Download The semantics of the canonical commutation relation

The semantics of the canonical commutation
relation
B. Zilber
August 31, 2014
1
Introduction
1.1 This paper is a part of a much broader project which aims to establish,
for a typical ’co-ordinate algebra’ A in the sense of non-commutative geometry, or a scheme, a geometric counterpart VA . Here “geometric” should
also be understood in some broad but well-defined sense. We believe this is
possible in the setting of model theory, where a crucial new experience in generalisation of semantics of algebraic and analytic geometry, and applications
of this, has been accumulated in the last 50 years.
In fact, we aim to establish a rigorous mathematical duality
AV ←→ VA
(1)
between algebras and structures of geometric flavour.
Let us recall that a project with similar aims, but mainly concentrated
around classical structures of mathematical physics, has been suggested by
C.Isham and J.Butterfield and is being developed by C.Isham, A.Döring and
others, see e.g. [3] and later publications. In that approach VA is supposed
to be a topos.
Our approach is different in many details: VA is a multi-sorted structure,
each sort of which is a Zariski geometry in the sense of [13]. An essential
component of the multi-sorted structure VA is the set of morphisms between
sorts functorially agreeing with embeddings between certain sublagebras of
A, which makes the left-to right arrow a functor between a category of those
subalgebras A and sorts VA of VA . This functor, in fact, defines a quite rich
1
sheaf over the category of sublagebras. This makes an interesting point of
contact of our approach with the topos-theoretic approach to foundations of
physics.
1.2 At this stage of our project we decided to keep the general context
to minimum and instead concentrate on developing and illustrating the approach in a very special case of Heisenberg-Weyl algebras and the canonical
commutation relation. And even there we quite quickly drop the generality of the n-th Heisenberg-Weyl algebras for the sake of developing rather
sophisticated calculus with one position and one momentum operators.
1.3 The n-th Heisenberg-Weyl algebra A(n) has generators P1 , . . . , Pn , Q1 , . . . , Qn ,
the “co-ordinates” of n-dimensional quantum mechanics, with commutation
relations
Ql Pk − Pk Ql = 2πiδlk I,
(2)
where the Pk and Ql are seen as self-adjoint operators, I is the “identity
operator” and δkl the Kronecker symbol. (We have chosen the multiplier 2πi
for convenience of further notations and to mean that exp 2πiI = I.)
The infinite-dimensional Heisenberg algebra is
A(ω) = ∪n A(n) .
The representation theory of A(n) is quite complicated, even for n = 1,
due to the fact that the algebra can not be represented as an algebra of
bounded operators on a Hilbert space, not to speak about finite-dimensional
representations.
However, as suggested by Herman Weyl, we may instead consider the
representation theory of algebras generated by the Weyl operators which can
be formally defined as
Ukak = exp iak Qk , Vlbl = exp ibl Pl
(3)
for ak , bl ∈ R. These are unitary (and so bounded!) operators if the Pk and
Ql are self-adjoint, and the following commutation relation holds:
Ukak Vlbl = qkl Vlbl Ukak
where qkl = exp 2πiak bl δkl
(4)
Given a choice of real a1 , . . . , an , b1 , . . . , bn the complex C ∗ -algebra generated by Weyl operators V1±a1 , . . . , Vn±an , U1±b1 , . . . , Un±bn ,
A(a1 , . . . , an , b1 , . . . , bn ) = C[V1±a1 , . . . , Vn±an , U1±b1 , . . . , Un±bn ]
2
is called (in this paper) a Weyl algebra.
The all-important Stone – von Neumann theorem states that the representation theory of all the Weyl algebras A(a1 , . . . , an , b1 , . . . , bn ) together
is equivalent to the Heisenberg commutation relation (2). One can say
that A(n) can be fully replaced by the entirety of its Weyl subalgebras
A(a1 , . . . , an , b1 , . . . , bn ) which have a good Hilbert space representation theory.
Especially nice finite-dimensional representations have Weyl algebras for
which the multipliers e2πiak bl are roots of unity. This is the case when all the
ak , bl are rational. We call such algebras rational Weyl algebras.
In a certain sense, made precise in the paper, the rational Weyl algebras
approximate the full Heisenberg-Weyl algebra just as rational points approximate points of the n-dimensional Euclidean space. In fact, the paper uses
the correspondence between rational Weyl algebras A and certain geometric
objects VA . The structure VA , encapsulates the representation theory of A,
and is at the same time a well-understood object of model-theoretic studies,
a quantum Zariski geometry, see [11].
The quoted paper established the duality (1) between quantum algebras
A at roots of unity (rational Weyl algebras in our case) and corresponding
quantum Zariski geometries VA , which extends the classical duality between
commutative affine algebras and affine algebraic varieties. The new step
implemented in the present work is to extend the duality on algebras A
approximated by rational Weyl algebras. In order to achieve this we need
to develop a notion of approximation on the side of structures VA . Such a
notion, of structural approximation, has essentially been developed in [12].
We use it here with small modifications.
1.4 A few introductory words about structural approximation:
We assume that we are given a set {Ai : i ∈ I} of rational Weyl algebras
and an ultrafilter D on I. Consider the ultraproduct of algebras {Ai : i ∈ I}
modulo D, call it Ã. This is an algebra containing operators U a V b , for certain
a, b ∈ ∗ Q, the field of non-standard rational numbers. The collection of
these U a V b in à depends on I and D. In fact, à is “generated” by a pair of
operators U α and V β , for non-standard rationals α and β, if we are allowed
to use pseudo-finite linear combinations.
We consider then the corresponding ultraproduct modulo D of Zariski
geometries VAi , denote it VÃ .
3
Now we are ready to define a limit object, the structure
lim D VAi
and the procedure of taking the limit,
lim : VÃ → lim D VAi
of the family VAi modulo D. By definition in [12] this is a homomorphic image
of VÃ , and a homomorphism here means a map preserving basic relations of
the language (and all their positive Boolean combinations).
We strengthen the definition of [12] by requiring that the equivalence
relation induced by the homomorphism lim on VÃ is positively type-definable
in the basic language (a model-theoretic notion).
This definition is very sensitive to the choice of the basic (primitive)
relations of the language and is not preserved under the transition to an
inter-definable language. The fact that we construct our structures as Zariski
geometries plays a key role here: the notion of homomorphism in the above
definition agrees with the notion of morphisms in the category of respective
Zariski geometries.
1.5 Once the approximation procedure is determined we can try and define
an object corresponding to the algebra A, the union of all rational Weyl
algebras. It is natural for construction of such VA to start with the family
{Ai : i ∈ I} of all the rational Weyl algebras and take D to be a Fréchet
filter with respect to the partial ordering, that is for each A,
{i ∈ I : A ⊂ Ai } ∈ D.
This leads to à being pseudo-finitely generated by a pair of operators
1
U and V ν , for non-standard integers µ and ν with the property that both
integers are divisible by all the standard integers.
Consequently we will get that A ⊂ Ã for any (standard) rational A, which
is one of the desirable properties.
But now we will be able to see that the properties and the invariants of
the limit object depend essentially on the ratio
1
µ
h=
4
µ
.
ν
When, h is a standard rational we get a perfect approximation to the classical
representation theory of (2). This is the case we study in sections following
section 7 and the notation à and Và will be fixed for this algebra and the
structure. In fact, ultimately, quantum mechanics is represented in in the
limit of one particular Ã-module of VÃ , the principal module (see 1.10 for
more details). We call this limit structure the space of basic states.
We leave open two important questions.
1. To what extent the structure VÃ depends on the choice of a Fréchet
ultrafilter? Is its elementary theory (positive theory) uniquely determined?
2. Is the limit object (the space of basic states) unique?
1.6 Note that another very interesting example of a limit representing the
algebra pseudo-generated by U α and V β is when αβ is a nonstandard approximation
√ to a standard real quadratic irrational number τ, e.g. of the form
τ = n for some standard positive integer n. We recall here that Yu.Manin
suggested an aprroach to attack a long standing number-theoretic problem
[7] via a non-commutative geometry approach using the Weyl algebra with
the commutation coefficient q = exp 2πiτ, for τ real quadratic irrationality.
1.7 The structure V∗ A , or rather an important part of it, is a substitute
for the Hilbert space of states of quantum mechanics with an action of appropriately defined P, Q and other formal self-adjoint operators on it. An
important result, see 8.3, is the proof that the canonical commutation relation
holds in lim V∗ A . This can be seen as another version of Stone - von-Neumann
Theorem.
As a matter of fact the above P, Q are initiall defined on proper subsets
of V∗ A . However, the domains DP and DQ of definition are dense in V∗ A in
the sense that lim DP and lim DQ cover all of the limit structure.
Our analysis explains why the action of P and Q has to be partial even in
the context of our approach: the value of P|xi for some finite norm elements
|xi of V∗ A (or Q|pi) becomes unbounded in terms of standard numbers and
so can not be accommodated in lim V∗ A . This is in close correspondence with
the classical reason that the canonical commutation relation (2) implies that
P and Q can not be realised as bounded operators.
1.8 Let us turn now to more detailed description of the notions indicated
above.
For a rational Weyl algebra A the object VA cab be seen as a bundle (not
locally trivial!) of what we call “algebraic-Hilbert” irreducible A-modules
5
over an algebraically closed field, say C. “Algebraicity” of the modules first
of all reflects the fact that the modules are of finite dimension nA . More
serious algebraic feature of VA is in the fact that the submodule over a much
smaller subfield, R0 [i] ⊂ C, where R0 is the field of algebraic totally real
numbers, has an inner product structure with values in R0 [i]. To express the
sesquilinearity of the inner product we use the formal complex conjugation
on R0 [i], which does not require definability of the reals. This is crucial since
we want to keep the structure VA a Zariski structure (hence ω-stable). In
fact, for each particular A we need just a finitely generated subfield of R0 [i].
With the inner product and a formal complex conjugation defined we can
also speak of adjoint operators, unitarity and so on.
1.9 The construction of the global object lim V∗ A , the space of basic states,
plays the crucial role in delivering calculation tools.
The field over which VÃ is given is the field ∗ C of non-standard complex numbers, and lim : ∗ C → C is the well-known standard part map. In
particular, lim ∗ R0 = R and we get the complete reals.
The dimension nà of the modules in Và is a non-standard integer, which
allows both to treat the new “Hilbert space” as both infinite-dimensional and
pseudo-finite dimensional space.
1.10 In fact, for each rational algebra A all the “physics” is modelled in
one particular module chosen in the bundle VA of algebraic-Hilbert modules.
This module, denoted VA (1), which we also call the principal A-module, is
characterised by the property that its U a - and V b - eigenvalues are roots of
unity q, that is the application of formal complex conjugation gives q ∗ = q −1 .
Consequently these operators act on it as unitary operators. It implies that P
and Q act as self-adjoint operators. This finite dimensional space effectively
represents what physicists call discrete, or finite-dimensional, Dirac calculus,
see e.g. [9], sections 12.1 - 12.3.
One can interpret the functor
VA 7→ VA (1)
as “determining the real part of the complex structure”. The inverse is the
“complexification” of the discrete Dirac model. Note that one can not reduce
the whole theory to just working with the discrete Dirac models VA (1), since
the functor
A 7→ VA (1)
6
is not invertible - one can not reconstruct the algebra A from its action on
one finite-dimensional representation.
1.11 Of course, discrete models have been in permanent use throughout
the history of quantum physics as a heuristic tool, see the reference [9] above
or more advanced treatment in similarly defined spaces in [4]. The analysis
of such a discrete model usually concludes with the phrase “now we pass
to continuous limit”, which actually is an ill-defined notion and a source of
main troubles of quantum physics.
Perhaps the main achievement of this paper is the approximation procedure lim suggesting a rigorous interpretation of the above.
1.12 The key to practical calculations (rigorous Dirac calculus) is the fact
that an inner product equation of the most general form
he|Xf i = r,
where e, f are vectors of modulus 1, X ∈ A an operator and r ∈ R0 [i],
is given by a positive quantifier-free formula. Such formulae are preserved
by homomorphisms. Hence a caclulation in these terms carried out in a
pseudo-finite dimensional space VÃ (1) can be carried over by lim to the
infinite-dimensional space VA (R).
1.13 We demonstrate our method by giving proofs by calculation to:
• the identity (2), the canonical commutation relation, thus proving a
form of Stone - von Neumann theorem;
• the formula for the Feynman propagator for the free particle;
• the formula for the Feynman propagator for the quantum harmonic
oscillator;
• the formula for the trace of the Feynman propagator for the quantum
harmonic oscillator.
2
Preliminaries
2.1 Weyl operators (3) generate so called Heisenberg group H(a1 , . . . , an , b1 , . . . , bn ).
Most of the time will be interested in the case n = 1 and the Heisenberg group
H(a, b), which is a group of rank 2. More generally a Heisenberg group of rank
2 is a subgroup H of H(a, b) such that the quotient H/[H, H] is non-cyclic.
7
2.2 Lemma. A Heisenberg group of rank 2 is isomorphic to H(c, d) for
some c, d. If H ⊆ H(a, b) for rational a, b, then c, d are rational as well.
Proof. Easy. 2.3 A general Heisenberg group of rank 2n is by definition a group of the
form
H1 × . . . × Hn
where the Hi are Heisenberg groups of rank 2. In this definition we also
assume that the generators of the Hi are of the form Uig11 Vig12 and Uig21 Vig22 ,
for real g11 , g12 , g21 , g22 . We write, correspondingly,
Hi = hUig11 Vig12 , Uig21 Vig22 i.
We also will consider an algebraically closed field F of characteristic 0 (e.g.
the field of complex numbers C) and the Weyl algebras, which are defined as
a group algebras FH of the respective Heisenberg groups. In particular,
A(a1 , . . . , an , b1 , . . . , bn ) := FH(a1 , . . . , an , b1 , . . . , bn ),
where
H(a1 , . . . , an , b1 , . . . , bn ) = hU a1 , . . . , U an , V b1 , . . . , V bn i.
2.4 By 2.2 any Heisenberg group of rank 2n is isomorphic to one of the
form H(a1 , . . . , an , b1 , . . . , bn ) and every Weyl algebra is isomorphic to one of
the form A(a1 , . . . , an , b1 , . . . , bn ).
2.5 A rational Weyl F-algebra of rank 2n is a Weyl algebra isomorphic
to one of the form A(a1 , . . . , an , b1 , . . . , bn ) with rational non-zero a1 , . . . , an ,
b1 , . . . , b n .
In this case the qkk in (4) are roots of unity. Such algebras, and more
generally quantum algebras at roots of unity, can be seen as “coordinate
algebras” of corresponding Zariski geometries. This has been studied in [11].
2.6 Theorem. Let A = A(a1 , . . . , an , b1 , . . . , bn ) be a rational Weyl Falgebra. Then
(i) A and the centre Z(A) are prime noetherian PI rings. Let nA be the
PI degree of A.
8
(ii) Every maximal ideal of A is of the form α̃ = Aα, for α ⊆ Z(A),
α ∈ Spec(Z(A)), a maximal ideal of the centre.
(iii) The unique irreducible A-module VA (α) with annihilator Ann(V) = α̃
is of dimension nA over F. The irreducible representations of A are
in bijective correspondence with points α ∈ Spec(Z(A)), the maximal
spectrum of Z(A), which can be identified with (F-points of ) an affine
irreducible variety.
(iv) There exists an isomorphism A/α̃ ∼
= M(nA , F), the full nA × nA -matrix
algebra.
(v) The irreducible A-module VA (α) is isomorphic to a tensor product of
corresponding irreducible Ak -modules VAk (αk ),
VA (α) ∼
=
n
O
VAk (αk )
k=1
for Ak = A(ak , bk ) and some maximal ideals αk of Z(Ak ).
Proof. For (i)-(iv) see [1], III.1.1 and III.1.6.
(v) follows from the fact that
H(a1 , . . . , an , b1 , . . . , bn ) =
n
Y
H(ak , bk ).
(5)
k=1
2.7 Note that the isomorphism in (iv) is given by a representation ja : A →
M(nA , F), which in its own right is determined by a choice of a maximal ideal
a ⊂ Z(A) and a basis e in the module VA (a), so j = ja,e .
2.8 Definition. Given the Heisenberg group H(a, b) with rational a, b, it
is easy to describe its automorphism group Aut H(a, b). It consists of automorphisms ξg,n,m , for g = (gij ) ∈ SL(2, Z), n, m ∈ 0, 1, . . . , N − 1 where N is
the order of defined on the generators of H(a, b) the root of unity q = e2πiab :
ξg,n,m :
U a 7→ q n U g11 a V g12 b
V b 7→ q m U g21 a V g22 b
9
Note that by isomorphism the commutator of the image of U a and V b is
equal to the same number q, hence det(gij ) = 1.
Set ΓH(a,b) to be the subgroup of Aut(H(a, b)) which fixes H(a, b)/Z(H(a, b)),
the quotient of the Heisenberg group modulo its centre.
