional integral $$ \int d \theta d \bar{\theta} e^{-\bar{\theta}M \theta +\Lambda \theta + \bar{\theta} J } $$ where $\theta$ is a $N$-dimensional ...
3 votes
1 answer
254 views

Question on Majorana Path Integral

I'm studying Shankar's Quantum Field Theory and Condensed Matter and got stuck in the issue related to changing measure in Majorana path integral. In section 9.4, the Euclidean action for the ...
2 votes
1 answer
333 views

Discretized derivation of Majorana path integral

Shankar's QFT book gives an overview for deriving a path integral representation for Majorana fermions. In the derivation, he works directly in continuous imaginary time, sweeping issues of ...
-2 votes
2 answers
834 views

Fermion Determinant [closed]

When we calculate fermion determinant for either Majorana or Weyl spinors, why do we get an extra factor of half in the coefficient of the determinant as compared to the Ghost determinant?
5 votes
1 answer
110 views

How important are purely imaginary finite action solutions for first-order instanton contributions?

I am working on a physics problem where I have to calculate instanton contributions for a non-relativistic Hamiltonian $$H=-\frac{1}{2}\frac{d^2}{dx^2}+\frac{1}{2}x^2+\frac{1}{6}g^2x^6 \tag 1$$ for ...
0 votes
1 answer
98 views

Fourier transform of the Gaussian action for the real scalar bosonic field

In my current homework, we have to get familiar with quadratic theory in order to reach $\phi^4$-theory. So the starting point is $$Z = \int Dx e^{-S[\phi]}$$ with the action for the real scalar ...
4 votes
1 answer
319 views

Propagator in massive QED/Schwinger model

I'm trying to integrate out the fermions from the path integral in the massive QED/Schwinger model $S = \int_{\mathbb{R}^d}d^{dx} \left( - \frac{(F_{\mu\nu})^2}{4} + \bar{\psi} \left( i\gamma^\mu D_\...
9 votes
1 answer
1k views

Measure of Feynman path integral

Feynman path integral for non-relativistic case is defined as: $$\int\mathcal{D}[x(t)]e^{iS/\hbar}$$ where $$\int \mathcal{D[x(t)]}=\lim_{N\rightarrow\infty}\Pi_{i=0}^{i=N}\bigg(\int_{-\infty}^{\infty}...
0 votes
1 answer
54 views

Chern-Simons theory: Connection between Thermal and Quantum Partition Function

I have been reading the Quantum Hall Effect from Prof. David Tong's notes. In the section on Chern-Simons theory, he describes the connection between the Thermal Partition Function and the Quantum ...
2 votes
1 answer
68 views

Free scalar field deriving Ehrenfest using the path integral

In his lecture notes on String theory, David Tong derives Ehrenfest theorem using the path integral: $$S = \frac{1}{4\pi \alpha'}\int d^2\sigma\ \partial_\alpha X\ \partial^\alpha X\tag{4.19}$$ $$ 0 =...
1 vote
1 answer
93 views

What's the minima of the quantum effective action?

Consider the vacuum expectation value of a (for simplicity scalar) field $\phi$, we know that its vacuum expectation value can be expressed as $$\langle\phi\rangle=\frac{1}{\mathcal{Z}}∫\mathcal{D}\...
2 votes
1 answer
91 views

The definition on vacuum-vacuum amplitude with current in chapter of External Field Method of Weinberg's QFT

I'm reading Vol. 2 of Weinberg's QFT. As what I learnt from both P&S and Weinberg, the generating function is defined as $$ Z[J] = \int \mathcal{D}\phi \exp(iS_{\text{F}}[\phi] + i\int d^4x\phi(x) ...
0 votes
2 answers
212 views

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 ...
6 votes
1 answer
1k views

Classical limit of the path integral formulation of quantum mechanics

It is well-known that if $S \gg \hbar$, then the classical path dominates the Feynman path integral. But is there some to show that if $S\gg\hbar$, then the particle's trajectory will approach the ...
0 votes
1 answer
310 views

Faddeev-Popov trick in QED Peskin and Schroeder

On page 297 of Peskin and Schroeder, the book obtains the propogator $$\tag{9.58} \tilde{D}_F^{\mu\nu}(k)=\frac{-i}{k^2+i\epsilon}\bigg(g^{\mu\nu}-(1-\xi)\frac{k^\mu k^\nu}{k^2}\bigg).$$ The book then ...
1 vote
0 answers
69 views

Majorana Boson Coherent States

Consider $a$ be a bosonic operator, and we define $\Phi = a+a^{\dagger}$ and it is clear that $\Phi^{\dagger}=\Phi$ that implies "Majorana Boson". Now, i want to find the coherent states for ...
2 votes
3 answers
104 views

How do vacuum bubbles "dress" terms in the $S$-matrix numerator?

