We apply De Haro's Geometric View of Theories to one of the simplest quantum systems: a spinless particle on a line and on a circle. The classical phase space M = T * Q is taken as the base of a trivial Hilbert bundle E ∼ = M × H, and the familiar position and momentum representations are realised as different global trivialisations of this bundle. The Fourier transform appears as a fibrewise unitary transition function, so that the standard position-momentum duality is made precise as a change of coordinates on a single geometric object. For the circle, we also discuss twisted boundary conditions and show how a twist parameter can be incorporated either as a fixed boundary condition or as a base coordinate, in which case it gives rise to a flat U (H)-connection with nontrivial holonomy. These examples provide a concrete illustration of how the Geometric View organises quantum-mechanical representations and dualities in geometric terms.
One of the central questions in the philosophy of science concerns the structure of scientific theories: Are they best understood as sets of sentences in a language or as families of models satisfying those sentences [1][2][3]? Recent work suggests that both answers are too "flat". They underdescribe the formal and interpretative structure that is actually at work in contemporary physics. The emerging "structured" or "geometric" views of theories insist that additional topological and geometric structure on the space of models is needed, especially once inter-theoretic relations such as dualities, reductions, and approximations are taken seriously [3][4][5][6]. In this spirit, De Haro's geometric view of theories (GV) proposes that a physical theory should be seen as an algebraic-geometric object, such as a bundle or manifold equipped with further structure, rather than merely a set of models [5,6].
Within GV, a particularly concrete realisation is the notion of a model bundle, which is a bundle whose fibres collect the models of what, in the more traditional schema, would be called a single theory T = ⟨S, Q, D⟩ of states, quantities, and dynamics. Dualities then appear as globally defined transition functions on this bundle, while quasi-dualities appear as transition functions defined only on overlapping regions of the base. In this way, we can say what the underlying geometric object is whose coordinates we are changing, and how the dual descriptions sit as different trivialisations of one and the same bundle.
The aim of this paper is to work out this picture in one of the simplest quantum-mechanical settings: A spinless particle on a circle and a line. This system is standardly used in physics to illustrate basic quantum features such as momentum quantisation, Fourier duality between the position and momentum representations, and the role of boundary conditions. It is therefore a natural test case for the geometric view. If that view cannot be made precise here, it would be hard to trust it in the more demanding cases. At the same time, the system is rich enough to exhibit both exact duality (between position and momentum representations) and mild generalisations (twisted boundary conditions) that resemble quasi-dualities in the sense of De Haro and Butterfield [6].
In Section 2, we construct the model bundle for a spinless particle on a line and show how the standard L 2 (R) and ℓ 2 descriptions are realised as global trivialisations, related by the Fourier transform viewed as a transition function. In Section 3, we repeat the construction for a particle on a circle, with base M = T * S1 , and in Section 4, we extend this to twisted boundary conditions. There we distinguish two cases: (A) a fixed twist, which affects only the boundary conditions and spectrum; and (B) a twist promoted to a base parameter, where a flat but nontrivial connection encodes the holonomy in the twist direction. In doing so, the paper illustrates how the duality between position and momentum can be treated, not merely as an abstract isomorphism of Hilbert spaces, but as a genuine change of coordinates on a geometric object. The conclusion summarises what these examples illustrate about the Geometric View and indicates some directions for further work.
Throughout the text, we are going to use the language of fibre bundles, so let us introduce some definitions before we go on. 1 Let M be the base, G a (Lie) group, and H a separable Hilbert space with a left unitary representation
Let P → M be any principal G-bundle with right action P × G → P , (p, g) → p • g. Define an equivalence relation on P × H by
The associated Hilbert bundle is then
These are all the definitions we need to know before attempting the physics.
Further information on the concepts is given in the paper when needed.
2 The Model Bundle for a Spinless Particle on a Line
, where the cotangent bundle of the configuration space
and since it is isomorphic to the 2D Euclidean space, m = (q, p) ∈ R2 .
The fibre F is the abstract Hilbert space, H, defined over the field of complex numbers with an inner product 2
where ψ ∈ H is defined as an equivalence class, ψ := [ψ] with respect to the equivalence relation ∼, under which the functions in [ψ] are the same except at a region of Lebesgue measure zero3 :
The structure group G of the bundle is the unitary group
where the elements h also satisfy linearity and are bijective over H. As usual, h ∈ U (H) act H on the left.
The principal fibre bundle (PFB) associated with it is trivial
Therefore, the associated fibre bundle (AFB)
is automatically trivial, as
where m ∈ R 2 , h ∈ U (H), and ψ ∈ H. Here G = U (H) acts on P × F by
and [(m, h), ψ] denotes the corresponding equivalence class in E,
where (m, h) ∈ R 2 × U (H). The logic behind the equivalence in eq.( 7) is that every class admits a canonical representative with principal part (m, id), for
by the definition in eq.( 2). 4 Therefore, we can set 5
The projection π E : E → M maps elements in E as (m, ψ) → m. The fibres are defined as
So, we have a copy of the Hilbert space over every point of the phase space M = R 2 .
Since the bundle is trivial, we can define them over all of M , as
where H(dx) = L 2 (R, dx) represents a Hilbert space with the Lebesgue measure dx:
4 That is, the elements (m, h, ψ) and (m, id, hψ) are both in the same set, which can be denoted as [(m, h), ψ] or [(m, id), hψ]. 5 That means, for every element m ∈ M , the element [(m, h), ψ] ∈ E can be represented by the element with the identity element of the group G. Therefore, we can set the equivalence relation
or just
from which the triviality of AFB E follows.
for x = p or x = q. 6 The first of them acts as
and the second one as
where the position and momentum space wavefunctions are related by the Fourier transform:
with F ∈ U (H) being a unitary operator on the Hilbert space.
The transition function
, whereby acting on M as an identity operator and on the fibres as the Fourier transform.
In this case, the configuration space is a circle, Q = S 1 , with elements θ ∈ [0, 2π), θ ∼ θ + 2π, and radius R. The cotangent space at θ associated with it is
Therefore, the phase space associated with it is
That means, the base of the model bundle in this case should be M = S 1 ×R. 6 These global trivialisations play a similar role to polarisations in geometric quantisation (GQ) [11,12]. Apart from this and the bundle structure, there hardly seems to be any similarities between the two frameworks, both in language and purpose, as in GQ one’s goal is to construct a quantum theory from the classical phase space, whereas in GV the main goal is to construct a geometric model for theories and models.
The fibre and the structure group stay the same. The PFB and AFB are defined as
where (m, ψ) = (θ, L), ψ ∈ E. The projection stays as defined above.
To define the local trivialisations, we need to look at the system a little closer:
In the position representation, the time-independent Schrödinger equation is7
This is an eigenvalue problem,
with periodic boundary conditions8
which yields a complete orthonormal basis {ϕ n }, for the general solution
That means one may also describe the quantised momentum labels n ∈ Z by replacing T * S 1 ∼ = S 1 × R with the disjoint union S 1 × Z ∼ = n∈Z S 1 n , i.e. a countable family of circles indexed by n. In this description, the bundle is nontrivial only in the θ-direction, while the Z-direction merely labels different copies of the same bundle.
The coefficients c n and ψ(θ) are related by the Fourier transform
Therefore, the position representation of the Hilbert space is
with the inner product
The momentum representation is
with the inner product
The trivialisations, which are again global, thus become
and the transition function between these two trivialisations is, again, the Fourier transform
For each base point m ∈ M , h θL (m) is a unitary operator on the fibre E m ∼ = H which relates the same abstract state to its two representations, that is, momentum coefficients c ∈ ℓ 2 (Z) and position wave-function ψ(θ) ∈ L 2 (S 1 , dθ/2π). 9 Concretely,
9 A (smooth) section of the Hilbert bundle π E : E → M is a map
In the present trivial situation E ∼ = M × H, a section can be written as
for some map ψ : M → H. Thus a section is a choice, for each phase space point m ∈ M , of a state vector in the corresponding fibre. In particular, a fixed state ψ ∈ H corresponds Thus h θL acts fibrewise, that is, as the identity on the base point m and as the Fourier transform on the fibre.
