dirac operators on manifolds and applications to physics · 1 introduction this report develops the...
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Dirac Operators on Manifolds and
Applications to Physics
Angus AlexanderSupervised by Adam Rennie and Alan Carey
University of Wollongong
Vacation Research Scholarships are funded jointly by the Department of Education and Training
and the Australian Mathematical Sciences Institute.
Abstract
This work was completed in the summer of 2017-18 as part of an AMSI vacation research scholarship
under the supervision of Adam Rennie and Alan Carey. The aim of the project was to study differential
operators on manifolds and applications to physics, in particular to understand the APS boundary conditions
presented by Bar and Strohmaier in [2] and apply them to the case of a cylinder. We further attempt to
combine the work in [2] with that of Connes’ in [3] and develop some possible candidates for a trace in
relativistic quantum mechanics.
1 Introduction
This report develops the prerequisite material for studying Dirac operators on Lorentzian manifolds and applies
the theory to the case of a cylinder with APS boundary conditions as found in [2].
In Section 2 we develop the theory of elliptic differential operators on manifolds using unbounded operator
theory for Hilbert spaces. In particular, we extend the notion of an adjoint to a differential operator and prove
some results relating to boundary conditions and self-adjointness.
In Section 3 we develop the algebraic prerequisites required for understanding Dirac operators. The section
is focused on Clifford algebras, modules and bundles with a number of examples. The concepts of spinor bundles
and connections are also introduced.
In Section 4, we define a Dirac operator for a vector bundle over a manifold and provide numerous Euclidean
examples, as well as the Dirac operator on a Riemannian cylinder.
In Section 5, the theory developed in the previous sections is utilised to determine the Dirac operator on a
Lorentzian cylinder. The eigenvalues for the operators are computed, as well as some candidates for a trace in
an attempt to combine the work in [2] and that of Connes’ in [3].
2 Elliptic Differential Operators
The material in this section is mostly based on [5, Chapter 10], with some elements drawn from [10] and [7]. We
develop the theory of differential operators on smooth vector bundles over smooth manifolds. For a refresher
on vector bundles and manifolds, see [4] and [6].
2.1 First Order Differential Operators
Definition 2.1. Let M be a smooth manifold and let p ∶ E → M be a smooth vector bundle over M . Let
C∞(M,E) be the space of smooth sections of E. A first order linear differential operator on E is a complex
linear map D ∶ C∞(M,E)→ C∞(M,E) which satisfies the following two properties:
1. If u1 and u2 are smooth sections of E which agree on an open set U ⊆M then Du1 and Du2 agree on U ,
and
1
2. For each coordinate patch U ⊆M , if we choose coordinates xj in U and a trivialisation for the bundle E
over U , then D can be represented in local coordinates by
Du =∑j
Aj∂u
∂xj+Bu,
where the Aj and B are smooth matrix valued functions on U .
We define the support of D to be the complement of the largest open set U for which the restriction of D to U
is zero.
The functions Aj and B above are dependent on the bundle trivialisation, however retain their form. Let
ΦU ∶ E∣U → U × Ck and ΦV ∶ E∣V → V × Ck be bundle trivialisations. The transition function gUV ∶ U ∩ V →
GLk(C) is defined as
gUV = ΦU ○Φ−1V .
Given coordinates ϕU ∶ U → Rn and u ∈ C∞(M,E∣U), we have for x ∈ Rn that
ΦU(Du)(x) =∑j
Aj(x)∂
∂xjΦU(u) ○ ϕ−1U (x) +BΦu(u) ○ ϕ−1U (x).
Let y = ϕV ○ ϕ−1U (x) and compute that
gV U (ΦU(Du)) (ϕ−1U (x)) = ΦV (Du) (ϕ−1V (y))
=∑j
(gV UAj∂
∂xj(ΦU ○Φ−1
V ○ΦV u)) (ϕ−1V ○ ϕV ○ ϕ−1U (x))
+ (gV UBΦU ○Φ−1V ○ΦV u) (ϕ−1v ○ ϕV ○ ϕ−1U (x))
=∑j
(gV UAj∂gUV∂xj
(ϕ−1U (x)) (ΦV (u)) (ϕ−1V (y)))
+∑j
gV UAjgUV∂yk
∂xj∂ΦV (u)∂yk
○ ϕ−1V (y) + (gV UBgUV ΦV (u)) (ϕ−1V (y))
=∑j
Aj∂ΦV (u)∂yj
(ϕ−1V (y)) + Bu (ϕ−1V (y)) .
From this expression we see that in the new trivialisation we still have a differential operator, however the
functions Aj and B have changed.
Definition 2.2. Define the symbol of D, σD ∶ T ∗M → End(E) by
σD(x, ξ) = i∑j
Ajξj .
Remark 2.3. Definition 2.1 only refers to first order differential operators. We may more generally consider
an order k differential operator by replacing condition (2) with
Du = ∑∣α∣≤k
Aα∂ ∣α∣u
∂xα,
2
where α = (α1, . . . , αn) ∈ Nn is a multi-index and ∣α∣ =n
∑j=1
αj . We may also extend Definition 2.2 to an order k
operator as
σD(x, ξ) = ik ∑∣α∣=k
Aαξα.
Definition 2.1 can also be extended to differential operators between different vector bundles E and F as
D ∶ C∞(M,E) → C∞(M,F ), where in local coordinates D is given by the same formula as above. The symbol
is defined in this case in the same manner, however σD ∶ T ∗M → Hom(E,F ). Note also that from the definition,
it is clear that if D1 and D2 are differential operators, then
σD1○D2(x, ξ) = σD1(x, ξ) ○ σD2(x, ξ).
A number of differential operators are already familiar from vector calculus and Riemannian geometry.
Example 2.4. Recall that the Laplace-Beltrami operator ∆ ∶ C∞(M,E)→ C∞(M,E) is given in local coordi-
nates as (for f ∈ C∞(M,E))
∆f = 1√g∑j,k
∂
∂xj(√ggjk ∂f
∂xk)
=∑j,k
(gjk ∂2f
∂xj∂xk+ 1
√g
∂√g
∂xj
∂f
∂xk).
So we calculate that for (x, ξ) ∈ T ∗M ,
σD(x, ξ) = i2∑j,k
gjkξjξk
= − ∥ξ∥2 .
Definition 2.5. If the vector bundle E has a Hermitian metric (⋅, ⋅) and M has a smooth measure µ, then we
may define an inner product on C∞c (M,S) by
⟨u, v⟩ = ∫M
(u(x), v(x))dµ(x).
Using this we define L2(M,S) to be the completion of C∞c (M,S) relative to this inner product.
Proposition 2.6. Let D ∶ C∞(M,E) → C∞(M,E) be a differential operator. There is a unique differential
operator D� ∶ C∞(M,E)→ C∞(M,E) such that
⟨Du, v⟩ = ⟨u,D�v⟩
for all u, v ∈ C∞c (M,E). The symbols of D and D� are related by the formula
σD�(x, ξ) = σD(x, ξ)∗.
We call D� the adjoint of D.
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Proof. By using a partition of unity, it suffices to show existence in the case where D is supported in a single
coordinate patch. Then we compute that on this patch U we have
⟨Du, v⟩ = ∫U
⎛⎝∑j
Aj∂u
∂xj+Bu, v
⎞⎠
dx
=∑j∫U(Aj
∂u
∂xj, v)dx + ∫
U(Bu, v)dx
=∑j∫U( ∂u∂xj
,A∗j v)dx + ∫
U(u,B∗v)dx
=∑j∫U[ ∂
∂xj(u,A∗
j v) − (u, ∂
∂xj[A∗
j v])]dx + ∫U(u,B∗v)dx
= −∑j∫U(u,
∂A∗j
∂xjv +A∗
j
∂v
∂xj)dx + ∫
U(u,B∗v)dx
= ∫U
⎛⎝u,∑
j
−A∗j
∂v
∂xj+⎡⎢⎢⎢⎣B∗ −∑
j
∂Aj
∂xj
⎤⎥⎥⎥⎦v⎞⎠
dx,
where the second last equality follows from the fact that D is zero in a neighbourhood of the boundary of U .
For uniqueness, suppose that D1 and D2 are both formal adjoints for D. Define L =D1 −D2 and compute that
for any u, v ∈ C∞c (M,E) we have
⟨u,Lv⟩ = ⟨u, (D1 −D2)v⟩
= ⟨u,D1v⟩ − ⟨u,D2v⟩
= ⟨Du, v⟩ − ⟨Du, v⟩
= 0.
Since the range of L is orthogonal to every u ∈ C∞c (M,E), we have L = 0 as required. Note also that from the
above formula for D�, we see that for (x, ξ) ∈ T ∗M we have
σD�(x, ξ) = i∑j
−A∗j ξj
=⎛⎝i∑j
Ajξj⎞⎠
∗
= σD(x, ξ)∗.
Example 2.7. Let Mg denote the operator of multiplication by g in C∞(M,S). Then we calculate that for
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any smooth section u of E
([D,Mg])u =D(gu) − gDu
=∑j
(Aj∂(gu)∂xj
) +Bgu − g⎛⎝∑j
Aj∂u
∂xj+Bu
⎞⎠
=∑j
(Aj∂g
∂xju +Ajg
∂u
∂xj− gAj
∂u
∂xj) +Bgu − gBu
=⎛⎝∑j
Aj∂g
∂xj
⎞⎠u.
Evaluating this computation at x ∈M , we have
i[D,Mg]u(x) = σD(x,dg)u(x).
Henceforth we shall refer to a first order differential operator as simply a differential operator.
2.2 Symmetric and Self-Adjoint Differential Operators
In this section we develop the Hilbert space theory of differential operators, although much care is needed since
these operators are usually not bounded. We consider a differential operator D as an unbounded operator
initially defined on C∞c (M,E) ⊂ L2(M,E). We begin with some technical results.
