Geometry in Physics the non-Abelian generalization of Berry's phase and black hole thermodynamic geometry

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University of Amsterdam

MSc Physics

Theoretical Physics

Master Thesis

Geometry in Physics

the non-Abelian generalization of Berry’s phase and black hole

thermodynamic geometry

by

Jonas J. van der Waa

6232078/10002077

July 2015

54 ECTS

1 September to 15 July

Supervisor:

Vladimir Gritsev Dr.

Examiner:

Vladimir Gritsev Dr.

Second Reader:

Ben Freivogel Dr.

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Abstract

In this Master thesis, we will discuss some of the basic aspects of geometry. We will introduce objects like the metric, connections and holonomies. We start by introducing this in the context of statistics and giving a number of examples. After that, we delve into the notion of quantum geometry: here we build the formalism of quantum distances and quantum holonomies. This formalism can be used to analyze the critical points con-nected to phase transitions of a system, without having prior knowledge of any of the intrinsics of the system, such as characteristic length scales.

We then generalize this formalism to mixed state quantum systems. In this setting we discuss the geometry of a manifold of mixed density matrices. We introduce the method, developed by Uhlmann, which gives the Bures metric and the Uhlmann connection. We derive some useful formula, especially for the 2x2 case, which describes two-level quantum systems.

We analyze these objects to see if they are the appropriate ones to choose from many. We find that the Bures metric is effective and has all the correct properties that we want it to have. The Uhlmann connection is also useful, but not as promising, since it appears to give trivial Chern numbers.

We calculate these geometric objects explicitly for a simple qubit and the quantum XY-model. In the XY-model, we see that both the pure state and the mixed geometry, given by the Bures metric and the Uhlmann connection, can be effectively used to analyze the phase diagram.

Finally, we touch upon the subject of thermodynamic geometry. We show that this is linked to the statistical geometry introduced in the beginning. This formalism, in-troduced by Weinhold and Ruppeiner, can be used to study phase diagrams in thermo-dynamic systems. Ruppeiner’s conjecture states that the curvature of these systems is intimately connected to the amount of interactions

We try to extend this formalism to black hole thermodynamics. In this context, we see if it is helpful to study the geometry of black hole thermodynamics. This is difficult, but promising, especially if we consider Ruppeiner’s conjecture.

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Popular Summary

Geometrie beschrijft de kromming van ruimte. We kunnen afstanden defineren binnen gekromde ruimte. Kromming kan gebruikt worden in de natuurkunde om verschillende aspecten te beschrijven.

We kunnnen phase diagrammen van quantum systemen beschrijven door de kromming van deze systemen te bestuderen. Voor het geval van geen temperatuur is dit formal-isme goed bestudeert. Als we temperatuur toe laten in de systemen, moeten we nieuwe manieren bedenken om deze geometrie te beschrijven. Het is niet helemaal zeker hoe we deze definieren.

Naast quantum systemen, kunnen we deze technieken ook gebruiken om thermody-namische systemen te bestuderen. Daarnaast bekijken we ook of we dit ook kunnen gebruiken om zwarte gaten te bestuderen.

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Acknowledgements

I like to thank my supervisor Dr. Vladimir Gritsev, who often helped and guided me, but also let me pursue my own interests within the field. I would also like to thank professor Hans Maassen, who was an enormous help in dissecting and understanding geometry in an intuitive way. Lastly, I want to thank all the people in Master room who were always there for support or help.

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Chapter 1 Introduction

Geometry is, put simply, the study of curvature and space. It is an interesting branch within mathematics, since it does not talk about how objects interact with each other, but rather how the space these objects inhabit acts.[1]

Geometry has always had a special role in physics. From Foucault’s pendulum in classical mechanics [2][3] to the Ahranov-Bohn effect in quantum mechanics [4] and the general theory of relativity, which accurately describes gravity.[5]

The geometric interpretation of physics is interesting, since it tells us things that would not have been obvious from any other context. A perfect example of this is Berry’s phase.[6] From the perspective that physicists had at that time, it was a complete surprise and no-one seemed to understand where it came from.

However, it was obvious from a geometric point of view and opened up a whole new field of quantum information geometry.[7]

Nowadays, people study the geometry mainly as a way of analyzing critical behavior, i.e. analyzing phase transitions, and as a way of classifying different systems.[8, 9] It is interesting to see where this geometric approach to quantum mechanics comes from, and therefore we will analyze the field of statistical geometry. An extension of this sta-tistical geometry, is that thermodynamic geometry and it is also interesting to study the basics of this field.

Since we discuss different metrics, which are not related to the metric used in general relative, we will not use any Greek, but only Italic indexes to avoid confusion.

This thesis is constructed as introductory to the field of quantum geometry in both the pure and mixed state, as well as introductory to the field of thermodynamic geometry.

We will discuss some of the important aspects of geometry. No preliminary knowledge is necessary apart from some elementary mathematics. On the other we will not discuss the mathematics in full detail. The same holds for the context of quantum geometry and mixed state geometry: we will introduce the geometric interpretation of quantum mechanics, but there are obvious a lot of caveats and subtleties. We will discuss some of these, but a complete overview is out of the scope of this thesis and we refer to the literature.[8, 9, 10]

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Part I

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Chapter 2 Mathematical description of

geometry

2.1 A short introduction to differential geometry

The field within mathematics that describes the curving of analytical spaces is known as differential geometry.

In this part, we will briefly give the most important aspects and objects, for a more in depth introduction, there is of course a surplus of literature, that goes a lot deeper than we will.[1, 8, 9, 10] These should be consulted if the following sections are not satisfactory. When discussing geometry, there are two important concepts: these are distances and geometric phases. Distances are measured by looking at the difference between two sets of values, e.g. points or vectors or even matrices.

When we talk about curved spaces, it is less obvious how distance is defined. On a sphere, we say that the distance between two points is a line between them, on the sphere. This distance is a circle and the difference between other definitions, like a curved line into some embedding space, is that this distance is minimal.[8]

This will be our definition: we define some distance function on our space and by minimizing this distance we construct the metric tensor gij. A measure of distance on

a space ∆l2 can then be written as

∆l2 = gij∆xi∆xj (2.1)

for some coordinate system {xi_}

With the introduction of this metric tensor, we can unleash the whole formalism of Riemannian geometry, where we define various objects, see Appendix A for a short overview.[5]

Geometric phases are a completely different story. Everything we discuss will be in the context of fiber bundles. A fiber bundle can be defined by three spaces (E, B, F ). ◦ We have the base space B, which we could view as our starting point in the con-struction of the geometry. For example, in physics, the base space is often something like the Minkowski spacetime or the space defined by the ground state ψ(~λ) of a quantum

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(a) a ”fiber bundle”

(b) locally it looks like the product of two spaces, but globally there is some strange transition

Figure 2.1: Two examples of fiber bundles mechanical system, with all the configurations of the parameters ~λ.

◦ At each point x ∈ B we associate a fiber Fx, which adds an extra level of

struc-ture to B. In addition, we could define a G-bundle, if there is a group strucstruc-ture G acting on x.

For example, for each point x ∈ B, we could an arbitrary phase factor eiα_{, α ∈ R.} This phase factor is related to a U (1) transformation and hence, we can move through the fiber at x, by applying a group multiplication

◦ The total space E then describes the global structure of both the base space and all its fibers.

◦ Lastly, there is also a projection π, which projects all different values in a fiber Fx

back to the point x

π : E → B

Fx → x (2.2)

The name comes from the idea that you have a space (the brush stick, so to say) with a different space attached to it at each point (the bristles), see Figure 2.1a

There are two more important notions:

◦ We say that E locally looks like a product space between a subsection of B and the associated fibers, but that globally can be more complex, see Figure 2.1b.

This means that if we zoom in locally on our total space to an open subsection U ⊂ B, we would find that it looks like U × FU, where FU is the collection of all fibers at U .