4 Twisted Boundary Conditions on M = T * S 1
As an example of a more interesting system, we are going to analyze systems with twisted boundary conditions. These kind of systems are introduced famously in the seminal papers [13,14]. To learn more about the mathematics of the subject, see chapters 16 and 17 in [9], and 10 in [8].
Before we start, let us introduce some basic concepts about connection and curvature in differential geometry [8,9].
As usual, let E be a Hilbert bundle with typical fibre H and base M . On overlaps U ∩ V ̸ = ∅, where U, V ⊂ M , the trivialisations are related by a transition function
as we discussed earlier.
Now, a connection on a vector (or Hilbert) bundle E is, by definition, an operator that differentiates sections of E, i.e. smooth choices of a vector in the fibre over each base point. Formally, a section is a map
so in a trivialisation E ≃ M × H we can write
to the constant section s ψ (m) = (m, ψ). When we speak of “wave-functions” in what follows, we mean such local representatives ψ(•) of sections in a chosen trivialisation. For instance, in the position trivialisation T θ , this reads
where ψ(θ; m) ∈ L 2 (S 1 ) may depend on m = (θ, L).
A connection D is then a map
which locally takes the form
where ψ : U → H is the local representative of a section, dψ is its ordinary differential, and A is an operator-valued one-form. Thus, connections and their curvature naturally act on sections. They describe how a state assigned to each point of M changes as we move in the base. In our trivial Hilbert bundle E ≃ M × H, a fixed state ψ ∈ H corresponds to the constant section s ψ (m) = (m, ψ), so we will not distinguish notationally between ψ ∈ H and the associated section.
A covariant derivative,
with an operator-valued 1-form,
defines a curvature as
where u(H) is the Lie algebra associated with the group U (H). On overlaps
These transformation rules express the fact that D and F are globally welldefined, even though the local representatives A U depend on the choice of trivialisation.
As in the previous section, we are going to consider a particle on a circle, with coordinate θ ∈ [0, 2π) and θ ∼ θ + 2π. Instead of strict periodicity, we allow the twisted boundary condition
We distinguish two situations:
(A) α is fixed once and for all, (B) α is treated as a classical parameter.
In Case A, the twist α is fixed. The base is purely classical,
and α is not a coordinate on M .
For fixed α, the position representation is
with inner product
Writing φ := α/(2π), the shifted modes
satisfy the boundary condition (45) and form an orthonormal basis of
In this basis, the angular momentum operator
so the spectrum is L = (n + φ)ℏ. For the free particle of mass m on a circle of radius R, the corresponding energy levels are
The momentum representation is the sequence space the previous case,
with inner product ⟨c, d⟩ = n∈Z c n d n .
Since the base space is the same, the AFB E is trivial again, as
Now, we choose two global trivialisations:
(position frame with fixed twist).
The transition function between these frames is the constant map
where
is the shifted Fourier transform
Thus h θL acts trivially on the base point m and unitarily on the fibre.
Because h θL (m) is independent of m (as it was in the previous sections),
we can choose the connection to be flat in both frames. In the momentum frame, let us set
This corresponds to a gauge fixing and does not change the underlying physics. Transforming to the position frame with (43) gives
So, with α fixed and not part of the base, there is no nontrivial holonomy arising from the connection as we move in M . The twist is entirely encoded in the boundary condition and the shifted Fourier basis.
As it is seen, apart from an extra constant factor α ∈ R, there is essentialy not much of a difference between the boundary conditions ( 22) and (45). The following case is more interesting in that respect.
In Case B, we promote α to a classical parameter and include it in the base,
where S 1 α parametrises α modulo 2π. A point of M ′ is (θ, L, α).
For each α, the position space is H (α) θ as in (47). As α varies, we thus obtain a family of Hilbert spaces
all of which are unitarily isomorphic to one another, but with different twisted boundary conditions (45). On the other hand, the momentum space
We again take E ′ → M ′ to be trivial and use two global frames
where in the second line we emphasise the α-dependence of the position space. The transition function is now α-dependent:
with F α still given by (56) but now viewed as a smooth family of unitary operators parametrised by α. We then have a family of quantum systems labelled by α, with a corresponding family of position spaces H
θ and a smooth family of unitary maps F α relating them to the momentum representation.
