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In Yr 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebs of the property market. Units are snapped up in document time and at document breaking prices. Developers are having fun with overwhelming demand and consumers want more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp duty imposed on residential properties, traders switch their focus to business and industrial properties. I consider every property purchasers need their property investment to appreciate in worth. 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.

This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors.

Lagrangian density

The field involved is the 4-potential A^μ = (φ/c, A), where φ is the electric potential and A is the magnetic potential. The Lagrangian density is given by:

${\mathcal {L}}=-{\frac {1}{16\pi }}(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu })(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })+{\frac {m^{2}c^{2}}{8\pi \hbar ^{2}}}A^{\nu }A_{\nu }.$

where c is the speed of light, ħ is the reduced Planck constant, and ∂^μ is the 4-gradient.

Equation

The Euler–Lagrange equation of motion for this case, also called the Proca equation, is:

$\partial _{\mu }(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu })+\left({\frac {mc}{\hbar }}\right)^{2}A^{\nu }=0$

which is equivalent to the conjunction of

$\left[\partial _{\mu }\partial ^{\mu }+\left({\frac {mc}{\hbar }}\right)^{2}\right]A^{\nu }=0$

with

$\partial _{\mu }A^{\mu }=0\!$

which is the Lorenz gauge condition. When m = 0, the equations reduce to Maxwell's equations without charge or current. The Proca equation is closely related to the Klein–Gordon equation, because it is second order in space and time.

In the more familiar vector calculus notation, the equations are:

$\Box \phi -{\frac {\partial }{\partial t}}\left({\frac {1}{c^{2}}}{\frac {\partial \phi }{\partial t}}+\nabla \cdot \mathbf {A} \right)=-\left({\frac {mc}{\hbar }}\right)^{2}\phi \!$

$\Box \mathbf {A} +\nabla \left({\frac {1}{c^{2}}}{\frac {\partial \phi }{\partial t}}+\nabla \cdot \mathbf {A} \right)=-\left({\frac {mc}{\hbar }}\right)^{2}\mathbf {A} \!$

and $\Box$ is the D'Alembert operator.

Gauge fixing

The Proca action is the gauge-fixed version of the Stueckelberg action via the Higgs mechanism. Quantizing the Proca action requires the use of second class constraints.

They are not invariant under the electromagnetic gauge transformations

$A^{\mu }\rightarrow A^{\mu }-\partial ^{\mu }f$

where f is an arbitrary function, except for when m = 0.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Textbooks

• W. Greiner, "Relativistic quantum mechanics", Springer, p. 359, ISBN 3-540-67457-8
• Supersymmetry P. Labelle, Demystified, McGraw–Hill (USA), 2010, ISBN 978-0-07-163641-4
• Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
• Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9

See also

• Maxwell's equations
• Photon
• Vector boson
• Electromagnetic field
• Quantum electrodynamics
• Quantum gravity