Weyl Poisson quantization ☳ Incepta Physica Mathematica literature

Poisson Brackets,
Weyl Algebras
and
Quantization
 ☳
   Hans Tilgner

  
Abstract

 Poisson brackets on the symmetric algebra over a fi­ni­te-di­men­sio­nal real sym­plec­tic vec­tor space are de­fi­ned in­va­ri­ant­ly by di­rec­tio­nal de­ri­va­tiv­es and sym­plec­tic struc­tures and com­pa­red to Weyl al­ge­bras. The most na­tu­ral pow­er-grad­ed li­ne­ar map be­tween the­se po­wer-grad­ed al­ge­bras is no iso­mor­phism of Lie al­ge­bras with res­pect to Pois­son bra­ck­ets resp. com­mu­ta­tors: As long as at least one ele­ment in a Pois­son bra­cket is at most of po­wer less than three ( other­wise not ), we have the same Lie bra­cket re­la­tions in both al­ge­bras. Thus left Lie bra­cke­ting de­fi­nes in both al­ge­bras an in­fi­ni­te se­ries of (ir­re­du­cib­le?) re­pre­sen­ta­tions of the sym­plec­tic Lie al­ge­bra, the first be­ing the self- and the se­cond the ad­joint re­pre­sen­ta­tion. There al­so is a self-ad­joint re­pre­sen­ta­tion. Kil­ling forms for sym­plec­tic and pseu­do‌-‌or­tho­go­nal Lie al­ge­bras are ex­pres­sed by the de­fi­ning bi­li­ne­ar forms. Quan­ti­za­tion is for­mu­la­ted in a ca­te­go­rial set up, dis­tin­gui­sh­ing be­tween ob­ser­vab­les and in­vari­ance groups, us­ing the con­cepts of self‌-‌ad­joint re­pre­sen­ta­tion, sym­me­tric spaces and Jor­dan al­ge­bras.

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first published
24. Sep 2011

revised upload
Real
Symplectic
Vector Spaces
Let  (EE,σ<,>)be a non-trivial, real, finite-dimensional symplectic vector space, i.e. σ<,> a non-de­ge­ne­rate skew bilinear form on EE,the dimension of which neces­sari­ly is even, say 2n. De­fine
σx ( y )  :=  σ<x,y> ,
i.e. σx is an element of the dual space  EE*of linear forms on EE.
σ : x  ι σx,σ : EE →  EE*
is an isomorphism of vector spaces ( since σ<,> is non-degenerate ). Whence we use in­stead of  EE*the no­ta­tion σEE.The dual space becomes a symplectic vector space as well by the de­fi­ni­tion
σ<σx,σy>  :=  σ<x,y>
wherefrom we can write without confusion  σ  for both symplectic forms. By con­struc­tion the iso­mor­phism σ is an isomorphism of symplectic vector spaces.
 A subspace of EEis called a symplectic subspace if the restriction of the symplectic bi­li­near form σ<,> is symplectic again. The skew symmetry is clear, but the non-de­ge­ne­rate­ness usu­al­ly needs a verification, since there are subspaces which are not sym­plectic ones - for in­stance spaces of odd dimension.
 Another class of subspaces are the isotropy sub- (necessarily) vector spaces
{ x ∈ EE /  σ<x,y> = 0∀ y ∈ EE} .
Isotropy is exclusive to the symplectic concept. A maximal isotropic subspace, necessarily of di­men­sion n, is call­ed Lagrangian. Ex­amp­les are given be­low along with Darboux bases by the li­ne­ar span of the q's resp. the p's.
Symplectic vector spaces are a category, the structure theory of which being deter­mi­ned com­ple­te­ly by its dimension. Con­trary to the case of symmetric bilinear forms there is no con­cept of sig­na­ture, positive‌- or indefiniteness, concepts which solve the struc­ture theory for sym­me­tric bi­li­near forms.
Two elements p,q ∈ EEare called canonical conjugate if  σ<p,q> = 1 holds. Then they are li­near­ly in­de­pen­dent. Since σ is non-degenerate every vec­tor  p ≠ 0  has ca­no­ni­cal con­ju­gates.
σ‌-‌ad­joint endomorphisms are defined as usual for non-degenerate bilinear forms by
σ<Ax,y> = σ<x,Ay>∀ A ∈ end(EE) ,
where  A ιA is a well-known involutive antiautomorphism of end(EE) .

are the
real numbers

typipcally
the vectors of  EE
are
q, p, u, v, w, x, y, z
∈ EE

 σEE  =EE*

is our notation
for the dual

we abbreviate
σ< , >  by  σ
The
Four Basic
Symplectic Categories
The importance of symplectic vector spaces for the theory of simp­le Lie groups is given by the de­ri­va­tion Lie algebra ( with the commutator as Lie alge­bra com­po­si­tion ) of  (EE,σ<,>)
(sp) ss p( EE, σ )  =  { B ∈ end(EE,ℝ / σ<Bx,y>+σ<x,By> = 0  ∀ x,y ∈ EE}
and the automorphim group of the symplectic vector space  (EE,σ<,>)
(Sp)SSp( EE, σ ) =  { G ∈ Aut(EE,ℝ)  / σ<Gx,Gy> = σ<x,y>  ∀ x,y ∈ EE} .
A linear transformation of this symplectic group is called symplectic. Both are n(2n+1)-‌di­men­sio­nal, the Lie al­ge­bra  ss p( , )as a real vec­tor space, the Lie group  SSp( , )as a real dif­fe­ren­tiab­le ma­ni­fold. Usu­al­ly the Lie al­ge­bra  ss p( , )is iden­ti­fied to the tan­gent space at the neu­tral ele­ment  idE ∈ SSp( , ).Here Aut means the set of in­vertible en­do­mor­phisms, most­ly writ­ten  GGℓ( , )in­stead. The lat­ter is ge­ne­ra­ted by ex­po­nen­tia­tion since the po­wer se­ries exp exists and con­ver­ges for en­do­mor­phisms, but the group in general is not ex­po­nen­tial in the sense, that one needs only one B from the Lie al­ge­bra to re­pre­sent a group element in the form  G = exp(B). (sp) re­sults from (Sp) by in­ser­ting the group element  G = exp(αB)  and col­lecting the fac­tor of  α ∈ ℝ  in the re­sul­ting series. A typi­cal, ge­ne­ra­ting li­near­ly ele­ment of  ss p( , )is gi­ven by the li­ne­ar trans­for­ma­tion
spx,y  : u ι σ<y,u>x + σ<x,u>y  with  spx,y = − spx,y = − spy,x ,
which is symmetric in x and y and fullfills the commutation relations of the symplectic Lie al­gebra with respect to the commutator [ , ] of two linear transformations
[spx,y , spu,v] = σ<x,u> spy,v + σ<y,u> spx,v + σ<x,v> spy,u + σ<y,v> spx,u ,
which even can be shortened to
[spx,x , spu,u] = 4 σ<x,u> spx,u    ∀ x,y,u,v ∈ EE ,
since by linearizing this identity  x ιx+y and  u ιu+v  twice we get the original com­mu­ta­tion re­la­tions back. The sym­plec­tic Lie al­gebra is a sub­algebra of the Lie al­ge­bra of the spe­cial li­ne­ar Lie al­ge­bra of all trace­less transformations, and the sym­plec­tic group is a sub­group of the spe­cial li­ne­ar Lie group of all ele­ments of de­ter­mi­nant 1.
In addition the linear space of σ-self-adjoint transformations
(s-ad)ss p+( EE, σ ) { A ∈ end(EE,ℝ / σ<Ax,y> = σ<x,Ay>  ∀ x,y ∈ EE} ,
is a Jordan algebra of (real) dimension (2n)2−n(2n+1) = n(2n-1) with respect to the anti­com­mu­ta­tor {A,B} = AB+BA  of two en­do­mor­phisms A,B, which is the series C clas­si­cal simp­le real Jor­dan al­ge­bra. A typical, ge­ne­rating element of this Jor­dan al­gebra is given by the li­near trans­for­ma­tions
spx,y+ : u ι σ<y,u>x − σ<x,u>ywithspx,y+ =  spx,y+ = − spy,x+ ,
which is skew in its two indices. Hence the linearization trick here doesn't work. It is easy to ve­ri­fy the symplectic anticommutation relations
{spx,y+ , spu,v+ }= − σ<x,u> spy,v+ + σ<y,u> spx,v+ + σ<x,v> spy,u+ − σ<y,v> spx,u+ .
a Lie group,
its Lie algebra,
a domain of positivity
and
its Jordan algebra
dinw( , )Define a Lie bracket t for a given vector t ∈ EE,two (running) vectors x,y and  ઠ = 0  or  ઠ = 1  ( on­ly these two cases, for other choices we get skew algebras without Jacobi identity, which, by the way, pro­ves that skew symmetry does not imply the Jacobi identity ) by
x t y = σ<t,x>y − σ<t,y>x − ઠ σ<x,y>twith   t t y = −(1 + ઠ ) σ<t,y>t .
Denote these Lie algebras by  dinw(EE,σ,t,ઠ )for the two cases. In the first case t is an ele­ment of the cen­ter, but in the se­cond case the cen­ter is trivial. Using
trace ( u ι➥ σ<x,u> y ) = σ<x,y> ,
their Killing forms κil become
κil(x,y) = (2n −1 + ઠ2) σ<t,x> σ<t,y> ,
which in both cases are degenerate. Hence these Lie algebras are not semi-simple, which also is clear from the fact, that their dimensions are 2n, which in general does not fit in­to the four stan­dard se­ries of real simp­le Lie al­ge­bras, lest in­to the exceptional ones. Since their com­mu­ta­tor al­ge­bras
EE t  EE
are not nilpotent, they too are not solvable. So it remains to find their Iwasawa decomposi­tions, i.‌e. their se­mi‌-‌simp­le Le­vi fac­tors and their solvable ideals, the radicals.
Changing the sign in the definition of  ▱ one gets a series of 2n-dimensional Jordan com­po­si­tions
x t y = σ<t,x>y + σ<t,y>xwitht t y = σ<t,y>t
without neutral element. This implies that it cannot be semi-simple [ Kch p 63]. However, a neu­tral ele­ment e can be associated to every Jor­dan al­ge­bra with­out, to give a Jordan composition on  EE  ℝe
( xαe ) t ( yβe ) = σ<t,x>y + σ<t,y>x + βx + αy αβ e  .
Denote the resulting Jordan algebras with neutral element by  dinw+(EE,σ,t) ,which should be cal­led  t h e Jor­dan al­ge­bra of the sym­plec­tic vec­tor spa­ce. By the same di­men­sion­al ar­gu­ment, they do not fit in­to the stan­dard se­ries of real simple Jor­dan al­ge­bras, lest the simp­le ex­cep­tio­nal one.
There are corresponding results for symmetric bilinear forms.
das ist nicht wahr,

das kann doch
nicht wahr sein

we need a
structure theory
of these algebras,
Dinw( , ) groups
resp.
symmetric spaces

there are
concepts, corresponding
to
Weyl- and Clifford-
algebras
The Three
Standard
Representations
of the
Symplectic Group
For the fourth standard symplectic category define a connected but not necessari­ly simp­ly con­nec­ted sub­manifold of Aut(EE,ℝ)by
SSp+( EE, σ ) { exp(α1A1)⋅⋅⋅exp(αkAk / αk ∈  , Ak self-adjoint, k = 1,2, ... }
i.e. a manifold generated multiplicatively by the exponentials of self-adjoint linear trans­for­ma­tions. This gives rise to the diagram
SSp+( EE, σ )×  SSp( EE, σ )=  Aut(EE,ℝ)
exp ↑exp ↑exp ↑
ss p+( EE, σ )   ss p( EE, σ )=  end(EE,ℝ)
with the vertical arrows representing the tangent functors. In addition to the above sym­plec­tic com­mu­ta­tion relations there are the com­mu­tation relations
[spx,y+ , spu,v+]= − σ<x,u> spy,v + σ<y,u> spx,v + σ<x,v> spy,u − σ<y,v> spx,u