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(a) [0,1] by [0,1] sheet with orientation(b) a simple cylinder, constructed from the sheet

Figure 2.2: The bundle space of the example ◦ The other concept, is that we can take a section

f : B → E (2.3)

which is adamant to saying that we choose a single value in each fiber above B. This means that if we have a path defined on B, we can lift this path to a path in E, by assuming points in the associated fibers.

Let us look at some simple examples.

Trivial bundle: cylinder

The easiest example is a trivial bundle, which is a bundle that is globally a product space of the base space and its fibers. An example of such a trivial bundle, is the cylinder.

We take

S1 = [0, 1] (2.4)

F = [0, 1] (2.5)

E = [0, 1] × [0, 1] (2.6) We can construct this trivial strip by gluing two sides of a sheet together paying attention to the correct orientation (placing the arrows parallel), see Figure 2.2a. We have the projection

π : [0, 1] × [0, 1] → [0, 1] (2.7) Taking a section in this example, would be nothing more than choosing a specific point on each fiber.

This basically tells us that the group structure working on this space and bringing us from the base space to the total space is trivial, meaning G = {e}, with e the identity element.[8]

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(a) [0,1] by [0,1] sheet with anti-parallel orientation

(b) the M¨obius strip, constructed from the sheet

Figure 2.3: The bundle space of the example

Non-trivial bundle: the M¨

obius strip

The simplest example of a non-trivial fiber bundle is the M¨obius strip. We take the sample setup as in the previous example, but now we take the orientation of the two parts of the sheet to be anti-parallel, see Figure 2.3a. We can again identify the base space and the fibers as the intervals, and locally the total space is going to look like the product of these two, but globally this is no longer the case.

We have the same base space and fibers, but the total space is no longer a product space of these two.

You can walk along the M¨obius strip, and try to keep the band straight, but at a certain point you are going to make a cross-over from one side to the other.

The structure group governing the map from the base space to the total space is now Z2 = {e, g}, since there are now two options to be sent to from the, either the positive

orientation, or the negative one.[8]

Parallel transport and the connection

So it is not obvious how we should deal with the total space of the M¨obius strip. One thing we know, however, is that locally, it is a product space. This can be seen in Figure 2.3a. The way we can describe the total space, is by describing how to move from one local patch to another.

The way that this is done, is by the notion of parallel transport. We can clearly see this in the example of the tangent bundle on S2. In this example, the base space is the sphere, where at each point in B, we can define a fiber, which is the two-dimensional tangent space, see Figure 2.4a This means that locally, the tangent bundle on the two sphere is the product space of a flat piece of space times the space of all the different orientations that a tangent vector can have at that patch of the two-sphere. Taking a

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(a) local piece of S2

(b) holonomy on S2

Figure 2.4: The tangent bundle on S2

section f in this space, is simply choosing a specific tangent vector in each fiber.

In this context, the notion of parallel transport is quite natural: when we move from one local patch to another, defined by a path on the base space, all we have to do, is follow the curved motion of path and we see that we can go from one tangent path to another. this becomes even more clear if we take a section f : we see that we can parallel transport a tangent vector, by tilting it over the sphere, see Figure 2.4b.

The other interesting thing we see in this picture, is that if move over a sphere and return to original place, the orientation has changed with respect to its original orienta-tion. This mis-match is known as the holonomy.

The holonomy is a well known phenomenon in geometry. It is attributed to the fact that we have had to tilt our vectors in order to link all the different local patches. The object that quantifies the parallel transport, is known as the connection.

A different way to define this object, is to decompose the space E in vertical and horizontal subspaces. We actually decompose the tangent space of the fibers T E, but these carry the same information.[8] We say that a vector v in the tangent space is vertical at a point p, if it is tangent to the fiber passing through at p, meaning that v is parallel to Fx, see Figure 2.5. We can describe the vertical subspace as

Vp = {v ∈ TpE | Tpπ(v) = 0}

since, if v is vertical, i.e. parallel to Fx, it will project down to B to zero. We then define

horizontal vectors in the following way: we define the connection one-form A, which has values in the vertical subspace

Ap(u) := vertical(u) ∈ Vp(u) (2.8)

In this notation, it is obvious that we can write the horizontal subspace of TpE as

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Figure 2.5: moving a tangent vector around a sphere [9]

since, if u ∈ Vp it means that u is parallel to the fiber Fx, meaning that it is vertical, and

thus orthogonal with respect to the horizontal subspace.

Using A we can define objects, such as the covariant derivative D (or sometimes ∇) and the curvature two-form F , which is defined as

F = dA + [A, A] = dA + A ∧ A (2.10) if [ , ] is a Lie bracket. F satisfies the Bianchi identity [9]

DF = 0 (2.11)

For example, well known in general relativity or gauge theories, we write ∇νxµ = ∂νxµ+ Γµανx

α _(2.12)

where Γ is the Levi-Civita or Christoffel connection.

we can also define the connection and its associated curvature form in the local patches. For a local patch Ui and a section

fi : Ui → E (2.13)

we write

A : = f∗A (2.14)

F : = f∗F (2.15)

which are respectively the local connection one-form and local curvature two-form, mean-ing we can write

A = Aidxi (2.16)

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for some local coordinates xi_.

Of course, a question we could ask is: how are the different local forms related to each other? the answer to this question is given by the transition functions, which move from the local subsection Ui ⊂ B to the bundle space. In physics, we associate the fiber

with a group G. this means that the transition functions are defined by acting on the local subspaces with elements g ∈ G.

By a local gauge transformation, we mean a map

g : Ui → G (2.18)

and a local section f is known as a local gauge. Since two different sections f and f0 only differ in the sense that we choose different elements in the fiber and since we can move from one element in the fiber to another by acting on it with elements in G, i.e.

f0(x) = g(x)f (x), x ∈ Ui (2.19)

we find that the connection and curvature forms in different sections (gauges) are related by [9]

A0 = g−1· A · g + g−1· dg (2.20)

F0 = g−1· F · g (2.21)

which are well known formula in gauge theories.

Finally, we can express the holonomy in terms of these local forms. For a path C on B, we can write Φf[C] := P exp Z C A = P exp Z S F (2.22) where P denotes a path ordering and we use Stokes theorem to go from the line integral to a surface integral for C = ∂S.[9] Any closed path gives rise to a holonomy and these are obviously gauge dependent. The correspondence between the same curve, but a different gauge, is given by

Φf[γ] = g(x0)−1· Φf0[γ] · g(x₀) (2.23)

An interesting point we can make, is that the trace of the holonomy is invariant of the gauge. This object, given by

Tr (Φf[γ]) = Tr (Φf0[γ]) (2.24)

is known to physicists as a Wilson Loop, for a local gauge f .

In the next sections, we will discuss a few important examples of fiber bundles

2.2 Examples of fiber bundles

Electrodynamics

As stated in the previous section, we can accurately describe gauge theories in the context of fiber bundles. The base space B will denote the Minkowski space, describing the space-time coordinates.

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We assume a principle bundle with a U (1) group structure. we can define a one-form A on an open subsection U on B. A local gauge transformation

g(x) : U → G = U (1) (2.25) can be represented as

g(x) = eiλx (2.26)

for x ∈ U and

λ : U → R (2.27)

with Equation 2.20, this gives

A0 = A + idλ (2.28)

If we define the connection as

A = iAµdxµ (2.29)

we see we get the familiar gauge transformation

A0_µ = Aµ+ ∂µλ (2.30)

We also find the familiar curvature two-form F = DA = dA + 1

2A ∧ A = dA = i

2(∂µAν − ∂νAµ) dx

µ_{∧ dx}ν _(2.31)

where we used the Einstein summation notation and the second part drops out since U (1) is an Abelian group. This also means that Equation 2.21 simplifies to

F0 = F (2.32)

Lastly, we see that dF = d2A = 0, which gives us a way to write the Bianchi identity as dF = 0

→ ∂µFνλ+ ∂λFµν + ∂νFλµ = 0 (2.33)

which describes all of Maxwell’s equations.[9]

Yang-Mills theory

With the same construction, but by taking G to be the non-Abelian generalization of U (1), denoted SU (N ), we can describe Yang-Mills theory.

