THE TOPOLOGY AND GEOMETRY OF SELF-ADJOINT AND ELLIPTIC BOUNDARY CONDITIONS FOR DIRAC AND LAPLACE OPERATORS

# The Topology and Geometry of Self-Adjoint and Elliptic Boundary Conditions for Dirac and Laplace Operators

## Abstract

The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators.

The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space of such extensions is shown to have a canonical group composition law structure.

The obtained results are compared with von Neumann’s Theorem characterising the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated.

The geometry of the submanifold of elliptic self-adjoint extensions is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian . The topology of is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change.

Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results.

A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved.

The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grasmmannian is obtained.

Finally, an interpretation of spontaneous symmetry breaking is offered from the point of view of the theory of self-adjoint extensions.

Self-adjoint extensions, elliptic boundary conditions, Dirac operators, Laplace operators.
\catchline{history}

## 1 Introduction

The construction and discussion of quantum mechanical systems requires a detailed analysis of the boundary conditions (BC) imposed on the system. Often such boundary conditions are imposed by the observers and their experimental setting or are inherent to the system. This is a common feature of all quantum systems, even the simpler ones.

The outcomes of measurable quantities of the system will change with the choice of BC because the spectrum of the quantum observables will vary with the different self-adjoint extensions obtained for them depending on the chosen BC.

The Correspondence Principle provides a useful guide to the analysis of quantum mechanical systems, but it is not obvious how it extends to include boundary conditions both in classical and quantum systems. For instance, Dirichlet’s boundary conditions corresponds to impenetrability of the classical walls determining the boundary of the classical system in configuration space, but, what are the corresponding classical conditions for mixed BC? Then, we are facing the problem of determining the classical limit of quantum BC. Conversely, we can address the question of ‘quantizing’ classical boundary conditions. In particular we can ask if the classical determination of BC is enough to fully describe a ‘quantized’ system.

As the experimental and observational capabilities are getting more and more powerful, we are being forced to consider boundary conditions beyond the standard ones, Dirichlet, Neumann, etc. For instance, in Condensed matter, models with ‘sticky’ boundary conditions are seen to be useful to understand certain aspects in the Quantum Hall effect [Jo95]; in quantum gravity, self-adjoint extensions are used to understand signature change [Eg95]. Even more fundamental, topology change in quantum systems is modelled using dynamics on BC’s [Ba95]. Physical implications of the problem were already analyzed in [As12].

Following Dirac’s approach, we can develop a canonical quantization program for classical systems with boundary. Such program requires a prior discussion on the dynamics of Hamiltonian systems with boundary.

Without entering a full discussion of this important problem, we may assume that a classical Hamiltonian system with boundary is specified by a Hamiltonian function defined on the phase space of a configuration space with boundary , and a canonical transformation of the symplectic boundary , obtained by considering the quotient of the restriction of the cotangent bundle of to and then, taking its quotient by the characteristic distribution of the restriction of the canonical symplectic structure on to it. Thus, classical boundary conditions (CBC’s) are defined by canonical maps

 S:T∗∂Ω→T∗∂Ω, (1)

and form a group, the group of symplectic diffeomorphisms of 1.

Dirac’s quantization rule will be stated as follows: Given a CBC , and two classical observables on , determine a quantum boundary condition (QBC) and two self-adjoint operators , depending on , such that

 [^fS,^gS]=iℏˆ{f,g}S, (2)

and

 ^S∘^R=ˆSR, (3)

where the composition on the left is a group composition on the space of QBC’s to be discussed later on. It is obvious that as in the boundaryless situation such quantization rule could not be implemented for all observables and all CBC’s. So, one important question for such program will be how to select subalgebras of classical observables and subgroups of CBC’s suitable for quantization.

Before embarking in such enterprise, some relevant aspects of the classical and the quantum picture need to be clarified. For instance, we have to understand the structure of the self-adjoint extensions of the operator corresponding to a classical observable. The most important class of operators arising in the first quantization of classical systems are first and second order elliptic differential operators: the Laplace-Beltrami operator when quantizing a classical particle without spin, the Dirac operator for the quantization of particles with spin.