[spx,y , spu,v+]  =   σ<x,u> spy,v++ σ<y,u> spx,v+− σ<x,v> spy,u+− σ<y,v> spx,u+ ,
which mean that we get Loos' [ Loo ] typical commutation relations of subspaces of end(EE)
[ ss p( , ) , ss p( , ) ] ⊂ ss p( , ) ,   [ ss p+( , ) , ss p+( , ) ] ⊂ ss p( , ) , [ ss p( , ) , ss p+( , ) ] ⊂ ss p+( , )
of an  involutive automorphism  of the  general linear group  ( see below for the most ge­ne­ral des­crip­tion by O. Loos ), leading to a symmetric de­com­po­si­tion. Since there is no doubt, that the  Lie trip­le  de­fined by these com­mu­ta­tion re­la­tions co­in­ci­des with the Lie trip­le, gi­ven by this Jor­dan al­ge­bra of σ-self‌-‌ad­joint trans­for­ma­tions, we con­clu­de that we have con­structed by this dia­gram the iden­ti­ty com­po­nent of a sym­me­tric space, the com­po­si­tion of which is given by Loos' sym­me­tric com­po­si­tion for the in­ver­tib­le ele­ments of a clas­si­cal real simp­le Jor­dan al­ge­bra.
  Remark: We started by defining the sym­plec­tic Lie group and Lie al­ge­bra by its self‌-‌re­pre­sen­ta­tion on the sym­plec­tic vec­tor space given, getting the ad­joint re­pre­sentation by the first com­mu­ta­tion relations given above, and getting a third re­presentation mo­dule of the sym­plec­tic Lie al­ge­bra and Lie group by the last com­mu­ta­tion re­la­tions herein, the best name of which be­ing self‌-‌ad­joint re­pre­sentation. It cer­tain­ly is under-in­vestigated. For instance it has to be shown that it is ir­re­du­cib­le, and whe­ther the Lie group, ge­ne­ra­ted by ex­po­nen­tia­tion, is iden­ti­cal to the sym­plec­tic group or on­ly re­la­ted to it by cover­ing, i.‌e. by a short exact se­quen­ce with dis­cre­te ker­nel. It exists for pseu­do‌-‌or­tho­go­nal vec­tor spaces as well, to be dis­cus­sed be­low, by the same con­struc­tion.
Our formulation is such that if dropping the letter σ and assuming the non-de­gene­rate bi­li­near form <,> to be sym­me­tric one gets near­ly word by word the struc­ture theo­ry of pseu­do‌-‌or­tho­go­nal vec­tor spa­ces, groups, Lie and Jordan alge­­bras. 
still
underinvestigated
in
classical dynamics,
relativity and
quantum mechanics

self ~

self-adjoint ~

adjoint ~
Projectors
and
Involutions
A  projector on EEis an idempotent endomorphism, i.e.
P : x ι Px,P : EE →  EElinear andP2 = P .
It projects onto the subspace im­age(P). Its  com­plementary projector  idEE − Ppro­jects on­to a com­ple­men­tary sub­space kern(P) of di­men­sion dim EE− dim image(P),such that we have the de­com­po­si­tion
(+)x = x − Px  +  Px,EE =  kern(P)  image(P) ,

(+)kern(idEE− P)= image(P),image(idEE− P)= kern(P) .
Note that non-trivial projectors ( trivial projectors are 0, pro­jec­ting onto the null space, and idE, pro­jecting onto the whole vector space EE )cannot be invertible and hence are neither in the sym­plec­tic group nor in the symplectic Lie al­ge­bra.
As an example define for canonical conjugate p, q ∈ EE
Pp,q : x ι σ<x,q>p,Pp,q : EE →  EEwithPp,q2 = Pp,q .
It projects onto the one-dimensional subspace spanned by p, which cannot be a sym­plec­tic sub­space since it is odd-dimensional.
Given a projector P, define its associated involution on EEas the involutive auto­mor­phism
JP := idEE− 2 PhenceJP2  =  idEEandJP− 1 = JP .
In fact it is a reflection at image(P) which leaves kern(P) invariant, i.e. reading elementwise,
JP( image(P) ) = − image(P),JP( kern(P) ) = kern(P)
results. This means that image(P) = eigenspace of eigenvalue −1 and kern(P) = eigenspace of eigen­value +1 of JP.
Clearly one can invert this construction, starting from an involution J , to get by
PJ  :=  ½ ( idEE− J )
the associate projector. Note that if J  is an involution −J  is an involution as well, called com­ple­men­ta­ry to J, such that its associated projector is complementary to PJ. Then
association
 JP
complementation
− Jid − P  ,
i.e. association and complementation commute. Note that involutions are not exponentials of pro­jec­tors, at least not for real parameters. Special involutions are the  reflections at a subspace LL
J t = − t  for  t ∈ LLbutJ t = t  for  t ∈ LL ⊥
for non-vanishing vectors t. Perpendicular applies if there is a non-degenerate bi­li­near form on the un­der­ly­ing vec­tor space. Examples are given below for these cases.
For the special example Pp,q above J changes the sign of p, leaving the linearly in­de­pen­dent rest of  EEinvariant. This projector is not σ-self-adjoint. More general,
this construction holds for the general case. In case of a sym­plec­tic bi­li­near form we have in ad­di­tion
(iff)JP ∈ SSp( EE, σ )P ∈ ssp+( EE, σ ).
Moreover, in this case we get with , the σ-orthogonal direct vector space sum,
()x = x − Px    Px,EE =  kern(P)    image(P) .
which can be proved as equivalent to this iff-statement as well. The proof is simple, σ‌-‌or­tho­go­na­li­ty fol­lowing for instance from the σ-self-adjointness of P
(⊥)σ< (id − P) x,Py > = σ<x,Py> − σ<Px,Py> = 0  .
Clearly we can interchange the role of p and q in Pp,q to get the same results. Then, mere­ly drop­ping the com­ma in the definition,
Ppq : x  ι σ<x,q>p − σ<x,p>q,Ppq : EE →  EEwithPpq2 = Ppq .
is a projector as well. And this one is σ-self-adjoint. Hence its associated involution is a sym­plec­tic trans­for­ma­tion.
 In quantum mechanics projectors P are called statistical operators, pro­jec­ting on­to states or rays, which are called pure, mathematicians would say primitive, for one-di­men­sio­nal im­age(P). In ge­ne­ral re­la­ti­vi­ty Singer & Thorpe have used the Weyl- and the Ein­stein-pro­jec­tors to de­ve­lop an ele­gant struc­ture theo­ry of the (ter­rib­le) high di­men­sion­al space of cur­va­ture struc­tures.
Remark: This theory of involutions and projectors has a generalization to the theory of groups and symme­tric spaces, given by Loos in sections II ( example 6° ) and IV [ Loo p 72ff]. 
are the
elegant approach
towards a
symplectic
structure theory
Complex
Structures
There is no such construction, leading to projectors, for a com­plex struc­ture, de­fin­ed as an en­do‌(ne­ces­sa­ri­ly auto)­mor­phism C subject to
(cs) C 2  =  − idEE    ( hence  C− 1 = − C ) .
There are complex structures in the symplectic group, which can be proved easily using the  ma­trix re­pre­sen­ta­tion  of the symplectic group.
 A complex structure C has two invertible square roots
± C  :=  ± 1/√2 ( idEE + C )   with    ( ± C)−1 = ± 1/√2 ( idEE − C )
and corresponding to (iff) we have
    C ∈ SSp( EE, σ ) C ∈ ssp( EE, σ )
which, however, will not be used in the following.
Note that multiplying projectors, involutions, complex structures and square roots with a sca­lar all four types of endo­mor­phisms have exponential series in terms of tri­go­no­me­tric resp. hy­per­bo­lic se­ries of the sca­lar fac­tors. In case of self-ad­joint en­do­mor­phisms, ex­po­nen­tia­tion gives one‌-‌pa­ra­me­ter sym­me­tric sub­spaces of  do­mains of po­si­tivi­ty, like we get one‌-‌pa­ra­me­ter sub­groups for ex­po­nen­tia­ted skew‌-‌ad­joint endo­mor­phisms.
We think, complex structures lead to emdedding unitary and pseudo-unitary Lie al­ge­bras in­to the ca­nonical for­ma­lism similar to the embedding of the gener­al li­ne­ar al­ge­bras by means of La­gran­gian de­com­po­si­tions in­to Lagrangian subspaces.
 In addition grouping complex structures into symplectic equivalence clas­ses should lead to the clas­si­fi­ca­tion of com­pact unitary and non-compact pseu­do-uni­tary Lie al­ge­bras.
are used
for
embedding
pseudo-unitary
Lie algebras
???
ExamplesThe standard example of a symplectic vector space is given in terms of a sym­met­ric non-de­ge­ne­rate bi­linear form <,> on a pseu­do-or­thogonal vec­tor space  VVof di­men­sion n: Then
      EE := V<,>VV,σ< x+<,>y, z+<,>w > := <,>w(x) −<,>y(z) = <w,x> − <y,z>   for x,y,z,w ∈ VV
( with the above notation, i.e. <,> substituted for σ ) is a symplectic vector space. Here every­thing goes through if the  con­fi­gura­tion space VVis cut down to an open subset of  VVonly. For in­stan­ce 1/r-po­ten­tials ( one point source or several ones ) are defined only on such a poin­ted con­fi­gu­ra­tion space. In this spe­cial case the mo­men­tum space  <,>VVre­mains the full dual space and the whole ma­ni­fold of doub­le di­men­sion, the phase space of classi­cal me­cha­nics, may be looked at as the co­tan­gent bund­le of this con­fi­gu­ra­tion space. But even in the more ge­ne­ral case of a closed sub­ma­ni­fold of  VVone can try to use this vec­tor space ap­proach by re­stric­ting maps and de­ri­va­tives ( for in­stance ) to this sub­ma­ni­fold, with­out ap­ply­ing the more in­vol­ved ma­chi­ne­ry of dif­fe­ren­tiab­le ma­ni­folds and -‌geo­me­try.
 Another piloting example is given in terms of a skew invertible  2n×2n  matrix  A  ( or such en­do­mor­phisms ). Then row × A × column is a symplectic bi­li­near form on 2n with respect to ma­trix mul­ti­pli­ca­tion ×. Often this is taken for definition of symplectic vec­tor spaces. But this is mis­lea­ding be­cause even in phy­sics there are ba­ses not of this form, for in­stance the real vec­tor spa­ces of the com­plex Pau­li-, Di­rac- and Duf­fin-Kem­mer-Pe­tiau ma­tri­ces in quan­tum me­cha­nics. In fact, ( to­ge­ther with the 2-di­men­sio­nal iden­ti­ty mat­rix ) the three Pau­li-ma­tri­ces, and these real vec­tor spa­ces of ma­tri­ces be­come Min­kow­ski spaces with res­pect to the ca­no­ni­cal sym­me­tric bi­li­near form
≪A,B≫  =  ½ ( trace(AB) − trace(A) trace(B) )
for ( endomorphisms or ) square ma­tri­ces A and B. Because of this we would prefer on Lie al­ge­bras to take for canonical form the negative of the Killing form and generalize this to any ca­te­go­ry of al­ge­bras. But what then holds for Jor­dan al­ge­bras?
not to be
mixed up
with the
general case
BasesDarboux's theorem guarantees the existence of symplectic bases on  EEand its dual in the form
 q1, ... , qn, p1, ... , pn ∈ EE ,  σq1, ... , σqn, σp1, ... , σpn ∈ σEE ,
 