Yang-Mills theory is the generalization of electrodynamic and describes gauge theories that are more complicated than Maxwell’s theory, but have the same structure. Examples in physics are the weak interaction, which describes the interactions of the so-called Z

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and W bosons, and the strong interaction, which describes the interactions of the quarks and gluons.[11, 12]

We start by the matrices {La}, which form a basis for the Lie algebra su(N ) (which,

if exponentiated gives the group SU(N)). These matrices have the property

[La, Lb] = fabcLc (2.34)

where f_abc is known as the structure constant. With this construction, we can write Aµ = AaµLa with A = Aµdxµ (2.35)

Fµν = Fµνa La with F = Fµνdxµ∧ dxν (2.36)

and we see that the local connection and curvature forms are non-Abelian, since they are proportional to La, which do not commute.[9]

We then find the expression for the curvature form F = DA = dA + 1 2[A, A] → Fa µν = ∂µAaν− ∂νAaµ+ f a bcA b µA c ν (2.37)

and we can define the covariant derivative as

D = 1∂µ− Aµ (2.38)

which satisfy the Bianchi identity [9] DF = 0

→ DµFνλ+ DνFλµ+ DλFµν = 0 (2.39)

In the quantized theories of Maxwell and Yang-Mils, the gauge potentials Aa

µ (or just

Aµ for Maxwell) correspond to the different generators of SU (N ) and a massless, spin-1

boson, the so-called gauge bosons.[9]

2.3 _{Parallel transport on projective space CP}

1

This next example will be far more relevant to actual quantum mechanical systems, so we will discuss it in greater detail. Let us start by discussing some properties of quantum mechanics. An important thing to understand is how we identify physical states. For a given quantum mechanical state ψ, the state λψ, with λ ∈ C and λ 6= 0 represent the same physical state.

In order to correctly describe the space of physical states, we have to introduce the concept of projective spaces. Essentially, we have to describe the space of physical states of a quantum mechanical state ψ as the equivalence class {λψ | λ ∈ C, λ 6= 0}. These equivalence classes are also called rays.[10]

The Hilbert spaces we describe are complex, and in general, for a finite, n-dimensional Hilbert space, we write

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So let us discuss a specific physical system. A two-dimensional Hilbert space, has a space of quantum mechanical states, or vectors given by C2_{, and associate space of physical}

states given by CP1, which is also known as the complex projective line.[10]

The fiber bundle associated to this space is the following: we have a base space B, described by the complex projective line CP1 and a total space E, which is the space of quantum mechanical states C2_{. The fiber F is given by C.}

In general, we only discuss normalized states, meaning that since the norm of a state ψ is constant that λ is a complex number of unit length. These equivalence classes now contain only the different phases associated to the respective state. The group G con-nected to this fiber, is the group of commutative, unitary transformations U(1).

An interesting observation, is that there is another way to describe the space of physical states of a two-dimensional Hilbert space. An example of a system with a two-dimensional Hilbert space is the Stern-Gerlach experiment. In this example, we have a fully lo-cated spin, which can point ”up” in some direction in space. This is of course a simplified model. In reality, The Stern-Gerlach experiment describes a more complicated object.

The idea, however, is that we can represent each physical state, with a direction in real space. The space of physical states is equivalent to the two-dimensional sphere S2.

On this sphere, any point is associated to a particular physical state, meaning a particular direction the state can be ”up” in. Antipodal points are ”up” in the opposite direction. Thus we two states are orthogonal to each other if the points that represent them on the sphere are antipodal.[10]

So now we can discuss the fiber bundle. To recap

E = C2 (2.41)

F = C (2.42)

B = CP1 ' S2 (2.43)

G = U (1) (2.44)

Let us look a little closer at the isomorphism between the complex projective line CP1

and the two-sphere S2. The fact that these two were connected, we motivated through

the example of Stern-Gerlach experiment, but there is a more mathematical approach. The isomorphism, is established by the projection

P (x, y, z) : C2 → C2

This is given by a projection on to the complex projective line. Such a projection, has to satisfy the conditions: [1]

1)P = P† (2.45)

2)P2 = 1 (2.46)

P is a projection, so it has to be represented by a matrix in C2_{, which we can write}

P = a b c d = V 1 0 0 0 V†, _{{a, b, c, d} ∈ R} (2.47)

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for some diagonalizing matrix V , since a projection either maps to a parallel (eigenvalue 1) or an orthogonal plane (eigenvalue 0). This means that P has determinant 0 and trace 1. These conditions imply

a, d ∈ R (2.48)

c∗ = b (2.49)

d + a = 1 (2.50)

Using this, we can write P (x, y, z) = 1 2 1 + z x − iy x + iy 1 − z = 1 2(1 + xσ1+ yσ2 + zσ3) (2.51) with {x, y, z} ∈ R and find

Det(P ) = 1 4(1 − x

2_{− y}2_{− z}2_{) = 0} _(2.52)

→ 1 = x2 + y2+ z2 (2.53) which describes the two-sphere.

This is the Bloch sphere parametrization.[10] In this context, we now have asso-ciated every point in the complex projective line with a point on the two-sphere.

S2 can be described by

S2 =(x, y, z) ∈ R3 | x2+ y2+ z2 = 1

(2.54) Since the projection was unto the complex projective line CP1_{, a vector in CP}1 _{will be}

invariant under the projection. Elements in E can be described by two complex numbers (η, ζ) and there is a continuous, surjective projection

π : E → B

(η, ζ) → (x, y, z) (2.55) with the property that the vectors in CP1 _{are invariant under this projection, i.e.}

1 2 1 + z x − iy x + iy 1 − z η ζ =η ζ (2.56) since these vectors lie on the complex projective line that Equation 2.51 projects on.

One of the conditions of it being a fiber bundle, is that E is locally a product space of B and the associated fibers. This means that any x ∈ B has a neighborhood U , such that

π−1(U ) ' U × FU (2.57)

where FU denotes the fibers at U

We will see that we need two open covers U±, which decompose E into two local

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is trivial.[8] Since in our case G = U (1), we already see that we cannot take a global section. However, let us look at it more carefully.

In order to satisfy Equation 2.56, we find the explicit expression for the vectors in E η ζ = λ 1 + z x + iy or λx − iy 1 − z (2.58) with λ ∈ C. We already see that there something interesting happening at z = ±1, which denote the poles of S2.

Let us choose one of the expressions of the vector and decompose it in polar coordi-nates. We get

η = |η||λ|eiφ1_, _{ζ = |ζ||λ|e}iφ2_, _φ

1, φ2 ∈ R (2.59)

|η|2_{+ |ζ|}2 _{= |λ|}2 _{(1 + z)}2_{+ |x + iy|}2_{= 2|λ|}2_{(1 + z)}

→ |λ|2 ₌ |η|

2 _{+ |ζ|}2

2(1 + z) (2.60)

which we see is singular at z = −1. This is not entirely watertight, since at z = 1, η and ζ might cancel this singularity.

Topologically, this is obvious: we cannot continuously deform the surface of the sphere to a single point, which we can do for spaces that enjoy a global section. If we puncture a single hole in the surface, we can continuously the space to a single point. This tells us that we need two different local sections to cover the S2 and that the space has a non-trivial geometry.

We can cover the space of S2 into two subspaces, by removing one of the poles at z = ±1.

We get

U± =

(x, y, z) | z 6= 1 (x, y, z) | z 6= −1 Let us look at the isomorphism π−1(U±) = U±× F . We get

π−1(U−) = λ 1 + z x + iy | λ ∈ C π−1(U+) = λx − iy 1 − z | λ ∈ C which satisfy Equation 2.56

The fibers on these subspaces are given by the complex number λ.

We want to describe what happens when we parallel transport a vector in the base space around a closed interval. Due to the Bloch parameterization, we can associate this by moving along a path on S2, see Figure 2.6a. We will move a point (x(t), y(t), z(t))

∈ C2 _{around S}

2, and at the same time describe what happens to the complex lines in the

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(a) the path C on S2

C

(b) The fibers defined at each point on S2: lines in C2

Figure 2.6: The bundle space of the example

At all times, we impose the parallel transport condition, given by

hψ(t), ∂tψ(t)i = 0 (2.61)

where t parametrizes the path.

