quantization from the algebraic viewpoint

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Quantization from the algebraic viewpoint L. 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 Downloaded 30 Sep 2013 to 128.233.210.97. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

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Page 1: Quantization from the algebraic viewpoint

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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Page 2: Quantization from the algebraic viewpoint

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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Page 3: Quantization from the algebraic viewpoint

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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Page 4: Quantization from the algebraic viewpoint

. ( . 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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