quantization from the algebraic viewpoint
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Quantization from the algebraic viewpointL. Abellanas and L. Martinez Alonso Citation: J. Math. Phys. 17, 1363 (1976); doi: 10.1063/1.523084 View online: http://dx.doi.org/10.1063/1.523084 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v17/i8 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors
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Quantization from the algebraic viewpoint L. Abellanas and L. Martinez Alonso
Departamento de FIsica Teorica. Universidad Complutense de Madrid, Madrid. Spain (Received 13 February 1976)
We use the Weyl quantization W in a general context valid for any finite-dimensional Lie algebra G. to derive an explicit fonnula for (PI' P2)= W -1([ W( PI), W( P2)]), PI' P2 polynomials. In the particular case of the Heisenberg Lie algebra, this fonnula reduces to the familiar Moyal bracket.
1. THE VON NEUMANN QUANTIZATION
Let G be a complex Lie algebra with commutation relations [AI,A}] =iLkC1.0-k' in a given basis {;l}t' Its symmetric algebra S, isomorphic to the polynomial ring a: [au' •• ,an] in n commuting variables {a }}~, admits a Poisson structure l defined by the Poisson brackets:
{ }_" k aPI aP2 PI> P2 = L...J cljak -~ - -~ -, Pu P2 F S.
i,J,k ual Uaj
On the other hand, the universal enveloping algebra2 U of G includes all (noncommutative) polynomials in the elements of G, and has natural Lie algebra structure given by
[u, u] =uv - vu, u, VE" U.
Suppose a quantization rule a} - A j' j = 1, ..• ,n, is fixed. Following von Neumann, 3 we are interested in finding all algebraic quantization prescriptions compatible with it, in the following sense:
Definition 1: A linear map p : S - U, defined on S is said to be a von Neumann G-quantization if and only if
(1) p(a j ) =A j , 'ItJ}'
(2) p(P"') = (P(P»'", 'ltJPr-c S, 'ltJm E IN,
It turns out, however, that these requirements are incompatible. Therefore, as it will be explicitly shown below, a given Lie algebra G does not necessarily admit any such quantization .. In particular it is so for the Heisenberg Lie algebra {p, Q, 1}.
It is because we propose a weaker version of von Neumann notion:
Definition 2: A linear map a: S - U, defined on S will be called a weak von Neumann G-quantization for G if and only if it verifies
a«L~jaj)m)=(.0~.0-j)m, 'ItJ~ = (~I>"" ~n)r-c a:n•
j
Proposition 1: For a given G there exists a unique weak von Neumann G-quantization. Furthermore, it is characterized by the symmetrization:
1 ')' a(a. a· ... a· ) = - LJA . A· ... A· )1 3 2 Jr r! I' ).".(1) 3 .. (2) JlI'(r)'
where the summation runs over all permutations rr of r objects.
Proof: It is easily verified that (1) defines a weak von Neumann G-quantization. Uniqueness follows from the fact that every polynomial P(aI> .•• ,an) can
(1)
1363 Journal of Mathematical Physics, Vol. 17, No.8, August 1976
be written as a linear combination of terms of the form (Lj ~jaj)m (QED).
We are already in a position to prove that the Heisenberg Lie algebra GH={AUA2,A3}, lAI>A 2 ]=iAa, lAuAa]=[A2,A3]=0, does not admit any G-quantization of the type included in Definition 1. In fact, as a simple calculation shows,
a(a~a~) =A~~ + 2iA00I - iA~ = (a(aI a2»2 - iA~.
2. THE WEYL QUANTIZATION
At this point we make some useful notation conventions:
a=(au ... ,an)ElNn, a! =a l !a2 !'" an!, lal s.0a}, J
~A =.B~.0- j' C' = ~fl~;2' .. ~:n, 'ItJ ~ F a:n,
al",1 _ al",1 a~'" = a~fIa~;2" . a~~n .
As it is well known2 U= (f'lmU'"' where U'" stands for the linear subspace of U generated by the elements a(a "') , la I=m. Let us consider the set U* of all formal series in the a(a"'). As a typical example we quote
The underlying idea in what follows is to use the right-hand side expression in order to freely manipulate the coefficients without interference of the noncommutative part a(a"'). Thus, given a set of complex-valued functions F,,: Rn - a:, a E N n, we define an associated map:
F:Rn_ U*, F(~)=.0F,,(~)a(a"'),
'" which assigns to each point ~ F R n a formal series in U*. Furthermore, if our F", are good enough, we can apply tempered distributions T to get
(T, F) =.0 (T, F ,,)a(a"').
'" (3)
This produces a formal series with complex coefficients. It is in this context that Weyl quantization 4 makes sense:
Definition 3: W:S-U. W(P) = (27T)-n/2 (P,e lfA), PES, where P(O=(27T)-n/2f P(a)e-ifada is the Fourier transform of P. But since P was a polynomial, P is a finite order distribution with point support ~ =0. Therefore, W(P) = (2rr)-n/2 L" (il "'1/ a! )(P, ~ "')a(a"') E U [observe that W(P) is nothing but a polynomial in the Aj]'
Copyright © 1976 American Institute of Physics 1363
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Proposition 2: W{P) = a{P), 'fIPE S.