σ<qi,qk> = 0 ,   σ<qi,pk> = ઠik ,  σ<pi,pk> = 0∀ i,k = 1, ... , n ,
where all basis vectors are dimensionless, since physical dimensions are attached to the real co­ef­fi­cients. But they are useless, in­fact if there are formulations involving co­or­di­nates, the prob­lems can't be important. Even the three basic dynamical systems of clas­sical resp. quan­tum me­cha­nics, the  free par­tic­le, the  har­mo­nic os­cil­la­tor  and the  (rigid) ro­ta­tor  can be for­mulated en­tire­ly in terms of the pre­cee­ding or­tho­go­nal vec­tor space examp­le, even if some per­tuba­tion terms of high­er or­der are add­ed. To con­fuse things, one can drop the pre­in­dex  σ  for the dual ba­sis and move the in­dex up resp. down in­stead. But then there is the prob­lem, that one has to do the con­verse for the real co­ef­fi­cients.
The construction of such a basis is by induction: Take a vector  q1≠ 0 . This im­plies that the di­men­sion of  EEis two or higher. Since σ is non-de­ge­ne­rate there is a vec­tor y such that  σ<q1,y>≠ 0 . Then for the vec­tor  p1 = y / σ<q1,y>  we get  σ<q1,p1> = 1  ( ex­actly here we could get a mi­nus sign in­stead for use in spe­cial re­la­ti­vi­ty's Min­kow­ski space ). Here q1 and p1 are li­ne­arly in­de­pen­dent by con­struc­tion and there­fore span a two‌-‌di­men­sio­nal sub­space  EE1of EE.For any vec­tor x ∈ EEtri­vial­ly we have
( ⊞ )x  =  x − ( σ<x,q1>p1− σ<x,p1>q1 )  + ( σ<x,q1>p1− σ<x,p1>q1 )
and the proof consists in showing that the plus makes a σ-ortho­gonal sum  ⊞ . For this we use (⊥) and theorem (iff) above. This implies that the re­stric­tion of the sym­plec­tic form on both sub­spaces is sym­plec­tic again ( only the non‌-‌de­ge­ne­rate property is not clear; in fact, it is easy to prove on the li­near span  EE1of q1 and p1, by in­ser­ting spe­cial y's; for the re­stric­tion of σ to the com­ple­men­ta­ry space of  EE1one has to use in the con­di­tion of non‌-‌de­ge­ne­rate­ness, the σ-or­tho­go­na­li­ty (⊥) and the non‌-‌de­ge­ne­rate­ness of σ on the whole space ). There­fore we can re­peat this con­struc­tion n times ( if n>2, for n=2 eve­ry­thing is done ) to get the above ba­sis of ca­no­ni­cal co­or­di­nates.
 In () the projector
P1 : x  ι σ<x,q1>p1− σ<x,p1>q1,P1 : EE →  EE ,P1 ∈ sp+(EE,σ)
projects onto the subspace image(P1, span­ned by q1 and p1. Its complementary projector  idE−P1 projects on­to its com­ple­men­tary sub­space kern(P1) of di­men­sion  dim EE− dim image(P1),here equal to 2n−2, such that
( )EE =  linear span of q1 and p1   kern(P1) .
is a σ-orthogonal direct sum decomposition. Repeating this n-times we get such a de­com­posi­tion with n non-trivial pro­jectors onto 2-di­men­sional sym­plec­tic sub­spaces sub­ject to
P1+ ... + Pn =  idE,PiPk = 0  if  i ≠ k ,σ<Pkx,y> = σ<x,Pky>∀ i,k=1,...,n .
However, a Lagrangian subspace is not given by a sum of these projectors, for them one has to add up non-self-ad­joint projectors of the type Pp,q.
Applying a symplectic group automorphism to these coordinates, the transformed co­or­di­nates are of the Dar­boux type ( i.e. canonical ) again. It remains to show, that there is on­ly one equi­va­len­ce class of such ca­no­ni­cal coordinates under the action of the sym­plec­tic group.
In case of introduction of a Darboux basis one gets matrix representations and often writes for the sym­plec­tic matrix group SSp(2n,)resp. ss p(2n,)for the symplectic matrix Lie algebra, not to be mi­xed up with the ge­ne­ral case of an arbitrary sym­plec­tic vector space.
are of no use
The
Symmetric
Algebra
of
Polynomials
Thereon
 The vector space structure on the dual is taken from the following definition of point­wise: For real‌-‌va­lued func­tions on EE,which are the observables of classical me­chan­ics, most­ly ta­ken in­fi­nite times differen­tiable from  ℭ(EE),point­wise addition  + , scalar mul­ti­pli­ca­tion and mul­ti­pli­ca­tion  ⋅  are de­fin­ed by
(pw)(αf+βg)(x) = α f(x) + β g(x),(f⋅g)(x) = f(x) g(x),α,β ∈ ,  x ∈ EE ,
which makes this set of real-valued functions on EEan infinite dimensional, com­mu­ta­tive, as­so­cia­tive real al­ge­bra. Cer­tain­ly the polynomials, i.e. linear combinations of  mo­nomi­als of power m  of the form
σx1⋅ ⋅⋅⋅ ⋅σxm,σxk ∈ σEE ,
and even some rational functions on EE,are contained in this al­gebra. Elements of pow­er 0 are defined as the real numbers, multiplied with the neutral element 1 of this as­so­ciative al­gebra as the constant map  x ι1 , which usu­al­ly is drop­ped. Thus we get the real infinite-di­men­sion­al, po­wer‌-‌graded, asso­cia­tive, commutative  sym­me­tric al­gebra  sym(σEE)of poly­no­mi­als over EE
(sym)  sym(σEE) =  ⋅1  ⊞   σEE 1 ⊞   σEE 2  ... ⊞   σEE m  ⋅⋅⋅
with  ⋅1 = σEE o, σEE 1= σEE , σEE 2= σEE⋅σEEand so on.
Mathematically this algebra can be enlarged by a standard procedure of taking quotients to be­come the  al­ge­bra of real rational functions on EE
    at( EE ) =  quotient( symσEE )=  quotient( polynom( EE ) ) .
However, already one of the most basic dynamical systems of mechanics, the harmonic os­cil­la­tor, needs an­other ex­ten­sion of this algebra, since its Hamiltonian contains square roots.
 The symmetric algebra  sym(EE)usually is defined in an universal way, by fac­tor­ising the  ten­sor al­ge­bra  ten(EE)over EE[ Bou ] by a two-sided ideal of the form  ((f⊗g − g⊗f))  to get a com­mu­ta­tive fac­tor al­gebra of the ten­sor al­ge­bra. This al­ge­bra can be written in the form
sym(σEE) =  symo(σEE) ⊞  sym1(σEE) ⊞  sym2(σEE) ⊞  …  ⊞  symm(σEE) ⊞  …
with exactly the same subspaces of power m as in (sym)
symm(σEE)= σEE m ,dim(σEE m ) = ( m −1+ dim(EE)over  m ) .
The above pointwise way has leapfrogged the uni­versal construction by giving a di­rect and ra­ther sim­ple approach.

In the following we are interested in the elements of first, second and third power, i.e. ele­ments of the dual in the form  σx  and of the form  σx⋅σy , σx⋅σy⋅σz, and we keep the mid­dot, which in most pre­sen­ta­tions is dropped.
((f-g)) =
{ h⊗(f-g)⊗k
/
all elements in the
ten­sor al­gebra }




 sym(σEE) ⊏ ℭ(EE)
Directional
Derivatives
and
Gradients
To get to Poisson brackets we have to start with the directional derivative: A real valued func­tion f on EEis said to be  differentiable at x in the direction of u  if for  τ∈ a map [ Kch I 6]
1
(∇)u  ι ∇f(x) u  =lim
( f( x + τ u ) − f( x ) )
 τ → 0 τ
exists. Hence  (∇f )(x) = ∇f(x)  is an element of  σEE, i.e. a linear form on  EE,but

∇f : x  ι (∇f )(x)   ,   ∇f : EE →  σEE

not necessarily is linear. Then from Witt's theorem for linear and non-degenerate bi­linear forms the­re is a uni­que vec­tor  σgrad f (x) ∈ EE such that
(grd)(∇f )(x) u = σ< σgrad f (x) , u >
which is called the  gradient of the function f at the position x. In the following we of­ten drop the pre­in­dex σ in the no­ta­tion of the gra­dient. Note that the­re is a pro­duct‌- and a chain rule for di­rec­tio­nal de­rivatives and gra­dients from which we get the distributive law
(∇1)grad(f⋅g) (x) = f(x) grad g(x) + g(x) grad f (x)  ∀ f,g ∈ ℭ(EE) .
Max Koecher discusses product- and chain-rule and their implications for gradients [ Kch I 6]. Es­pe­ci­ally, if the gradient is taken with respect to a symplectic bilinear form, div grad va­ni­shes iden­ti­cal­ly, which means that there is no concept of Laplace- and quabla-operator. Hence the theo­ry of 2‌nd or­der par­tial differential equa­tions is much poorer than in the case of sym­me­tric bi­li­near forms.
 ∇f(x)  = ∇xf

σgrad f (x) 
= σgradxf 

  σℙ(f,g)(x) 
  = σx(f,g)


are
equivalent
notations
Examples
for
Directional
Derivatives
Here it is convenient to calculate examples:
  • 0-differentials are given by differentiation the above constant map 1: Clearly the right hand side of the de­fi­ni­tion vanishes, hence we get the 2n-dimensional ge­ne­rali­za­tion of the one-di­men­sio­nal re­sult zero and the gradient becomes the null vector.

  • Linear forms: Given the linear form  f(x) := σy(x)  we get  ∇σy (x) = σy , i.e. the con­stant map, and  grad σy (x) = y  for all x.

  • Quadratic forms: Given  f(x) := (σy⋅σy)(x)  we get  ∇(σy⋅σy)(x) = 2σ<y,x>σy  i.e. a linear form on EEand for the gradient the vector  grad(σy⋅σy)(x) = 2σ<y,x>y ∈ EE.

  • Cubic forms: Given  f(x) := (σy⋅σy⋅σy)(x)  we get by a tedious, but straight­for­ward cal­cu­la­tion which is left to the reader.
Note that one can switch easily from quadratic to bilinear, resp. from cubic to tri­linear, by po­la­ri­zing, i.‌e. by sub­stituting y + z for y, applying then these formulas.
are easy to
evaluate
in the
basis-free way
Poisson
Brackets
Gradients lead to Poisson brackets by the definition [ Sou p 86]
σℙ( f , g )(x) = σ< σgrad f (x) , σgrad g (x) > ,
where again, since there is only one σ, the prefix is and will be dropped in the fol­low­ing. It is clear that the Pois­son bracket is skew in f and g, and well-known that it fulfills the Jaco­bi‌-‌iden­ti­ty ( from the chain rule ). Since more­over it is distributive with res­pect to the real num­bers, it de­fines a real Lie al­gebra struc­ture on the in­fi­nite-di­men­sio­nal li­near space of dif­fer­en­tiab­le real‌-‌va­lued func­tions on EE,and in­side there the po­ly­no­mials of sym(σEE)are a Lie sub­al­ge­bra. Most im­por­tant is that it al­so has the dis­tri­bu­tive pro­per­ty with res­pect to the point­wise mul­ti­pli­ca­tion ⋅ , which is a di­rect con­se­quence of that for gradients ( from the pro­duct rule ):
(∇2)ℙ(f⋅g, h) = f ⋅ℙ(g,h) + ℙ(f,h) ⋅g   ∀ f,g,h ∈ ℭ(EE) .
Specialising to  f = σx  and  g = σy  and inserting the above example (L) into the defini­tion of the Pois­son bra­cket we get the famous canonical relations
(PoiB)ℙ( σx , σu ) = σ<x,u>   ∀ x,u ∈ EE
of classical mechanics ( we didn't write explicitely the con­stant map from  EEon­to the real num­ber 1 on the right hand side ). It defines the Heisenberg Lie al­gebra of di­men­sion 2n+1 on EE ⊞ ℝ.For proof one has to insert the gra­dients of the special case (O) above in­to the de­fi­ni­tion of the Pois­son bra­cket. It was stu­died in [ Til ] as a nil­po­tent sub­al­gebra of a sol­vab­le  spec­trum gene­rat­ing ( but not in­va­ri­ance )  Lie al­ge­bra for 2‌nd po­wer‌-‌Ha­mil­to­nians. Its im­por­tance in Har­monic Ana­ly­sis is des­cri­bed in [ How ]. For an­other des­crip­tion of clas­si­cal me­cha­nics see [ God ].
 As is seen from these Poisson bracket relations, the infinte-dimensional (polyno­mi­al) Lie al­ge­bra  ( sym(σEE) , ℙ ) is  power-graded , i.e. for a pair of natural numbers  i, k
(pg)ℙ( σEE i , σEE k )σEE i + k − 2,
where the powers are taken with respect to the pointwise multiplication. (PoiB) is the special case with i = 1 and k = 1, in fact the case of the power generating linear forms of  sym(σE). Its a litt­le bit te­dious to write this explicitely for ele­ments
(pg) ℙ( σx1σxi , σu1σuk)  =  σ<xj,ulσx1σxj -1σxj +1σxiσu1σul -1σul +1σuk
with the sum over  j = 1, ..., i , l = 1, ..., k . Here (∇2) and the commutativity of the point­wise mul­tipli­cation is used frequently. There are special cases which describe some facts of the sym­plec­tic Lie al­gebras. But since we get cor­res­ponding algebraic equa­li­ties in the Weyl al­ge­bras ( which is the central problem here ), those will be discussed below.
the
universal approach
to
symmetric algebras
is not needed here
Weyl
Algebras
Weyl algebras are a non-commutative version of the symmetric algebras with res­pect to a sym­plec­tic vec­tor space as above. They were studied in detail in the two papers [ Til ], which we follow here. Its the uni­ver­sal en­ve­lop of the Hei­sen­berg Lie al­ge­bra, which was rea­lized above by Pois­son brackets. But since the cen­tral ele­ment 1 on the right-hand side of (PoiB) is not the iden­ti­ty ele­ment  1ue of that infinite dimen­sio­nal, as­so­cia­tive, power-graded, but non‌-‌com­mu­ta­tive al­ge­bra it must be identified with it. This is done by fac­tor­ising the uni­ver­sal en­ve­lop by a two-sided ideal of the form ((1ue − 1)). Then one has the Lie bracket, especially the Pois­son bra­cket, realized as a com­muta­tor  [ , ] of an asso­cia­tive mul­ti­pli­ca­tion, but in the case of Pois­son bra­ckets this is not rea­li­zed by de­ri­va­tives resp. gra­di­ents.
Thus one gets the famous canonical commutation relations of quan­tum mecha­nics ( in quan­tum field theo­ry those of Bose-Einstein crea­tion / annihila­tion ope­ra­tors ), reproducing the Pois­son bra­ck­et re­la­tions (PoiB) above,
(CCR)[ x , u ] = xu − ux =  σ<x,u> 1∀  x,u ∈ EE ,
where the non-commutative, associative multiplication in the Weyl algebra is writ­ten with­out a sign, 1 is its neu­tral ele­ment, and x,u ( and y,v in the following ) are taken from the em­bed­ding of  EEin­to the Weyl algebra: De­fine
0EE 1 ,1EE= EE ,

