Abstract
In the last chapter, we found a functional integral representation for the correlation functions of areal scalar field.
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Notes
- 1.
Equivalently, we could constraint the current by which we take functional derivatives to be a conserved current, that is, so that it obeys \(\partial _\mu J^\mu (x)=0\). This would mean that functional derivatives of it are constrained to produce only correlations of the transverse components of the vector potential field, \(t^{\mu \nu }A_\nu (x)\), where the transverse and longitudinal projection operators are defined by
$$ t^{\mu \nu }(x,y)\equiv \left( \eta _{\mu \nu }-\frac{\partial _\mu \partial _\nu }{\partial ^2}\right) \delta (x-y) \equiv \int \frac{dk}{(2\pi )^4}e^{ikx}\left( \eta _{\mu \nu }-\frac{k_\mu k_\nu }{k^2}\right) $$$$ \ell ^{\mu \nu }(x,y)\equiv \left( \frac{\partial _\mu \partial _\nu }{\partial ^2}\right) \delta (x-y) \equiv \int \frac{dk}{(2\pi )^4}e^{ikx}\left( \frac{k_\mu k_\nu }{k^2}\right) $$obey
$$ t^{\mu \nu }(x,y)+\ell ^{\mu \nu }(x,y)=\eta ^{\mu \nu }\delta (x-y) $$$$ \int dy~ t^{\mu \nu }(x,y)t_\nu ^{~\lambda }(y,z)=t^{\mu \lambda }(x,z),~ \int dy~ \ell ^{\mu \nu }(x,y)\ell _\nu ^{~\lambda }(y,z)=\ell ^{\mu \lambda }(x,z) $$$$ \int dy~ t^{\mu \nu }(x,y)\ell _\nu ^{~\lambda }(y,z)=0=\int dy ~\ell ^{\mu \nu }(x,y)t_\nu ^{~\lambda }(y,z) $$Then, the transverse part of the vector potential is gauge invariant,
$$ t^{\mu \nu }A_\nu (x) = - \int dy \int \frac{dk}{(2\pi )^4} \frac{e^{ik(x-y)}}{k^2} \partial ^\mu F_{\mu \nu }(y) $$ - 2.
The co-factor here is \(D_{1\ell }\) and the minor is the determinant of the matrix that would be obtained by removing the first row and the \(\ell \)’th column of D.
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Semenoff, G.W. (2023). More Functional Integrals. In: Quantum Field Theory. Graduate Texts in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-99-5410-0_12
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DOI: https://doi.org/10.1007/978-981-99-5410-0_12
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