Quantum Harmonic Oscillator as a Zariski Geometry

0.21 MB·

by  Vinesh Solanki

Quantum Harmonic Oscillator as Zariski Geometry
9
0 Boris Zilber and Vinesh Solanki
0
2
September 24, 2009

p
e
S
1 Introduction
4
2
We describe the structure QHO = QHON (dependent on the positive integer number N) on the
O] universe Lwhich is a finite cover,oforderN, ofthe projectivelineP=P(F),F analgebraicallyclosed
field of characteristic 0. We prove that QHO is a complete irreducible Zariski geometry of dimension
L
1. We also prove that QHO is not classical in the sense that the structure is not interpretable in an
.
h algebraically closed field and, for the case F=C, is not a structure on a complex manifold.
t
a There are several reasons that motivate our interest in this particular example. First, this Zariski
m geometry differs considerably from the series of examples in [HZ] which all are based on the actions
[ of certain kinds of noncommutative groups as the groups of Zariski automorphisms of the structures
constructed. Inthepresentcasewerepresentthewell-knownnoncommutativealgebrawithgenerators
1
P and Q satisfying the relation
v
QP PQ=i, (1)
5
−
1
as the bundle of eigenspaces of the Hamiltonian H = 1(P2 +Q2), the system known as the simple
4 2
harmonic oscillator.
4
. The algebra considered is the famous Heisenberg algebra, historically the first example of a quan-
9
tisation of a classical Hamiltonian system and an important source of noncommutative geometry. In
0
9 [Zil2] the first author developed a construction that puts in correspondence to an arbitrary quantum
0 algebra at roots of unity a Zariski structure, similar to the correspondence between polynomial alge-
: bras and affine varieties. This construction is a source of many new non-classical Zariski structures,
v
i but there the new examples start from dimension 2. The Heisenberg algebra differs from the class of
X
algebrasconsideredin [Zil2] by virtue of it notbeing an algebraatroots of unity. Indeed, whereasthe
r irreducible modules for the algebras considered in [Zil2] were finite-dimensional, irreducible modules
a
for the Heisenberg algebra are necessarily infinite-dimensional. Consequently, this paper constitutes
an extension of the construction and method of proof in [Zil2] to a wider class of noncommutative
algebras.
Some remarks about the methods of noncommutative geometry and possible interactions with
model theory are in order. Noncommutative geometry provides something of a union of operator
theory (specifically C∗-algebras) and algebraic topology. To this end, noncommutative geometers
invoke a dictionary by which concepts in topology can be translated into concepts in operator theory
and vice versa. A feature of noncommutative geometry is that concrete constructions of geometric
counterparts to the algebras studied isn’t carried out, the operator methods in themselves sufficing
for any “geometric” arguments one may need to produce. The Gelfand-Naimark and Serre-Swan
correspondences provide the means by which such a philosophy is justified.
It is our belief, that though operator methods are very powerful in their own right, the absence of
geometric counterpartscorrespondingto (noncommutative)operatoralgebrasresults in a picture that
isincomplete. Thispaper,andthepaper[Zil2]representstepstakeninthedirectiontowardsfillingthis
gap. Furthermore, given that the notion of a Zariski geometry provides an abstract characterization
1
1 INTRODUCTION 2
of the geometry on an algebraic variety, model-theoretically one has the means of proving that a non-
classicalgeometricstructureassociatedtoaspecificoperatoralgebrais‘rich’(i.e. thatonehasameans
of developing algebraic geometry on the structure).
It should be noted that the Heisenberg algebra is a -algebra: it is an algebra equipped with an
∗
additional operation which associates to each element X of the algebra an element X∗, seen (by
∗
analogy with Hilbert space theory) as the adjoint to X. It is not a C∗-algebra: any representation
of the Heisenberg algebra as an algebra of operators on a Hilbert space must, by the nature of the
defining relations, result in at least one of the operators being unbounded. Of course, an important
theorem in the representation theory of C∗-algebras is that any C∗-algebra can be represented as an
algebra of bounded operators on some Hilbert space. Consequently, one does not have many of the
methodsavailabletonon-commutativegeometerstostudythisalgebradirectly. TheWeyl algebra(the
‘exponential’ of the Heisenberg algebra) is a C∗-algebra and is consequently the favoured object of
study. Whatis interestingabout the approachdevelopedinthis paperis that this apparentissue with
theHeisenbergalgebrahasnotmanifesteditselfgeometrically: thecorrespondinggeometryisstillrich.