Lemma 2.8. Let D ∶ C∞c (M,E)→ C∞
c (M,E) be a differential operator. Then D is closable.
Proof. Suppose that vj → 0 in C∞c (M,E) and Dvj → w ∈ L2(M,E). For any u ∈ C∞
c (M,E) we have that
⟨u,w⟩ = limj→∞
⟨u,Dvj⟩
= limj→∞
⟨D�u, vj⟩
= ⟨D�u,0⟩
= 0.
Since w is orthogonal to C∞c (M,E), a dense subspace of L2(M,E), we have that w = 0 and so D is closable.
The overline indicating closure of an operator will be omitted when the distinction between the two is not
necessary. Recall that an operator T is symmetric if ⟨Tu, v⟩ = ⟨u,Tv⟩ for all u, v ∈ Dom(T ). For a differential
operator on a manifold without boundary this amounts to being self-adjoint, and these are the primary operators
of interest to us.
Definition 2.9. Let D ∶ C∞c (M,E) → C∞
c (M,E) be a symmetric differential operator. The minimal domain
of D is Dom(D) and the maximal domain of D is Dom(D∗).
Note that if D is symmetric then D ⊆ D ⊆ D∗ and all three of these operators may be different from one
another.
5
Example 2.10. Consider the manifold M = (0,1). Let Dom(D) = C∞c (M,C) and define the differential
operator D ∶ Dom(D)→ L2([0,1]) by D = i d
dx. Recall that
Dom(D∗) = {ξ ∈ L2([0,1]) ∶ there exists η ∈ L2([0,1]) s.t for all ψ ∈ Dom(D), ⟨Dψ, ξ⟩ = ⟨ψ, η⟩} .
Then we may check that ξ(x) = x is in Dom(D∗), since for any ψ ∈ Dom(D),
⟨Dψ, ξ⟩ = i∫1
0
dψ
dxxdx
= i∫1
0
d
dx[ψ(x)x]dx − i∫
1
0ψ(x)dx
= ⟨ψ,−i⟩.
However ξ ∉ Dom(D). To see this, first fix u ∈ Dom(D) and calculate that
⟨Du,1⟩ = i∫1
0
du
dxdx
= 0,
then extend by continuity in the graph norm (∥ξ∥Gr = ∥ξ∥ + ∥Dξ∥) to see that this holds for all u ∈ Dom(D)
also. Now note that
⟨Dξ,1⟩ = i∫1
01 dx
= i
≠ 0.
Hence D∗ and D are distinct operators.
The example above suggests that failure of an operator to be self-adjoint is related to boundary conditions
on a non-compact manifold M . We will make this more precise soon, but some technical results are required.
Elements of the proof of the following result were adapted from [10, Proposition 3.2].
Lemma 2.11. Let K ⊂ M be compact. For sufficiently small t > 0, there exist operators Ft ∶ L2(K,E) →
L2(M,E) such that
1. ∥Ft∥ < 1,
2. for each u ∈ L2(K,E), Ftu→ u in L2(M,E) as t→ 0,
3. for each u ∈ L2(K,E), Ftu is smooth, with compact support, and
4. the commutator [D,Ft] extends to a bounded operator from L2(K,E) to L2(M,E), whose norm is
bounded independent of t.
We call such a family Friedrichs mollifiers on M .
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Proof. Fix ε > 0. Let ϕ ∶ Rn → R be a smooth positive function with compact support and total mass 1
with supp(ϕ) ⊆ B1(0). There are many choices of such a function, one of which is ϕ(x) = cne− 1
1−∥x∥2 on
B1(0) extended to 0 on Rn, where cn is a normalising constant. Define Ft ∶ L2(Rn) → L2(Rn) by Ftu(x) =
t−n ∫Rnu(y)ϕ(x − y
t)dy. Note that Ftu(x) = (u ⋆ ϕt)(x), where ϕt(x) = t−nϕ(x
t).
To estimate the norm, we apply Young’s inequality to see that
∥Ftu∥2 = ∥u ⋆ ϕt∥2
≤ ∥u∥2 ∥ϕt∥1
= ∥u∥2 .
Hence taking the supremum over all u with ∥u∥2 = 1, we have that ∥Ft∥ ≤ 1.
Next, we aim to show L2 convergence. To this end, we first show that Ftw → w uniformly for w ∈ Cc(Rn).
Note that supp(ϕt) ⊆ Bt(0) and so we may find s > 0 such that sup{∣w(x − y) −w(x)∣ ∶ ∣y∣ ≤ t} < ε for t < s. So
fix t < s and calculate that
∣Ftw(x) −w(x)∣ = ∣∫Rnw(x − y)ϕt(y)dy − ∫
Rnw(x)ϕt(y)dy∣
≤ ∫Rn
∣w(x − y) −w(x)∣ϕt(y)dy
≤ sup{∣w(x − y) −w(x)∣ ∶ ∣y∣ ≤ t}
< ε.
Find v ∈ Cc(Rn) such that ∥u − v∥ < ε3
. Applying property 1 we have that
∥Ftu − Ftv)∥2 = ∥Ft(u − v∥2
≤ ∥u − v∥2
< ε3.
Choose t > 0 such that ∥Ftv − v∥2 <ε
3(possible since uniform convergence implies L2 convergence) and calculate
that
∥Ftu − u∥2 ≤ ∥Ftu − Ftv∥2 + ∥Ftv − v∥2 + ∥u − v∥2
≤ 2ε
3+ ∥Ftv − v∥
< ε.
Hence Ftu→ u in L2.
By definition, we have that supp(Ftu) ⊆ {x + y ∶ x ∈ supp(u) and y ∈ Bt(0)} which is bounded. Hence
7
supp(Ftu) is compact. To check smoothness, we calculate that
limh→0
Ftu(x + hei) − Ftu(x)h
= limh→0∫Rn
ϕt(x + hei − y) − ϕth
(x − y)u(y)dy
= ∫Rn
∂ϕt(x − y)∂xi
u(y)dy
= u ⋆ ∂ϕ
∂xi.
So we see inductively that Ftu is smooth, since ϕt is smooth. Note also that in a similar manner to the above
calculation, we can show that∂
∂xj(u ⋆ ϕt) = ∂u
∂xj⋆ ϕt = u ⋆
∂ϕt∂xj
. So we calculate the commutator for any
u ∈ C∞c (Rn) as
[D,Ft]u(x) =D(Ftu) − Ft(Du)
=∑j
Aj∂
∂xj(u ⋆ ϕt) +B(u ⋆ ϕt) −∑
j
(Aj∂u
∂xj) ⋆ ϕt − (Bu ⋆ ϕt)
=∑j
Aj (u ⋆∂ϕt∂xj
) +B(u ⋆ ϕt) −∑j
(Aj∂u
∂xj) ⋆ ϕt − (Bu ⋆ ϕt),
where we have moved the derivative from u to ϕ as discussed before the calculation. We now write the
convolution in integral form and apply integration by parts, giving
[D,Ft]u(x) =∑j
(Aj(x)∫Rnu(y)∂ϕt
∂xj(x − y)dy − ∫
RnAj(y)
∂u
∂yjϕt(x − y)dy)
+B(x)∫Rnu(y)ϕt(x − y)dy − ∫
RnB(y)u(y)ϕt(x − y)dy
=∑j∫Rn
[(Aj(x) −Aj(y))u(y)∂ϕt∂xj
(x − y) −Aj(y)∂
∂yj(u(y)ϕt(x − y))]dy
+ ∫Rn
(B(x) −B(y))u(y)ϕt(x − y)dy.
We now rearrange terms and use the fact that∂ϕt∂xj
(x − y) = −∂ϕt∂yj
(x − y) to obtain
[D,Ft]u(x) =∑j∫Rn
(Aj(x) −Aj(y))u(y)∂ϕt∂xj
(x − y)dy + ∫Rn
(B(x) −B(y))u(y)ϕt(x − y)dy
=∑j
1
t∫Rn
(Aj(x) −Aj(y))u(y)∂ϕ
∂xj(x − y
t)dy + ∫
Rn(B(x) −B(y))u(y)ϕt(x − y)dy.
Using local coordinates and a partition of unity, it is possible to graft this onto an arbitrary manifold to
construct a family of Friedrichs mollifiers there. The proof and technique for this are quite involved and will
not be presented here.
We will need another technical lemma before being able to prove the result about ‘boundary conditions’,
however the proof is omitted. The proof can be found in [5, Lemma 1.8.1].
Lemma 2.12. Let T be a closable operator on a Hilbert space H. Then u ∈H belongs to the minimal domain
if and only there is a sequence (uj) in the domain of T such that uj → u while ∥Tuj∥ remains bounded.
8
Lemma 2.13. Let D be a symmetric differential operator on a bundle E over a manifold M and suppose that
u is a compactly supported element of L2(M,E). Then u belongs to the minimal domain of D if and only if it
belongs to the maximal domain.
Proof. We use the Friedrichs’ mollifiers developed in Lemma 2.11. Suppose u ∈ Dom(D∗) is supported in K.
We know that Ftu→ u as t→ 0 and Ftu is compactly supported and smooth. We also have that
DFtu = FtDu + [D,Ft]u,
which is bounded in norm by Lemma 2.11. Hence by Lemma 2.12, we have that u belongs to the maximal
domain if and only if belongs to the minimal domain.
Corollary 2.14. Every compactly supported symmetric differential operator on an open manifold is essentially
self-adjoint.
Proof. Applying Lemma 2.13, we see that Dom(D) = Dom(D∗) = Dom(D∗). Hence D is self-adjoint and D is
essentially self-adjoint.