This family of operators are certainly the most fundamental of all elliptic operators, and in fact, in a sense all elliptic operators are obtained from them. Thus for Dirac and Laplace operators we will like to understand their self-adjoint realizations in terms of CBC’s. For that we need to understand first their self-adjoint realizations in terms of QBC’s.

Von Neumann developed a general theory of self-adjoint extensions of symmetric operators [Ne29]. Such theory is often presented in the realm of abstract Hilbert space theory that causes that the relevant features attached to the geometry of the operators is lost. Thus we will proceed following a direct approach to the theory of self-adjoint extensions exploiting the geometry of first and second order elliptic differential operators, and obtaining a fresh interpretation of von Neumann’s theory directly in terms of boundary data. A consequence of this analysis will be an interpretation of QBCs in terms of the unitary group at the boundary, that provides the group composition law needed for the implementation of Dirac’s quantization rule Eq. (3).

Elliptic differential operators have been exhaustively studied culminating with the celebrated Atiyah-Singer index theorem that relates the analytical index of such operators with the topological invariants of the underlying spaces [At68]. Such analysis extends to the boundary situation provided that appropriate elliptic boundary conditions are used. A remarkable example is provided by the global elliptic boundary conditions introduced in [At75], or APS conditions, that allow to describe the index of Dirac operators on manifolds with boundary. Such extensions have been adequately generalized for higher order elliptic operators giving rise to interesting constructions of boundary data [Fr97]. We should also recall here the important contributions by Lesch and Wojciechowski on the theory of elliptic boundary conditions and spectral invariants and the geometry of the elliptic Grassmannian (see for instance [Le96]).

However, from a physical viewpoint another family of boundary conditions has been considered for quark fields in bag models of quark confinement in QCD. The theory of chiral boundary conditions is not well developed although many physical applications have been analyzed from this perspective [Ch74], [Rho83], [As13], [As15].

The program sketched so far concerns exclusively with first quantization of classical systems, but second quantization is needed to truly deal with their physical nature. First quantization of classical systems, requires to consider the quantization of boundary conditions, which leads automatically to consider a collection of QBC’s for the first quantized system. Even for very simple systems, like a fermion propagating on a disk we need to consider ‘all’ self-adjoint extensions of Dirac’s operator on the disk. Thus, to proceed to second quantization we need to understand the global structure of such space of extensions. We will show that such space lies naturally in the infinite dimensional Grassmannian manifold and defines a Lagrangian submanifold of it, that will be called the self-adjoint Grassmannian.

Such infinite dimensional Grassmannian was introduced in the study of the KdV and KP integrable hierarchies of nonlinear partial differential equations [Se85]. It represents a ‘universal phase space’ for a large class of integrable evolution problems. Lately, such infinite dimensional Grassmannian was introduced again as the phase space of string theory and its quantization was discussed [Al87], [Wi88].

Our approach here is different, the infinite dimensional Grassmannian appears as the natural setting to discuss simultaneously all QBC’s for a first quantized classical system of arbitrary dimension. In fact, the relevant QBC’s are contained in a Lagrangian submanifold of the elliptic Grassmannian that should be subjected to second quantization. Lagrangian submanifolds of symplectic manifolds play the role of ‘generalized functions’, thus such program would imply quantizing a particular observable of the Grassmannian, making contact again with string theory. We should stress here that string theory is genuinely 2D whereas we are dealing with a classical point-like theory for classical objects in arbitrary dimensions.

The realization of such program would eventually introduce quantum dynamical effects on the space of QBC’s, suggesting that QBC’s could actually change (as suggested in [Ba95]), or that there is the possibility of non-vanishing probability amplitudes between states characterized by different boundary conditions. This observation imply that probably our first statement concerning the structure of classical systems with boundary is not totally correct, and we should extend it as follows: In the boundaryless situation, a Hamiltonian function defines a flow of symplectic diffeomorphisms on the phase space of the system. We must replace then the specification of CBC’s by a fixed symplectic diffeomorphism on the boundary and to allow for a flow of symplectic diffeomorphisms on the boundary too, i.e., by a boundary Hamiltonian . Upon quantization such Hamiltonian will define a quantum Hamiltonian on the space of QBC’s that eventually will lead to a propagator on second quantization.