I am self-studying QFT using the book "A modern introduction to quantum field theory" by Maggiore. On page 124-125 he's doing the calculation in the interaction picture for a process with ...
1 vote
1 answer
47 views

Fermionic propagator [closed]

Given the fermionic generating functional $$Z[\eta]=\ det^{\frac{1}{2}}(K_{ij})e^{-\frac{i}{2}\eta_{i}G^{ij}\eta_{j}},\tag{1}$$ where $$G^{ij}=K^{-1}_{ij}$$ is the Green function of our theory, then ...
0 votes
1 answer
115 views

The definition of the path integral

I still have big conceptual questions about the path integral. According to (24.6) of the book "QFT for the gifted amateur" from Lancaster & Blundell the path integral is equal to $$Z =\...
13 votes
2 answers
3k views

Physical meaning of partition function in QFT

When we have the generating functional $Z$ for a scalar field \begin{equation} Z(J,J^{\dagger}) = \int{D\phi^{\dagger}D\phi \; \exp\left[{\int L+\phi^{\dagger}J(x)+J^{\dagger}(x)}\phi\right]}, \end{...
3 votes
0 answers
93 views

What are exactly the loop correction to the potential? [duplicate]

I am struggling with the concept of quantum effective action, but first recall the definition : given a Wilsonian effective action $W[J]$ of our theory, the quantum effective action is just $$\Gamma[\...
1 vote
0 answers
55 views

Derivation of massive photon propagator

I'm trying to derive the massive photon propagator using the path integral formalism for a theory with $$ \mathcal{L} = -\dfrac{1}{4} F_{\mu\nu} F^{\mu\nu} + \dfrac{1}{2} m^2 A_\mu A^\nu, \text{with } ...
1 vote
2 answers
208 views

Fermionic measure in path integral

When writing the fermionic path integral one arrives at an expression containing $\mathcal{D}\bar{\psi}$ and $\mathcal{D}\psi$: $$ \int \mathcal{D}\bar{\psi} \mathcal{D}\psi e^{iS} $$ Usual ...
2 votes
1 answer
80 views

Proof of Batalin-Fradkin-Vilkovisky (BFV) theorem using BRST operator and graded Poisson bracket algebra

In the proof of Batalin-Fradkin-Vilkovisky (BFV) theorem one has to determine how the path integral measure changes. The set of canonical variables $$\varphi = (Q^A;P_{A}) =(q,\eta;\pi,\mathscr{P})\...
0 votes
1 answer
47 views

Can any meaning be given to a path integral with no fixed end point?

A path integral has the interpreted as the probability a particle goes from $A$ to $B$ in time $t$. Such a path integral is given by $$\langle x_B, t|x_A, 0\rangle = \frac{1}{Z} \int_{\textrm{paths } ...
5 votes
1 answer
53 views

Onsager-Machlup functional and the Boltzmann distribution

I've been looking into path integral representations of stochastic processes lately and came across the Onsager-Machlup functional description of the Langevin equation. In the overdamped case, where ...
1 vote
1 answer
87 views

Problem solving for Wilsonian Effective Action

I'm currently doing some basic questions on renormalisation group, but I've ran into a wall when it comes to one particular step in an answer. The question is as follows: This problem is a toy model ...
1 vote
1 answer
76 views

Why does a singularity imply the need for a distribution?

I am following Section 11 of Prof. Etingof's MIT OpenCourseWare notes on "Geometry And Quantum Field Theory" in which he says: ...for $d = 1$, the Green's function $G(x)$ is continuous at $...
12 votes
2 answers
5k views

Understand "Quantum effective action" in Weinberg's book "The quantum theory of fields"

In Weinberg's book "The Quantum theory of fields", Chapter 16 section 1: The Quantum Effective action. There is an equation (16.1.17), and several lines of explanation, please see the Images....
8 votes
4 answers
581 views

Why can we shift the field $\phi$, so that $\langle \Omega | \phi(x) | \Omega \rangle = 0$?

Problem Introduction In different derivations of the LSZ reduction formula the author makes a shift of the field $\phi(x)$ $$ \phi'(x) = \phi(x) - \langle \Omega | \phi(x) | \Omega \rangle, $$ and ...
1 vote
1 answer
83 views

Does path intergral formula only works in perturbative situation?

