Download Canonical Quantum Gravity as a Gauge Theory with Constraints

Canonical Quantum Gravity as a
Gauge Theory with Constraints
Kuan-Jung (John) Lai
Department of Physics, Theoretical Physics Group
Imperial College London
This dissertation is submitted in partial fulfillment of the
requirements for the award of the degree of
Master of Science
September 2016
Acknowledgements
The author would like to acknowledge the guidance provided by Professor Joao
Magueijo throughout the production of this dissertation.
Additional thanks are due to Jake Gordon, Jimmy Kim, and Chang Li, for their
input, feedback, and company, but also for their assistance in regards to the “Tokyo
Disappointment,” which occurred as the written form of this dissertation was in its
initial stages.
A final word of gratitude must also be expressed to Hsien-Tsang Lai, without
whom none of this would have been possible.
Table of contents
Notation
vii
1 Introduction
1.1 Formulation of a General Relativistic Theory . . . . . . . . . . . . .
1.1.1 Diffeomorphism Invariance . . . . . . . . . . . . . . . . . . .
1.1.2 Background Independence . . . . . . . . . . . . . . . . . . .
2 Gauge Theory
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Ehresmann Connection . . . . . . . . . . . . . .
2.2.1 Local Trivializations of ω . . . . . . . . . . . .
2.3 Horizontal Lifts . . . . . . . . . . . . . . . . . . . . .
2.4 Derivatives on Associated Fibre Bundles . . . . . . .
2.4.1 Example: Frame Bundles . . . . . . . . . . . .
2.5 Curvature . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 General Relativity on Frame Bundles: Preliminaries
3 Gravity
3.1 Lagrangian Gravitation (I): Einstein-Hilbert
3.2 Hamiltonian Gravitation: ADM formalism .
3.2.1 ADM Formalism (I): Statics . . . . .
3.2.2 ADM Formalism (II): Dynamics . . .
3.2.3 ADM Formalism (III): Constraints . .
4 Quantum Gravity
4.1 Lagrangian Gravitation (II): Frame Fields .
4.1.1 Self-Duality, Complexification . . .
4.2 Ashtekar’s Variables . . . . . . . . . . . . .
4.2.1 Quantum Constraints . . . . . . .
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vi
Table of contents
5 Loop Quantum Gravity
5.1 The Base Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The Gauge Invariant Space . . . . . . . . . . . . . . . . . . . . . . . .
5.3 The Diffeomorphism-Invariant Space . . . . . . . . . . . . . . . . .
63
66
69
71
6 Conclusions
75
References
79
Notation
Indices
A, B, C, D
matrix group indices
a, b, c, d
abstract indices
i, j, k
3D spatial indices
I, J, K, L
Lie algebra (matrix) indices, or Minkowski indices
µ, ν, σ, ρ
coordinate indices
s, t, u, v
Lie algebra basis indices
Math
[·, ·]
algebra commutator
d
exterior derivative
δab
Kronecker delta
det
determinant function
δ(x)
Dirac delta distribution
(3)
coordinate 3-form
e
√
qe
volume form
ε
Levi-Civita symbol
η = diag(−1, 1, 1, 1)
Minkowski metric
gl(n, V )
general linear algebra (real or complex)
Notation
viii
GL(n, V )
general linear group (real or complex)
L(G)
Lie algebra of G
£
Lie derivative
∇
covariant derivative
Ω
curvature 2-form
ω
connection 1-form
⊕
direct sum of spaces
⊗
tensor product
σi
Pauli matrices
{·, ·}
Poisson brackets
Ric
Ricci tensor
R
Ricci scalar
R
Riemann curvature tensor (see equation 2.31)
sl(n, V )
special linear algebra (real or complex)
SL(n, V )
special linear group (real or complex)
so(n)
special orthogonal algebra
SO(n)
special orthogonal group
⋆
Hodge star
su(n)
special unitary algebra
SU (n)
special unitary group
TM
Tangent space of M
T ∗M
Cotangent space of M
T (p,q) M
(p, q)-tensor space of M
Tr
trace function
Notation
∧
ix
exterior product
Units
c=1
speed of light
~=1
Planck’s constant
8πG = 1
Newton’s constant
Chapter 1
Introduction
There presently exist two very successful theories of nature, from which the entirely of observed physical phenomena is thought to be derived. One of these, the
Standard Model of particle physics, lords over the domain of subatomic physics;
its subjects are the fundamental constituents of matter: leptons, quarks and force
bosons. Far down the other extremity, in the realm of stars and galaxies, the general
theory of relativity reigns as chief sovereign, dictating the large-scale distribution
of matter and the structure of the cosmos.
Both these theories are very well-tested experimentally in the regions where
they hold sway. With the additional features of a cosmological constant and neutrino mass, they have withstood all direct challenges posed by experimental and
observational tests thus far.
But even so, the two theories seem to be in direct conflict conceptually: general
relativity produces a dynamic, malleable geometry wherein physical quantities
are well-defined at any given instant, while the world-view espoused by quantum
mechanics postulates a rigid background spacetime, and contains non-commuting
observables that cannot be simultaneously determined to arbitrary precision, even
in principle.
In the regimes currently accessible to us, the two sets of laws seem to adopt
a policy of “peaceful coexistence” where their borders overlap. For instance in
the derivation of Hawking radiation, the black hole geometry is treated as a fixed
background, and any back-reaction to the gravitational field could be ignored on
grounds of being “small”. Similarly, the laws of quantum statistics may determine
Introduction
2
the internal composition of dense stars, but their motions through the cosmos, the
choreography of the interstellar dance, is given by orbital mechanics.
We cannot, however, expect this harmony to continue deep into the very high
energy scales, such as in the Planck epoch of cosmology. We will have to resolve
this issue sooner or later if we wish to obtain a complete description of all physical
phenomena, united under one theory. We will need a theory of gravitation that is
compatible with quantum mechanics: a quantum theory of gravity.
The approach to this problem that we will present, termed canonical quantum
gravity, is distinguished from most other approaches to quantum gravity in that
we treat general relativity on the same footing as quantum mechanics− namely,
as two separate paradigms of physical theories sharing a common ancestor in
Newtonian mechanics. In this context, the gravitational field is not just “one more
field” to be quantized under the framework of QFT.
Commonly, what the latter means is to write the spacetime metric g in the form
g =η+h
(1.1)
for “small” perturbations h around a simple background metric η, a known solution
to the Einstein equations. This expression would then be plugged into the EinsteinHilbert action, and terms polynomial in h and its derivatives would emerge under
the integral sign. These are to be interpreted as the Lagrangian of a spin-2 boson,
the graviton, with self-interacting terms. However, it is well-known that such methods lead to non-renormalizable theories that diverge badly in the ultraviolet regime
[27]. To remedy this, a plethora of additional structure is introduced: compactified
extra dimensions, supersymmetric partners, particles as strings...
That such treatments are doomed to failure (or at least, plagued with mathematical complexities,) should come as no surprise in our view, like taxonomizing the
banana as a tropical breed of apple. This is because such treatments disregard the
truly revolutionary character of GR: that the gravitational field is in fact the same
entity as spacetime itself ; it is the ocean and everything contained in its depths,
and not merely the waves that ripple across its surface.
3
A necessary consequence of treating GR with full respect, then, is the fact that
our theory must be non-perturbative. That is, we do not, and indeed cannot,
posit any prior geometry η around which we base our theory. This will require a
conceptual overhaul of our formalism, but we will be rewarded for our efforts with
the following “good” features [20]:
1. four spacetime dimensions, and no need for more,
2. no ultraviolet divergence,
3. no need for supersymmetry, and
4. manifest background independence.