Given a rational Weyl algebra A = A(a1 , . . . , an , b1 , . . . , bn ) we set
ΓA = ΓA1 × . . . × ΓAn
for Ak = A(ak , bk ), k = 1. . . . , n, with the action on A = A1 ⊕ . . . ⊕ An
coordinate-wise.
2.9 Lemma. For A = A(a, b) the group ΓA is generated by the two automorphisms µ and ν defined on the generators of A as follows
µ:
U a 7→ qU a
V b 7→ V b
ν:
U a 7→ U a
V b 7→ qV b
(6)
ΓA and ΣA fix the centre Z(A) point-wise.
ΓA ∼
= (Z/N Z)2 .
Proof. By definition any γ ∈ ΓA acts by U a 7→ q k U a and V b 7→ q m V b ,
for some integers k and m. Then γ : U a V b U −a V −b 7→ U a V b U −a V −b , that is
γ(qI) = qI for all γ ∈ ΓA .
Clearly, µ, ν ∈ ΓA . Also γ −1 µk ν m acts as identity on U a , V b , hence γ =
k m
µ ν . The transformation φ fixes qI by (4).
Finally, it is immediate from (6) that the generators µ, ν of ΓA commute
and are of order N. 2.10 In what follows we will extensively use (5) and reduce the study of
n-th Weyl algebras to the study of 1-st Weyl algebras.
2.11 Lemma. The centre Z(A) of rational A = A(a1 , . . . , an , b1 , . . . , bn )
is of the form A(N1 a1 , . . . , Nn an , N1 b1 , . . . , Nn bn ), where Nk is the order of
the root of unity exp 2πiak bk , that is equal to the denominator of the reduced
rational number ak bk .
Proof. Immediate. 10
2.12 Lemma. Suppose
A(a01 , . . . , a0n , b01 , . . . , b0n ) ⊆ A(a1 , . . . , an , b1 , . . . , bn )
and A(a01 , . . . , a0n , b01 , . . . , b0n ) is commutative and maximal among commutative.
Then
(i) a0k = Mk ak , b0k = Lk bk , for some Mk , Lk ∈ Z such that Mk Lk is equal
to the denominator of the reduced fraction ak bk , k = 1, . . . , n.
(ii) In particular, A(N1 a1 , . . . , Nn an , b1 , . . . , bn ) and A(a1 , . . . , an , N1 b1 , . . . , Nn bn )
are maximal commutative subalgebras of A(a1 , . . . , an , b1 , . . . , bn ).
(iii) Maximal commutative subalgebras A(Nk ak , bk ) and A(ak , Nk bk ) of
A(a1 , . . . , an , b1 , . . . , bn ) are 1-generated by V bk and U ak , respectively, over
the centre of the algebra.
Proof. (i) We may assume that n = 1 and we consider A(a0 , b0 ) ⊆
A(a, b) maximal commutative. Then a0 = M a, b0 = Lb, for some M, L ∈ Z
by the embedding assumption. Commutativity requires that M Lab ∈ Z.
Maximality implies the last of the conditions.
(ii) Immediate from (i).
(iii) Just note that U Nk ak and V Nk bk are in the centre of the algebra. 2.13 Lemma. Suppose the rational n-th Weyl algebra A = A(a1 , . . . , an , b1 , . . . , bn )
is commutative. Then there is finitely many commutative rational n-th Weyl
algebras B extending A.
Proof. If B = A(a01 , . . . , a0n , b01 , . . . , b0n ) then commutativity implies that
a0k bk , ak b0k ∈ Z for all k. This puts bounds on the denominators of a0k and b0k .
On the other hand, since A(a1 , . . . , an , b1 , . . . , bn ) ⊂ A(a01 , . . . , a0n , b01 , . . . , b0n ),
the ak and bk are multiples of a0k and b0k , respectively. This bounds the
numerators of a0k and b0k . 2.14 Lemma. Given a commutative rational n-th Weyl algebra
C = A(c1 , . . . , cn , d1 , . . . , dn ), let C ↑ be the algebra generated by all the
commutative rational n-th Weyl algebras B extending C. Then
(i) C ↑ = A( Nc11 , . . . , Ncnn , Nd11 , . . . , Ndnn ), where Nk = ck dk , k = 1, . . . , n.
(ii) C ↑ is generated by two maximal commutative subalgebras
A( Nc11 , . . . , Ncnn , d1 , . . . , dn ) and A(c1 , . . . , cn , Nd11 , . . . , Ndnn ).
11
(iii) If C is the centre of A = A(a1 , . . . , an , b1 , . . . , bn ), then C ↑ = A.
Proof. (i) It is clear that C ↑ contains A( Nc11 , . . . , Ncnn , d1 , . . . , dn ) and
A(c1 , . . . , cn , Nd11 , . . . , Ndnn ) as the two subalgebras are commutative and contain
C.
(ii) It is easy to see that both are maximal commutative subalgebras of
↑
C as presented in (i). It is obvious from the explicitly given parameters that
the two sublagebras generate C ↑ .
(iii) By 2.11 C = A(N1 a1 , . . . , Nn an , N1 b1 , . . . , Nn bn ), where the Nk are
denominators of the reduced rational numbers ak bk . Hence, by setting ck =
Nk ak and dk = Nk bk we get by (i) C ↑ = A. 2.15 Corollary. The functors A →
7 Z(A) and C 7→ C ↑ between categories
of rational n-th Weyl algebras and commutative rational n-th Weyl algebras
are inverse to each other.
2.16 Lemma.
The family of all the commutative Weyl subalgebras C of a rational Weyl
algebra A containing the centre Z(A) of A is finite.
Proof. It is enough to note that the generators Ukak , Vkbk of A are periodic
modulo the centre. Hence the images of canonical generators Ukck , Vkdk of C
under the γ’s modulo Z(A) take only finitely many values. 2.17 Definition.
Let, for a rational Weyl algebra A = A(a, b),
O(A) := {C : maximal commutative 1-generated over Z(A)}.
Set for a rational Weyl algebra A = A(a1 , . . . , an , b1 , . . . , bn ),
O(A) := {C1 × . . . × Cn : Ck ∈ O(A(ak , bk )), k = 1, . . . , n}.
Remark. Note that A(a, N b) and A(N a, b) are in O(A).
2.18 Lemma. Suppose now A = A(a, b) and C ∈ O(A). Then C =
A(a, N b)γ for some γ ∈ AutH(a, b).
Proof. By definition C is generated by W ∈ A over Z(A), where W =
U g11 a V g12 b , g11 , g12 ∈ Z, co-prime modulo N. Choose g21 , g22 ∈ Z such that
g11 g22 − g12 g21 = 1 and consider W 0 := U g12 a V g21 b . We see that
γ : U a 7→ W V b 7→ W 0
is an automorphism of H(a, b). 12
Categories Af in and Cf in.
3
3.1 Definition. We fix F to be an algebraically close field of characteristic
zero.
Af in is the category of all the rational Weyl F-algebras with canonical
embeddings as morphisms.
(n)
Af in is the subcategory of n-th Weyl algebras of Af in
(n)
Cf in and Cf in are the subcategories of commutative algebras of Af in and
(n)
Af in respectively.
3.2 Properties.
It is easy to check:
(i) Af in and Cf in are small categories of unital F-algebras with at most
one morphism between two objects.
(ii) The functor A 7→ Z(A) is an isomorphism between Aop
f in and Cf in . We
↑
will describe the inverse functor Z(A) 7→ A : Z 7→ Z below.
(iii)
\
Z(A) = F.
A∈Af in
Proof. (i) and (iii) are immediate from definitions. (ii) is just 2.15.
3.3 Subfield F0 with complex conjugation.
For most of our purposes it is suficient to work within a subfield F0 of F.
We set R0 ⊂ F to the subfield of the totally
real algebraic numbers and
√
F0 = R0 [i], the extension of R0 by i = −1. In particular, F0 ⊆ Qalg , the
algebraic closure of Q.
Note that µ, be the group of all roots of unity in F, is a subset of F0 . Note
also that q + q −1 is a totally real number for q ∈ µ.
We will think of µ as the group of “elements of modulus 1” and introduce
an formal (complex) conjugation x 7→ x∗ , an automorphism of the field
F0 , defined as
∗
: r + is 7→ r − is, for r, s ∈ R0 .
In particular, u∗ = u−1 , for u ∈ µ and r∗ = r for r ∈ R0 .
In particular, given n × n matrix X over F0 , we define an adjoint X ∗
as the matrix obtained by transposition and formal conjugation. We call X
F0 -unitary if X ∗ X = 1.
13
3.4 Remark. Suppose F is of cardinality continuum (which may be assumed without loss of generality). Then F is abstractly isomorphic to the
field of complex numbers. So, in this case F0 ⊂ C and by construction the
involution x 7→ x∗ coincides on F0 with the complex conjugation. This is
immediate from the definition.
3.5 Proposition. An A ∈ Af in is an affine prime algebra over F, finitely
generated over its centre Z(A) as a module. A is finitely generated as an
F-algebra with the field of definition Q[q1 , . . . , qn ], for qk = exp 2πiak bk k =
1, . . . , n.
Every maximal ideal of A is regular.
Proof. Immediate.
Such algebras are considered in [1] which contains in particular the following statement.
3.6 Proposition. Let A, B ∈ Af in . Then Z(A) ∩ Z(B) ∈ Cf in and so
(Z(A) ∩ Z(A))↑ = hA, Bi ∈ Af in .
Proof. Immediate by 2.14. 3.7 Notation. Given A ∈ Af in we write SpA for SpecMax(Z(A)).
3.8 Lemma. Suppose A, B ∈ Af in , B ⊆ A, so Z(A) ⊆ Z(B). Given
β ∈ SpB , a maximal ideal of Z(B), we get a maximal ideal β ∩ Z(A) of
Z(A). The map
b 7→ β ∩ Z(A), πBA : SpB → SpA
is surjective.
Proof. Follows from 2.6(ii). 3.9 Lemma. Let A ∈ Af in and C ∈ O(A). Let B = C ↑ . Then
SpC = SpB and πBA : SpB → SpA
is unramified of order nA .
Proof. Since C = Z(B), we have by definition SpC = SpB .
Let n = nA . Recall that by 3.2 the image of Z(A) in A/α̃ is F · 1, the
subalgebra of scalar matrices. By 2.17(ii) the image of C in the algebra of
matrices is the diagonal subalgebra, that is of the form F · i1 + . . . + F · in ,
where i1 , . . . , in are orthogonal idempotents.
14
P
Note that every maximal ideal of C/α̃ ∩ C is of the form j6=k F · ij , for
some k = 1, . . . , n. It follows that every maximal ideal of C is of the form
X
ξk =
F · ik + α̃ ∩ C, where ik ∈ C, ik = ik + α̃ ∩ C.
j6=k
On the other
P hand, ξk ∩Z(A) = α. Indeed, (ξk ∩Z(A))/(α̃∩C ∩Z(A)) = 0
since F · 1 ∩ j6=k F · ik = 0. Hence ξk ∩ Z(A) ⊇ α̃ ∩ C ∩ Z(A) = α. The
inverse inclusion is obvious by maximality of α.
Hence πBA : c 7→ α = c ∩ Z(A) is an n−to−1 map SpB → SpA . 3.10 Corollary. Suppose B ⊆ A, A, B ∈ Af in . Then
(i)
πBA : SpB → SpA
is an unramified projection of order nA : nB .
−1
(ii) for a generic α ∈ SpA and β ∈ πBA
(α) the extension (Q(β) : Q(α))
is Galois, and the Galois group of the extension acts transitively on
−1
πBA
(α).
Proof. (i) is immediate from 3.9.
−1
(ii) follows from the fact that SpB is irreducible (see 3.2(ii)) and so πBA
(a)
is irreducible over Q(α). 3.11 Remark. The projection πBA can be seen in a graphical way if we
take into account that Z(B) is an extension of Z(A) by a finite set of operators X. Correspondingly, any element of SpB can be determined by αa ξ
(concatenation) where α ∈ SpA and ξ correspond to eigenvalues of operators
X ∈ Z(B). Now
πBA : αa ξ 7→ α.
(7)
3.12 Now suppose C ∈ O(A), α ∈ SpA . Then α = Z(A) ∩ α̃ ⊆ C ∩ α̃.
Consider the irreducible A-module VA (α) introduced in 2.6(iii). Let n =
nA = dimF VA (α). So there is a basis {e1 (α), . . . , en (α)} of VA (α) consisting
of common eigenvectors of all the operators in C.
On the other hand, given such a basis {e1 (α), . . . , en (α)}, α ∈ SpA , we
will have
C = {X ∈ A : ∀α ∈ SpA ∃xi ∈ F
nA
^
i=1
15
Xα ei (α) = xi ei (α)},
where Xα denotes the image of X in the representation A/α̃.
We call such a basis a C-basis.
3.13 Now suppose A is the Weyl algebra A(a, b) and C = A(a, N b), a
maximal commutative Weyl subalgebra as defined in 2.12(ii). Recall that
A(a, N b) is generated over the centre by U a . Maximal ideals α of the centre
A(N a, N b) of A(a, b) are generated by two elements U aN − ûI and V bN − v̂I
of the centre, where û, v̂ ∈ F× and I is the unit of A(a, b).
Choose two elements of F× , suggestively denoted ua and v b , such that
û = (ua )N and v̂ = (v b )N . Let q = exp 2πiab, and let N be the order of the
root of unity q.
We define a canonical A(a, N b)-basis {u}α for VA (α),
α = hU aN − uaN I, V bN − v bN Ii,
a,b a b
a,b
a b
a,b N −1 a b
ua,b
u , v )}
VA (α) = {u (u , v ), u (qu , v ), ..., u (q
to satisfy
U a : ua,b (q k ua , v b ) 7→ q k ua ua,b (q k ua , v b )
V b : ua,b (q k ua , v b ) 7→ v b ua,b (q k−1 ua , v b )
(8)
We often abbreviate the terminology and call such a basis a U a -basis.
We also introduce symmetrically a canonical A(N a, b)-basis, or V b -basis,
for the same module,
a,b
vV
= {va,b (v b , ua ), va,b (qv b , ua ), ..., va,b (q N −1 v b ), ua }
A (α)
to satisfy
V b : va,b (q k v b , ua ) 7→ q k v b va,b (q k v b , ua )
U a : va,b (q k v b , ua ) 7→ ua va,b (q k+1 v b , ua )
(9)
3.14 Suppose now A = A(a, b) and C ∈ O(A). Then by 2.18 there is an
automorphism γ of H(a, b) such that U aγ = S, V bγ = T and S generates C
over Z(A). A canonical C-basis {s}α of VA (α) is defined following (3.13) as
S : s(q k s, t) 7→ q k s s(q k s, t)
T : s(q k s, t) 7→ t s(q k−1 s, t)
(10)
3.15 In the general case of A = A(a1 , . . . , an , b1 , . . . , bn ) we will also consider
maximal commutative subalgebras of the form C = A(a1 , . . . , an , N1 b1 , . . . , Nn bn ),
generated by U a1 , . . . , U an over the centre. Using the fact that an irreducible
16
A-module VA can be represented as a tensor product of modules VAk (αk ) for
Ak = A(ak , bk ) and the corresponding αk .
A canonical A(a1 , . . . , an , N1 b1 , . . . , Nn bn )-basis (or hU a1 , . . . , U an i-basis)
is defined correspondingly as the tensor product of the A(ak , Nk bk )-bases.
3.16 Below, unless indicated otherwise, we assume A = A(a, b), Ǔ = U a ,
V̌ = V b and the A module VA (α) is determined by parameters u = ua , v =
v b . We abbreviate the notation for canonical Ǔ -bases as u = ua,b .
3.17 Lemma. (i) Any two canonical bases {u(u, v), u(qu, v), ...u(q N −1 u, v)}
and {u0 (u, v), u0 (qu, v), ...u0 (q N −1 u, v)} given by the condition (8) can only
differ by a scalar multiplier, that is for some c 6= 0,
u0 (q k u, v) = c u(q k u, v),
k = 0, . . . , N − 1.
(ii) Given also a canonical base {u(u, vq m ), u(qu, vq m ), ...u(q N −1 u, vq m )},
there is a c 6= 0 such that
u(q k u, vq m ) = cq km u(q k u, v),
k = 0, . . . , N − 1.
(iii) Let V̂ ∈ H(a, b) be such that
Ǔ V̂ = q V̂ Ǔ
and v̂ an V̂ -eigenvalue in the module VA (α) and let {û(u, v̂), û(qu, v̂), ..., û(q N −1 u, v̂)}
be a canonical Ǔ -basis in VA (α) with regards to Ǔ and V̂ . Then there is a
c 6= 0 and an integers n such that
û(q k u, v̂) = cq −n
k(k+1)
2
u(q k u, v),
k = 0, . . . , N − 1.