As before, let us choose the momentum frame so that the connection is flat and trivial:
Transforming to the position frame using (43) gives
Here d includes differentiation with respect to α, since F α depends on α. A short calculation using
shows that A ′ θ acts, in the position frame, as multiplication by iθ/(2π) along the α-direction:
Thus the covariant derivative along α is
The curvature remains zero,
In this paper we have worked out the Geometric View of quantum mechanics for a spinless particle on a line and on a circle. In both cases the classical phase space M = T * Q serves as the base of a trivial Hilbert bundle For the particle on a circle, we analysed in more detail how momentum quantisation and boundary conditions fit into this framework. In the untwisted case, the phase space T * S 1 and the Hilbert bundle E remain trivial, while the discrete momentum labels n ∈ Z appear in the choice of basis and in the operator algebra acting on the fibres. We then introduced twisted boundary conditions and distinguished two situations. In Case A, the twist parameter α is fixed and does not enter the base; the effect of the twist is entirely encoded in the shifted Fourier basis and the resulting spectra, and one can work with a flat, trivial connection. In Case B, the twist is promoted to a parameter and included in an enlarged base M ′ = T * S 1 × S 1 α . The Hilbert bundle remains trivial as a bundle, but an α-dependent family of Fourier transforms induces a flat connection with nontrivial holonomy along the α-circle. In this way the twist is geometrically realised as holonomy in parameter space.
On the mathematical side, these examples illustrate how the Geometric View organises familiar quantum-mechanical data (states, representations, spectra and boundary conditions) into the language of bundles, trivialisations, transition functions and connections. Philosophically, they illustrate several claims of the geometric view of theories. First, they make precise the idea that a duality is a “change of coordinates” on a single underlying object: The duality between the position and momentum descriptions is now literally a change of trivialisation of a bundle with fibre H over M . In this simple case, the bundle is trivial, as GV predicts for genuine dualities; yet this triviality does not make the structure philosophically trivial. Rather, it shows that the usual semantic picture of a theory as a set of models corresponds to the special case of a globally trivial model bundle with identity structure group. The bundle formalism thus sharpens, rather than replaces, the semantic insight.
Second, the construction clarifies the status of representations in quantum mechanics. On the schematic view of dualities developed by De Haro and Butterfield, a theory is given by a triple T = ⟨S, Q, D⟩, and its models or representations are ways of instantiating this triple. The present example studies a simple version of how such representations can be organised in a geometric object. Position and momentum representations are then just different coordinate systems on this object, related by a duality map that is implemented in the bundle as a globally defined transition function.
Third, the example supports a cautious form of realism about some of the global structure captured by the geometric view. On its own, the model bundle is not a substitute for interpretive work, and nothing in the formalism forces distinct representations to be automatically equivalent. However, once a common core has been identified and the duality verified, the bundle provides a natural home for realist commitments about the shared structure and its parameter space. In the present case, the triviality of the bundle is itself physically and philosophically informative: It encodes the fact that position and momentum representations are fully interchangeable, with no obstructions or singularities in the parameter space.
Finally, the analysis suggests several directions for further work. One is to move beyond exact dualities to quasi-dualities, by considering systems where different semi-classical regimes are only locally related, so that the model bundle is genuinely non-trivial. 10 One could treat configuration spaces with higher topology, such as Q = S 2 , or systems with non-Abelian internal degrees of freedom, where nontrivial vector subbundles and Berry curvatures are expected to appear. One could also enlarge the fibre to include, in addition to the Hilbert space, the algebra of quantities acting on it, thereby more fully implementing the ⟨S, Q, D⟩ scheme of De Haro and Butterfield [6]. We leave these developments to future work.
For more on the fibre bundles and their uses in physics, see[7][8][9].
See[10] to learn more about the quantum mechanical parts in the paper in more detail.
See the footnote 3 in the chapter
4.1 in [6]
Although we only need the boundary conditions and not the dynamics, we started with the Schrödinger equation for the sake of completeness.
For the case twisted boundary conditions, see the next section.
For the role of quasi-dualities as local transition functions on moduli spaces, see[5] sect. 3.