2EE{ linear combinations of  1/2!( xy+yx ) } ,

3EE{ linear combin. of  1/3!( xyz+zxy+yzx+xzy+yxz+zyx ) } ,

4EE{ linear combin. of  1/4!( all 24 permutations of x,y,u,v ) } ,
and so on for higher powers, the factor in front of the sum of permutations being  1 / m!  in the case of a more general monomial x1…xm. Let us write  Λx1…xm  for the total­ly sym­me­tri­zed de­fin­ing ele­ments in the subspace of totally symmetrized mo­no­mials [ Til ]. Then
weyl(EE,σ)  =   1    1EE   2EE  ...   mEE  ... .
is a direct sum decomposition, which even is σ-orthogonal, whenever this skew bi­li­near form can be lif­ted to the whole algebra in an appropriate way.
contrary to the
case of
Poisson brackets
we have to use a
universal concept
here
Commutation
Relations
Commutation relations for the Heisenberg- and symplectic Lie al­ge­bras can be for­mu­la­ted with the help of sym­metric and Weyl al­ge­bras in a basis-free and in­va­ri­ant, hence ele­gant way [Til] for x,y,z,u,v ∈ EE
Λxy  =  1/2! ( xy + yx )  =  1/2( (x+y)2 − x2 − y2) ,

Λxyz   =  1/3! ( xyz + zxy+ yzx+ xzy+ yxz+ zyx ) ,

Λxyuv = 1/4!( xyuv+xvyu+xuvy+xyvu+xuyv+xvuy  +  yxuv+yvxu+yuvx+yxvu+yuxv+yvux
+ uxyv+uvxy+uyvx+uxvy+uyxv+uvyx  +  vxyu+vuxy+vyux+vxuy+vyxu+vuyx ) ,
where the construction implies that no two terms coincide. Below we need the spe­cial case  y = x  and  v = u
3! Λxxuu  =  x2u2 + xu2x + u2x2  +  xuux + xuxu + uxux ,
where the last three elements easily can be brought to the form of the first three ones, us­ing (CCR) and   uxxu+xuxu+uxux = xxuu+xuux+uuxx − σ<x,u>2 . Inserting we get
3 Λxxuu  =  x2u2 + xu2x + u2x2  − ½ σ<x,u>2 .
We need this below to disprove the isomorphism property, which entirely is due to the ex­isten­ce of the coda  ∈0EEin this equation behind the minus sign.
One easily derives the following commuta­tion re­lations, which hold exact­ly in the same way if the com­mu­ta­tor  [ , ] is substituted by the Poisson bracket
(SRp)   [ Λxy,u ] = σ<x,u> y + σ<y,u> x

(CRs)  [ Λxy,Λuv ] = σ<x,u>Λyv + σ<y,u>Λxv + σ<x,v>Λyu + σ<y,v>Λxu .
The second shows that 2EEis a Lie subalgebra in the  (n+1)(2n+1) -dimensional Lie al­gebra of ele­ments up to the second power, and the first shows that it is iso­mor­phic to the n(2n+1)-di­men­sional Lie al­gebra sp(EE,σ).To see this write as usual for the Lie al­ge­bra­ic left mul­ti­pli­ca­tion
(ad)ad( X ) Y  =  ℙ( X, Y ),ad( X ) Y  =  [ X, Y ]
for elements  X,Y  out of the symmetric resp. the Weyl algebra. In fact (SRp) shows that these adjoint re­pre­sen­ta­tions are in  sp(EE,σ),when ad is restricted to EE.Then one proves that the ad?(Λxy) have the same com­mu­ta­tion re­lations as the Λxy them­selves. Since these gene­rate the spaces of second power linearly, the Lie al­ge­bras are iso­morphic [ Til ].
Clearly the ad? restricted to EEare the self-, restricted to  EE⋅EEresp. to 2EEthe ad­joint re­pre­sen­tations of the symplectic Lie al­gebra. Both are irreducible.
Note that it suffices to verify the shorter commutation relations instead
(SRp)  [ Λxx , z ] = 2 σ<x,z> x

(CRs)[ Λxx , Λzz ] = 4 σ<x,z>Λxz ,
since by polarizing  x ιx+y  and  z ιz+w  one gets back the more general relations. But this some­times is more tedious than the direct verification!
the same
algebraic relations
hold for classical
Poisson brackets
and
quantum mechanical
Weyl algebras


but only if
one of the entries
in the
commutation relations
is of power
less than three
Higher Order
Polynomials
and the
Isomorphy Problem
Commutation relations for (symmetrized) monomials of higher power than two can easi­ly be for­mu­la­ted in a ba­sis-free way. Define for arbitary monomials of power m a linear map
Λ : σx1⋅ ⋅⋅⋅ ⋅σxm ιΛ x1…xm,Λ : symm( σEE )mEE
which preserves the power-grading. Continuing this linearly to the whole al­ge­bras, we get an iso­mor­phism of in­fi­nite dimensional, power-graded vector spaces, but also of Lie al­ge­bras? This is the main problem of this ar­tic­le. But this does not follow from uni­ver­sa­li­ty, since the Pois­son bra­cket Lie struc­ture on the sym­me­tric al­ge­bra has nothing to do with its as­so­cia­tive mul­ti­pli­ca­tion. In fact this as­so­cia­tive algeba is com­muta­tive, where­as the Weyl al­gebra is not! Hence for the Lie algebras this must be dis­proved di­rect­ly. For this note, that the distri­bu­tive law (∇2) holds in the Weyl algebra as well, but is destroyed by the sym­me­tri­za­tion pro­ce­ture there­in. How­ever, here the polari­za­tion trick helps.
The main conjecture on Poisson and quantum Lie brackets is: The linear map
Λ : ( sym( σEE ) , ℙ ) → ( weyl( EE,σ ) ,  [ , ] )
is no isomorphism of infinite dimensional power-graded Lie algebras.
Remark: But even if isomorphism could be proved, quantum mechanics would be more in­volved ( even in the simple case of poly­nomial Hamilton­ians ) since physical re­pre­sen­ta­tions must be self-adjoint, i.e. in Hilbert spaces, which runs into severe domain questions for in­fi­nite di­men­sional representations.
The Λ-isomorphism condition for monomials of power i resp. k becomes
[Λx1...xi , Λu1...uk ] =  σ<xj,ulσx1σxj -1σxj +1σxiσu1σul -1σul +1σuk

[Λ x1...xi ,Λu1...zk ]  =   σ<xj,ul> Λx1⋯xj -1xj +1⋯xi u1⋯ul -1ul +1⋯uk
by evaluating the isomorphism Λ on both sides with j,l in the summation as in (pg) above. This still is an awful identity to prove, especially since we don't know any proof of an in­duc­tion theo­rem with two running indices. Note that this condition gives (SRp) and (CRs) for the above spe­cial cases. Identifying all the x and all the u we get the much simpler con­di­tion for powers

(nc)[ xi , uk ] = i k σ<x,u> Λ(xi -1uk -1) ,

which again gives the simpler forms of (SRp) and (CRs) for the above special cases. This is a ne­ces­sa­ry con­di­tion for Λ to be an isomorphism of Lie algebras.
Let us note more equations, where the first implies the second by polarization
(ad23)  [xx , Λuvw]  =  2 ( σ<x,w>Λxuv + σ<x,v>Λxuw + σ<x,u>Λxvw )

(ad23)[Λxy , Λuvw] =  σ<x,w>Λyuv + σ<y,w>Λxuv  +
ad23σ<x,v>Λyuw + σ<y,v>Λxuw  + σ<x,u>Λyvw + σ<y,u>Λxvw ,
which shows that like for Poisson brackets, also monomials of power three are closed un­der left ad-ac­tion by ele­ments of the sym­plectic Lie al­ge­bra, i.e. these mono­mi­als too span a re­pre­sen­ta­tion mo­dule of the sym­plec­tic Lie al­ge­bra ( a simple, but leng­thy veri­fica­tion gives the same result for power four monomials ). For dy­na­mical sys­tems in phy­sics this means, that as long as a Hamilton­ian has its in­vari­ance Lie al­ge­bra in the sym­plec­tic Lie al­gebra there is no al­ge­bra­ic dif­ference in clas­si­cal and quan­tum re­ali­za­tions.
The corresponding equations can be shown to hold for power four polynomials, which gives rise to the con­jec­ture, that there is an induction proof of this result for all powers.
In addition one proves in the same way, that for monomials of power one the left ad‌-‌ac­tion is a de­ri­va­tion ( as for any Lie al­gebra ) which lowers the po­wer by one. So the ques­tion ari­ses, whe­ther the re­sult (pw) is valid in the whole Weyl algebra. To in­vesti­gate this com­pu­ter prob­lem one has to study
[Λxyz , Λuvw] =  linear combination of 4 monomials ( and of lower powers? ) ,
where lower powers on the right hand side would destroy the isomorphism proper­ty in ques­tion. Since we are looking for a negative result it suffices to treat the special case
(MR)   [Λx3, Λu3]  =  [x3,u3]  =  9 σ<x,u>Λxxuu  ⊞  3/2 σ<x,u>3 .
The proof, first given by the well-known mathematician Brute Force, consists in using the dis­tri­bu­ti­ve law and (CCR) for the left hand side to get
[Λx3, Λu3]  =  [x3,u3]  =  3 σ<x,u> ( x2u2 + xu2x + u2x2 ) .
The three terms in this bracket on the right hand side where found above in  Λxxuu  be­fore the coda.
is
quantization
a
representation process
of different,
or only one of
different representations
of the same
algebraic structure?
A
Generalization
Let us assume a slighlty more general point of view, in which the footprint of the po­wer-gra­ded map in σEEis symplectic. Instead of considering the linear map Λ, take a more ge­ne­ral li­ne­ar map
Γ : σx1⋅ ⋅⋅⋅ ⋅σxm ιΓ( Λ x1…xm ),Γ : symm( σEE )mEE
preserving the power-graduation given by the symmetrization. The condition for Γ to be al­so a Lie al­gebra iso­mor­phism is on σEE
[ Γ(σx) , Γ(σu) ] = Γ( ℙ(σx,σu) )    σ<Γ(σx),Γ(σu)> = σ<x,u> Γ(1) ,
which means that Γ restricted to σEEis a symplectic map and after identification of the un­der­ly­ing sym­plec­tic vec­tor spaces even an element of the symplectic group. We may write Γ(σx) =: Gx for some  G ∈ SSp(EE,σ).Universality of the Weyl algebra says that there is a uni­que au­to­morph­ism ex­ten­ding this G to the whole algebra. This is given by
G • Λx1 ⋅⋅⋅ xn = ΛGx1 ⋅⋅⋅ Gxn ,
i.e. Γ commutes with symmetrization. Applying this to (MR) we get a contradiction – the coda is there for commutators but not for Poisson brackets. Hence there is no iso­mor­phism be­tween the­se two infin­ite-di­men­sional Lie al­ge­bras which pre­ser­ves the power-gra­dua­tion gi­ven by sym­me­tri­za­tion.
Naturally one can try to find other graduations, even given by po­wers, involving low­er or high­er po­wers. But such gra­duations must coincide with symmetrization up to mo­no­mials of se­cond po­wer, be­cause in the ten­sor al­ge­bra every se­cond po­wer monomial can be written as a sum of a an­ti­sym­me­tri­zed and a sym­me­tri­zed mo­no­mi­al and pro­jection on­to the Weyl al­ge­bra iden­ti­fies the first one with a multiple of the unit element. So the re­main­ing sym­me­trized mo­no­mi­al sur­vi­ves as gott­ge­ge­ben. But even for high­er po­wer mo­no­mi­als such gra­dua­tions are high­ly un­natur­al: It is clear that for k ∈ ℕ
Λx1…xk  =  1/k ( x1Λx2⋅⋅⋅xk + + xkΛx1⋅⋅⋅xk-1 )
is the rule to form symmetrized monomials of power k by means of a symmetrization of tho­se of power k-1. This was used above to write down those of power up to four. In words: The ac­tion of the per­mu­ta­tion group of k objects is given by that of the per­mu­ta­tion sub­group of k-1 ob­jects, fol­lo­wed by sym­me­tri­zation. Substituting this last symmetrization for something else is un­na­tu­ral.
So the mathematical meaning of the canonical commutation relations is that to­tal sym­me­tri­za­tion of po­ly­no­mials is the natural quantization, which usually is attributed to Hermann Weyl. Non-equi­va­lent as­so­cia­tions of ordering polynomials are un­ma­the­ma­ti­cal. Especially in­tro­du­cing ba­ses is mis­lea­ding. This can­not be seen if one uses ba­ses for the ca­no­ni­cal com­mu­ta­tion re­la­tions!
This corresponds to the physical meaning of the canonical commutation relations, which are Hei­sen­bergs un­cer­tain­ty re­lations and their experimental implications in the tun­nel ef­fect.
is
there any
Lie algebra
isomorphism
of classical
and quantum
polynomial
observables?