Proof. (i) We will have u0 (u, v) = c u(u, v) for some c since the space of
Ǔ -eigenvectors with a given eigenvalue u in an irreducible (Ǔ , V̌ )-module is
one-dimenional. The rest follows from the definition (8) of the action by V̌
on the bases.
(ii) We have u(u, vq m ) = c u(q k u, v) and so by induction on k, applying
V̌ −1 and (8) to the both parts we get the desired formula.
(iii) First note that from assumptions V̂ = q l V̌ Ǔ n for some integers n
and l. In particular, it follows v̂ = q r+l un v for some integer r.
17
Assume the equality holds for a given k. Apply V̂ −1 to both sides:
V̂ −1 : û(q k u, v̂) 7→ v̂ −1 û(q k+1 u, v̂)
V̂ −1 : cq −n
k(k+1)
2
u(q k u, v) 7→ cq −n
k(k+1)
2
q −n(k+1) v̂ −1 u(q k+1 u, v).
Compairing we get the required. 3.18 Lemma - definition. Set
N −1
1 X −mk
v(vq , u) := √
q
u(uq k , v), m = 0, . . . , N − 1.
N k=0
m
The system
{v(v, u), . . . , v(vq N −1 , u)}
is a canonical V̌ -basis of VA(a,b) satisfying (9)
Proof. One checks directly that the system satisfies (9). 3.19 Groups ΓA (α).
For A ∈ Af in , α ∈ SpA , we define the F-linear transformations of VA (α)
given in a canonical Ǔ -basis as:
µα : u(q m u, v) 7→ u(q m−1 u, v),
and
να : u(uq m , v) 7→ q −m u(uq m , v).
We define ΓA (α) to be the group generated by µα and να .
3.20 Lemma. The definition of µα and να depend on the choice of an
Û -basis {u(q m u, v)} in the following way:
(i) any other basis is of the form
u(q m u0 , v 0 ) := c · γ u(q m u, v), m = 0, 1 . . . , N − 1
for γ ∈ ΓA (α) and c ∈ F;
(ii) for µ0α and να0 corresponding to the basis {u(q m u0 , v 0 )} we will have
µ0α = µγα , να0 = ναγ ;
(iii) the group ΓA (α) does not depend on the choice of the canonical Û basis, that is the group generated by µ0α , να0 is ΓA (α);
18
(iv) moreover, ΓA (α) ∼
= ΓA (β) for any β ∈ SpA .
Proof. (i) is by 3.17. (ii) is immediate and (iii) follows from the fact that
the change of basis corresponds to the inner automorphism of the groups.
(iv) For any β ∈ SpA a vector space isomorphism VA (β) → VA (α) sending
an Û -basis to Û -basis induces, by (ii), a group isomorphism ΓA (α) → ΓA (β).
Given ξ ∈ ΓA (α) and X ∈ A let Xα be the image of X in A(α), an
operator acting on VA (α), and
Xαξ := ξXα ξ −1 .
3.21 Lemma. We have
Ǔανα = Ǔα
V̌ανα = q V̌α
Ǔαµα = q Ǔα
V̌αµα = V̌α
3.22 Corollary. The map hα : ΓA → ΓA (α) defined on the generators as
hα :
µ 7→ µα
ν 7→ να
extends to a surjective homomorphism of groups.
3.23 Remarks. (i) For A commutative we have ΣA (α) = 1, for all α ∈ SpA .
Indeed, nA = 1 in this case.
(ii) In a canonical Ǔ -basis the group ΣA (α) is represented by a subgroup
SU(nA , F0 ) of F0 -unitary matrices. This is immediate by definition.
3.24 Lemma. (i) If ξ ∈ ΣA (α) and Xαγ = Xα for all X ∈ A, then ξ = gI,
for some g ∈ F[nA ], root of unity of order nA .
(ii) ΣA (α) is finite.
Proof. (i) Suppose Xa ξ = ξXa , for all X ∈ A. Let e ∈ VA (a) be an
eigenvector of ξ with eigenvalue g. Then ξ 0 := ξ−gI is a linear transformation
of VA (a) which commutes with all the Xa and
ξ 0 VA (a) = W ⊂ VA (a), W 6= VA (a).
19
But Xa W = Xa ξ 0 VA (a) = ξ 0 Xa VA (a) = ξ 0 VA (a) = W, that is W is an Asubmodule of VA (a). By irreducibility of the latter, W = 0 and so ξ = gI.
But since det ξ = 1, g nA = 1.
(ii) By 2.16 ΣA contains a subgroup Σ0A of finite index which fixes setwise
O(A).
By definition elements of Σ0A (α) send a C-basis to a C-basis, for any C ∈
O(A). So, a possibly smaller subgroup of finite index preserves C-eigenvalues,
for all C ∈ O(A). We may assume Σ0A (α) has this property.
Hence any ξ ∈ Σ0A (α) commutes with any Xα , for X ∈ C. By 2.6(ii) ξ
commutes with A/α̃. By (i) ξ = gI, g ∈ F[nA ]. 3.25 U -frames EUA (α). Given α ∈ SpA , choose a Ǔ -eigenvector e ∈ VA (α)
and set
EUA (α) := ΓA (α) · ec .
We call EUA (α) an U -frame of VA (α).
Clearly, e can be included in a canonical basis
{u(uq m , v) : m = 0, . . . , N − 1} as, say, e = u(uq k , v). Then
U
{u(uq m , v) : m = 0, . . . , N − 1} = {µm
α e : m = 0, . . . , N − 1} ⊆ EA (α).
By 3.17 and the fact that να ∈ ΓA (α), any other canonical Ǔ -basis is a subset
of EUA (α) of the form ∆e for a cyclic subgroup ∆ ⊂ ΓA (α).
3.26 Inner product. Let F0 [EuA (α)] stand for the F0 -linear span of EUA (α),
that is the F0 -vector subspace of VA (α) span over EUA (α). By above {u(uq m , v) :
m = 0, . . . , N − 1} is a basis of the vector space F0 [EUA (α)].
We introduce an inner product on F0 [EUA (α)] by declaring the basis
orthonormal. By 3.23(ii) the action of ΓA (α) on F0 [EUA (α)], and so on
VA (α), is given by unitary F0 -matrices. Since any other canonical basis can
be obtained by a transformation in ΓA (α), the inner product structure does
not depend on the choice of the initial basis. We denote the inner product
between two elements f, g ∈ F0 [EUA (α)] by
hf |giα .
We usually omit the subscript.
Following physics tradition we write the property of F0 -sesquilinearity as
hf |τ gi = τ hf |gi = hτ ∗ f |gi
for τ ∈ F0 .
20
3.27 Example. The definition of a V basis in terms of U -basis in 3.18
gives
1
hu(uq k , v)|v(vq m , u)i = √ q km .
N
3.28 We call each instance of VA (α) endowed with an U -frame EUA (α) an
algebraic-Hilbert space and denote it VA (α). Note EUA (α) gives rise to the
unique inner product on F0 [EUA (α)].
3.29 Remark. As in the remark in 3.3 assume F = C. Then we may use the
same canonical basis to introduce an inner product on C[EUA (α)] = VA (α).
This gives VA (α) a structure of a Hilbert space which extends the algebraicHilbert space structure on F0 [EUA (α)].
3.30 We call S ∈ A(a1 , . . . , an , b1 , . . . , bn ) a pseudo-unitary operator if
S ∈ H(a1 , . . . , an , b1 , . . . , bn ). We set a formal adjoint of S to be S −1 .
3.31 We call P ∈ A(a1 , . . . , an , b1 , . . . , bn ) a pseudo-selfadjoint operator
if
n
X
P =
ai Si , some pseudo-unitary Si and ai ∈ F0
i=1
such that
∗
P = P :=
for formal conjugates
a1 , . . . , an ∈ k.
a∗i
and adjoints
n
X
a∗i Si∗ ,
i=1
Si∗ of
ai and Si respectively.
3.32 Suppose S ∈ A = A(a1 , . . . , an , b1 , . . . , bn ) is pseudo-unitary, α ∈ SpA
and s is an eigenvalue of S acting on VA (α). Then inner product in F0 [EUA (α)]
is invariant under the action of s−1 S, that is for any f, g ∈ F0 [EUA (α)]
hs−1 Sf |s−1 Sgiα = hf |giα .
In particular, the matrix of s−1 S in a canonical U -basis is F0 -orthogonal.
Proof. We may assume without loss of generality that n = 1. By skewlinearity we just need to check the identity for f, g belonging to a given
canonical U -basis, so we may assume f = u(uq m , v) and g = u(uq k , v).
Clearly, if the statement is true for S = Ǔ and for S = V̌ , then it is true for
any group word of these, so is true for S. The lemma follows. 21
3.33 For each rational Weyl algebra A one particular module corresponding
to the the ideal α = h1, 1i =: 1 (i.e. uN = 1 = v N ) will play at the end a
special role. We call the module VA (1) and the respective algebraic-Hilbert
space VA (1) the principal A-module and the principal algebraicHilbert space. The main property of this module is that the eigenvalues of
U a and V b are roots of unity.
Again, the notions introduced above have natural extension to rational
Weyl n-algebras.
3.34 Proposition. Let VA (1) be the principal algebraic-Hibert space, S a
pseudo-unitary and P a pseudo-selfadjoint operators. Then
(i) there is an orthonormal basis of S-eigenvectors of the space;
(ii) eigenvalues of S are in F0 [N ] for some N (i.e. roots of unity);
(iii) there is an orthonormal basis of P -eigenvectors of the space;
(iv) eigenvalues of P are totally real algebraic numbers.
Proof. (i) and (ii) is immediate by 3.32.
Note also that by the argument in 3.32
hP f |gi = hf |P gi
for any f, g ∈ F0 [EA (1)]. Thus P is a self-adjoint operator in the (formal)
Hilber space over field F0 ⊂ C with complex conjugation. (iii) and (iv) follow.
3.35 Suppose A = A(a, b), C ∈ O(A) and e = {e0 , . . . , eN −1 } is a canonical
C-basis of VA (α) such that he0 |e0 i = 1. Then the basis is orthonormal and
the transition matrix from a canonical U -basis to e is F0 -unitary.
Proof. By 3.14 there are pseudo-unitary generators S, T of A such that C
is generated by a pseudo-unitary S and the basis e is of the form {s(sq m , t) :
m = 0, . . . , N − 1} satisfying (10). Hence it is orthonormal.
By definition S = q l Ǔ k V̌ m for some integer l, k and m such that k and m
are co-prime modulo N. We may also assume without loss of generality that
l = 0. Then by (8) we may set s = uk v m .
Denote
wn,p = hs(sq n , t)|u(uq p , v)i, p = 0, 1, . . . , N − 1.
22
Apply s−1 S to both parts of the pairing. Then by 3.35 we get
wn,p = hs(sq n , t)|u(uq p , v)i = hq n s(sq n , t)|q (p−m)k u(uq p−m , v)i =
= q (p−m)k−n hs(sq n , t)|u(uq p−m , v)i = q (p−m)k−n · wn,p−m .
Using the fact that Ǔ = q l S c T d , for some integer l, c and d, and applying
u−1 Ǔ to the initial pairing one gets similarly
wn,p = q r wo,p
for some integer r depending on l, c, d, n, p and o.
We thus get an F0 -linear homogeneous system of equation for the transition matrix W = {wnp }. We may thus assume that W is an F0 -matrix,so
conjugation is applicable, and since it takes an F0 -orthonormal basis to an
F0 -orthonormal basis, W is F0 -orthogonal. 3.36 Remark. It is not hard to write down the matrix W in terms of F0 .
3.37 Corollary. The binary relation e ∈ EUA (α) (between e and α) is definable if and only if the N + 1-ary relation “{e0 , . . . , eN −1 } is a canonical
orthonormal U -basis of VA (α)” is definable.
3.38 Definitions. Let C ∈ O(A). Set EC (α) for each α ∈ SpA to be the
set of C-eigenvectors in F0 [EUA (α)] of norm 1. Set
[
EA (α) :=
EC
A (α).
C∈O(A)
3.39 Lemma. Let C ∈ O(A) and suppose that the binary relation e ∈
EUA (α) is definable.
Then the binary relation e ∈ EC
A (α) (between e and α) is definable.
Proof. By 3.14 and 3.35 we can pick a canonical U -basis u(α) in EUA (α),
for each α ∈ SpA , and a F0 unitary matrix R such that e(α) = Ru(α) is a
canonical orthonormal C-basis. So, e(α) ⊆ EC
A (α) for each α. By 3.17 there
is a finite group of F0 -unitary transformations (independent on α) such that
any element in EC
A (α) can be obtained by applying these transformations to
e(α). 3.40 Corollary. e ∈ EA (α) is definable if and only if e ∈ EUA (α) is definable.
23
3.41 Lemma-definition For all α ∈ SpA , the algebraic-Hilbert spaces
VA (α) = (VA (α), EA (α)) are of the same isomorphism type. We denote
it
VA = (VA , EA ).
Proof. Immediate by 3.20 and definitions. 3.42 Lemma. Suppose F = C and consider a Hilbert space structure
on VA (α) extending the algebraic-Hilbert space structure on F0 [EA (α)] (see
3.29).
Let C ∈ O(A), Cα be the image of C in the representation A/α̃ of A on
VA (α) and let X ∈ Cα . Then the adjoint Xα∗ ∈ Cα .
Proof. By 2.6(iii) in any canonical C-basis the algebra Cα is represented
as the algebra of the diagonal complex matrices. This is closed under taking
adjoints. 3.43 Lemma. Let B ⊆ A, B ∈ Af in and let Bα ⊆ A/α̃ be the image of B
in A/α̃. Then Bα is a semi-sipmle algebra.
Proof. Without loss of generality we may assume that F = C. Then
each C ∈ O(B) ⊆ O(A), Cα is closed under taking adjoints. Since Bα is
generated by all such Cα (see 2.6(ii)), Bα is closed under taking adjoints, in
particular the involution Xα 7→ Xα∗ is an anti-automorphism of the ring Bα .
This implies that the Jacobson radical of Bα is closed under the involution.
It remains to invoke a well-known argument that under the above condition the Jacobson radical J of the matrix algebra Bα is trivial, that is Bα is
semisimple. Indeed, suppose Y ∈ J. Then Y ∗ Y ∈ J and is nilpotent, that
is (Y ∗ Y )k = 0 for some k. By positive-semidefiniteness of Y ∗ Y , this implies
Y ∗ Y = 0. So the inner product hY v|Y vi = 0, for any vector v. Hence Y = 0.
3.44 The bundle of algebraic-Hilbert A-modules
We treat the family
{(VA (α) : α ∈ SpA }
of A-modules as a bundle
VA :=
[
{VA (α) : α ∈ SpA }
with the projection map ev : VA → SpA , ev(v) = α iff v ∈ VA (α), the
algebraic-Hilbert space structure and the A-module structure on each fibre.
24
Note that as a bundle of algebraic-Hilbert spaces it is trivial, but as a
bundle of A-modules, in general it can be highly non-trivial. See [11] for
a statement and a proof of the fact that the bundle of A-modules is not
definable in F even when EA is omitted.
3.45 Let A, B ∈ Af in , B ⊆ A. Hence VA (α) can also be considered a
B-module, in general, reducible. By 3.43 VA (α) splits into a direct sum
of irreducible Bα -modules, so irreducible B-modules. The number of the
irreducible components is determined by dimensions nA and nB , so is equal
to nA : nB . Invoking 3.10 we obtain,
M
VA (α) =
VAB (β), VAB (β) ∼
= VB (β),
πBA (β)=α
for uniquely determined B-submodules VAB (β) ⊆ VA (α) for each maximal
ideal β ⊆ Z(B), α ⊆ β.
3.46 We will describe the embeddings of 3.45 in terms of the canonical
bases.
Let as before A be generated by Ǔ = U a and V̌ = V b , nA = N and
{u(uq m , v) : m = 0, . . . , N − 1} be a canonical orthonormal Ǔ basis in a
A-module VA (α), where α is determined by uN and v N .
Then B ⊂ A is the algebra hǓ n , V̌ k i generated by Ǔ n and V̌ k for some
k, n ∈ N. In fact, we may assume that B = hǓ n , V̌ i or B = hǓ , V̌ n i, since
any B can be reached by such two steps.
In case B = hǓ n , V̌ i a B-submodules VAB (β) can be constructed by
choosing an Ǔ n -eigenvector un,1 (un , v) of modulus 1 and then generating a
whole canonical Ǔ n -basis by applying V̌ according to (8). Since Ǔ N acts on
the module as a scalar, we may assume that n divides N.
It is easy to see that a canonical V̌ -basis in VAB (β) can be identified as
follows:
N
(11)
v1,n (vq pn , un ) = v(vq pn , u), p = 0, . . . , − 1,
n
where on the right we have elements of the V̌ -basis in VA (α).