This parallel transport condition tells us two things: the real part of Equation 2.61 tells us that the length of ψ(t) must stay the same. This means that λ can only be a phase.

The complex part basically tells us that the added phase cannot tilt ψ(t) too much: we must keep ψ(t) ”as parallel as possible”.

We obviously cannot keep ψ(t) entirely parallel to itself, since we are going to move from one complex line to another, but the complex part of Equation 2.61 will prevent the tilting of the orientation to be too extreme.[6]

We are only going to move over one part of the hemisphere, so we avoid one of the poles. The subspace on which the path is defined, is covered by U−. Let us start at the

top of the sphere, denoted by {x, y, z} = {0, 0, 1}. With Equation 2.51, we get P (0, 0, 1) = 1 0

0 0

(2.62) for the coordinate on S2. At this point of S2, the vector in the fiber space is chosen to be

ψ(0) =1 0

(2.63) where we took λ = 1 and we normalized the vector. When we follow path A, which we parametrize with t → π/2 (a quarter of a circle), we come to the coordinates {x, y, z} = {1, 0, 0}, which gives P (0, 1, 0) = 1 2 1 1 1 1 (2.64)

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and describes the normalized vector √1 2

1 1

on E. We see that we need to make the transition 1 0 → √1 2 1 1 (2.65) in E and the transition

  0 0 1  →   1 0 0   (2.66)

on S2. To parametrize this, we take x = sin (t), z = cos (t) and y = 0, and we see

C2 3 ψA(t) = cos (t/2) sin (t/2) =            1 0 for t = 0 1 √ 2 1 1 for t = π/2 (2.67)

This vector in E has to satisfy the projection for the Bloch parameterization, given by Equation 2.56. We see

P (sin (t), 0, cos (t))cos (t/2) sin (t/2) = 1 2 1 + cos (t) sin (t) sin (t) 1 − cos (t) cos (t/2) sin (t/2) = 1 2

cos (t/2) + cos (t) cos (t/2) + sin (t) sin (t/2) sin (t/2) + sin (t) cos (t/2) − cos (t) sin (t/2)

=cos (t/2) cos

2_{(t/2) + cos (t/2) sin}2_(t/2)

sin (t/2) cos2_{(t/2) + sin}2_{(t/2) sin (t/2)}

=cos (t/2) sin (t/2)

(2.68) where we used the identities

1

2sin (t) = cos (t/2) sin (t/2) 1 2(1 + cos (t)) = cos 2_(t/2) 1 2(1 − cos (t)) = sin 2_(t/2)

We see that ψA(t) has length one, and automatically satisfies the parallel transport

con-dition.

Now, let us move the vector over the path B. We get {x, y, z} = {0, 1, 0}, and we have P (0, 1, 0) = 1 2 1 −i i 1 ∈ S2 (2.69)

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on S2 and √λ₂

1 i

on E. It is not obvious how we can write the parametrization. Naively, we could write C2 3 ψB(t) = 1 √ 2 1 eit =            1 √ 2 1 1 for t = 0 1 √ 2 1 i for t = π/2 (2.70)

which has length one, and satisfies the boundary conditions, but it does not satisfy the parallel transport condition

hψB(t), ∂tψB(t)i =

i

2 6= 0 (2.71)

We can fix this by adding a phase to parametrization. This amounts to changing the orientation of the ”line” in C2. We take ψB(t) = √1₂

1 eit

eiθ(t) and impose the parallel transport condition hψB(t), ∂tψB(t)i = 1 2(2∂tθ + 1)i = 0 → θ = − t 2 This gives us ψB(t) = √1₂ e−it/2 eit/2 = √1 2 1 eit

e−it/2. This added phase does not change the length of the vector, but only the orientation. This means that we had to adjust the orientation, while moving over the base point, which makes sense, since we moved into the direction of y, which is paramount to rotating in the complex plane.

At the final point of the path of B, t equals π/2. This means that the added phase, which we will have to take along on the path C will be e−iπ/4. So the correct parametriza-tion is C2 3 ψB(t) = 1 √ 2 1 eit e−it/2 =            1 √ 2 1 1 for t = 0 1 √ 2 1 i e−iπ/4 for t = π/2

Now, lastly, we want to get back to our original place, following path C. So, now again, we have {x, y, z} = {0, 0, 1} and

P (0, 0, 1) = 1 0 0 0

(2.72) on S2 and we can parametrize the vector as

C2 3 ψC(t) = cos (t/2) i sin (t/2) e−iπ/4=            e−iπ/41 0 for t = 0 e−iπ/4 1√ 2 1 i for t = π/2

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The length of this vector is again one and it satisfies the parallel transport condition, but we see something remarkable happening: we get to the same initial position, but the orientation has changed with a phase of π/4.

In summary, we have described the parallel transport of a unit vector ψ in the fiber, while moving the base point over the surface of the unit sphere S2. We did this by

split-ting the path into three pieces and describing what the vector is in these three pieces. We found that when we trace out an octant of the sphere, we end up at the same position on the sphere, but with a different orientation. This is the emergence of the holonomy, γ.

We can write the geometric phase γ(t), in terms of the summation of the path, as an integral

γ(t) = I

C

hψ(t), ∂tψ(t)i ∈ iR (2.73)

Furthermore, it is notable that the phase is also comparable to the surface of the path, which was an octant of the sphere, see figure Figure 2.6. We can expand on this remark: We imposed the parallel transport condition, given by Equation 2.61. However, we already noticed that the total space had some non-trivial structure, which made us split the space S2 into the open subspaces U±. The parallel transport condition is satisfied

locally, when we took our steps in the path. Globally, we end up with a geometric phase. We define the one form

A =X µ Aµdxµ= X µ ihψ, ∂µψidxµ = X µ i Imhψ, ∂µψidxµ (2.74)

for normalized vectors ψ. The surface of the path is given as S, i.e. ∂S = C, and we can use Stokes theorem to get

γ(t) = I C X µ Aµdxµ= I S dX µ Aµdxµ = I S d (A0dx0+ A1dx1) = I S (∂1A0dx0∧ dx1+ ∂0A1dx1∧ dx0) = I S (∂1A0− ∂0A1) dx0∧ dx1 (2.75)

where x1and x0are coordinates on S2that parametrize the path and we used the identities

d2xi = 0 = dxi ∧ dxi, for any i. The left hand side of Equation 2.75 has to be equal to

the surface of the path, which was π/4.

We can can take the vector write the vector 1 + z

x + iy

(2.76) which has an orientation of λ = 1. We can write

x = sin (θ) cos (φ) (2.77) y = sin (θ) sin (φ) (2.78)

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which gives ˜ ψ =1 + cos (θ) eiφ_{sin (θ)} (2.80) which has normalization

q ˜

ψ†_{ψ =}˜ p

2(1 + cos (θ)) =p2 cos2_{(2θ) =}√_{2 cos (2θ)} _(2.81)

and we find the normalized vector ψ = √ 1 2 cos (2θ) 1 + cos (θ) eiφ_{sin (θ)} = cos (θ/2) eiφ_{sin (θ/2)} =ψ0 ψ1 (2.82) where we used the fact that

sin (θ)

cos (θ/2) = 2 sin (θ/2) (2.83) Using this expression for the vector, we find for the integrand of Berry’s phase

dA = dX

µ

i Im(hψ, ∂µψi) = i Im (dψ0∗∧ dψ0+ dψ1∗∧ dψ1)

= i Im d(cos (θ/2)) ∧ d(cos (θ/2)) + d(e−iφsin (θ/2)) ∧ d(eiφsin (θ/2)) = i sin (θ/2)cos (θ/2)

2 (−dφ ∧ dθ + dθ ∧ dφ) = i

2sin (θ)dθ ∧ dφ (2.84)

where all the other terms drop due to the anti-symmetry of the wedge product. We see that this is exactly i/2 times the Jacobian, so if we integrate this over the surface (in our example, θ ∈ {0, π/2} and φ ∈ {0, π/2}) we find i/2 times the surface:

γ(S) = Z S dA = i 2 Z π/2 0 dθ Z π/2 0 dφ sin (θ/2)dθ ∧ dφ = i 2 π 2 = i π 4 (2.85) which is exactly the holonomy we found in our extensive analysis.

