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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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### 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) <\Omega|\mathcal{T}\...
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10 votes
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### Does a good path integral exist in Loop Quantum Gravity?

The Hamiltonian operator of Loop quantum gravity is a totally constraint system $$H = \int_\Sigma d^3x\ (N\mathcal{H}+N^a V_a+G)$$ Here, $\Sigma$ is a 3-dimensional hypersurface; a slice of ...
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### Different features of Gravity and Yang-Mills

I am reading a famous paper by S.Hawking - "Quantum gravity and path integrals" https://doi.org/10.1103/PhysRevD.18.1747. On the third page left column there is a statement, after the ...
9 votes
0 answers
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### Lattice QFT of the Jones Polynomial

Start with a gauge theory with Chern-Simons action $$S[A] = \frac{k}{4 \pi} \text{Tr} \intop_{M} \left( A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right)$$ and a Wilson loop observable in the ...
9 votes
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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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8 votes
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### Definition of gravity path integral?

In a non-abelian gauge theory there is a "fundamental" gauge field $A_\mu^a$ with gauge index $a$ often called connection. Although $A_\mu^a$ is not gauge invariant, gauge invariant quantities can be ...
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8 votes
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### Is it possible to do a path integral between two boundaries analytically on a quantum lattice?

I have been trying to perform some path integral between two boundaries for a massless scalar field $$\int_{\varphi(t_a, \vec{x})}^{\varphi(t_b, \vec{x})} \mathcal{D}\varphi(x)e^{iS[\varphi(x)]}$$ ...
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8 votes
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6 votes
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5 votes
1 answer
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### 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 ...
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5 votes
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216 views

### Transition amplitude between field configurations from the path integral

In the path integral formulation of QFT, we should in principle be able to calculate the transition amplitude from a classical field configuration $\phi_{in}(x,t=0)$ to $\phi_{out}(x,t=T)$ using the ...
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5 votes
1 answer
117 views

### What is the meaning of the function of $x$ and $p$ that you get when you cut open the phase space path integral?

When we "cut" an ordinary path integral, we obtain a state in the position representation. That is, if we fix some initial position $x_i$, then the path integral $$\int_{x_i}^{x_f}Dx e^{-S}$$...
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5 votes
2 answers
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### 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 ...
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### Schwinger-Keldysh contour and $i\epsilon$ prescription

In Tom Hartman's notes on path integrals, he describes the Schwinger-Keldysh (or "in-in") formalism for calculating vacuum correlators in QFT. He explains that Lorentzian time-ordered vacuum ...
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### 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 S = \displaystyle \int d^4 x \; \mathcal L \, \left( \partial_{\mu} \phi , \...
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### New symmetries upon quantization

In standard field theory texts, a “classical symmetry” is defined to be a transformation $\phi\to\phi’$ such that the corresponding action is left invariant. The symmetry is said to survive ...
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### Physicist path integral and cylinder set measures

Path integral via discretization So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is just what I understood!). Let a quantum system ...
• 36k
5 votes
0 answers
913 views

### Normal ordering in path integral of QFT

In QFT, we use normal ordering to eliminate infinity from hamiltonian. In path integral formulation of QFT though, since what we integrate over is "classical field configuration", instead of operators,...
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5 votes
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166 views

### Path integral and gauge redundancy for slave particle

In the slave boson, we have $c^\dagger = b f^\dagger$ where $b$ is boson and $f$ is fermion. There is also a local constraint $b^\dagger b+f^\dagger f=1$ to retrict the Hilbert space and a $U(1)$ ...
• 301
5 votes
2 answers
376 views

### What's the path of least action for fermions off-shell?

The Lagrangian of fermions is first order both in space-derivatives and time-derivatives. In the variation of the action one usually fixes both the initial point and end point. I have the following ...
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5 votes
1 answer
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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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### Thermodynamic free energy of interacting system

This question concerns an interacting system's thermodynamic free energy $\Omega$. Generally speaking, The action $S$ for an interacting system has the following form: S(\phi,\psi) = ...
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### Is the path integral emergent?

I have recently read a couple of papers on lattice QCD and found that there is a well-established connection between Boltzmann distribution and the path integral in QFT (disclaimer: I am not a huge ...
4 votes
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### Question on the implication of Lebesgue integration in Einstein-Hilbert action

Suppose I have two Riemannian manifolds in two dimensions, a 2-sphere $\mathbb{S}^2$ and a disk $\mathbb{D}$. I would like to know whether there is a redundancy in Hawking's path integral approach to ...
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4 votes
1 answer
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