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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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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Modern relevance of canonical quantisation [closed]

In some modern field theory texts such as Siegel's Fields it is claimed that canonical quantisation of fields is obsolete as it is not used it modern research papers. Thus, it should be removed from ...
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Is Feynmann gauge reduce always physical gauge?

Is Feynmann gauge reduce always physical gauge? I heard in QCD, Feynmann gauge does not always give correct physics. The lecture says, "Fenymann gauge gives physical gauge, if the theory contains ...
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Doubt in Path integral equation

In Pokorski's "Gauge Field Theories" book, page 108 we find equation (2.87) $$\int{}\mathcal{D}\phi{}e^{iS_0[\phi]+iS_I[\phi]+i\int{}d^4y\,\phi(y)J(y)}=e^{iS_I[-i\frac{\delta}{\delta{}J(x)}]}\int{}\...
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Green function of squared chiral pseudoscalar in QCD

I need to compute the Green function $$ G(0) \equiv \int d^{4} x\int D[\text{QCD}]\bar{q}\gamma_{5}Mq(x)\bar{q}M\gamma_{5}q(0)e^{iS_{QCD}} \equiv $$ $$ \tag 1 \equiv \int d^{4}x\langle 0|T( \bar{q}\...
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Why use coherent state path integral? What is its motivation or goal?

In almost all textbooks of quantum field theory for high energy, they insert the position and momentum eigenstate to formulate the path integral. While in condensed matter field theory, they insert ...
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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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Time-ordered product vs path integral

Suppose we have the Green function $$ G(k) \equiv \tag 1\int d^4x e^{ikx}\langle 0| T\left(\partial^{x}_{\mu}A^{\mu}(x)B(0)\right)|0\rangle , $$ which in path integral approach is equal to $$ \tag 2 ...
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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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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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48 views

Derivation of Aharonov Bohm effect for Quasiparticles

I've noticed the following: Observation: Central results in the condensed matter physics rely on Aharonov Bohm-type arguments involving quasiparticles with fractional charge. However, I can't ...
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Books on path integral methods [duplicate]

Are there advanced books on applications to physics of the method of path integral? I am aware of some of the standard textbooks on QFT, but looking for more advanced applications of the method.
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Determinant of a propagator

Say I have a path integral $\int D \phi \exp(i S_0)$. $S_0$ is the usual free action $$S_0=\frac{1}{2}\int\phi (-\Box-m^2) \phi=\frac{1}{2}\int \phi G^{-1} \phi,$$ and at the moment I'm not ...
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Lagrangian vs Hamiltonian and symmetry of a theory

It is said that since the path-integral formulation of quantum mechanics/or quantum field theory uses the Lagrangian rather than the Hamiltonian, as the fundamental quantity, it preserves all the ...
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How to deal with boundary conditions for path integrals?

For non-relativistic quantum mechanics, the boundary conditions are rather simple to deal with, they are just \begin{equation} \langle x_1, t_1 \vert x_2, t_2\rangle = \int_{x_1(t_1)}^{x_2(t_2)} \...
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192 views

Why does Feynman say that Huygens' principle is not correct for optics?

I am reading Feynman's famous paper Space-Time Approach to Non-Relativistic Quantum Mechanics, and in section 7 he says: Actually, Huygens' principle is not correct in optics. It is replaced by ...
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61 views

Equivariant cohomology formula

I'm studying equivariant cohomology on three references: Szabo's review about equivariant localization (S); Libine's note on equivariant cohomology (L); Berline, Getzler, Vigne's book "Heat Kernels ...
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184 views

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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49 views

recovering numerical wave function using path integral after Wick rotation

I have written two different path-integral codes, PATHINT and PATHTREE, to numerically solve some classical-physics problems in nonlinear systems, finance and neuroscience. They work just fine. My ...
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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 \...
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Does “sum over all paths” in the path integral imply “sum over all paths” in momentum space when one Fourier-transforms?

How is the Fourier-transformed-field path integral interpreted? Is it still a "sum of all paths" in momentum space? Just that with another action? Consider for instance the (Euclidean) partition ...
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162 views

Mathematical proof that $\exp(-1/|g|)$ is always related with formation of bound states through scales?

I know that this function ($g$ means coupling) is non-analytical in $g=0$, so this function is only appreciable under non-perturbative calculations, so is a non-perturbative phenomena. This function ...
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${\phi}^4$ 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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Bohmian trajectories vs. Feynman path integrals, continuous paths?

After reading some of the other posts online, I'm clear on the fact that Bohmian trajectories (of the de Broglie Bohm formulation) and the paths of the Feynman path integral formulation are very ...
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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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82 views

How to separate an exponential with a Hamiltonian with both momentum and position operators?

Statement of exercise On a page 11 of A.Zee's book QFT in a Nutshell, he derives Dirac's formulation of the path integral formulation of QM for a free particle. This starts with the free particle ...
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153 views

General formula for expanding wave function in terms of orthogonal states?