One of the consequences of canonical quantum gravity, which we will not have
time to cover, is the extraordinary result that spacetime has a granular, or discrete,
structure at very short length scales (≈ 10−35 metres). In this way, we resolve the
problem of ultraviolet divergences by depriving them of the space to exist, through
a cutoff scale imposed by nature itself. The first and third points are also extremely
attractive from a reductionist point of view, especially in light of the lack of evidence for supersymmetry in the LHC.
Also of note is the fact that our program does not aim directly towards a final
theory of “everything.” Unification with the other forces of nature is something
left to another day and another theory. We focus solely on the limited objective of
quantizing gravity.
The purpose of this publication is to present a straight path to the doorstep of
canonical quantum gravity, by developing the prerequisite tools along the road in
a simple but thorough manner. To keep our narrative focused, we will always set
the matter content of spacetime, Tab , and the cosmological constant, Λ, to zero.
We will begin with a Hamiltonian formulation of GR, obtained through purely
classical considerations, and continue in the tradition of Dirac and postulate that
our canonical variables, given by some pair (q, p), promote to quantum operators
satisfying a generalization of the identity
[ q̂ , p̂ ] = i~
(1.2)
Introduction
4
Our first candidate for a choice of q would perhaps most logically be the “spatial
part” of the spacetime metric g. With some additional conditions on spacetime,
this can always be unambiguously defined. This choice, however, leads to some
very formidable non-polynomial terms which impede quantization, forcing us to
go down a different path.
Using instead the “square root” of the metric, a quartet of (0, 1)-tensors eI such
that
X
ηIJ eIa eJb = gab
(1.3)
I,J
will do the trick for the role of the conjugate momentum p. The role of q will instead be played by a generalised version of the Christoffel symbols, called the spin
connection.
Thusly, our theory will be formulated most naturally in terms of a gauge theory,
a class of field theories that are in some ways generalizations of Maxwell’s electrodynamics. Taking the place of the matter fields in this gauge theory will be the
“field of frames” eI , or, a choice of four arrows at each spacetime point representing
perpendicular directions. The role of the gauge potential will be played by the spin
connection, and the corresponding gauge symmetry will be the equivalence of
frames eI that are related by a Lorentz transformation.
The precise mathematical formulation of gauge theories already exists, and is
described by the theory of principal bundles. This motivates our summary of the
theory of principal bundles and Ehresmann connections in chapter 2. Since this is
a mathematically terse subject that needs to be covered, we get it out of the way
first, so that we may talk freely about physics afterwards without being impeded by
considerations of rigour.
In our third chapter, we will briefly summarize the (classical) canonical formulation of general relativity given by the ADM formalism, which we will take as the
starting point of our incursion into quantum gravity. The ADM formalism presents
GR as a totally constrained Hamiltonian system, and leads to the vanishing of
“time” in the physical picture. This timelessness is a general feature of canonical
formulations of GR, and will pop up at various points, though it will not present any
problems to us, at the level of our coverage. We will also briefly discuss the general
1.1 Formulation of a General Relativistic Theory
5
theory of constrained Hamiltonian systems in that chapter, using Yang-Mills theory
as an example.
After that, we introduce the formalism that allows us to write the GR action
in terms of the frame field, as a gauge theory. The canonical coordinates that
result from this formulation are called the Ashtekar variables, and they allow us
to quantize the theory with a minimal amount of effort. We will also give a brief algebraic account of the Ashtekar variables, particularly how they produce a Lorentz
algebra. After that, we will see how to recover the statements of diffeomorphismand gauge-invariance from the classical constraints, as we must.
In the fifth chapter, we finally set the stage for a quantum theory of gravity.
Using the tools developed thus far, we will construct a manifestly diffeomorphismand gauge-invariant inner product space, which will serve as the space of physcally
permissible states in Loop Quantum Gravity.
In the presentation that follows, we will assume that the reader has some familiarity with the language of tangent bundles in differential geometry, on about the
level of volume one of Spivak’s book, [24]. Apart from that, a level of knowledge of
physics at about the level of the QFFF MSc. course at Imperial College London will
also be assumed.
1.1
Formulation of a General Relativistic Theory
We will begin our voyage with a minor prelude, a qualitative discussion of some
features that distinguish GR from most other theories of nature, which we would
like to see manifest in our quantum theory. These are diffeomorphism invariance
and background independence.
1.1.1
Diffeomorphism Invariance
Perhaps the most distinctive feature of general relativity is the property of diffeomorphism invariance. It is also a very easily-misunderstood principle, because it is
not well-expressed in purely mathematical language, and is often confused as a
statement on coordinate systems. As such, it would seem a good idea to present a
Introduction
6
detailed discussion of this symmetry before we begin.
Suppose we have a robot, H, who has attained such a mastery of chess that
it can play a perfect game on any chessboard we can give it: three-dimensional
boards, toroidal boards, whatever. Given a board, H maps a configuration of pieces
Ψ0 uniquely to another configuration Ψ1 = H[Ψ0 ], and another, Ψ2 = H ◦ H[Ψ0 ],
and so on, ad infinitum, one move at a time. Suppose we give H a chessboard
without boundary, Z2 , say, and an initial configuration Ψ0 of chess pieces. Or, we
can give it the same board Z2 and the same initial configuration, but with all the
pieces moved one square to the right, R[Ψ0 ]. Since H has the perfect strategy, and
since the squares of the board have no intrinsic character, H would end up playing
the same game. In other words,
Hn ◦ R[Ψ0 ] = R ◦ Hn [Ψ0 ]
(1.4)
We could even warp the board so that there is a smooth kink bending it slightly
to the left, or make the squares progressively smaller in one direction, shrinking
as 2−n as we go upwards the board. None of these specifics matter to the robot, so
long as we tell it how to modify how each of the chess pieces move, say by telling
rooks to turn slightly to the right as they pass the kink, or to scale their moves in
proportion to the size of the squares. The robot will still play the exact same game,
evolving analogously to 1.4.
However, if we change the topology of the board, say by removing a square, or
by wrapping the board into a cylinder, then the robot will have to revise its strategy,
because some previously legal moves will have become illegal, and vice versa. So
then we no longer have any relation analogous to 1.4. This is all to say that the
specific points of the board are of no intrinsic importance to H’s strategy− only the
relative positioning of the pieces and the set of legal moves have any real substance.
The structure of the board only enters the strategic considerations by determining, say, whether or not pawns can return to their starting squares (only possible
on a toroidal board), or if there are squares inaccessable to a bishop, (there will be
unless the edges of the board are “twisted” and identified,) and how far a rook can
move in one direction before running into a wall, and so on.
1.1 Formulation of a General Relativistic Theory
7
Fig. 1.1 What’s in a point? That which we call chess, with any other choice of
immersion, would still be the same game.
In the continuum limit, we upgrade the board to a manifold M , R to a diffeomorphism φ, and the evolution given by H is replaced with the Hamiltonian, the
generator of time evolutions. To say that our theory is diffeomorphism-invariant,
then, is to say that 1.4 holds in the continuum limit− that the action of diffeomorphisms commutes with time evolution. Meanwhile, the transformation law on
vectors
v → φ∗ v
⇐⇒
v a → ∂b φa v b
(1.5)
corresponds to the infinitesimal, pointwise, direction-wise modification to each of
the chess pieces’ movements.