NO !

at least
if it preserves
the power-graduation
An
Infinite
Series of
Symplectic
Representations
As mentioned above, the subspaces of symmetrized elements of  weyl(EE,σ)are clos­ed un­der the left ad-ac­tion of 2EE,i.e. of the sym­plectic Lie al­ge­bra sp(EE,σ) .This can be di­rect­ly cal­cu­la­ted for the powers three and four, in which cases we find no coda, but must be pro­ved by in­duc­tion for ar­bitrary mEE .
However, it is very unlikely that, although the self- and the adjoint representations are ir­re­du­cib­le, this also holds for higher power representations. A series of finite, but ever in­crea­sing di­men­sions for this stan­dard real simp­le Lie al­ge­bra of series C is un­like­ly. On the other hand, if they were re­du­cib­le there must be a struc­ture theo­ry for these re­pre­sen­ta­tion mo­dules of po­wer m, to be formulated in terms of pro­jectors and Clebsch-Gor­don co­effi­cients and in­va­ri­ant un­der the action of the sym­plec­tic Lie al­ge­bra and - group.
This prob­lem doesn't exist for the pseudo-orthogonal counterpart ( the stan­dard sim­ple Lie al­ge­bras of se­ries B and D ), given by the exterior = Grass­mann and the Clif­ford al­ge­bra, with the ex­te­rior algebra cor­res­pon­ding to the sym­me­tric, and the Clif­ford to the Weyl al­ge­bra. The­se two al­ge­bras have the same finite di­men­sions, hence there on­ly is a fi­nite num­ber of re­pre­sen­ta­tion mo­du­les ge­ne­rated by powers, making the quest for ir­redu­cibili­ty more likely.
irreducibility
is
unlikely

at least for
powers
higher than three
Lagrangian
Subspaces
and
ggℓ(n,)
Since in this section we don't use monomials of higher than second power, every­thing re­mains true, if we con­si­der Pois­son bra­ck­ets in the sym­me­tric alge­bra over EEinstead.
Given two Lagrangian subspaces LL1and LL2of EE,we have the following theorem
(L≠)LL1 ∩ LL2{0}     LL1 = LL2.

 (L=)LL1 ∩ LL2 = {0}    LL1 LL2 = EE ( direct sum )
and in this case there is a Darboux basis, such that

LL1 is the linear span of the q's and LL2 that of the p's .
The existence of the second statement is clear from Darboux' construction, although a direct proof is as involved as that of the first statement. In the light of this theorem write
LL1=: LLq   and   LL2 =: LLp,
to get a Lagrangian or canonical decomposition
(Ld)x = xq + xp  ∈  EE= LLq LLp.
of EE.Note that both subspaces are not symplectic and the direct sum cannot be σ-or­tho­go­nal. Con­verse­ly every element in one has a canonical conjugate in the other.
Moreover, in the latter case there is an involutive automorphism Jqp of EEinter­chang­ing the q's and p's mu­tu­al­ly. This involution as well should be called Lagrangian or ca­noni­cal. It re­mains to find under which con­di­tion an involution con­verse­ly de­ter­mines a La­grang­ian de­com­posi­tion and that a Lagrangian involution is self-ad­joint but not symplectic.
Lagrangian subspaces are Cartan ( i.e. maximal abelian ) sub­al­ge­bras of the Hei­sen­berg Lie al­ge­bra and (therefore?) give rise to ½n(n+1)-dimensional Cartan sub­alge­bras in the sym­plec­tic Lie algeba 2EE .
In the fol­low­ing write
LLqq{ linear span of  ½( xqyq+yqxq ) } = { linear span of  xqyq} ,
LLqp{ linear span of  ½( xqyp+ypxq ) } ,
LLpp{ linear span of  ½( xpyp+ypxp ) } = { linear span of  xpyp} ,
where only in the second mixed subspace LLqp the generating elements x and y do not com­mu­te and there­fore do not collapse to a single one. In this notation we have
(Ld2) Λxy  =  xqyq  +  Λxqyp+Λyqxp  +  xpyp  ∈  LLqq   LLqp   LLpp ,
a direct sum decomposition which extends to a direct sum of Lie subalgebras: Specia­liz­ing in (CRs) we get the com­mu­ta­tion re­la­tions
(CRgl)[ Λxqyp,Λuqvp ] = σ<yp,uq>Λxqvp + σ<xq,vp>Λypuq ,
(CRgl)[ Λxqyp,Λuqvq ] = σ<yp,uq>Λxqvq + σ<yp,vq>Λxquq ,
(CRgl)[ Λxqyp,Λupvp ] = σ<xq,up>Λypvp + σ<xq,vp>Λypup ,
(CRgl)[ Λxqyq,Λupvp ] = σ<xq,up>Λyqvp + σ<yq,up>Λxqvp + σ<xq,vp>Λyqup + σ<yq,vp>Λxqup .
The three subalgebras certainly are not Lie algebraic ideals - since the symplectic as a simple Lie algebra has none: In fact in 2EEthese commutation re­la­tions mean
[ LLqq,LLqq] = [ LLpp,LLpp] = {0},  [ LLqp,LLqp] ⊂ LLqp,

[ LLqp,LLqq] ⊂ LLqq , [ LLqp,LLpp] ⊂ LLpp , [ LLqq,LLpp] ⊂ LLqp .
The two Cartan subalgebras LLqqand LLppleave exactly from the total n(2n+1) di­men­sions of the symplectic Lie algebra of second power monomials unaccounted for. To show that the com­muta­tion re­la­tions (CRgl) are those of  ggℓ(LLq,)we have to spe­cia­lize the com­mu­ta­tion re­la­tions (SRp) to this decom­posi­tion into La­grangi­an sub­spaces
(SRgl)[ Λxqyp,uq ] = σ<yp,uq> xq  i.e.  [ LLqp,LLq ] ⊂ LLq,

(SRgl)[ Λxqyp,up ] = σ<xq,up> yp  i.e.  [ LLqp,LLp ] ⊂ LLp.
These commutation relations show that we have five representation spaces for the ge­ne­ral li­ne­ar Lie al­gebra  ggℓ(LLq,),the isomorphism being given by the re­stric­tion of the ad­joint re­pre­sen­ta­tion ad to LLqresp. LLpand the three subspaces of second power in the Weyl al­ge­bra.
 (CRgl) and (SRgl) are not the most economic ways to write ggℓ(LLq,)as a Lie algebra struc­ture. To do this in an unpolarized version, use the canonical involution J ( drop­ping the in­dex qp ), gi­ven above: 
(CRgl)[ ΛxqJxq,ΛuqJuq ]=σ<Jxq,uq>ΛxqJuq + σ<xq,Juq>ΛJxquq ,
(SRgl)[ ΛxqJxq,uq ]=σ<Jxq,uq> xq .
Altogether, by (CRgl) we got elegant basis-free representation of the com­muta­tion re­la­tions of the gene­ral li­ne­ar Lie al­gebras. Inside these Lie al­gebras there is a cen­tral ele­ment map­ping by the res­tric­tion of ad on­to the trace times the iden­ti­ty, such re­pre­senting even the sl(n,) Lie al­ge­bras of trace­less elements.
From the Darboux construction it follows, that two Lagrangian subspaces are sym­plec­ti­cal­ly equi­va­lent, which gives on­ly one iso­mor­phy class of the general and special linear Lie al­ge­bras.
have
group and
Lie algebraic
implications
Higher
Powers
These constructions generalizes to arbitrary powers m in mEEif we find the higher di­men­sion of these sub­spaces of mo­no­mi­als in the Weyl al­ge­bra. Certainly the concept of a Car­tan sub­al­ge­bra can be over­ta­ken li­teral­ly - there are two of them, spanned by pure mo­no­mials. What turns out dif­fer­ent­ly are the sub­spaces of mixed mo­no­mials, which no longer can be Lie sub­al­ge­bras. But they re­main re­pre­sen­ta­tion spaces of the above Lie al­ge­bras in the sense dis­cussed above. It re­mains to show, that these mixed re­pre­sen­ta­tion spa­ces con­tain no more abe­li­an Lie sub­al­ge­bras.have a
structure theory
resembling
that of
simple Lie algebras?
The
Killing Form
of a
Symplectic Lie Algebra
Weyl algebras give an elegant expression of the Killing form of the sym­plec­tic Lie al­ge­bra ( Λ2EE , [,]-)en­tire­ly in terms of the symplectic structure σ of the underlying sym­plec­tic vec­tor space. De­fine a bi­linear form κil on the space of symmetrized elements of 2‌nd po­wer in the Weyl al­ge­bra by
(κ)κil‹ Λxy,Λuv ›  =  σ<x,u> σ<y,v> + σ<x,v> σ<y,u>
and linear continuation to the whole of 2EE.Then it is straightforeward to show, that this bi­li­ne­ar form is sym­me­tric, non-degenerate and invariant under the left ad-action, i.e. that
(inv)κil‹ ad(Λwz) Λxy,Λuv ›  + κil‹ Λxy,ad(Λwz) Λuv ›   =  0 ∀ x,y,u,v,w,z ∈ EE.
Since the symplectic Lie algebra is simple and for simple Lie algebras every invariant sym­me­tric non-de­ge­ne­rate bi­linear form is the Killing form up to a multiplicative constant, (κ) is that re­mar­kab­le simple closed form of the Killing form.
Generalizing (κ) to arbitrary powers there arises the question, whether these bilinear forms, skew and sym­me­tric in­terchanging, are likewise symplectic and pseudo-orthogo­nal re­pre­sen­ta­tions of the symplectic Lie al­ge­bra.
It makes sense to name the 4-form ( with dropping all multiplicative factors, containing the di­mension of the underlying vector space - from taking traces explicitely in the defi­ni­tion of the Kil­ling form )
σ)κσ( x,y,u,v )  =  σ<x,u> σ<y,v> + σ<x,v> σ<y,u>
on EEthe  Killing form of the symplectic vector space. By construc­tion it is sym­me­tric in­ter­chan­ging x,y resp. u,v and the pair x,y by the pair u,v. Moreover (inv) translates in a fourth sym­me­try, in­vol­ving six vectors.
expressed by
σ
alone
The
Pseudo-Orthogonal
Alternative
The above notation was chosen such that dropping the letter σ one gets nearly the same re­sults and cor­res­pon­ding formulas for the pseudo-orthogonal ( i.e. non-dege­ne­rate sym­me­tric bi­li­near forms of some sig­na­ture +,...,+,−,...− ) case. Ac­tual­ly in both papers [ Til ] this was ful­ly ex­ploi­ted, with the Clif­ford al­ge­bra instead of the Weyl al­ge­bra uni­versally con­struc­ted above the pseu­do-or­tho­gonal vec­tor space and the Grassmann or exteri­or al­ge­bra in­stead of the sym­me­tric al­gebra.
Instead of symmetrizing the elements of these two real associative finite-dimensional uni­ver­sal algebras with unit ele­ment 1, we have to an­ti-sym­me­trize ( take the alterna­tive in­stead of the symme­tric group for for­ming power gra­duations ), for in­stance
0EE =  1,1EE = EEwith  dim EE = nnot necessarily even,

2EE = { linear combinations of 1/2!( xy−yx ) },

3EE = { linear combin. of 1/3!( xyz+zxy+yzx−xzy−yxz−zyx ) },

4EE = { linear combin. of 1/4!( all 24 alternations of x,y,u,v ) }  , ... ,
but there are no unpolarized versions here. Especially the Lie algebra of the anti-sym­me­tri­zed ele­ments of 2‌nd po­wer is the pseu­do-or­tho­go­nal Lie al­gebra, with its ( here ba­sis-free, well-known in the dusty index notation ) com­mu­ta­tion re­la­tions given in [ Til ]
(SRp)[ Vxy , u ]  = <x,u> y + <y,u> x