3 Algebraic Prerequisites
We wish to work with a specific class of differential operators, called Dirac operators. However before defining
Dirac operators, we must first develop some algebraic prerequisites from the theory of Clifford algebras and
vector bundles. The material in this section is primarily found in [7, Chapter I].
3.1 Clifford Algebras
Definition 3.1. Let V be a finite dimensional vector space and q ∶ V ×V → C a non-degenerate inner product.
Define an ideal Jq in the tensor algebra T (V ) =⊕n≥0
V ⊗n generated by elements of the form v ⊗ v + q(v, v)1 for
v ∈ V . We define the Clifford algebra Cliff(V, q) to be the quotient
Cliff(V, q) = T (V )/Jq.
Note that Cliff(V, q) is generated by 1 and v,w ∈ V satisfying
v ⋅w +w ⋅ v = −2q(v,w).
Hence if v and w are orthogonal they anticommute in Cliff(V, q). Clifford algebras also satisfy a universal
property as follows.
Lemma 3.2. If A is a unital associative algebra and c ∶ V → A is a linear map satisfying
c(v)c(v) = −q(v, v)1A
for all v ∈ V , there exists a unique algebra homomorphism c ∶ Cliff(V, q)→ A extending c.
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Proof. Firstly, there exists a unique algebra homomorphism ϕ ∶ T (V )→ A such that ϕ○ι = c, where ι ∶ V → T (V )
is the inclusion map, defined by ϕ(v1 ⊗⋯⊗ vn) = c(v1)⋯c(vn). Next, check that for v ∈ V we have
ϕ(v ⊗ v + q(v, v)) = ϕ(v ⊗ v) + ϕ(q(v, v)
= c(v) ⋅ c(v) + q(v, v)1A
= 0.
Hence ϕ vanishes on Jq and so descends uniquely to c on Cliff(V, q).
Example 3.3. Note that if A ∈ O(V, q) = {A ∶ V → V ∶ q(Av,Av) = q(v, v) for all v ∈ V }, then we may define a
linear map c ∶ V → Cliff(V, q) by c(v)w = Av ⋅w. Since this satisfies
c(v)c(v) = Av ⋅Av
= −q(Av,Av)
= −q(v, v).
Hence by Lemma 3.1, c extends uniquely to an automorphism c ∶ Cliff(V, q) → Cliff(V, q). If we consider the
map α ∶ V → V defined by α(v) = −v, we have an involution on Cliff(V, q) and thus we may decompose into the
eigenspaces for ±1 as
Cliff(V, q) = Cliff0(V, q)⊕Cliff1(V, q).
Since α(v1 ⋅ ⋅ ⋅ ⋅ ⋅ vn) = (−1)nv1⋯vn we see that Cliff(V, q) decomposes into sums of even and odd products of
vectors. Since α is an algebra homomorphism, we see that
Cliffi(V, q) ⋅Cliffj(V, q) ⊆ Cliffi+j(V, q),
where the addition of indices is mod 2. Algebras satisfying this property are said to be Z2-graded.
3.2 Clifford Modules and Bundles
Definition 3.4. A C-representation of the Clifford algebra Cliff(V, q) is an R-algebra homomorphism ϕ ∶
Cliff(V, q) → EndC(W ) into the algebra of complex linear transformations of a finite dimensional complex
vector space W . A vector space W together with a Cliff(V, q) representation is called a Cliff(V, q) module.
For v ∈ Cliff(V, q) and w ∈W , we write
v ⋅w = ϕ(v)(w)
and refer to this as Clifford multiplication of w by v. We may define R-representations and H-representations
in a similar manner.
10
Definition 3.5. A C-representation is called reducible if there exists a decomposition W =W1 ⊕W2 such that
ϕ(v)(Wj) ⊂ Wj for j = 1,2 and all v ∈ Cliff(V, q). In this case we may write ϕ = ϕ1 ⊕ ϕ2. A representation is
irreducible if there are no proper invariant subspaces. Two representations are equivalent if there exists a C
linear isomorphism F ∶W1 →W2 such that Fϕ1(v)F −1 = ϕ2(v) for all v ∈ Cliff(V, q).
It is a non trivial fact that every complex Cliff(V, q) module decomposes uniquely as a sum of irreducibles.
Example 3.6. Define ϕ ∶ Cliff(V, q) → Cliff(V, q) by ϕ(v)w = v ⋅ w. Then we see that Cliff(V, q) is trivially a
Cliff(V, q) module.
Example 3.7. For a more involved example, consider the exterior algebra Λ∗V . Let (v1, . . . , vn) be an ortho-
normal basis for V . Define exterior multiplication by vj as ej and interior multiplication by vj as ij . Recall
that ejik + ikej = δjk. Define ϕj = ej − ij and calculate that
ϕj ○ ϕk + ϕk ○ ϕj = (ej − ij)(ek − ik) + (ek − ik)(ej − ij)
= ejek − ejik − ijek + ijik + ekej − ekij − ikej + ikij
= −2δjk.
So for a fixed vj , define a linear map ϕ ∶ V → End(Λ∗V ) by ϕ(v) = ϕj(v). We can uniquely lift this to
ϕ ∶ Cliff(V )→ End(ΛV ) by the universal property (Lemma 3.1). Hence Λ∗V is a Cliff(V, q) module.
We now wish to extend the properties of Clifford algebras to vector bundles, in order to apply the results to
manifolds and their tangent and cotangent bundles.
Recall that by Example 3.3, for any A ∈ SOn we have a unique extension to Cliff(Rn). Hence we have a
representation ρn ∶ SOn → Aut(Cliff(Rn)) defined by lifting the transformation to the Clifford algebra. This
representation will allow us to construct bundles of Clifford algebras.
Definition 3.8. The Clifford bundle of the oriented Riemannian vector bundle E over the manifold M is the
bundle
Cliff(E) = PSO(S) ×ρn Cliff(Rn).
Note that we could also have defined Cliff(E) as the quotient bundle
Cliff(E) = T (E)/I(E),
where T (E) = ⊕n≥0
E⊗n and I(E) is the bundle of ideals whose fibre at x ∈ M is the two-sided ideal I(Ex) in
T (Ex) generated by elements v ⊗ v + ∥v∥2 for v ∈ Ex. From this definition we see that Cliff(E) is a bundle of
Clifford algebras over M . By defining fibrewise multiplication for sections u, v ∶M → Cliff(E) as
(u ⋅ v)x = u(x)v(x)
11
we may give an algebra structure to the space of sections, Γ(E). Note that many notions for Clifford algebras
carry through to the theory of Clifford bundles. For example, if we consider the bundle automorphism
α ∶ Cliff(E)→ Cliff(E)
defined by extending the map α in Example 3.1, then the eigenbundles for ±1 gives a corresponding decompo-
sition
Cliff(E) = Cliff0(E)⊕Cliff1(E).
3.3 Spinor Bundles
Definition 3.9. The spin groups are defined, for n ∈ N, as
Spin(n) = {v1⋯vk ∶ vi ∈ Rn, ∥vi∥ = 1 for 1 ≤ i ≤ k and k is even}.
Note that for a unit vector v ∈ Rn we have v ⋅ v = −1 and v ⋅ (−v) = 1 we see that {1,−1} ⊂ Spin(n). We also see
that by definition Spin(n) ⊆ Cliff0(Rn).
For n ≥ 3 there exists a universal covering homomorphism ξ0 ∶ Spin(n) → SO(n). The kernel of this
homomorphism is {−1,1}.
Definition 3.10. Suppose n ≥ 3. Then a spin structure on E is a principal Spin(n) bundle PSpin(E) together
with a 2-sheeted covering
ξ ∶ PSpin(E)→ PSO(E)
such that ξ(pg) = ξ(p)ξ0(g) for all p ∈ PSpin(E) and all g ∈ Spin(n).
When n = 2, the definition remains the same except that ξ0 ∶ SO2 → SO2 is now the connected double cover.
When n = 1, PSO(E) ≅M and a spin structure is a double cover of M .
Definition 3.11. Let E →M be an oriented Riemannian vector bundle with a spin structure ξ ∶ PSpin(E) →
PSO(E). A real spinor bundle of E is a bundle of the form
S(E) = PSpin(E) ×µ L,
where L is a left module for Cliff(Rn) and µ ∶ Spinn → SO(L) is the representation given by left multiplication
by elements of Spinn ⊂ Cliff0(Rn).
Likewise, a complex spinor bundle of E is a bundle of the form
SC(E) = PSpin(E) ×µ LC,
where LC is a complex left module for Cliff(Rn)⊗C. If L (or LC) is Z2-graded, we say the corresponding spinor
bundle is Z2-graded.
Example 3.12. Consider Cliff(Rn) as a module acting on itself by left multiplication µ. This gives a ’principal
Cliff(Rn) bundle‘ with a free right action α ∶ Cliff(Rn) × S(Cliff(Rn))→ S(Cliff(Rn)).
12
3.4 Connections
Definition 3.13. Let M be a Riemannian manifold and let E be a smooth (R or C) vector bundle over M of
dimension n. A connection on E is a linear map ∇ ∶ C∞(M,E)→ C∞(M,T ∗M ⊗E) such that
∇(fu) = df ⊗ u + f∇u
for all f ∈ C∞(M) and u ∈ C∞(M,E). Given a smooth vector field X ∶ M → TM we thus have a map
∇X ∶ C∞(M,E) → C∞(M,E), called the covariant derivative with respect to X. Note that this covariant
derivative satisfies
∇X(fu) =X(f)u + f∇Xu.
Definition 3.14. Let (v1, . . . , vn) be a local orthonormal basis for Ex and (e1, . . . , em) a basis for TxM . Then
we define the connection coefficients by
∇eivj = Γkijvk.
