CONSTITUTIVE RELATIONS IN
CLASSICAL OPTICS IN TERMS OF GEOMETRIC ALGEBRA
Adolfas Dargys
Semiconductor Physics Institute, Center for Physical Sciences
and Technology, A. Goštauto 11, LT-01108 Vilnius, Lithuania
E-mail: adolfas.dargys@ftmc.lt
Received 10 February 2015; revised 30 March 2015; accepted 15 June
2015
To have a closed system, the Maxwell
electromagnetic equations should be supplemented by
constitutive relations which describe medium properties and
connect primary fields (E, B) with secondary
ones (D, H). J.W. Gibbs and O. Heaviside
introduced the basis vectors {i, j, k}
to represent the fields and constitutive relations in the
three-dimensional vectorial space. In this paper the
constitutive relations are presented in a form of Cl3,0
algebra which describes the vector space by three basis
vectors {σ1, σ2, σ3}
that satisfy Pauli commutation relations. It is shown that the
classification of electromagnetic wave propagation phenomena
with the help of constitutive relations in this case comes
from the structure of Cl3,0 itself.
Concrete expressions for classical constitutive relations are
presented including electromagnetic wave propagation in a
moving dielectric.
Keywords:
electrodynamics, constitutive relations, light propagation in
anisotropic media, geometric algebra, Clifford algebra
PACS: 03.50.D, 42.25.B,
77.22.C, 78.20.E
SANDAROS RYŠIAI KLASIKINĖJE
OPTIKOJE GEOMETRINĖS ALGEBROS POŽIŪRIU
Adolfas Dargys
Fizinių ir technologijos mokslų centro Puslaidininkių fizikos
institutas, Vilnius, Lietuva
Kad Maxwello lygčių sistema būtų
uždara, ją reikia papildyti sandaros ryšiais, nusakančiais
terpės, kurioje sklinda elektromagnetinė banga, savybes ir
susiejančiais pirminius elektromagnetinius laukus su antriniais.
Straipsnyje pateikti sandaros ryšiai užrašyti Cl3,0
algebros, vadinamosios Cliffordo algebra, kalba. Nuo
standartinio vektorinio skaičiavimo, plačiai taikomo
elektrodinamikoje, ši algebra skiriasi tuo, kad Euklido erdvę
sudarantys trys ortai joje tenkina tuos pačius komutacinius
sąryšius kaip ir Paulio matricos. Kadangi Cl3,0
algebra yra izomorfiška reliatyvistinės Cl1,3
algebros lyginiam poalgebriui, manoma, kad Cl3,0
algebros matematinis aparatas teisingiau aprašo trimatę Euklido
erdvę nei daugiau kaip prieš 100 metų J.W. Gibbso ir O.
Heaviside pasiūlyti ortai {i, j, k} ir su
jais susietas vektorinis skaičiavimas. Be to, Cl1,3
ir Cl3,0 algebros aiškiau suskirsto
elektrodinamiką į reliatyvistinę ir klasikinę. Straipsnyje
nagrinėjami sandaros ryšiai klasikinės elektrodinamikos
požiūriu, kai aplinkos atsakas yra tiesinis sužadinimo atžvilgiu
ir be vėlinimo. Parodyta, kad tokiu atveju elektromagnetinių
bangų sklidimo savybių klasifikacija išeina iš pačios Cl3,0
algebros vidinės sandaros ir todėl sandaros ryšiams suformuluoti
nėra reikalingi jokie kiti papildomi apribojimai. Pateiktos
konkrečios sandaros sąryšių matematinės išraiškos Cl3,0
algebros kalba, taip pat jų pagalba išspręstas elektromagnetinės
bangos sklidimo judančiame dielektrike uždavinys.
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