2πi
Set, for ` = 0, . . . , n − 1, m = 0, . . . , Nn − 1, ξ = e n ,
n−1
u
n,1
n mn
(u q
n−1
1 X
q m` X k`
, vq ) := √
u(uq m ξ k , vq ` ) = √
ξ u(uq m ξ k , v)
n k=0
n k=0
`
25
(12)
The second equality follows from 3.17(ii).
One checks immediately that for a fixed ` this is a canonical orthonormal Ǔ n -basis for the B-submodule corresponding to the maximal ideal β` of
N
N`
N
Z(B) = hǓ N , V̌ n i generated by Ǔ N − uN I and V̌ n − v n I. There are n
distinct such β and respective non-isomorphic B-modules.
In case B = hǓ , V̌ n i set, for ` = 0, . . . , n − 1,
u1,n (uq nm+` , v n ) := u(uq nm+` , v), m = 0, . . . ,
N
−1
n
(13)
This is a canonical orthonormal Ǔ n -basis for the B-submodule corresponding
N
N
N`
to the maximal ideal β` of Z(B) = hǓ n , V̌ N i generated by Ǔ n − u n I and
V̌ N − v N I.
3.47 Lemma. Suppose as above A = hǓ , V̌ i, B = hǓ n , V̌ k i. Let VAB (β) ⊂
VA (α) be a B-submodule and {uB (u0 qBm , v 0 ) : 0 ≤ m < nB } and {vB (v 0 qBp , u) :
0 ≤ p < nB } be canonical Ǔ n - and V̌ k -bases in VAB (β). Then
huB (u0 qBm , v 0 )|vB (v 0 qBp , u)iVA (α) = huB (u0 qBm , v 0 )|vB (v 0 qBp , u)iVAB (β)
In particular, the inner product is preserved in the embedding VAB (β) →
VA (α).
Proof. As above it is enough to check the case k = 1. Then qB = q n ,
where now n = nA : nB . We can rewrite
uB ((u0 qBm , v 0 ) = u1,n (un q mn , v)
and
vB ((u0 qBm , v 0 ) = vn,1 (vq mn , un ).
By 3.27
1
1
huB (u0 qBm , v 0 )|vB (v 0 qBp , u0 )iVAB (β) = √ qBmp = √ q nmp .
nB
nB
By (12) and (11) we can calculate
n−1
nA
1 X
hu1,n (un q nm , v)|vn,1 (vq np , un )iVA (α) = √
hu(uq m+ n k , v)|v(vq np , u)i =
n k=0
√
1 nmp
1
n
= √ · n√ q
= √ · q nmp
nA
nA
n
(taking into account that q is of order nA .)
It remains to note that
26
3.48 Keeping the notation and assumptions of 3.45 we will consider Bmodule isomorphisms pβBA , abbreviated to pβ ,
pβ : VB (β) → VAB (β) ⊂ VA (α)
such that
pβ (EB (β)) ⊆ EA (α) ∩ VAB (β).
In case B = hǓ n , V̌ i we define the maps on the bases, according to (12),
as
n−1
β
p :u
n,1
n mn
(u q
N
1 X
N
, vq ) 7→ √
u(uq m+ n k , vq ` ), m = 0, . . . , − 1,
n
n k=0
`
(14)
where un,1 (un q mn , vq ` ) is now naming elements of a canonical basis of VB (β).
In case B = hǓ , V̌ n i
N
−1
n
Note that in terms of the V̌ bases (14) can be rewritten as
pβ : u1,n (uq nm+` , v n ) 7→ u(uq nm+` , v), m = 0, . . . ,
pβ : vn,1 (vq nm+` , un ) 7→ v(vq nm+` , u), m = 0, . . . ,
N
−1
n
(15)
(16)
(just use the symmetry between V̌ and Ǔ ).
Finally, we would like to remark that the definition of pβ does depend on
the choice of the canonical orthonormal bases in VB (β) and VA (α).
3.49 Lemma. Let pβ and p0 β , β ∈ SpB be two isomorphisms VB (β) →
VAB (β). Then there is a gβ ∈ F∗ [nB ] such that pβ = p0 β gβ . In particular,
there are exactly nB choices for pβ for a given β ∈ SpB .
Proof. Consider (p0 β )−1 pβ . This is a linear transformation commuting
with operators from B/β̃, since pβ and p0 β are B-module isomorphisms. By
the argument in 3.24(i) (p0 β )−1 pβ = gβ I for some gβ ∈ F∗ [nB ]. 3.50 Definition A map pβ satisfying 3.48 will be called a local morphism
at β of VB → VA .
We call the collection pβ of all the local morphism at β a fibre of a
morphism VB → VA at β.
We call the family of fibres
pBA = {pβ : β ∈ SpB }
the morphism VB → VA .
27
3.51 Remark. Lemma 3.49 implies that VA is determined by A uniquely
0
up to isomorphism, that is given VA and VA
both satisfying the definition
in 3.48, there is a family
0
pα : VA (α) → VA
(α), α ∈ SpA
of local isomorphisms establishing an isomorphism between the two structures.
3.52 Continuing the notation of 3.45, set
ΓAB (β) = {γ ∈ ΓA (α) : γVAB (β) = VAB (β)}.
Let
γ 7→ γ̌, ΓAB (β) → GL (VB (β))
be the group homomorphism defined as follows. For γ ∈ ΓAB (β) set γ̌ to be
the unique Af in -transformation γ̌ of VB (β) such that
γpβ v = pβ γ̌ v, for all v ∈ VB (β)
(the transformation on VB (β) induced from VAB (β) by the isomorphism pβ ).
nA
In general, det γ̌ does not have to be equal to 1 (in fact, (det γ̌) nB = 1),
so we define
− n1
∗
πBA
(β) : γ 7→ (det γ̌)
B
· γ̌, γ̌ = (pβ )−1 γpβ
a 1-nB -correspondence, a multivalued map into ΓB (β), or a homomorphism
∗
πBA
(β) : ΓA (α) → ΓB (β)/F∗ [nB ] ⊆ PGL (VB (β)) ,
with the domain ΓAB (b) ⊂ ΓA (α). Here F∗ [nB ] is the group of roots of
unity of order nB , which is also the centre of ΓB (β). We write PΓB (β) for
ΓB (β)/F∗ [nB ] below for any B ∈ Af in .
Note that by definition, for α ∈ F∗ [nA ], αγ
ˇ = αγ̌, so in fact we have
equivalently defined
∗
πBA
(β) : PΓA (α) → PΓB (β),
a homomorphism of the projective groups.
28
3.53 Remark. Recall that the action of ΓA (α) on the algebra A/α̃ reduces
to the action of PΓA (α) on the algebra. So, in this regard we do not loose
any information switching to the projective groups.
3.54 VA as structures.
For each A ∈ Af in the bundle VA can be described as a two sorted structure in a language LV . First sort, named F will stand for (the universe of)
the field F. The language LV will have names for all Zariski closed subsets of
Fn , all n, including the ternary relations for graphs of + and ·, the field operations. One of the closed subsets of Fm , where m is the number of generators
of the affine algebra A, will correspond to the spectrum SpA = Spec(Z(A))
of the centre Z(A) of A : for each of the generating operators X1 , . . . , Xm of
Z(A) and for each maximal ideal α of Z(A) we put in correspondence the
m-tuple hx1 , . . . , xm i ∈ Fm such that X1 − x1 I, . . . , Xm − xm I generate the
ideal α. We call this closed subset SpA .
The second sort S
named V will stand for VA , that is as a universe (set of
points) it is the set {VA (α) : α ∈ SpA } described in 3.44.
There is in LV a symbol ev for a map ev : VA → SpA ⊆ Fm which is
interpreted as described in 3.44. In particular, for each α ∈ SpA , ev−1 (a) =
VA (a) is definable.
The language LV also contains a binary relation RX (v, w) for each X ∈ A,
which is interpreted on sort VA as X · v = w for v, w ∈ VA (α), for α ∈ SpA .
There is also a ternary relation S(ξ, v, w) between ξ ∈ F, v, w ∈ VA (a)
allowing us to say that ξ · v = w.
A ternary relation v +A u = w on VA will be saying that there exists α
such that u, v, w ∈ VA (α), and that v + u = w in the sense of the module
structure on VA (α).
Finally, LV contains a binary symbol E U , which distinguishes in VA a
relation
E U (e, α) ≡ e ∈ EUA (α) & α ∈ SpA
as well as the binary symbols IPr , for non-zero r ∈ F0 , which are interpreted
on VA as
IPr (e1 , e2 ) ≡ ∃α ∈ SpA e1 , e2 ∈ EA (α) & he1 |e2 iα = r,
the value of the inner product (see 3.25).
3.55 Note that by 3.40 e ∈ EUA (α) is definable if and only if e ∈ EA (α) is
definable if and only if it is definable that for some C ∈ O(A), e ∈ EC
A (α).
29
Note also that defining e ∈ EA (α) from e ∈ EUA (α) may require formulas
which depend on A.
3.56 Remark. The language LV has no symbols for ΓA or group actions.
The group plays a role in the theory of VA through axioms describing canonical bases in EA (a) and the matrices of linear transformations from one base
to another, see subsection 2.1 of [11].
3.57 Definition. Let N and M be structures in languages LN and LM ,
correspondingly. We say that a structure N is almost interpretable in M
if,
(i) there is a structure N0 in a language L0N , which is interpretable both
in M and in N;
(ii) N is prime over N0 , in particular, any automorphism of N0 can be
lifted to an automorphism of N;
(iii) N ⊆ aclN (N 0 ), that is the universe N of N is in the algebraic closure,
in the sense of N, of N 0 .
Remark. [5], [11] and [8] discuss examples of Zariski structures which
are almost interpretable but not interpretable in an algebraically closed field
F.
3.58 Theorem. Given an A ∈ Af in ,
1. VA is almost interpretable in the field F, in particular, the first order
theory of the structure VA is categorical in uncountable cardinalities.
2. VA is a Zariski geometry in the topology given by the positive formulas
with all quantifiers restricted to the predicate E.
Proof. 1. The field F is definable in VA as the sort F by definition, see
3.54. It is also clear by the construction that aclVA (F ) = VA . Theorem A of
[11] proves that VA is categorical in uncountable cardinalities and that VA
is prime over F.
2. This is a special case of Theorem B of [11]. 3.59 Corollary. Given Af in , A, B ∈ Af in , VB is almost interpretable in
VA . .
3.60 Proposition. Assume B ⊂ A. Then VB is definable in VA .
We may assume that A is the Weyl algebra generated by Ǔ and V̌ , N =
nA , and B is generated by Ǔ n and V̌ m , n, m divide N. Let hǔ, v̌i ∈ F2 define
30
α as in 3.13. We deduce that ǔ = uN , v̌ = v N , for some u, v ∈ F× (following
N
N
notations of 3.15) and β is determined by hu m , v n i.
The subbundle {VAB (β) : β ∈ SpB }, of the bundle {VA (α) : α ∈ SpA } as
α = πBA (β) is defined by the formula
N
N
N
N
N
N
“w ∈ VB (u m , v n )” := Ǔ m w = u m w & V̌ n w = v n w.
Since VA (α) are B-modules, the VAB (β) get the structure of B-modules
too.
Finally, we define
EAB (β) := EA (α) ∩ VAB (β) for α = πBA (β).
Thus we get the bundle of algebraic-Hilbert B-modules on {VAB (β) : β ∈
SpB }.
This is canonically (but not uniquely) isomorphic to the bundle of algebraicHilbert B-modules on {VB (β) : β ∈ SpB } by fibrewise isomorphisms pβBA
between VB (β) and the corresponding submodules of VAB (β) as defined in
3.48. 3.61 Remark. The image of a 0-definable relation on VB under pBA does
not depend on the choice of a representative in pBA since definable relations
are invariant under isomorphisms.
Moreover, pBA sends Zariski-closed relations on VB to Zariski-closed ones
on VA , that is this is a Zariski-continuous Zariski-closed morphism.
3.62 Proposition. The categories Af in (F) and Vf in (F) are isomorphic. In
other words, the functor
A 7→ VA
is invertible.
Proof. In fact, even the weaker functor A 7→ VA is invertible. We
only need to show how to recover A from VA . Since VA has the structure
of a bundle of A-modules, we are already given the action of A on each
module VA (α). It remains to see that the annulator of all the modules VA (α),
α ∈ SpA , is trivial. This is immediate from the classification of irreducible
representations of A, see [1]. 31
4
The categories A∗f in and Vf∗in.
4.1 Define the category A∗f in (C) as consisting of algebras A ∈ Af in (C)
(that is F = C) endowed with an involution X 7→ X ∗ , which is defined on
the pseudo-unitary operators by the rule
∗
: U a V b 7→ V −b U −a ,
and on C is defined as complex conjugation. The morphisms of A∗f in (C)
will be the same as of Af in (C), which of course preserve the involution by
definition.
Given A ∈ Af in (C), we will write A∗ for the algebra in A∗f in (C) obtained
by introducing the involution. We consider the functor
FA : A 7→ A∗ ; Af in (C) → A∗f in (C).
4.2 Lemma. The functor FA is an isomorphism of the categories Af in (C)
and A∗f in (C).
Proof. Immediate. 4.3 The objects of the category Vf∗in (C) will be defined as the real-coordinate
parts of the Zariski geometries VA , for A = A(a, b) ∈ A∗f in (C). We define
the real-coordinate part of SpA to be
SpA (R) = {huN , v N i : |u| = |v| = 1},
that is the eigenvalues of the generators U a and V b of the algebras on the
module VA (α) are of modulus 1 when α ∈ SpA (R).
Note that for α ∈ SpA (R), pseudo-unitary operators act in the algebraicHilbert module VA (α) as unitary, if one considers VA (α) as a Hilbert space
as described in
Now define the object
VA (R) = {VA (α) : α ∈ SpA (R)}
as the bundle of Hilbert spaces VA (α) as defined in 3.29, with the distinguished bases EA (α).
Clearly, VA (R) is a substructure of VA .
The morphisms in the category Vf∗in (C) are given by the same maps as
in Vf in (C) restricted to the substructures VA (R).
We consider the functor of taking the real co-ordinate part
FV : VA 7→ VA (R); Vf in (C) → Vf in (C)
32
4.4 Lemma. FV is an isomorphism of the categories Vf in (C) and Vf∗in (C).
Proof. By 4.2 and 3.62 it suffices to prove that the functor
A 7→ VA (R)
is invertible.
Suppose not. Then there is a non-zero operator X ∈ A which annihilates
all the modules VA (α) with α ∈ SpA (R).
Note that the set
Null(X) = {α ∈ SpA : ∀v ∈ VA (α) Xv = 0}
is definable in the Zariski structure VA . Moreover, it is Zariski closed since
N
N
Null(X) = {α ∈ SpA : ∃u, v hu , v i = α &
N
−1
^
u(uq k , v) ∈ EUA & Xu(uq k , v) = 0}
k=0
is given by a core-formula (see [11]).
Our assumption implies that SpA (R) ⊆ Null(X). But the real 2-torus
SpA (R) is Zariski dense in the complex 2-torus SpA = C× × C× , so
Null(X) = SpA
that is X annihilates all the irreducible modules. Then X = 0. The contradiction. 4.5 Model-theoretic commentary. Note that SpA (R) is a real 2-torus
and as such can be defined in the field R. Also each Hilbert space VA (α)
with the distinguished bases EA (α) is definable in R using parameters (to
fix the frame, the family of distinguished bases). The bundle VA (R) is not
interpretable in R (prove it). However, one can see that VA (R) is prime over
its definable substructure SpA (R).
5
Regular unitary transformations.
In this section we work with a fixed Weyl algebra A = A(a, b) generated by
Û = U a and V̂ = V b . We keep using the notation of the previous section. In
partucular, every α ∈ SpA is of the form α = α(u, v) = huN , v N i, for some
u, v ∈ F× .
33
5.1 Let
L = {Lα,α0 : VA (α) → VA (α0 ), α, α0 ∈ SpA }
be a family of F-linear transformations on the bundle VA . We say that L is
regular unitary if there is a rational invertible map
hw1 , w2 i : F× × F× → F× × F×
and a positive quantifier-free formula Λ(b, b0 , u, v) (in variables b, b0 ∈ EA
and u, v ∈ F× ) with the following conditions satisfied:
(i) There are subalgebras B = hS, T i and B 0 = hS 0 , T 0 i of A, with S, T
and S 0 , T 0 pseudo-unitary generators, and an isomorphism
σ : B → B0;
S σ = S 0, T σ = T 0.
(ii) Given u, v ∈ F× consider α = α(u, v) = huN , v N i and α0 = α0 (u, v) =
hw1 (u, v)N , w2 (u, v)N i. Then there are β ∈ SpB and β 0 ∈ SpB 0 , with β =
β(u, v) and β 0 = β(u, v) rational functions of u, v such that VB (β) ⊆
VA (α), VB 0 (β 0 ) ⊆ VA (α0 ) and
Lα,α0 : VB (β) → VB 0 (β 0 )
is an isomorphism between the representation of B to the representation
of B 0 preserving inner product. The image of a canonical base is a
canonical base and for any X ∈ B and e ∈ VB (β)
Lα,α0 (Xe) = X σ (Lα,α0 e).
