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Quantum Harmonic Oscillator as a Zariski Geometry

We carry out a model-theoretic analysis of the Heisenberg algebra. To this end, a geometric structure is associated to the Heisenberg algebra and is shown to be a Zariski geometry. Furthermore, this Zariski geometry is shown to be non-classical, in the sense that it is not interpretable in an algebraically closed field. On assuming self-adjointness of the position and momentum operators, one obtains a discrete substructure of which the original Zariski geometry is seen as the complexification.


Introduction
We describe the structure QHO = QHO N (dependent on the positive integer number N ) on the universe L which is a finite cover, of order N, of the projective line P = P(F), F an algebraically closed field of characteristic 0. We prove that QHO is a complete irreducible Zariski geometry of dimension 1. We also prove that QHO is not classical in the sense that the structure is not interpretable in an algebraically closed field and, for the case F = C, is not a structure on a complex manifold.
There are several reasons that motivate our interest in this particular example. First, this Zariski 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. In the present case we represent the well-known noncommutative algebra with generators P and Q satisfying the relation as the bundle of eigenspaces of the Hamiltonian H = 1 2 (P 2 + Q 2 ), the system known as the simple harmonic oscillator.
The algebra considered is the famous Heisenberg algebra, historically the first example of a quantisation of a classical Hamiltonian system and an important source of noncommutative geometry. In [Zil2] the first author developed a construction that puts in correspondence to an arbitrary quantum algebra at roots of unity a Zariski structure, similar to the correspondence between polynomial algebras and affine varieties. This construction is a source of many new non-classical Zariski structures, but there the new examples start from dimension 2. The Heisenberg algebra differs from the class of algebras considered in [Zil2] by virtue of it not being an algebra at roots of unity. Indeed, whereas the irreducible modules for the algebras considered in [Zil2] were finite-dimensional, irreducible modules 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 counterparts corresponding to (noncommutative) operator algebras results in a picture that is incomplete. This paper, and the paper [Zil2] represent steps taken in the direction towards filling this gap. Furthermore, given that the notion of a Zariski geometry provides an abstract characterization of the geometry on an algebraic variety, model-theoretically one has the means of proving that a nonclassical geometric structure associated to a specific operator algebra is 'rich' (i.e. that one has a means 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 methods available to non-commutative geometers to study this algebra directly. The Weyl algebra (the 'exponential' of the Heisenberg algebra) is a C * -algebra and is consequently the favoured object of study. What is interesting about the approach developed in this paper is that this apparent issue with the Heisenberg algebra has not manifested itself geometrically: the corresponding geometry is still rich.
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 canonical commutative 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 Namely the energy levels of the Hamiltonian are quantized. Our Zariski geometry 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 H and the creation and annihilation of these, it is still a rather limited example as far as mathematical physics is concerned. One fails to see the interdependence between eigenstates of H, P and Q, expressed mathematically in the form of inner product. Such issues will be addressed in the near future.

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: 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: We redefine the algebra in terms of two operators a and a † and these satisfy the following relations: [a, a † ] = 1 H = aa † + 1 2 Define: One easily sees that [N, It follows that if e a is an eigenvector of N with eigenvalue a, then Na † e a = (a † N + a † )e a = (a + 1)e 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 a , it gives a new eigenvector with an eigenvalue a + 1 (respectively a − 1). If this algebra is represented represented as an algebra of linear operators on a Hilbert space, with P and Q assumed self-adjoint and the eigenvectors e a normalised, then a † is adjoint to a and the inner product satisfies (e a , a a † e a ) = (e a , {a † a + 1}e a ) = (e a , {N + 1}e a ) = a + 1.
In other words a † e a = be a+1 , b 2 = a + 1.
Similarly, a e a = be a−1 , b 2 = a. Now observe that since H is the sum of squares of self-adjoint operators, its eigenvalues a+ 1 2 are real non-negative. But a , applied to an eigenvector e a lowers its eigenvalue by 1. It must therefore follow that after finitely many applications of a one obtains an eigenvector e 0 so that Ne 0 = 0 (referred to 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.

The structure
Definition 2.1. We consider the two-sorted theory T N with sorts L and F in the language L = L r ∪ {∞, π, ·, A, A † } subject to the following axioms: 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 L 2 × F) obey the following property: for every γ ∈ F[N ].
6. For N even, we postulate the following additional properties for A, A † : Evidently the theory T N is first-order. We will denote models of T N by QHO N . The additional constant symbol ∞ is required to define the equivalence relation to be introduced below. As is wellknown, one can identify 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 of the ǫ k · 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 = ǫ k · x for 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: We have a structure which is induced from QHO N and has universe (L × P)/ ∼ (so it lives in QHO eq N ), which we refer to as Q N . One can define the following additional operations of addition and scalar multiplication on Q N (by abuse of notation, we write (e, x) for the equivalence classes [(e, x)]): 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 a = {[(e, x)] : π(e) = a ∧ x ∈ F} then we see that for each a ∈ F, V a is a one-dimensional vector space over F with the above operations. Put H = a∈F V a .
We can now introduce the linear maps a, a † on Q N : for each a ∈ F and e ∈ π −1 (a), a(e, 1) := (e ′ , b) where A(e, e ′ , b) in the structure QHO N and we extend this linearly. Similarly a † (e ′ , 1) := (e, b) where A † (e ′ , e, b), also extended linearly.
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 additional condition on A for the even case. Similarly for A † .
Proposition 2.1. The theory T N is consistent and, for even N, is categorical in uncountable cardinals. Moreover, if F and F ′ correspond to the field sort in two models QHO and QHO ′ of theory T N and there exists i : F → F ′ , a ring isomorphism, then i can be extended to an isomorphismî : QHO → QHO ′ . In particular the only relations on F induced from QHO are the initial relations corresponding to the field structure.
Proof. First we construct a model of T N . For each a ∈ F choose e a ∈ L(QHO) such that p(e a ) = a and choose arbitrarily √ a, a square root of a. Now for every a define a e a := √ a e a+1 and a † e a := √ a − 1 e a−1 . Extend this linearly to maps V a → V a+1 and V a → V a−1 correspondingly. This is well defined for all a and so defines a and a † on a model Q N according to the axioms of T N . One then sees that we have a corresponding model QHO of T N .
To prove categoricity, consider two models of T N with isomorphic fields. We may assume that F = F ′ and i is the identity. Partition P into the orbits of the action of the additive subgroup Z ⊆ F: where S is some choice of representatives, one for each orbit (∞ + m = ∞ for each m ∈ Z, so we have a one 1-element orbit). For each s ∈ S choose first e s ∈ L(QHO) and e ′ s ∈ L(QHO ′ ). Now for each n ∈ Z choose arbitrarily (s + n) 1 2 . By the axioms there is e ∈ L ∩ p −1 (s + 1) such that a e s = ǫ(s + n) 1 2 e, for some ǫ ∈ {1, −1}. Define e s+1 := ǫe which is in L since N is even and ǫ ∈ F[N ]. Similarly define e ′ s+1 ∈ L(QHO ′ ). By induction we can define e s+n and e ′ s+n for all n ≥ 0 so that in the induced Q N , Q ′ N of the two models, a e s+n := (s + n) 1 2 e s+n+1 (and the corresponding relations for e ′ s+n ). Note that by axioms we also have, for all n > 0, a † e s+n = (s + n − 1) 1 2 e s+n−1 . By the similar inductive procedure for all n > 0 define e s−n so that a † e s+1−n = (s − n) 1 2 e s−n and the same in the second model. Again by axioms this determines the action of a on e s−n and e ′ s−n , for all n > 0. Hence we have constructed the bijective correspondence e a → e ′ a , a ∈ F so that it extends to the action of the linear maps a and a † on Q N in the corresponding structures, therefore inducing an isomorphism QHO → QHO ′ .
Lemma 2.1. Assume that F = C. Then for each N one can construct QHO N definable in R.
Proof. Consider C as R+ iR, definable in R. Choose an R-definable complex function z → z 1 2 satisfying (z 1 2 ) 2 = z. Define L to be C and p to be the map x → x N . One can now define Q N and then define a as the only linear map V z → V z+1 such that for each e z ∈ p −1 (z) there is e z+1 ∈ p −1 (z + 1) such that a e z := z 1 2 e z+1 . As observed before this also defines a † .
Proposition 2.2. QHO N is not definable in an algebraically closed field.
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.
Let F 0 be the minimal (finitely generated) subfield of F which contains parameters for the definition 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 0 and consider an element e ∈ p −1 (a) (identified with (e, 1)) which by the assumption of definability can be identified with a tuple in F and also F 0 (e) ⊇ F 0 (a). The orbit O(e) of e under the action of the Galois group Gal(F 0 (e) : F 0 (a)) is a subset of p −1 (a) and also 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 under the action of a subgroup Γ of F[N ]. Moreover, if σ ∈ Gal(F 0 (e) : F 0 (a)) fixes e then it fixes ǫe for all ǫ ∈ F[N ], so Gal(F 0 (e) : F 0 (a)) ∼ = Γ and is cyclic.
Let k be the order of Γ, which is also the order of the cyclic Galois extension (F 0 (e) : F 0 (a)). Since all roots of 1 of order k are in F 0 , by theory of cyclic extensions there exists an element b ∈ F 0 (e) such that b k = a and F 0 (e) = F 0 (b). In particular e = f (b) for some rational function f over F 0 and also b = g(e) for some other such function. It follows that we can assume that in our interpretation of QHO in F for all but finitely many a ∈ F the set p −1 (a) contains all k solutions of the F 0 (a)-irreducible equation x k = a and e is one of these. Also, the action e → γe for γ ∈ Γ is the action by Galois automorphisms, so for the solutions of x k = a can be identified with the multiplication e → ǫe, ǫ ∈ F[k].
Clearly r = r(e, e ′ ) is a definable, hence rational, function of e, e ′ . Since {y ∈ F : y k = a+1} ⊆ p −1 (a+1) by axioms e ′ = ǫb, for some b ∈ E, ǫ ∈ F[N ], b k = a + 1. Redefining q(e, b) := ǫr(e, e ′ ) and writing x, y for e, b we have Denote by C the set of all the (x, y) ∈ F 2 which satisfy (3). Up to finitely many points C is a plane curve over F 0 . We see that for generic x there is an y such that y k = x k + 1 and (x, y) ∈ C.
Since y k = x k + 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 y k = x k + 1 and so in C.
For γ 2 = 1 this contradicts the first two equations of (3). So, we conclude k = 1 or k = 2 and ǫ 2 = 1. Now in the first case q(x, x + 1) 2 = x for all generic x -contradiction. In the second case q(x, y) 2 = x 2 on the curve y 2 = x 2 + 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 N (C) is not Zariski-isomorphic to a structure on a complex space, with relations given by analytic subsets.
Proof. If QHO N (C) were a complex space it will have to be an unramified finite cover of the projective 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 N (C). Consider the extra assumption that P and Q are selfadjoint operators. Then the analysis in section 1.1 shows that the eigenvalues of a † a must be nonnegative 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 (N) as the real part of QHO N (C). Conversely, the latter is the complexification of the former.

Definable sets
We can view Q N as a two-sorted structure (H, F), where H is defined as before. Introduce the projection map p : H → F where p : (e, x) → π(x). Note that p is definable. We wish to pick "canonical basis" elements in each fiber V a which we regard as having modulus one. In our terminology, these 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 H × F 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 H. Let Σ ⊆ {(i, j) : 1 ≤ i, j ≤ s}. We suppose that g is tuple of variables from H of length |Σ|, α is an s-tuple of variables from F, γ is a tuple of variables from F of length |Σ|, as is b. Suppose further that e is an n-tuple of variables from H, λ is an n-tuple of variables from F and that a is an m-tuple of variables from F. Define the following formulas: It is clear from the definition that each c (i,j) is an n (i,j) tuple of variables from the sort F. The first conjunct of G Σ (f, g, b, γ) ensures that the b (i,j) are products of square roots, i.e. that the g (i,j) is the canonical basis element chosen in accordance with repeated applications of the map a to f i . It will become clear that G Σ only becomes significant in the case where f i , f j lie in the same coset of the additive subgroup Z.
Note that we have arranged a particular enumeration of the tuple of variables e = (e 1 , . . . , e n ): the n elements are enumerated as {e ij : 1 ≤ i ≤ s, 1 ≤ j ≤ s i } where s 1 + . . . s s = n. We call such an enumeration a partitioning enumeration. It is evident from the notation established that we have a corresponding partitioning enumeration for λ. Let R(α, γ, b, λ, a) define a Zariski constructible set in F q where q = s + 2|Σ| + n + m (over a parameter set C ⊆ F). We define a core formula over R to be: So this is a formula with free variables (e, a) over C ⊆ F. Denote by ctp(e, a/C) the set of core formulas over R (over C) with free variables (e, a).
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 ∃f R. For further purposes, we would like to determine the effects of conjunction and negation on core formulas. Indeed, for core formulas ∃f R 1 , ∃f R 2 we would like to show that: Fix a core formula ∃f R where R(α, γ, b, λ, a) defines a Zariski constructible set in F q . For δ = (δ 1 , . . . , δ s ) ∈ F[N ] s , we wish to define R δ (α, γ, b, λ, a) (which we regard as the action of δ on R). First assume that R defines an irreducible set. Put: We define R δ to be the Zariski closure of the following set: Remark 3.1. Suppose we have a tuple (e, a) such that Q N |= ∃f R(e, a). Then we obtain canonical basis elements f i ∈ p −1 (α i ) for 1 ≤ i ≤ s and λ ij such that e ij = λ ij f i and similarly for some Σ we also have the relations g (i,j) = γ (i,j) f j holding. We wish to examine the effect of transforming f i → f ′ i = δ i f i on these relations. We find that the g (i,j) get transformed to g i . So the set R δ gives those tuples (α, γ, b, λ, a) for which R still holds after the transformation of basis elements.
If R is Zariski closed, we can decompose R into a finite union of irreducibles: Proof. Recall the type p = tp F (α, γ, b, λ, a) obtained in the proof of quantifier-elimination for core formulas. For P ∈ p, we can assume that either P is a system of equations or the negation of a 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 δ∈F[N ] s ¬S δ ∈ p then δ∈F[N ] s ¬S δ ≡ ¬T and for every ǫ ∈ F[N ] s we have ¬(T ǫ ) = ¬T . So T is F[N ]-invariant, ¬S |= ¬T and we can replace P by ¬T . So suppose that δ∈F[N ] ¬S δ ∈ p. Then there is a maximal subset ∆ ⊆ F[N ] s such that: 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 ¬T δ ∈ p. As p is complete, it follows that T δ ∈ p and so: Now note that: The first disjunct is clearly in p and the last disjunct is equivalent to ¬T |= ¬S as 1 ∈ ∆. So we take We may also assume that R is F[N ]-invariant by replacing R with R 1 = δ∈F[N ] s R δ . It is clear that ∃f (R ∧ ¬S) implies ∃f (R 1 ∧ ¬S). We use the above lemma to show the converse: suppose that Q N |= ∃f (R 1 ∧ ¬S)(e, a). Then there are f, g, α, γ, b, λ such that Q N |= A Σ (f, g, e, α, γ, b, λ) and Q N |= R δ (α, γ, b, λ, a). So Q N |= R(α, γ ′ , b, λ ′ , a) (as in the definition of R δ ) and by the remark following the definition we also have Q N |= A Σ (f ′ , g ′ , e, α, γ ′ , b, λ ′ ). By the F[N ] invariance of S we then obtain that Q N |= ∃f (R ∧ ¬S)(e, a).
Lemma 3.2. Suppose that ∃f R 1 and ∃f R 2 are core formulas and that R 2 is F[N ]-invariant. Then: Proof. Left-to-right in 1 is obvious. Conversely, suppose that Q N |= ∃f R 1 (e, a) and Q N |= ∃f R 2 (e, a).
