Spinors in Hilbert Space

¡ Springer Science & Business Media
āĻ‡-āĻŦā§āĻ•
91
āĻĒā§ƒāĻˇā§āĻ āĻž
āĻ°ā§‡āĻŸāĻŋāĻ‚ āĻ“ āĻ°āĻŋāĻ­āĻŋāĻ‰ āĻ¯āĻžāĻšāĻžāĻ‡ āĻ•āĻ°āĻž āĻšā§ŸāĻ¨āĻŋ  āĻ†āĻ°āĻ“ āĻœāĻžāĻ¨ā§āĻ¨

āĻāĻ‡ āĻ‡-āĻŦā§āĻ•ā§‡āĻ° āĻŦāĻŋāĻˇā§Ÿā§‡

1. Hilbert Space The words "Hilbert space" here will always denote what math ematicians call a separable Hilbert space. It is composed of vectors each with a denumerable infinity of coordinates ql' q2' Q3, .... Usually the coordinates are considered to be complex numbers and each vector has a squared length ~rIQrI2. This squared length must converge in order that the q's may specify a Hilbert vector. Let us express qr in terms of real and imaginary parts, qr = Xr + iYr' Then the squared length is l:.r(x; + y;). The x's and y's may be looked upon as the coordinates of a vector. It is again a Hilbert vector, but it is a real Hilbert vector, with only real coordinates. Thus a complex Hilbert vector uniquely determines a real Hilbert vector. The second vector has, at first sight, twice as many coordinates as the first one. But twice a denumerable in finity is again a denumerable infinity, so the second vector has the same number of coordinates as the first. Thus a complex Hilbert vector is not a more general kind of quantity than a real one.

āĻ‡-āĻŦā§āĻ•ā§‡ āĻ°ā§‡āĻŸāĻŋāĻ‚ āĻĻāĻŋāĻ¨

āĻ†āĻĒāĻ¨āĻžāĻ° āĻŽāĻ¤āĻžāĻŽāĻ¤ āĻœāĻžāĻ¨āĻžāĻ¨āĨ¤

āĻĒāĻ āĻ¨ āĻ¤āĻĨā§āĻ¯

āĻ¸ā§āĻŽāĻžāĻ°ā§āĻŸāĻĢā§‹āĻ¨ āĻāĻŦāĻ‚ āĻŸā§āĻ¯āĻžāĻŦāĻ˛ā§‡āĻŸ
Android āĻāĻŦāĻ‚ iPad/iPhone āĻāĻ° āĻœāĻ¨ā§āĻ¯ Google Play āĻŦāĻ‡ āĻ…ā§āĻ¯āĻžāĻĒ āĻ‡āĻ¨āĻ¸ā§āĻŸāĻ˛ āĻ•āĻ°ā§āĻ¨āĨ¤ āĻāĻŸāĻŋ āĻ†āĻĒāĻ¨āĻžāĻ° āĻ…ā§āĻ¯āĻžāĻ•āĻžāĻ‰āĻ¨ā§āĻŸā§‡āĻ° āĻ¸āĻžāĻĨā§‡ āĻ…āĻŸā§‹āĻŽā§‡āĻŸāĻŋāĻ• āĻ¸āĻŋāĻ™ā§āĻ• āĻšā§Ÿ āĻ“ āĻ†āĻĒāĻ¨āĻŋ āĻ…āĻ¨āĻ˛āĻžāĻ‡āĻ¨ āĻŦāĻž āĻ…āĻĢāĻ˛āĻžāĻ‡āĻ¨ āĻ¯āĻžāĻ‡ āĻĨāĻžāĻ•ā§āĻ¨ āĻ¨āĻž āĻ•ā§‡āĻ¨ āĻ†āĻĒāĻ¨āĻžāĻ•ā§‡ āĻĒā§œāĻ¤ā§‡ āĻĻā§‡ā§ŸāĨ¤
āĻ˛ā§āĻ¯āĻžāĻĒāĻŸāĻĒ āĻ“ āĻ•āĻŽā§āĻĒāĻŋāĻ‰āĻŸāĻžāĻ°
Google Play āĻĨā§‡āĻ•ā§‡ āĻ•ā§‡āĻ¨āĻž āĻ…āĻĄāĻŋāĻ“āĻŦā§āĻ• āĻ†āĻĒāĻ¨āĻŋ āĻ•āĻŽā§āĻĒāĻŋāĻ‰āĻŸāĻžāĻ°ā§‡āĻ° āĻ“ā§Ÿā§‡āĻŦ āĻŦā§āĻ°āĻžāĻ‰āĻœāĻžāĻ°ā§‡ āĻļā§āĻ¨āĻ¤ā§‡ āĻĒāĻžāĻ°ā§‡āĻ¨āĨ¤
eReader āĻāĻŦāĻ‚ āĻ…āĻ¨ā§āĻ¯āĻžāĻ¨ā§āĻ¯ āĻĄāĻŋāĻ­āĻžāĻ‡āĻ¸
Kobo eReaders-āĻāĻ° āĻŽāĻ¤ā§‹ e-ink āĻĄāĻŋāĻ­āĻžāĻ‡āĻ¸ā§‡ āĻĒāĻĄāĻŧāĻ¤ā§‡, āĻ†āĻĒāĻ¨āĻžāĻ•ā§‡ āĻāĻ•āĻŸāĻŋ āĻĢāĻžāĻ‡āĻ˛ āĻĄāĻžāĻ‰āĻ¨āĻ˛ā§‹āĻĄ āĻ“ āĻ†āĻĒāĻ¨āĻžāĻ° āĻĄāĻŋāĻ­āĻžāĻ‡āĻ¸ā§‡ āĻŸā§āĻ°āĻžāĻ¨ā§āĻ¸āĻĢāĻžāĻ° āĻ•āĻ°āĻ¤ā§‡ āĻšāĻŦā§‡āĨ¤ āĻŦā§āĻ¯āĻŦāĻšāĻžāĻ°āĻ•āĻžāĻ°ā§€āĻ° āĻ‰āĻĻā§āĻĻā§‡āĻļā§āĻ¯ā§‡ āĻ¤ā§ˆāĻ°āĻŋ āĻ¸āĻšāĻžā§ŸāĻ¤āĻž āĻ•ā§‡āĻ¨ā§āĻĻā§āĻ°āĻ¤ā§‡ āĻĻā§‡āĻ“ā§ŸāĻž āĻ¨āĻŋāĻ°ā§āĻĻā§‡āĻļāĻžāĻŦāĻ˛ā§€ āĻ…āĻ¨ā§āĻ¸āĻ°āĻŖ āĻ•āĻ°ā§‡ āĻ¯ā§‡āĻ¸āĻŦ eReader-āĻ āĻĢāĻžāĻ‡āĻ˛ āĻĒāĻĄāĻŧāĻž āĻ¯āĻžāĻŦā§‡ āĻ¸ā§‡āĻ–āĻžāĻ¨ā§‡ āĻŸā§āĻ°āĻžāĻ¨ā§āĻ¸āĻĢāĻžāĻ° āĻ•āĻ°ā§āĻ¨āĨ¤