(iii) For α, α0 , β, β 0 of (ii) the formula Λ(b, b0 , u, v) defines the restriction of
the map Lα,α0 to EB (β) :
Lα,α0 : b 7→ b0 ; EB (β) → EB 0 (β 0 ).
(iv) There is a positive integer M such that, given u, v and α, α0 , β, β 0 as in
(ii) and (iii) and given u0 , v 0 ∈ F× such that β(u, v) = β(u0 , v 0 ) there is
a ζ ∈ F0 such that ζ M = 1 and the formula Λ(b, b0 , u0 , v 0 ) defines on
VA (α) the partial map
ζ · Lα,α0 : b 7→ b0 = ζ · Lα,α0 (b), EB (β) → EB 0 (β 0 ).
34
We write
L : VA → VA
and call VB the domain and VB 0 the range of L.
5.2 Commentary. In the second half of the paper we will only work
with principal modules. There is only one such B-submodule and one Dsubmodule in VA (1). In this context L is a partial map on VA (1), that is its
domain of definition and its range may be proper subspaces of the algebraicHilbert spaces.
5.3 Remark. Let V̄B (β) and V̄B 0 (β 0 ) stand for the quotients of VB (β)
and VB 0 (β 0 ) respectively, by the equivalence relation
e ≈ e0 iff ∃ζ ζ M = 1 & e0 = ζ e.
Then the formula
∃u, v : β(u, v) = β & Λ(b, b0 , u, v)
(17)
defines for every β a well-defined map
V̄B (β) → V̄B 0 (β 0 )
for a corresponding unique β 0 = β 0 (u, v).
5.4 Let L be regular on VA . Let Nc = nB = nB 0 , c a positive integer. Then
given u, v we will have by (ii) and (iii) of 5.1 α(u, v), α0 (u, v), β(u, v) and
β 0 (u, v) and respective canonical S- and S 0 -bases {s(sq ck , t), 0 ≤ k < Nc }
and {s0 (s0 q ck , t0 ), 0 ≤ k < Nc } of VB (β) and VB 0 (β 0 ) correspondingly, with
s = s(u, v) and s0 = s0 (u, v), such that
N
− 1,
c
By 3.39 and 3.17 there is a finite group Γ of F0 - unitary Nc ×
γ so that any b ∈ EB (β) has the form
Lα,α0 : s(sq ck , t) 7→ s0 (s0 q ck , t0 ), k = 0, . . . ,
N
b=
−1
c
X
γ(1, k) · s(sq ck , t).
k=0
35
(18)
N
-matrices
c
Then clearly
N
Lα,α0 (b) = b0 =
−1
d
X
γ(1, k) · s0 (s0 q ck , t0 ).
k=0
We thus have the representation of Lα,α0 in the form
N
N
Lα,α0 (b) = b0 ⇔
_
b=
γ∈Γ
−1
c
X
γ(1, k)·s(sq ck , t) & b0 =
k=0
−1
c
X
γ(1, k)·s0 (s0 q ck , t0 ).
k=0
In fact, for the same reason applied in VA (α) we have equalities
s(sq ck , t) =
N
−1
X
cn,k u(uq n , v)
n=0
and
0
0 ck
0
s (s q , t ) =
N
−1
X
c0n,k u(w1 q n , w2 )
n=0
cn,k , c0n,k
for some fixed coefficients
∈ Q[q] and w1 = w1 (u, v), w2 = w2 (u, v)
defined in 5.1.
Hence the above can be represented in the form
Lα,α0 (b) = b0 ⇔
N
_
γ∈Γ
b=
−1 N −1
c
X
X
(19)
N
γ(1, k) · cn,k u(uq n , v) & b0 =
k=0 n=0
−1 N −1
c
X
X
γ(1, k) · cn,k u(w1 q n , w2 )
k=0 n=0
5.5 Proposition. (i) Any Λ(b, b0 , u, v) defining a regular unitary transformation can be equivalently represented in the form (19)
(ii) The formula (17) defines a Zariski closed binary relation (between b
and d).
Proof. (i) is proved just above.
(ii) The formula (17) with Λ(b, b0 , u, v) in the form (19) is a canonical
form of a Zariski closed relation (a core formula) as defined in [11], section
3. 36
5.6 Lemma. To every regular L : VA → VA one can associate a rational
2×2-matrix gL = (glk ) of determinant 1 and an automorphism of the rational
Heisenberg groups H(Q, Q).
Proof. Given L, by definition we are given S, T, S 0 , T 0 ∈ H(a, b) and the
isomorphism σ between subgroups of H(a, b) generated by S, T and S 0 , T 0 .
This isomorphism corresponds to an isomorphism of Lie subalgebras of the
Lie algebra hP, Qi over the rationals with generators corresponding to S, T
and S 0 , T 0 .
This can be naturally extended to the automorphism of the Lie algebra
hP, Qi which induces an automorphism
σ̌ : H(Q, Q) → H(Q, Q)
of the rational Heisenberg groups of rank 2. We thus will have
σ̌ :
7 cU Û g11 V̂ g12
S→
7 S 0 Û →
;
T →
7 T 0 V̂ →
7 cV Û g21 V̂ g22
for some rational matrix (gkl ) and cS , cT , cU , cV ∈ Z(H(Q, Q)).
The fact that σ̌ is an automorphism implies that det(gkl ) = 1 (see a
remark in 2.8). 5.7 Proposition. To every regular L : VA → VA one can associate an
(1)
automorphism σ̌ of the category Af in such that for any non-zero rational
r, s,
s(s−1)
r(r−1)
σ̌ : hÛ r , V̂ s i → hq − 2 cU Û rg11 V̂ rg12 , q − 2 cV Û sg21 V̂ sg22 i.
The same determines an automorphism of the algebra
[
A(1) =
A.
(1)
A∈Af in
Proof. This follows by 5.6. The formula for σ̌ is obtained by using the
Campbell - Hausdorff formula for the Lie exponentiation. 5.8 Question. Prove that, conversely to 5.6, given A, any matrix (gkl ) ∈
SL(2, Q) gives rise to a regular unitary transformation L of VA .
37
5.9 Let q, N and the invariants of A-modules be determined in agreement
with 3.16. In particular, α = huN , v N i and, conversely, u, v are defined by
α up to roots of unity of order N. Denote α0 = hv N , uN i. The Fourier
transform ΦA = Φ : VA (α) → VA (α0 ) is defined on the A-modules as
N −1
1 X −mk
q
u(vq k , u),
Φ : u(uq , v) 7→ √
N k=0
m
(20)
or, according to (3.18),
Φ : u(uq m , v) 7→ v(uq m , v),
and the domain and range of Φ is the whole of VA (α).
Applying Φ twice we easily get
Φ : u(uq m , v) 7→ u(uq −m , v).
Note that for the principal A-module
ΦA = Φ : VA (1) → VA (1).
5.10 Proposition. Φ is a regular unitary transformation associated with
the matrix
0 1
−1 0
and the automorphism of the category
σ̂ :
Ur →
7
Vr
.
Vs →
7
U −s
Proof. (i)-(iii) of the definition
5.11 Gaussian transformation. We introduce a special transformation
1
GA = G of the bundle VA . We set q 2 to be a root of order 2 of q.
Set, in the given module
N −1
a X (l−m)2
G u(uq , v) = √
q 2 u(uvq l , v)
N l=0
m
(21)
where a ∈ F0 is a constant of modulus 1 to be determined later and u(uq m , v)
and u(uvq l , v) are assumed (by the corresponding formula Λ(b, b0 , u, v) ) to
be from EUA .
38
5.12 Lemma. G is unitary. Moreover,
{G u(uq m , v) : m = 0, 1, . . . , N − 1}
is a canonical S-basis, for
1
S := q − 2 Ǔ V̌ −1 .
More precisely, for s(uq m , v) := G u(uq m , v) we have
S : s(uq m , v) 7→ uq m s(uq m , v)
V : s(uq m , v) 7→ vs(uq m−1 , v)
(22)
SGu(uq m , v) = uq m Gu(uq m , v) = GǓ u(uq m , v).
(23)
and
Proof. (22) is by direct application of S to (21). (23) is a corollary of
(22).
Since S is unitary by construction, the basis is orthogonal, so Lemma
follows. 5.13 We can calculate the action of G on the basis {v(vq n , u) :
0, 1, . . . , N − 1}.
Substituting (21) in formula (3.18) for v(vq n , u) we get
n =
N −1
N −1
N −1
a X nm X (l−m)2
1 X nm
q Gu(uq m , v) =
q
q 2 u(uvq l , v) =
Gv(vq n , u) = √
N
N m=0
m=0
l=0
=
N −1 N −1
a X X (l−m)2 +nm
q 2
{
}u(uvq l , v)
N l=0 m=0
Note that (l −m)2 +2nm = [−n2 +2nl]+(m+n−l)2 . Hence we may continue
the above
N −1
N −1
N
−1
X
n2
a X − n2 +nl X (m+n−l)2
a
l
−
=
q 2
{
q nl u(uvq l , v)
q 2 }u(uvq , v) = G(N )q 2
N l=0
N
m=0
l=0
where
G(N ) =
N
−1
X
q
(m+n−l)2
2
m=0
=
N
−1
X
m=0
39
q
m2
2
is a quadratic Gauss sum, see [6]. The last expression by (3.18) can be
rewritten as
n2
a
= √ G(N )q − 2 v(vq n , uv).
N
We have proven that
n2
a
G : v(vq n , u) 7→ √ G(N )q − 2 v(vq n , uv),
N
in particular, for v = 1 the v(vq n , u) are eigenvectors of G with eigenvalues
2
n
√a G(N )q − 2
N
.
5.14 We set
√
a=
N
.
G(N )
By 5.13 this is equivalent to the assumption that
n2
Gv(vq n , u) = q − 2 v(vq n , uv)
(24)
Moreover, reversing the calculation in 5.13 we see that (24) is equivalent to
(21).
5.15 Proposition. Assume that N is even. Then
π
a = e−i 4 .
Proof. Recall the Gauss formula, for integers a and L,
r
L−1
a−1
X
X
|aL|
a 2
L 2
L
πi L
n
πi 4aL
e
eπi a n ,
·
= | |·e
a
n=0
n=0
(25)
see [6]. The statement follows, if we take a = 1 and L = N even. 5.16 In applications a general Gaussian transformation is given by the associated matrix defined in 5.6, of the form
1 − db
0 1
40
for some positive integers b and d.
This corresponds to
B = hÛ d , V̂ b i = B 0 = hÛ d V̂ −b , V̂ b i.
σ̂ :
Û d → Û d V̂ −b
V̂ b → V̂ b
5.17 Proposition. GB is a regular unitary transformation.
Proof. For simplicity (but without loss of generality) we consider the
example with b = 1.
We assume A = hÛ , V̂ i and correspondingly
B = hÛ d , V̂ i = B 0 = hÛ d V̂ −1 , V̂ i.
Let β ∈ SpB and α be an element of SpA such that VB (β) ⊆ VA (α). We
may identify α with a pair huN , v N i ∈ F× × F× , and according to 3.45-3.46,
N
β = huN , v d i, so we have exactly d distinct B-submodules in the A-module.
A canonical Û d basis of VB (β) can be represented in the form
{ud,1 (ud q dm , v) : m = 0, . . . ,
N
− 1}
d
defined by the formula (12). In this notation the definition (21) becomes
r
GB ud,1 (ud q dm , v) = a
N
−1
d
(l−m)2
d X
q d 2 ud,1 (ud vq dl , v) =: s(ud q dm , v)
N l=0
(26)
where s(ud q dm , v) is an element of a canonical Uˆd V̂ −1 -basis as described in
(22).
So, the formula (26) defines, for the given choice of parameters u and v,
the map
VB (β) → VB 0 (β 0 ), β 0 = β
between the submodules of VA (α). This verifies conditions (i),(ii) and (iii)
of the definition 5.1.
To check (iv) note that β(u, v) = β(u0 , v 0 ) if and only if u0 = uq k and
0
v = vq dn for some k, n ∈ Z.
41
Suppose u = u0 and consider the substitution v 0 = vq dn for v in (26). By
3.17 ud,1 (ud q dm , vq dn ) = c1 q dnm ud,1 (ud q dm , v) and
ud,1 (ud vq dl , vq dn ) = c2 q dnl ud,1 (ud vq dl , v) with c1 , c2 roots of unity of order N
(since the choice of the bases is within EB ). Write G0B for the transformation
defined by Λ(b, b0 , u0 , v 0 ).
We will have
r
G0B ud,1 (ud q dm , vq dn )
r
= c2 a
:= a
N
−1
d
(l−m)2
d X
q d 2 ud,1 (ud vq d(l+n) , vq dn )
N l=0
N
−1
d
(l−m)2
d X
q d 2 q dn(l+n) ud,1 (ud vq dl , v)
N l=0
and
G0B ud,1 (ud q dm , vq dn ) = c1 q dnm G0B ud,1 (ud q dm , v).
Hence
r
G0B ud,1 (ud q dm , v) = c−1
1 c2 a
= c−1
1 c2 q
n2
2
r
a
Set ζ = c−1
1 c2 q
N
−1
d
(l−m)2
d X
q d 2 q dn(l+n−m) ud,1 (ud vq d(l+n) , v) =
N l=0
N
−1
d
(l+n−m)2
n2
n2
d X
−1
d dm
2 = c
2 s(u q
q d 2 ud,1 (ud vq d(l+n) , v) = c−1
, v).
1 c2 q
1 c2 q
N l=0
n2
2
. Clearly, ζ 2N = 1. 5.18 Remark. If we only require the coefficient ζ to be an element of F0
of modulus 1, this can be shown by a simpler argument:
Note that the substitution u := u0 and v := v 0 by 3.17 transforms the basis
{ud,1 (ud q dl , v) : 0 ≤ l < Nd } by the application of a transformation of the form
cγ, where γ is a F0 -unitary matrix and c a non-zero constant. It follows that
the basis {GB ud,1 (ud q dl , v) : 0 ≤ l < Nd } under the substitution transforms by
the application of a F0 -unitary matrix γ 0 to a basis {s0 (ud q dl , v) : 0 ≤ l < Nd }
satisfying (22). Now 3.17 imply that s0 (ud q dl , v) = ζ · s(ud q dl , v) for some
ζ ∈ F0 of modulus 1. 42
6
6.1
The sheaf on Af in
Category Vf in .
6.1 The category Vf in consists of objects VA , A ∈ Af in , and morphisms
pBA : VB → VA , B ⊆ A.
Note that pBA determines both
πBA : SpB → SpA
and
∗
πBA
: PΓA → PΓB .
By definition the functor A → VA from Af in to Vf in is a presheaf on the
category Af in and so a presheaf on the category Cfopin .
We use the same name Vf in for the functor.
6.2 Remark. We have associated with an A ∈ Af in the (finite) set O(A)
of maximal Cf in -algebras.
It is appropriate to think about the collection O(Af in ) of all maximal C ∈
Cf in as a Grothendieck site with subsets O(A) providing ”open subsets“ of
the Grothendieck topology. Any finite covering will be considered admissible.
6.3 Remark. Given A, B ∈ Af in , B ⊆ A there is a unique morphism
pBA : VB → VA .
This follows from Lemma 3.49.
6.4 Theorem. Vf in is a sheaf over the site O(Af in ).
Proof. Let B1 , . . . , Bk ∈ Af in and VB1 , . . . , VBk , the corresponding objects of the presheaf. By the remark above the morphisms between the object
are uniquely determined. By 3.6 and 3.2 there is A ∈ Af in , Z(A) ∈ Cf in such
Z(A) = Z(B1 ) ∩ . . . ∩ Z(Bk ), equivalently, O(A) = O(B1 ) ∪ . . . ∪ O(Bk ).
We need to prove that there is a unique object VA in the category with the
corresponding morphisms pBi A : VBi → VA . For this just take the uniquely
defined VA and the unique morphisms pBi A . 6.5 Sheaf Vf in as a structure. We define the structure Vf in as a multisorted structure with sorts VA , for each A ∈ Af in , in the language LV
expanded by unary predicates for each of the sorts VA .
In this definition, each sort VA is a multi-sorted structure, and it is
assumed that all the sorts have a common field F.
43
6.6 Theorem.
(i) The theory of Vf in is categorical in uncountable cardinalities. Moreover, every two models over a field F are isomorphic over F.
(ii) For any A, B ∈ Af in , B ⊆ A, the morphism pBA is definable. More
precisely, there is a definable sort PBA and a definable relation
PBA ⊆ PBA × SpB × VB × VA
in Vf in , such that for every p ∈ PBA and β ∈ SpB the binary relation on
VB × VA ,
PBA (p, β, v1 , v2 )
is the graph of a local morphism pβBA , and every local morphism can be obtained in this way.
