Electrodynamics in geometric algebra Sylvain D. Brechet1 Institute of Physics Station 3 Ecole Polytechnique F ed erale de Lausanne - EPFL CH-1015

2025-04-24 0 0 446.07KB 91 页 10玖币

侵权投诉

Electrodynamics in geometric algebra

Sylvain D. Brechet1

Institute of Physics, Station 3, Ecole Polytechnique F´ed´erale de Lausanne - EPFL, CH-1015

Lausanne, Switzerland

Abstract

We consider the electrodynamics of electric charges and currents in vacuum

and then generalise our results to the description of a dielectric and magnetic

material medium : ﬁrst in spatial algebra (SA) and then in space-time algebra

(STA). Introducing a polarisation multivector ˜

P=˜p−1

c˜

Mand an auxiliary

electromagnetic ﬁeld multivector G=ε0F+˜

P, we express the Maxwell equation

in the material medium in SA. Introducing a bound current vector ˜

J=J−

c∇ · ˜

Pin space-time, the Maxwell equation is then expressed in STA. The

wave equation in the material medium is obtained by taking the gradient of

the Maxwell equation. For a uniform electromagnetic medium consisting of

induced electric and magnetic dipoles, the stress-energy momentum vector is

written as ˙

T˙

∇=1

cJ·F=fwhere fis the electromagnetic force density

vector in space-time. Finally, the Maxwell equation in the material medium can

be written in STA as a wave equation for the potential vector A.

Contents

1 Introduction 3

2 Maxwell equations in vector space (VS) 7

3 Maxwell equations in spatial algebra (SA) 8

4 Maxwell equation in vacuum (SA) 11

Email address: sylvain.brechet@epfl.ch (Sylvain D. Brechet)

Preprint submitted to Journal of L

X Templates October 12, 2022

arXiv:2210.05601v1 [physics.class-ph] 11 Oct 2022

5 Electromagnetic waves in vacuum (SA) 14

6 Electromagnetic energy and momentum in vacuum (SA) 15

7 Poynting theorem in vacuum (SA) 19

8 Electric and magnetic potentials (SA) 20

9 Maxwell equation in matter (SA) 20

10 Electromagnetic waves in matter (SA) 24

11 Electromagnetic energy and momentum in matter (SA) 25

12 Poynting theorem in matter (SA) 27

13 Maxwell equation in vacuum (STA) 29

14 Maxwell equation in matter (STA) 32

15 Electromagnetic waves in vacuum (STA) 35

16 Electromagnetic waves in matter (STA) 36

17 Electromagnetic ﬁelds (STA) 38

18 Stress energy momentum in vacuum (STA) 46

19 Stress energy momentum in matter (STA) 52

20 Electromagnetic potential (STA) 55

21 Conclusion 57

A Spatial algebra (SA) 59

B Duality in spatial algebra (SA) 64

C Diﬀerential duality in spatial algebra (SA) 70

D Algebraic identities in spatial algebra (SA) 71

E Diﬀerential algebraic identities in spatial alge-

bra (SA) 73

F Space-time algebra (STA) 78

1. Introduction

In his seminal paper of 1865 entitled “A dynamical theory of the electro-

magnetic ﬁeld” [1], James Clerk Maxwell obtained a fully self consistent ﬁeld

description of electromagnetic phenomena. The electromagnetic ﬁelds he used

were introduced by Michael Faraday in order to accurately present the results of

his experiments. In his earlier attempts to establish a theory of electromagnetic

phenomena, Maxwell ﬁrst used mechanical analogies to explain these phenom-

ena in mechanical terms [2]. With great insight, he understood that since the

mechanical scaﬀolding of his theory was neither relevant nor needed, he could

simply get rid of it. By doing so, Maxwell revealed a beautiful ﬁeld theory that

served as a template for numerous other ﬁeld theories. It was a paramount

paradigm shift in the history of physics that paved the way for the discovery of

special relativity presented by Albert Einstein as “the electrodynamics of mov-

ing bodies” [3]. The prediction of electromagnetic waves played historically a

key role for the foundations of quantum mechanics that resulted from the syn-

thesis of the wave mechanics of Erwin Schr¨odinger [4] and the matrix mechanics

of Werner Heisenberg, Max Born and Pascual Jordan [5, 6]. The importance

of Maxwell’s work was clearly stated by Einstein on the centenary of Maxwell’s

birth : “We may say that, before Maxwell, physical reality, in so far as it was to

represent the process of nature, was thought of as consisting in material parti-

cles, whose variations consist only in movements governed by partial diﬀerential

equations. Since Maxwell’s time, physical reality has been thought of as rep-

resented by continuous ﬁelds, governed by partial diﬀerential equations, and

not capable of any mechanical interpretation. This change in the conception of

Reality is the most profound and the most fruitful that physics has experienced

since the time of Newton.” [7] Maxwell understood the importance of his theory

of electromagnetism and shared his enthusiasm with his cousin Charles Cay : “I

have also a paper aﬂoat, containing an electromagnetic theory of light, which,

till I am convinced to the contrary, I hold to be great guns”. [8]