Apart from the difficulties and physical implications of such ideas, we must point out an immediate consequence related to the topology of the systems studied. Changing the classical and quantum boundary conditions implies that possibility of changes in the topology of the system. For instance the quantization of a fermion moving on a disk with changing boundary conditions can change the topology of the disk and evolve into a surface of higher genus. Such process was analyzed by Asorey, Ibort and Marmo [As05] and it was shown that in the first quantized scheme such change can only occur if the trajectory in the space of self-adjoint extensions of the system cuts a submanifold where the spectrum of the operator diverges. Such submanifold is called the Cayley submanifold and describes its topology. Thus, in this sense the Cayley submanifold acts as a ‘wall’ in the space of QBC’s for a first quantized system with boundary, even though this will not preclude tunnelling in second quantization (see also [Sh12] for arguments along similar lines).

There are many other physical phenomena involving the ‘boundary’ of physical systems that it would be simply impossible to enumerate here. We would just like to mention a few recent contributions related to different physical problems: the analysis of boundary conditions and the Casimir effect in [As06], [As07], the role of boundary conditions in topological insulators [As13], and boundary conditions generated entanglement [Ib14b].

In this work we will try to offer a panorama of the field by presenting a discussion on some fundamental aspects of the theory of self-adjoint extensions of Laplace and Dirac-like operators, as well as a variety of related ideas and problems still on development. There is a vast literature on the subject covering its mathematical flank and it would be impossible to reproduce here. We must cite however, apart from the reference to the work by von Neumann already quoted, the pioneering work by Weyl [We09], Friedrichs’ construction [Fr34] or the contributions by Krein and Naimark[Kr47, Na43] to the general theory as a few historical landmarks.

On the other hand the theory of extensions of elliptic operators has attracted a lot of attention and, apart from the index theorem related works cited above, there is a number of fundamental results on the fields. We will just quote the analysis of non-local extensions of elliptic operators by Grubb [Gr68] and the theory of singular perturbations of differential operators by Albeverio and Kurasov [Al99] because of their influence on this work (see also [Ib14] for a quadratic forms based analysis of the extensions of the Laplace-Beltrami operator and [Ib12] where the reader will find a reasonable list of references on the subject).

Thus, we will proceed by reviewing first the geometric theory of self-adjoint extensions of Dirac and Laplace operators where the role of the Cayley transform at the boundary will be emphasized and the group structure of the space of self-adjoint extensions will be described explicitly. This will be the content of Sects. 2 and 3. Afterwards, Section 4 the relation with von Neumann’s theorem will be discussed and the one-dimensional situation will be discussed thoroughly. The spectral function for arbitrary boundary conditions will be obtained explicitly and the quantum analogue of Kirchoff’s laws will be obtained.

In Section 5 we will begin to discuss the semiclassical theory of self-adjoint extensions and its implementation in the path integral representation of quantum systems. Section 6 will be devoted to study the global structure of the space of extensions both elliptic and self-adjoint. The infinite-dimensional Grassmannian will be discussed as well as the Lagrangian submanifold of elliptic self-adjoint extensions.

We will introduce the study of dissipative quantum systems via non-self-adjoint boundary conditions in Section 7 where a unitarization theorem for dissipative systems will be stated.

The problem of dealing with symmetries of quantum systems will be addressed in Section 8, that is, if the quantum system has a symmetry, which are the self-adjoint extensions compatible with it. This problem will be stated and partially solved (see also [Ib14c]) and some examples will be exhibited. Particular attention will be paid to the space of self-adjoint extensions of the quotient Dirac operator of a Dirac operator with symmetry and its characterization as the symplectic manifold of fixed points of the self-adjoint Grassmannian.

It could also happen that even if we have a symmetry of a symmetric operator, the self-adjoint extension describing the corresponding quantum system will not share the same symmetry. We will say then that there is a spontaneous breaking of the symmetry. This situation will be succinctly dealt with in Section 9.

## 2 Self-adjoint extensions of first order elliptic operators: Dirac operators

### 2.1 Dirac operators

As it was indicated before, Dirac operators constitute an important class of first order elliptic operators, to the extent that most relevant elliptic operators arising in geometry and physics are directly related to them. Let us set the ground for them. We will consider a Riemannian manifold with smooth boundary . We denote by the Clifford bundle over , defined as the algebra bundle whose fibre at is the Clifford algebra generated by vectors in with relations

 u⋅v+v⋅u=−2η(u,v)x,∀u,v∈TxΩ.

Let be a -complex vector bundle over , i.e., for each , the fibre is a -module, or in other words, there is a representation of the algebra on the complex space . We will indicate by the action of the element on the vector . We will also assume in addition that the bundle carries a hermitian metric denoted by such that Clifford multiplication by unit vectors in is unitary, that is,

 (u⋅ξ,u⋅ζ)x=(ξ,ζ)x,∀ξ,ζ∈Sx,u∈TxΩ,||u||2=1. (4)

Finally, we will assume that there is a Hermitean connection on such that

 ∇(V⋅ξ)=(∇ηV)⋅ξ+V⋅∇ξ, (5)

where is a smooth section of the Clifford bundle , and denotes the canonical connection on induced by the Riemannian structure on .

A bundle with the structure described above is commonly called a Dirac bundle [La89] and they provide the natural framework to define Dirac operators. Thus, if is a Dirac bundle, we can define a canonical first-order differential operator by setting:

 /Dξ=ej⋅∇ejξ, (6)

where is any orthonormal frame at .

There is a natural inner product on the space of smooth sections of the Dicrac bundle induced from the pointwise inner product on by setting

 ⟨ξ,ζ⟩=∫Ω(ξ(x),ζ(x))xvolη(x),

where is the volume form defined by the Riemannian structure on . We will denote the associated norm by and is the corresponding Hilbert space of square integrable sections of .

The Dirac operator is defined on the Frechet space of smooth sections of , , however this is not the largest domain where it can be defined. The largest domain in where this operator can be defined consists of the completion of with respect to the Sobolev norm defined by

 ||ξ||2k,2=∫Ω(ξ(x),(I+∇†∇)k/2ξ(x))xvolη(x), (7)

with , where is the adjoint operator to in . In fact, such space is the space of sections of possessing first weak derivatives which are in . This Hilbert space will be denoted by (see for instance [Ad03]).

The operator defined on is not self-adjoint as we will discuss in what follows. However it is immediate to check that the Dirac operator is symmetric in , the space of smooth sections of with compact support contained in , the interior of . In fact, after an integration by parts we obtain immediately,

 ⟨/Dσ,ρ⟩=⟨σ,/Dρ⟩,∀σ,ρ∈Γ0(∘S). (8)

The operator with this domain is closable on and its closure is the completion of with respect to the norm . Such domain will be denoted by . The elements of are sections vanishing on with -weak derivatives. Notice that both and are dense subspaces in .

If we denote by the closure of on , it is simple to check that with domain , where denotes the adjoint of in . The domains of the different self-adjoint extensions, if any, of will be linear dense subspaces of containing such that the boundary terms obtained in the integration by parts procedure will vanish. We will denote in what follows by the domain of the operator and by its range, then the symmetric extensions of will be defined on subspaces such that

 H10(Ω,S)=Dom(/D0)⊂Dom(/Ds)⊂Dom(/D†s)⊂Dom(/D)=H1(Ω,S),

and for any . A self-adjoint extension of is a symmetric extension of such that .

We will characterize such self-adjoint extensions by using first the geometry of some Hilbert spaces defined on the boundary of and, later on, we will compare with the classical theory of self-adjoint extensions of densely defined symmetric operators developed by von Neumann [Ne29]. To do that, we will repeat first the well-known integration by part process used to derive formula (8).

Let and an orthonormal frame in a neighborhood of so that at for all . If , are sections of , then they define a vector field in a neighborhood of by the condition

 η(X,Y)=−(ξ,Y⋅ζ),∀Y.