Proof. (i) Immediate from 3.58.
(ii) By construction. 6.7 Gluing maps and pairing. Given B, D ∈ Af in we introduce a pairing
[· | ·] : EB × ED → R0 .
First, we recall that the algebra A = hB ∪ Di is in Af in and so there are
morphisms pBA : VB → VA and pDA : VD → VA as defined in 3.50.
Let β ∈ SpB , δ ∈ SpD . For e ∈ EB (β) and f ∈ ED (δ) set
|hpβ (e)|pδ (f )iα |2 , if πBA (β) = α = πDA (δ)
[e|f ] =
0, otherwise
(the absolute value of the inner product).
6.8 Lemma. The pairing [e|f ] is well-defined and does not depend on the
choice of pβ and pδ .
Proof. Note that if πBA (β) = α = πDA (δ), we will have pβ (e) ∈ EA (α),
pδ (f ) ∈ EA (α) and so the inner product hpβ (e)|pδ (f )ia is defined.
It does not depend on pβ and pδ by 3.49. 6.9 Remark. The “physical” interpretation of the pairing [· | ·] should
be “probability”, whereas the inner product would correspond to “amplitude”. We have means to define amplitude locally, on each module VA (α),
but globally we have to use probability because of non-uniqueness in the
choice of embedding pβ . The latter forced upon us if we want uniqueness for
our whole construction (Theorem 6.6), in fact, by model-theoretic stability
assumptions.
44
6.10 Clearly, the pairing is symmetric,
[e|f ] = [f |e],
and
[se|f ] = |s|2 · [e|f ] and [e|tf ] = |t|2 · [e|f ]
for any s, t ∈ F0 , roots of unity of orders nB and nD respectively.
6.11 Note also that in this case
[e|f ] = s · s∗ for s = hpβ (e)|pδ (f )iα
We call an orthonormal canonical basis of VA a collection
[
e ⊆ {EA (α) : α ∈ SpA }
with the property that e ∩ EA (α) is an orthonormal canonical C-basis in
VA (α) for some C ∈ O(A).
6.12 Lemma. Let B, D ∈ Af in and e be an orthonormal basis of VB .
Then, for any δ ∈ SpD and f ∈ ED (δ),
(i) there are finitely many e ∈ e such that [e|f ] 6= 0 and
P
(ii)
e∈e [e|f ] = 1.
Proof. Note that g := pBA (e) is an orthonormal basis of VA , more
precisely,
[
g ⊂ {VAB (α) : α ∈ SpA }.
Now (i) follows by definition.
For (ii) note that, assuming α = πDA (δ) and using the fact that pδ (f ) ∈
EA (α),
X
X
X
[e|f ] =
|hg|pd (f )ia |2 =
hpd (f )|gia ·hg|pd (f )ia = hpd (f )|pd (f )ia = 1.
e∈e
g∈g
g∈g∩EA (a)
45
6.13 Dirac rescaling. We still have a problem with using the pairing between the elements of the VB with distinct B. Namely, the absolute value
becomes typically vanishingly small as the dimension nA of the ambient modules approaches infinity, since the absolute value of he|f i in VA (α) is proportional to √1N , for N = nA .
In later sections we will concentrate on a large enough, in terms of inclusions, algebra A and use a rescaled Dirac inner product. The rescaling
depends on the number nA as well as on a “parametrisation” by a variable
x based on the following data.
Let B, D ⊆ A be two Weyl subalgebras of A with pseudo-unitary generators B = hR, R† i and D = hS, S† i.
Suppose also that there is a regular unitary transformation L of the bundle VA (α) such that for each α ∈ SpA the domain of Lα is a B-submodule
VB (α) and the range of Lα is a D-submodule VD (α) of VA (α), both of
dimension nB , the domain is spanned by the R-eigenvectors
{r(rq dk , r† ) : k = 0, 1, . . . , nB − 1}
and the range by S-eigenvectors
{s(sq dm , s† ) : m = 0, 1, . . . , nB − 1}
canonical R and S bases of VB (α) and VD (α) respectively, such that
Lα : r(rq dk , r† ) 7→ s(sq dk , s† ), k = 0, 1, . . . , nB − 1.
(27)
Our next step is to define a rescaling parameter called ∆k.
If R and S commute, then r = s and we define
r
d
,
∆k = c
N
where c is a fixed parameter to be defined later, independent of A and L.
Otherwise we assume that RS 6= SR.
Let b < N be the minimal positive integer with the property
RS = q b SR or SR = q b RS.
Let aR , aS be the maximal positive integers with the property that
1
1
R aR ∈ A and S aS ∈ A.
46
Set
bc
√ .
aR aS N
We can now define the Dirac delta-function, which depends on the
parametrisation, that is on A and L.
1
, if p = 0
∆k
δ(p) =
0, otherwise.
∆k =
Further set the rescaled product, dependent on the chosen parametrisation:
hr(rq dk , r† )|s(sq dm , s† )iDir =
1
hr(rq dk , r† )|s(sq dm , s† )i
∆k
(28)
6.14 An example. Let A = hǓ , V̌ i.
L be the Fourier transform Φ : VA → VA , which by definition is a
bijection of VA -modules. That is B = D = A.
We respectively will have R = Ǔ = S† , S = V̌ = R† , and hB, Di = A.
Hence ∆k = c√1N
hu(uq k , v)|v(vq m , u)iDir = c ·
√
N hu(uq k , v)|v(vq m , u)i = cq km
and does not depend on N.
6.15 A non-example. Consider the transform
ΦB(e,f ) : ue,f (ue q ek , v f ) 7→ vf,e (v f q f k , ue ).
This is not a restriction of ΦA , so 6.16 is not applicable. Moreover, condition
(27) for parametrisation is not satisfied.
6.16 Proposition. Let e, f be positive integers and
B 0 = hRe , R†f i, D0 = hS e , S†f i,
subalgebras of B and D of 6.13 respectively. Consider the restriction L0 of L
sending B 0 -submodules onto D0 -submodules
L0α : re,f (re q dek , r†f ) 7→ se,f (se q dek , sf† ), k = 0, f, 2f, . . . ,
47
N
− f.
def
Then
hre,f (rq dek , r†f )|se,f (se q dem , sf† )iDir = hr(rq dk , r† )|s(sq dm , s† )iDir ,
where the left-hand side is rescaled with respect to L0 , and in the case RS =
SR both sides of the equality take value equal to δ(k − m), with respective
interpretations of this symbol on the left- and on the right-hand side.
Proof. It is enough to prove the statement in the two cases:
Case 1. e = 1.
and
Case 2. f = 1.
The relation between the re,f and r (and between se,f and s) has been
analysed in 3.46.
In the first case, the respective bases of the submodules are just subsets
of the initial bases, so nothing changes and we have the required equality.
In the second case, the new bases are different, however calculating with
formula (12) we get
hre,f (rq dek , r†f )|se,f (se q dem , sf† )i = hr(rq dk , r† )|s(sq dm , s† )i.
Now we look at the new ∆k. Clearly, b0 = e2 b, a0R = eaR and a0S = eaS . Hence
∆k remains the same and so the required equality holds. 6.17 We can now introduce the Dirac rescaled pairing (probability measure) just replacing the inner product in 6.7 by the Dirac rescaled inner
product.
[e|f ]Dir := (
1 2
) · [e|f ].
∆k
6.18 Lemma – definition. Let B ⊆ A, A, B ∈ Af in and C ∈ O(A). Then
the commutative subalgebra CB = hC ∩ B, Z(B)i (generated by C ∩ B and
Z(B)) is in O(B). Moreover, the map
∗
πBA
: O(A) → O(B), C 7→ CB ,
is surjective.
6.19 Lemma – definition. For a pseudo-unitary S define
C(S) := hS a : a ∈ Qi,
48
1
that is the subalgebra of A(1) generated by all elements S k , k ∈ Z.
C(S) is a maximal commutative subalgebra of A(1) .
This has natural generalisation to maximal commutative subalgebra of
(n)
A .
Proof. Immediate. 7
Limit objects
Our aim here is to extend the categories A∗f in (C) and Vf∗in (C) by adding
some limit objects. This follows the idea that objects in A∗f in (C) and Vf∗in (C)
approximate objects from a larger categories A∗ (C) and V ∗ (C) in a functorial
correspondence to the way the rational Weyl algebras approximate (in the
sence of normed algebras) subalgebras of the whole Weyl algebra.
The approximation will be carried out in terms of structural approximation described in [12], sections 3 and 4.
One particular limit object of Vf∗in (C) will be called the space of basic
states and will take the role of the rigged Hilbert space of mathematical
physics.
We assume F0 ⊂ C when F0 is mentioned and will mostly drop referring
to C below.
7.1 We define
A :=
[
A.
A∈A∗f in
This we call a weakly universal limit object for the category A∗f in . Obviously, every object of A∗f in embeds into A. However, A is not closed with
regards to a natural topology which we study below.
Clearly,
\
Z(A) =
Z(A) = C.
A∈A∗f in
and thus SpA consists of just one point. We write
SpA = {1}.
7.2 We fix an ultrapower ∗ Z (equivalently, a saturated enough model) of
the ring of integers.
49
Fix a non-standard integer µ ∈ ∗ Z with the divisibility property:
µ is divisible by any standard m ∈ Z
(29)
We also fix for the rest of the paper a (standard) positive rational number
h.
1
The definitions and statements of previous sections are applicable in the
new context, with ’rational’ and ’integer’ replaced by more general ’nonstandard rational’ and ’non-standard integer’. In particular, in the definitions we consider ’non-standard’ Heisenberg group H̃( µ1 , µh ) consisting of nonstandard pseudo-unitary operators generated (in the non-standard sense) by
h
1
U µ and V µ . More precisely, H̃( µ1 , µh ) can be identified with the ultraprodh
uct of the groups H( m1 , m
), m ∈ N, along an ultrafilter Ddiv which for every
positive integer n contains the set
{m ∈ N : n|m}.
We use, for a positive d ∈ ∗ Z the notation “much less”
d << µ
with the meaning that
d
µ
is an infinitesimal and d divides µ.
7.3 A universal nonstandard-rational Weyl algebra. Define the alh
gebra à to be the ultraproduct of the algebras A( m1 , m
), m ∈ N, along the
ultrafilter Ddiv . One may think of à as being generated, in a pseudo-finite
1
h
sense, by U µ and V µ . A pseudo-unitary operator W in à is by definition
any element of H̃( µ1 , µh ).
7.4 We call a pseudo-unitary W ∈ Ã finite if it can be presented in the
form
2πihν
ρ
hτ
W = e µ2 U µ V µ
(30)
for nonstandard integers ν, ρ, τ such that |ν| < µ2 and µρ , µτ ∈ ∗ Qf in .
Define Ãf in to be the subalgebra of à generated by all the finite pseudounitary W.
Note that
A ⊂ Ã
(31)
and so à is an upper bound for Af in .
The theory which follows is also applicable to a non-standard rational h ∈ ∗ Qf in
provided the numerator of h is “much less” than µ and divides µ.
1
50
7.5 We set
~ = 2πh
(32)
and the constant introduced in 6.13
√
c=
2π~.
(33)
7.6 We are interested in the physically meaningful Ã-module corresponding
h
1
to U µ - and V µ -eigenvalues u = 1 and v = 1, which is the principal module
VÃ (1) (see also 3.33). We abbreviate the latter to V∗ (1).
It follows from definitions that V∗ (1) is the ultraproduct of VA (1), A =
1
h
hU m , V m i along the ultrafilter Ddiv . In order to understand V∗ (1) we study
some construction on the VA (1).
7.7 Let N be the denominator of the reduced fraction
h
.
µ2
By definitions
N = dim V∗ (1).
By divisibility of µ we have
µ2
2πµ2
N=
=
.
h
~
(34)
If not stated otherwise, we denote
q=e
2πih
µ2
i~
= e µ2 .
7.8 The global language LGlob
V .
We are interested in switching to a language which is, on each VA( 1 , h ) ,
m m
interdefinable with our original language but has the advantage not to refer
to the parameter m, (equivalently N), or similar parameters which change
with the “size” of VA .
For each C ∈ O(A) the language LGlob
contains the unary predicate EC ,
V
which we also may denote ES if C = C(S). In each VA (α) the interpretation
A
of e ∈ EC is e ∈ EC
A (α).
The unitary predicate E is interpreted in each VA as
_
E≡
EC .
C∈O(A)
51
We are going to “forget” the names of pseudo-unitary predicates such
h
1
names for pseudo-selfadjoint
as U m and V m and instead introduce to LGlob
V
operators R = RW , for each W ∈ Ãf in , or more generally of the form
W =e
2πihn
m2
r
U mV
ht
m
where n, m, r and t can be standard or non-standard integers.
Interpret RW in VA as a linear operator
RW =
W − W −1
.
2i{| mr | + | mt |}
(35)
In particular, we use special names for operators
V ah − V −ah
1
U a − U −a
and P =
, a= .
2ia
2ia
m
Note RW are in Ãf in .
Clearly, for each A, the Q is interdefinable with U a and P with V ah .
However the interpretation is given by different formulas depending on A.
A concrete algebraic Hilbert space VA (α) is given in terms of global
as the two sorted structure (E, F) with
language LGlob
V
Q=
• unary predicates EC , for each C ∈ O(A);
• pseudo-selfadjoint operators RC for each C ∈ Cf in ;
• regular unitary transformations L interpreted for each A by the formula
in (19) and a fixed associated matrix gL (see 5.6).
As before F will stand in LGlob
for the field sort with addition and multipliV
cation defined on it.
The formal linear equalities Eq(x1 , . . . , xn , e1 , . . . , en ) :
x1 e1 + . . . + xn en = 0, x1 , . . . , xn ∈ F, e1 , . . . , en ∈ E
are expressible in LGlob
V , and are interpreted in each VA (α) according to the
values taken by the linear combination in the F-vector space. Note that one
can interpret arbitrary elements of the vector space VA (α) in terms of Eq.
Later we will add to the global language predicates encoding the Dirac
inner product (see 6.13 and 6.17).
We call these globally defined operators and predicates.
52
7.9 Below we standardise our notation for the generators of Ã,
h
1
Û := U µ , V̂ = V µ .
More generally, an important role will be played by pairs of pseudo-unitary
generators of Ã, Ŝ and T̂ conjugated to Û and V̂ by a γ ∈ Aut(H( µ1 , µh ) :
Ŝ = Û γ and T̂ = V̂ γ
(see 3.14). Canonical Û , V̂ and generally Ŝ-bases for V∗ (1) will be denoted
u(q k ), v(q k ) and s(q k ), 0 ≤ k < N, respectively. (See the definition (10) for
s).
We will also often work in subalgebras of à of the form B(a, b) = hÛ a , V̂ b i,
with a, b, c, d ∈ ∗ Z, 0 < a, b < µ. The corresponding canonical bases for the
principal module VB(a,b) (1) will be referred to as ua,b (q m ), vb,a (q m ), and
sa,b (q m ), respectively.
7.10 We define a pseudo-metric (that is the distance between distinct point
can be 0) on the group H̃( µ1 , µh )f in of all the finite pseudo-unitary elements of
Ãf in .
For W and W 0 in the form (30) define the pseudo-distance,
dist(W, W 0 ) := lim {|
ρ ρ0
τ
τ0
ν
ν0
− | + | − | + | 2 − 2 |},
µ µ
µ
µ
µ
µ
where ν 0 , ρ0 and τ 0 are the respective parameters of W 0 in the representation
(30) and lim is just the standard part map ∗ C → C.
Note that
dist(Û , 1) = 0 = dist(V̂ , 1).
(36)
Let
Ã0f in = {RW ∈ Ãf in : ν = 0 &
ρ τ ∗
, ∈ Qf in }
µ µ
which we call the basic pseudo-selfadjoint operators.
The pseudo-metric on Ã0f in is given by
dist(RW , RW 0 ) := lim {|
ρ
ρ0
τ
τ0
− 0
|+|
−
|}+dist(W, W 0 ).
|ρ| + |τ | |ρ | + |τ 0 |
|ρ| + |τ | |ρ0 | + |τ 0 |
53
7.11 Define
W ≈ W 0 iff dist(W, W 0 ) = 0,
an equivalence relation on the group H̃( µ1 , µh )f in . It is immediate from the
definition that ≈ is invariant under the group operation. Set
1 h
HR = H̃( , )f in /≈ .
µ µ
This is a real Heisenberg group.
Define on Ã0f in
R ≈ R0 iff dist(R, R0 ) = 0.
The elements of the quotient
Ã0f in / ≈
will be called the basic selfadjoint operators. The pseudo-metric becomes
a metric on the set of basic selfadjoint operators.
7.12 Conjecture. The metric on the space of basic selfadjoint operators
can be extended to a norm on the C-algebra generated by the basic selfadjoint
operators.
Problem.Define a pseudo-metric on as large subalgebra B of à as possible
so that B/ ≈ is a normed C-algebra.