It must be emphasized that the character of our diffeomorphism φ is distinct
from a “mere” coordinate transformation. The diffeomorphism φ : M → M maps
points on the manifold M to other points, and corresponds to an active motion
on the fields on M . To say that the fields φ∗ ω produce the same physics as ω is the
statement that ω is diffeomorphism invariant.
On the other hand, a coordinate transformation is a passive diffeomorphic
transformation y ◦ x−1 : Rn → Rn , and corresponds to a change in us, changing
our description of the same object. That our laws of physics are invariant under
coordinate transforms means that there “really is” some field on M , ω ∈ T ∗ M , say,
whom the coordinate charts are talking about in a consistent manner.
In light of equation 1.5, it is quite easy to see the cause for confusion. After
all, the distinction between the two “invariances” would vanish by simply setting
(∂y µ/∂xν ) = (∂φa/∂xb ). However, to do so would be a confusion of the roles of the
Introduction
8
abstract indices a, b, c, d and the coordinate indices µ, ν, σ, ρ, since (∂y µ/∂xν ) is not
a “true” geometric object, and cannot be expressed with abstract indices. It also
ignores the distinction of the active/passive roles of φ and y ◦ x−1 . For a more
in-depth discussion of the abstract index notation, see section 2.4 of [28].
1.1.2
Background Independence
Background independence is a perhaps less subtle concept, but it is no less important. It is a central feature of our approach to quantum gravity, which distinguishes
it from perturbative methods. Consider the Klein-Gordon Lagrangian density in
Minkowski space, written in manifestly coordinate-independent notation:
L(φ, dφ) =
1
η(dφ, dφ) − m2 φ2
2
(1.6)
or the Maxwell equation, written similarly:
d⋆ F = 4π ⋆ j
(1.7)
In these equations, there exists a privileged (0, 2)-tensor η (whose presence is implied in the second equation through the Hodge dual operator). η is privileged
in the fact that it is a tensor field that is “exempt” from the selective process of
physical evolution− that is, it is not the solution to some field equation. η possesses
a place in the equations only by virtue of its “noble birth,” through the existence of
a prior geometry.
General relativity demonstrates its truly egalitarian character by stripping the
Minkowski metric of its special status. It then institutes a field equation, the Einstein field equations, which qualifies general symmetric, Lorentz-signature (0, 2)
tensors for the position previously held by η.
Henceforth, the tensor field η, on M , cannot enter any physical equation except
as a particular solution to the EFEs. This is the property of background independence. This property does not exclusively discriminate against the Minkowski
metric, but any prior choice of metric: the anti-de Sitter metric, the Schwarzschild
metric, etc. This statement must hold true even in the limit “at infinity.” That is to
say that we cannot a priori postulate that the spacetime metric approaches some η
at spacelike infinity, or similar.
Chapter 2
Gauge Theory
We begin by summarizing some fundamental results from the theory of principal
bundles in differential geometry. Gauge theories are expressed most naturally in
the mathematical arena of principal bundles. But this fact may be occluded by
the dry and abstract mathematical definitions involved in the construction of that
arena. To remedy this, we will pepper our presentation of the subject with many
physically-relevant examples and constructions.
The correspondence between the mathematical subjects we cover in this chapter and their physics counterparts is summarized thusly:
Connection 1-forms (2.2)
Horizontal lifts (2.3)
Covariant derivative (2.4)
Curvature 2-forms (2.5)
→
→
→
→
Christoffel symbols, gauge fields
Parallel transport
Gauge-covariant derivatives
Riemann curvature, Field strength tensor
Our treatment of the subject will demonstrate a strong preference for brevity
over rigour, omitting the mathematical details of uniqueness, existence, etc. Those
results may be found in [12, 15, 24], among others.
2.1
Preliminaries and Definitions
Definition 2.1.1 Fibre bundles: A differentiable fibre bundle is a collection (E, π, M, F, G)
such that:
1. E, the total space is a differentiable manifold,
2. M , the base space is also a differentiable manifold,
Gauge Theory
10
3. π, the projection is a surjection π : E → M ,
4. F , the typical fibre satisfies π −1 (p) := Fp ∼
= F for each p ∈ M ,
5. G, the structure group, which we will always take to be a finite-dimensional
Lie group, acts on F on the left,
6. E is locally trivial, that is, there exists an open covering {Ui } of M and an
associated set of diffeomorphisms {φi } with φi : Ui × F → π −1 (Ui ), satisfying
π ◦ φi (p, f ) = p for each f ∈ F ,
7. for Ui ∩ Uj ̸= ∅, we have φj (p, f ) = φi (p, tij (p)f ), where tij := φ−1
i,p ◦ φj,p for
φi,p := φi (p, ·) : F → Fp , and {tij (p) |φi , φj } = G. In other words, “changing
coordinates” is governed by G.
Topologically speaking, we say that E is a trivial bundle when E ∼
=M ×F
As an example, the familiar tangent bundle T Rn from multivariate calculus is a
(trivial) fibre bundle with base space the manifold Rn , and has as its typical fibre
the vector space (Rn , +), and structure group GL(n, R), which defines the familiar
change-of-coordinates on tangents, (p, v)x ∼ (p, D(y ◦ x−1 )v))y .
Our definition of the tij necessitates, straightforwardly:
tii (p) = e
tij (p) = t−1
ji (p)
tij (p)tjk (p) = tik (p)
(2.1a)
(p ∈ Ui ∩ Uj )
(2.1b)
(p ∈ Ui ∩ Uj ∩ Uk )
(2.1c)
Definition 2.1.2 Sections: A section, on Ui ⊂ M , is a smooth σi : Ui → E, such that
π ◦ σ = idUi . We shall denote the set of all sections on Ui as Γ(Ui , F ).
We define σi locally instead of globally because a smooth global section may not
exist. (Since F may not be a vector space, a “zero vector field” may not exist.)
We will often call a section a vector field, even if the fibres F are not vector spaces,
because the two terms are equivalent on tangent bundles.
2.2 The Ehresmann Connection
11
In what follows, we shall deal exclusively with two special cases of fibre bundles,
endowed with some additional structure. One of these cases will have F ∼
= G, and
will be what is termed a principal bundle, the natural setting for gauge theories.
The other case is defined alongside a (separate) principal bundle, and will have
an F endowed with vector space structure, and is known as an associated bundle,
which may be used to characterize matter fields in Yang-Mills theories.
Definition 2.1.3 Principal bundles: A principal fibre bundle, or just a principal
bundle, P (M, G) is a fibre bundle with typical fibre equal to its structure group G,
such that G acts freely (that is, without fixed point) on P on the right: (u, g) ∈ P × G
is mapped to ug := Rg u ∈ P , with ug = u ⇐⇒ g = e
In the context of trivializations, then, this requires that φi (p, gi )a = φi (p, gi a). Often
we may call P simply a G-bundle on M .
We observe that, with this additional structure, a local section σi : Ui → P very
naturally defines a local trivialization on π −1 (Ui ). After all, we can define a map
φi : M × G → P with φi (p, g) = Rg ◦ σi (p). In other words, in this trivialization
we associate σi (p) with (p, e), and multiply on the right by the parametre gi (u) in
φ−1
i (u) = (π(u), gi (u)), to “reach” the other elements on each fibre Gp . This map is
smooth because σi is, and is bijective because we postulated that G act freely on P .
From this we can readily see that the existence of a single smooth section
defined everywhere on M implies that P ∼
= M × G is trivial. Conversely, if P is
trivial, then any global trivialization defines a global section.
2.2
The Ehresmann Connection
Recall that for any differentiable manifold M , we defined Tp M , the set of tangent
vectors at p using an equivalence class of smooth curves γ : [−τ, τ ] → M satisfying
γ(0) = p, and then we stitched the resulting fibres together with local triviality
and structure group axioms, analogous to axioms 6 and 7 in definition 2.1.1. The
principal bundle P (M, G) is also a differentiable manifold, and we can consider
the bundle T P (M, G), a tangent bundle defined on top of a principal bundle.
12
Gauge Theory
Because of the quite abstract definition, it would perhaps be instructive to
picture the principal bundle P as a field of seeding dandelions: in this picture, the
field (i.e., “the ground”) represents M , and we take the (spherical) head of each
dandelion to be isomorphic to G, the individual spores in each head representing
group elements.
Consider a little man whose physical dimensions are comparable to the dandelion spores, living on top of one of the spores. The man may slide down the stalk of
his individual dandelion to reach an unambiguously defined point on the ground
(the projection π). The man may also walk from his spore to any other spore on
the same dandelion by reading off an address (right action of G on Gp ), or he may
walk from his spore to a spore on some other dandelion, in some unprescribed
way, which corresponds to a general path in P . Up to an equivalence, the set of all
paths that the man may traverse, then, characterizes T P .
Fig. 2.1 The action of a vertical vector field is associated with movement along a
single dandelion head, while a horizontal movement moves across the field.
Roughly speaking, the fibres of T P will contain vectors that are in the Lie algebra of G, L(G). These are simply the tangents of a path that stays within a single
fibre/dandelion. However, T P also contains vectors from T M , which result when
we walk between fibres, “horizontally” with respect to the plane of “the ground.”
Intuitively, then, we may well think that T P ∼ T M ⊕ L(G), a direct sum of the two
2.2 The Ehresmann Connection
13
vector spaces. In what follows, we shall elaborate on this relation.
Before we may begin, we must put into precise terms the L(G) structure on T P :
For each u ∈ P we can define a vector field isomorphism # : L(G) → Γ(P, T P )
with
d #
(2.2)
A = Rexp(tA)∗ dt t=0
In other words, for differentiable f : P → R
d
A# f (u) = f (u exp(tA)) dt
t=0
(2.3)
The A# so defined is called the fundamental vector field generated by A. This
isomorphism can also be taken pointwise to parametrize the set of vectors tangent
to Gp at each u, which we shall term to be the vertical subspace of Tu P , Vu P . Note
that A# does not trivialize T P , because the latter is not a principal bundle.
Definition 2.2.1 An (Ehrenfest) connection Γ on a principal bundle is then a choice
of a horizontal subspace of Tu P , labeled Hu P , that depends smoothly on u, satisfying:
1. Tu P = Vu P ⊕ Hu P
2. Every smooth vector field X may be decomposed into smooth vector fields
X = XV + XH such that XV is vertical everywhere and XH is horizontal
everywhere.
3. Ra∗ Hu P = Hua P
Equivalently, and perhaps more commonly (in physics applications,) we see an
alternative definition of a connection with an L(G)-valued 1-form, ω, called the
connection 1-form.
Definition 2.2.2 A linear ω : T P → L(G) is a connection 1-form if it satisfies:
1. ω(A# ) = A
2. Rg∗ ω = Adg−1 ω := g −1 ωg
Because Rg∗ A# = (adg− 1 A)# , the third axiom in the first definition and the second
axiom in the second are required for logical consistency. Intuitively we may think
of ω as dim(L(G))-many 1-forms on T P which may be expanded in a basis on L(G):
Gauge Theory
14
ω = gi ω i , so that ω may still have a “direction” after contraction with a vector.
In this definition, Hu P is defined pointwise in u as the subspace of Tu P that annihilates ω. i.e.,
Hu P = {X ∈ Tu P | ωu (X) = 0}
(2.4)
so that ω is then a projection from Tu P to Vu P ∼
= L(G). Assuming X H ∈ Hu P, from
the second definition we have
Rg∗ ωug (X H ) = g −1 ωu (X H )g = 0
(2.5)
so that ω defines horizontal subspaces that are invariant under the right action, as
in the first definition.
One may be misled to think that the definition of Vu P , uniquely specified by the
definition of P , also uniquely specifies Hu P . After all, we could simply call Hu P the
set of tangents that are orthogonal to Vu P . This would be putting the cart before
the horse, because it is the connection itself that prescribes this orthogonality to
begin with. For orthogonality is not defined without a metric, which, as we will see
later on, is in some sense prescribed “up to a gauge” by ω. Without a connection,
all we have to work with is linear independence to choose our basis vectors, which
leaves some freedom of choice for the horizontal basis.
Returning to our dandelion analogy: right translation (or, the finite movement
generated by vertical vectors) provided our little man an unambiguous way of
moving between spores on a single dandelion. Horizontal movement is not well
defined a priori, but we shall see that the definition of a connection resolves this.
We may then think of a choice of connection in analogy to the “bending” of the
stalks of each dandelion on the field in some direction, as though in the wind. If
our little man wishes to traverse the field purely horizontally with respect to the
ground, then his path must take into consideration the choice of connection/the
swaying of the stalks.
2.2.1
Local Trivializations of ω
Here we shall illustrate how the connection 1-form relates to gauge theories. By
the local triviality of P , we may consider a local trivialization on Ui ⊂ M with σi (p).
2.2 The Ehresmann Connection
15
In the context of gauge theories, the gauge field Ai is then defined as
Ai = σi∗ ω ∈ L(G) ⊗ T ∗ Ui
(2.6)
i
Let us make the connection with Yang-Mills theory. In SU (2) theory, Aµ = Aiµ σ2i
is an su(2)-valued 1-form on R4 . In entry-level QFT, we (implicitly) assumed that
the principal bundle was trivial, so that a single trivialization sufficed to define the
gauge field on all of spacetime. We also postulated that Aµ transform as:
Aµ (p) → U (p)(Aµ (p) + ∂µ )U (p)−1
(2.7)
where U : R4 → SU (2) is called a gauge transformation. This transformation is a
passive one, and is the result of changing between trivializations on P .
In our generalized theory, we seek to reproduce 2.7. Indeed, a series of straightforward calculations show that on Ui ∩ Uj ̸= ∅, we must have,
Aj (p) = t−1
ij (p) (Ai (p) + d) tij (p)
(2.8)
where, as before, the tij (p) ∈ G are transition functions.
As a matter of practicality, though, ω is almost always defined in the direction
opposite to our presentation: by stitching together 1-forms {Ai } on open neighbourhoods {Ui }, and then defining their relation to each other via equation 2.8 on
the overlap. We can thusly construct non-trivial bundle structure in a straightforward way, by “twisting” the gauge forms “by hand” on the overlap of the charts.
An example of this can be seen in 3-dimensional electrostatics, in the case of
the Dirac monopole: there we could define, in polar coordinates
AN (x) = ig(1 − cos θ)dφ
θ ∈ UN := [0, π/2 + ε]
(2.9a)
AS (x) = −ig(1 + cos θ)dφ
θ ∈ US := [π/2 − ε, π]
(2.9b)
where g ∈ R, and ε > 0 so that the charts overlap, and on the overlap:
AN = AS + d(2gφ)
(2.10)
Gauge Theory
16
so that tN S = exp(2igφ) in equation 2.8. Furthermore, periodicity of the φ coordinate requires 2g ∈ Z.
The punchline of this construction is revealed by defining B = dA to be the
magnetic field, and integrating B over a sphere of radius r0 ,
Z
S2
B=
Z
θ>π/2
=
=
Z
dAN +
AN −
S1
Z 2π
Z
S1
Z
θ<π/2
dAS
As
(2.11)
d(2gφ)
0
= 4πg
The first line is true because for small ε, the overlap UN ∩ US is of arbitrarily small
size. The second line uses Stokes’ theorem, and the minus sign occurs because of
the choice of orientation. But this implies that1
Z
r<r0
dB =
Z
S2
B
(2.12)
= 4πg
which is independent of r0 , so that clearly divB = 4πgδ (3) (x), and the configuration
of fields corresponds to a magnetic charge. This rather straightforward example
illustrates the wealth of physical scenarios that may be described elegantly in the
language of principal bundles.
Inverting the definition, we may obtain an ω ∈ L(G) ⊗ T ∗ P , working “from the
bottom up," by stitiching together (as we did just now) an appropriate collection of
Ai on open patches Ui , and using the associated trivializations {Ui , σi , gi } to define:
ω|Ui = gi−1 π ∗ Ai gi + g −1 dgi
(2.13)
this satisfies equation 2.6, because by definition gi ◦ σi = e identically on Ui , and
σ ∗ π ∗ = (π ◦ σ)∗ is the identity map on T ∗ P .
1
B is a 2-form, so divB = dB
2.3 Horizontal Lifts
2.3
17
Horizontal Lifts
Our current situation is quite similar to that in introductory general relativity, where
we did not have any prior means of comparing the vector spaces Tp M and Tq M
for p ̸= q. In the presence of a metric, though, we defined a transportation of a
vector v ∈ Tp M → Tq M that depended on a choice of path γ : [−τ, τ ] → M and the
metric gab , through the Levi-Civita connection. In other words, v ∈ Tp M would be
transported to Xq , the value at q of a vector field X satisfying ∇γ̇ X = 0, and Xp = v.
In a coordinate system:
d
(X µ ◦ γ) + γ̇ σ (Γµ νσ X ν ) ◦ γ = 0
dt
(2.14)
This is the parallel transport equation. By the fundamental theorem of ODEs, this
equation always has a unique, suitably differentiable solution. We would like to
import this technique into the theory of principal bundles, to compare fibres Gp
and Gq . This shall be done in the following:
Definition 2.3.1 Horizontal lift: Let P (M, G) be a principal fibre bundle over M
with group G. Consider a continuously differentiable curve γ : [0, 1] → M with
γ(0) = p. Then a curve γ̃ : [0, 1] → P with π ◦ γ̃ = γ is called a horizontal lift of γ if
its tangent γ̃∗ dtd is horizontal everywhere.
In other words, for a given curve γ, our dandelion-man γ̃ walks along the
“shortest possible path” in P (that is, without any extraneous vertical motion). His
“shadow,” projected on M traces the curve γ, with the same velocity as γ. The
horizontal lift defines a unique transport between the fibres of P , Gp and Gq :
Theorem 2.3.2 Given a γ : [0, 1] → M with γ(0) = π(u) there exists a unique
horizontal lift of γ such that γ̃(0) = u.
To prove this, let us work on a coordinate patch (Ui , σi , gi ). It would be no loss to
write γ̃(t) = vit ait , where vit := σi (γ(t)) is a curve with vi0 = u and ait is a curve in
the structure group G with ai0 = e. Then we write
d
γ̃(t) = v̇it ait + vit ȧit
dt
by the Leibniz rule on T P , and q̇(t) denotes the tangent of q at t.
(2.15)
Gauge Theory
18
(For convenience, and since we are working exclusively with matrix groups, we
may as well take G to be its fundamental representation on GL(n, C) and define
all the relevant products there. In general, however, there is no need for such
heavy-handed methods, and the proof works for finite-dimensional Lie groups
that are not matrix groups, too. [12])
We then impose the horizontal condition:
0 = ω(v̇it )
= ω(Rait ∗ v̇it ) + ω(vit ȧit )
d
ait
= ada−1 ω(v̇it ) + a−1
it
it
dt
!
d
d ∗
+ a−1
= ada−1 σi ω γ∗
ait
it
it
dt t=0
dt
d
= Ai (X)ait + ait
dt
(2.16)
The first line follows by the definition of the connection, the second by linearity,
the third by the definition of the connection again, and the fact that vit ȧit is actually
#
(a−1
it ȧit ) at vit . The fourth line is true because vit = σi ◦ γ(t), and in the fifth line we
defined X = γ∗ dtd , the tangent field to γ, and right-multiplied the whole expression
by ait .
With our embedding into GL(n, C), the result is a linear system of n2 coupled
first-order ODEs in ait , which has a unique and suitably differentiable solution. In
fact, we can write an expression for the solution right away:
(
ait = ai0 I −
∞ Yn
X
˜
n=1
Z ti−1
i=1
dti
0
)
Yn
˜
i=1
(Ai (X(ti )))
(2.17)
QN
where t0 := t, and ˜ i=k (Ti ) is a glyph that represents an instruction to write down
Tk Tk+1 · · · TN , or just 1, if k > N . This is not necessarily a multiplicative product,
but is purely formal.
This is reminiscent of Dyson’s formula for scattering in QFT [18]:
(
U (t, T ) = I −
∞ Yn
X
˜
n=1
i=1
Z ti−1
T
dti
Yn
˜
i=1
)
(iHI (ti ))
(2.18a)
2.3 Horizontal Lifts
19
which is written more compactly as
U (t, T ) = T exp −i
Z t
T
dτ HI (τ )
(2.18b)
The exact same combinatorial argument allows us to write
ai1 = ai0 P exp −
= P exp −
Z
γ
Z
Ai
[0,1]
Ai
d
γ∗
dt
!!
(2.19)
where P is the path-ordering symbol, a glyph that indicates that products in the
power-series expression are to be path-ordered, in analogy to the time-order with
the path parametre replacing the time.
So, finally, the horizontal lift of γ is2
γ̃(t) = σi (γ(t)) P exp −
Z
[0,t]
Ai
d
γ∗
dt
!!
(2.20)
This expression extends straightforwardly to cases where γ extends outside of
Ui : simply transform according to equation 2.8 on the overlap region, and apply
equation 2.20 again in the new coordinate patch. For any u′ = ug in the same fibre as u, γ̃(t)g is a horizontal lift of γ that starts at u′ . And thus the theorem is proved.
Equation 2.20 should be independent of a choice of trivialization, because
γ̃ ⊂ T P is− but it depends quite clearly on a choice of trivialization σi ; we will see
how this works out later, when we have developed a suitable vocabulary.
One might think that we could have written the parallel transport on (psuedo-)
Riemannian manifolds as a curve γ̃ in T M such that π ◦ γ̃ = γ. Indeed, as we shall
see later, the parallel transport may indeed be formulated this way, but γ̃ will be a
curve on a GL(n, R)-bundle on M , rather than in T M itself.
When γ is a loop, so that γ(1) = γ(0) := p, we generally do not have γ̃(1) =
γ̃(0), so that γ̃ maps Gp to Gp via right multiplication by an element in G. This is
commonly called the holonomy group at p, a subgroup of G, and its elements are
2
This is well defined because the path-ordered exponential converges: see page 235 of [3] for a
simple proof.
Gauge Theory
20
often written
U (ω, γ) = P exp −
I
γ
Ai
(2.21)
We may then define a “group” of loops, then, with multiplication γ2 ◦ γ1 defined
by traversing γ1 first, then γ2 and γ −1 to be the loop γ traversed in the opposite
direction. The direction of γ matters, but not its parametrization. In that case,
U (ω, γ2 ◦ γ1 ) = U (ω, γ2 )U (ω, γ1 )
(2.22a)
U (ω, γ −1 ◦ γ) = e
(2.22b)
the first identity can be seen by first transporting along γ1 , then γ2 , which is the
same as transporting along γ2 ◦ γ1 . The second identity is true because reversing
the direction of γ reverses the sign on the integral, and eA e−A = id. But this is not
truly a group, though, because γ −1 ◦ γ is not the identity curve. Generally, the word
holonomy is used even for cases where γ is not a loop. We shall be liberal with our
terminology, and adopt this usage when it is convenient.
2.4
Derivatives on Associated Fibre Bundles
In classical Yang-Mills theory, the gauge fields A began as book-keeping devices
to preserve the form of the physical equations, as an answer to the postulate of
invariance under local gauge transformations. The matter fields were vectors fields
whose vectors belonged to a vector space of a certain sort, which had a left action
of local rotations by the so-called gauge group.
In SU (2) Yang-Mills theory, this vector space was C2 , on which SU (2) acted
through its fundamental representation. We defined a covariant derivative on
these matter fields ψ in a rather ad-hoc way, by insisting that the action of the
covariant derivative operator D transform as Dψ → U Dψ under local gauge transformations U . The gauge field A was then tacked on to the definition of D, and
postulated to transform according to equation 2.7 in order to cancel the inhomogeneous term in the local transform. Thus we obtained D = d + A.
We now present an alternative but ultimately equivalent definition of the gauge
derivative− one that in fact generalizes to the covariant derivative from classical
GR.
2.4 Derivatives on Associated Fibre Bundles
21
Definition 2.4.1 Associated fibre bundle: Let P (M, G) be a principal bundle and F
a manifold (in particular a vector space, for us) on which G acts on the left. Consider
the product manifold P × F , with an equivalence relation (u, ξ) ∼ (ua−1 , aξ). The
quotient space of P × F by this equivalency , E(M, F, G, P ), is called the fibre bundle
associated with P . Sometimes we will also write E = P ×G F .
As a fibre bundle, E has base space M , typical fibre F , which we will always take to
be endowed with a vector space structure, and structure group G. It inherits a projection πE from P . To see this, consider a trivialization on P, Ui , so that the product
manifold has the local equivalence(p, g, ξ) ∼ (p, ga−1 , aξ), so that the intermediary
G on the trivialization becomes extraneous, and πE ((p, g, ξ)) has πE−1 (Ui ) ∼
= Ui × F .
Immediately from this definition, we can define a map from P × F to E, with
the right product given by uξ = [(u, ξ)], the right-hand side being the equivalence
class on E. This product is well defined because u(aξ) = (ua)ξ for all a ∈ G.
The elaborate and rather opaque construction above was to formulate the idea
of a parallel transport unambiguously.
Definition 2.4.2 Parallel transport: Consider a curve γ on M . We say that a section
of E, s, is parallel transported along γ if s(t) has a representative (γ̃(t), ξ(t)) for
which ξ is constant, and γ̃ is any horizontal lift of γ.
We showed earlier that horizontal lifts of the same curve differ by right multiplication by some constant a ∈ G, so that the specific choice of lift does not matter. i.e.,
for γ̃ ′ = γ̃(t)a, we have (γ̃(t)a, ξ(t)) ∼ (γ̃(t), aξ(t)), and aξ(t) is constant if ξ(γ(t))
is. In other words, ξ(t) is parallel transported if its “relative angle” to a γ̃ remains
constant, i.e., ξ rotates with the holonomy along γ.
Definition 2.4.3 Covariant derivative: Consider any curve γ : [−τ, τ ] → M with
tangent X and p = γ(0). Let s be a section of E defined on some neighbourhood
of p. We can write s|γ with representative s(γ(t)) = (γ̃(t), η(γ(t))), where γ̃ is any
horizontal lift of γ. Then s0 (t) = [(γ̃(t), η(p))] is a parallel transported section along
γ. Then the (gauge) covariant derivative is defined as
DX s = lim [s(γ(h)) − s0 (h)] /h
h→0
"
=
d
γ̃(0), η(t)
dt
t=0
!#
(2.23)
Gauge Theory
22
where addition is inherited from F , acting on the second argument. In other words,
we take nearby values of s and parallel transport it back to p with γ to compare the
two vector spaces, and then take their first-order difference. This definition does
not depend on the specific choice of γ nor its horizontal lift. The first claim is fairly
obvious, and the second is evident from the equivalence class.
The covariant derivative D : T M × Γ(M, E) → Γ(M, E) is “actually” a derivative operator. That is, it is linear in its first and second arguments, and obeys a
Leibniz rule when the second argument is multiplied by a non-constant function. Thus we could instead choose to interpret the derivative operator as a
D : Γ(M, E) → T ∗ M × Γ(M, E).
As far as practical calculations are concerned, the covariant derivative is far
more conveniently expressed in local coordinates. To this end, Let us choose
a basis {ea } of F , the typical fibre of E and a corresponding basis {ea } of F ∗ so
that ea (eb ) = δ a b , a local trivialization σi , and coordinate chart {xµ } on Ui . With
respect to this trivialization we can unambiguously write an explicit representative
s = (σi , ξi ) = (σ, ξia ea ). Without proof,
"
DX s|p =
σi (p),
d a
ξ ◦ γ + X µ Aiµ a b ξib
dt i
!
!#
ea
(2.24a)
t=0
The L(G) coefficient of the connection is interpreted as a linear Aiµ : V → V , defined by the representation of L(G) on V , so this map is defined unambiguously as
an endomorphism on V .
Oftentimes we will forgo the cumbersome notational structure on E, and write
simply:
DX ξi = X µ ∂µ ξia + Aiµ a b ξib ea
(2.24b)
so that
DX ea = X µ Aiµ b a eb
(2.24c)
where the local trivialization σi is implicitly understood. And so finally, we can
write in abstract index notation,
(Dµ ξi )a = ∂µ ξia + Aiµ a b ξib
(2.24d)
2.4 Derivatives on Associated Fibre Bundles
23
The same expression is true in a different trivialization, given by σj = σi tij :
DX (tij ξj ) = X µ tij ∂µ ξja + Aiµ a b ξjb ea
(2.25)
= tij DX ξj
and (σi , DX ξi ) = (σi , tij DX ξj ) ∼ (σj , DX ξj ).
We may now make explicit the coordinate-independence of equation 2.20.
Writing γ̃(t) = σi (γ(t))ait with the same specifications as in equation 2.20, and a
section σj := σi tij :
(γ̃(t), ξ) ∼ (σi (t)ait , ξi )
= (σj (t)t−1
ij (t)ait , tij (0)ξj )
(2.26)
∼ (σj (t)t−1
ij (t)ait tij (0), ξj )
!
= (σj (t)ajt , ξj )
So
aj0 P exp −
Z
[0,t]
Aj
d
γ∗
dt
!!
=
t−1
ij (t)ai0 P
exp −
Z
[0,t]
Ai
d
γ∗
dt
!!
tij (0)
(2.27)
As a demonstration of this mathematical machinery in action, we immediately
apply it to an example, defining along the way some crucial concepts we will use
later on in our quest for a quantum theory of gravity.
2.4.1
Example: Frame Bundles
Let M be some n-dimensional differentiable manifold, and T M its tangent bundle.
At every point p ∈ M , we can choose a basis of n linearly independent vectors
{Xi }ni=1 ⊂ Tp M . Such a u = {Xi }ni=1 is called a linear frame at p. Denote the set
of all [choices of linear frames at all points in M ] as F (M ). We will construct a
principal fibre bundle out of this manifold:
Definition 2.4.4 Frame bundles: Consider a coordinate system {xµ } on an open
patch Ui ⊂ M , which naturally gives a local section of F (M ), σi = {∂/∂xµ }. We
P
may expand the vectors in any linear frame on Ui as Xi |p = µ Xiµ |p (∂/∂xµ ). Clearly
then Xiµ ∈ GL(n, R) parametrizes F (M )p for each p, so that the fibres of F (M ) are
Gauge Theory
24
diffeomorphic to GL(n, R). Changing coordinates to {y µ } shows that the structure
group is GL(n, R).
Finally, we endow F (M ) with a GL(n, R) action from the right, ua = { i ai j Xi }nj=1 .
π
The result is that F (M ) → M is a principal GL(n, R)-bundle, which we call the
frame bundle over M .
P
A linear frame u at p ∈ M has a natural left action on vectors i v i ei ∈ Rn , given by
P
P
u( i v i ei ) = i v i Xi |p ∈ Tp M , where ei ∈ Rn are basis vectors. We may then consider the tangent bundle T M as an Rn -bundle associated with F (M )(M, GL(n, R)).
In other words, T M ∼
= F (M )×GL(n,R) Rn with the isomorphism [(u, ξ)] → uξ defined
earlier.
P
By extension, we can also use F (M ) to contextualize the tensor bundles T (p,q) M
of (p, q)-tensor fields, with linear combinations of the action on basis elements:
a ...a
ueb11...bqp → Xa∗1 · · · Xa∗r Xb1 · · · Xbs ⊂ Tp(r,s) M .3
With these isomorphisms, then, a local section u|Ui of F (M ) may instead be defined as a linear u : Rn → Γ(Ui , T Ui ). We will often make use of this more versatile
notation over the formal definition.
These elaborate definitions allow us to recover some already-familiar results
from elementary differential geometry: consider a gl(n, R)-valued 1-form on M =
Rn , Γ, which defines the connection on F (M )(M, GL(n, R)). The covariant derivative of a vector field v ∈ Γ(M, T M ) can be written in a coordinate system with
equation 2.24d:
(Dµ v)ν = ∂µ v ν + Γµ ν σ v σ
(2.28)
And from this, the representation of L(G) on tensor products of T M gives the
correct expression for the covariant derivative of tensor fields of arbitrary valence.
Furthermore, by equation 2.8 we straightforwardly find the transformation of the
′
′
′
connection under coordinate change {xµ } → {y µ }, so that txy µ µ = ∂y µ /∂xµ :
Γµ ν σ
3
∂xν
→ ν′
∂y
′
Γµ′
ν′
′
∂y σ
∂ ∂y σ
+
σ′
∂xσ
∂xµ ∂xσ
!
∂xµ
∂y µ′
(2.29)
Of course, this is just the action of the representation of the gl(n, R) algebra on the tensor-ified
r−2
s−2
space T M ⊗ · · · ⊗ T M ⊗ T ∗ M ⊗ · · · ⊗ T ∗ M .
2.5 Curvature
25
The 1-form index µ plays a very distinct role from that of the gl(n, R) indices ν, σ so
we now see see that Γ only resembles a tensor superficially. Note, though, that we
have not imposed either the metric-compatibility nor the torsion-free condition,
and so that the above conditions hold for general connections Γ.
2.5
Curvature
Consider Maxwell theory on Rn . Here, the gauge group is abelian U (1): the group
elements all commute, and Ai is a general imaginary-valued 1-form. We can then
rewrite equation 2.21 with Stokes’ equation:
I
Z
U (ω, γ) = P exp −
γ
= P exp −
Γ
Ai
Fi
(2.30)
where Γ is any 2-surface with boundary γ, and Fi is the Maxwell tensor dAi . If Ai is
a gradient (pure gauge) along γ, then Fi vanishes identically, and the holonomy
group becomes trivial.
In the section above, we briefly described parallel transport on (psuedo-) Riemannian manifolds in terms of a connection on F (M )(M, GL(n, R)). In that context, it should be obvious that parallel transport on a (psuedo-) Riemannian manifold is trivial if and only if the curvature is uniformly zero, i.e., the metric is flat.
Indeed, the definition of the Riemann curvature tensor [15, 28]
(∇X ∇Y − ∇Y ∇X ) Z − ∇[X,Y ] Z = R(X, Y )Z
(2.31a)
or, in abstract index notation,
Rabc d ωd = (∇a ∇b − ∇b ∇a ) ωc
(2.31b)
is exactly the measure of the change in fields when parallel transported along a
small loop.
Clearly, then, the Maxwell tensor plays a role similar to the curvature tensor
in U (1) gauge theory. In what follows, we will strive to make this similarity more
precise.
Gauge Theory
26
Definition 2.5.1 Covariant exterior derivative: Consider an L(G)-valued r-form η
on a principal bundle P (M, G), η ∈ L(G) ⊗ Ωr (P ). We define the covariant exterior
derivative D as follows:
Dη(X1 , · · · , Xr+1 ) = dη(hX1 , · · · , hXr+1 )
(2.32)
where d is the exterior derivative on P , hX is the horizontal component of the vector
field X, and dη := gi dη i
However, the covariant exterior derivative is not “actually” a covariant derivative,
because it generally does not satisfy D2 η = 0 in general.
Definition 2.5.2 Curvature 2-form: Let ω be a connection 1-form on a principal
bundle P (M, G). The curvature 2-form Ω ∈ L(G) ⊗ Ω2 (P ) is defined by
Ω = Dω
(2.33)
From the definition of the covariant exterior derivative, and the definition of ω
(specifically, the right-invariance of Hu P and the right action on ω, Rg∗ ω = g −1 ωg)
we have:
Rg∗ Ω = g −1 Ωg
(2.34)
Borrowing the Lie algebra structure, we can define a commutator between an
L(G)-valued p-form η = η s τs and L(G)-valued q-forms ζ = ζ s τs :
[ η , ζ ] = [ τs , τ t ] η s ∧ ζ t
(2.35)
where [·, ·] is the Lie bracket on L(G), ∧ is the exterior product on P , and τ s is a
basis on L(G). When a product can be defined between elements of L(G), (e.g.,
via the fundamental matrix representation,) we may just as well define an exterior
product:
η ∧ ζ := η s ∧ ζ t τs τt
(2.36)
or,
(η ∧ ζ)I K = η I J ∧ ζ J K
(2.37)
where I, J are L(G) indices, is the exterior product of a “matrix of forms”. Note that
[ η , η ](X, Y ) = 2[ η(X), η(Y ) ]
2.5 Curvature
27
The curvature 2-form satisfies one of the two Cartan structure equations,
Ω = dω + ω ∧ ω
(2.38)
which can be proven by decomposing its input vectors X1 , X2 into horizontal and
vertical components, then using linearity and skew-symmetry to reduce into the
proof into three distinct cases. [15]
Now consider a local trivialization σi . The curvature in this coordinate system is
Fi : = σi∗ Ω
= dAi + Ai ∧ Ai
(2.39a)
Then by introducing a coordinate system on M , we can write
Aiµν = ∂µ Aiν − ∂ν Aiµ + [ Aiµ , Aiν ]
(2.39b)
which is just the familiar Yang-Mills field strength tensor when G is a gauge group.
