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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1answer
2k views

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 to calculate the path integral (or partition function) of a set of phonons with continuous frequencies?

I have a system of phonons with continuous frequencies so that the Hamiltonian of this system is $H=\int_0^{k_0} h(k) a_k^\dagger a_k \mathrm{d}k$. How do I calculate the partition function of this ...
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8answers
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Why is the contribution of a path in Feynmans path integral formalism $\sim e^{(i/\hbar)S[x(t)]}$?

In the book "Quantum Mechanics and Path Integrals" Feynman & Hibbs state that the probability $P(b,a)$ to go from point $x_a$ at time $t_a$ to the point $x_b$ at the time $t_b$ is $P(b,...
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2answers
395 views

Detailed calculation of the scattering amplitude between two particles in $\phi^4$-perturbation theory using the path integral formalism

I'm looking for detailed example/calculation of the scattering amplitude between two particles (mesons) in $\phi^4$-perturbation theory using the path integral formalism. Do anyone of you guys have a ...
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1answer
154 views

Coherent state basis of (relativistic) particle Fock space

For a neutral scalar bosonic particle of mass $m$, I consider a Fock space with an orthonormal basis of momenta eigenstates \begin{equation}\label{Fock-p-states} \left|p_1p_2\cdots p_n\right\rangle=\...
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Path Integral on Feynman Hibbs: Interaction of EM field and matter, how can we get to equation (9.68) from (9.67)?

On Feynman Hibbs "Quantum Mechanics and Path Integrals", the equation (9.67) describes the transition amplitude of the matter (for example an atom) to go from the state $M$ to the state $M$ ...
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1answer
322 views

Difference between green's function and density matrix

What is the basic difference between green's function or propagator of given system and density matrix (in the position basis) of the same system ? Can some one explain the difference between with ...
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3answers
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Does a path integral necessarily mean there is a quantum mechanical description?

Given a path integral for a system $$Z(\phi) = \int [D\phi] e^{-S[\phi]},$$ where I am working in the Euclidean signature, necessarily mean that the system described is quantum mechanical? In the ...
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1answer
180 views

One-loop effective action for scalar field on the curved background in large potential

I hope to compute a functional integral $Z=\int \mathcal{D}\phi\,\, e^{-S[\phi]}$ with an action $$S[\phi]=\int d^2x \sqrt{g}\Big((\nabla \phi)^2+\frac{1}{\lambda}M^2(x) \phi^2\Big)$$ The scalar field ...
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1answer
49 views

Are there other possible options to represent the amplitude in the path integral formalism?

The path intgral formalism of quantum mechanics states that the amplitude to go from $\left(x_i,t_i\right)$ to $\left(x_f,t_f\right)$ is $$K\left(x_f,t_f,x_i,t_i\right) = \int \mathcal{D}x\quad e^{i\...
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0answers
52 views

Path integral for free fermion: meaning of boundary terms

In article Deriving Projective Hyperspace from Harmonic there is discussion of JWKB (page 7) path integral for free fermion. Here I briefly rederive the statement: If one start with Lagrangian: $$ L = ...
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26 views

How to change the integration measure between vectors and complex numbers?

When using $CP(1)$ representation, we will write the unit vector $\boldsymbol{n}$ as the the spinor form, i.e. complex number: $$\boldsymbol{n}=\left(\begin{array}{cc}z_{1}^{\dagger} & z_{2}^{\...
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1answer
51 views

Meaning of capital pi symbol in sum over histories integral

This question is primarily mathematical in nature. I have been reading Quantum Field Theory for the Gifted Amateur and I am reading about Feynman’s path integral approach. The definition of the “sum ...
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1answer
186 views

Constraints in path integral and the Lagrange multiplier

I was reading some references on the slave-particle approach to the Kondo problem and Anderson model. It is known that the slave-particle is introduced in the large Hubbard $U$ limit of the system so ...
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Proof that the effective/proper action is the generating functional of one-particle-irreducible (1PI) correlation functions

In all text book and lecture notes that I have found, they write down the general statement \begin{equation} \frac{\delta^n\Gamma[\phi_{\rm cl}]}{\delta\phi_{\rm cl}(x_1)\ldots\delta\phi_{\rm cl}(x_n)}...
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1answer
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Partition function in quantum field theory

Why does the partition function include current term in free scalar field $$Z[J] = \int \mathcal{D}\phi \, e^{i \left(S[\phi] + \int d^4x \,J(x) \phi(x) \right)}~$$
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1answer
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Conjugate momentum in the vacuum functional for the fermionic oscillator

The vacuum functional for the fermionic oscillator is given by $$ Z[0] = N\int\mathcal{D}\overline{\psi}\mathcal{D}\psi \exp\left(i\int_0^Tdt\left(i\overline{\psi}\psi-w\overline{\psi}\psi \right)\...
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1answer
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How can I understand the tunneling problem by Euclidean path integral where the quadratic fluctuation has a negative eigenvalue?