An interesting observation, is the fact that this is i/2 times the surface. From a Riemann connection, we would not have found this extra factor. It seems that the tangent bundle, associated with a a real fiber bundle over S2 _{lacks a i/2 factor compared}

the bundle over CP1_.

This difference is well known in quantum mechanics, where we have objects with spin-1₂ that require a 4π rotation to be back to its initial position.

We will see that this holonomy in quantum mechanics also exists, and is called Berry’s phase, after its discoverer.

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Chapter 3 Geometric interpretation of

statistical, classical information

3.1 Statistical mathematics

Let us look at some of the interesting properties of statistical mathematics. Given the probability P of N possible outcomes, then we can define the vector ~p with N components and pi ≥ 0 and

P

ipi = 1.[10]

We can define the Shannon entropy as SS(P ) = −k

N

X

i

pilog pi (3.1)

where k ∈ R≥0, which we take to be 1. The Shannon entropy effectively measures the

uncertainty of knowing what the outcome is going to be. If the Shannon entropy is 0, this means that the outcome is certain, since then there is only one pi, which is 1 and

gives log pi = 0.[10]

When comparing two different probability distributions, we use the Relative en-tropy, or the Kullback-Leibner enen-tropy, or the information divergence. For two proba-bility distributions, P and Q, the Relative entropy is defined as

SR(P ||Q) = N X i=1 pilog pi qi (3.2) which effectively measures how different P and Q are.

Another important concept, is that of Monotonicity. For any Markovian chain T , monotonicity implies

SR(T P ||T Q) ≤ SR(P ||Q) (3.3)

this essentially says that the distinguishability between P and Q decreases with time, i.e. a Markovian chain. A function F , that has the property

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is said to be monotonic.

For continuous distributions, we have equivalent functions. Similar to the Shannon entropy, we have the Boltzmann entropy, given by

SB = −

Z ∞

−∞

dxp(x) log p(x) (3.5) Now that we have set the stage for statistical models, it is time to look what an geometric interpretation can add to our understanding of statistical theories.

3.2 Fisher-Information matrix

We represent a probability distribution p(x; θθθ), with x the stochastic variable living on a space X and θθθ = (θ1_{, θ}1_{, , ...) the internal parameters that characterize the microscopic}

model.[13] It is a continuum model, so the probability distribution has the property that Z

X

p(x; θθθ)dx = 1 (3.6)

For example, if we assume a normal distribution, we get p(x; µ, σ) = √1 2πσ exp −(x − µ) 2 2σ (3.7) with mean µ and standard deviation σ. In this context, the function p(x; θ) is known as the likelihood, where people try to make guesses to what kind of probability distribution is attributed to a set of given parameters θ.[10]

We introduce the notation O = Z X dx p(x; θθθ)O(x; θθθ) = Z dx0p(x0; θθθ)O(x; θθθ) (3.8) with O(x0; θθθ) = O(x; θθθ) and p(x0; θθθ) = J−1p(x; θθθ), with some Jacobian. To counter the possible problems one might get from the log, we can define the Boltzmann entropy with some gauge function m(x; θθθ)

SB(p) = −

Z ∞

−∞

dx p(x; θθθ) log p(x; θθθ)

m(x; θθθ) (3.9) where the function m(x; θθθ) depends on how you take the continuous limit from the Shan-non entropy.[10] However, we will not use this in later discussions.

The Log-likelihood is defined as

l(x; θθθ) = log p(x; θθθ) (3.10) Or we can define the spectrum

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The score vectors are defined by

la(x; θθθ) = ∂al(x; θθθ) (3.12)

and we can represent the von Neumann entropy as S(θθθ) = − Z X p(x; θθθ) log p(x; θθθ) = Z X p(x; θθθ)γ(x; θθθ) =γ (3.13) One thing we immediately see, is that the expectation of the score vectors vanish:

la = Z dx p(x)la(x) = Z dx p(x) ∂ ∂xalog p(x) = Z dx ∂ ∂xap(x) (3.14) = ∂ ∂xa Z dx p = ∂ ∂xa(1) = 0 (3.15)

This means we can use the score vectors to represent the tangent vectors, instead of using the partial derivatives ∂a, i.e. for any tangent vector A, we can write A(x) = Aala(x).[10]

What this means, is that we can describe our sets of probability distributions as manifolds, where the tangent vectors are defined by the score functions, which have zero expectation value. So, we have defined our manifold of probability distributions. Can we define a metric on the space as well?

It turns out that the answer is yes. The metric on this space is known as the Fisher-Rao metric, or Fisher Information matrix.[10][13] The Fisher-Fisher-Rao metric is defined as

ds2 =X i,j gijdpidpj with gij = δij pi (3.16) we can write the Fisher-Rao metric in terms of the spectrum, or equivalently in terms of the log-likelihood gij(θθθ) = Z X p(x; θθθ)∂γ(x; θθθ) ∂xi ∂γ(x; θθθ) ∂xj dx =∂iγ∂jγ (3.17) This is equivalent to saying that the Fisher-Rao metric is the Hessian of the Shannon entropy, i.e.

gij ∼ ∂i∂jS (3.18)

which is an interesting observation, since the metric apparently measures the difference in statistical information, which is given by the Shannon entropy.

Traditionally, the Fisher-Rao metric can be found by computing the Kullback-Leibler divergence, given by [13]

DKL = Z X p(x; θθθ) log p(x; θθθ) − Z X p(x; θθθ) log p(x; θθθ + dθθθ) (3.19)

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By expanding this to second order, we find DKL = Z X p(x; θθθ) log p(x; θθθ)dx − Z X p(x; θθθ) log p(x; θθθ) + 1 p(x; θθθ) ∂p(x; θθθ) ∂θi − 1 2 p(x; θθθ) ∂p(x; θθθ) ∂θi ∂p(x; θθθ) ∂θj + O(dθ 3₎ dx = − ∂i Z X p(x; θθθ)dx − 1 2 Z X 1 p(x; θθθ) ∂p(x; θθθ) ∂θi 1 p(x; θθθ) ∂p(x; θθθ) ∂θj dθ i dθj + O(dθ3) = 1 2gij(θθθ)dθ i_dθj_{+ O(dθ}3₎ _(3.20)

where we used the fact that 1 p(x; θθθ) ∂p(x; θθθ) ∂θi = − ∂γ(x; θθθ) ∂θi (3.21)

Pulling all the strings together, we find the practical use of the Fisher-Rao metric and its interpretation: for a given set of parameters θθθ, we obtain a physical state with an associated entropy. By changing the parameters slightly by dθθθ, we find a new physical state, with its associated entropy.

The Fisher-Rao metric measures the distance between these states in the space where we store all the information from the entropy of the given states. This classical spacetime has coordinates θθθ.[13]

The Fisher-Rao metric is an extremely important object in statistical analysis: in the context of doing measurements on real systems (i.e., real probability distributions) the Fisher-Rao metric essentially tells you how many measurements are necessary to deter-mine if a probability is useful.

For discrete systems, we define similar equations. We know that for a discrete prob-ability distribution pn(θθθ), we have

X

n

pn(θθθ) = 1 (3.22)

and we can define the expectation value of a function On(θθθ) as

O =X

n

pn(θθθ)On(θθθ) (3.23)

We can also define the discrete equivalent of the spectrum

γn(θθθ) = − log pn(θθθ) (3.24)

and use it to represent the Shannon entropy S(θθθ) = −X n pn(θθθ) log pn(θθθ) = X n pn(θθθ)γn(θθθ) (3.25)

and the Fisher-Rao metric gij(θθθ) = X n pn(θθθ) ∂γn(θθθ) ∂θi ∂γn(θθθ) ∂θj =∂iγ∂jγ (3.26)

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One interesting point about the spectrum, is the fact that its expectation value is zero, just as in continuous case. This means that

∂iγn(θθθ) = − X n pn(θθθ)∂i(log pn(θθθ)) = ∂i X n pn(θθθ) = 0 (3.27)

and we can also write the Fisher-Rao metric as ∂j∂iγ = ∂i∂jγ = − X n pn(θθθ)∂i∂j(log pn(θθθ)) = −X n pn(θθθ) 1 pn(θθθ) ∂2_p n(θθθ) ∂θi_∂θj − 1 p2 n(θθθ) ∂pn(θθθ) ∂θi ∂pn(θθθ) ∂θj =X n 1 pn(θθθ) ∂pn(θθθ) ∂θi ∂pn(θθθ) ∂θj =(∂iγ)(∂jγ) = gij(θθθ) (3.28)

where we should make clear that the identity is not satisfied, outside of the average, i.e. (∂iγ)(∂jγ) 6= ∂i∂jγ (3.29)

If we make the transformation

Xi =ppi _{→ dX}i ₌ dp i

2ppi (3.30)

we see that the Fisher-Rao metric simplifies to ds2 =

N

X

i=1

dXidXi (3.31)

with the condition

N X i=1 pi = 1 → N X i=1 XiXi = 1 (3.32) which tells us that the Fisher-Rao metric actually describes the geometry on the unit N-sphere, and therefore the geodesics are great circles.[10]

The geodesic distance, is also known as the Bhattacharyya distance, given by cos DBhatt = N X i p pi_qi _{= B(P, Q)} _(3.33)

between two probability distributions denoted P and Q, which is just the distance between the vectors pi _{and q}i_{. The right-hand side of Equation 3.33 is known as Bhattacharyya}

coefficient, and its square is known as the classical Fidelity, given by F (P, Q) = N X i=1 p pi_qi !2 (3.34)

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Figure 3.1: Continuous deformation of a torus into a cup. Topologically, they are the same

which looks very similar to what we often calculate in quantum mechanics, and we will see the quantum mechanical generalizations for both the pure state and the mixed state case.[10, 14]

Alternatively, we can also define the distance between two points, by the distance between the two points in the flat embedding space. This is known as the Hellinger distance and given by

DH N X i=1 p pi ₋p_qi2 !1/2 (3.35) The Fisher-Rao metric is an important tool for distinguishability measures, which tells you how to distinguish probability distributions from each other. This is necessary since in practice we have to make decisions based on an imperfect set of data.

The quantum case is even more complicated, since doing measurements affects the quantum states them self. This means that a quantum state can give rise to many different classic probability distributions, depending on what the measurement procedure is. The Fisher-Rao is monotonic and this property is what ensures that the metric measures how easily two probability distributions can distinguished.[10]

The broad question is, what the quantum version of this metric might be. We will want it to be monotonic and roughly speaking, we are going to compare it to the Fisher-Rao metric. Before we do that, however, we will discuss a few properties of geometry and topology in general to see what might be the interesting things to look for.

3.3 Topological invariants

Since we are discussing geometry, we should also talk about topology. In topology, we have the notion that spaces can be topologically equivalent, meaning that there is a con-tinuous deformation from one space to the other, see Figure 3.1.

Now the question is, how to distinguish between these different spaces. The full an-swer to this question is not clear, but we have fairly good substitute. We say two spaces are equivalent, when they share the same topological invariants.[8] A topological in-variant is essentially a quantity that is inin-variant under homeomorphism, i.e. continuous deformations.

A simple example of a topological invariant would be the number of holes. Speaking very roughly, the cup and the torus have an equal number of holes and therefore be-long to the same equivalence class of homeomorphisms. In general, however, these invariants are a bit more abstract. For example, the closed line element [−a, a] is not

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homeomorphic to the open interval (−a, a), since the first one is compact and the second is not.

Finding the topological invariants is necessary to describe the different equivalence classes of homeomorphism, meaning we can classify the different space by their topolog-ical invariants: number of holes (the genus), compact or non-compact etc. In general, we can describe the topological invariants as numbers. Some of these are

1) Euler characteristic 2) Chern number 3) Pontryagin number 4) Stiefel-Whitney numbers .. .

We will briefly discuss the first two topological invariants, which are also important in many physical systems.[8]

3.4 The Euler characteristic

The Euler characteristic is the easiest to imagine, since there is a connection with geometric objects in R3_{. Let us start with a simple example. We can define the Euler}

characteristic of polyhedron, as a linear combination of edges, faces and vertexes.

In general, an object X that is homeomorphic to K ∈ R3 has an Euler characteristic, defined by

χ(K) = ]vertexes + ]faces − ]edges (3.36) what polyhedron K is, does not matter as long as X is homeomorphic to K.[8].

For example, the two sphere is both homeomorphic to the tetrahedron and the cube. We find for both that

χ(tetrahedron) = 4 + 4 − 6 = 2 (3.37) χ(cube) = 8 + 6 − 12 = 2 (3.38) This is a famous theorem by Poincar´e and Alexander.[8]

We can also look at more abstract cases. For example the Klein bottle cannot be realized in three dimensions, but we can describe it as a rectangle where the edges are identified in a certain matter, see Figure 3.2 we cannot even visualize the Klein bottle, but from the rectangle in Figure 3.2b, we can calculate

χ(Klein bottle) = 4 + 0 − 4 = 0 (3.39) since the Klein bottle has no well-defined orientation, the number of faces is zero. The reason why the Euler characteristic is such a successful quantity, is due to the fa-mous Gauss-Bonnet theorem. This theorem unveils an important connection between the Euler characteristic and the curvature scalar defined on the associated Riemannian manifold.

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(a) The Klein bottle pro-jected to R3

Klein bottle

(b) The Klein bot-tle as a rectangle where the sides are twisted

Figure 3.2: Representations of the Klein bottle

Figure 3.3: A polyhedron that is homeomorphic to the torus

In general, for a 2n orientable, compact manifold M without boundary, this theorem can be written as

χ(M ) = Z

M

(2π)−nPf(F ) (3.40) with the curvature two-form F and where the Pfaffian is defined as

Pf(F )2 = Det(F ) (3.41) An example is the torus T2_{. We can define a global trivialization, since there is no}

twisting like in the Klein bottle. This means that we can define a flat connection and thus the curvature tensor and the Pfaffian are both zero. This tells us that the right hand side of Equation 3.40 is zero.

On the other hand, we see that the torus is homeomorphic to the polyhedron in Figure 3.3, and we can calculate

χ(T2) = 16 + 16 − 32 = 0 (3.42) The Gauss-Bonnet theorem thus connects the Euler characteristic with an integral over the curvature two-form over the whole manifold on which the form is defined.

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The Euler numbers have strong connections with the Pontryagin numbers and even the Chern numbers in certain cases (number of dimensions of the manifold etc). In all these different cases, there are different ways of explicitly writing the Gauss-Bonnet theorem. An important one is for a 4-dimensional manifold M . In that case, we can write

χ(M ) = 1 32π Z M |Rµνρσ|2− 4|Rµν|2+ R2 (3.43) where Rµνρσ denotes the Riemann tensor, Rµν the Ricci tensor and R the Ricci scalar.

The operation | . | means to sum over all possible ways of contraction in order to obtain a scalar, i.e. [8]

|Rµν|2 = RµνRµν = RµνRρσgµρgνσ (3.44)

3.5 Chern Classes

For a fiber bundle (P, M, π, G) we could define the curvature form F . we could also define a local curvature form F = f∗F on a local section U . If we have two different bundles P1, P2, over M with the same group G, then we call the bundles equivalent iff there is a

map

φ : P1 → P2 (3.45)

which preserves the bundle structure, meaning it takes a fiber in P1 defined at x to a

fiber in P2

φ(π−1₁ (x)) = π₂−1(x) (3.46) We can define Characteristic classes for the local curvatures in respectively P1 and P2,

given by [f (F1)] and [f (F2)]. If the two bundles are equivalent, it turns out that

[f (F1)] = [f (F2)] (3.47)

for any invariant polynomial f .[9]

This essentially tells us, that if we find an appropriate function for f , we can use the characteristic classes to different bundles.