By the postulates of quantum mechanics, P and Q are considered to be self-adjoint (self-adjoint
operators have real eigenvalues). Consequently, so is H. The Zariski structure considered does not
originally witness the -structure on the Heisenberg algebra,and so it produces, for F=C, essentially
∗
a (non-classical)complex geometry. The assumption of self-adjointness,in the canonicalcommutative
context, leads to cutting out the real part of a complex variety. The result of the same operation with
our structure QHO is the discrete substructure, (the finite cover of) the infinite set
1
n+ :n=0,1... (2)
{ 2 }
Namely the energy levels of the Hamiltonian are quantized. Our Zariskigeometry should therefore be
seen as the complexification of (2), obtained from the same noncommutative coordinate algebra. This
complexification exposes the true geometry of the discrete structure.
Finally, we would like to note that although the construction of QHO represents the eigenstates of
Handthe creationandannihilationofthese,itisstilla ratherlimited exampleasfarasmathematical
physicsisconcerned. OnefailstoseetheinterdependencebetweeneigenstatesofH,PandQ,expressed
mathematically in the form of inner product. Such issues will be addressed in the near future.
1.1 Background
In this subsection, we outline the appropriate background concerning the analysis of the quantum
harmonic oscillator. Any physical system has a corresponding Hamiltonian H given by:
P2
H= +V(Q)
2m
The first quantity on the right-hand side is kinetic energy. The latter quantity, the potential V(Q),
is typically a polynomial expression in Q. The Hamiltonian, as a physical quantity, is conserved and
represents the total energy of the physical system. In the case of the quantum harmonic oscillator,
P and Q satisfy the canonical commutation relation (1) (written [Q,P] = i, taking ~ = 1). The
Hamiltonian H takes the form:
1
H= (P2+Q2)
2
We redefine the algebra in terms of two operators a and a† and these satisfy the following relations:
1
[a,a†]=1 H=aa†+
2
2 THE STRUCTURE 3
Define:
1
N:=H =a†a.
− 2
One easily sees that
[N,a†]=a† and [N,a]= a.
−
It follows that if e is an eigenvector of N with eigenvalue a, then
a
Nae =(aN a)e =(a 1)e ,
a a a
− −
Na†e =(a†N+a†)e =(a+1)e .
a a a
For this reason, a and a† are referred to as ladder operators (respectively annihilation and creation
operators in the broader context of quantum field theory). When a† (respectively a) acts on an
eigenvector e , it gives a new eigenvector with an eigenvalue a+1 (respectively a 1).
a
−
If this algebra is representedrepresented as an algebra of linear operators on a Hilbert space, with
PandQassumedself-adjointandtheeigenvectorse normalised,thena† isadjointtoa andthe inner
a
product satisfies
(e ,aa†e )=(e , a†a +1 e )=(e , N+1 e )=a+1.
a a a a a a
{ } { }
In other words
a†e =be , b2 =a+1.
a a+1
Similarly,
ae =be , b2 =a.
a a−1
NowobservethatsinceHisthesumofsquaresofself-adjointoperators,itseigenvaluesa+1 arereal
2
non-negative. But a, applied to an eigenvector e lowers its eigenvalue by 1. It must therefore follow
a
that after finitely many applications of a one obtains an eigenvector e so that Ne = 0 (referred to
0 0
as the ground state). So the spectrum of N consists of all the non-negative integers and the spectrum
of the Hamiltonian H is the set (2) above.
2 The structure
Definition 2.1. We consider the two-sorted theory T with sorts L and F in the language =
N
L
,π, ,A,A† subject to the following axioms:
r
L ∪{∞ · }
1. F is an algebraically closed field of characteristic 0.
2. P is the projective line over F.
3. π :L P is surjective.
→
4. We have a free and transitive group action : F[N] L L on each of the fibers π−1(a) for
a P. · × →
∈
5. The ternary relations A, A† (on L2 F) obey the following property:
×
( a F)( e π−1(a))( b F)( e′ π−1(a+1))(b2 =a A(γ e,γ e′,b) A†(γ e′,γ e,b))
∀ ∈ ∀ ∈ ∃ ∈ ∃ ∈ ∧ · · ∧ · ·
for every γ F[N].
∈
6. For N even, we postulate the following additional properties for A,A†:
′ ′
A(e,e ,b) A(γ e, γ e , b)
→ · − · −
A†(e′,e,b) A†(γ e′, γ e, b)
→ · − · −
2 THE STRUCTURE 4
Evidently the theory T is first-order. We will denote models of T by QHO . The additional
N N N
constant symbol is required to define the equivalence relation to be introduced below. As is well-
known, one can id∞entify P with F . Each of the fibers π−1(a) has size N: take x π−1(a) and
ǫ a generator of F[N]. Then each o∪f{t∞he}ǫk x for 0 k <N are distinct (as the action∈is free) which
· ≤
implies that π−1(a) N. If π−1(a) > N then there would be y π−1(a) such that y = ǫk x for
| | ≥ | | ∈ 6 ·
any 0 k <N, contradicting transitivity.
≤
We now use this structure to define a line bundle over F (not claiming local triviality), achieved
by introducing the following equivalence relation on L P:
×
(e,x) (e′,x) (π(e)=π(e′) ( γ F[N])(γ e=e′ γ−1x=x′) x=x′ =0 x=x′ = )
∼ ⇔ ∧ ∃ ∈ · ∧ ∨ ∨ ∞
We haveastructure whichis inducedfromQHO andhasuniverse(L P)/ (soit livesinQHOeq),
N × ∼ N
which we refer to as Q . One can define the following additional operations of addition and scalar
N
multiplication on Q (by abuse of notation, we write (e,x) for the equivalence classes [(e,x)]):
N
λ(e,x):=(e,λx) (e,x)+(e,y):=(e,x+y)
One sees that we have compatibility with the group action: (γ e,x)=(e,γx)=γ(e,x) for γ F[N].
If we set V = [(e,x)] : π(e) = a x F then we see that fo·r each a F, V is a one-dime∈nsional
a a
vector space ove{r F with the above∧ope∈rati}ons. Put = V . ∈
H Sa∈F a
We can now introduce the linear maps a,a† on Q : for each a F and e π−1(a), a(e,1) := (e′,b)
N
where A(e,e′,b)inthe structureQHO andweextendthis linea∈rly. Similar∈lya†(e′,1):=(e,b)where
N
A†(e′,e,b), also extended linearly.
Remark2.1. Supposewehavethata(e,1)=(e′,b). Thenforanyγ F[N], itfollows thata(γ e,1)=
′ ′ ∈ ·
γa(e,1)=γ(e ,b)=(γ e ,b). So we should have that A(γ e,γ e,b)in the structureQHO . Similarly
· · · N
for a†.
Remark 2.2. Suppose that N is even and that a(e,1) = (e′,1). We then have that 1 F[N] and
a( e,1)=a(e, 1)=(e′, b). As ( b)2 =a, this explains why we stipulated the addit−iona∈l condition
− − − −
on A for the even case. Similarly for A†.
Proposition2.1. ThetheoryT is consistentand, foreven N,iscategorical inuncountablecardinals.
N
Moreover, ifFandF′ correspondtothefieldsortintwomodels QHOandQHO′ of theoryT andthere
N
exists i:F F′, a ring isomorphism, then i can be extended to an isomorphism ˆi:QHO QHO′. In
particular t→he only relations on F induced from QHO are the initial relations correspondin→g to the field
structure.
Proof. First we construct a model of T . For each a F choose e L(QHO) such that p(e ) = a
N a a
∈ ∈
and choose arbitrarily √a, a square root of a. Now for every a define ae := √ae and a†e :=
a a+1 a
√a 1e . Extend this linearly to maps V V and V V correspondingly. This is well
a−1 a a+1 a a−1
− → →
defined forallaandsodefines a anda† onamodelQ accordingto the axiomsofT . One thensees
N N
that we have a corresponding model QHO of T .
N
To prove categoricity, consider two models of T with isomorphic fields. We may assume that
N
F=F′ and i is the identity. Partition P into the orbits of the action of the additive subgroup Z F:
⊆
P= [ s+Z
s∈S
whereS issomechoiceofrepresentatives,oneforeachorbit( +m= foreachm Z,sowehavea
one1-elementorbit). Foreachs S choosefirste L(QHO∞)ande′ ∞L(QHO′).No∈wforeachn Z
∈ s ∈ s ∈ ∈
2 THE STRUCTURE 5
choose arbitrarily (s+n)12. By the axioms there is e L p−1(s+1) such that aes = ǫ(s+n)21e,
for some ǫ 1, 1 . Define e := ǫe which is in L∈since∩N is even and ǫ F[N]. Similarly define
s+1
e′ L(Q∈HO{′).−By} induction we can define e and e′ for all n 0 so t∈hat in the induced Q ,
Qs′N+1o∈fthetwomodels,aes+n :=(s+n)21es+n+1s+(nandthesc+onrrespondin≥grelationsfore′s+n). NotethNat
by axioms we also have, for all n>0, a†es+n =(s+n 1)21es+n−1.
−
By the similar inductive procedure for all n>0 define es−n so that a†es+1−n =(s n)21es−n and
−
the same in the secondmodel. Againby axioms this determines the action of a on e and e′ , for
s−n s−n
all n > 0. Hence we have constructed the bijective correspondence e e′, a F so that it extends
a 7→ a ∈
to the action of the linear maps a and a† on Q in the corresponding structures, therefore inducing
N
an isomorphism QHO QHO′.
→
Lemma 2.1. Assume that F=C. Then for each N one can construct QHO definable in R.
N
Proof. ConsiderCasR+iR,definableinR.ChooseanR-definablecomplexfunctionz z21 satisfying
(z21)2 =z. Define L to be C and p to be the map x xN. One can now define QN an7→d then define a
7→
as the only linear map V V such that for each e p−1(z) there is e p−1(z+1) such that
z z+1 z z+1
→ ∈ ∈
aez :=z21ez+1. As observed before this also defines a†.
