Bose Algebras: The Complex and Real Wave Representations by Torben T. Nielsen

By Torben T. Nielsen

The arithmetic of Bose-Fock areas is equipped at the thought of a commutative algebra and this algebraic constitution makes the speculation beautiful either to mathematicians without history in physics and to theorectical and mathematical physicists who will straight away realize that the usual set-up doesn't vague the direct relevance to theoretical physics. the well known advanced and genuine wave representations look right here as normal outcomes of the fundamental mathematical constitution - a mathematician conversant in class idea will regard those representations as functors. Operators generated via creations and annihilations in a given Bose algebra are proven to provide upward push to a brand new Bose algebra of operators yielding the Weyl calculus of pseudo-differential operators. The ebook may be invaluable to mathematicians drawn to research in infinitely many dimensions or within the arithmetic of quantum fields and to theoretical physicists who can cash in on using an efficient and rigrous Bose formalism.

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I=O n-1 2 (½) k : (x+y*)n-1-2k) : + (n-1)'- ~ kl. in_2k)! -(n-2i)! - ~ i=0 n-1 2 :(x+y*)n-2i I< )k * n-1-2k * n - 2 k (z y , x > :(x+y ) : y n k:. (n-2k) ! - (n-2k) ! : ) : k=0 n-1 2 + n!. - (n-2k) ! : ( x + y ) k=0 and s i n c e [½n] = ½(n-l) for [½n]~ (~lk n! The the reader. : (x+y O L k=0 case The n odd . " (n-2k) ! of even proof n of , the which second is v e r y m u c h identity is the left same, to the is left to reader as well. 5: For x,y6H exp(x+y*) we d e f i n e = ~ on F0H the o p e r a t o r s (x+y*)n/n!

16]) and on elements For every :exp(x+y*): of F0K , x,yeH are and on the series which absolutely F0K we summable have the when operator identity exp(x+y*) Proof: For n>m We consider = e ½ elements :exp(x+y*): of the form o am a6H i i we have :(x+y*)n:(am) = n ~ (~)~+(xn-k)y*k(am) = k=0 m k=0 n)a+(xn-k ) m! (k (m-k)! = m ~ (~)~+(xn-k)y*k(am) k'a m-k k=0 m = ~ (~)( n! -k-x -a k=0 We estimate n •'-~" :(x+y * )n:(am) n=m <_ ~ n=m n=m o~ n=m oo m m m -< ~ ~ (k)" n=m k=0 ~(n-k)! am-k m ~ (~).

N+m-2k)! ]½ (n-k)! k=O Ixln-k. ixln-k. lalm-k (n-k)! ,, Ixln-k k=0 for n=m ~n-~j ! k=I,2 .... ,m . (½)½P(½p)! *)n: (am) p=0 n=0 co co (½) ½p. : (x+y*)n: (am ) = ~ ~ 6p(½p)! n! n=0 p=0 = ~ [in] ~ (½< exp y * e+(exp and the remaining identity of proposition x) follows.

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