Chern number

We can define the invariant polynomial ck, through

det 1 + i 2πF =: n X k=0 ck(F ) (3.48)

ck(F ) is called the kth Chern form and Ck(P ) = [ck(F )] is called the kth Chern class.

We can write the determinant ck(F ) = (−1)k (2πi)k j1...jk i1...ikF i1 j1 ∧ ... ∧ F ik jk (3.49)

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using j1...jk

i1...ik, the anti-symmetric Levi-Civita tensor. We can then write [9]

C0(P ) = 1 (3.50) C1(P ) = i 2πTr(F ) (3.51) C2(P ) = 1 2 i 2π 2 {Tr(F ) ∧ Tr(F ) − Tr(F ∧ F )} (3.52) .. . (3.53) Cn(P ) = i 2π n det(F ) (3.54)

For a U (1) bundle we find

C1 =

i

2πTr(F ) = i

2πF (3.55)

which characterizes the bundle. For an SU (2), we can write the local curvature as F = LaFa=

σa

2iFa (3.56)

where σa are the Pauli matrices, which satisfy

σaσb = δab+ iabcσc (3.57)

and have zero trace (and the delta’s have trace two). We find

C1(P ) = 0 (3.58) C2(P ) = − 1 8π2 0 + 1 2 Fa∧ Fa = − 1 16πF a_{∧ F} a (3.59)

so we see that we need the second Chern class to characterize SU (2).

For a 2n orientable dimensional manifold M , we can also define the Chern number of the bundle P as the value of the integral

Z M Cn(P ) = Z M cn(F ) (3.60)

There are n of these numbers. It turns out that these Chern numbers are actually integers. For such a manifold, the Chern number is defined as

n = 1 2π

I

S

F (3.61)

where we integrate the curvature two-form over the surface S. To see that this number is an integer, let us discuss what happens when we apply Stokes theorem to the Holonomy. For a path C that splits the surface S into two pieces S1 and S2, the holonomy must

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be invariant under the choice of which surface we take. This means that for the two different choices, we have

Φ(C) = I C A = I S1 F + 2πm1 = I S2 F + 2πm2 (3.62)

with m1, m2 ∈ R, since the exponent of these expressions have to be equal. This gives

n = 1 2π I S F = I S1 F − I S2 F = m1 − m2 (3.63)

where the minus sign comes from the fact that the two surfaces have opposite orienta-tions. This shows that the Chern number n, is an integer.[15, 8, 16, 17, 18]

The Chern number is a topological invariant. This is not difficult to imagine: we could continuously deform the surface S, but when splitting into two different surfaces, the same rules will apply.

In the next chapters, we will discuss quantum geometry. We will discuss how the quan-tum phase space can be described as having geometric aspects. We will analyze gapped system, i.e. systems that have a definite energy difference in their spectrum between the ground state and the excited states.

Differentiation and integration will be with respect to the parameters λ of the Hamil-tonian H(λ). Continuous deformation can be translated as smoothly changing the pa-rameters of H or the surfaces S. This leads to smoothly changing the energy levels of the bands, without closing the gap.

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Part II

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Chapter 4 Metrics and holonomies in quantum

mechanics

4.1 Second order quantum phase transitions

In this part of the story, we will only discuss zero temperature systems. Phase transitions are a well known phenomena in nature. However, many of the examples we see in everyday life are connected to a notion of temperature: ice will melt, when a certain degree of energy, in form of heat, is injected into the system.

These are classical phase transitions (CPT). In the case of the phase transition be-tween ice and water, theres is a discontinuity in the heat capacity, C = ∂U/∂T . Such a phase transition is called a first order phase transition. CPTs cannot happen at zero temperature, mainly because entropy is not defined at T = 0 for classical systems. Quantum mechanically, a phase transitions can happen without thermal fluctuations. For example, a one-dimensional quantum crystal will melt, entirely due to quantum fluctuations.[19] These are called second order phase transitions.[20]

In general, we say that a quantum phase transition (QPT) can be described as transitions where the characteristic energy scale vanishes. For example, we will discuss gapped systems, meaning that there is an energy difference ∆ between the ground state and the first excited states, see Figure 4.1.[20]

In general, we refer to the non-analytic behavior of any observable as the critical E

xc x

∆

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points, and moving from one of these points to another will result to a phase transition. States in different phases will often have a very different structure can easily be distin-guished by acting on them with some appropriately chosen observable.[21]

Another important object, is the presence of a characteristic length scale, denoted ξ. This objects diverges when we reach the critical points. A simple example would be a liquid. For a specific particle in the liquid, a characteristic length scale would be the mean free path: the path the particle could follow without interacting with another par-ticle in the liquid. However, when the liquid vaporises, the mean free path essentially becomes infinite.[22] Quantitatively, this ξ is can be connected to the length scale that determines the exponential decay in equal-time correlations This characteristic length scale also decays as a power law, close to the critical points

ξ ∼ Λ|xi− xi_c|ν (4.1) with Λ an inverse length scale. Λ is often referred to as the momentum cutoff, which in condensed matter systems can be compared to the lattice spacing. The ratio of these two close to the critical points, z is known as the dynamical critical exponent and has the relation

∆ ∼ ξ−z (4.2)

This sort analysis is sometimes referred to as the Landau-Ginzburg paradigm. The problem with this method, is that sometimes certain aspects of a system are not clear. For example, it could be that we cannot identify any characteristic length scale or other order parameters.[20]

A solution to this problem could lie in analyzing the quantum version of the fidelity, introduced in section 3.2. In this approach, the QPTs are analyzed by looking at the overlap of two ground states with slightly displaced parameters xi. This overlap is cal-culated by the scalar product of the two ground states. QPTs are observed when the fidelity falls off and shows scaling behavior.

No knowledge of any of the order parameters is needed, and thereby no knowledge of important observables or symmetry breaking patterns.[21]

4.2 Quantum fidelity and natural distance

The idea of defining a distance in quantum phase space was proposed by an interesting article by Provost and Vall´ee in 1980, before Berry discovered his famous anomalous phase.[23]

We assume a set of vectors {ψ(x)} that depend smoothly on an n-dimensional co-ordinate x in some Hilbert space. We have a a well defined inner product and norm, given by respectively h , i and || ||. The idea is that we want to look at the distance in coordinate space x between a vector ψ(x) and vector that is infinitesimally displaces ψ(x + dx), ∆ψ = ψ(x) − ψ(x + dx).

The overlap of any two vectors ψ1(x) and ψ2(x) is known as the quantum fidelity

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as a generalization of the fidelity defined by Equation 3.34. The fidelity measures how much two vectors are similar. We can expand the fidelity of the difference of the vectors ψ(x) and ψ(x + dx) up to second order, to find

F (∆ψ, ∆ψ) = ||ψ(x) − ψ(x + dx)|| ∼ h∂iψ, ∂jψidxidxj (4.4)

where, in the last expression we suppressed the dependence on x. We can split the inner product into the real and imaginary parts, to get

h∂iψ, ∂jψi = γij + iσij (4.5)

We see that γij is symmetric and σij is anti-symmetric in its indexes, due to the

sesquilin-ear nature of the inner product. Since the product dxi_dxj _{is symmetric, the overlap}

reduces to

F (∆ψ, ∆ψ) = γijdxidxj + O(dx3) (4.6)

This function already looks like an infinitesimal line element in quantum phase space. It is obvious that the function γij transforms as a tensor, when we change coordinates

x → x0, see Appendix A. Furthermore, it is interesting to notice that if the displacement dx was zero, i.e. there was zero distance between the vectors, γij would also be zero.

However, we did not take into account that in quantum mechanics, a physical state is a ray, and that any vector in the Hilbert space can be multiplied with a phase factor to obtain the same physical state.