We are not going to use any of the above in the present work. Nevertheless
the construction of the limit object for Vf in depends on the pseudo-metrics
on Ãf in .
7.13 Test-equivalence on E. Let a, b, c, d ∈ ∗ Z, 0 < a, b, c, d << µ.
For e ∈ ∗ E(1)B(a,b) (1) we define We to be the unique element of H̃( µ1 , µh )f in
with the properties:
• ν = 0 in the form (30) for We ,
• e is an eigenvector of We ,
• ρ > 0 or ρ = 0 and τ > 0,
• |ρ| + |τ | is minimal.
54
Given e, e0 ∈ EB(c,d) (1) let W = We , and W 0 = We0 and R = RW ,
R = RW 0 be the corresponding pseudo-selfadjoints.
Define the two elements to be test-equivalent,
0
e ≈ e0
if
R ≈ R0 and he|Rei ∼ he0 |R0 e0 i.
7.14 Note that
e ≈ re,
for r a root of unity. In other words, ≈ identifies elements representing the
same physical state.
7.15 Lemma. f1 ≈ f2 is an equivalence relation.
Proof. (i) Immediate by definition.
7.16 Extending the global language by ≈ . We introduce for each
positive ∈ Q a new binary predicate e1 ≈ e2 , interpreted in each EA as a
statement:
“e1 , e2 ∈ E and there are pseudo-selfadjoint operators R1 , R2 ∈ A such
that for some w1 , w2 ∈ F,
R1 e1 = w1 e1 & R2 e2 = w2 e2 & |w1 − w2 | < ”.
This is definable by a positive quantifier-free formula since for every choice
of A the existential quantifier runs through finitely many choices.
Note that the relation ≈ on ∗ E(1) is definable by a type consisting of
positive formulas (positively type-definable):
^
e1 ≈ 1 e2 .
e1 ≈ e2 ≡
n
n∈N
7.17 Lemma. Let R = RW ∈ Ãf in be a selfadjoint operator and e ∈ ∗ E(1)
its eigenvector. Then
e ≈ Û e and e ≈ V̂ e.
Proof. Clearly, e is a W -eigenvector too. That is W e = we for some
w ∈ ∗ F×
0.
55
Let W 0 := Û W Û −1 . Then, using the commutation identity (4),
2πih
W0 = e
τ
µ2
W
and
W 0 e0 = Û W Û −1 Û e = we0 .
It follows that
RW 0 e0 =
1 2πih µτ2
−2πih τ2 −1
µ w
w−e
e0 ,
e
iσ
for σ = | µρ | + | µτ | as defined in (30).
We claim that
1
1 2πih µτ2
−2πih τ2 −1
µ
e
w−e
w
∼ (w − w−1 )
σ
σ
that is the RW 0 -eigenvalue of e0 is infinitesimally close to the RW -eigenvalue
of e. Indeed, this is immediate from the fact that
1 τ
·
σ µ2
is infinitesimal. This proves the statement for e0 = Û e.
The proof for e0 = V̂ e is similar. 7.18 Lemma. Let k, l ∈ ∗ Z, µk , µl ∈ ∗ Qf in . Then
ua,b (q ak ) ≈ u(q k ),
(37)
vb,a (q bl ) ≈ v(q l ).
(38)
and
Proof. Note that for finite values of
hua,b (q ak )| P ua,b (q ak , 1)i =
k
µ
calculating in the B(a, b)-module
q ak − q −ak
q k − q −k
k~
∼
∼
.
2ia/µ
2i/µ
µ
And in the Ã-module
hu(q k )|Pu(q k )i =
q k − q −k
k~
∼
.
2i/µ
µ
This proves the first equivalence. The second and third equivalences follows
by the same argument. 56
7.19 Remark. Under the above assumptions we have a|l and b|k by definition of a canonical basis. We then have by calculation in 6.14 in the ambient
module B(a, b) :
hua,b (q ak )|vb,a (q bl )iDir = c q kl
(39)
7.20 Lemma. Let k be as in 7.18 and d ∈ ∗ Z, 0 ≤ d << µ. Then
u(q k+d ) ≈ u(q k )
(40)
v(q k+d , 1) ≈ v(q k )
(41)
and
Proof. Using the same same calculations as in 7.18 we would need to
prove
q k+d − q −(k+d)
q k − q −k
∼
.
2i/µ
2i/µ
Recall that q k = exp i~k
, q k+d = exp i~(k+d)
and so
µ2
µ2
eiα − 1
q k+d − q k
= qk µ
2i/µ
2i
where αµ is infinitesimal. The equivalence follows. 7.21 The space of basic states and the bra-ket notations.
We define the space of basic states as the quotient space
E(R) := ∗ E(1)/ ≈
and the canonical quotient map as
lim : ∗ E(1) → E(R).
(42)
Note that by definition ∗ E(1) is a pseudo-finite union of definable subsets of
the form EC , C = C(S), S ∈ Ã.
Keeping with physics notation we set
|xi = lim u(e
57
ix0
µ
)
ip0
|pi = lim v(e µ )
for
x0 , p0 ∈
~ ∗
· Z, x0 , p0 ∈ ∗ Qf in ,
µ
(43)
µ µ
Note that these assumptions imply that x0 , p0 ∈ [− 2~
, 2~ ) and lim x0 ∈ R,
lim p0 ∈ R, taking all possible values in R.
These notation rely on distinguishing between the so called position
eigenstates |xi and the momentum eigenstates |pi just by the letters x
or p used to denote possibly the same number, the eigenvalue of the corresponding operator (to be explained below).
Often we simply right
ix
|xi = lim u(e µ )
and
ip
|pi = lim v(e µ )
assuming that the passage from the right-hand side to the left is clear.
We will also use the more general notation
is
|si = lim s(e µ )
7.22 We extend the domain of lim to ∗ EB(a,b) (1) and define
lim ua,b (e
iax
µ
lim vb,a (e
ibp
µ
and
lim sa,b (e
ias
µ
) = |xi
(44)
) = |pi
(45)
) = |s(s)i
(46)
This agrees with ≈ by (37) and (38).
7.23 Remark. Note that ∗ EB(a,b) (1) ⊂ V∗ (1) (see 3.45) and in a noncanonical way ∗ EB(a,b) (1) can be embedded into ∗ E(1). In this sense we say
that the bases ua,b , vb,a and sa,b are dense in u, v and s respectively.
7.24 Dirac product on E(R). This is defined in accordance with 6.13
with usually fixed parametrisation in variables x or p.
58
ix
We will work in the situation when we are given canonical bases u(e µ ),
ix
ix
ix
ix
r(e µ ), and s(e µ ) such that both r(e µ ) and s(e µ ) are parametrised by regular
transformations
ix
ix
ix
ix
Lr : u(e µ ) 7→ r(e µ ) and Ls : u(e µ ) 7→ s(e µ ).
This will define ∆u, which we following above notations will write as ∆x.
We will also use the notation
ix
ix
lim r(e µ ) = |r(x)i and lim s(e µ ) = |s(x)i.
Now set
1
hr(x1 )|s(x2 )i := lim
∆x
x01
max
x01 ∼x1 , x02 ∼x2
x02
hr(e
ix01
µ
)|s(e
ix02
µ
)i
(47)
ix01
µ
ix02
µ
Here the quantifier max chooses and so that the modulus of hr(e )|s(e )i
reaches maximum. The argument of this expression depends continuously on
x01 and x02 (see the definition and 3.27). Hence the right-hand side of (47) is
well-defined.
On the left-hand side of (47), the use of bra-ket notation assumes that
we rescale the product.
Similarly one defines the parametrisation on variable p, just replacing
ip
ix
u(e µ ) by v(e µ ).
We will have to use respectively ∆p in the definition of the Dirac product.
But note, that the application of the Fourier transform will produce the
equality
∆p = ∆x.
7.25 Example. Reinterpreting the calculation in 6.14 and taking into account (33) we have the classical
1
eixp .
hx|pi = √
2π~
7.26 The space of basic states as structure.
After we defined the set of states E(R), our aim here is to define a structure on it, which we call the (Hilbert) space of basic states and denote
op
V(R). This is a two sorted structure (E(R), C) in the extension LGlobT
of
V
Glob
the global language LV by:
59
(i) unitary operators of the form
S = uU s V t , u ∈ C, s, t ∈ R, |u| = 1;
(ii) unary predicates EC for C = C(S), S as above;
(iii) binary predicates IPc,,Dir (e1 , e2 ), for each non-zero c ∈ C and positive
∈ Q interpreted as
∃c0 6= 0 he1 |e2 iDir = c0 & |c − c0 | ≤ .
(iv) binary predicates e1 ≈ e2 for each positive ∈ Q.
We define the interpretation of the predicates and operations by constructing a covering surjective map
lim : (∗ E(1), ∗ F0 ) → (E(R), C)
The image of lim of ∗ F0 is C∪{∞}, and we identify this lim with the standard
part map. 2
The non-standard elements of ∗ F0 which map into ∞ will be called infinite, and those mapping into 0, infinitesimals. An x ∈ ∗ F0 is said to be
finite if it is not infinite. Given x1 , x2 ∈ ∗ F0 , we write x1 ∼ x2 if x1 − x2 is
infinitesimal.
On ∗ E(1) the map is defined totally by (42).
Note that on E(R)
^
e1 ≈ 1 e2 ↔ e1 = e2 ,
n
n∈N
(see 7.16), in particular, E(R) is the quotient of ∗ E(1) by a positive typedefinable equivalence relation and for each n the formula e1 ≈ 1 e2 defines a
n
neighbourhood of e1 in a Hausdorff topology.
7.27 We also use the Dirac delta in V∗ (1) as defined in 6.13. Since the
dimension of the module is pseudo-finite the ∆x and δ are nonstandard
integers-valued. In the limit structure on E(R) we introduce the Dirac δ(x)
We may assume that ∗ F0 ⊆ ∗ C, the non-standard complex numbers. It is easy to see
and is explained in [2] that any reasonable specialisation map lim : ∗ C → C is equivalent
to the standard part map.
2
60
as an “infinite constant”-valued function, or symbol, along with symbol dx
and set
lim ∆x := dx,
lim δ(x) := δ(lim x).
7.28 Using integration notation. It is helpful to re-write the definition
(47) in the following suggestive terms.
From the notations agreement above we get in particular,
dx hr(x1 )|s(x2 )i = hr(e
ix1
µ
)|s(e
ix2
µ
)i.
(48)
In functional Hilbert spaces elements |r(x)i and |s(x)i are understood as a
family of functions ρ(x, p) and σ(x, p), with variable, say p. Now a summation
formula
X
iy
iy
ix
ix
hr(e µ )|s(e µ )is(e µ )
r(e µ ) =
y
corresponds to the integral
Z
dy hr(x)|s(y)i · σ(y, p).
ρ(x, p) =
y∈R
E.g. the summation formula for the Fourier transform (20) becomes,
ixp
1
e ~ , an eigenvector of the operator f 7→ −i~ df
, and δ(x − p), an
for √2π~
dx
eigenvector of the operator f 7→ xf,
Z
iyp
iyx
1
1
δ(x − p) =
dy √
e− ~ √
e~ ,
2π~
2π~
y∈R
i.e.
Z
1
dy
δ(0) =
2π~ y∈R
Conversely,
1
√
eixp =
2π~
Z
dy
y∈R
1 iyx
e δ(p − y),
2π~
7.29 Note also that by definition the integral reinterpretation of definitions
6.13 yield
Z
δ(x)dx = 1.
R
61
7.30 Topology on the space of basic states. Note that the construction of the space of basic states (E(R), C) as the limit (42) of the pseudofinitary space V∗ (1) = (∗ E(1), ∗ F0 ) introduces a topology on the pseudoHibert space. Namely, C canonically acquires the metric topology from the
standard part map lim : ∗ F0 → C. The predicates e1 ≈ 1 e2 define topology
n
on E(R). More generally, on Cartesian products of E(R) and C the topology
is given by declaring closed subsets those which are definable by positive
extended by metric relation
quantifier free formulas in the language LGlob
V
|x| ≤ 1 on C. All such relations are preserved by map lim .
7.31 Theorem. The object V(R) is universally attracting for the category
Vf in . In other words, for every VA ∈ Vf∗in there is a unique morphism
pA : VA → V(R).
Proof. First we consider the trivial morphism
πA : SpA (R) → 1,
the right-hand side being the spectrum of V(R), by definition.
Now we need to define the embeddings
pβA : VA (β) → V(R)
for every β ∈ SpA (R).
h
Since A ⊆ A( m1 , m
) for m ∈ Z large enough (in terms of divisibility), such
2
m
h
that h is an integer, we may assume A = A( m1 , m
). Recall that N = nA = mh
is the dimension of VA (β).
By definition, β = huN , v N i for some u, v ∈ C of modulus 1, that is
x
p
u = ei m , v = ei m , for some unique real x, p ∈ [0, 2mπ).
µ
consider
m
1,b k
For b =
the subalgebra B(1, b) = hÛ , V
h
m
i and a canonical
2πih
mµ
mµ
h
Û -basis {u (q ) : k = 0, . . . ,
− 1} of VB(1,b) (1), q = e .
Since µ is an infinite integer, for k = 0, . . . , mµ − 1, lim q k takes all the
complex values of modulus 1. Let k0 be such that
lim q k0 = u
62
and denote
|uq bn i = lim u1,b (q k0 +bn ) :
By definition
n = 0, 1, . . .
m2
− 1.
h
1
U m : |uq bn i 7→ uq bn |uq bn i
and
V
h
m
: |uq bn i 7→ |uq b(n−1) i.
1
2
Write the U m -basis of VA (β) as {u(uq bn , v) : n = 0, 1, . . . , mh − 1} and
define pβA on the basis as
pβA : u(uq bn , v) 7→ v −n |uq bn i, n = 0, 1, . . . ,
m2
− 1.
h
1
It is clear that pβA commutes with U m . One also checks that
V
h
m
: pβA u(uq bn , v) = v −n |uq b(n−1) i = vpβA u(uq b(n−1) , v),
which proves that pβA commutes with V
tween the A-modules.
8
h
m
. Hence it is an isomorphism be-
Some basic calculations.
8.1 We calculate with operators Q and P defined in 7.8.
ix
−1
Q|xi = lim Û −2iÛ u(e µ ) =
= lim
µ
ix
µ
µ
(e
2i
−e
− ix
µ
(49)
ix
µ
)u(e ) = x|xi.
It is clear, that this calculation is invariant under replacing x by x0 , with
x ∼ x0 .
Hence
Q|xi = x|xi.
On the other hand
V̂ −V̂
P|xi = lim
u
n
−1
ix
(e µ ) =
( ix
− i~2 )
µ
µ
= lim 2i
u(e
µ
63
( ix
+ i~2 )
µ
) − u(e
µ
o
)
(50)
is ill-defined since the vector in the bracket is an infinitesimal and µ is
infinite.
Similarly,
P|pi = p|pi
but
Q|pi is undefined.
8.2 Comments. The real reason for P and Q to be only partially defined
operators on E is that these are unbounded operators, that is
||P|| = ∞ = ||Q||.
For Q this follows from the calculation (49) and similarly can be checked for
P.
In mathematical physics this is the main difficulty for treating the Heisenberg algebra as a C ∗ -algebra.
8.3 Let us calculate the application of QP − PQ to |xi and |pi. To simplify
the computation, we rewrite
1
1
Q = (Q0 + Q00 ), P = (P0 + P00 )
2
2
where
Q0 =
Û − 1
1 − Û −1
, Q00 =
,
i/µ
i/µ
P0 =
V̂ − 1
1 − V̂ −1
, P00 =
.
i/µ
i/µ
First we calculate
0
0
0
0
ix
µ
2
ix
(Q P − P Q )u(e ) = −4µ Û V̂ − V̂ Û u(e µ ) =
ix i~
ix
i~
i~
ix
−
− i~
−µ2 1 − e µ2 Û V̂ u(e µ ) = −µ2 1 − e µ2 e µ µ2 · u(e µ µ2 )
Now note that by definition and (40)
ix
ix
lim u(e µ ) = |xi = lim u(e µ
64
− i~2
µ
),
so, applying lim to the initial and final term of the equality and using that
ix
i~
lim µ2 (e µ2 − 1) · e µ
− i~2
µ
= i~
we get
(Q0 P0 − P0 Q0 )|xi = i~|xi.
Similarly we get the same result for (Q0 P00 −P00 Q0 ), (Q00 P0 −P0 Q00 ) and (Q00 P00 −
P00 Q00 ). It follows
(QP − PQ)|xi = i~|xi.
Analogously one gets
(QP − PQ)|pi = i~|pi.
In fact, these are the instances of the more general fact.
8.4 Theorem. The Canonical Commutation Relation (2) holds in E(R) :
QP − PQ = i~I.