Proposition 2.2. QHO is not definable in an algebraically closed field.
N
Proof. Suppose towards a contradiction it is definable in an algebraically closed field F′. Since any
infinite field definable in an algebraically closed field is definably isomorphic to it, and in case of
characteristic zero by a unique isomorphism, we may assume that F′ =F. We may also assume that F
is of infinite transcendence degree.
LetF betheminimal(finitelygenerated)subfieldofFwhichcontainsparametersforthedefinition
0
of L, p, a, a† and other operations in the definition of QHO as well as all elements of F[N].
Let a be a generic element of F over F and consider an element e p−1(a) (identified with (e,1))
0
whichbythe assumptionofdefinabilitycanbeidentifiedwithatuplein∈FandalsoF (e) F (a).The
0 0
orbit O(e) of e under the action of the Galois group Gal(F (e):F (a)) is a subset of p−1⊇(a) and also
0 0
if ǫe O(e) for ǫ F[N] then for all e′ O(e), ǫe′ O(e). Since by definition each e′ O(e) is of the
form∈ǫe, it follows∈that O(e) is also the∈orbit unde∈r the action of a subgroup Γ of F[N∈]. Moreover, if
σ Gal(F (e):F (a)) fixes e then it fixes ǫe for all ǫ F[N], so Gal(F (e):F (a))=Γ and is cyclic.
∈Letkb0etheor0derofΓ,whichisalsotheorderofth∈ecyclicGaloisext0ension0(F (e)∼:F (a)).Sinceall
0 0
rootsof1oforderk areinF ,bytheoryofcyclicextensionsthereexistsanelementb F (e)suchthat
0 0
bk =a andF (e)=F (b).Inparticulare=f(b)forsomerationalfunctionf overF ∈andalsob=g(e)
0 0 0
for some other such function. It follows that we can assume that in our interpretation of QHO in F
for allbut finitely manya F the set p−1(a) containsall k solutions of the F (a)-irreducible equation
0
∈
xk =a ande isone ofthese. Also,the actione γeforγ Γ is the actionby Galoisautomorphisms,
so for the solutions of xk =a can be identified w7→ith the mu∈ltiplication e ǫe, ǫ F[k].
By the axioms of T there is e′ p−1(a+1) and ρ F such that 7→ ∈
N
∈ ∈
a(e,1)=(e′,r), a†(e′,1)=(e,r), r2 =a.
Clearlyr =r(e,e′)isadefinable,hencerational,functionofe,e′.Since y F:yk =a+1 p−1(a+1)
by axioms e′ =ǫb, for some b E, ǫ F[N], bk =a+1. Redefining q({e,b∈):=ǫr(e,e′) an}d⊆writing x,y
∈ ∈
for e,b we have
a(x,1)=(y,q(x,y)), a†(y,1)=(x,q(x,y)ǫ−2) q(x,y)2 =ǫ2xk. (3)
Denote by C the set of all the (x,y) F2 which satisfy (3). Up to finitely many points C is a plane
curve over F . We see that for generic∈x there is an y such that yk =xk+1 and (x,y) C.
0
∈
Since yk =xk+1 defines an irreducible curve we conclude that all but finitely many points of this
curve belong to C. In particular, for every γ F[k], the point (x,γy) is in yk =xk+1 and so in C.
∈
3 DEFINABLE SETS 6
For γ2 = 1 this contradicts the first two equations of (3). So, we conclude k = 1 or k = 2 and
6
ǫ2 =1.
Nowinthe firstcaseq(x,x+1)2 =xforallgenericx–contradiction. Inthe secondcaseq(x,y)2 =
x2 on the curve y2 = x2 + 1. This implies that q(x,y) x or q(x,y) x on the curve. But
≡ ≡ −
q(x, y)= q(x,y) by the first equation of (3) – contradiction.
− −
Corollary 2.1. QHO (C) is not Zariski-isomorphic to a structure on a complex space, with relations
N
given by analytic subsets.
Proof. If QHO (C) were a complex space it will have to be an unramified finite cover of the projec-
N
tive line P(C), so a 1-dimensional compact manifold. But every such manifold is biholomorphically
isomorphic (so Zariski isomorphic) to a complex algebraic curve – contradiction.
Remark 2.3. The real part of QHO (C). Consider the extra assumption that P and Q are self-
N
adjoint operators. Then the analysis in section 1.1 shows that the eigenvalues of a†a must be non-
negative and so the only points in P that survive this extra condition are the non-negative integers N.
The corresponding points in L form the N-cover of N, so we get the discrete structure QHO (N) as
N
the real part of QHO (C). Conversely, the latter is the complexification of the former.