Maxwell’s equations are usually presented in textbooks as a set of four vec-

torial equations [9, 10]. This was not the direct result of Maxwell’s work. It

was in fact Oliver Heaviside [11] who gathered Maxwell’s equations into a set

of four vectorial equations in 1884 using the vector calculus he coformulated

with Josiah Willard Gibbs [12]. In his initial paper published in 1865, Maxwell

wrote 20 equations in components. Subsequently, in his “Treatise on Electric-

ity and Magnetism” [13], Maxwell recast his set of 20 equations in terms of

quaternions discovered by William Rowan Hamilton in 1843. Since, the vector

space of Heaviside and Gibbs is ideally suited to describe translations, it was

adopted historically as the main geometric framework of physics. However, the

quaternion algebra His far more suited for the description of rotations than the

vector space R3where pseudovectors need to be introduced in order to treat

rotations. This naturally prompts the question : “Do we really need to choose

or is it possible to take advantage of both framework at the same time ?” In

fact, the good news is that both frameworks belong to a broader mathematical

framework called either geometric algebra (GA) G3or Cliﬀord algebra C`3(R)

that was discovered by William Kingdom Cliﬀord in 1878 in an article entitled

“Applications of Grassmann’s extensive algebra” [14].

In geometric algebra (GA) G3, there are four types of natural geometric en-

tities. First, there are geometric entities of dimension 0 called scalars, which

are oriented points deﬁned by their magnitude and orientation (e.g. positive or

negative). Second, there are geometric entities of dimension 1 called vectors,

which are oriented lines deﬁned by their magnitude and orientation. Third,

there are geometric entities of dimension 2 called bivectors, which are oriented

surfaces deﬁned by their magnitude and orientation. Fourth, there are geomet-

ric entities of dimension 3 called trivectors or pseudoscalars, which are oriented

volume deﬁned by their magnitude and orientation (e.g. positive or negative).

The geometric algebra (GA) G3is a vector space consisting of multivectors,

which are linear combinations of scalars, vectors, bivectors and pseudoscalars.

The quaternion algebra H, which consists of linear combination of scalars and

bivectors is the even subalgebra of the geometric algebra (GA) G3. The vector

space R3, which consists of linear combination of vectors, is a subspace of the

geometric algebra (GA) G3. The geometric algebra (GA) G3is a vector space

endowed with two composition laws : the inner and outer product of two mul-

tivectors. The algebraic sum of the inner and outer product of two vectors is

called the geometric product, as explained in Appendix A.

The geometric algebra (GA) G3is also called the spatial algebra (SA). Since

special relativity was based on electromagnetism, it is relevant to generalise

the spatial algebra (SA) to include an additional temporal dimension. The

generalisation of the spatial algebra (SA) G3to space-time is the geometric

algebra (GA) G1,3or the Cliﬀord algebra C`1,3(R) called the space-time algebra

(STA) developed by Hestenes [15]. The spatial algebra (SA) G3is isomorphic, or

equivalent in structure, to the even subalgebra of the space-time algebra (STA)

G1,3. Formally, bivectors consisting of the outer product of a space vector and

a time vector in G1,3are isomorphic to spatial vectors in G3, the outer product

of these bivectors in G1,3are isomorphic to spatial bivectors in G3and the

pseudoscalar Iin G3is isomorphic to the pseudoscalar in G1,3.

Geometric algebra in its various forms, namely spatial algebra (SA) G3or

space-time algebra (STA) G1,3, appears to be the most appropriate language

to describe physical phenomena since its structure is based on the geometry

underlying the natural world. In his memoir entitled “Reapings and Sowings”,

Alexander Grothendieck, one of the greatest mathematician of the 20th century

states [16] : “If there is one thing in mathematics that fascinates me more than

anything else (and doubtless always has), it is neither ‘number’ nor ‘size,’ but

always form. Among the thousand-and-one faces whereby form chooses to reveal

itself to us, the one that fascinates me more than any other and continues to

fascinate me, is the structure hidden in mathematical things.”

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ElectrodynamicsingeometricalgebraSylvainD.Brechet1InstituteofPhysics,Station3,EcolePolytechniqueFederaledeLausanne-EPFL,CH-1015Lausanne,SwitzerlandAbstractWeconsidertheelectrodynamicsofelectricchargesandcurrentsinvacuumandthengeneraliseourresultstothedescriptionofadielectricandmagneticmaterialmedi...

展开>> 收起<<

Electrodynamics in geometric algebra Sylvain D. Brechet1 Institute of Physics Station 3 Ecole Polytechnique F ed erale de Lausanne - EPFL CH-1015.pdf

共91页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Electrodynamics in geometric algebra Sylvain D. Brechet1 Institute of Physics Station 3 Ecole Polytechnique F ed erale de Lausanne - EPFL CH-1015

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: