Geometrical viewpoint of quantum mechanics: wavefunctions on curved spaces

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GEOMETRICAL VIEWPOINT OF QUANTUM MECHANICS

Wavefunctions on curved spaces

Bachelor of Science Thesis Faculty of Engineering and Natural Sciences (ENS) Supervisors: Prof. Lasse Laurson, Andun Skaugen Ph.D December 2020

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ABSTRACT

Khai Phan: Geometrical viewpoint of quantum mechanics Bachelor of Science Thesis

Tampere University

International Degree Program in Science and Engineering December 2020

This thesis attempts to present an existing idea on formulating quantum mechanics from a geometrical viewpoint. Although the main propositions can be found in literature, some mathematical details are missing in derivations and constructions, hence this thesis aims to fill in those gaps.

In addition, the momentum operators in generalized coordinates are redefined using the notion of covariant derivatives and connection 1-forms so that the operators are Hermitian. Finally, there are two examples where we essentially work out the momentum operators in the 3-dimensional Euclidean space in spherical coordinates and in 2-dimensional curved spaces with constant curvature.

Keywords: quantum mechanics, manifolds, associated fibre bundle, covariant derivative, connection 1-forms, Yang-Mills field

The originality of this thesis has been checked using the Turnitin OriginalityCheck service.

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PREFACE

My initial goal in choosing this thesis topic was to learn how objects in quantum mechanics can be described in the language of differential geometry. The process of writing this thesis helped me realize some important relations between quantum theory and the classical counterpart as well as understand the physical and mathematical origins of the indispensable axioms of quantum mechanics and their corollaries.

I would like to thank my supervisors for encouraging me to choose this thesis topic which is outside of my undergraduate physics curriculum and their usual topic of research. Had it not been for Audun’s patience and interest to investigate the mathematical details, many subtle issues would have been overlooked and remained unaddressed. I really enjoyed the conversations I had with Audun while doing this thesis because of his willingness to listen to my unrigorous lines of thoughts and especially his insightful queries which so often led to better understanding. I also benefited greatly from professor Lasse’s comments on the writing as well as the overall structure of this thesis.

Tampere, 30th December 2020

Khai Phan

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1 INTRODUCTION

In a quantum mechanical system, pure states can be represented by wavefunctions, which are elements of a Hilbert vector space. Any choice of such a Hilbert (vector) space reflects certain mathematical features, which are directly related to some corresponding physical properties of nature that we believe to be true such as rotational symmetry, translational symmetry, and Poincare invariance. A Hilbert space encodes these axioms through the admission of operators which obey the corresponding symmetry. In particular, standard textbook quantum mechanics constructed the Hilbert vector space using complex square-integrable functions. ¹This choice has achieved success in providing us a set of Hermitian observables (i.e. quantum operators) with suitable Lie algebras which respect the spatial symmetries of quantum systems in Cartesian coordinates. The Hilbert space presented in this thesis maintains the symmetries while allowing us to work in generalized coordinates.

As one naively attempts to apply by analogy the standard structure to describe quantum pure states in some other coordinate systems, one quickly runs into technical issues, such as the non-Hermiticity of momentum operators. With some pondering, one realizes that the formalism lacks geometrical coherence at the level of wavefunctions and quantum operators. In particular, the construction as such only refers to the relationship between different coordinates via coordinates transformations. To put simpler, different coordinate variables are related to each other via the fact that they encode the same points in real space, but the actual quantum objects (i.e. pure states and operators) do not inherit any geometrical link. Mathematically, the structure of wavefunctions as complex functions on real space is not enough to establish a coordinate-independent picture of quantum objects.

This way of dissecting the problem hinted at a way to modify the theory, namely by further geometrize it. The theoretical sections in this thesis are devoted to summarizing the mathematical constructions of the Hilbert space and operators acting on it from a geometrical point of view. This turns out to not only be a fix of the technical incompatibility existed in the standard quantum theory, but also gives a framework that allows us to carry out analysis of quantum systems with constraints, because the structures remain valid in

1Note that there is an intentional use of the term real space, to distinguish that from a point in Euclidean spaceR³, which may or may not be suitable to model our “real” space.

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any orientable² manifold and general coordinates. In particular, we might be interested in studying systems of particles that are confined in loops and surfaces³, which clas- sically could be mathematically modeled using generalized coordinates via Lagrangian and Hamiltonian formalisms. The advantage of formalisms which allow generalized coordinates is that by carrying out a coordinate transformation, hidden symmetries⁴ can be revealed at the cost of working with lower-dimensional curved spaces⁵.

The goal of this project is to summarize the construction of the mathematical structures necessary to formulate components of a quantum system using differential geometry.

From there, we then define pairs of position and momentum operators such that they satisfy the canonical commutation relation, i.e. Lie brackets algebra, and are both self- adjoint. Next, from a physical point of view, we hope to get a consensus of predictions between observers in different coordinate charts which requires some compatible conditions. Thus, we will identify those conditions with explicit constraint equations during construction. For completeness, we finally go through some concrete example applications, one case with a flat space in polar coordinates and one with curved space that has constant curvature.

Existing works that formulate Quantum Mechanics on manifolds impose manifold structures on either phase space or configuration space of physical systems. Some papers that introduce geometrical properties of real space with the prior approach are Kibble 1979, Bayen et al. 1978, Batalin and Tyutin 1990, in which the Hilbert space and state vectors are not introduced. This thesis is concerned with the latter approach, particularly by constructing an associated bundle on the configuration space of a single particle and declaring that state vectors are sections of the associated bundle.

In literature, the idea of taking sections of some associated bundle to be elements of the underlying Hilbert space in Quantum Mechanics has been presented by, for instance, Tanimura 1993, and Schuller 2013-2014. In those references, the specific choices of the associated bundles differ as they admit distinct underlying groups, their actions and associated vector spaces. In this thesis, we focus on the associated bundle which was presented in the lecture 26 of the lecture series (Schuller 2013-2014). Most of chapter 3 and all of chapter 4 are dedicated to presenting in detail the ideas and statements made in the lecture, except sections 3.2 and 3.5 where we added points that were not part of the lecture. In particular, in section 3.2, we define a specificleft actionso that the construction is coherent and rigorous. In section 3.5, we define an inner product⁶on the

2Orientability is crucial as we need the notion of an inner product of wavefunctions, which requires a global volume form to be integrated over.

3Say, an electron moving in a circle under influence of magnetic field

4Say, rotational symmetry of group SO(3)

5For instance, model a particle on a sphere using intrinsic geometry, i.e. ignore the 3-dimensional embedding, requires working with a 2-dimensional curved space.

6In other words, we pick a different Borel measure, which specifies a volume form on the base manifold, to define the Hilbert space inner product.

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Hilbert space which differs from that in the lecture by a factor of the metric’s determinant in order to obtain a well-defined inner product, i.e. its form agree for all choices of coordinate system. In addition, the condition for a Yang-Mills fields which allow Hermitian momentum operators in chapter 5 is also different from that in the lecture due to the extra factor in the inner product.

A list of foundational mathematical concepts are mentioned in the appendix section, most details can be found in the reference books (Hamilton 2017, Fecko 2006).

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2 MATHEMATICAL FOUNDATION

The theory of manifolds generalizes the intuitive notion of Euclidean space by starting from the most fundamental level, mathematical sets, then re-build on them structures such as topologies, differentiability, and metrics (figure 2.3). At each level, one is free to decide which structures to equip the manifolds with, which makes its application in modeling nature ubiquitous, from various quantum fields in quantum field theory to the fabric of space-time in Einstein’s theory of relativity. Furthermore, the way the theory is formu- lated intrinsically without referring to an exterior Euclidean embedding gives a powerful and elegant language to describe self-referencing systems, which ultimately includes the universe itself.

This section is supposed to be a prelude, hence semi-rigorous definitions are given. It aims to give readers some ways to intuitively think of some of the necessary geometrical objects. Therefore, mathematical rigor will be neglected to some degree. Readers who seek formal constructions and proofs shall be referred to the references and the appendix section.

2.1 Sets, topologies and manifolds

In physics, modeling either physical or abstract objects requires very different mathematical structures (fig. 2.3). Roughly speaking, the number of features needed to capture a class of objects varies, in a hierarchical fashion. For example, apples and oranges belong to a class of discreet and enumerable objects (kinds of fruits) so counting them can be done withset theory. Some other phenomena require notions of convergence and conti- nuity, meaning that we care about whether there are ways to get to a certain element of a set, at least as a limit, via linking "vicinity neighborhoods", i.e. open sets. In topological spaces, one defines atopology on a set by choosing a subset of the power set (i.e. set of all possible subsets), which captures exactly the mentioned idea.

Amanifold can be thought of as a topological space that locally resembles a Euclidean space with the standard topology, i.e. the topology where an open set refers to an open ball of some radius. Though strictly unnecessary, a manifold is often equipped with an atlas, which is a set of charts. Each chart is a function that maps an open subset of the

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manifold onto a Euclidean space with equal dimensions¹. Because some open subsets of the manifold may be covered by multiple charts (the overlapped patches), it is reasonable to sometimes demand that the transition functions from some chart to others be k-times differentiable². In physics, we often demand that k be infinity, in which case we have a smooth manifold. The introductory ideas of topologies and manifolds can be found in chapter 1 (Fecko 2006).

2.2 Tangent spaces and coordinate-induced basis

At each pointpon a manifoldM, we can define atangent spaceT_pM which is a vector space consisting of functions acting on differentiable functions on the manifold, i.e. X ∈ T_pM =⇒ X :C^∞(M)→C^∞(M)that satisfy the following conditions:

• Leibniz rule: ∀f, g ∈ C^∞(M), X ∈ T_pM : X(f(p)·c(p)) = (X(f))(p)·g(p) + f(p)·(X(g))(p),

• Linearity: X(f+g) = X(f) +X(g), andX(λf) =λX(f)for any numberλ. Having satisfied those conditions, elements of a tangent space are called derivations and the space T_pM itself is a vector space. Pictorially and intuitively, we can visualize the vectors in the tangent space at some point p on M as a set of Euclidean vectors making up a hyper plane tangent to a surface representing the manifold M as in figure 2.1. We can justify this intuitive picture to be appropriate by introducing a Euclidean ambient space that has higher dimension than M to embed the manifold in, which is always possible for any smooth manifold M (Whitney embedding theorem). Given a coordinates chartx= (x¹, x², ..., x^d)onM, we associate to each coordinate variablexⁱ a function _∂x^∂i :C^∞(M)→C^∞(M), which is defined as follows:

∀f ∈C^∞(M) : (︃ ∂

∂xⁱ )︃

p

(f) = (︃ ∂

∂xⁱ(f ◦x⁻¹) )︃

(x(p)), (2.1)

note that on the right hand side, we have the normal partial derivative acting on functions on a Euclidean space since f ◦x⁻¹ : R^d → R. It can be shown that each of these functions at a point is an element of the tangent space at that point, i.e. (︁ _∂

∂xⁱ

)︁

p ∈ T_pM and they are linearly dependent. As a result, they together make up a basis that spans the tangent vector space which is called thecoordinate-induced basis. More details can be found in chapter 2.2 (Fecko 2006).

1Hence called "charts" and "atlas". The open subset that is covered by a chart is called a patch.

2Transition functions are just functions from one Euclidean space to another, so the differentiability is well-defined.

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2.3 Pull-back of a function on a manifold

LetM, N be two smooth manifolds andf : M →N be an injective map. We define the mapf^∗ :C^∞(N)→C^∞(M)simply as:

∀p∈M, ϕ∈C^∞(N) : (f^∗(ϕ))(p) =ϕ(f(p)). (2.2) The mapf^∗is then called the pull-back off. Note that in more generality, pull-backs can act onk-forms, not just on functions (which are0-forms). However, only the special case mentioned above is discussed in this thesis, so the full definition of pull-backs andk-forms are left out and can be found in chapter 6.1 (Fecko 2006).

2.4 Metric and metric manifolds

If distances and angles are needed, we wish to have apseudoor a proper inner product on every tangent space of the manifold. Such a distribution of (pseudo or proper) inner products constitutes a metric. A metric manifold is a manifold equipped with a metric. We have that an inner product is a bi-linear function that maps any two vectors to a number field (complex, real or else). Formally, we have that an inner product(., .) : V ×V →F over a vector spaceV must satisfy

• Bi-linearity:(ax, y) =a(x, y)and(x+z, y) = (x, y) + (z, y)∀x, y, z∈V.

• Symmetric property:(x, y) = (y, x)∀x, y ∈V.

• Positive-definiteness:∀x, y ∈V : (x, y)≥0and(x, x) = 0 ⇐⇒ x= 0. A pseudo inner product has a weaker conditions, namely

• Bi-linearity:(ax, y) =a(x, y)and(x+z, y) = (x, y) + (z, y)∀x, y, z∈V.

• Symmetric property:(x, y) = (y, x)∀x, y ∈V.

• Non-degeneracy:∀and(x, x) = 0 ⇐⇒ x= 0.

We can see that because a metric may give negative valued results when we input some two vectors, we are allowed to talk about "negative distances", as is the case of the Lorentzian metric that shows up in Einstein’s relativity. A metric on a manifold can be understood as a distribution of pseudo-inner products, one for each tangent space of the manifold. For notation, we denote components of a metricg in some coordinatesxas:

g_(x)ij =g (︃ ∂

∂xⁱ, ∂

∂x^j )︃

. (2.3)

More details can be found in chapter 2.6 (Fecko 2006).

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2.5 Bundle

A bundle of manifolds is a triplet (E, π, M), where both E,M are manifolds which are called thetotal spaceand thebase space manifolds respectively, andπ is a continuous functionπ :E →M which is called the bundle’sprojection. In addition, at each point on the base space, the pre-image of the projection π of that point is called the fibre at that point (figure 2.2). For instance, tangent bundle is a commonly seen concept in physics, in which each fibre is a tangent space at its corresponding point. The concept of fibre bundle is addressed in detail in chapter 17.2 (Fecko 2006). In addition, tangent and cotangent bundles, which are two important types of bundles in physics, are presented clearly and rigorously in chapter 17.1 of the same book.

2.6 Lie group and its Lie-algebra

In physics, we are often interested in expressing continuous symmetries through the actions of some corresponding groups. The concept of Lie groups appears naturally in many contexts in physics precisely for that reason, as it essentially captures both the differentiability and the group structure, which are two aspects of a typical continuous symmetry.

Formally, a Lie group G is defined as a differentiable manifold that is equipped with a smooth mapµ, which could be understood as a function that performs a group operation

◦, that satisfies the group axioms:

• Closure: ∀g₁, g₂ ∈G:µ(g₁, g₂) = g₁◦g₂ ∈G,

• Associativity: ∀g₁, g₂, g₃ ∈G: (g₁◦g₂)◦g₃ =g₁◦(g₂◦g₃),

• Identiy: ∃e∈G: ∀g ∈G:e◦g =g◦e=g,

• Inverse: ∀g ∈G:∃g⁻¹ ∈G:g◦g⁻¹ =g⁻¹◦g =e.

In some cases, we demand a commutative Lie group, i.e. µ(g₁, g₂) =µ(g₂, g₁)∀g₁, g₂ ∈ G.

On the other hand, linearization of a Lie group gives rise to a corresponding Lie algebra, whose elements constitute a vector space, hence are convenient to work with. It is suffi- cient for the purpose of this thesis to consider elements of the Lie-algebra induced by a Lie-groupGas vectors on the tangent plane at the identity ofG, i.e. as a vector space, the Lie-algebra is isomorphic toT_eG. In addition, to be called a Lie-algebra,T_eGis equipped with an anti-symmetric bi-linear map[., .] :T_eG×T_eG→T_eGwhich satisfies:

• anti-symmetricity: ∀X₁, X₂ ∈T_eG: [X₁, X₂] =−[X₂, X₁],

• bi-linearity: ∀X₁, X₂, X₃ ∈ T_eG : [X₁ +X₂, X₃] = [X₁, X₃] + [X₂, X₃], and

∀f ∈C^∞(G) : [f X₁, X₂] =f[X₁, X₂],

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Figure 2.1. Intuitive view of a tangent space.

Figure 2.2.Bundle and fibres.

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• Jacobi identity: ∀X1, X2, X3 ∈TeG:

[X₁,[X₂, X₃]] + [X₃,[X₁, X₂]] + [X₂,[X₃, X₁]] = 0.

Chapter 1 (Hamilton 2017) deals with the concept of Lie groups and Lie algebras in the context of physical theories in physics, whereas generic mathematical details related to Lie groups and the relation to its Lie algebras can be found in chapter 11 (Fecko 2006).

In particular, step-by-step construction of the Lie algebra of some Lie group, both as left- invariant tensor fields and as tangent vectors at the Lie group’s identity are demonstrated from chapter 11.1 to 11.5 (Fecko 2006).

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Figure 2.3.Hierarchy of mathematical structures.

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3 THE HILBERT SPACE

We begin our discussion by constructing a Hilbert space whose elements are wavefunctions that represent quantum pure states. In particular, a wavefunction is defined to be a section of an associated C-line bundle to the frame bundle on the base manifoldM, whereM is the real space. The following sections are dedicated to unfolding the meaning of that expression, as well as pointing out choices made when a degree of freedom arises during construction.

3.1 The frame bundle on M

For each option of coordinates system, we have a canonical basis which is a set of basis vectors that span the entire tangent spaces. To encode abstract vectors by some elements ofR^d, we decompose the vectors using one of those basis, which result in distinct elements in R^d depending on the choice of basis, each contains vector components. In linear algebra, we say that as we apply a basis transformation T, the components of a vector as an element ofR^d is transformed by the inverse matrix T⁻¹, so that the result represents “the same” abstract vector. It is precisely this idea that motivates us to build a structure consisting of two parts: the basis (traditionally calledframein this context), and an element ofR^d (i.e. vector components in that basis). From the definition of the frame bundle, we see that it allows us to capture the idea of a set of frames (basis) at each point and transformations from one to another via some linear transformation. The associated bundle to the frame bundle, roughly speaking, consists of pairs of a frame and an element inR^d^.

At each point in M, we associate a set of all possible choices of basis vectors (at that point). On top of the set structure, we wish to define maps that take one choice of basis to another. Those maps constitute a group isomorphic to the general linear groupGL(Rⁿ), i.e. group of invertible matrices. Any frame (basis) can be obtained from another frame by acting on it via the right group action (fig. 3.1).

As a result, we have a bundleP → M via projection mapπ, whereP is the total space (frame bundle itself), each point on which can be regarded as a pair of of a point inM, and a choice of basis for the tangent space at that point (i.e. a point inM and an element

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Figure 3.1.The right action of groupG≃GL(R^d).

inGL(R^d)). We alternatively denote the bundle as a triplet: (P, M, π) (diagram in fig. 3.2).

3.2 The associated bundle to the frame bundle

The associated bundle P_F is constructed alongside with the frame bundle (hence the name) in order to capture the equivalence relation between pairs of a basis (frame) and an element in some vector space (one dimensional complex vector spaceCin this case).

In general, such an equivalence relation captures some invariance in physics by allowing us to define appropriate transformations on the vector space (e.g. C) when we change coordinates system so that the resulting pair remains unchanged (the quotient space induced by the equivalence relation is the set of pairs that are physically identical).

First we consider a Cartesian product ofP and a vector space, specifically a 1-dimensional complex vector spaceF =C. We denote this set asS ={(q, z)|q ∈P, z ∈C}. On this we define an equivalence relation as follows:

(q₁, z₁)∼(q₂, z₂), (3.1)

if and only ifq₂ =q₁◁ a, and z₂ =a⁻¹▷ z₁ for some elementa∈G. We can see that in order for this to work, we need a notion of the left action ▷: GL(R³)×C →C^{. In other} words, we need to define how an element of the group acts on the vector spaceF =C^.

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Figure 3.2.The (principal) frame bundle and its section (diagram).

We define the left action of an element of the groupGon a complex number as follows:

g ▷ z := det(g)·z, (3.2)

where the operation · is simply the usual multiplication of 2 complex numbers. From the quotient PF = S/ ∼¹, we can construct a bundle so-called associated bundle (P_F, M, π_F)²whose diagram is illustrated in fig. 3.3.

3.3 Hilbert space elements (wavefunctions)

On the constructed associated bundle, we define the notion of wavefunctions as elements of the set: S = {σ :M → P_F|π_F ◦σ = 1}. In other words, a wavefunction is a section of the associated bundle. Because there exists a bijective map from the set S to a subset of functionsC^∞(P)³, we can associate such a sectionσ with its corresponding function ψ_σ :P →C. Thus, we can equivalently define wavefunctions as functions{ψ_σ :P →C} (proof in appendix section A).

1We also denote an element of this quotient set by a representative [q,z], with the square brackets.

2The mapπFcan be canonically defined (appendix).

3It is only bijective if we put some restriction on functions inC^∞(P), details can be found in appendix.

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Figure 3.3. The associated bundle to the frame bundle and its section (diagram).

3.4 Local representation of wavefunctions

On each patch of(P, U_i, π), we have a canonical section which is given by the coordinate basis, i.e. the basis {_∂x^∂i|_p} at every pointpon M that is induced by the choice of coordinate chartx. We denote the induced section ϕ_x :U_i → P. In other words,ϕ_x takes a point onU_iand gives us the set of basis vectors induced by coordinatexat that point.

We then define the local representation ofψσ as follows:

ψ =ϕ^∗_x(ψσ), (3.3)

whereϕ^∗_xis the pullback given by sectionϕx. In other words, using the canonical section ϕ_x, we have pulled back the function ψ_σ : P → C to a function ψ : U_i → C^{, whose} domain is on the base manifold M. We notice that this is the familiar definition of a wavefunction in standard quantum mechanics on Euclidean space.

3.5 Hilbert space inner product 3.5.1 Case 1: single global section

In order to define a suitable inner product of the Hilbert space, we first need to show how various objects transform under general coordinate transformations.

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Metric determinant

First and foremost, a metric is a (0,2)-tensor, so it does not have a well-defined determinant as a geometrical object. Nonetheless, we may define the determinant of matrices⁴ which carry the metric components in certain coordinates. Therefore, this definition of metric determinant is coordinate-dependent⁵. In particular, we can show that the transformation law of the metric determinant is

det(g_(x)ij) = det (︃

g_(y)mn∂y^m

∂xⁱ

∂yⁿ

∂x^j )︃

= det(g_(y)ij) [︃

det (︃∂yⁱ

∂x^j )︃]︃2

.

(3.4)

Note that _∂x^∂yⁱj is the Jacobi matrix, which is an endomorphism (i.e. (1,1)-tensor), so its determinant is chart-independent.

Figure 3.4.Two sections induced by two different coordinate charts.

Local representation of wavefunctions

We know that the coordinate induced sectionsϕ_x and ϕ_y are related via the right action by some group elementa: (fig. 3.4)

ϕ_x(p) = ϕ_y(p)◁ a, (3.5)

4Matrix in the sense of 2 dimensional array with rows and columns, not anendomophismon a vector space.

5while determinant of an actual endomorphism matrix is coordinate-independent.

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wherea∈G∼=GL(d, R), andaⁱ_j = _∂x^∂yⁱj. By eqn A.2, we have

ψ_σ(ϕ_x(p)) = a⁻¹▷ ψ_σ(ϕ_y(p))

= 1

det(a) ·ψσ(ϕy(p)). ^(3.6) Note that, by definition of a pull-back on functions,ψ_σ(ϕ(p)) = (ϕ^∗(ψ_σ))(p).

Inner product on the Hilbert space

We now are ready to define the suitable inner product of the Hilbert space, and show that it is independent of the choice of coordinates.

< ψ_σ₁|ψ_σ₂ >=

∫︂

M

[det(g_(x))]^3/2(ϕ^∗_x◦ψ_σ₁)^∗(x)·(ϕ^∗_x◦ψ_σ₂)(x)·d⁴x

=

∫︂

M

[︂√︂

det(g_(x))d⁴x]︂

(ϕ^∗_x◦ψ_σ₁)^∗(x)·(ϕ^∗_x◦ψ_σ₂)(x)·det(g_(x)).

(3.7)

Applying the laws of transformations under a change of coordinates system in equations 3.4 and 3.6, we have

< ψ_σ₁|ψ_σ₂ >=

∫︂

M

[︂√︂

det(g_(y))d⁴y]︂[︃

1

det(a)(ϕ^∗_y◦ψ_σ₁)^∗(x) ]︃

· [︃ 1

det(a)(ϕ^∗_y ◦ψ_σ₂)(x) ]︃

·det(g_(y)) [det(a)]²

=

∫︂

M

[det(g_(y))]^3/2(ϕ^∗_y◦ψ_σ₁)^∗(y)·(ϕ^∗_y◦ψ_σ₂)(y)·d⁴y.

(3.8)

Note that the volume form is independent of the choice of charts⁶, hence√︁

det(g_(x))d⁴x=

√︁det(g_(y))d⁴y. We can see that the inner product above is well-defined as its form agrees for all coordinate charts.

3.5.2 Case 2: multiple charts.

If the base manifold is non-trivial, there exists no single chart that covers the entire base manifold, hence no global section exists. As a result, we are forced to define the inner product via integrals on multiple patches of the manifold, each with its own chart. First, we need the notion of a partition of unity. A partition by unity is a set of functions{P_i|P_i : M → R⁺^,∑︁n

i=1P_i(p) = 1 ∀p ∈ M and supp(P_i) ⊂ U_i. The inner product is then

6Because the square root of the metric determinant transforms in the opposite way to the differential under change of chart.

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defined as

< σ_ψ₁|σ_ψ₂ >=

n

∑︂

i=1

∫︂

Ui

(P_i◦x_i⁻¹)(x_i)·[det(g_(x_i₎)]^3/2(ϕ^∗_x

i◦ψ_σ₁)^∗(x_i)·(ϕ^∗_x

i◦ψ_σ₂)(x_i)·d⁴x_i, (3.9) where xi : M → R^d is the chart map of the patch Ui and P is any partition by unity. It can be shown that the inner product does not depend on the choice of partition and the proof is exactly analogous to the case of integration on manifolds.

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4 THE MOMENTUM OPERATORS

We claim that by defining the momentum operator of the i^thcoordinate of the chartxto be

pˆ =_i −iℏ∇ ∂

∂xi

, (4.1)

where ∇ ∂

∂xi

is the covariant derivative in the _∂x^∂i direction, we can make it self-adjoint by choosing an appropriate connection 1-form ω. We will see during this section that an (affine) covariant derivative contains three degrees of freedom, i.e. three choices we need to make to uniquely define it, namely the section of the frame bundle, the left action of the underlying Lie group, and the connection 1-form¹.

4.1 Construct the covariant derivative from connection 1-form

Ultimately, in order to have a covariant derivative, we need to decide a subspace called horizontal subspace of the tangent space of the frame bundle that "connects" elements of the frame bundle in different fibres. In other words, such a choice in horizontal subspace is often needed to transport a vector at a point to some other next vector at an infinitesimally close point in a parallel fashion². Although the mentioned horizontal subspace is a choice, the vertical subspace, i.e. its complementary space, is not. Thus, it is convenient to first define the vertical subspace.

4.1.1 Bijection between Lie-algebra and a subspace of T

_q

P

It turns out that we could construct the vertical subspace directly via the Lie-algebra of the groupG, and the way this relation coincides with the formal definition of vertical subspace may give some insightful intuition (fig. 4.1).

First we define a mapping from an element of the Lie-algebraT_eGto a tangent vector of the frame bundleTqP as follows

i(A) =X_A, (4.2)

1Or we shall see, equivalently, the horizontal subspace of the frame bundle’s tangent space.

2Refering to the notion of parallel transport

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Figure 4.1. The vertical subspace at a point on the frame bundle.

whereA∈T_eGandX_A∈T_qP.X_Ais defined by its action on a functionf :P →R

XAf = d dt

⃓

⃓_t=0

[f ◦(q ◁exp(tA))], (4.3)

wheret∈Ris the real number line that parametrize both curves in figure 4.1.

Remarks: expis a map from a Lie-algebra element to a Lie-group element defined via integral curves of the left invariant vector field generated by the Lie-algebra element.

Thus,exp(tA)∈Gis a parametrized curveγGinG, andq ◁exp(tA)is a curveγP in the frame bundle generated by the curveγ_G. The vectorX_A is then defined as the tangent vector to frame bundle P through which the curve γ_P passes. Note: by definition the curveγ_P³lies entirely in the same fibre at some pointp∈ M, thus the tangent vector to the curve points "vertically" along the fibre (fig. 4.1). The vertical notion will be justified further in the next section, when we show that those vectors lie entirely in the vertical subspace which are projected to the nullspace at the base point on M. Details can be found in chapter 11.4 (Fecko 2006).

3since it is generated by the dragging action of elements inG.

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4.1.2 Vertical subspace of T

_q

P

We now formally define the vertical subspace of the tangent space of the frame bundle at q∈P as follows:

V_qP ={X_q∈T_qP|π∗(X_q) = 0 ∈T_qM}. (4.4) In other words, the vertical subspace atqis the kernel of the push-forward by the projection mapπ. It can be shown that any vector inVqP can be generated by some element of the Lie-algebra, and each Lie-algebra element generates a unique vector in the vertical subspace by equation 4.3. Hence, we have

dim(V_qP) =dim(T_eG) = d², (4.5) where dis the dimension of the base manifoldM. Recall that the elements on the total space of the frame bundle can be interpreted as pairs of a point on M and a choice of basis at the corresponding point, hencedim(P) = d+d². Thus, dimension of the full tangent spacedim(T_qP) = dim(P) = d²+d.

4.1.3 Horizontal subspace of T

_q

P

A subspaceH_qP ⊂T_qP is a horizontal subspace of the tangent spaceT_qP if:

H_qP ⨁︂

V_qP =T_qP, (4.6)

and it satisfies the compatible condition when moving between any two elements of the same fibre:

(◁g)_∗H_qP =H_q◁gP, (4.7)

and the unique decomposition condition:

Xq =ver(Xq) +hor(Xq), (4.8)

where by definition ver(Xq)∈VqP and hor(Xq)∈HqP.

Remark: Since we have no notion of angles (since no inner product is defined on the total space of the frame bundle), the choice of such horizontal subspace is arbitrary. Before the choice of such a horizontal space is established, neither the horizontal component nor the vertical component of a vector (that does not lie entirely inV_qP) is defined, although the vertical subspace is well-defined without the existence of a horizontal subspace (fig.

4.2).

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Figure 4.2.Two different horizontal subspaces yield different vertical components.

4.1.4 Connection 1-form

Though the notion of horizontal subspace as defined is intuitively pleasing, practically it is very difficult to work with an abstract choice of subspace when constructing covariant derivatives and other geometrical structures. It turns out that we can conveniently encode the choice of horizontal subspaces in a differential 1-forms without any loss of information.

We now define the Lie-algebra valued connection 1-formω :T_qP →T_eGto be:

ω=i⁻¹◦ver. (4.9)

In other words, a connection 1-form takes a vector in the tangent space of the frame bundle, projects it onto the vertical subspace (i.e. take the vertical component), then map it to the corresponding Lie-algebra element through the mapi. We can see that each connection 1-form encodes a unique choice of horizontal subspace (fig. 4.3). In particular, from a choice of connection 1-form ω, we can obtain the corresponding horizontal subspace by taking its kernel, i.e. H_qP = ker(ω_q) :={X ∈T qP |ω_q(X) = 0}.

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Figure 4.3.One choice of horizontal subspace at a point on the frame bundle.

4.1.5 A derivative on frame bundle

We define a derivative on the total space of the frame bundle D : C^∞(P) → T_q^∗P as follows:

Dψ_σ =dψ_σ◦hor, (4.10)

recallψ_σ : P → Cis a Lie algebra-valued function, and T_q^∗P is the cotangent space at q∈P. It can be shown that (proof comes later) that:

Dψ_σ(X) = dψ_σ(X) +ω(X)▷ ψ_σ, (4.11) whereX ∈T_qP.

Roughly speaking, if we are provided with a vectorX, eqn. 4.11 gives us a global version of the covariant derivative on functions onP. The only remaining step is to define a local version of the covariant derivative via a pull-back.

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4.1.6 Covariant derivative

We can define the local covariant derivative that acts on the local representation of wavefunctions which live on the base manifoldM as follows. Supposeϕ^∗ψ_σ =ψ, we define

∇_Xψ = (ϕ^∗(Dψ_σ))(X) (4.12)

=dψ(X) + (ϕ^∗ω)(X)▷ ψ. (4.13) where the pullback of the connection 1-formϕ^∗(Dψ_σ)is called the Yang-Mills field, which encodes the same information as the connection 1-form, although it is a 1-form living on the base manifold. Thus, the Yang-Mills field will be the central object whose value we will derive in the next section, such that the momentum operators are self-adjoint.

4.2 Canonical commutation relations

We now show that the canonical commutation relations hold for our newly defined momentum operators.

[pˆ_i, xˆ_i]ψ =−iℏ∇ ^∂

∂xi

(xⁱ·ψ)−xⁱ·(−iℏ)∇ ^∂

∂xi

(ψ) (4.14)

=−iℏ (︃

d(xⁱψ) (︃ ∂

∂xⁱ )︃

+ω_ixⁱψ−xⁱdψ (︃ ∂

∂xⁱ )︃

−xⁱω_iψ )︃

(4.15)

=−iℏ (︃ ∂

∂xⁱ(xⁱψ)−xⁱ∂ψ

∂xⁱ )︃

(4.16)

=−iℏψ. (4.17)

Other commutation relations can be shown to be zero.

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5 EQUATION FOR THE YANG-MILLS FIELD

Letψandϕbe the local representatives of two wavefunctionsσ_ψandσ_ϕrespectively. We define the inner product of two wavefunctions in their Hilbert space as follows:

< σ_ψ|σ_ϕ >=

∫︂

M

ψ^∗·ϕ·[det(g)]^3/2d^dx, (5.1)

wheregis the Riemannian metric on the base manifoldM. Note that although the integral is written in a specific chartx, it can be shown that this integral over the entire manifold is independent of the choice of chart. If we denotedet(ϕ^∗(ω)(eˆⁱ)) =ωi, whereeˆi is thei^th coordinate basis vector inT_pM of the current chart, then we have:

< ψ|pˆ_jϕ >=

∫︂

M

ψ^∗(x)(−i)(∂_j +ω_j)ϕ(x)·[det(g)]^3/2d^dx

=

∫︂

M

i[∂_j(ψ·[det(g)]^3/2)]^∗ϕd^dx+

∫︂

M

(−i)ψ^∗ω_jϕ·[det(g)]^3/2d^dx

=

∫︂

M

[−i∂jψ]^∗ϕ·[det(g)]^3/2d^dx+

∫︂

M

i∂j[det(g)]^3/2ψ^∗ϕ

+

∫︂

M

(−i)ψ^∗ω_jϕ·[det(g)]^3/2d^dx

=

∫︂

M

[−i(∂_j+ω_j)ψ]^∗ϕ·[det(g)]^3/2d^dx−i

∫︂

M

ω^∗_jψ^∗ϕ·[det(g)]^3/2d^dx

−i

∫︂

M

ω_jψ^∗ϕ·[det(g)]^3/2d^dx+i

∫︂

M

∂_j[det(g)]^3/2ψ^∗ϕd^dx,

(5.2) and

< pˆ_jψ|ϕ >=

∫︂

M

[−i(∂_j +ω_j)ψ]^∗ϕ·[det(g)]^3/2d^dx. (5.3)

In order to make momentum operators self-adjoint, we require< ψ|pˆ_jϕ >=< pˆ_jψ|ϕ >, i.e.

∂_j[det(g)]^3/2 = [det(g)]^3/2(ω_j^∗+ω_j)

= 2[det(g)]^3/2Re(ω_j). ^(5.4)

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6 EXAMPLE APPLICATION ON SPHERICAL COORDINATES (FLAT SPACE)

Suppose we have a flat physical space and we want to work in spherical coordinates. The components of the base manifold metric in spherical coordinates are:

g_spherical =

⎡

⎢

⎣

1 0 0

0 r² 0

0 0 r²sin²(θ)

⎤

⎥

⎦

. (6.1)

Thus, we havedet(g_spherical) =r⁴sin²(θ).

We can apply the result from the previous section, i.e. eqn. 5.4 to find the appropriate components of the Yang-Mills field.

∂_r(r⁶sin³(θ)) = 2r⁶sin³(θ)Re(ω_r) (6.2)

=⇒ Re(ω_r) = 3

r, (6.3)

∂_θ(r⁶sin³(θ)) = 2r⁶sin³(θ)Re(ω_θ) (6.4)

=⇒ Re(ωθ) = 3

2arctan(θ), (6.5)

∂_ϕ(r⁶sin³(θ)) = 2r⁶sin³(θ)Re(ω_ϕ) (6.6)

=⇒ Re(ω_ϕ) = 0. (6.7)

Since we are free to choose the imaginary parts of the Yang-Mills field’s components, we can set them to zero, which yields:

ω_i = [︃3

r,3

2arctan(θ),0 ]︃

. (6.8)

Note that the derivation goes similarly for multiple patches, and the resulting equation 6.8 remains unchanged. As a result, the momentum operators in spherical coordinates take

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the form:

p

ˆ_rψ =−iℏ (︃

∂_rψ+3 rψ

)︃

, (6.9)

p

ˆ_θψ =−iℏ (︃

∂_θψ+ 3

2arctan(θ)ψ )︃

, (6.10)

p

ˆ_ϕψ =−iℏ∂ϕψ. (6.11)

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7 EXAMPLE APPLICATION ON A CURVED SPACE: QM ON A SPHERE

Suppose we have a 2-dimensional curved space with constant curvature everywhere, there exists a chart(x)in which the metric takes the form¹:

g_(x)=

⎡

⎣

a 0

0 asin²(x¹)

⎤

⎦, (7.1)

whereais a constant dictating the global curvature of the manifold.

Thus, we havedet(g) =a²sin²(x¹). The Yang-Mills field’s components can be obtained by solving:

= 2a³sin³(x¹)Re(ω_x¹) (7.2)

=⇒ Re(ω_r) = 1

acot(x¹) csc(x¹), (7.3)

∂_x²(a²sin²(x¹)) = 2a³sin³(x¹)Re(ω_x₂) (7.4)

=⇒ Re(ω_x₂) = 0. (7.5)

The Yang-Mills field with real components is then:

ω_i = [︃1

acot(x¹) csc(x¹),0 ]︃

. (7.6)

The momentum operators in this coordinate system are:

p

ˆ_x1ψ =−iℏ (︃

∂_x¹ψ+ 1

acot(x¹) csc(x¹)ψ )︃

, (7.7)

p

ˆ_x2ψ =−iℏ∂_x²ψ. (7.8)

1In fact, this coordinate system can be interpreted as the spherical coordinates with the radial dimension removed. In other words, with the 3-dimensional Euclidean embedding, the manifold can be seen as a sphere.

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8 DISCUSSIONS

8.1 Summary

Let us summarize the ideas introduced in this thesis. We have gone as far as giving up the notion that wavefunctions are complex-valued functions so that we can define covariant derivatives and take geometrical properties of real space seriously. In the new framework, we observe that various queries regarding properties of quantum systems have to be posed differently to be meaningful at all. For example, spatial translations of a wavefunction now refer to operations that occur in the total space of the associated bundle and the frame bundle. Similarly, momentum operators are an instance of objects that manifest themselves as operators acting on the sections of the associated bundle and only locally reduce to operators on smooth functions via pull-backs. Even though we many times turn to the Cartesian coordinates system as a sanity check, we can see from the constructions that in no place did it stand out as special. As a result, we have shown that the Cartesian coordinates play no bigger role than a psychological guiding light to make sure that results shall reduce to those established in standard quantum mechanics.