Definition 3.15. A connection ∇ on a Riemannian bundle E is called Riemannian if it satisfies the inner
product rule
X⟨u,u′⟩ = ⟨∇Xu,u′⟩ + ⟨u,∇Xu′⟩
for all X ∈ C∞(M,TM) and u,u′ ∈ C∞(M,S).
4 Dirac Operators
We give a motivation for the development of Dirac operators as presented in [7, Chapter 2] The study of Dirac
operators begins historically with physicist Paul Dirac, who was searching for a first order differential operator
that was Lorentz invariant and squared to the Klein-Gordon operator,
1
c2∂2
∂t2−
3
∑j=1
∂2
∂x2j+ m
2c2
h2.
Thus he was essentially looking for an operator D satisfying D2 = ◻ (the d’Alembertian) compatible with special
relativity, and guessed the form
D =n
∑k=1
ψk∂
∂xk.
Dirac realised that the ψk must be matrices and satisfy what we now recognise as the generating relations for
a representation of Cliff(Rn). Generalisations of these operators will be the differential operators of interest to
us.
In modern physics we are often working in 3D-spacetime, called Minkowski space, rather than Euclidean
space. Minkowski space is defined as the inner product space (Rn+1, q) where for x = (x0, x1, . . . , xn) and
13
y = (y0, y1, . . . , yn) in Rn we have
q(x, y) = −x0y0 +n
∑k=1
xkyk.
This is the framework used for the study of special relativity. Just as Euclidean space can be generalised
to Riemannian manifolds, Minkowski space can be generalised to an object called a Lorentzian manifold. A
Lorentzian manifold is defined in the same manner as a Riemannian manifold however the metric is no longer
required to be positive definite, but is required to have signature (1, n) where (n + 1) is the dimension of the
manifold. For an in depth discussion on pseudo-Riemannian and Lorentzian geometry, see [8] and [1, Appendix
2].
4.1 Definition
Definition 4.1. Let M be a Riemannian manifold of dimension n and E a smooth complex vector bundle
over M . Write left Clifford multiplication as c ∶ C∞(M,TM ⊗ E) → C∞(M,E) and let ∇ ∶ C∞(M,E) →
C∞(M,T ∗M ⊗ E) be a connection. Let φ ∶ C∞(M,T ∗M ⊗ E) → C∞(M,TM ⊗ E) denote the isomorphism
between vector and covector fields. We define the Dirac operator associated to the bundle E and connection ∇,
D ∶ C∞(M,E)→ C∞(M,E), as D = c ○ φ ○ ∇. Note that this also depends on the representation chosen.
Thus for an orthonormal frame (ek) of (TM)x, with dual frame (ek) of (T ∗M)x we have that
Du(x) = c ○ φ(n
∑k=1
ek ⊗∇eku∣x)
= c(n
∑k=1
ek ⊗∇eku∣x)
=n
∑k=1
ek ⋅ ∇eku∣x.
Example 4.2. Choosing normal coordinates at x ∈ M so that∂
∂xj(x) = ej(x), we may compute the Dirac
14
Laplacian D2 as
D2u(x) =n
∑k=1
ek(x) ⋅ ∇ek(x) (Du)
=n
∑k=1
ek(x) ⋅ ∇ek(x)⎛⎝
n
∑j=1
ej(x) ⋅ ∇ej(x)u⎞⎠
=n
∑k=1
n
∑j=1
[ek(x) ⋅ ej(x) ⋅ ∇ek(x)∇ek(x)u + ek(x)∇ek(x)ej(x) ⋅ ∇ej(x)u]
=n
∑k=1
n
∑j=1
[−gjk ⋅ ∇ek(x)∇ej(x)u + ek(x)∇ek(x)ej(x) ⋅ ∇ej(x)u]
=n
∑k=1
n
∑j=1
[−gjk ⋅ ∇ ∂
∂xk∇ ∂
∂xjulfl + ek(x)∇ek(x)ej(x) ⋅ ∇ ∂
∂xju]
=n
∑k=1
n
∑j=1
[−gjk ⋅ (∇ ∂
∂xj( ∂ul
∂ej(x)fl) + ul∇ ∂
∂xjfl) + ek(x)Γlkjfl ⋅ ∇ ∂
∂xj(ulfl)]
=n
∑k=1
n
∑j=1
[−gjk ⋅ ( ∂2ul
∂xkxjel +
∂ul
∂xj∇ ∂
∂xjvl + ulΓmjlvm) + ek(x)Γmkjvm ⋅ ( ∂u
l
∂xjvl + ul∇ ∂
∂xj)vl)]
=n
∑k=1
n
∑j=1
[−gjk ⋅ ( ∂2ul
∂xkxjvl +
∂ul
∂xjΓmklvm + ek(x)ulΓmjlvm) + Γmkjvm ⋅ ( ∂u
l
∂xjvl + ulΓpjlvp)]. (4.1)
Note that everything in this computation is evaluated at x ∈M . From this computation we see that
σD2(x, ξ) = −n
∑j=1
n
∑k=1
gjkξjξk
= − ∥ξ∥2 .
Thus D2 is a Laplace type operator, as expected.
4.2 Examples
We will now examine some Euclidean examples of Dirac operators. Consider the simple manifold M = Rn with
the standard inner product and let Cliff(Rn) be it’s Clifford algebra. Let W be a complex vector space of
dimension m and ϕ ∶ Cliff(Rn) → EndC(W ) be a representation of Cliff(Rn). Let E =M ×W have the trivial
Levi-Civita connection. We consider the Dirac operator D ∶ C∞(M,E)→ C∞(M,E). Then from Definition 4.1
we have for u ∈ C∞(M,E) and {e1, . . . , en} the standard basis vectors for Rn that
Du =n
∑k=1
ek ⋅ ∇eku
=n
∑k=1
ϕ(ek)∂u
∂xk
=n
∑k=1
ϕk∂u
∂xk
where ϕ(ek) ∶= ϕk ∶W →W are linear maps satisfying
ϕk ○ ϕj + ϕj ○ ϕk = −2δkjIdW . (4.2)
15
So identifying EndC(W,W ) ≅ Mm(C) and applying Equation 4.2 with j = k shows that ϕk ∈ GLm(C) for all
1 ≤ k ≤ n. Note also that applying 4.1 and the fact that gjk = δjk and Γkij = 0 for all i, j, k gives
D2u = −n
∑k=1
∂2u
∂xkIdW .
Example 4.3. Let n = 1, so Cliff(R) = C and let W = C. Then we have a single linear ϕ ∶ C → C satisfying
ϕ ○ ϕ = −1. So we have two choices, ϕ(x) = ±ix. The usual convention is ϕ(x) = −ix for the Dirac operator to
be self adjoint. Thus the Dirac operator on R is
D = 1
i
d
dx.
Example 4.4. Let n = 2 and W = C ⊕C, so that Cliff(R2) = H ≅ C ⊕C, as R-linear spaces (not as algebras).
Note that the decomposition of H into C⊕C corresponds to the Z2 grading Cliff(R2) = Cliff0(R2)⊕Cliff1(R2).
To see this, define α1 ∶ Cliff0(R2) → C and α2 ∶ Cliff1(R2) by α0(x + ye1e2) = x + iy = α2(xe1 + ye2). Define
ϕ0 = ϕ(1) = Id. To find ϕ1 and ϕ2, we note that we need two 2×2 matrices who square to −Id and anticommute.
Hence we recall two of the three Pauli matrices as
ϕ1 =⎛⎜⎝
0 −1
1 0
⎞⎟⎠
and
ϕ2 =⎛⎜⎝
0 i
i 0
⎞⎟⎠.
Note that the third Pauli matrix is generated by the relation
ϕ(e1e2) = ϕ(e1)ϕ(e2)
=⎛⎜⎝
−i 0
0 i
⎞⎟⎠.
Then we have for u ∈ C∞(R2, S) that
Du = ϕ1∂u
∂x1+ ϕ2
∂u
∂x2
=⎛⎜⎝
0 − ∂u∂x1 + i ∂u∂x2
∂u∂x1 + i ∂u∂x2 0
⎞⎟⎠
=⎛⎜⎝
0 −∂u∂z
∂u∂z
0
⎞⎟⎠.
Thus we see that on R2, D is the Cauchy-Riemann operator.
A more interesting example to consider is the Dirac operator on a cylinder.
Example 4.5. Let M = R × S1 be the infinite cylinder with the product metric g = dt2 + dθ2. Let C =
[t1, t2] × S1 ⊂M be the finite cylinder with metric g = g∣C and S = C ×C2. The Levi-Civita connection on C is
16
flat. Note that Cliff(T ∗M,g) = H and we aim to find a representation for this algebra. So we aim to find an R
algebra homomorphism ϕ ∶ H→ EndC(C2). Define ϕ ∶ H→ EndC(C2) by
ϕ(α + iβ + jγ + kδ) = α⎛⎜⎝
1 0
0 1
⎞⎟⎠+ β
⎛⎜⎝
0 −1
1 0
⎞⎟⎠+ γ
⎛⎜⎝
0 i
i 0
⎞⎟⎠+ δ
⎛⎜⎝
i 0
0 −i
⎞⎟⎠.
Note that a basis for Cliff(T ∗M) is {1,dt,dθ,dtdθ} and so we see that we are identifying ϕ(dt) =⎛⎜⎝
0 −1
1 0
⎞⎟⎠
and
ϕ(dθ) =⎛⎜⎝
0 i
i 0
⎞⎟⎠
.