Given a wave function $\psi(x) = \langle \psi | x \rangle$. It can be expanded in terms of orthogonal states: $$ \langle \psi | x \rangle = \sum_n \langle \psi | n \rangle \langle n |x \rangle $$ ...
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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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How to apply cutoff in path integral?

I am working on harmonic oscillator for quantum fluctuations (apart from clasical part), path may written as $$ S_q=\int_0^Tdt[(\partial_tq)^2+w^2q^2] $$ This may written as $$ S_q=\int dt(q\Delta q) ...
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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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Work done against gravity comes out positive

Given two points in 3-space, $A = (9, 1, 5)$ and $B=(2,8,7)$, the work done by the gravitational field $\bf{F}$ when an object is moved from point $A$ to point $B$ comes out positive, even though it ...
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A quantum particle moving from A to B will take every possible path from A to B at the same time

If a quantum particle can take an unlimited number of paths to get from point A to point B wouldn't a quantum particle never get from point A to point B? A quantum particle takes every path at the ...
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What is the interpretation of coefficients in path integral

Say we want the amplitude for particles starting at x,y with distribution f(x,y) and ending up at w,z with distribution w,z. Given a functionals: $F[\phi] = \int f(x,y)\phi(x,t)\phi(y,t) dx^3 dy^3$ ...
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What is the connection between Hilbert Space and path integrals?

Given a space of states $|\rangle$, $|x\rangle$, $|x,y\rangle$, with the creation operators such as $\hat{\phi}(x)|y,z\rangle=|x,y,z\rangle$ for creating a particle at position $x$ and so on. How ...
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Gaussian path integrals and convergence

The Hamiltonian path integral in quantum mechanics, for a particle with coordinate $q$ and momentum $p$ and Hamiltonian $H=p^2/2m+V(q)$, is $\int \mathcal{D}q(t)\mathcal{D}p(t)e^{i\int_0^T(p\dot{q}-\...
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Boundary conditions in holomorphic path integral

Consider the holomorphic representation of the path integral (for a single degree of freedom): $$ U(a^{*}, a, t'', t') = \int e^{\alpha^{*}(t'') \alpha(t'')} \exp\left\{\intop_{t'}^{t''} dt \left( -a^...
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Chern-Simons function

The Chern-Simons function on the space of connections, mod the gauge transformations, on a 3-manifold can be defined by an integral. I study mathematics as profession, so I want to know what is the ...
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104 views

Regularization of the 1-dimensional Laplacian

Disclaimer: this is a technical question about regularization of functional determinants which comes from a person with (relatively) strong background in QFT, string theory and path integrals, who ...
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77 views

Probability amplitude for motion from $x_i$ to $x_f$ in Heisenberg picture

In M. Nakahara's book Geometry, Topology and Physics on page 19, the probability amplitude for a particle to move from $x_i$ at time $t_i$ to $x_f$ at time $t_f$ is given as $$ \tag{1} \langle x_f, ...
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162 views

Harmonic Oscillator propagator

I'm reading the book on Quantum Field Theory by Anthony Duncan, and I'm a little lost with something of propagators. He first define the propagator $K(q_f,T;q_i,0)$ as the amplitude of detecting a ...
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Path Integral Simulation

I am trying to do a simulation with path integrals. I previously simulated with a step potential by time evolving the eigenvectors of the Hamiltonian. See code and video. Notice is is a bit weird, ...
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Lattice propagator computation

I am reading this lecture on random surface theory by Thordur Jonsson. I feel like I should describe the problem a little for those who don't want to read the lecture. We are ultimately interested in ...
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Why integrate of metrics and not inverse metrics?

In quantum gravity it is generally said that the partition function is written as: $$ Z[J] = \int e^{ i S[g] +J.g } D[g] $$ My question is why do we functionally integrate over all possible metrics ...
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Conceptual questions on the path integral formulation of QFT

I'm currently trying to teach myself the path integral formulation of QFT (having studied the canonical approach previously), but I'm having some conceptual difficulties that I hope I can clear up ...
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148 views

Completeness relations of eigenstates in the Heisenberg picture

I've been reading Srednicki's introduction to path integrals and I'm slightly unsure of the notation that he uses for the completeness relation of position eigenstates in the Heisenberg picture. In ...
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constructing string path integral through Markov chain procedure

We all know the textbook way of constructing the path integral for a point particle. It goes approximately like this (if relativistic). Say we want the propagator $\Delta(x_\text{in},x_\text{out}) = \...
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Lagrangian from Path Integral

Suppose I somehow know propagator for a given quantum mechanical system but I don't happen to know either the Lagrangian or Hamiltonian. (For simplicity, assume that this is non-relativistic.) Is ...
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123 views

Path integral for boson vs fermion (soft derivation + use )

I have been looking around for a soft derivation with a bit of detail for boson and fermion path integrals that I could understand. I have a passing knowledge generally of what a path integral is in ...
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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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In path integrals, why is phase defined the way it is? [duplicate]

In reading from the book, there's a peculiar definition of the phase, stating: \begin{align} K[a,b] &= \sum_{\text{All paths from $a$ to $b$}} \phi [x(t)]\\ \phi [x(t)] &= \text{const} \...