In other words, given |si ∈ E(R)
(QP − PQ)|si = i~|si.
s
s
Proof. By definition |si = lim s(ei µ ), for some s(ei µ ) ∈ ∗ E(1), an element
of a canonical S-basis.
s
Now we apply (Q0 P0 − P0 Q0 ) to s(ei µ ) and get by the calculation in 8.3
i~
s
s
(Q0 P0 − P0 Q0 )s(ei µ ) = µ2 e µ2 − 1 Û V̂ s(ei µ ).
s
iµ
s
iµ
By 7.17 Û V̂ s(e ) ≈ s(e ). Also µ
2
e
i~
µ2
− 1 ∼ i~. Hence
s
s
(Q0 P0 − P0 Q0 )s(ei µ ) ≈ i~ s(ei µ )
and hence,
s
s
(QP − PQ)s(ei µ ) ≈ i~ s(ei µ ).
65
8.5 By the argument in 8.3 we also have for E(R)
Q0 = Q and P0 = P.
8.6 The Fourier transform is defined on E(R) by applying lim to (20):
Φ|xi = |pi, for p = x.
Applying Φ twice we get
Φ2 : |xi 7→ | − xi.
8.7 The structure on ∗ E(1) in the global language plays a central role in the
theory below. This is a pseudo-finite structure which can be seen as a model
of “huge finite universe” of quantum mechanics. Its specialisation (image under lim ), the “standard” structure E(R), the space of basic states, is a close
analogue of the standard Hilbert space where operators of quantum mechanics are being represented. The calculations in the next sections demonstrate
that we may carry out rigorous quantitative analysis of processes in the “huge
finite universe” and then, specialising these to the standard space of basic
states, obtain well-defined formulas in complete agreement with traditional
methods of quantum mechanics.
8.8 Problem. Study ∗ E(1) from model-theoretic point of view:
(i) Does the elementary theory of (∗ E(1), ∗ C) depend on the choice of the
Fréchet ultrafilter Ddiv ?
(ii) Does the elementary theory of (∗ E(1), ∗ C) interpret arithmetic?
8.9 Problem. Is the structure E(R) determined uniquely by the construction?
9
The simplest time evolution operator.
9.1 Let t be a rational number. We introduce a regular unitary transformation Kfrt on the space of states, which corresponds to the so called time
evolution operator for the free particle. In terms of the Heisenberg-Weyl
algebra this operator could be defined as
P2
e−it 2~ .
66
Equivalently, this is the operator acting on the position eigenstates as
p2
P2
e−it 2~ |pi = e−it 2~ |pi.
(51)
Our Kfrt satisfies the same property.
As above Kfrt will first be defined on the pseudo-finitary space V∗ (1) and
then transferred to E(R) by lim .
9.2 In fact, we can only fully define Kfrt on a submodule of V∗ (1) the
construction of which depends on t.
We assume that t = db , where b, d ∈ ∗ Z, |b|, |d| << µ.
We calculate over the subalgebra B(b, d) of Ã.
By 3.45 the Ã-module V∗ (1) splits into the direct sum of d of its B(b, d)submodules. We define Kfrt on the submodule VB(b,d) (1).
Set
ibd~
N
µ2
q̌ = e µ2 = q bd and Ň =
= .
bdh
bd
Clearly,
Ň = dim VB(b,d) (1).
Choose a canonical Û d -basis of VB(b,d) (1) in the form
{ǔ(q̌ m ) : m = 0, . . . , N − 1}
satisfying
Û d : ǔ(q̌ m ) 7→ q̌ m ǔ(q̌ m )
V̂ b : ǔ(q̌ m ) 7→ ǔ(q̌ m−1 )
Set Kfrt on the U -basis by:
Ň −1
a X (l−m)2
Kfrt ǔ(q̌ m ) = √
q̌ 2 ǔ(q̌ l ) =: s(q̌ m )
Ň l=0
(52)
where a is defined in 5.15.
One sees immediately that Kfrt is the Gaussian transformation with the
domain of definition and range VB(b,d) (1). So we may use 25 - 5.17.
67
9.3 An alternative definition of Kfrt . According to quantum mechanics
P2
the time evolution operator for the free particle is Kfrt := e−it 2~ and the expression on the right makes sense in the operator algebra Then the canonical
commutation relation (2) gives
[Q, Kfrt ] = tPKfrt .
Hence, QKfrt − Kfrt Q = tPKfrt and
Kfrt iQKfr−t = i(Q − tP).
Note that by the Baker-Campbell-Hausdorff formula, for any r ∈ R
1
U r · V −rht = exp irQ · exp{−irtP} = exp(irQ − irtP + [irQ, −irtP]) =
2
2
= exp(irQ − irtP + iπr2 th) = eπihr t K t U r K −t .
Letting t =
b
d
and r =
b
µ
as above, we get
1
Kfrt Û d Kfr−t = S, S = q̌ − 2 Û d V̂ −b
(53)
which is (23) in corresponding notation.
Another property that follows from the form of the time evolution operator, is its commutation with P and so with V̂ b :
K t V̂ b K −t = V̂ b
(54)
(53) and (54) can be taken as the defining properties of Kfrt . Note that
these two algebraic properties alone do not allow to determine the scalar
coefficient a.
9.4 Now we define Kfrt on E(R) by continuity, that is set
lim Kfrt s(q̌ m ) = Kfrt lim s(q̌ m ),
for all s ∈ ∗ E(1)B(d,b) .
By (45), remembering that
b d
v̌(q̌ n ) = v µ , µ (e
i~nbd
µ2
68
) = vb,a (e
ibp
µ
),
where
p=
nd~
µ
satisfies (43). We have by definition
b d
|pi = lim v̂ µ , µ (e
We also note that
q̌ −
n2
2
ibp
µ
).
p2
b
= e−t 2 , t = .
d
By (24) as
p2
Kfrt |pi = e−it 2~ |pi.
(55)
Respectively, in (52) we set
x1 =
mb~
lb~
, x2 =
,
µ
µ
we get by definition and recalling that
lbd~
ǔ(q̌ l ) = ub,d (e µ2 ) = ub,d (e
ǔ(q̌ m ) = ub,d (e
mbd~
µ2
dx1
µ
) = ub,d (e
|x1 i = lim ǔ(e
dx1
µ
and
Kfrt |x2 i = lim s(e
),
dx2
µ
),
)
dx2
µ
).
9.5 Our aim now is to determine the value of
hǔ(e
dx1
µ
)|s(e
dx2
µ
)iDir ,
for which we need to determine ∆x. This was determined in 6.13 taking into
account (33) and gives for B = D = B(d, b)
∆x =
b~
.
µ
By (48) and (52)
hǔ(e
dx1
µ
)|s(e
dxx
µ
)iDir
√
1
bdh −πi (l−m)2
l
m
=
hǔ(q̌ )|s(q̌ )i =
e 4 q̌ 2
∆x
µ∆x
69
which gives us by application of lim :
r
i(x1 −x2 )2
πi
dh 1 i(x1 −x2 )2
1
hx1 |Kfrt x2 iDir = e− 4
· e 2
=√
e 2t~
(56)
b ~
2πi~t
The left-hand side of (56) is called the kernel of a time evolution
operator (Feynman propagator for the corresponding particle).
10
Harmonic oscillator.
10.1 In the Heisenberg Weyl algebra consider the operator
H=
P2 + Q2
,
2
the Hamiltonian for the Harmonic oscillator, where we suppressed some parameters of physical significance).
We are interested in the time evolution operator for the Harmonic oscillator, i.e. the operator
H
t
KHO
:= e−it ~ .
Calculations in a Banach algebra lead to well-known formulae
2 πih sin t cos t
−t
t
KHO
U a KHO
= e−a
U a cos t V −ah sin t .
(57)
and
2 πih sin t cos t
−t
t
KHO
V ah KHO
= ea
U a sin t V ah cos t .
(58)
10.2 We will assume that t ∈ R is such that sin t and cos t are rational, and
even allow these to be non-standard rational.
For the rest of this section we fix
f
e
sin t = , cos t = ,
c
c
for some nonzero e, f, c ∈ ∗ Z, 0 ≤ |e|, |f |, c << µ.
We work in the principal Ã-module V∗ (1) and in the space of basic states
E(R) and want to constract, as suggested by (57) and (58) an operator K t
so that
1
K t Û c (K t )−1 = q − 2 ef Û f V̂ −e
70
(59)
1 3
2
(K t )V̂ ce (K t )−1 = q 2 e f Û e V̂ f e
where
(60)
πih
1
q 2 = e µ2 .
10.3 Denote
1
1
1
2
St := q − 2 ef Û f V̂ −e , Rt := q 2 ef Û e V̂ f and check that Rtm := q 2 ef m Û em V̂ f m .
Note that the algebra hSt , Rte i is isomorphic to hÛ c , V̂ ce i by the isomorphism given by conjugation by K t , so its irreducible modules are of the same
dimension cN2 e .
Clearly, St and Rt are pseudo-unitary. Recall that with our notation
dim VB(1,e) (1) =
N
.
e
Note also that the Û basis u1,e (q el ) can be identified with the subset {u(q k ) :
e|k} of the Û -basis of V∗ (1).
10.4 Define the operator
K t : VB(c,ce) (1) → V∗ (1)
by the action on an U -basis,
N
K t : uc,ce (q
c2 em
) 7→ s(q
c2 em
r
) := C0
−1
e
l2 −e2 m2
e X
3
q ef 2 −e ml u(q e(l+mf ) ), (61)
N l=0
where C0 ∈ F0 is a constant of modulus 1, the exact value of which we
determine at the end of this section.
10.5 Lemma. For any integer m,
2
2
2
St : s(q c em ) 7→ q c em s(q c em )
2
2
Rte : s(q c em ) 7→ s(q c e(m−1) ).
71
Proof. One checks that
1
2
St : u(q e(l+mf ) ) 7→ q ef (l+ 2 )+emf u(q e(l+1+mf ) )
and thus, using f 2 + e2 = c2 ,
St : q ef
l2 −e2 m2
−e3 ml
2
u(q e(l+mf ) ) 7→ q c
2 em
· q ef
(l+1)2 −e2 m2
−e3 m(l+1)
2
u(q e(l+1+mf ) ).
This implies the first statement.
For the second statement first note that
N
r
s(1) := C0
−1
e
l2
e X
q ef 2 u(q el )
N l=1
and that
Rt−em : u(q el ) 7→ q k(e,f,m,l) u(q el+emf )
for
k(e, f, m, l) = −e3 f
m2
− e3 ml.
2
It follows
N
Rt−em : s(1) 7→ C0
r
−1
e
l2 −e2 m2
e X
3
2
q ef 2 −e ml u(q e(l+mf ) ) = s(q c em ).
N l=1
10.6 For n, m ∈ ∗ Z we have, expressing uc,ce (q c
huc,ce (q c
2 en
2 em
)|s(q c
2 en
) in terms of u (see 3.46),
)i =
N
C0
=√ ·
c
r
−1
c−1
e
X
l2 −e2 m2
e X
3
cen+k N
c )|
h
u(q
q ef 2 −e ml u(q e(l+mf ) )i
N k=0
l=0
Set M := Nc and lk = kM + l0 , for k = 0, . . . , c − 1.
2
2
Note that the necessary condition for the product huc,ce (q c en )|s(q c em )i
to be non-zero is that
cen + kM = el + emf for some l = 0, . . . , M − 1, k = 0, . . . , c − 1.
72
(62)
Let l0 = cn − mf .
Claim.
f l0 − e2 m ≡ 0 mod c.
Note that f l0 − e2 m = f cn − mf 2 − me2 = f cn − mc2 . Claim proved.
So
cn + kM = lk + mf.
Recall that 0 < |c|, |e| << N which implies that
M 2 ≡ 0 mod N.
Now we can continue
huc,ce (q c
r
2 en
2 em
)|s(q c
)i =
c−1
2 2
l2
N
e X
3
k −e m
hu(q cen+k c )|q ef 2 −e mlk u(q e(lk +mf ) )i =
Nc k=0
r
r
c−1
c−1
e X ef l2k −e2 m2 −e3 mlk
e ef l20 −e2 m2 −e3 ml0 X ef l0 kM −e3 mkM
2
2
= C0
q
q
= C0
q
Nc k=0
Nc
k=0
= C0
Now note that ef l0 kM − e3 mkM = ekM (f l0 − e2 m) is divisible by N by
Claim above. Hence the last sum is equal to c and we have finally
r
ec ef l20 −e2 m2 −e3 ml0
c,ce c2 en
c2 em
2
q
.
(63)
hu (q )|s(q
)i = C0
N
10.7 Note by direct substitution and using f 2 + e2 = c2 that
l02 − e2 m2
1
ef
− e3 ml0 = ec2 {(n2 + m2 )f − 2cnm}.
2
2
(64)
Now rename
cen~
cem~
, x2 :=
.
µ
µ
Then, using 7.3, 7.7 and (64), the equality (63) becomes
√
(x2 + x22 )f − 2x1 x2 c
ech
2
2
huc,ce (q c en )|s(q c em )i = C0
exp i 1
=
µ
2e~
√
ech
(x2 + x22 ) cos t − 2x1 x2
= C0
exp i 1
.
µ
2~ sin t
x1 :=
73
(65)
10.8 We rescale the last product according to 6.13 and (28). We have
2
hÛ c , V̂ ce i in place of hR, R† i, and hSt , Rte i in place of hS, S† i. N = N = µh .
We have
SR = q ce RS
and aR = c, the maximal with the property
1
R c ∈ Ã,
while for S the corresponding number aS = 1.
Hence, by 6.13 and (33)
√
e h
e~
∆x = c
= .
µ
µ
Thus
µ c,ce c2 en
2
hu (q )|s(q c em )i =
e~
√
2
2
ech
(x + x2 ) cos t − 2x1 x2
C0
µ
(x2 + x22 ) cos t − 2x1 x2
exp i 1
=√
.
= C0
exp i 1
e~
µ
~ sin t
2~ sin t
2π~ sin t
huc,ce (q c
2 en
)|s(q c
2 em
)iDir =
10.9 For the Feynman propagator we need to apply the limit
hx1 |K t |x2 i = lim huc,ce (q c
2 en
)|s(q c
2 em
)iDir
which by the final formula of 10.8 above gives us
r
(x2 + x22 ) cos t − 2x1 x2
1
t
exp i 1
.
hx2 |KHO
|x1 i = C0
2π~ sin t
2~ sin t
This is in accordance with the well-known formula [10], p552, if we set
πi
C0 = a = e− 4 .
(66)
Our definition can only determine K t up to the coefficient C0 of modulus
1.
t
10.10 In this subsection we will motivate a stronger definition for KHO
which will have (66) as a corollary.
First we need to formulate and prove several claims. Note, that in the
definition (61) C0 may depend on t. We need to have reasons to claim that
it does not.
74
Claim 1. Given t1 and t2 as in 10.2 we have the following operator identity
K t1 K t2 = K t1 +t2
where K t1 , K t2 and K t1 +t2 are defined by (61) with the same C0 (but with
different values of e and c, which depend on t).
Proof of the Claim. TODO.
Corollary. For all t1 ∈ Q · t,
C0 (t1 ) = C0 (t).
Now we may use the fact used by physicists and motivated by simple algebraic (the Trotter product formula) and continuity arguments which imply
that the physicists’ values of the operator must satisfy
t
x2 i
hx1 |KHO
lim
= 1.
t→0 hx1 |K t x2 i
Fr
t
t
as
and KFr
Postulating that this identity holds for the operators KHO
defined above and comparing the corresponding formulas we conclude that
(66) holds.
11
Trace of the time evolution operator for
the Harmonic oscillator.
11.1 First note that by (62)
huc,ce (q c
2 en
2 en
)|s(q c
)i =
6 0 ⇔ en(c−f )+kM = el for some l = 0, . . . ,
Hence, the n on the right-hand side can only take values
n = 0, 1, . . . ,
N
−1
ec(c − f )
11.2 Note for n = m (64) is equal to
−ec2 (c − f )n2 .
75
N
−1, k = 0, . . . , c−1.
e
Hence, by (63) and 11.1
r
ec
Tr(K t ) := a
N
Denote L =
N
−1
ec(c−f )
X
2 (c−f )n2
q −ec
.
n=0
N
.
ec2 (c−f )
We can rewrite the above as
s
cL−1
X 2πi 2
1
t
Tr(K ) = a
e− L n .
c(c − f )L n=0
By Gauss quadratic sums formula (25)
L−1
X
− 2πi
n2
L
e
− πi
4
r
= 2e
n=0
Hence
cL−1
X 2πi 2
√
√
L
− πi
4
=e
2L;
e− L n = c 2L.
2
n=0
s
√
Tr(K t ) = c a2 2L
1
= a2
c(c − f )L
s
2
.
(1 − fc )
Returning to the original notation 10.2 we have
s
s
2
2
1
=
.
=
(1 − cos t)
| sin 2t |
(1 − fc )
Since a2 = −i, we finally get,
t
Tr(KHO
)=
1
.
i| sin 2t |
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