Then, at , we get:

 ⟨/Dξ,ζ⟩=⟨ej∇ejξ,ζ⟩=−∇ej⟨ξ,ejζ⟩+⟨ξ,/Dζ⟩,

but, by definition , hence,

 div(X)=∇ejη(X,ej)−η(X,∇ejej)=−(ej⋅ξ,ej⋅ζ).

Namely,

 Missing dimension or its units for \raise

Integrating both parts of the equation, we will find,

 ⟨/Dξ,ζ⟩−⟨ξ,/Dζ⟩=∫Ωdivx(X)volη(x)=∫Ωd(iXvolη)=∫∂Ωi∗(iXvolη),

where we denote by the canonical inclusion. If denotes the inward unit vector on the normal bundle to , the volume form can be written as , where is an extension of the volume form defined on by the restriction of the Riemannian metric to it, and is a 1-form such that . Then we easily get,

 iXvolη=(iXθ)vol∂η=η(X,ν)vol∂η=(ξ,ν⋅ζ)vol∂η,

and, finally, we get

 ⟨/Dξ,ζ⟩−⟨ξ,/Dζ⟩=∫∂Ωi∗(ν⋅ξ,ζ)vol∂η. (9)

### 2.2 The geometric structure of the space of boundary data

We will denote by the restriction of the Dirac bundle to , i.e., , for any , . Notice that becomes a Dirac bundle over . It inherits an inner product induced from the Hermitean product on as well as an Hermitean connection , defined again by restricting the connection on to sections along . Thus the boundary Dirac bundle , carries a canonical Dirac operator denoted by .

Notice that is a manifold without boundary, thus the boundary Dirac operator is essentially self-adjoint and has a unique self-adjoint extension (see [La89], Thm. 5.7). We shall use this fact later on.

We will denote as before by the Hilbert space of square integrable sections of and by its Hilbert product structure:

 ⟨ϕ,ψ⟩∂Ω=∫∂Ω(ϕ(x),ψ(x))xvol∂η(x),ψ,ϕ∈L2(∂Ω,S∂Ω). (10)

Because of Lions trace theorem [Li72] the restriction map , , extends to a continuous linear map:

 b:H1(Ω,S)→H1/2(∂Ω,S∂Ω).

The Hilbert space will be called the Hilbert space of boundary data for the Dirac operator and will be denoted in what follows by . It carries an interesting extra geometrical structure induced by the boundary form defined by Green’s formula (9) responsible for the non self-adjointeness of the Dirac operator in .

 Σ(ϕ,ψ)=∫∂Ω(ν(x)⋅ϕ(x),ψ(x))xvol∂η(x). (11)

The normal vector field defines an automorphism of the Dirac bundle and an isomorphism:

 ν:Γ(S∂Ω)→Γ(S∂Ω),

by , for all , . Such automorphism extends to a continuous complex linear operator of denoted now by . Because in the Clifford algebra, such operator verifies . In addition, because of the Dirac bundle structure, eq. (4), is an isometry of the Hilbert space product,

 ⟨Jϕ,Jψ⟩∂Ω=⟨ϕ,ψ⟩∂Ω,     ∀ϕ,ψ∈HD, (12)

i.e., defines a compatible complex structure on . In general, a complex Hilbert space with inner product and a compatible complex structure defines a new continuous bilinear form by setting,

 ω(φ,ψ)=⟨Jφ,ψ⟩,∀φ,ψ∈H.

Such structure is skew-hermitian in the sense that

 ω(φ,ψ)=−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ω(ψ,φ).

(Notice that in the real case will define a symplectic structure on .) We will call such structure symplectic-Hermitean and the corresponding linear space a symplectic hermitian linear space (see for instance [Ha00] for a discussion of finite-dimensional Hermitian symplectic geometry).

The compatible complex structure allows to decompose the Hilbert space as

 H=H+⊕H−, (13)

where are the eigenspaces of , that is if . The subspaces are orthogonal as the following computation shows.

 ⟨ϕ+,ϕ−⟩=⟨Jϕ+,Jϕ−⟩=⟨iϕ+,−iϕ−⟩=−⟨ϕ+,ϕ−⟩

The Hilbert space will carry in this way a natural decomposition in two orthogonal infinite dimensional closed subspaces, i.e., a polarization. Notice that the Hilbert space carries already another complex structure, denoted by , multiplication by , and both complex structures are also compatible in the sense that . because of the bundle is a complex bundle. In addition this implies that

 J(H+±iH−)⊂H±.