(CRs)    [ Vxy , Vuv ] = −<x,u>Vyv + <y,u>Vxv + <x,v>Vyu − <y,v>Vxu
for the self- and the adjoint representation of the pseudo-orthgonal Lie algebras. The Clif­ford al­ge­bra can be seen as the universal en­ve­lop of a Jor­dan al­gebra, de­fined by the ca­no­ni­cal an­ti-com­mu­ta­tion re­la­tions in terms of the sym­me­tric bi­li­near form  <,> 
(CAR)[ x , u ]+ = ½ (xu + ux) = <x,u> 1   ∀  x,u ∈ EE,
the neutral element of this Jordan algebra identified ( factorise by a two-sided ideal ) to the unit ele­ment 1.
However, there is a huge difference in the pseudo-orthogonal case - there is a  symmetric Pois­son bracket  here
<,>ℙ◀f,g▶ (x) = <<,>grad f (x)  , <,>grad g (x) > ,
but on the symmetric instead of the Grassmann algebra. Herein the  <,>-gradient is defined as in (grd) above. Without danger of confusion the sup-in­di­ces can be dropped. Like in the sym­plec­tic case above it is easy to verify ( for a moment we abbreviate the sup-index  <,> = τ )
(sPoiB)ℙ◀τx ,τu ▶ (z) = <x,u> 1 (z)∀ z ∈ EEτx,τu ∈ <,>EE ( the dual space ) ,
with 1 being the constant function which associates 1 to all x. So we get the (CAR). Likewise the right hand side vanishes if we insert a constant function in the left hand side. Therefore 1 is not the identity ele­ment of this algebra. It remains to show, whether this sym­me­tric Pois­son bracket  ℙ◀,▶ makes the sym­me­tric al­ge­bra a Jor­dan al­gebra ( using the chain rule for di­rec­tional de­ri­va­tives and gra­dients ), but since it is infi­nite‌-‌di­men­sio­nal there is no point in com­pa­ring it to the Clif­ford ( or Grass­mann ) al­ge­bra. There­fore it does not suit for a clas­si­cal ca­no­ni­cal theo­ry for fer­mi­ons. How­ever, one can try to set up a Lagrangian theo­ry ( mi­mi­cking the Ha­mil­to­nian one gi­ven by a sym­plec­tic struc­ture on a co­tan­gent bund­le ) in 2n di­men­sions, gi­ven by a tan­gent bund­le over a n‌-‌di­men­sio­nal (pseudo‌-)‌or­tho­go­nal struc­ture, re­pre­sen­ting con­fi­gu­ra­tion space, the tan­gent fib­res be­ing ve­lo­ci­ties.
Using this and the product rule for gradients, we get ( re­mem­ber that the point­wise mul­ti­pli­ca­tion of func­tions on EEis written without a symbol )
(sSRp)   ℙ◀τxτy,τu ▶ = <x,u> τy + <y,u> τx

(sACR)ℙ◀τxτy,τuτv ▶ = <x,u> τyτv + <y,u> τxτv + <x,v> τyτu + <y,v> τxτu .
The space of monomials of 2nd power in the symmetric algebra is ½ n(n+1) -, where­as the space of mo­no­mials of 2‌nd pow­er in the Grass­mann algebra is ½ n(n−1)-dimensional. Thus the ele­ments of 2‌nd pow­er are a Jor­dan sub­algebra with respect to this sym­me­tric Pois­son bra­cket, re­ali­sing the  pseu­do‌-‌or­tho­go­nal Jor­dan al­ge­bras  of <,>‌-‌sym­me­tric trans­for­ma­tions.
Instead one can try to lift the symmetric Poisson bracket to the Grass­mann al­gebra, and then try to prove, that the re­sul­ting po­wer-graded Jordan al­ge­bra is not isomorphic to the  Clif­ford al­ge­bra, seen as a Jordan al­ge­bra in terms of the an­ti-com­mu­ta­tor.
Here we are interested only to show, that the pseudo-orthogonal Lie algebras have their Kil­ling form cor­res­pon­dingly to the symplectic case
(κ)κilVxy,Vuv ›  =  <x,u> <y,v> + <x,v> <y,u>
and linear continuation to the whole of 2EE.The proof is simple - verify
(inv)κil‹ ad(Vwz) Vxy,Vuv ›  + κil‹ Vxy,ad(Vwz) Vuv ›   =  0   ∀ x,y,u,v,w,z ∈ EE .
It makes sense to name the 4-form ( again dropping all multiplicative factors, containing the di­men­sion of the underlying vector space )
τ)κ<,>( x,y,u,v )  =  <x,u> <y,v> + <x,v> <y,u>
on EEthe  Killing form of the pseudo-orthogonal vector space. By construction it is skew in­ter­chan­ging x,y resp. u,v and symmetric interchanging the pair x,y by the pair u,v. Moreover (inv) translates in a fourth form of skew-symmetry.
 For Jordan algebras one usually defines left multiplication by  L(x)y := ℙ◀x,y▶, where ℙ◀,▶ de­no­tes an ar­bi­trary Jordan bracket, and an (invariant) symmetric bilinear form ( play­ing the same role in the struc­ture theo­ry as the Kil­ling form in that of Lie al­gebras ) by
  trace L(ℙ◀x,y▶) .
Then it is straightforward to verify the two symmetry conditions
L(τxτy) τu,τv › = τu,L(τxτy) τv ›      L(τxτy) ∈ oo+( <,>EE , <,> )

κil L(τxτy) τuτv,τwτz › = κilτuτv,L(τxτy) τwτz ›      L(τxτy) ∈ oo+( sym2(<,>EE) , κil ,  ) ,
where oo+( , )means the self-adjoint linear transformations with respect to the now sym­me­tric bi­li­ne­ar form on the right hand side. Hence it turns out that the Kil­ling form of the pseu­do-or­tho­go­nal Lie al­ge­bra is the ca­no­ni­cal bi­linear form of the pseu­do-or­tho­go­nal Jor­dan al­ge­bra as well, if one can prove, that a bi­li­ne­ar form on a Jor­dan al­ge­bra, ful­fil­ling the sym­me­try con­di­tion, up to a con­stant fac­tor is the ca­no­ni­cal one, as it is the fact for Lie al­ge­bras. Pro­bab­ly the proof is the same.
Remark: This rises the question whether, given an arbitrary algebra with composition  ◊  and left mul­ti­pli­ca­tion  L(x) y = x ◊ y , the canonical (invariant?) bilinear form
    ≪x,y≫  =  trace L( x ◊ y ) − trace L(x) trace L(y)
( of endomorphisms like in the examples above ) isn't the better choice as the Kil­ling form of this al­ge­bra? In the case of Lie al­ge­bras, the theorem of Ado-Iwasawa ( that every Lie al­ge­bra has a finite-di­men­sio­nal faith­ful ma­trix re­pre­sentation ) im­plies, that the first term va­ni­shes. Hence the rest be­comes the ne­ga­tive Kil­ling form, with the ad­van­tage, that com­pact simple Lie al­ge­bras are cha­rac­ter­ized by a po­si­tive de­fi­nite bi­li­near form in­stead of a ne­ga­tive one.
runs
equivalently
Universal Envelops
of
Symplectic
Lie Algebras
The symplectic Lie algebra, realized above as second powers in the symmetric and the Weyl al­ge­bra, has an uni­ver­sal envelopping al­ge­bra of its own. It is used for in­stance in the stu­dy of Ca­si­mir ope­ra­tors, which span the cen­ter of this free ( i.e. free of re­stric­tions other than be­ing an as­so­cia­tive al­gebra subject to the com­muta­tion re­la­tions of the embedded Lie al­gebra ) al­ge­bra. Here the main problem is to identify the product  Λxx = xx = x²  with the as­so­cia­tive pro­duct of the uni­ver­sal en­ve­lop of the sym­plec­tic Lie al­ge­bra, in or­der to get an em­bed­ding of this uni­ver­sal en­ve­lop in­to the spaces of even powers in the Weyl al­ge­bra. For in­stance, ele­ments of zero po­wer are the mul­tip­les of the identity element, the ele­ments of the Lie algebra itself map in­to those of se­cond power in the Weyl al­gebra, those of se­cond uni­versal power map on­to those of fourth pow­er and so on. From the pre­cee­ding it fol­lows, that if fourth powers are involved, dif­fer­ences of the em­beddings in­to the clas­si­cal al­ge­bra with Pois­son bra­ckets resp. the quan­tum Weyl al­gebra with commutators are to be ex­pec­tedembedded in
the symmetric and
the Weyl algebra
but with no
correspondence
for
Clifford algebras
Generalized
Unitary
Groups
Given a group GGwith identity element e and an involutive automorphism æ of GG,the set of fix­ed points of (GG,æ)
    GGæ  = { g ∈ G / æ(g) = g }
is a subgroup, which we call  generalized unitary group UU(GG,æ)of(GG,æ).A second subset is giv­en by the symmetric ele­ments of (GG,æ)
    GGæ  =  { gæ(g)–1 / g ∈ GG },
called the symmetric space of (GG,æ).In general it is not a sub­group. It is easy to show that æ co­in­ci­des with in­ver­sion when re­stric­ted to the sym­me­tric ele­ments and by de­fi­ni­tion with the iden­ti­ty on the ge­ne­ra­lized uni­tary group. It is clear that
    GGæ ⋂ GGæ{e},and  GGæ ∪ GGæ = GG
is the  generalized unitary decomposition  into kernel and image of æ ( with respect to the group mul­ti­pli­ca­tion )
    g = g æ(g)–1æ(g)  ,GG= GGæ GGæ  i.e. GG = GGæ UU(GG,æ).
The symmetric space composition on  GGæis that of the group GG
(g₪)  f ₪ g = f g–1 f f,g ∈ GGæ
and æ is an automorphism of this symmetric composition also. To prove this one only has to find a group ele­ment h such that
    (f æ(f)−1) ₪ (g æ(g)−1) = h æ(h)−1 ,   which is the case for   h = f æ(f)−1æ(g) .
Loos even shows, that every symmetric space is of this type [ Loo p 74].
For a Lie group GGits Lie algebra íℯ GGdecomposes into a direct vector space sum
(ℓ)  x = íℯ æ (x)  ⊞  x − íℯ æ (x) ,íℯ GG =  íℯ GGæ  íℯ GGæ
with respect to the induced involutive automorphism íℯ æ of this Lie algebra. Here the first vec­tor space on the right hand side, given by the tangent functor, is the complement of the second one, i.‌e. of the Lie al­ge­bra of the fixed point group of æ. The proof uses that the two in­vo­lu­tive au­to­mor­phism com­mute with the ex­po­nen­tial map, then col­lecting the li­near terms in the ex­po­nen­tial ex­pan­sion. In case of an em­bed­ding of the Lie group and its Lie algebra in­to a vec­tor space, both in­vo­lu­tions co­in­cide. The Lie algebraic version of the ele­ment h, with f = exp(λx) and g = exp(λy) be­comes by linearizing, i.‌e. using the exponential se­ries and col­lec­ting the terma li­near in λ
(ℓíℯ)  x − íℯ æ (x)   ⊞   íℯ æ (y)  ,
which defines a(n associative?, but) non-commutative algebra structure on the whole Lie al­ge­bra?
Specialization of the group GGto the general linear group  Gℓ(VV,CC)and of the in­vo­lu­tive au­to­mor­phism æ gives the fol­low­ing re­ali­za­tions of the clas­si­cal ma­trix (Lie) groups:
  • For the (real) pseudo-orthogonal cases of relativity one has a non-degenerate sym­me­tric ( po­si­tive-de­fi­nite or indefinite ) bi­linear form <,> with res­pect to which the ad­joint is ta­ken. The in­vo­lu­tive auto­mor­phism æ is given by
     æ(g) = g†−1  .
    Symmetric space and generalized unitary group of fixed points are
    GG†-1( Gℓ(VV,RR) , †−1 )=: +O( VV ,<,>)and  UU( Gℓ(VV,RR) , †−1 )= O( VV ,<,>)
    resp., the latter being the pseudo-orthogonal group in this special case, which con­tains re­la­ti­vi­ty if the di­men­sion is 4 and the sig­na­ture of the bilinear form <,> is a Min­kow­ski one of type (+,-,-,-).
Standard example:  For x, y ∈ VVwith  <y,y> ≠ 0 , i.e. for elements outside the null-cone
    (VV,<,>){ l ∈ VV/ <l,l> = 0 } ,
define a Loos product  by [ Loo see remarks down there]
<x,y><x,x>
(o₪) x ₪ y  =  2 
x  −
 y  .
<y,y><y,y>
The proof of the four axioms of a symmetric space is given in [ Loo ].
Here there even is a side step into the theory of finite groups ( supp­lied with the dis­cre­te to­po­lo­gy in or­der to satisfy the 4th iso­la­tion axiom of a sym­me­tric space ): The sub­group of all per­mu­ta­tions of the ba­sis vec­tors, in ma­trix lan­guage those of all ma­tri­ces with exact­ly one 1 in eve­ry row and eve­ry co­lumn, the others 0, is the sym­me­tric group of per­mu­ta­tions of  dimVV = nob­jects. Then the same æ ... .
  • For the (real) symplectic case everything runs equivalently for the same form of in­vo­lu­tion æ if we de­fine - like above - the ad­joint with res­pect to the sym­plec­tic bi­li­ne­ar form σ<,>. Then the space of sym­me­tric ele­ments and the fixed point group for this in­vo­lu­tion are
    GG†-1( Gℓ(EE,RR) , †−1 )=: +Sp( EE , <,> ) and  UU( Gℓ(EE,RR) , †−1 )= Sp( EE , σ )
    resp. - classical dynamics in phase space must be formulated in these categories. We do not know any stan­dard example in this symplectic case.
  • For the pseudo-unitary cases the ground-field of the linear group are the com­plex num­bers, the de­fi­ning non-de­ge­nerate form is a ses­qui-li­near ≺|≻ one and se­mi-sym­me­tric in the sen­se that ≺y|x≻ is the conjugate complex of ≺x|y≻, not ne­ces­sa­ri­ly po­si­tive-de­fi­nite. For con­ve­ni­ance we abbreviate the real part of ≺|≻ by
      ℜℓ≺x|y≻  =  ½ ( ≺x|y≻ + ≺y|x≻ ) .
    The involutive automorphism æ again is defined as the adjoint with respect to ≺|≻, fol­lo­wed by in­ver­sion ( hence for the physical case we need algebras with involution and not  CC*‌-‌al­ge­bras,the latter car­ry­ing the an­ti­in­vo­lu­tion * ), and the spa­ces of sym­me­tric ele­ments and ge­ne­ra­lized uni­ta­ry fix­ed point groups are
    GG†-1( Gℓ(HH,CC) , †−1 )=: +U( HH ,≺|≻) and  UU( Gℓ(HH,CC) , †−1 )=U( HH ,≺|≻)
    resp., the latter being the (ordinary) pseudo-unitary groups. This gives the play-ground for quan­tum me­cha­nics in Hilbert space ( HH, ≺|≻),but in or­der to quan­tize non‌-‌li­ne­ar dy­na­mi­cal sys­tems use in­stead of (local) her­mi­tian, i.‌e. self‌-‌ad­joint ope­ra­tors those of the above (glo­bal) spa­ce of sym­me­tric ele­ments ( Stone‌'‌s theo­rem in in­fi­ni­te‌-‌di­men­sional Hilbert spaces must be pro­ven  for the spa­ce of sym­me­tric ele­ments ).
 Standard example: For x, y ∈ HHwith  ≺y|y≻ ≠ 0 , i.e. for elements outside the null-cone
    (HH,<,>){ l ∈ HH/ ≺ l | l ≻ = 0 } ,
define a  Loos product  by
≺x|y≻ + ≺y|x≻≺x|x≻ℜℓ≺x|y≻≺x|x≻
(u₪)  x ₪ y  = 
x  −
y=  2  
x −
y .
≺y|y≻≺y|y≻≺y|y≻≺y|y≻
The proof is a direct check of the first three axioms of a symmetric space. For the 4th (iso­la­tion) axi­om note that both co­ef­fi­cients on the right hand side of are real, whe­re­from  x ₪ y = y  im­plies that x and y are linear­ly de­pen­dent, i.e.  x=λy  for some real λ. Hence the proof of the 4th axi­om runs in the same way as that gi­ven by Loos for the pseu­do-or­tho­go­nal spe­cial case. A se­cond spe­cial case for this com­po­si­tion u₪ fol­lows from
 ≺x|x≻²
(u₪)≺ x ₪ y | x ₪ y ≻  = ——––for (real)≺x|x≻ = ≺y|y≻ = r²,
≺y|y≻
generalizing spheres and hyperboloids from the pseudo-orthogonal to the pseu­do-uni­ta­ry case.
This is only a slight ge­nera­lization to the uni­ta­ry case of Loos' pseu­do-or­tho­go­nal sym­me­tric pro­duct (o₪), which we use for spa­ce-time ap­pli­ca­tions.
There is one property of this ca­se which is not shared by the other ones: Both spa­ces have the same ( real and complex ) di­men­sion dim VV.And exactly this ( which is not true in the real pseu­do‌-‌or­tho­go­nal case ) allows in the as­so­cia­ted tan­gent spa­ces to switch ca­te­go­ries mere­ly by mul­ti­ply­ing with the com­plex unit: If A is ≺|≻‌-‌self‌-‌ad­joint then iA is ≺|≻‌-‌skew‌-‌ad­joint and con­versely, i.‌e. the vec­tor spa­ce iso­mor­phism be­tween the­se two comp­lex vec­tor spa­ces of en­do­mor­phisms is a ra­ther simp­le one. Ap­ply­ing the ex­po­nen­tial se­ries to these two spa­ces we get a bi­jec­tion of the two ma­ni­folds ge­ne­ra­ted by ex­po­nen­tials, which is just mul­ti­ply­ing with the comp­lex num­ber exp(i). Hence both have the same to­po­lo­gi­cal struc­ture and di­men­sion. exp(i) idHHis the most na­tu­ral choi­ce for base point in the sym­me­tric space of sym­me­tric ele­ments. Clear­ly both may have a dif­fe­rent num­ber of con­nec­ti­vi­ty com­po­nents. For quan­ti­za­tion we do not need the group of in­ver­tib­le ele­ments but this sym­me­tric spa­ce of sym­me­tric ele­ments in­stead. And we have to prove that if a con­fi­gu­ra­tion space is non‌-‌com­pact the space of sym­me­tric ele­ments ne­ces­sa­ri­ly is in­fi­ni­te‌-‌di­men­sio­nal.
  • Likewise for the special linear group ( of any ground-field ) as fixed point group the in­vo­lu­tion æ is gi­ven by the com­ple­men­ta­ry ope­ra­tor fol­lowed by an in­ver­sion. Sin­ce is gi­ven by de­ter­mi­nant and in­version, on­ly the de­ter­mi­nant sur­vi­ves in æ. Hen­ce