For 2, right-to-left is obvious. If Q N |= ∃f (¬R 2 ) then there are some elements witnessing this. If there was some witness to ∃f R 2 , then by F[N ]-invariance of R 2 we could transform the latter elements 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 We now consider a more general class of formulas over parameters in H and C ⊆ F. This time e is an (n + r)-tuple of variables from H and we define A Σ on the first n elements of e as before. Suppose that h = (h 1 , . . . , h t ) is a tuple of parameters from H where each h i is a canonical basis element. Suppose that µ is an r-tuple of variables from F. For a partitioning enumeration of the remaining r variables in e, {e s+i,j : 1 ≤ i ≤ t : 1 ≤ j ≤ t i } We define: We define the following formulas: So p is a tuple of variables of length |∆ 1 | from H, m, δ are tuples of variables of length |∆ 1 | from F and F[N ] respectively, q is a tuple of variables of length |∆ 2 | from H and o, ǫ are tuples of variables of length |∆ 2 | from F, F[N ] respectively.
Proof. Suppose that φ is a formula with free variables (e, a) over a finite tuple of parameters l = (l 1 , . . . l p ) in H. Then φ is equivalent to ψ(e, l, a) where ψ contains no parameters in H. By the previous 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.
We carry out the substitution and re-name the v variables as e ij and arrange a partitioning enumeration of the e variables so that the substitution occurs in the variables {e ij : s < i ≤ s + t, 1 ≤ j ≤ q i } where q i ≤ p i and we have that p(l ij ) = p(l km ) if and only if i = k. There exists β i such that p(l ij ) = β i for each s < i ≤ s + t. For each fiber p −1 (β i ), there are only finitely many possible choices of canonical basis elements h i . Once a h i has been chosen, we obtain fixed λ 1 ij and for those h i , h j for which β i , β j lie in the same coset of Z, we have fixed b 1 (i,j) , γ 1 (i,j) and g 1 (i,j) . It follows that ∃f R v:=l is equivalent to: ∃f ∃g∃α∃γ∃b∃λ(A Σ (f, g, e, α, γ, b, λ) ∧ R(α, γ, b, λ, a)) (e l ,α l ,g l ,b l ,γ l ):=(l,β,g 1 ,b 1 ,γ 1 ) Here, by α l we mean those α variables corresponding to l, and similarly for the other variables. After substitution, we rename the remaining λ variables λ ij for s < i ≤ s + t, q i < t ≤ p i as µ i−s,j−qi . R then becomes a constructible predicate in α, γ, b, λ, µ, a over a parameter set C ∪ {β, γ 1 , b 1 }. We now deal with the formula A Σ (f, g, e, α, γ, b, λ) (e l ,α l ,g l ,b l ,γ l ):=(l,β,g 1 ,b 1 ,γ 1 ) . Some conjuncts trivially hold, i.e. E(h i , β i ), all l ij = λ 1 ij h i and: Here we have some Σ 1 ⊆ {(i, j) : s < i, j ≤ s + t}. So we delete these. The remaining conjuncts are then: i>s j>qi We combine (4), (5) and (7) to give A Σ2 (f, g, e, α, γ, b, λ) for some Σ 2 ⊆ {(i, j) : 1 ≤ i, j ≤ s}. We recognize (6) as B(e, h, µ) after re-indexing h i−s as h i and µ i−s,j−qi as µ ij . In (8), the index set We replace the occurrences of α j in E(g (i,j) , α j ) with p(h j ) and after the re-indexing of h j , ∆ 1 becomes a subset of {(i, j) : 1 ≤ i ≤ s, 1 ≤ j ≤ t}. By replacing g (i,j) , b (i,j) and γ (i,j) with p (i,j−s) , m (i,j−s) and δ (i,j−s) respectively, we obtain D 1 (f, p, h, m, δ). Similarly in (9), after re-indexing of the h j we get that ∆ 2 ⊆ {1 ≤ i ≤ t, 1 ≤ j ≤ s} and we replace g (i,j) , b (i,j) and γ (i,j) with q (i−s,j) , o (i−s,j) and ǫ (i−s,j) respectively to get D 2 (f, q, h, o, ǫ, α). Furthermore, note that by renaming the variables in (8), (9), R becomes a constructible predicate R(α, γ, δ, ǫ, b, m, o, λ, µ, a).
A similar analysis to that carried out for core formulas will give that we can assume R to be F[N ]invariant: one specifies additional conditions for (i, j) ∈ ∆ 1 , ∆ 2 in the definition of R η , namely the following: We introduce a topology on Q N by defining the basic closed sets as those defined by general core formulas over R where R are F[N ]-invariant systems of polynomial equations. We will denote the corresponding basic closed set byR. Closed sets are defined as finite unions and arbitrary intersections of basic closed sets.