We see that the construction of γij is such that if we transform

ψ(x) → ψ0(x) = eiα(x)ψ(x)

γij → γij0 = Re(∂iψ0, ∂jψ0) = γij + i∂jα(∂iψ, ψ) − i∂iα(ψ, ∂jψ) + (∂iα)(∂jα)

= γij + βi∂jα + βj∂iα + (∂iα)(∂jα) (4.7)

where we defined the functions

βj = −i(ψ, ∂jψ) = i(∂jψ, ψ) (4.8)

which transform as

βj → βj0 = βj + ∂jα (4.9)

So we see that γij is not invariant under the transformation and in that sense not

physi-cally meaningful. We can, however write a new function

Gij(x) := γij(x) − βi(x)βj(x) = h∂iψ| 1 − |ψihψ||∂jψi (4.10)

where we introduced the bra-ket notation representing the inner product. This function is invariant under multiplication of a phase factor and transforms as a tensor, since both γij and βi transform as tensors. The tensor Gij is also positive definite and therefore can

be regarded as measuring a distance on the manifold of states.[23]

With this, we now have a notion of distance between state vectors in a Hilbert space. There is a more useful and perhaps more illuminating way to define a metric on a quantum phase space, which we will look at later. First, let us look at some more conventional notions of geometry in quantum mechanics.

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4.3 Berry’s phase and Simon’s interpretation

Adiabaticity

The discovery of Berry’s phase was the first real start to adding a geometric aspect to quantum mechanics.

To understand how this property was added to the analysis, we should understand how Berry’s phase was introduced and how it was later interpreted. First, let us introduce the Adiabatic Theorem. This theorem basically says that if we have a system in a state and we change the external properties of the system slowly enough, the system will stay in that same state.

To be able to discuss fast and slow, we obviously need some intrinsic notion of time. In quantum mechanics, this notion is met by the various gaps that a system can have. If the spectrum of a system can be split into two or more different parts (bands), then we call this jump from one part to another a gap.

An intuitive way to see this, is through the uncertainty relation

∆E∆T ≥ ~ (4.11)

if we evolve a system and want to keep it in the same state and the system has a gap, then the time ∆T has to be large enough that the energy fluctuations ∆E are lower than the gap.

A system that has a non-degenerate spectrum, i.e. no overlapping energy eigenvalues, automatically has a gap.[9]

Let us rescale the physical time t, with a timescale T . Defining s = t/T we get

i∂sψT(s) = T HψT(s) (4.12)

for the Schr¨odinger equation.

The adiabatic theorem tells us that if we start with a state ψT(0) in one part of the

spectrum, we will after a time t later end up in the state ψT(t) due to unitary evolution,

within the same part of the spectrum up to an error O(T−1). In the adiabatic limit of T → ∞, this error goes to zero.[9][7]

This means that the initial state and the state after adiabatic evolution can only differ a phase factor.

Berry’s phase

With the adiabatic theorem in mind, let us start by introducing the anomalous phase the way Berry found it.

Given a curve C on a manifold of parameters M

t −→ xt∈ M (4.13)

and that the adiabatic evolution is described by the parameters x of the Hamiltonian H(x), meaning that the time dependence of the Hamiltonian is solely due to the time

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dependence of the parameters. If we assume that the Hamiltonian H(x) has a discrete spectrum ∀x ∈ M , we can write

H(x)|n(x)i = En(x)|n(x)i (4.14)

for orthonormal states

hn(x)|m(x)i = δnm (4.15)

Lastly, we will assume that we can locally define the map

M 3 x → |n(x)i ∈ H (4.16) with H the Hilbert space. If we assume that the nth eigenvalue En(x) is non-degenerate,

we can define the nth eigenspace Hn(x) as

Hn(x) := Range(Pn(x)) = {eiα|n(x)i | α ∈ R} (4.17)

with Pn(x) the one dimensional projection onto the nth eigenspace of H, defined as

Pn(x) = |n(x)ihn(x)| (4.18)

Of course, changing the eigenstate

|n(x)i → |n0(x)i = eiαn_|n(x)i _(4.19)

will still satisfy Equation 4.14 and Equation 4.15. Such a phase transformation does not, however, change Pn(x).

If we start with ψ(0) = |n(x0)i, and adiabatically evolve to ψ(t), the adiabatic

theo-rem tells us that at time t, the state is again in the nth eigenspace of H and we can write ψ(t) = |n(xt)i.

This means that for an adiabatically evolved system from one point on x0 ∈ M back

to the same, xT = x0, we find

ψ(T ) = eiγψ(0) (4.20) for some T > 0. Formally, this condition can also be written as [7]

hn(x(t))|n(x(t + δt))i ∼ 1 + O(δt2₎

→ lim

T →∞hn(x)|dn(xT)i = 0 (4.21)

Any state ψ(t) obviously has to satisfy the time-dependent Schr¨odinger equation

i∂tψ(t) = H(x(t))ψ(t) (4.22)

Now the question is, what γ is. The obvious guess would be γ = −

Z T

0

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since it would satisfy the time-independent Schr¨odinger equation, i.e. Equation 4.14. However, this is wrong. This is not a solution of Equation 4.22, since the derivative of t will also hit the parameters x(t) and this will add a term ∼ ∂/∂xi|n(x(t))i.

To solve this, we add an extra term, which gives γ = − Z T 0 En(t)dt + iφn (4.24) so we can write ψ(t) = exp −i Z t 0 En(t0)dt0+ iφn(t) |n(xt)i (4.25)

Equation 4.22 then tells us i ˙ψ(t) =nEn(t) − ˙φn(t)

|n(x)i + i| ˙n(x)ioexp −i Z t 0 En(t0)dt0 + iφn(t) = H(x)ψ(x) = En(x)|n(x)i exp −i Z t 0 En(t0)dt0+ iφn(t) → ˙φn(x) = ihn(x)| ˙n(x)i (4.26)

which allows us to define the one-form on M , known as the Berry potential one-form A(n)= ihn|d|ni

= −Im(hn|d|ni) (4.27)

since the real part of hn|d|ni is zero due to the normalization. We can also write this as A(n) = A(n)_i dxi

= ihn|∂i|nidxi (4.28)

Integrating over ˙φn, we get

φn(t) = i Z t 0 hn(t0)| ∂ ∂t0|n(t 0 )i = Z C A(n) (4.29) and we define γn(C) := φn(T ) = I C A(n) (4.30)

and we find the final expression for the phase γ(C) = − Z T 0 En(t0)dt0 | {z } dynamical phase + γn(C) | {z } geometric phase (4.31)

Geometry in Physics the non-Abelian generalization of Berry's phase and black hole thermodynamic geometry

University of Amsterdam

MSc Physics

Theoretical Physics

Master Thesis

Geometry in Physics

the non-Abelian generalization of Berry’s phase and black hole

thermodynamic geometry

by

Jonas J. van der Waa

6232078/10002077

July 2015

54 ECTS

1 September to 15 July

Supervisor:

Vladimir Gritsev Dr.

Examiner:

Vladimir Gritsev Dr.

Second Reader:

Ben Freivogel Dr.

Acknowledgements

Contents

I

Statistical geometry

7

II

Quantum geometry

36

III

Thermodynamic geometry

90

Chapter 1

Introduction

Part I

Chapter 2

Mathematical description of

geometry

2.1

A short introduction to differential geometry

Trivial bundle: cylinder

Non-trivial bundle: the M¨

obius strip

Parallel transport and the connection

2.2

Examples of fiber bundles

Electrodynamics

Yang-Mills theory

2.3

Parallel transport on projective space CP

Chapter 3

Geometric interpretation of

statistical, classical information

3.1

Statistical mathematics

3.2

Fisher-Information matrix

3.3

Topological invariants

3.4

The Euler characteristic

3.5

Chern Classes

Chern number

Part II

Chapter 4

Metrics and holonomies in quantum

mechanics

4.1

Second order quantum phase transitions

4.2

Quantum fidelity and natural distance

4.3

Berry’s phase and Simon’s interpretation

Adiabaticity

Berry’s phase

_{Parallel transport on projective space CP}