I came across the S. Coleman's seminal papers 'Fate of the false vacuum' (http://dx.doi.org/10.1103/PhysRevD.15.2929, http://dx.doi.org/10.1103/PhysRevD.16.1762) where he describes the tunneling ...
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Applying the heat-kernel method?

I was wondering if the strategy I am going to try is correct? Am I missing anything important? I am currently trying to apply the stationary phase approximation to a path integral. I need to find the ...
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2answers
57 views

Why the classical configuration always static when applying saddle point (semi-classical) approximation?

For an Green function/partition function: $$\int D[\phi]e^{\frac{i S[\phi]}{\hbar}}$$ We can make saddle point approximation and gives classical configuration: $$\delta \mathcal{S}=0\Longrightarrow \...
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2answers
87 views

Harmonic oscillator partition function via Matsubara formalism

I am trying to understand the solution to a problem in Altland & Simons, chapter 4, p. 183. As a demonstration of the finite temperature path integral, the problem asks to calculate the partition ...
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1answer
58 views

Generating Functional for Complex Scalar Theory

The generating functional for a free complex scalar field theory is given by: $$W[J,J^*]=\int D\phi D\phi^* \exp (i \int_{}^{} d⁴ x [(\partial_{\mu}\phi)^*(\partial^{\mu}\phi) -m^2\phi^*\phi + J^*\...
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1answer
212 views

Why can the time-ordered exponentials be brought to the right?

Having worked through almost all calculations in section 4.2 of Peskin & Schroeder's An Introduction to QFT, I still don't get why we can get to Eq. (4.31) \begin{equation} <\Omega|\mathcal{T}\...
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1answer
147 views

Relation between worldsheets and worldlines

If we want to find the propagator of say a zero-spin particle, the formal inverse can be computed by introducing $$-i\int_{0}^{\infty}e^{i(p^2 - m^2)s}\mathrm{d}s$$ We can rewrite this in path ...
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1answer
448 views

Why does the classical path of a particle give the dominant contribution in the path integral?

Why is it that the classical path of a particle gives the dominant contribution in the quantum mechanical path integral? How do we understand this?
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1answer
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Confusions on expectation value for $\hbar$ going to zero

In Matthew D. Schwartz's QFT book, Chapter 28, the author claims when $\hbar \rightarrow 0$, the following equality (eq 28.4) holds: So how can I see the second "$=$" holds? It seems the ...
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4answers
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What is the path integral exactly?

I asked a question here about path integrals and QFT. I just want to confirm something. Is the path integral in quantum field theory a mathematical tool only? I thought the path integral meant that ...
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0answers
27 views

For the QFT path integral action for a pair of sources and sinks $W_{ij}(J_1,J_2)$, is $W_{12} = W_{21}$ ever NOT exactly upheld?

The path integral's action $W_{ij}$ as a function of a pair of sources and sinks $J = J_1 + J_2$ can be written $$W_{ij} = -\frac{1}{2} \int \frac{d^4k}{(2 \pi)^4}J_j^*(k)\frac{1}{k^2-m^2+i\epsilon}...
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1answer
20 views

Stationary Phase approximation with multiple coordinates?

The stationary phase approximation can be used to find an approximate value for the path integral \begin{equation}\int Dx e^{-S[x]} \approx e^{-S[\bar{x}]} \left(\det{\frac{\hat{A}}{2 \pi}}\right)^{-1/...
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Chern-Simons Path integral restricting to small gauge transformations

How does one compute the Chern-Simons path integral in 2+1 dimensions considering only small gauge transformations? Is this even a well-defined theory?
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2answers
131 views

Does the quadratic mass term $\phi^2$ belong to the free Lagrangian or is it an interaction term?

$$L = -\frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{m^2}{2}\phi^2.$$ Why is the $\phi^2$ term in the scalar Lagrangian not considered a self-interaction?
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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}...
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1answer
68 views

Derivation of transition amplitude probability between two adjacent space points for general time independent hamiltonian

I am studying Srednicki book of Quantum Field theory. In chapter 6 regarding the path integral there was derived equation of transition probability for hamiltonian of type: $$H(\hat{P},\hat{Q})= \frac{...
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48 views

What is the entropy and/or equation of state of a partition function such as $Z=\int D\phi \exp (i S[\phi]/\hbar)$?