N
3 Definable sets
We can view Q as a two-sortedstructure ( ,F), where is defined as before. Introduce the projec-
N
tion map p : F where p : (e,x) π(xH). Note thatHp is definable. We wish to pick “canonical
H → 7→
basis” elements in each fiber V which we regard as having modulus one. In our terminology, these
a
canonical basis elements are exactly the elements (e,1) in each fiber. Note that there are N possible
choices in each fiber: if γ F[N], then (e,γ)= (γ e,1). We introduce a predicate E(f,α) on F
∈ · H×
which says “f is a canonical basis element of the fiber p−1(α)”.
We follow the analysis in [Zil2]. Suppose that f is an s-tuple of variables from . Let Σ (i,j) :
H ⊆ {
1 i,j s . We suppose that g is tuple of variables from of length Σ, α is an s-tuple of variables
fro≤m F,≤γ is}a tuple of variables from F of length Σ, as isHb. Suppose|fu|rther that e is an n-tuple of
variables from , λ is an n-tuple of variables from| |F and that a is an m-tuple of variables from F.
H
Define the following formulas:
n(i,j) n(i,j)
GΣ(f,g,b,γ,α):=∃c(i,j)( ^ (ck(i,j))2 =π(fi+k)∧ Y ck(i,j) =b(i,j)∧
k=1 k=1
^ (E(g(i,j),αj) an(i,j)fi =b(i,j)g(i,j) (a†)n(i,j)g(i,j) =b(i,j)fi g(i,j) =γ(i,j)fj)
∧ ∧ ∧
(i,j)∈Σ
s s si
AΣ(f,g,e,α,γ,b,λ):= ^E(fi,αi) GΣ(f,g,b,γ) ^ ^eij =λijfi
∧ ∧
i=1 i=1j=1
It is clear from the definition that each c(i,j) is an n tuple of variables from the sort F. The first
(i,j)
conjunct of GΣ(f,g,b,γ) ensures that the b are products of square roots, i.e. that the g is the
(i,j) (i,j)
canonical basis element chosen in accordance with repeated applications of the map a to f . It will
i
become clear that GΣ only becomes significant in the case where f ,f lie in the same coset of the
i j
additive subgroup Z.
Note that we have arranged a particular enumeration of the tuple of variables e = (e ,...,e ): the
1 n
n elements are enumerated as e : 1 i s,1 j s where s +...s = n. We call such an
ij i 1 s
{ ≤ ≤ ≤ ≤ }
3 DEFINABLE SETS 7
enumeration a partitioning enumeration. It is evident from the notation established that we have
acorrespondingpartitioningenumerationforλ. LetR(α,γ,b,λ,a)define aZariskiconstructiblesetin
Fq where q =s+2Σ +n+m(overa parametersetC F). We define acore formulaoverR to be:
| | ⊆
f g α γ b λ(AΣ(f,g,e,α,γ,b,λ) R(α,γ,b,λ,a))
∃ ∃ ∃ ∃ ∃ ∃ ∧
Sothisisaformulawithfreevariables(e,a)overC F. Denotebyctp(e,a/C)thesetofcoreformulas
⊆
over R (over C) with free variables (e,a).
Proposition 3.1. If ctp(e1,a1/C)=ctp(e2,a2/C) then tp(e1,a1/C)=tp(e2,a2/C).
Proof. Assume that Q is -saturated. We show that there is an automorphism σ such that σ :
N 0
ℵ
(e1,a1) (e2,a2). Fix a partitioning enumeration of e1 so that p(e1 ) = p(e1 ) if and only if i = k.
→ ij kl
Then 1 i s for some s and there exist α1 F such that p(e1 )=α1. By re-enumerating the α1 if
≤ ≤ i ∈ ij i i
necessary, we suppose that α1 <α1 for i<j. Now we construct a subset Σ (i,j):1 i,j s as
i j ⊆{ ≤ ≤ }
follows: put (i,j) Σ if there is an n >0 such that α1+n =α1 (so α1 and α1 lie in the same
∈ (i,j) i (i,j) j i j
coset of the additive subgroup Z). We can choose a canonical basis element f1 in each fiber p−1(α1)
i i
and so there exist λ1 in F such that:
ij
s si
QN |= ^ ^e1ij =λ1ijfi1
i=1j=1
For (i,j) Σ, by repeated application of a to f1 (n times) we obtain b1 and g1 , where the
∈ i (i,j) (i,j) (i,j)
latter is a canonical basis element of the fiber p−1(α1). As f1 is also a canonical basis element of the
j j
fiber p−1(α1), thereisγ1 F[N]suchthatg1 =γ1 f1. Doingthis foreach(i,j) Σ, weobtain
j (i,j) ∈ (i,j) (i,j) j ∈
tuples g1,b1,γ1 such that Q =GΣ(f1,g1,b1,γ1). It follows that:
N
|
Q =AΣ(f1,g1,e1,α1,γ1,b1,λ1)
N
|
We consider the following type in variables f,e,α,γ,b,λ, which by assumption is consistent:
q = AΣ(f,g,e2,α,γ,b,λ) R(α,γ,b,λ,a2):Q =AΣ(f1,g1,e1,α1,γ1,b1,λ1) R(α1,γ1,b1,λ1,a1)
N
{ ∧ | ∧ }
By -saturation, q is realized by f2,g2,α2,γ2,b2,λ2. In particular, by quantifier elimination for
0
algeℵbraically closed fields, in the language of rings tpF(α1,γ1,b1,λ1,a1)=tpF(α2,γ2,b2,λ2,a2). So by
saturation of F there is an automorphism σ of F such that:
σ :(α1,γ1,b1,λ1,a1) (α2,γ2,b2,λ2,a2)
7→
PartitionF= r+ZwherethesetofrepresentativesRcontainsasmanyoftheα1 aspossible. For
Sr∈R i
eachi,weextendσtothefiberp−1(α1)byσ :µf1 σ(µ)f2. Clearly,itthenfollowsthatσ(e1 )=e2.