Hence, the Hilbert space of boundary data for the Dirac operator is a polarized Hilbert space carrying a compatible complex structure and the corresponding symplectic-hermitian structure . Using these structures the boundary form is written as,

 Σ(ξ,ζ)=ω(b(ξ),b(ζ))=⟨Jb(ξ),b(ζ)⟩∂Ω,∀ξ,ζ∈H1(S). (14)

From this characterization we immediately see that self-adjoint extensions of will be obtained in domains such that their boundary image, will be isotropic subspaces of , thus vanishing the r.h.s. of Eq. (14). Moreover to be self-adjoint, such domains must verify that , thus, they must be maximal subspaces with this property. We have thus proved the first part of the following theorem,

{theorem}

Self-adjoint extensions of the Dirac operator are in one to one correspondence with maximally closed -isotropic subspaces of the boundary Hilbert space . The domain of any of these extensions is the inverse image by the boundary map of the corresponding isotropic subspace. Moreover, each maximally closed -isotropic subspace of defines an isometry , called the Cayley transform of and conversely.

{proof}

Let be a closed -isotropic subspace of . Then, is a closed subspace of containing . Let be the extension of defined on and compute . If , then because is isotropic. This shows that . If there were , then, the same computation shows that for all , and the subspace will be -isotropic, which is contradictory. Thus and the extension is self-adjoint. The converse is proved similarly.

Let us consider now a closed maximal -isotropic subspace . Let us show that is transverse to . Let , then , then . Then, the subspace defines the graph of a continuous linear operator and vectors have the form . Then, the -isotropy of implies,

 0 = ωD(ϕ++Uϕ+,ψ++Uψ+) = ⟨iϕ+−iUϕ+,ψ++Uψ+⟩=−i⟨ϕ+,ψ+⟩+i⟨Uϕ+,Uψ+⟩,

for every , that proves that is unitary.

### 2.3 The Cayley transform on the boundary

Theorem 2.2 is the boundary analogue of Von Neumann’s description of selfadjoint extensions by means of isometries between the deficiency spaces (see later on). In spite of the inherent interest of this result, it is well-known that sometimes it is more useful to have an alternative description of such extensions in terms of selfadjoint operators. For that we will use the Cayley transform. We shall define the Cayley transform on the polarized boundary Hilbert space as the isomorphism defined by , for every , . The complex structure is transformed into and the symplectic-hermitian structure is transformed into the bilinear form

 σD(ϕ1,ϕ2;ψ2,ψ2)=2i(⟨ϕ2,ψ2⟩−⟨ϕ1,ψ2⟩). (15)

Let be a unitary operator such that is invertible. Then we have and using the Cayley transform, , we will obtain that,

 ϕ1=iI+UI−Uϕ2,

that defines an operator . In general it will be more convenient to consider graphs of operators because acts on , then it actually maps subspaces of in subspaces of .

If is a subspace of , then the adjoint of will be the subspace defined by setting,

 W†={(ψ1,ψ2)∈H+⊕H−∣⟨ϕ1,ψ2⟩=⟨ϕ2,ψ1⟩,  ∀(ϕ1,ϕ2)∈H+⊕H−}.

The subspace is said to be symmetric if and self-adjoint if . (see [Ar61] for more details on operational calculus with closed subspaces of a Hilbert space). We will say that an operator is selfadjoint if its graph is a selfadjoint subspace of . In this sense it is obvious that the Cayley transform operator of is selfadjoint. Moreover, it is clear that self-adjoint subspaces are maximally isotropic subspaces of the bilinear form given by Eq. (15). But is the transformed bilinear form on by the Cayley transform, then, maximally -isotropic subspaces correspond to maximally -isotropic subspaces, that is the Cayley transform maps one-to-one graphs of isometries into self-adjoint subspaces of .