So applying Definition 4.1 we have for a local basis {v1, v2} of C that
Du =2
∑j=1
2
∑l=1
∂
∂xj⋅ ∇ ∂
∂xj(ulvl)
=2
∑j=1
2
∑l=1
∇ ∂
∂xj( ∂u
l
∂xjvl)
=2
∑j=1
2
∑l=1
dxj ( ∂ul
∂xjvl)
=2
∑l=1
(dt(∂ul
∂tvl) + dθ (∂u
l
∂θvl))
=⎛⎜⎝
0 −∂tu2
∂tu1 0
⎞⎟⎠+⎛⎜⎝
0 i∂θu2
i∂θu1 0
⎞⎟⎠
=⎛⎜⎝
0 −∂t + i∂θ∂t + i∂θ 0
⎞⎟⎠
⎛⎜⎝
u1
u2
⎞⎟⎠.
Note that this has the same form as the Dirac operator on R2 in Example 4.2.
5 A Dirac Operator on the Lorentzian Cylinder R × S1
In physical problems, we are often working with a Lorentzian manifold. We will now consider the Dirac operator
on a cylinder with a Lorentzian metric. We will impose APS boundary conditions as found in [2] and solve
the eigenvalue problem for D. The aim of this section is to apply the results of [2] to a simple situation and
compute some traces with the goal of combining this work with that presented by Connes’ in [3], generalising
to the setting of Lorentzian rather than Riemannian manifolds.
5.1 Constructing the Dirac Operator
Let M = R×S1 be the infinite cylinder with the Lorentzian metric g = −dt2+dθ2. Let C = [t1, t2]×S1 ⊂M be the
finite cylinder with metric g = g∣C and S = C ×C2 be a spinor bundle over C. The Levi-Civita connection on C
is flat. Note that since the metric is Lorentzian, D will no longer be an elliptic operator, rather a generalisation
of a hyperbolic operator. We use the following definition from [1].
17
Definition 5.1. Let M be a Lorentzian manifold and let E →M be a real or complex vector bundle. A second
order linear differential operator D ∶ C∞(M,E) → C∞(M,E) is called normally hyperbolic if its principal
symbol is given by the metric,
σD(x, ξ) = − ∥ξ∥2 ⋅ IdEx ,
for all (x, ξ) ∈ T ∗M .
Example 5.2. The computation in Example 4.2 works for a metric of any signature, hence for a Dirac operator
D on a Lorentzian manifold we have that D2 is a normally hyperbolic operator rather than an elliptic operator.
Note also that Cliff(T ∗M,g) = M2(R) and we aim to find a representation for this algebra. So we aim to
find an R-algebra homomorphism ϕ ∶M2(R)→ EndC(C2).
Construct a basis of M2(R) as
⎧⎪⎪⎪⎨⎪⎪⎪⎩Id =
⎛⎜⎝
1 0
0 1
⎞⎟⎠, e1 =
⎛⎜⎝
0 1
1 0
⎞⎟⎠, e2 =
⎛⎜⎝
0 1
−1 0
⎞⎟⎠, e1e2 = e3 =
⎛⎜⎝
−1 0
0 1
⎞⎟⎠
⎫⎪⎪⎪⎬⎪⎪⎪⎭.
Note that this basis satisfies
Id2 = e21 = −e22 = e23 = 1.
So we define ϕ ∶ M2(R) → EndC(C2) by inclusion, which is clearly an R-algebra homomorphism. So we
identify dt = e1 and dθ = e2 as in the previous example.
So applying Definition 4.1 we have for a local basis {v1, v2} of C that
Dψ =2
∑j=1
2
∑l=1
∂
∂xj⋅ ∇ ∂
∂xj(ψlvl)
=2
∑j=1
2
∑l=1
∇ ∂
∂xj(∂ψ
l
∂xjvl)
=2
∑j=1
2
∑l=1
dxj (∂ψl
∂xjvl)
=2
∑j=1
dxj∂ψ
∂xj
=⎛⎜⎝
0 ∂ψ∂t
∂ψ∂t
0
⎞⎟⎠+⎛⎜⎝
0 −∂ψ∂θ
∂ψ∂θ
0
⎞⎟⎠
=⎛⎜⎝
0 ∂ψ∂t
− ∂ψ∂θ
∂ψ∂t
+ ∂ψ∂θ
0
⎞⎟⎠.
5.2 APS Boundary Conditions
As is usual with physical problems, we would like to impose boundary conditions on our operator D. The
boundary conditions we will work with are Atiyah-Patodi-Singer (APS) boundary conditions, as described in
18
[2]. We define the APS spaces as
C∞APS(C,S+) = {ψ ∈ C∞(C,S+) ∶ P≥0(t1)ψ = 0 = P≤0(t2)ψ},
C∞APS(C,S−) = {ψ ∈ C∞(C,S−) ∶ P≤0(t1)ψ = 0 = P≥0(t2)ψ}.
These definitions were adapted from those given in [2, Section 1.3]. Here we have P≥0(t) = χ[0,∞)(∂θ ∣t) and
P≤0(t) = χ(−∞,0](∂θ ∣t), spectral projections onto the non-negative and non-positive eigenspaces for the operator
∂θ respectively. Physically, C∞APS(C,S+) represents particles whose energy is non-positive at time t1 and is
non-negative at time t2. Likewise for C∞APS(C,S−). To see what this means explicitly, first note that we can
decompose any function ψ ∶ C → C in a Fourier basis as
ψ(t, θ) = ∑n∈Z
ψn(t, θ).
A particle has entirely non-positive energy at t1 if ψn(t1, θ) = 0 for all n ≥ 0 and non-negative energy at time t2
if ψn(t2, θ) = 0 for all n ≤ 0. Hence we find that
D =⎛⎜⎝
0 D−
D+ 0
⎞⎟⎠
=⎛⎜⎝
0 ∂t − iA
∂t + iA 0
⎞⎟⎠,
where A = 1
i∂θ, D+ = ∂t + iA and D− = ∂t − iA. Note also that Dom(D+) = C∞
APS(C,S+) and Dom(D−) =
C∞APS(C,S−).
We consider the Dirac operator acting on spinors ψ which take the form ψ =⎛⎜⎝
ψ+
ψ−
⎞⎟⎠
, where ψ+ represents a
particle and ψ− represents an antiparticle.
5.3 Determining the Eigenvalues
As is standard when working with Dirac operators, we first find the eigenvalues of the associated Laplacian.
Compute that
D2 =⎛⎜⎝
D−D+ 0
0 D−D+
⎞⎟⎠
=⎛⎜⎝
∂2t − ∂2θ 0
0 ∂2t − ∂2θ
⎞⎟⎠.
Note that although D+D− and D−D+ have the same expression, they are in fact different operators, since
Dom(D+D−) = {ψ ∈ Dom(D−) ∶D−ψ ∈ Dom(D+)} and Dom(D−D+) = {ψ ∈ Dom(D+) ∶D+ψ ∈ Dom(D−)}.
As mentioned, a solution ψ will be particle-antiparticle pair of the form ψ =⎛⎜⎝
ψ+
ψ−
⎞⎟⎠
. As usual with a linear
PDE, we apply the method of separation of variables. First, suppose ψ+(t, θ) = T (t)Θ(θ). Applying this to the
19
eigenvalue problem we see that
T ′′(t)T (t)
− Θ′′(θ)Θ(θ)
= λ2.
So we have that Θ′′(θ) = −kΘ(θ) and T ′′(t) = (−k+λ2)T (t) for some constant k ∈ C. Since Θ(θ) ∈ S1 we require
that Θ(0) = Θ(2mπ) for all m ∈ Z and so this has solution if and only if√k ∈ Z. Hence for each n ∈ Z we have
the solution Θn(θ) = einθ. Next, we see that for each n ∈ Z solutions Tn(t) will be of the form
Tn(t) = αnei√n2−λ2(t−t1) + βne−i
√n2−λ2(t−t1)
for some αn, βn ∈ C when λ2 ≠ n2 and
Tn(t) = αn(t − t1) + βn
when λ2 = n2. Suppose for now that λ2 ∈ R.
We now apply our APS boundary conditions to determine possible values of λ. The APS boundary conditions
tell us that Tn(t1) = 0 for n ≥ 0 and Tn(t2) = 0 for n ≤ 0. So for n ≥ 0 and n2 ≠ λ2 we have αn = −βn and for
n ≥ 0 and n2 = λ2 we have βn = 0. For n ≤ 0 we have Tn(t2) = 0. This forces αn = −e−2i√n2−λ2(t2−t1)βn when
n2 ≠ λ2 and αn = 0 when n2 = λ2. Hence we have eigenvectors
ψ+n,λ(t, θ) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
(ei√n2−λ2(t−t1) − e−i
√n2−λ2(t−t1)) einθ, if n ≥ 0 and λ2 ≠ n2,
(ei√n2−λ2(t−t1) − e−i
√n2−λ2(t−2t2+t1)) einθ, if n ≤ 0 and λ2 ≠ n2,
0, if λ2 = n2.
We find similarly for antiparticles that
ψ−n,λ(t, θ) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
(ei√n2−λ2(t−t1) − e−i
√n2−λ2(t−t1)) einθ, if n ≤ 0 and λ2 ≠ n2,
(ei√n2−λ2(t−t1) − e−i
√n2−λ2(t−2t2+t1)) einθ, if n ≥ 0 and λ2 ≠ n2,
0, if λ2 = n2.
= ψ+−n,λ(t, θ).
Now we attempt to solve the eigenvalue problem
(∂t + ∂θ)ψ+n,λ = λψ−n,λ.
With n ≥ 0 and n2 ≠ λ2 this leads to
0 = [(i√n2 − λ2 + in − λ) ei
√n2−λ2(t−t1) + (i
√n2 − λ2 − in + λe2i
√n2−λ2(t2−t1))] einθ.
Comparing coefficients and using linear independence we see that
i√n2 − λ2 + in = λ
20
and
i√n2 − λ2 − in = −λe2i
√n2−λ2(t2−t1).