i i 7→ i ij ij
Take some α R. By the axioms, there is c F and h p−1(α1+1) such that af1 =ch. Similarly,
there is d iF∈and l p−1(σ(α1)+1) such∈that af2 =∈dl. Asic2 = α1 and d2 =iα2 we have that
∈ ∈ i i i i
σ(c) = ǫd where ǫ 1,1 . So we extend σ to p−1(α1+1) by mapping µh σ(µ)ǫl (note that we
∈ {− } i 7→
have assumedN is even). We continue this process inductively to extend σ to every fiber p−1(α1+n)
i
for n>0 as in the proofof categoricity. If we haveα1 =α1+n for some n >0 everything still
j i (i,j) (i,j)
works by construction. Similarly, extend σ in the other direction and repeat the construction for each
coset.
It follows (by compactness) that any formula with free variables (e,a) over C F is a finite
⊆
disjunction of a conjunction of core formulas and their negations. Denote a core formula over R by
fR. For further purposes, we wouldlike to determine the effects ofconjunction and negationon core
∃
formulas. Indeed, for core formulas fR , fR we would like to show that:
1 2
∃ ∃
3 DEFINABLE SETS 8
f(R R ) fR fR .
1 2 1 2
• ∃ ∧ ≡∃ ∧∃
f( R ) fR .
1 1
• ∃ ¬ ≡¬∃
Fix a core formula fR where R(α,γ,b,λ,a) defines a Zariski constructible set in Fq. For δ =
(δ ,...,δ ) F[N]s,∃we wish to define Rδ(α,γ,b,λ,a) (which we regard as the action of δ on R).
1 s
∈
First assume that R defines an irreducible set. Put:
V := α Fs : γ b λ aR(α,γ,b,λ,a)
R
{ ∈ ∃ ∃ ∃ ∃ }
We define Rδ to be the Zariski closure of the following set:
s si
{(α,γ,b,λ,a):α∈VR∧∃γ′∃λ′( ^ γ(′i,j) =δiγ(i,j)δj−1∧ ^ ^λ′ij =λijδi−1)∧R(α,γ′,b,λ′,a)}
(i,j)∈Σ i=1j=1
Remark 3.1. Suppose we have a tuple (e,a) such that Q = fR(e,a). Then we obtain canonical
N
| ∃
basis elements f p−1(α ) for 1 i s and λ such that e = λ f and similarly for some Σ we
i i ij ij ij i
∈ ≤ ≤
also have the relations g = γ f holding. We wish to examine the effect of transforming f
(i,j) (i,j) j i
′ ′ 7→
f = δ f on these relations. We find that the g get transformed to g = δ g . Consequently,
i i i (i,j) (i,j) i (i,j)
g′ =δ γ f and so we put γ′ =δ γ δ−1 so that the relations g′ =γ′ f′ hold. Similarly,
(i,j) i (i,j) j (i,j) i (i,j) j (i,j) (i,j) j
e = λ′ f′ where λ′ = λ δ−1. So the set Rδ gives those tuples (α,γ,b,λ,a) for which R still holds
ij ij i ij ij i
after the transformation of basis elements.
If R is Zariski closed, we can decompose R into a finite union of irreducibles: R=R ...R . In
1 k
this case,weput Rδ :=Rδ ... Rδ. We saythat Ris F[N]-invariantif Rδ =R forevery∪δ F[N]s.
1∪ ∪ k ∈
Lemma 3.1. We may assume that the core formulas in ctp(e,a/C) are over (R S)(α,γ,b,λ,a)
where R and S are systems of equations and S is F[N]-invariant. ∧¬
F
Proof. Recall the type p = tp (α,γ,b,λ,a) obtained in the proof of quantifier-elimination for core
formulas. For P p, we canassume that either P is a systemofequationsor the negationofa system
∈
of equations. If P = R a system of equations, we are done. So we deal with the case that P = S
¬
where S is a system of equations.
