What is the Difference Between Lorentz Gauge and Coulomb Gauge

Thekey difference between Lorentz gauge and Coulomb gaugeis that Lorentz gauge is related to the Minkowski space, whereas Coulomb gauge is related to the Euclidean space.

Generally, Minkowski space is a 4D (four-dimensional) real vector space. This is equipped with a nondegenerate, symmetric bilinear form. It also occurs on the tangent space at each point in spacetime. Euclidean space, on the other hand, is a fundamental in classical geometry. It is a 3 D (three-dimensional) space.

内容

1.Overview and Key Difference
2.What is Lorentz Gauge
3.What is Coulomb Gauge
4.Lorentz Gauge vs Coulomb Gauge in Tabular Form
5.Summary – Lorentz Gauge vs Coulomb Gauge

What is Lorentz Gauge?

Lorentz gauge is a partial gauge fixing of the electromagnetic vector potential. This concept was first described by Ludwig Lorenz. This term mainly has its applications inelectromagnetism. Generally, we can use the Lorentz gauge in electromagnetism for the calculation of time-dependent electromagnetic fields through the related potentials.

最初，当Ludwig Lorenz的作品出版时，Maxwell并没有很好地收到它。此后，他消除了库仑electrostatic他的电磁波方程的推导力。这是因为他在库仑仪上工作。更重要的是，Lorentz仪表与Minkowski空间有关。

What is Coulomb Gauge?

Coulomb gauge is a type of gauge that is expressed in terms of instantaneous values of the fields and densities. It is also known asa transverse gauge. This concept is very useful in quantum chemistry and condensed matter physics. We can define it using gauge condition, or more precisely, using gauge fixing condition.

This Coulomb gauge is particularly useful in semi-classical calculations that come inquantum mechanics. Here, the vector potential is quantized, but Coulomb interaction is not. In the Coulomb gauge, we can express the potentials in terms of instantaneous values of the fields and densities.

Moreover, gauge transformations can retain the Coulomb gauge condition, which may be formed with gauge functions that satisfy the concept. However, in regions that are far from the electric charge of the scalar potential, the Coulomb gauge becomes zero, and we call it the radiation gauge. This electromagnetic radiation was first quantized in this gauge.

Furthermore, the Coulomb gauge admits a natural Hamiltonian formulation of the evolution equations (concerning the electromagnetic field) of the electromagnetic field interacting with a conserved current. This is an advantage of the quantization of the theory. More importantly, Coulomb gauge is related to the Euclidean space.

What is the Difference Between Lorentz Gauge and Coulomb Gauge?

Lorentz Gauge和Coulomb Gauge是两个在量子化学中很重要的概念。Lorentz仪表是电磁载体电势的部分量规固定，而Coulomb仪表是一种按照磁场和密度的瞬时值表示的仪表。Lorentz Gauge和Coulomb仪表之间的主要区别在于Lorentz仪表与Minkowski空间有关，而库仑仪与欧几里得空间有关。Minkowski空间是4D（四维）真实矢量空间，而欧几里得空间为3 d（三维）空间，这也是经典几何的基本。

以下是Lorentz Gauge和Coulomb Gauge以表格形式进行并排比较之间的差异的摘要。

Summary – Lorentz Gauge vs Coulomb Gauge

We can distinguish the Lorentz gauge and Mikowskin gauge depending on the dimensions. The key difference between Lorentz gauge and Coulomb gauge is that Lorentz gauge is related to the Minkowski space, whereas Coulomb gauge is related to the Euclidean space. Minkowski space is a 4D (four-dimensional) real vector space, while Euclidean space is a fundamental of the classical geometry and is a 3 D (three-dimensional) space.

Reference:

1. Yann Picand, Dominique Dutoit. “Lorenz Gauge Condition.”Sensagent.

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1.“”Coord system CA 0”豪尔赫Stolfi – Own work (Public Domain) via Commons Wikimedia
2. “World line” By SVG version: K. Aainsqatsi at en.wikipediaOriginal PNG version: Stib at en.wikipedia – Transferred from en.wikipedia to Commons.(Original text: self-made)(CC BY-SA 3.0)via Commons Wikimedia