Combining these gives
n +√n2 − λ2
n −√n2 − λ2
= −e2i√n2−λ2(t2−t1). (5.1)
If n2 > λ2 then the left hand side is real, forcing
2√n2 − λ2(t2 − t1) =mπ
for some m ∈ Z. Rearranging we see that the eigenvalues satisfy
λ2 = n2 − m2π2
4(t2 − t1)2.
If n2 < λ2, let√n2 − λ2 = iρn where ρn ∈ R and so
n + iρnn − iρn
= e2ρn(t2−t1).
The left hand side is real and so we must have ρn = 0, a contradiction. A similar analysis for n ≤ 0 gives the
same result. Hence we have eigenvalues
λn,m =
¿ÁÁÀn2 − m2π2
4(t2 − t1)2,
where n,m ∈ Z and n2 > m2π2
4(t2 − t1)2. From this expression we see that finding an analogue of Weyl’s theorem in
this case will be difficult, as the eigenvalues are indexed by two variables.
Next, consider the case λ2 ∉ R. Then we must have λ = α + iβ for some 0 ≠ α,β ∈ R. Note that all of the
analysis up to and including Equation 5.1 still applies.
5.4 Action Principles
Many physical problems require us to find and solve equations of motion. To do this, a common technique is to
develop an action principle, which usually involves finding minima of functions called actions. For an excellent
overview of material regarding action principles in physics, refer to [11].
Example 5.3 (Fermat’s Principle). In optics, Fermat’s principle states that light will take the path of shortest
time. Mathematically, this says that light will minimise the integral
∫t2
t1
ds
v.
Example 5.4 (Hamilton’s Principle). In classical mechanics, Hamilton’s principle states that the actual path
taken by a particle will minimise the action of the Lagrangian, that is it will minimise the integral
S(x, x) = ∫t2
t1L(t, x(t), x(t))dt,
where L = T −U is the Lagrangian, T is the kinetic energy and U is the potential energy.
21
Physicist Richard Feynman attempted to extend this to quantum mechanics with his path integral formali-
sation, however this is often poorly defined and difficult to work with, so another approach is needed.
5.5 Connes’ Spectral Action Principle
In the 1990’s, Connes’ attempted to develop an action principle for compact Riemannian manifolds defined in
terms of a trace of a differential operator, found in [3].
Let H be a Hilbert space, A an algebra of bounded operators on H and D a self-adjoint operator on H.
Consider an operator on H of the form DA =D+A+JAJ∗, where J is an anti-linear isometry. Connes’ spectral
spectral action principle is define in terms of the trae
Trace(χ(D2A
Λ2)) ,
where χ ∶ σ(DA) → R is measurable and satisfies (χ(λ) = 1 for λ ≤ 1, and Λ ≥ 0 is a spectral cutoff. For large
values of Λ this can be computed using a heat kernel expansion, with the coefficients containing information
regarding the geometry.
However spacetime is neither compact nor Riemannian, so more work is required for this to apply to a non-
idealistic physics problem. Compactness can be overcome by multiplying by a compactly supported function,
however it is not so simple to move to the Lorentzian setting. To do this, we would like to be able to define a
trace on such Dirac operators D.
5.6 Existence and Analysis of Trace Functionals
Before being able to determine any useful geometric or physical information from these traces, we must first do
some analysis to determine that the integrals involved converge. In this section we present some candidates for
a trace and some calculations involving them.
A simple trace we would like to compute is
Trace (Mf (1 +D2APS,ε,R)
− s2 ) ,
where we have imposed an energy cutoff R and a lower limit ε > 0 for our operator. To compute this, we will
require some technical results.
Lemma 5.5. Let f ∶ Rn → C be rotationally and reflectionally symmetric. Then for any v ∈ Rn with ∥v∥ = 1
we have that
∫Rnf(x)dx = Vol (Sn−1)∫
∞
0f(rv)rn−1 dr.
Proof. First, recall the form of the Beta function β(x, y) = 2∫π2
0sin2x−1 (θ) cos2y−1 (θ)dθ. Then we compute
22
that
∫Rnf(x)dnx = ∫
2π
0∫
π
0⋯∫
π
0∫
∞
0f(rv)rn−1 sinn−2 (φ1) sinn−3 (φ2) . . . sin (φn−2)dr dφ1 dφ2 . . .dφn−1
= (∫2π
0dφ1)(∫
π
0sinn−2 (φ2)dφ2) . . .(∫
π
0sin (φn−2)dφn−2)∫
∞
0f(rv)rn−1 dr
= 2π (2∫π2
0sinn−2 (φ2)dφ2) . . .(2∫
π2
0sin (φn−2)dφn−2)∫
∞
0f(rv)rn−1 dr
= 2πβ (n − 1
2,1
2)β (n − 2
2,1
2) . . . β (1,
1
2)∫
∞
0f(rv)rn−1 dr
= 2πΓ (n−1
2)Γ ( 1
2)Γ (n−2
2)Γ ( 1
2)⋯Γ(1)Γ ( 1
2)
Γ (n2)Γ (n−1
2)⋯Γ ( 3
2) ∫
∞
0f(rv)rn−1 dr
=2πΓ ( 1
2)n−2
Γ (n2) ∫
∞
0f(rv)rn−1 dr
= 2πn2
Γ (n2) ∫
∞
0f(rv)rn−1 dr
= Vol(Sn−1)∫∞
0f(rv)rn−1 dr.
Lemma 5.6. For any r ∈ Z, define fr = ure−u. Then for k ∈ N we have
∂k
∂ukfr =
k
∑j=0
(−1)j+k Γ (r + 1)Γ (r − j + 1)
⎛⎜⎝
l
j
⎞⎟⎠∂k−l
∂uk−lfr−j .
Proof. We proceed by induction. The case l = 0 is clearly true. For 0 < l < k, we have that
l
∑j=0
(−1)j+l Γ (r + 1)Γ (r − j + 1)
⎛⎜⎝
l
j
⎞⎟⎠∂k−l
∂uk−lfr−j =
l
∑j=0
(−1)j+l Γ (r + 1)Γ (r − j + 1)
⎛⎜⎝
l
j
⎞⎟⎠((r − j) ∂
k−l−1
∂uk−l−1fr−j−1 −
∂k−l−1
∂uk−l−1fr−j)
=l
∑j=0
(−1)j+lΓ (r + 1)Γ (r − j)
⎛⎜⎝
l
j
⎞⎟⎠∂k−l−1
∂uk−l−1fr−j−1
−l
∑j=0
(−1)j+l Γ (r + 1)Γ (r − j + 1)
⎛⎜⎝
l
j
⎞⎟⎠∂k−l−1
∂uk−l−1fr−j .
We now make the change of indices n = j + 1 on the first sum and relabel the second sum. This gives
l
∑j=0
(−1)j+l Γ (r + 1)Γ (r − j + 1)
⎛⎜⎝
l
j
⎞⎟⎠∂k−l
∂uk−lfr−j =
l+1
∑n=1
(−1)n+l+1 Γ (r + 1)Γ (r − n + 1)
⎛⎜⎝
l
n − 1
⎞⎟⎠∂k−l−1
∂uk−l−1fr−n
−l
∑n=0
(−1)n+l Γ (r + 1)Γ (r − n + 1)
⎛⎜⎝
l
n
⎞⎟⎠∂k−l−1
∂uk−l−1fr−n.
23
Figure 1: Domain of Integration
We now combine these into a single sum so that we may use the identity⎛⎜⎝
l + 1
n
⎞⎟⎠=⎛⎜⎝
l
n − 1
⎞⎟⎠+⎛⎜⎝
l
n
⎞⎟⎠
. This gives
l
∑j=0
(−1)j+l Γ (r + 1)Γ (r − j + 1)
⎛⎜⎝
l
j
⎞⎟⎠∂k−l
∂uk−lfr−j =
l
∑n=1
(−1)n+l+1 Γ (r + 1)Γ (r − n + 1)
⎛⎜⎝
⎛⎜⎝
l
n − 1
⎞⎟⎠+⎛⎜⎝
l
n
⎞⎟⎠
⎞⎟⎠∂k−l−1
∂uk−l−1fr−n
+ Γ (r + 1)Γ (r − l)
∂k−l−1
∂uk−l−1fr−l−1 +
∂k−l−1
∂uk−l−1fr
=l+1
∑n=0
(−1)n+l+1 Γ (r + 1)Γ (r − n + 1)
⎛⎜⎝
⎛⎜⎝
l
n − 1
⎞⎟⎠+⎛⎜⎝
l
n
⎞⎟⎠
⎞⎟⎠∂k−l−1
∂uk−l−1fr−n
=l+1
∑n=0
(−1)n+l+1 Γ (r + 1)Γ (r − n + 1)
⎛⎜⎝
l + 1
n
⎞⎟⎠∂k−l−1
∂uk−l−1fr−n.
Note that in the last equality, we have used the fact that for the terms n = 0 and n = l+1 there are only a single
term in the sum.
Example 5.7. We now attempt to compute
Trace (Mf (1 +D2APS,ε,R)
− s2 ) .
We will be computing the trace over the domain of the eigenvalues, so we will integrate over the region displayed
in Figure 1.
Note that the two lines have gradient c and −c respectively, where c = π
2(t2 − t1). Since there are no zero
eigenvalues, we introduce a lower limit ε for λ2 and a spatial cutoff R. The kernel for this operator is
Ks(x, y) = 2[ d+12
](2π)−d+12 f(x)gs(x − y),
24
where g is defined as
gs(ξ) = (1 + ∥ξ∥2d − c2ξ2t )
− s2H (∥ξ∥2d − c
2ξ2t − ε2) .