If Sδ p then Sδ T and for every ǫ F[N]s we have (Tǫ) = T. So
Vδ∈F[N]s¬ ∈ Vδ∈F[N]s¬ ≡ ¬ ∈ ¬ ¬
T is F[N]-invariant, S = T and we can replace P by T. So suppose that Sδ p. Then
¬ | ¬ ¬ Vδ∈F[N]¬ 6∈
there is a maximal subset ∆ F[N]s such that:
⊆
T = ^ Sδ p
¬ ¬ ∈
δ∈∆
As S p, we have that 1 ∆. Put Stab(∆) = δ F[N]s : δ∆ = ∆ . As ∆ is maximal, for any
δ ¬F[N∈] Stab(∆), we have∈ Tδ p. As p is comp{let∈e, it follows that T}δ p and so:
∈ \ ¬ 6∈ ∈
^ Tδ p
∈
δ∈F[N]\Stab(∆)
Now note that:
_ Tδ ^ Tδ = _ Tδ
¬ ∧ | ¬
δ∈F[N]s δ∈F[N]s\Stab(∆) δ∈Stab(∆)
The first disjunct is clearly in p and the last disjunct is equivalent to T = S as 1 ∆. So we take
R= Tδ and S = Tδ, S is F[N]-invariant¬, an|d r¬eplace P∈by R S .
Vδ∈F[N]s\Stab(∆) ¬ 1 Wδ∈F[N]s¬ 1 ∧ 1
3 DEFINABLE SETS 9
We may also assume that R is F[N]-invariant by replacing R with R = Rδ. It is clear
1 Wδ∈F[N]s
that f(R S) implies f(R S). We use the above lemma to show the converse: suppose that
1
∃ ∧¬ ∃ ∧¬
Q = f(R S)(e,a). Then there are f,g,α,γ,b,λ such that Q = AΣ(f,g,e,α,γ,b,λ) and
N 1 N
Q |= R∃δ(α,γ∧,b¬,λ,a). So Q = R(α,γ′,b,λ′,a) (as in the definition|of Rδ) and by the remark
N N
follow|ing the definition we also h|ave Q = AΣ(f′,g′,e,α,γ′,b,λ′). By the F[N] invariance of S we
N
|
then obtain that Q = f(R S)(e,a).
N
| ∃ ∧¬
Lemma 3.2. Suppose that fR and fR are core formulas and that R is F[N]-invariant. Then:
1 2 2
∃ ∃
1. f(R R ) fR fR .
1 2 1 2
∃ ∧ ≡∃ ∧∃
2. f( R ) fR .
2 2
∃ ¬ ≡¬∃
Proof. Left-to-rightin1 isobvious. Conversely,suppose thatQ = fR (e,a)andQ = fR (e,a).
N 1 N 2
| ∃ | ∃
Thentherearef1,g1,α,γ1,b,λ1 witnessing fR andf2,g2,α,γ2,b,λ2 witnessing fR . Thereexists
1 2
δ F[N]s such that f1 = δf2 and so carryi∃ng out this transformation and noting∃that Rδ = R , we
∈ 2 2
obtain Q = f(R R )(e,a).
N 1 2
| ∃ ∧
For 2, right-to-left is obvious. If Q = f( R ) then there are some elements witnessing this. If
N 2
therewassomewitnessto fR ,thenb|yF∃[N]¬-invarianceofR wecouldtransformthe latterelements
2 2
∃
into the former, resulting in contradiction.
Combining the previous two lemmas with the previous proposition, we get that any formula with
free variables (e,a) over parameters C F is equivalent to a finite disjunction of core formulas over
R where the R are F[N]-invariant. ⊆
i i
We now consider a more general class of formulas over parameters in and C F. This time e
H ⊆
isan(n+r)-tupleofvariablesfrom andwedefineAΣ onthefirstnelementsofeasbefore. Suppose
H
that h = (h ,...,h ) is a tuple of parameters from where each h is a canonical basis element.
