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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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In the paper A Duality Web in 2 + 1 Dimensions and Condensed Matter Physics, on page 34, it talks about the periods of a closed $2$-form. Consider the following path integral, $$\int\mathcal{D}B\;\e …
asked Mar 12 '18 by The Last Knight of Silk Road
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The action of electromagnetic field is $$S=\int\left(-\frac{1}{2e^{2}}F\wedge\ast F+\frac{\theta}{8\pi^{2}}F\wedge F\right),$$ where $F=dA$ is the curvature $2$-form, and $A$ is the connection $1$- …
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In section 13.6 of Nakahara, the parity anomaly is in odd dimensional spacetime. From the paper Fermionic Path Integral And Topological Phases by Witten, the problem appears as one cannot define the …
asked Oct 25 '18 by The Last Knight of Silk Road
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I've been studying the paper A Duality Web in 2+1 Dimensions and Condensed Matter Physics. On page 22, starting with the Lagrangian $$|D_{b}\phi|^{2}+|D_{\hat{b}}\hat{\phi}|^{2}-V(|\phi|,|\hat{\phi}| …
asked Feb 28 '18 by The Last Knight of Silk Road
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If a theory is parity invariant classically, is its path-integral measure also invariant under parity?
asked Mar 27 '18 by The Last Knight of Silk Road
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A delta function can be written as $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dp\,e^{ipx}.$$ I have a very poor understanding of the Wick rotation technique used in quantum field theory. Doe …
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The Polyakov action of a point-particle is $$S[X,e]=\frac{1}{2}\int d\tau\left(\frac{\dot{X}^{2}}{e}-m^{2}e\right)$$ with the $(−,+,+,+)$ Minkowski sign convention. How to perform the path-integral …
asked Aug 15 '18 by The Last Knight of Silk Road
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Can the path-integral of Abelian Chern-Simons theory be valuated exactly? $$\int \mathcal{D}[A] \exp\left\{{\frac{i}{2\pi}\int A\wedge dA}\right\}$$ I found Witten's paper "Quantum Field Theory and …
asked Feb 28 '18 by The Last Knight of Silk Road
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My questions are about worldline path integrals from the book Gauge Fields and Strings of Polyakov. On page 153, chapter 9, he says Let us begin with the following path integral \begin{align} &\ …
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I want to do the following path integral. $$\mathcal{Z}=\int\mathcal{D}x e^{iS[\dot{x}]}$$ The action only denpends on $\dot{x}$. For some reason, I want to replace the integral measure $\mathcal{D} …
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This has a mathematically rigorous proof using graph-related group theory. You can find it from the MIT lecture notes MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY On page 13, theorem 3.4 ha …
answered Jul 5 '18 by The Last Knight of Silk Road
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From the Harvard lecture notes XY model: particle-vortex duality by Subir Sachdev, the path-integral of 1D XY-model is given by $$\mathcal{Z}=\int\mathcal{D}\theta\exp{\left\{-\frac{K}{2}\int \!dx~( …
asked Mar 25 '18 by The Last Knight of Silk Road
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In quantum field theory, the partition function of a free scalar is $$\mathcal{Z}=\int\mathcal{D}\phi\exp i\int d^{n}x\frac{1}{2}\left[(\partial_{\mu}\phi)(\partial^{\mu}\phi)-m^{2}\phi^{2}\right]$$ …
asked Oct 9 '18 by The Last Knight of Silk Road
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From the paper "Fermi-Bose Transmutations Induced by Gauge Fields" by Polyakov, http://inspirehep.net/record/22956 http://dx.doi.org/10.1142/S0217732388000398 the theory in 3D, $$\mathcal{L}=\sum_ …
asked Apr 10 '18 by The Last Knight of Silk Road
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The determinant was computed long time ago by Andrew Cohen, Gregory Moore, Philip Nelson, and Joseph Polchinski in An off-shell propagator for string theory. The parameter $T$ in your path integral is …
answered Jan 13 by The Last Knight of Silk Road