I'm learning quantum field theory. In Peskin & Schroeder, when they derive $$\int {D\phi(x)\phi ({x_1})\phi ({x_2})\exp [i\int {{d^4}x\mathcal{L(x)}] = \left\langle {{\phi _b}|{e^{ - iHT}}T\{ \phi ...
4 votes
2 answers
624 views

Ghosts in QCD Lagrangian

The QCD Lagrangian is $$ \mathcal{L}_{\text{QCD}} = -\dfrac{1}{2} \text{Tr}\, G_{\mu\nu}G^{\mu\nu} + \sum_i^{N_f} \bar{q_i} \left(i \gamma^\mu \mathcal{D}_\mu - m_i\right)\,q_i, \tag{1} $$ where $\...
9 votes
2 answers
2k views

Path integral for complex scalar field

I am taking a QFT course which focuses on the path integral formulation. At a certain point, I was confused because we saw that, when integrating over complex Grassmann fields for fermions, we defined ...
4 votes
1 answer
259 views

Deriving a path-integral expression for a thermal density matrix with position-dependent temperature

I've been fiddling with deriving a path-integral expression for a thermal partition function with a position-dependent temperature but I'm not sure how to get started on this. Concretely, I'm trying ...
3 votes
1 answer
327 views

Correct method for splitting path integral in two

In 3D point particle quantum mechanics we have that the propagator can be represented as a path integral \begin{equation} \langle x|e^{-iHt}|y\rangle =\int_{\gamma(0)=x}^{\gamma(t)=y}\mathcal{D}[\...
0 votes
0 answers
43 views

How to derive the gauge invariance of Yang-Mills action with external source?

In the Faddeev-Popov procedure of path integral of $$ Z[J] = \int [DA] e^{iS(A,J)}, \quad S(A,J)= \int d^4x [-\frac{1}{4}F^{a\mu\nu}F_{a\mu\nu} + J^{a\mu}A_{a\mu} ] $$ we have used that $S(A,J)$ is ...
2 votes
0 answers
36 views

Does a quantum field theory have an effective single-particle action in the single-particle subspace?

In non-interacting quantum field theories, the particle number is conserved so we can restrict to a given subspace of fixed particle number. On the single-particle subspace, the state will evolve ...
7 votes
1 answer
649 views

What happens to branching in the Many-Worlds Interpretation of quantum mechanics in the limit when Planck's constant goes to 0?

We learn from quantum mechanics courses that one recovers classical mechanics in the limit when Planck's constant goes to zero. This can be seen in the path integral formulation. This is why ...
0 votes
1 answer
94 views

In QFT when performing path integral, why don’t we divide it by the volume of Poincaré group, as what we did for gauge group?

When performing path integral in gauge theory, we naively want to compute $$ Z = \int DA \exp(iS[A]) $$ But we noticed, that because the action is the same for gauge equivalent conditions, we should ...
1 vote
1 answer
60 views

Time ordered correlator from path integral: equation of motion?

Consider a Lagrangian $L(\phi)$ for a field $\phi$ (assume it is a free real scalar for simplicity). Then the time ordered propagator can be expressed as a path integral $$ \langle\Omega|T\{ \phi(x) \...
0 votes
0 answers
26 views

Resources for Faddeev-Popov method. (Specifically for diffeomorphism gauge fixing.)

I am struggling to get the same result as this paper (eq. 3.10) for my ghost field when gauge-fixing diffeomorphisms in linearized gravity. I would appreciate it if someone could point me in the ...
0 votes
0 answers
37 views

What does it even mean?: "Perturbation theory developed around minima of potential (vacuum) is stable."

(Context: QFT, spontaneous symmetry breaking): What I have understood from reading the path integral version of the story is that this "vacuum" is actually the classical solution [i.e. which ...
3 votes
2 answers
116 views

Instantons and Spontaneous Symmetry Breaking

I'm following an introductory lecture on instantons by Hilmar Forkel. In a non-relativistic quantum mechanical setting we have the potential $$ V(x) = \dfrac{\alpha^2 m}{2 x_0^2} (x^2 - x_0^2)^2 \tag{...
1 vote
1 answer
234 views

Euclidean functional Integrals: Existence of zero eigenvalue due to time translation symmetry

In the chapter "Uses of Instantons" from the book "aspects of symmetry" by Sidney Coleman I have come across the euclidean version of the path integral in semi-classical ...
1 vote
1 answer
264 views

What is the gravitational path integral computing?

What is the gravitational path integral (which roughly goes like $\int [dg]e^{iS_{\text{EH}}[g]}$) computing? Usually, path integrals arise from transition amplitudes such as these: $\lim_{T\to\infty}\...
1 vote
1 answer
103 views

On Gravity and the Path Integral

The path integral, in the simplest case, usually attributes a classical action to every conceivable trajectory a particle can take between to points in spacetime. This assumes a flat, Minkowski ...

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