    GG‡-1( Gℓ(VV,KK) , ‡−1 )=: +Sℓ( VV,KK ) and  UU( Gℓ(VV,KK) , ‡−1 )=Sℓ(VV,KK)

    are space of symmetric elements and generalized unitary group of fixed points. The spa­ce of sym­me­tric ele­ments is the one-di­men­sio­nal space of di­la­ta­tions ( with­out 0 ), which even is a group. There is no phy­si­cal aspect in this case.
  • Besides some more real simple Lie groups in the symplectic case these are the clas­si­cal real sim­ple Lie groups. There is no doubt, that the ex­cep­tio­nal sim­ple Lie groups can be ob­tain­ed in the same way, by enlarging the ground‌-‌field  KKto qua­ter­ni­ons or Cay­ley num­bers.
Note that in all cases inverting and adjoining resp. complementing commutes.
 For physical reasons we call the categories on the left hand side observable and those on the right hand side the in­va­riance ca­te­go­ries. Con­tra­ry to what was stated 1972 in [ Til ], quan­ti­za­tion should not cross the di­vide between ob­ser­vab­les and in­va­ri­ance. In order to play with this di­vi­de, the fol­low­ing mi­micks ad­joint ac­tions and re­pre­sen­ta­tions of Lie groups: De­note the ad­joint ac­tion by
f • g = f g f −1∀  f,g ∈ GG,
which defines an (inner) automorphism of this group and an automorphism of the resul­ting sym­me­tric space with the composition . For g in the fixed point group we get
g • f æ(f)−1 = g f æ(f)−1 g−1 = (g f) æ(f −1) æ(g−1) = (g f) æ( f −1g−1) = (g f) æ((gf)−1) = (g f) æ(gf)−1 ,
i.e. the fixed point group  UU(GG,æ)  acts via the adjoint acion as a transformation groupon the spa­ce of sym­me­tric ele­ments +UU(GG,æ).This self-adjoint action de­fines the group mor­phism
g  ιg • ,UU(GG,æ) → Aut +UU(GG,æ)
and is a (non-linear) group representation. This induces a linear representation of the fixed point group in the tan­gent space at e, the self-adjoint representation, which in turn induces the above self-adjoint re­pre­sen­ta­tion of the Lie algebra of the fixed point group.
Like for groups the group algebra for symmetric spaces there should also be the concept of an uni­ver­sal as­socia­tive al­gebra, into which the given symme­tric space is embedded, free in the sen­se of  free from res­trictions other than those gi­ven by the sym­me­tric com­po­si­tion . This is a fac­tor al­ge­bra of the ten­sor al­gebra of the symmetric space with respect to the sym­me­tric com­po­si­tion law, such that it re­sults from the lar­ger group al­ge­bra of a group in­to which the sym­me­tric space is em­bed­ded via the con­struc­tion of sym­me­tric ele­ments. This lar­ger group al­gebra with an in­vo­lu­tion de­com­poses into this sym­me­tric space al­ge­bra ti­mes the group al­ge­bra of the fix­ed point group. The best name for these as­so­cia­tive al­ge­bras ( with unit ele­ment ) over sets  ℳ  with a mul­ti­pli­ca­tion  •  would be struc­ture al­gebra, wri­ting str( ℳ , •), since they are defined free of res­tric­tions ( other than those given by the un­der­ly­ing struc­ture  • ). Such an associative al­ge­bra must not exist, since there is the ex­cep­tio­nal Jor­dan al­ge­bra, which has no faith­ful fi­ni­te‌-‌di­men­tio­nal re­pre­sen­ta­tion, but has the sym­me­tric space of its in­ver­tib­le ele­ments. So the ques­tion ari­ses, whe­ther one has to drop the as­so­cia­tive­ness for its struc­ture al­ge­bra in fa­vor of the axi­oms of a sym­me­tric space.
More examples and their implications in classical dynamics are given by Fomenko [ Fmk p 306], al­though not in a basis-free form and without the use of Loos multiplications. How­ever, his use of in­vo­lu­tions and groups means that these examples can be reformulated easily.
this is a special
case of
O. Loos'
definition,
description
and
classification
of
symmetric spaces
p 73,
the four axioms
of which are

g ₪ g = g
g ₪ (g ₪ h) = h
 f ₪( g ₪ h ) = ( f ₪ g )₪( f ₪ h )

every g is an isola-
ted fixed point of Sg


where

Sgf = g ₪ f

is the left
multiplication

the 2nd and 3rd
axiom stating that
Sg
is an
involutive automorphism



with the
additive abelian

(₪a)x ₪a y  = 2x − y  

example, related to
(g₪)
via exponentiation
like
(u₪) and (o₪)
too



the group
generated by the

SfSg

is the

group of displacements

a  normal subgroup  of
non-linear transformations
of the
automorphism group

if  f,g  are taken from
a connected
symmetric space,
it is the
connectivity component
of the identity
of the
automorphism group
Quantization In any of these cases the tangent space in e is a Jordan algebra with res­pect to the an­ti-com­mu­ta­tor, i.e. that al­gebraic struc­ture with which Pas­cual Jor­dan axio­ma­ti­zed the ob­ser­vab­les of quan­tum me­chanics. From there one gets back to the sym­me­tric spa­ce by ex­po­nen­tia­tion.
  • Quan­ti­za­tion ( in Hilbert space ) is the dia­gram

    U( GG , æ )• str(+U(GG,æ), ₪ )U( HH ,≺|≻)• str(+U(HH,≺|≻), ₪ )

    embed ↑↓ embed ↑↓ 

    U( GG , æ )+U( GG , æ )U( HH , ≺|≻ )+U( HH , ≺|≻ )

      exp ↑↓ƒtanexp ↑↓ ƒtan

    u( GG , æ )+u( GG , æ ) u( HH , ≺|≻ )+u( HH , ≺|≻ )

    for some Hilbert space ( HH,≺|≻)and the tangent functor ƒtan ( it is desirable to ex­press this func­tor as a logarithmic series ). Like in the case of Lie groups we ex­pect that for com­pact spa­ces of sym­me­tric ele­ments fi­nite-di­men­sio­nal re­pre­sen­ta­tions exist.