Proposition 3.3. The topology on Q N is Noetherian.
Claim: If ∃f R 1 and ∃f R 2 are general core formulas over parameters h 1 , h 2 respectively and ∃f R 2 → ∃f R 1 , then we can assume that h 1 = h 2 .
We can therefore assume that the general core formulas ∃f R i have the same parameters from H. As there are only finitely many ways of partitioning the first n e-variables, we can assume that each of the ∃f R i have the same partitioning enumeration. Consequently, the question of whether the sequence of closed sets stabilizes reduces to the question of whether the sequence R 1 ⊇ R 2 ⊇ . . . stabilizes, which holds as algebraic varieties are Noetherian topological spaces.
We know (by quantifier elimination) that the projection of a constructible set is constructible. For further purposes, we require more information about what the formulas that define these projective sets look like.
Lemma 3.3. Suppose we have a general core formula ∃f R (with previously established notation). Let k ∈ {i : 1 ≤ i ≤ s + t} and let l be a subset of indices from {j : 1 ≤ j ≤ s k } if k ≤ s or a subset of indices from {j : 1 ≤ j ≤ t k } if k > s. Then the formula ∃e kl ∃f R is a general core formula over R ′ with R ′ equivalent to one of the following: 1. ∃µ k−s,l R.

∃m
Proof. The proof divides into four cases: 1. Suppose that k > s and that the indices l are not all of 1 ≤ l ≤ t k . The variables e ij do not occur in A Σ , D or R, so the formula ∃e kl ∃f R is equivalent to: To obtain this, we have noted that R is the only part of the formula that contains the variables µ k−s,l that we have deleted from B, so we can move the existential quantification over these variables to R.
2. Suppose that k > s and that the the l are all of 1 ≤ l ≤ t k . We then obtain a further reduction, namely that any formulas involving the parameter h k now become redundant. So we get a new formula respectively, where (i, j) ∈ ∆ ′ 1 for some ∆ ′ 1 ⊆ ∆ 1 and similarly q ′ , o ′ , ǫ ′ contain q (i,j) , o (i,j) , ǫ (i,j) respectively, where (i, j) ∈ ∆ ′ 2 for some ∆ ′ 2 ⊆ ∆ 2 . So we get that ∃e kl ∃f R is a general core formula over ∃m (i1,j1) ∃δ (i1,j1) ∃o (i2,j2) ∃ǫ (i2,j2) ∃µ k−s,l R where (i 1 , j 1 ) ∈ ∆ 1 \ ∆ ′ 1 and (i 2 , j 2 ) ∈ ∆ 2 \ ∆ ′ 2 (note that it is possible that some of the α j also get deleted from D, but we can't shift existential quantifications over these α j to R because these α j still occur in A Σ ). i∼Rj (q (k,i) = q (k,j) ∧ o (k,i) = o (k,j) ∧ ǫ (k,i) = ǫ (k,j) )) D is over the indexing set ∆ ′ 1 ∪∆ ′ 2 . The main difference now is that f and α are n-tuples of variables. It is easy to see that we can find Σ ′ , ∆ ′ 1 and ∆ ′ 2 so that ∃f R is equivalent toR: one carries out the stated linear re-enumeration of the first n e-variables and then duplicates and re-enumerates the remaining variables according to the established equivalence relation. We also have an analogous formulaS equivalent to ∃f S.
Theorem 3.1. The structure Q N and its substructure QHO N are Zariski geometries, i.e. dimension notion introduced satisfies the properties of good dimension and that (SP ), (sP S) and (EU ) hold (providing that the algebraically closed field F is uncountable). Furthermore, QHO N is a one-dimensional complete Zariski geometry.
Proof. Exactly as in [Zil2], which we do not replicate: all statements that need to be verified translate to corresponding statements about algebraic varieties. In particular, it follows that Q N is one-dimensional. That QHO N is a Zariski geometry follows from the observation that QHO N is a definable substructure of Q N (so the two structures are inter-definable) with closed one-dimensional irreducible universe and closed induced defining relations.