To quote Lychagin (Lychagin 1999): "the coordinates are more artificial than physical" as they are merely components of a chart. Hence, though coordinates are a necessary tool to do practical computations, serious physical questions must be posed with regards to abstract geometrical objects only.

8.2 Translation subgroups in generalized coordinates

For every self-adjoint operator, we can construct a family of operators by applying the exponential map. The resulting operators can be shown to be unitary (details in appendix A). In the case of Cartesian coordinate on a Euclidean space, the self-adjoint operators p_x, p_y, p_zgenerate 3 one-parameter groups of unitary operators. For instance,

T_x(s) = exp(pˆ_xs) (8.1)

are translation operators in thexˆdirection parametrized bys ∈R. The translation can be seen from the action:

T_x(s)ψ(x) =ψ(x−s). (8.2)

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In fact, in standard quantum mechanics, the derivative representation of the momentum operators (in position basis) was derived from the definition that momentum operators are generators of translations (details can be found in the popular textbook ’Modern Quantum Mechanics’ - Sakurai and Commins 1995). Since the momentum operator in 4.1 was derived differently, whether this relationship between momentum operators and translation groups holds is not obvious.

To check this, we exponentiate the momentum operators in 4.1 to find the unitary groups (appendix A). However, the procedure to spectrally decompose the momentum operators, which is necessary to construct the exponential map, is non-trivial. Additionally, we might even run into difficulties in the overlapping patches in case we have a general non- Euclidean manifold. The potential issue regarding spectral decomposition of operators in geometrical formalism is mentioned by Lychagin 1999. Author pointed out that the decomposition might only be possible if the manifold can be covered by one single chart, which is not always the case for curved spaces.

To summarize, although no proof was presented in this thesis, we expect that the momentum operators generate translations via the exponential map. The motivations behind this speculation include the intuition from the classical theory where the flow induced by the momentum observables generate spatial translation (classical mechanics on symplectic manifolds) and the conservation of momentum which would follow from it.

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REFERENCES

Kibble, T. W. (1979). Geometrization of quantum mechanics.Communications in Mathe- matical Physics65.2, 189–201.

Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. and Sternheimer, D. (1978). Deforma- tion theory and quantization. II. Physical applications. Annals of Physics 111.1, 111–

151.

Batalin, I. and Tyutin, I. (1990). Quantum geometry of symbols and operators. Nuclear Physics B345.2-3, 645–658.

Tanimura, S. (1993). Quantum mechanics on manifolds.arXiv preprint hep-th/9306144.

Schuller, F. P. (2013-2014). Lectures on the Geometric Anatomy of Theoretical Physics.

University Lectures - FAU. URL:https://www.video.uni-erlangen.de/course/

id/242.html.

Hamilton, M. J. (2017).Mathematical gauge theory. Springer.

Fecko, M. (2006).Differential geometry and Lie groups for physicists. Cambridge university press.

Lychagin, V. (1999). Quantum mechanics on manifolds. Acta Applicandae Mathematica 56.2-3, 231–251.

Sakurai, J. J. and Commins, E. D. (1995).Modern quantum mechanics, revised edition.

Richtmyer, R. D. (1978). Spectral Decomposition of Self-Adjoint and Unitary Operators.

Principles of Advanced Mathematical Physics. Springer, 158–189.

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A APPENDIX

A.1 Foundational concepts

Different structures of manifolds Manifold

Topological manifold Differential manifold Metric manifold

Tangent and cotangent spaces Pull-backs and push-forwards Differential forms

Bundle

Vector and covector fields Lie-algebras

Lie-groups and their Lie-algebras Hilbert spaces

A.2 The 1-to-1 correspondence between ψ

σ

and section σ on associated fibre bundle

Construct sectionσfromψ_σ

Given a mapψ_σ :P →F, we define the corresponding sectionσonP_F as follows:

σ(p) = [q, ψ_σ(q)], (A.1)

for anyq ∈ P andπ(q) = p ∈ M. Now it is obvious that we have to add some require- ment on the mapσ to guarantee that the equivalence relation on the associated bundle is maintained, namely we want: (q₁, σ(q₁)) ∼ (q₂, σ(q₂))for any q₁, q₂ ∈ π⁻¹(p). By definition of the quotient space, we achieve precisely that if and only if

ψ_σ(q₁◁ g) = g⁻¹▷ ψ_σ(q₁), (A.2)

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whereq2 =q1◁ g.

Construct the corresponding mapψ_σ from a sectionσ Given a sectionσonP_F, we constructψ_σ as follows:

ψ_σ(q) = z, (A.3)

such that (q, z) is a representative of σ(π(q)). Although this at first sight seems quite implicit as a way to describe ψ_σ, it can be constructed easily in practice. Starting by writing down an arbitrary representative(q^′, z^′)of the image of the sectionσ, we find the appropriate representative using group action:(q^′, z^′)∼(q^′◁g, g⁻¹▷z^′), whereq^′◁g=q. The value we are looking for is simplyψ_σ(q) =z =g⁻¹▷ z^′.

Checking if the two directions match

The remaining task is trivial, which is to check if we go through both constructions in either order, we would get back to where we start.

A.3 Exponential map and spectral decomposition

Given a Hermitian operator on a Hilbert spacepˆ ∈ O(H), there exists a mapdP :R → O(H)such that:

p ˆ =

∫︂ +∞

−∞

λdP(λ), (A.4)

where dP is often called a PVM (projection-valued measure). The exponential map of such a Hermitian operator is defined as follows:

U(1) = exp(pˆ) =

∫︂ +∞

−∞

e^λdP(λ). (A.5)

By multiplying the operator pˆ with different real factors, we can obtain a one-parameter family of unitary operators U(s). The constructive proof including how to find the PVM can be found in reference book (Richtmyer 1978).

Geometrical viewpoint of quantum mechanics: wavefunctions on curved spaces