At this link https://en.wikipedia.org/wiki/Partition_function_(mathematics), it is claimed that the following partition function: $$ Z=\int D\phi \exp (-\beta H[\phi]) \tag{1} $$ is a consequence of ...
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1answer
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Understanding the statement “orbifold theories are QFTs with finite gauge group”

I'd like to understand the equivalence of orbifold theories in string theory and (2D worldsheet) QFTs with finite gauge group, using the path integral. Suppose my action is $$S= \frac{1}{2\pi \alpha'...
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1answer
216 views

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} \,\dot{\...
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According to Hartle-Hawking state, could we build a sum over all possible metrics (including non-compact ones)?

Physicists Stephen W Hawking and James B Hartle 1 proposed that the universe, in its origins, had no boundary conditions both in space and time. To do that, they proposed a sum over all compact ...
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1answer
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Question about Faddeev-Popov gauge-fixing in Schwartz textbook

I am trying to understand equation (25.91) from Schwartz's Quantum Field Theory textbook. The goal is to gauge-fix the path integral for Quantum chromodynamics using the Faddeev-Popov trick. Briefly, ...
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1answer
189 views

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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2answers
78 views

Wick Rotation & Scalar Field Value & Mapping

Wick Rotation helps to solve the problem of the convergence of the path integral, by changing the integral contour in the complex plane. But my question is: In the scalar field path integral, the ...
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1answer
310 views

Dirac delta function conversion into gauge-fixing Lagrangian in the path integral

So, I am at the moment working on gauge-fixing a path integral. The procedure involves adding a delta function $\delta g$ to the path integral (together with the Faddeev-Popov determinant, but that is ...
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2answers
131 views

Why does a square root term make the quantisation of action difficult?

When going over my lecturer's notes on String Theory and trying to understand a particle as a theory of gravity in 1D, it is mentioned that the action $(1)$ is regularisation invariant, $$S=-m\...
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1answer
60 views

Eliminating residual gauge in BRST quantization of Yang-Mills theory

I would like to know if there is a procedure to completely fix a gauge, which I believe we must do in order to make sense of the path integral? In chapter 74 Sredniki introduces the Lagrangian $$ \...
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43 views

Expression for sum over paths

In an introductory lecture on the path integral formalism, I came across the following. Suppose that $\gamma$'s are paths such that a particle travelling along any of them reaches the position co-...
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55 views

Symmetries in quantum field theory and anomalies

Suppose we have a lagrangian quantum field theory, thus a theory where we can write an action in the form \begin{equation} S = \displaystyle \int d^4 x \; \mathcal L \, \left( \partial_{\mu} \phi , \...
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1answer
34 views

D'Alembert Operator on Fermionic Field in Path Integral

I am learning the Faddeev–Popov path integral formlism with Schwartz's QFT textbook. In the section 25.4.2 "BRST invariance", I came across the Lagrangian as: $$\mathcal{L}=-\frac{1}{4} F_{\mu \nu}^{2}...
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1answer
47 views

Fermionic Harmonic Oscillator Partition Function

I am reading Nakahara Geometry, Topology, and Physics. In the section on fermionic harmonic oscillator, after some math, the partition function is given by $$\begin{aligned} Z(\beta) &=\mathrm{e}^{...
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1answer
191 views

Deriving the path integral for periodic boundary conditions

I'm thinking about path integrals with the Euclidean time formalism, where I have partition function $Z=\operatorname{Tr} e^{-\beta \hat H}$. I'm used to the following derivation of the path integral: ...
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0answers
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Examples of path integral where path of extremal action does not contribute the most?

I have learnt that by doing a saddle point approximation in the path integral formulation of quantum mechanics, the classical action (extremal action where $\delta S=0$) is the one that contributes ...
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1answer
130 views

What is path ordered product? Please explain

What is path ordered product? I had heard this mentioned in my Quantum computation class but couldn't really find useful resources to understand it clearly.

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