Proca action

In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca action and equation are named after Romanian physicist Alexandru Proca. The Proca equation is involved in the Standard Model and describes there the three massive vector bosons, i.e. the Z and W bosons. This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors.

In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca action and equation are named after Romanian physicist Alexandru Proca. The Proca equation is involved in the Standard Model and describes there the three massive vector bosons, i.e. the Z and W bosons. This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors. \mathcal{L}=-\frac{1}{2}(\partial_\mu B_\nu^*-\partial_\nu B_\mu^*)(\partial^\mu B^\nu-\partial^\nu B^\mu)+\frac{m^2 c^2}{\hbar^2}B_\nu^* B^\nu, where c is the speed of light in vacuum, \hbar is the reduced Planck constant, and \partial_{\mu} is the 4-gradient. \partial_\mu \Bigl( \partial^\mu B^\nu - \partial^\nu B^\mu \Bigr) + \left( \frac{ m c }{\hbar} \right)^2 B^\nu = 0, which is conjugate equivalent to \left[ \partial_\mu \partial^\mu + \left( \frac{ m c }{ \hbar } \right)^2 \right]B^\nu = 0 and for m ≠ 0 implies \partial_\nu B^\nu = 0 , equivalent to a generalized Lorenz gauge condition. For the massive case however, this is a physical constraint rather than an optional gauge condition. For non-zero sources, with all fundamental constants included, the field equation is: c \mu_0 j^\nu = \left[ g^{\mu \nu } \left( \partial_\sigma \partial^\sigma + \frac{ m^2 c^2 }{ \hbar^2 } \right) - \partial^\nu \partial^\mu \right] B_\mu When , the source-free equations reduce to Maxwell's equations without charge or current, and the above reduces to Maxwell's charge equation. This Proca field equation is closely related to the Klein–Gordon equation, because it is second order in space and time. In the vector calculus notation, the source-free equations are: \Box \phi - \frac{ \partial }{\partial t} \left(\frac{ 1 }{ c^2} \frac{ \partial \phi }{ \partial t } + \nabla\cdot\mathbf{A} \right) = -\left(\frac{ m c }{\hbar}\right)^2 \phi \Box \mathbf{A} + \nabla \left( \frac{ 1 }{ c^2 } \frac{ \partial \phi }{\partial t} + \nabla \cdot \mathbf{A} \right) = -\left(\frac{ m c }{ \hbar }\right)^2 \mathbf{A} and \Box is the D'Alembert operator.

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