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.
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<A†x,y> = σ<x,Ay>∀ A ∈ end(EE) ,
where A ι➥A† is a well-known involutive antiautomorphism of 𝕖nd(EE) .
ℝ 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<,>)
A linear transformation of this symplectic group is called symplectic. Both are n(2n+1)-dimensional, the Lie algebra 𝕤𝕡( , ) as a real vector space, the Lie group 𝕊𝕡( , ) as a real differentiable manifold. Usually the Lie algebra 𝕤𝕡( , ) is identified to the tangent space at the neutral element id𝔼 ∈ 𝕊𝕡( , ). Here 𝔸ut means the set of invertible endomorphisms, mostly written 𝔾𝕝( , ) 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 𝕤𝕡( , ) 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
since by linearizing this identity x ι➥x+y and u ι➥u+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
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
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▱tEE
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 ▱t 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
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
𝕊𝕡+( EE, σ ) = { exp(α1A1)⋅⋅⋅exp(αkAk) / αk ∈ ℝ, Ak σ-self-adjoint , k = 1,2, ... }
i.e. a manifold generated multiplicatively by the exponentials of σ-self-adjoint linear transformations. This gives rise to the diagram
𝕊𝕡+( EE, σ )
×
𝕊𝕡( EE, σ )
=
𝔸ut(EE,ℝ)
exp ↑
exp ↑
exp ↑
𝕤𝕡+( EE, σ )
⊞
𝕤𝕡( 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
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
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,
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 toJ, such that its associated projector is complementary to PJ. Then
association
J
⇄
P
complementation
⇅
⇅
− J
⇄
id − 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 ???
Examples of Symplectic Vector Spaces
The 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 := VV ⊞<,>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 ℝ2n 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
Bases
Darboux's theorem guarantees the existence of symplectic bases on EEand its dual in the form
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
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,ℝ)resp. ss p(2n,ℝ)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 ∈ ℝ, 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 ι➥1 , which usually is dropped. Thus we get the real infinite-dimensional, power-graded, associative, commutative symmetric algebra sym(σEE)of polynomials over EE
with ℝ⋅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 onEE
ℝat( 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
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∈ℝ 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) = σℙx(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 ):
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
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= ℝ 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) = ℝ 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
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
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)adℙ( 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 ι➥x+y and z ι➥z+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 ι➥Λ 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
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
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
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
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 ∈ 𝕊𝕡(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 ( x1⋅Λx2⋅⋅⋅xk + … + xk⋅Λx1⋅⋅⋅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,ℝ)
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 LL2that 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
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
The two Cartan subalgebras LLqqand LLppleave exactly n² 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,ℝ)we have to specialize the commutation relations (SRp) to this decomposition into Lagrangian subspaces
These commutation relations show that we have five representation spaces for the general linear Lie algebra ggℓ(LLq,ℝ),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,ℝ)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,ℝ) 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
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 )
(κσ)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 = ℝ 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 ]
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 ∈ EE, τx,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 )
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)κil‹Vxy,Vuv › = <x,u> <y,v> + <x,v> <y,u>
and linear continuation to the whole of ∨2EE.The proof is simple - verify
It makes sense to name the 4-form ( again dropping all multiplicative factors, containing the dimension of the underlying vector space )
(κτ)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
where 𝕠+( , ) 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 expected.
embedded in the symmetric and the Weyl algebra but with no correspondence for Clifford algebras
Generalized Unitary Groups
Given a group 𝔾 with identity element e and an involutive automorphism æ : 𝔾 → 𝔾 of this group, the set of fixed points of (𝔾,æ)
𝔾æ = { g ∈ 𝔾 / æ(g) = g } for æ ∘ æ = id𝔾 with æ(g–1) = æ(g)–1,
is a subgroup, which we call generalized unitary group 𝕌(𝔾,æ) of (𝔾,æ). In case of a Lie group we denote its Lie algebra as 𝕦(𝔾,æ) . For non-trivial involutions in general it is not a normal subgroup
defining g = æ(h)h−1
gg = e implies (gg) f (gg)−1 = f ∀ f ∈ 𝔾
is a sufficient condition for 𝕀𝕞𝕒𝕘𝕖(æ) being a normal subgroup. This leads into the cosmos of commutative < nilpotent < solvable groups. A second subset is spanned by the symmetric elements of (𝔾,æ)
𝔾æ = { g æ(g)–1 / g ∈ 𝔾 } ,
called the symmetric space of (𝔾,æ). In general it is not a subgroup but a symmetric space with respect to the Loos multiplication
g ₪ h = g æ(g−1) æ(h) .