  • Classical dynamics ( in phase space ) is the diagram

    U( GG , æ )• str(+U(GG,æ), ₪ )Sp( EE ,σ<,>)• str(+Sp( EE,σ<,>), ₪ )

    embed ↑↓ embed ↑↓

    U( GG , æ )+U( GG , æ )Sp( EE ,σ<,> )+Sp( EE , σ<,> )

      exp ↑↓ƒtanexp ↑↓ ƒtan

    u( GG , æ )+u( GG , æ )sp( EE , σ<,> )+sp( EE , σ<,> ) ,

    where this may not be the most general case: If the configuation space of the dy­na­mi­cal sys­tem nei­ther is nor is embeddable into an open subspace of a vector space, its co­tan­gent bund­le is not re­pre­sen­table as a sym­plec­tic vec­tor space. In these cases sym­plec­tic vec­tor spaces must be sub­sti­tu­ted by sym­plec­tic ma­ni­folds. How­ever, quan­ti­za­tion is not touched.
Herein only the central layer is necessary for quantization of non-linear spaces, and even on­ly the sym­me­tric spa­ce with­out its in­va­rian­ce group, since there may be dif­fe­rent dy­na­mi­cal sys­tems with the same in­va­ri­ance group.
(GG,æ)entirely determines the time development of a dy­na­mi­cal sys­tem, which must be a sym­me­tric one­-pa­ra­me­ter sub­space in the space of sym­me­tric ele­ments, this given by a one-pa­ra­me­ter sub­group in the group of dis­place­ments - in­stead of using i × uni­tary ope­ra­tors.
as a
commutative diagram
of
algebraic structures
A Class of
Jordan Algebras
on
Hilbert-Spaces
Given ( HH, ≺|≻)like in (u₪) above and  t ∈ HHwe define a real t-dependent Jordan algebra com­po­si­tion on HHby
x  ▣t  y  = ½(≺x|t≻+≺t|x≻)y +½(≺y|t≻+≺t|y≻)x −½(≺x|y≻+≺y|x≻)t
 =ℜℓ≺x|t≻ y +ℜℓ≺y|t≻ x −ℜℓ≺x|y≻ t ,
the coefficients on the right hand side being real numbers like those in (u₪), in fact the real com­po­nents of the ses­qui-li­near forms. This generalizes a well-known Jor­dan al­ge­bra on pseu­do‌-‌or­tho­go­nal vector spaces to the pseudo-unitary case. The sym­me­try of this com­po­si­tion is clear, for the proof of the Jor­dan iden­ti­ty we have to use the spe­cial cases
    t  ▣t t = ≺t|t≻ t andx ▣t ( t ▣t y ) = ≺t|t≻ x ▣t y = t ▣t ( x ▣t y )
The dimension  dim HHof this Jordan algebra is the same as that of the symmetric space gi­ven by u₪. For  ≺t|t≻ ≠ 0  there is the neutral element
t1t
e =
( with ≺e|e≻ =
 , normalizing  ê :=
 to get  ≺ê|ê≻ = 1 )
≺t|t≻≺t|t≻ ≺t|t≻ 
and the inverse ( note that all factors are real )
≺x|t≻+≺t|x≻12 ℜℓ≺x|t≻1ℜℓ≺x|e≻≺e|e≻
x−1 =
t −
x =
t −
x = 2
e −
x
≺x|x≻≺t|t≻² ≺x|x≻≺t|t≻≺x|x≻≺t|t≻²≺x|x≻≺t|t≻≺x|x≻≺x|x≻
( needing closer inspection if ≺t|t≻ = 0 ). In phy­sics the po­si­tive-de­fi­nite case is well-known from the lad­der ope­ra­tors in quan­tum Fer­mi sta­ti­stics: To see this in­tro­duce a ≺|≻-or­tho­go­nal di­rect de­com­po­si­tion of HHby
    x = x − ≺ê|x≻ê    ≺ê|x≻ê  =: x    ≺ê|x≻ê,HH= HH  CC ê.
If the ground field is restricted to the real numbers and the (then) bilinear form ≺,≻ to a po­si­tive‌-‌de­fi­nite one, we get these Fermi an­ti-com­mu­ta­tion re­la­tions in a ba­sis-free ver­sion on the pseu­do‌-‌or­tho­go­nal vec­tor space (HH,≺,≻)in the form
 ≺x,y≻ 
 x ▣t y = − ≺x,y≻ t = −
e ,x ▣t e = e ▣t x = x ,e ▣t e = e ,
≺e|e≻
where physicists usually use an orthonormal basis and Kronecker symbols in the first equa­tion.
 Mathematical expectation: This Jordan composition t defines t h e algebraic superstructure, which has as its tri­vial com­plex 1‌-‌di­men­sio­nal spe­cial case the com­plex field it­self, as its first non‌-‌tri­vi­al 2‌-‌di­men­sio­nal case the quaternions, as the next well-known case for the 4‌-‌di­men­sio­nal spe­cial case the Cay­ley num­bers ( or oc­to­ni­ons ), as the next named field the sedenions of the com­plex 8‌-‌di­men­sio­nal case and so on. These named exam­ples are given by posi­tive‌-‌de­fi­nite ses­qui‌-‌li­near forms ≺|≻, in which case all non‌-‌va­ni­shing ele­ments are in­ver­tib­le and in fact a sym­me­tric space with res­pect to the Loos mul­ti­pli­ca­tion u₪ ( re­mem­ber: this be­ing in­de­pen­dend of t ). This is a glo­bal (de­fi­ning) ver­sion of com­plex -, qua­ter­nio­nic - and Cay­ley num­bers. It re­mains to show, that a con­nec­ti­vi­ty com­po­nent in the ma­ni­fold of in­ver­tib­le ele­ments of the Jor­dan al­ge­bra is ge­ne­ra­ted by ex­po­nen­tials, and if so es­pe­cial­ly by one on­ly. For all these al­ge­bras, ex­cept the tri­vial 1‌-‌di­men­sio­nal case, there exist pseu­do‌- struc­tu­res, which are de­fi­ned by in­de­fi­ni­te ses­qui‌-‌li­ne­ar forms ≺|≻, which, how­ever, are not ,fields' in the sen­se, that all ele­ments on the null‌-‌cone of which are not in­ver­tib­le. For in­stan­ce there are pseu­do‌-‌qua­ter­nions if the ma­trix of ≺|≻ is cho­sen to be diag(1,-1). For the Cay­ley num­bers there even are two more non‌-‌iso­mor­phic pseu­do‌-‌struc­tures, ob­tained in this way. For the po­si­tive‌-‌de­fi­nite cases the au­to­mor­phism groups, i.‍e. the ge­ne­ra­lized uni­ta­ry groups in the above sen­se, are com­pact, for the in­de­fi­ni­te, i.‌e. pseu­do- struc­tures non‌-‌com­pact.
 Question: Does the Clifford algebra over this real (pseudo-)orthogo­nal vec­tor space com­ple­xi­fy to the Clif­ford al­gebra over the general complex algebra, constructed from the uni­ver­sal en­ve­lop of this ge­ne­ral Jor­dan al­ge­bra by iden­ti­fy­ing the neu­tral ele­ment e with the uni­ty ele­ment ( like in the case of the Weyl al­ge­bra )? At least this Jor­dan al­ge­bra is spe­cial via an em­bed­ding in­to this free al­ge­bra.
 To show that this Jordan algebra com­po­si­tion is the lo­cal struc­ture of a sym­metric space one, to be used in quan­ti­za­tion as the bot­tom la­yer in the pre­cee­ding diagram, we have to use
the left ope­ra­tion of any Jordan algebra composition
    L(x)y  =  x  ▣ y
to calculate the quadratic representation
    Ƥ(x) y=  2 L(x)L(x) − L(x ▣ x )
    i.e.
    Ƥ(x) y−1 = x ₪ y  which is equivalent to  Ƥ(x) y = x ₪ y−1 = L(x) y−1
for the symmetric space composition u₪ ( even for any Jordan algebra ),
the proof following from
    Ƥ(x) y−1  =  2 x  ▣t ( x  ▣t y−1) − ( x  ▣t x )  ▣t y−1  =  x ₪ y   !
here. The last equation follows from tediously equating the middle term to
ℜℓ≺y|t≻1
2
( 2 ℜℓ≺x|t≻ x − ≺x|x≻ t )
( 2 x ▣t ( x ▣t y ) − ( x ▣t x ) ▣t y )
≺y|y≻≺t|t≻≺y|y≻≺t|t≻
Thus this Jordan composition  ▣t is the local structure of the symmetric one, where­in the t-de­pen­den­ce is can­cel­led and e is not an uni­ty ele­ment of . Powers in can be defined such that they co­in­cide with po­wers of the Jor­dan com­po­si­tion, this not be­ing as­so­cia­tive but po­wer‌-‌as­so­cia­tive [ Loo ]. This al­lows the con­cept of ex­po­nen­tial se­ries, which in the abo­ve or­tho­go­nal de­com­po­si­tion be­comes
    exp( λ x) = cosh( λ βx) e + βx−1 sinh( λ βx) xwhereinβx =  − ≺x,x≻ ≺e|e≻ .
Quantization now delivers a philosophical question. Are space and time observables to be re­pre­sen­ted by self‌-‌ad­joint ope­ra­tors - as is assumed in quan­tum phy­sics - or have they to be re­pre­sen­ted as un­ob­ser­vab­le states, i.‌e. as ele­ments of HH( or, equi­valent­ly, as  sta­ti­sti­cal ope­ra­tors = idem­po­tent endo­mor­phisms  on the Hil­bert space of states, pure states be­ing pri­mi­tive idem­po­tents ) - greek phi­lo­so­phers would as­su­me this, be­cause for them space exists on­ly in points where bo­dies touch, not exi­sting else­where? There­fore they were able to in­vent geo­me­try, but not analytic geo­me­try.
instead of
Lie triples
?
if so there are
physical
implications
mass spectrum 
Literature with comments
[Bou]N Bourbaki  Algebra I  Hermann, Paris [1970] ISBN 2 7056 5675 8  remains the standard in­for­ma­tion on universality in algebra and chap. III §6 that on sym­me­tric algebras.
[Fmk]A F Fomenko  Differential Geometry and Topology  Consultants Bureau, NY [1987] translated from Russian with more references.
[God]C Godbillon  Géometrie Différentielle et Mécanique Analytique  Hermann, Paris [1969]  de­fi­nes Pois­son brackets on p 125.
[How]R E Howe  On the Role of the Heisenberg Group in Harmonic Analy­sis  Bull. Am. Math. Soc 3 no 2 [1981] p 821-843
[How]R E Howe  Remarks on Classical Invariant Theory  Trans. Am. Math. Soc 313 no 2 [1989] p 539-570
[Kch]M Koecher  The Minnesota Notes on Jordan Algebras and Their Applications  Springer Lecture Notes in Mathematics 1710 [1999] His con­cept of mutations IV 2 still has to be included here.
[Loo]O Loos  Symmetric Spaces I + II  Benjamin, N.Y. [1969] no ISBN  Above only the first volume is used. We generalize his sym­me­tric mul­ti­pli­ca­tion from hy­per­bo­loids (spheres) to arbitrary elements outside null-cones, this being well-known to the Artin school, for in­stance gi­ven in an­oth­er ver­sion of [Kch], and this in turn to the complex case of pseudo-Hilbert spaces. One im­pli­ca­tion of Loos' theory is, that instead of stu­dy­ing  CC*-algebras in quantum me­cha­nics, one should study algebras with an involu­tion ( in­stead of the *-anti-involution).
[Sou]J-M Souriau  Structure des Systèmes Dynamiques  Dunod, Paris [1970] no ISBN  remains the clas­sic book uniting theoretical physics and mo­dern ma­the­ma­tics. There is an eng­lish trans­la­tion:
[Sou]J-M Souriau  Structure of Dynamical Systems  Birkhäuser, Basel [1997] ISBN 0 8176 3695 1  de­fines the Poisson bracket on p 88.
[Til]H Tilgner  A Class of Solvable Lie Groups and Their Relation to the Canonical Formalism  Ann. Inst. H. Poin­caré Sec. A Physique Théo­rique 13 no 2 [1972] p 103-127
[Til]H Tilgner  Graded Generalization of Weyl and Clifford Algebras  J. Pure Appl. Algebra 10 no 2 [1977] p 163-168 
start / topsym.algebrasexamplesPoisson bracketsWeyl algebrascomm.relationsnon-isomorphysympl.repres.Lagr.subsp.Killing formsquantizationJordan alg.literature

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