Let us denote by a compact operator, we denote by the subspace of given by and we will call it a compact deformation of . If we denote by the space of self-adjoint subspaces of . We will introduce a topology in as follows. We shall define base for the topology by the family of balls given by the sets of subspaces of the form . Hence becomes a topological space (in fact as we will see later a smooth manifold).

We can summarize the previous discussion in the following theorem.

{theorem}

The Cayley transform defines a homeomorphism between the space of isometries from to with the operator topology and the space of self-adjoint subspaces of . Moreover, the self-adjoint extensions of the Dirac operator are in one-to-one correspondence with the self-adjoint subspaces in .

### 2.4 Simple examples: The Dirac operators in 1 and 2 dimensions

Consider a 1-dimensional manifold with boundary, hence a connected component may be either the half-line or a closed interval in the compact case. We may assume without loss of generality that the metric is constant. Then the Clifford bundle becomes the trivial bundle complex line bundle over , as is algebra isomorphic to . Hence Dirac’s operator is simply , i.e., the momentum operator.

Notice that if is the half-line , there are no self-adjoint extensions of . The Hilbert space of boundary data is simply . Then, Eq. \eqrefint_part_D1 becomes:

 ⟨/Dξ,ζ⟩−⟨ξ,/Dζ⟩=−i¯ξ(0)ζ(0),

but, the only symmetric extensions are given by the condition , i.e., functions in , but such domain defines just a symmetric operator in full agreement with von Neuman’s theorem (see later Sect. 4).

However, if is a closed interval , then and

 ⟨/Dξ,ζ⟩−⟨ξ,/Dζ⟩=i(¯ξ(b)ζ(b)−¯ξ(a)ζ(a)).

Notice that and the space of self-adjoint extensions is given by unitary maps from to , that is (see also Sections 3.4 and 4.2 for a discussion on the relation between the self-adjoint extensions of and the Laplace operator )

Dirac operators in 2D have been the subject of exhaustive research both from the mathematical and physical side. We will consider to be now a two dimensional compact orientable manifold, i.e., a Riemann surface with boundary where . Consider now a Dirac bundle . Notice that because of the general previous considerations carries a representation of the Clifford algebra bundle .

Take a point and a local chart around it with local complex coordinates . The tangent space is spanned by the local vector fields , , and the two vectors can be taken to be orthonormal. Hence the Clifford algebra at is generated by with the relation:

 e1⋅e2+e2⋅e1=0,e21=−1,e22=−1.

The generators of the Clifford algebra act on the tangent bundle as the matrices

 σ1=(01−10),σ2=(0ii0).

This representation happens to be the spinor representation of the spin group which is a double covering of (the covering given by the map ). Thus if we take the complexified tangent space as a Dirac bundle then we will have for the Dirac operator:

 /D=e1⋅∇e1+e2⋅∇e2=(0¯∂+A∂+¯A0),

with , being the Cauchy–Riemann operators on the variables , and the homogeneous part of the Levi–Civita connection corresponding to the given Riemannian metric on .

The representation above is reducible because anticommutes with and . In fact, which has eigenvalues . The representation space of decomposes into two subspaces of eigenvectors of respectively. The operator is known as the chirality operator, and the spaces of sections of are the right-handed and left-handed fermions respectively. Notice that , and the chiral projectors, i.e., the orthogonal projectors into the orthogonal subspaces are given by . Thus given any section of , we have , . Notice finally that , where denotes the anticommutator, hence , and exchanges the irreducible representation spaces as it is apparent from the block structure of the matrix representing above.

Assuming that the boundary is connected, i.e., , then the space of boundary data is given as which is isomorphic to the space of functions of Sobolev class on with values in . Clifford multiplication by , the outward normal determines a polarisation as described in Eq. \eqrefH+H-. Thus, according with Thm. 2.2 the space of self-adjoint extensions of is given by the family of unitary operators . Notice that each one of the subspaces is isomorphic to , hence the space of self-adjoint extensions can be identified with the group of unitary operators of the Sobolev space .

## 3 Self-adjoint extensions of Laplace operators

### 3.1 The covariant Laplacian

We will start the discussion of the theory of self-adjoint extensions of second order differential operators considering a particular situation of big physical interest. Later on, we will extend this considerations to more general families of second order operators and we will find the relation with the theory of extensions for Dirac operators studied in the first part.