1 t i
Suppose that µ is an r-tuple of variables from F. ForHa partitioning enumeration of the remaining r
variables in e, e :1 i t:1 j t We define:
s+i,j i
{ ≤ ≤ ≤ ≤ }
t ti
B(e,h,µ):= ^ ^en+i,j =µi,jhi
i=1j=1
Suppose further that ∆ (i,j): 1 i s,1 j t and ∆ (i,j): 1 i t,1 j s . We
1 2
⊆ { ≤ ≤ ≤ ≤ } ⊆ { ≤ ≤ ≤ ≤ }
define the following formulas:
n(i,j) n(i,j)
D1(f,p,h,m,δ):=∃c(i,j)( ^ (ck(i,j))2 =π(fi+k)∧ Y ck(i,j) =m(i,j)∧
k=1 k=1
^ (E(p(i,j),p(hj)) an(i,j)fi =m(i,j)p(i,j) (a†)n(i,j)p(i,j) =m(i,j)fi p(i,j) =δ(i,j)hj)
∧ ∧ ∧
(i,j)∈∆1
n(i,j) n(i,j)
D2(f,q,h,o,ǫ,α):=∃c(i,j)( ^ (ck(i,j))2 =π(hi+k)∧ Y ck(i,j) =o(i,j)∧
k=1 k=1
^ (E(q(i,j),αj) an(i,j)hi =o(i,j)q(i,j) (a†)n(i,j)q(i,j) =o(i,j)hi q(i,j) =ǫ(i,j)fj)
∧ ∧ ∧
(i,j)∈∆2
So p is a tuple of variables of length ∆ from , m,δ are tuples of variables of length ∆ from F
1 1
and F[N] respectively, q is a tuple of v|aria|bles ofHlength ∆ from and o,ǫ are tuples of|var|iables of
2
| | H
3 DEFINABLE SETS 10
length ∆ from F, F[N] respectively.
2
| |
Put D := D D . Suppose that R(α,γ,δ,ǫ,b,m,o,λ,µ,a) defines a Zariski constructible subset
1 2
of Fq where q =∧s+2(Σ + ∆ + ∆ )+n+m+r. We define a general core formula overR with
1 2
| | | | | |
parameters h to be:
f g α γ δ ǫ b p q m o λ µ(AΣ(f,g,e,α,γ,b,λ) D(f,p,q,h,m,o,δ,ǫ,α) B(e,h,µ)
∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∧ ∧ ∧
R(α,γ,δ,ǫ,b,m,o,λ,µ,a))
Proposition 3.2. Every formula with parameters in is equivalent to a finite disjunction of general
H
core formulas.
Proof. Suppose that φ is a formula with free variables (e,a) over a finite tuple of parameters l =
(l ,...l ) in . Then φ is equivalent to ψ(e,l,a) where ψ contains no parameters in . By the pre-
1 p
H H
vious proposition, ψ(e,v,a) (where v is a tuple of variables replacing the parameters l) is equivalent
to a finite disjunction of core formulas. So it suffices to prove that a core formula (with free variables
(e,v,a)) is equivalent to a finite disjunction of general core formulas after the substitution v :=l.
Wecarryoutthesubstitutionandre-namethevvariablesase andarrangeapartitioningenumeration
ij
of the e variablesso thatthe substitution occursin the variables e :s<i s+t,1 j q where
ij i
{ ≤ ≤ ≤ }
q p and we have that p(l )=p(l ) if and only if i=k. There exists β such that p(l )=β for
i i ij km i ij i
≤
each s < i s+t. For each fiber p−1(β ), there are only finitely many possible choices of canonical
i
≤
basis elements h . Once a h has been chosen, we obtain fixed λ1 and for those h ,h for which β ,β
i i ij i j i j
lie in the same cosetof Z, we have fixed b1 ,γ1 and g1 . It follows that fRv:=l is equivalent to:
(i,j) (i,j) (i,j) ∃
_ f g α γ b λ(AΣ(f,g,e,α,γ,b,λ) R(α,γ,b,λ,a))(el,αl,gl,bl,γl):=(l,β,g1,b1,γ1)
∃ ∃ ∃ ∃ ∃ ∃ ∧
hi∈p−1(βi)∧E(hi,βi)
Here, by α we mean those α variables corresponding to l, and similarly for the other variables. After
l
substitution, we rename the remaining λ variables λ for s < i s+t, q < t p as µ . R
ij ≤ i ≤ i i−s,j−qi
then becomes a constructible predicate in α,γ,b,λ,µ,a over a parameter set C β,γ1,b1 . We now
∪{ }
deal with the formula AΣ(f,g,e,α,γ,b,λ)(el,αl,gl,bl,γl):=(l,β,g1,b1,γ1). Some conjuncts trivially hold, i.e.
E(h ,β ), all l =λ1 h and:
i i ij ij i
n(i,j) n(i,j)
∃c(i,j)( ^ (ck(i,j))2 =π(fi+k)∧ Y ck(i,j) =b1(i,j)∧
k=1 k=1
^ (E(g(1i,j),βj)∧an(i,j)hi =b1(i,j)g(1i,j)∧(a†)n(i,j)g(1i,j) =b1(i,j)hi∧g(1i,j) =γ(1i,j)hj)
(i,j)∈Σ1
Here we have some Σ (i,j) : s < i,j s+t . So we delete these. The remaining conjuncts are
1
⊆ { ≤ }
then:
E(f ,α ) (4)
i i
s si
^ ^eij =λijfi (5)
i=1j=1
^ ^ eij =µi−s,j−qihi−s (6)
i>sj>qi