Proof: From the familiar properties of Fourier transform,
01,,1 a"(~)=(21T)"/2il"l a~" 0(0,
we conclude that
Hence W=a on S, by linearity (QED).
3. CONVOLUTION VERSUS QUANTIZATION
Since a: S - U is a bijective map,2 the element
P, * P2 =' (J""(a(p, )a(P2», P2, P2 E S,
is well defined in S. The formal analogy between (4) and the convolution productf*g=]-'(Jf· ]g) in Fourier theory is obvious, and as a matter of fact.it will be made explicit in what follows,
Proposition 3:
(4)
where V"=,ojaa', V'''='ojoa'', and T(~,11)='I\.(~,11)-~-11 denotes the nonlinear part of the exponent in the BakerHausdorff-Campbell5 formula: eHAeinA = eiil.(f,n)A.
Remark: Since P" P2 are polynomials, the right-hand side in the previous formula has a finite number of terms, so it is also a polynomial.
Proof:
By expressing I\. "(~, 11) =' I\.t"(~' 11 )1\.:2(~, 11) •.. I\.:"(~, 11) as a (finite) Taylor series relative to the point ~ +1), we obtain
Thus we can write
where
D "a =' (21T)"n / 2( TB(~, 1) )PI (~)1'2(1), L aa(~ + 1)),
o ISII\." I Laa(x)=, a7 .
x.=x
(5)
In order to calculate D "a, we only need to recall the analycity of TB at the origin, 5 let us say
TB(~, 1) = .0 t",,,,, ~ "'1) a" a',a"
and some well-known properties of the Fourier transform:
1364 J. Math. Phys., Vol. 17, No.8, August 1976
D"a=(21T)"n/Z
X '£ t",,,"«~)(~)(i~)(1), LaB(~ +1)) ~~ .
= 1::: ta'a,,«IV0-::;:(;i")*(i~,),Letll) a'.O("
By substituting in (5) we get
a(PI )a(P2) 'Ietl = (21T)"n/2:0 _z _,
a 0'.
= «e laT (1<;", iV" ) PI (a')P2(a") I a'''''=a)" (~), elf A )
= a(elaT(IV',IV") P , (a')P2(a") I a'=a" ..,).
The rest of the proof is trivial (QED).
We will now define a new Lie structure on the symmetric algebra S as follows, If PH P2 are elements in S, also (PI' P2) =' PI * Pz - P2 * P , E S, and has the following properties:
(1) (P" Pz) = (f"l([ a(p, ), a(Pz)]) ,
(2) (P" P2 ) = - (p2 , P,),
(3) (P" I\.P2 + IJ. P3) = I\.(p" P2) + IJ. (P" P3),
(4) (P" (p2, P3» + (P2 , (P3 , P,» + (P3 , (p" P2» = 0.
Finally, by making use of the Proposition 3, we have the explicit formula
(p P.) = (elaT(lV', IV") _ elaT(IV", iV'»P (a')P. (a") I U 2 1 2 a':;;:a"=a-
(6)
4. THE CANONICAL EXAMPLE
Let GH be the Heisenberg Lie algebra. In this particular case we find:
I\.(~, 1) = (~, +1)" ~2 +1)2' ~3 +113 + t(~z1), - ~,1)2»
and thus
T(~, 1) = (0, 0, 'H~z1), - ~11JZ».
The formula (6) gives us the result
In quantum mechaniCS, where a, = q, a2 = p, a3 = Ii, this reads as follows:
L. Abellanas and L. Martinez Alonso 1364
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. ( . n (a a a a)) =2t Sln 2 aq' ap" - aq" ap'
x Pi (q' ,P')P2(q", p") 1.,=.,,= .. p'=p"=p
This is identical with the well-known expression obtained by Moyal, 6 which can also be written in the form
(Plo P2 )(q, p)
. '" ~l( n)2n+l (_ 1)n+k a2n+l Pi a2n+l P2 =2t:0L.J - 1(2 1)1 n=O /pO 2 k. n - k + . apkal"+l-k alap2n+l-k •
One easily sees that if one (or both) of the polynomials Plo P2 have degrees < 3, then
1365 J. Math. Phys., Vol. 17, No.8, August 1976
( )() 'n(aPl aP2 aPl ap2 ) 'n{P p.} Pi' P2 q, P = t aq ap - ap aq = t 10 2'
where {Pit P2} is the familiar Poisson bracket.
1M. Vergne, Bull. Soc. Math. France 100, 301 (1972). 2J. Dixmier, Algebres enveloppantes (Gauthiers-Villars, Paris, 1974).
3J. von Neumann, Mthematische Begriindung der Quanterrmechanik (Springer-Verlag, Berlin, 1932).
4H. Weyl Gruppentheorie und Quantenmechanik (HirzelVerlag, Leipzig, 1928).
5N. Jacobson, Lie algebras (Interscience, New York, 1962). 6J. E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949).
L Abellanas and L Martinez Alonso 1365
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