This follows from the fact that it is the Loos multiplication for groups
i.e. the two maps coincide with inversion when restricted to their mutual complementary images. For non-trivial æ this complementary map æᒼ in general is not an involution
æᒼ ∘ æᒼ (g) = ( æᒼ(g) )2 .
The bridge to invertible elements of a Jordan algebra and to reflections in pseudo-orthogonal vector spaces is given by the global fundamental formula
(gFF) æᒼ(g) æᒼ(h) æᒼ(g) = æᒼ( æᒼ(g) h ) .
It is clear that
𝔾æ ⋂ 𝔾æ ⊃ {e},and 𝔾æ ∪ 𝔾æ = 𝔾
is the generalized unitary decomposition into the images of the two mappings æ and æᒼ
g = g æ(g)–1 æ(g) ,𝔾 = 𝔾æ 𝔾æi.e. 𝔾 = 𝔾æ 𝕌(𝔾,æ) .
Loos even shows, that every symmetric space is realized by such a pair (𝔾,æ) [ Loo p 74]. However, a systematic 2nd cohomology for symmetric spaces, giving rise to short exact sequences and semi- and direct products still is underinvestigated. So we cannot decide whether the decomposition into the spaces of symmetric and unitary elements is a split or even a retract one.
For a Lie group 𝔾 its Lie algebra 𝕝ie𝔾 decomposes into a direct vector space sum
with respect to the induced involutive automorphism 𝕝ie 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 − 𝕝ie æ (x) ⊞ 𝕝ie æ (y) ,
which defines a(n associative?, but) non-commutative algebra structure on the whole Lie algebra?
Specialization of the group 𝔾 to the general linear group 𝔾l(𝕍,ℂ) 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 .
The symmetric space, which we denote by an high-preindex +, and the generalized unitary group of fixed points are
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 ∈ 𝕍 with <y,y> ≠ 0 , i.e. for elements outside the null-cone,
☓(𝕍,<,>) = { l ∈ 𝕍 / <l,l> = 0 } ,
defines 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 by Loos. 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 n = dim𝕍 objects. 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<,>. The space of symmetric elements and the fixed point group for this involution then are
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
resp., the latter being the (ordinary) pseudo-unitary groups. This gives the play-ground for quantum mechanics in Hilbert space ( ℍ, ≺|≻), 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 ∈ ℍ with ≺y|y≻ ≠ 0 , i.e. for elements outside the null-cone
☓(ℍ,<,>) = { l ∈ ℍ / ≺ 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≻ = ፫²
,
≺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 complexy) dimension dim𝕍. 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) idℍ is 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
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 𝕂 to quaternions or Cayley numbers.
Note that in all cases inverting and adjoining resp. complementing commute. 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 ∈ 𝔾 ,
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(g f)−1 ,
i.e. the fixed point group 𝕌(𝔾,æ) acts via the adjoint action as a transformation group on +𝕌(𝔾,æ) , the space of symmetric elements. This self-adjoint action defines the group morphism
g ι➥g • ,𝕌(𝔾,æ) → 𝔸ut +𝕌(𝔾,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 Pascal 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
for some Hilbert space ( ℍ,≺|≻) 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
for the tangent functor ƒtan. This may not be the most general case: If
the configuation space ( +𝕌(𝔾,æ), ₪ ) of the dynamical system
neither is flat 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. In this case Loos' concept of locally symmetric spaces may apply. Taking traces or determinants somewhere one proves, that for non-compact configuration spaces the Hilbert representation spaces necessarily are infinite-dimensional. However, finite-dimensional Hilbert spaces also describe physical problems. The lowest dimensional non-trivial Hilbert space in two (complex) dimensions leads to Carl-Friedrich von Weizsäcker's Ur-theory of the simple alternative and the (complex) three-dimensional Hilbert space to hadron physics.
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. (𝔾,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 acting on some element in this one-parameter subspace - instead of using i × unitary operators.
as a commutative diagram of algebraic structures
A Class of Jordan Algebras on Hilbert-Spaces
Given a Hilbert space ( ℍ, ≺|≻) like in (u₪) above and some t ∈ ℍ we define a real t-dependent Jordan algebra composition on ℍ by
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 ℍ of this Jordan algebra is the same as that of the symmetric space given by u₪. For ≺t|t≻ ≠ 0 there is the neutral element
t
1
t
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≻
1
2 ℜℓ≺x|t≻
1
ℜℓ≺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 ℍ by
x = x − ≺ê|x≻ê ⊞ ≺ê|x≻ê =: x⊥⊞ ≺ê|x≻ê,ℍ= ℍ⊥⊞ ℂ 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 (ℍ⊥,≺,≻) 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 element 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
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
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