Let cd = 2[ d+12
](2π)−d+12 , A = {(ξt, ξ1, . . . , ξd) ∈ Rd+1 ∶ ∥ξ∥2d − c2ξ2t > ε2} and B = {(r, p) ∈ R2 ∶ r2 − c2p2 > ε2}. Also
let ξ = (ξt, ξ1, . . . , ξd), x = (xt, x1, . . . , xd) and ∥ξ∥2d =d
∑k=1
ξ2k. Then we compute that
Trace (Mf (1 +D2APS,ε,R)
− s2 ) = ∫Rd+1
Ks(x,x)dx
= cd ∫Rd+1
Mf gs(0)dx
= cd ∫Rd+1
Mf(2π)−d+12 ∫
Rd+1(1 + ∥ξ∥2d − c
2ξ2t )− s2
dξ dx
= 2[ d+12
]
(2π)d+1 ∫Rd+1f(x)dx∫
A(1 + ∥ξ∥2d − c
2ξ2t )− s2
dξ.
We now carefully calculate the rightmost integral, using an analogue of Lemma 5.5, as
∫A(1 + ∥ξ∥2d − c
2ξ2t )− s2
dξ = 1
Γ ( s2) ∫
∞
0us2−1 ∫
Ae−u(1+∥ξ∥
2d−ξ
2t ) dξ du
= 1
Γ ( s2) ∫
∞
0us2−1∬
BVol (Sd−1) rd−1e−u(1+r
2−c2p2) dr dpdu.
We will need a change of coordinates. Let v = r− cp and w = r+ cp, so that v+w = 2r and w−v = 2cp. Calculate
the Jacobian as
J(r, p) = det⎛⎜⎝
∂v∂r
∂v∂p
∂w∂r
∂w∂p
⎞⎟⎠
= det⎛⎜⎝
1 c
−c 1
⎞⎟⎠
= 1 + c2.
Figure 5.7 demonstrates the coordinate transformation given above. Thus we have that
∬B
Vol (Sd−1) rd−1e−u(1+r2−c2p2) dr dp = 1
2 (1 + c2) ∫R
ε∫
w
ε2
w
(1
2(v +w))
d−1
e−uvw dv dw
+ 1
2 (1 + c2) ∫2R
R∫
2R−w
ε2
w
(1
2(v +w))
d−1
e−uvw dv dw
= 1
2d (1 + c2) ∫R
ε∫
w
ε2
w
d−1
∑k=0
⎛⎜⎝
d − 1
k
⎞⎟⎠vkwd−1−ke−uvw dv dw (5.2)
+ 1
2d (1 + c2) ∫2R
R∫
2R−w
ε2
w
d−1
∑k=0
⎛⎜⎝
d − 1
k
⎞⎟⎠vkwd−1−ke−uvw dv dw. (5.3)
We may now note that vke−uvw = (−1
w)k ∂k
∂uke−uvw and apply Lemma 5.6 to further compute the first integral
25
Figure 2: Coordinate transformation applied to the domain of integration
5.2 as (for 2k ≠ d − 1)
∫R
ε∫
w
ε2
w
wd−1−kvke−uvw dv dw = ∫R
ε∫
w
ε2
w
(−1)kwd−1−2k ∂k
∂uke−uvw dv dw
= (−1)k ∂k
∂uk∫
R
εwd−2−2k ( 1
u(e−uε
2
− e−uw2
))dw
= (−1)k ∂k
∂uk( 1
ue−uε
2
)( Rd−1−2k
d − 1 − 2k− εd−1−2k
d − 1 − 2k)
− (−1)k ∂k
∂uk∫
R
εwd−2−2k
e−uw2
udw.
To work with these integrals, we recall that they were inside an integral over u. Then for s large and using
integration by parts we find that
∫∞
0us2−1e−u ∫
R
ε∫
w
ε2
w
wd−1−kvke−uvw dv dw du = ∫∞
0(−1)ku
s2−1e−u
∂k
∂uk⎛⎝
e−uε2
u
⎞⎠
du( Rd−1−2k
d − 1 − 2k− εd−1−2k
d − 1 − 2k)
− ∫∞
0(−1)ku
s2−1e−u
∂k
∂uk∫
R
εwd−2−2k
e−uw2
udw du
=k
∑j=0
cj,k,s ∫∞
0us2−j−2e−ue−uε
2
( Rd−1−2k
d − 1 − 2k− εd−1−2k
d − 1 − 2k)du
−k
∑j=0
cj,k,s ∫R
ε∫
∞
0us2−j−2e−ue−uw
2
wd−2−2k dudw
=k
∑j=0
cj,k,s (1 + ε2)−s2+j+1 ( Rd−1−2k
d − 1 − 2k− εd−1−2k
d − 1 − 2k)
−k
∑j=0
cj,k,s ∫R
ε(1 +w2)−
s2+j+1wd−2−2k dw
where cj,k,s = (−1)j+kΓ ( s
2)
Γ ( s2− j)
⎛⎜⎝
k
j
⎞⎟⎠
. Using integration by parts we see that this is much more computable when
d − 2 − 2k is odd than even.
If 2k = d − 1 then the same calculation applies, however every term of the form ( Rd−1−2k
d − 1 − 2k− εd−1−2k
d − 1 − 2k)
26
becomes ln(Rε), so that
∫∞
0us2−1e−u ∫
R
ε∫
w
ε2
w
wd−1−kvke−uvw dv dw du =k
∑j=0
cj,k,s (1 + ε2)−s2+j+1 ln(R
ε)
−k
∑j=0
cj,k,s ∫R
ε(1 +w2)−
s2+j+1wd−2−2k dw
We compute this last integral as
∫R
ε(1 +w2)−
s2+j+1wd−2−2k dw = ∫
R
ε
1
Γ ( s2− j − 1) ∫
∞
0us2−j−2e−u(1+w
2)wd−2−2k dw du
= 1
Γ ( s2− j − 1) ∫
∞
0us2−j−2e−u(−1)
d−2−2k2
∂d−2−2k
2
∂u2−d−2k
2∫
R
εwP (d−2−2k)e−u(1+w
2) dw du
= 1
Γ ( s2− j − 1) ∫
∞
0
∂d−2−2k
2
∂ud−2−2k
2
(us2−j−1e−u)md,k(u)du
= 1
Γ ( s2− j − 1)
[ d−2−2k2
]
∑p=0
(−1)[d−2−2k
2]+p
Γ ( s2− j − 1)
Γ ( s2− j − 1 + p)
⎛⎜⎝
[d−2−2k2
]
p
⎞⎟⎠Md,k,s,p,
where P is the parity, giving 1 if odd and 0 if even,
md,k(u) =
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
√π2(erf (
√uR) − erf (
√uε)) if d − 2 − 2k is even,
12(e−uR
2
− e−uε2
) if d − 2 − 2k is odd.
and
Md,k,s,p = ∫∞
0us2−j−1−p−1e−umd,k(u)du.
We may now also compute in a similar manner the integral 5.3 as
∫2R
R∫
2R−w
ε2
w
wd−1−kvke−uvw dv dw = (−1)k ∫2R
R∫
2R−w
ε2
w
wd−1−2k∂k
∂uk(e−uvw)dv dw
= (−1)k ∂k
∂uk∫
2R
R
wd−1−2k
uw(e−uε
2
− e−uw(2R−w))dw
= (−1)k ∂k
∂uk⎛⎝
e−uε2
u
⎞⎠((2R)d−1−2k
d − 1 − 2k− Rd−1−2k
d − 1 − 2k)
− (−1)k ∫2R
R
wd−2−k
ue−uw(2R−w) dw
= (−1)k ∂k
∂uk⎛⎝
e−uε2
u
⎞⎠((2R)d−1−2k
d − 1 − 2k− Rd−1−2k
d − 1 − 2k) − Id,k.
Note that the case 2k = d−1 gives rise to a logarithm term, as in the previous integral. For further computation,
27
we again bring back the integral over u and apply integration by parts. This gives
∫∞
0us2−1e−uId,k du = (−1)k ∫
∞
0us2−1e−u
∂k
∂uk∫
2R
R
wd−2−2k
ue−uw(2R−2) dw du
= ∫∞
0
∂k
∂uk(u
s2−1e−u)∫
2R
R
wd−2−2k
ue−uw(2R−w) dw du
=k
∑j=0
(−1)j+kΓ ( s
2)
Γ ( s2− j)
⎛⎜⎝
k
j
⎞⎟⎠∫
∞
0us2−2−je−u(1+w(2R−w)) ∫
2R
Rwd−2−2k dw du
=k
∑j=0
(−1)j+kΓ ( s
2)
s2− j − 1
⎛⎜⎝
k
j
⎞⎟⎠∫
2R
R(1 +w(2R −w))−
s2+1+j wd−2−2k dw.
Combining these calculations together, we see that
∫A(1 + ∥ξ∥2d − c
2ξ2t )− s2
dξ =d−1
∑k=0
⎛⎜⎝
d − 1
k
⎞⎟⎠
k
∑j=0
(−1)j+kΓ ( s
2)
s2− j − 1
⎛⎜⎝
k
j
⎞⎟⎠(1 + ε2)−
s2+j+1 ( Rd−1−2k
d − 1 − 2k− εd−1−2k
d − 1 − 2k)
−d−1
∑k=0
⎛⎜⎝
d − 1
k
⎞⎟⎠
k
∑j=0
(−1)j+k⎛⎜⎝
k
j
⎞⎟⎠
Γ ( s2)
s2− j − 1
∫R
ε(1 +w2)−
s2+j+1wd−2−2k dw
+d−1
∑k=0
⎛⎜⎝
d − 1
k
⎞⎟⎠
k
∑j=0
(−1)j+k⎛⎜⎝
k
j
⎞⎟⎠
Γ ( s2)
s2− j − 1
(1 + ε2)−s2+j+1 ((2R)d−1−2k
d − 1 − 2k− Rd−1−2k
d − 1 − 2k)
−d−1
∑k=0
⎛⎜⎝
d − 1
k
⎞⎟⎠
k
∑j=0
(−1)j+kΓ ( s
2)
s2− j − 1
⎛⎜⎝
k
j
⎞⎟⎠∫
2R
R(1 +w(2R −w))−
s2+j+1wd−2−2k dw
=d−1
∑k=0
k
∑j=0
⎛⎜⎝
d − 1
k
⎞⎟⎠
⎛⎜⎝
k
j
⎞⎟⎠(−1)j+k
Γ ( s2)
s2− j − 1
(1 + ε2)−s2+j+1 ((2R)d−1−2k
d − 1 − 2k− εd−1−2k
d − 1 − 2k)
−d−1
∑k=0
⎛⎜⎝
d − 1
k
⎞⎟⎠
k
∑j=0
(−1)j+k⎛⎜⎝
k
j
⎞⎟⎠
Γ ( s2)
s2− j − 1
∫R
ε(1 +w2)−
s2+j+1wd−2−2k dw
−d−1
∑k=0
⎛⎜⎝
d − 1
k
⎞⎟⎠
k
∑j=0
(−1)j+kΓ ( s
2)
s2− j − 1
⎛⎜⎝
k
j
⎞⎟⎠∫
2R
R(1 +w(2R −w))−
s2+j+1wd−2−2k dw
and so
Trace (Mf (1 +D2)−s2 ) =
2[ d+12
]−dVol (Sd−1)(2π)d+1 (1 + c2) ∫Rd+1
f(x)dx∫A(1 + ∥ξ∥2d − c
2ξ2t )− s2
dξ.
We will now compute this trace for some low dimensional examples from d = 1 to d = 3. First we consider d
odd since the integrals are simpler.
Example 5.8 (Dimension d + 1 = 2). We will consider the expression for I(s) = ∫A(1 + ∥ξ∥2d − c
2ξ2t )− s2
dξ
computed in example 5.7. Note that there is a simple pole at s = 2. We compute the residue at this pole to
determine its behaviour.
Ress=2(I(s)) = ln(2R
ε) − (R − ε) − (2R −R)
= ln(2R
ε) − 2R + ε.
28
Example 5.9 (Dimension d + 1 = 4). We will consider the expression for I(s) = ∫A(1 + ∥ξ∥2d − c
2ξ2t )− s2
dξ
computed in example 5.7. Note that there are simple poles at s = 2, s = 4 and s = 6. We compute the residue
at these poles to determine their behaviour.
For the pole at s = 2, we only need to consider terms with j = 0. Hence we have
Ress=2(I(s)) = −2R2 + ε2
2− ln(2R
ε) + (2R)−1
2− ε
−1
2− (2R)−2 + ε−2.
For the pole at s = 4, we only need to consider terms with j = 1. Thus we have
Ress=4(I(s)) =⎛⎜⎝
2
1
⎞⎟⎠
⎛⎜⎝
1
1
⎞⎟⎠
Γ (2) (1 + ε2)0 ln(Rε) +
⎛⎜⎝
2
2
⎞⎟⎠
⎛⎜⎝
2
1
⎞⎟⎠(−1)Γ (2) (1 + ε2)0 ((2R)−2
−2− ε
−2
−2)
−⎛⎜⎝
2
1
⎞⎟⎠
⎛⎜⎝
1
1
⎞⎟⎠
Γ (2) (ln(Rε) + ln(2R
R)) + 2
⎛⎜⎝
2
2
⎞⎟⎠
⎛⎜⎝
2
1
⎞⎟⎠
Γ (2)∫R
εw−3 dw
= 4 ln(Rε) − 4 ln(R
ε) − 4 ln (2) + 4 (2R−2 − ε−2) − (R−4 − ε−4)
= −4 ln (2) + 8R−2 − 2R−4 − 4ε−2 + 2ε−4.
For the pole at s = 6, we only need to consider terms with j = 2. We compute this as
Ress=6(I(s)) =⎛⎜⎝
2
2
⎞⎟⎠
⎛⎜⎝
2
2
⎞⎟⎠(−1)2+2Γ (4)((2R)−1
−1− ε
−1
−1− ∫
R
εw−2 dw − ∫
2R
Rw−2 dw)
= 6 (−(2R)−1 + ε−1 − ε−1 + (2R)−1)
= 0.
We now compute an odd example with d + 1 = 3.
Example 5.10 (Dimension d + 1 = 3). We will consider the expression for I(s) = ∫A(1 + ∥ξ∥2d − c
2ξ2t )− s2
dξ
computed in example 5.7. Note that there is a simple pole at s = 2 and at s = 4. We compute the residue at
these poles to determine their behaviour.
For s = 2, we only need to consider terms with j = 0. Calculate the residue as
Ress=2(I(s)) =⎛⎜⎝
1
1
⎞⎟⎠
⎛⎜⎝
1
0
⎞⎟⎠
Γ (1) [ε−1 − (2R)−1 − ∫R
εw−2 dw − ∫
2R
Rw−2 dw]
+⎛⎜⎝
1
0
⎞⎟⎠
⎛⎜⎝
0
0
⎞⎟⎠[2R − ε − ∫
R
εw dw − ∫
2R
Rw dw]
= ε−1 − (2R)−1 − ε−1 + (2R)−1 + 2R − ε − 1
2(2R)2 + 1
2ε2
= 2R − ε − 1
2(2R)2 + 1
2ε2.
29
For s = 4, we need only consider terms with j = 1. This residue is calculated as
Ress=4(I(s)) = −⎛⎜⎝
2
2
⎞⎟⎠
⎛⎜⎝
2
1
⎞⎟⎠[ε
−2 − (2R)−2
2− ∫
R
εw−3 dw − ∫
2R
Rw−3 dw]
+⎛⎜⎝
2
1
⎞⎟⎠
⎛⎜⎝
1
1
⎞⎟⎠[ln(2R
ε) − ∫
R
εw−1 dw − ∫
2R
Rw−1 dw]
= 0.
It should be noted that in all of the above examples, taking the residue made the integral computations
significantly easier, removing the distinction between even and odd. We would also like to take the limit as
R → ∞, which is not expected to commute with the operation of taking a residue, hence these results can be
expected to have lost information about the problem.
Example 5.11. Another possible candidate for our trace is Trace (MfD2 (1 +D∗D)−
s2 ). Let ξ = (ξt, ξ1, . . . , ξd),
x = (xt, x1, . . . , xd) and y = (yt, y1, . . . , yd) then compute, using analogous results to those found in [9, Corollary
14], and Example 5.7 that
Trace (MfD2(1 +D∗D)−
s2 ) = 2[ d+1
2](2π)−(d+1) ∫
Rd+1∫Af(x)
⎛⎝− ∂2
∂x2t+
d
∑j=1
∂2
∂x2j
⎞⎠[e−i(x−y)⋅ξ (1 + ∥ξ∥2)
− s2 ]x=y
dξ dx
= cd ∫Rd+1∫A(f(x) (ξ2t − ∥ξ∥2d) (1 + ξ2t + ∥ξ∥2d)
− s2e−i(x−y)⋅ξ)
x=ydξ dx
= cd ∫A((1 + ξ2t + ∥ξ∥2d)
− s2 (2ξ2t + 1) − (1 + ξ2t + ∥ξ∥2d)− s2+1)dξ∫
Rd+1f(x)dx,
where cd = 2[ d+12
](2π)−(d+1) and A = {(ξt, ξ1, . . . , ξd) ∈ Rd+1 ∶ ∥ξ∥2d − c2ξ2t > ε2}. We will calculate these two
integrals separately. Let B = {(r, p) ∈ R2 ∶ r2 − c2p2 > ε2} and compute that
∫A(1 + ξ2t + ∥ξ∥2d)
− s2 (2ξ2t + 1)dξ = 1
Γ ( s2) ∫
∞
0∫Aus2−1 (2ξ2t + 1) e−u(1+ξ
2t+∥ξ∥
2d) dξ du
= Vol (Sd−1)∫∞
0∬
Bus2−1 (2p2 + 1) rd−1e−u(1+r
2+p2) dpdr du.
We will evaluate the integrals separately and recombine. First, consider the r integral and note that the region
of integration is D = (−∞,−√ε2 + c2p2) ∪ (
√ε2 + c2p2,∞). Then we have that
Id = ∫Drd−1e−ur
2
dr
= ∫D−r
d−2
2u
d
dr(e−ur
2
)dr
=⎛⎝−r
d−2e−ur2
2u
⎞⎠∣∂D + d − 2
2u∫Drd−2e−ur
2
dr
= e−u(ε2+c2p2) ((
√ε2 + c2p2)
d−2− (−
√ε2 + c2p2)
d−2) + d − 2
2uId−3.
30
So we calculate I2 as
I2 = ∫Dre−ur
2
dr
= −1
2(e−ur
2
) ∣∂D
= 0
and I1 as
I1 = ∫D
e−ur2
dr
=√π
uerfc (
√ε2 + c2p2) .
This integral appears quite difficult to evaluate and so we will not attempt to continue the computation.
6 Conclusion
In this paper we developed the theory of differential operators on manifolds and clifford algebras in order to
define and work with Dirac operators, of particular interest in physics. The main result of this report is that a
trace can be computed for the Dirac operator on a Lorentzian cylinder. Further work could include analysing
the trace in the limit as R →∞ before taking the residues and continuing with the integral computations found
in Example 5.11. A further extension would be to consider a compact manifold with more interesting topology,
such as a torus.
31
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