We will consider a physical system subject to the action of a Yang–Mills field. The configuration space of the system will be a compact connected smooth Riemannian manifold with smooth boundary 2. We will consider a hermitian vector bundle with hermitian product . The Yang–Mills potential will be a connection 1-form with values on . The space of smooth sections of will be denoted by and the covariant differential operator will map to , i.e., it will map sections of to 1–forms in with values on .

We will define the -product in as usual,

 ⟨ψ1,ψ2⟩=∫Ω(ψ1(x),ψ2(x))xvolη(x), (16)

where denotes the Riemannian volume defined by the metric . We define as the completion with respect to the norm of the space .

Similarly, we will define the product of two –forms , on with values on by the formula,

 ⟨α,β⟩=∫Ω(αi1…ik(x),βi1…ik(x))volη(x),

where we have used the metric to raise the subindexes on the components of . We define the Hodge operator as a map from to , defined by

 ⋆α(x)∧β(x)=(αi1…ik(x),βi1…ik(x))ηg(x),

and thus,

 ⟨α,β⟩=∫Ω⋆α∧β.

We will consider the completion of with respect to the norm and then define the adjoint to the operator with respect to this Hilbert space structure, i.e.,

 ⟨d†Aα,ψ⟩=⟨α,dAψ⟩,

for all (notice that is well defined because is dense in ).

The quantum Hamiltonian for a particle moving in the presence of the Yang–Mills potential is formally given by the second order differential operator,

 H=−12ΔA+V, (17)

where

is the covariant Laplace-Beltrami operator and is a smooth function on .

Clearly, the operator is closable and symmetric on . Its closure is defined on the domain , i.e., the completion of with respect to the Sobolev norm defined in Eq. (7) with . In fact, such space is the space of sections of vanishing on , such that they possess first and second weak derivatives which are in . We will denote this operator by , .

The operator has another extension, the largest space where it can be defined, the closure of with respect to . This domain is the Sobolev space and it is easy to check that the adjoint of is precisely with domain , thus , . On the other hand the smooth function defines a essentially self-adjoint operator on . If we denote by the operator defined by with domain it is possible to check that , i.e., the domain of the adjoint operator of is . Hence, the operator defined on is not self–adjoint in general. We will be interested in finding extensions of with domain such that

 H20(Ω,E)=DomH0⊂DomHs=DomH†s⊂DomH=H2(Ω,E),

and , for any . To do that, we will integrate by parts as follows.

 ⟨ψ1,Hψ2⟩=⟨Hψ1,ψ2⟩+12Σ(ψ1,ψ2), (18)

for any smooth sections , where by Stokes theorem and the parallelism of with respect to the connection defined by , we get:

 Σ(ψ1,ψ2)=∫∂Ωi∗[(⋆dAψ1,ψ2)−(ψ1,⋆dAψ2)]. (19)

Notice that is an –form on with values in , and then denotes the –form on obtained by taking the product of the values of and on the fibres of .

The boundary term has a relevant physical interpretation. It measures the net total probability flux across the boundary. If the operator were self-adjoint this flux would have to vanish: the incoming flux would have to be equal to the outcoming flux because the evolution operator in such a case would be unitary and preserve probability.

We will characterize first self-adjoint extensions of by using the geometry of some Hilbert spaces defined on the boundary of and later on, we will compare with the classical theory of von Neumann.

### 3.2 Complex structures on the Hilbert space of boundary data

We will denote again by the restriction of the bundle to . Let us denote by the Riemannian volume form defined by the restriction of the Riemannian metric to the boundary . Then we will denote by the Hilbert space of square integrable sections of the bundle and by its Hilbert product structure given by,

 ⟨ψ,φ⟩∂Ω=∫∂Ω(ψ(x),φ(x))vol∂Ω. (20)

To any section we can associate two sections as follows,

 i∗Ψ = φ i∗(⋆dAΨ) = ˙φvol∂η, (21)

in other words, the function is the restriction to of , and is the normal derivative of along the boundary.

Thus the restriction map , assigning to each section its Dirichlet boundary data,