Indefinite Inner Product Spaces
J. Bognar
Π΄Π΅ΠΊ. 2012β―Π³. Β· Springer Science & Business Media
4,0star
2 ΡΠ΅ΡΠ΅Π½Π·ΠΈΠΈreport
Π-ΠΊΠ½ΠΈΠ³Π°
226
Π‘ΡΡΠ°Π½ΠΈΡΠΈ
reportΠΡΠ΅Π½ΠΈΡΠ΅ ΠΈ ΡΠ΅ΡΠ΅Π½Π·ΠΈΠΈΡΠ΅ Π½Π΅ ΡΠ΅ ΠΏΠΎΡΠ²ΡΠ΄Π΅Π½ΠΈ Β ΠΠΎΠ·Π½Π°ΡΡΠ΅ ΠΏΠΎΠ²Π΅ΡΠ΅
ΠΠ° Π΅-ΠΊΠ½ΠΈΠ³Π°Π²Π°
By definition, an indefinite inner product space is a real or complex vector space together with a symmetric (in the complex case: hermi tian) bilinear form prescribed on it so that the corresponding quadratic form assumes both positive and negative values. The most important special case arises when a Hilbert space is considered as an orthogonal direct sum of two subspaces, one equipped with the original inner prod uct, and the other with -1 times the original inner product. The subject first appeared thirty years ago in a paper of Dirac [1] on quantum field theory (d. also Pauli [lJ). Soon afterwards, Pontrja gin [1] gave the first mathematical treatment of an indefinite inner prod uct space. Pontrjagin was unaware of the investigations of Dirac and Pauli; on the other hand, he was inspired by a work of Sobolev [lJ, unpublished up to 1960, concerning a problem of mechanics. The attempts of Dirac and Pauli to apply the concept and elemen tary properties of indefinite inner product spaces to field theory have been renewed by several authors. At present it is not easy to judge which of their results will contribute to the final form of this part of physics. The following list of references should serve as a guide to the extensive literature: Bleuler [1], Gupta [lJ, Kallen and Pauli [lJ, Heisen berg [lJ-[4J, Bogoljubov, Medvedev and Polivanov [lJ, K.L. Nagy [lJ-[3], Berezin [lJ, Arons, Han and Sudarshan [1], Lee and Wick [1J.
ΠΡΠ΅Π½ΠΈ ΠΈ ΡΠ΅ΡΠ΅Π½Π·ΠΈΠΈ
4,0
2 ΡΠ΅ΡΠ΅Π½Π·ΠΈΠΈ
ΠΡΠ΅Π½Π΅ΡΠ΅ ΡΠ° Π΅-ΠΊΠ½ΠΈΠ³Π°Π²Π°
ΠΠ°ΠΆΠ΅ΡΠ΅ Π½ΠΈ ΡΡΠΎ ΠΌΠΈΡΠ»ΠΈΡΠ΅.
ΠΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΈ Π·Π° ΡΠΈΡΠ°ΡΠ΅
ΠΠ°ΠΌΠ΅ΡΠ½ΠΈ ΡΠ΅Π»Π΅ΡΠΎΠ½ΠΈ ΠΈ ΡΠ°Π±Π»Π΅ΡΠΈ
ΠΠ½ΡΡΠ°Π»ΠΈΡΠ°ΡΡΠ΅ ΡΠ° Π°ΠΏΠ»ΠΈΠΊΠ°ΡΠΈΡΠ°ΡΠ° Google Play Books Π·Π° Android ΠΈ iPad/iPhone. ΠΠ²ΡΠΎΠΌΠ°ΡΡΠΊΠΈ ΡΠ΅ ΡΠΈΠ½Ρ
ΡΠΎΠ½ΠΈΠ·ΠΈΡΠ° ΡΠΎ ΡΠΌΠ΅ΡΠΊΠ°ΡΠ° ΠΈ Π²ΠΈ ΠΎΠ²ΠΎΠ·ΠΌΠΎΠΆΡΠ²Π° Π΄Π° ΡΠΈΡΠ°ΡΠ΅ ΠΎΠ½Π»Π°ΡΠ½ ΠΈΠ»ΠΈ ΠΎΡΠ»Π°ΡΠ½ ΠΊΠ°Π΄Π΅ ΠΈ Π΄Π° ΡΡΠ΅.
ΠΠ°ΠΏΡΠΎΠΏΠΈ ΠΈ ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠΈ
ΠΠΎΠΆΠ΅ Π΄Π° ΡΠ»ΡΡΠ°ΡΠ΅ Π°ΡΠ΄ΠΈΠΎΠΊΠ½ΠΈΠ³ΠΈ ΠΊΡΠΏΠ΅Π½ΠΈ ΠΎΠ΄ Google Play ΡΠΎ ΠΊΠΎΡΠΈΡΡΠ΅ΡΠ΅ Π½Π° Π²Π΅Π±-ΠΏΡΠ΅Π»ΠΈΡΡΡΠ²Π°ΡΠΎΡ Π½Π° ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠΎΡ.
Π-ΡΠΈΡΠ°ΡΠΈ ΠΈ Π΄ΡΡΠ³ΠΈ ΡΡΠ΅Π΄ΠΈ
ΠΠ° Π΄Π° ΡΠΈΡΠ°ΡΠ΅ Π½Π° ΡΡΠ΅Π΄ΠΈ ΡΠΎ Π΅-ΠΌΠ°ΡΡΠΈΠ»ΠΎ, ΠΊΠ°ΠΊΠΎ ΡΡΠΎ ΡΠ΅ Π΅-ΡΠΈΡΠ°ΡΠΈΡΠ΅ Kobo, ΡΠ΅ ΡΡΠ΅Π±Π° Π΄Π° ΠΏΡΠ΅Π·Π΅ΠΌΠ΅ΡΠ΅ Π΄Π°ΡΠΎΡΠ΅ΠΊΠ° ΠΈ Π΄Π° ΡΠ° ΠΏΡΠ΅ΡΡΠ»ΠΈΡΠ΅ Π½Π° ΡΡΠ΅Π΄ΠΎΡ. Π‘Π»Π΅Π΄Π΅ΡΠ΅ Π³ΠΈ Π΄Π΅ΡΠ°Π»Π½ΠΈΡΠ΅ ΡΠΏΠ°ΡΡΡΠ²Π° Π²ΠΎ Π¦Π΅Π½ΡΠ°ΡΠΎΡ Π·Π° ΠΏΠΎΠΌΠΎΡ Π·Π° ΠΏΡΠ΅ΡΡΠ»Π°ΡΠ΅ Π½Π° Π΄Π°ΡΠΎΡΠ΅ΠΊΠΈΡΠ΅ Π½Π° ΠΏΠΎΠ΄Π΄ΡΠΆΠ°Π½ΠΈ Π΅-ΡΠΈΡΠ°ΡΠΈ.