Switching to the trivialization σj = σi tij , it is quite straightforward to show that
Fj = t−1
ij Fi tij
(2.40)
from which we may recover the action of the field strength tensor under gauge
transformations. In fact, it can be shown that any L(G)-valued p-form φ satisfies
Dφ = dφ + [φ, ω]
(2.41)
For a proof of this, see page 79 of [12], and proceed by induction.
When P (M, G) is the frame bundle F (M ), then writing the connection 1-form
as Γµ I J with the gl(n, R)-indices I, J written out explicitly as before, we obtain:
(Rµν )I J = ∂µ Γν I J − ∂ν Γµ I J + Γµ I K Γν K J − Γν I K Γν K J
(2.42)
which is extremely similar to the familiar Riemann curvature tensor, but for the
fact that the two objects live in different spaces: Rabc d is a (1, 3)-tensor field over M ,
and RI J is a gl(n)-valued 2-form− but see the next section. This expression does in
fact transform like a tensor under coordinate change, because equation 2.40 gives
the correct GL(n) transformation.
Gauge Theory
28
We conclude this section by stating the Bianchi identity:
0 = dΩ − d2 ω − dω ∧ ω + ω ∧ dω
= dΩ + (ω ∧ Ω) − (Ω ∧ ω)
(2.43)
= DΩ
The first equation is from taking the (normal) exterior derivative of Ω, and the third
follows from the second because ω annihilates horizontal vectors. It can be shown
that on a frame bundle F (M ), this expression translates to the familiar Bianchi
identity on (psuedo-) Riemannian manifolds, ∇[e Rab]c d = 0
2.6
GR on Frame Bundles: Preliminaries
Let (M, g) be an n-dimensional Lorentzian manifold. We present here the formalism of general relativity in a “non-coordinate basis”: on some open cover {Ui } of
M , we may diagonalize the metric locally on each Ui with vector fields {eI }Ui so
that
g(eI , eJ )|Ui = ηIJ
(2.44a)
Where η = diag(−1, 1, 1, · · · ). Alternatively, we could have begun by defining {θI }Ui
the basis dual to {eI }Ui so that
g|Ui = ηIJ θI ⊗ θJ
(2.44b)
which is equivalent to equation 2.44a.
In a coordinate basis on M , we can expand eI = eµI ∂µ , and θI = eIµ dxµ , where the
eIµ , eµI are matrix inverses of each other, eIµ eµJ = δ I J , because the bases are defined
as duals of each other, θI (eJ ) = δJI .
Equations 2.44a, 2.44b are equivalent to a condition on {eIµ }:
eIµ eJν ηIJ = gµν
(2.45)
which determines {eIµ } up to an SO(n−1, 1)-rotation ΛI J satisfying
ΛI K ηIJ ΛJ L = ηKL
(2.46)
2.6 General Relativity on Frame Bundles: Preliminaries
29
such a frame is called a vierbien, or tetrad (resp. German and Greek) in four
dimensions, or vielbein in n-dimensions.
In the language of principal bundles formulated in this chapter, {eI }Ui is a local
section of the frame bundle F (M )(M, GL(n, R)) that in particular solves equation
2.44a. When g is a solution to the Einstein equations, we may very well confer
upon the vielbein {eI } and its dual {θI } the distinguished title of gravitational field,
instead of g.
Then instead of formulating general relativity as a theory about the evolution of
the metric g, we could cast {θI } in the starring role of a gravitational gauge theory
of frames. (In what follows, we will drop the explicit reference to coordinate charts
{Ui }.)
Co-starring in this re-formalism is the connection (gl(n)-valued) 1-form ω on F (M ),
which features in the other Cartan’s structure equation:
dθI + ω I J ∧ θJ = ΘI
(2.47a)
(which can be taken to be the definition of the torsion 2-form ΘI .) In particular, ω
is said to be torsion free if it solves ΘI = 0.
In a coordinate basis, a torsion-free connection solves the equation
∂[µ eIν] + eJ[ν ωµ] I J = 0
(2.47b)
Taking the exterior derivative of equation 2.47a and using 2.38 gives the other
Bianchi identity:
dΘI + ω I K ∧ ΘK = ΩI K ∧ θK
(2.48)
The operator d is the exterior derivative on the frame bundle, so the above equations are statements on the linear frame defined by {θI }, and not the dual vector fields θI individually. In the absence of torsion, it can be shown that the
Bianchi identity reduces to the familiar symmetry property of the Riemann tensor:
R[abc] d = 0. ([24], p.288)
The metric-compatibility condition manifests in the frame bundle formalism as
ηIK ω K J = −ηJK ω K I
ωIJ = −ωJI
(2.49)
which is to say that the values of ω are in particular restricted to the so(n − 1, 1)
subspace of gl(n), reflecting the symmetry imposed by equation 2.45. It should be
Gauge Theory
30
clear that the metric-compatibility and torsion free conditions really do correspond
to the familiar conditions ∇X (ηIJ θI θJ ) = 0 and [∇X , ∇Y ]f = 0 respectively, where
∇ is the covariant derivative operator associated with the connection ω on the
associated bundle T (r,s) M . The unique ([24], p. 237) metric-compatible, torsionfree connection is called the Levi-Civita connection. The connection 1-form ω
associated with it is often called the spin connection. In a coordinate basis, the
spin connection is related to the Christoffel symbols, in keeping with 2.13,
ωµ I J = eIν Γµ ν σ eσJ + eIν ∂µ eJσ
(2.50)
or we can solve for its components using metric compatibility and vanishing of
torsion:
1 ν
σ ν
ωµIJ =
eI ∂[µ eν]J + eσJ ∂[σ eµ]I − eK
e
e
∂
e
(2.51)
[σ
ν]K
µ I J
2
We can also explicitly write out the natural isomorphisms on the F (M )-associated
bundles that we defined in section 2.4.1, in terms of the frame components {eµI }
and its inverse {eIµ }. e.g., for vector fields,
v I → v I eI
(2.52a)
v µ → eIµ v µ
(2.52b)
ωI → ωI eI
(2.52c)
ωµ → eµI ωµ
(2.52d)
and for dual fields, too:
so we may write the frame field and its dual with abstract indices from now on, eaI , eIa ,
as “transformations between indices.” They change abstract indices on T M and
T ∗ M , {a, b, c · · · }, to so(n−1, 1) (Minkowski/internal) indicies on Rn , {I, J, K · · · },
and vice versa. This generalizes to tensors of arbitrary valence. e.g.,
Ωab I J edI eJc = Rabc d
(2.53)
so the Riemann tensor is “actually” the curvature, after all.Finally, we end this
section with the Ricci tensor and scalar written in terms of {eµI }:
RicIJ = Ωµν K I eµK eνJ
(2.54a)
2.6 General Relativity on Frame Bundles: Preliminaries
R = Ωµν IJ eµI eνJ
31
(2.54b)
which will come in handy later, when we resolve to express GR fully in terms of
frame bundles, in chapter 4.
Chapter 3
Gravity
Now that we have properly settled the matter of mathematical prerequisites, it is
now time to start talking about the main focus of this publication: a canonical
formulation of quantum gravity.
By the adjective “canonical”, we mean “pertaining to Hamiltonian mechanics,”
the machinery underlying time evolution. We can obtain a Hamiltonian function
describing gravitational fields by performing a Legendre transform on a suitable
Lagrangian function, provided to us by the Einstein-Hilbert action. We will briefly
review this in section 3.1.
The Hamiltonian function thus obtained is called the ADM Hamiltonian, which
we will discuss in section 3.2. It describes the evolution of a field qab , the spatial
3-metric, with respect to a fiducial time parametre τ . We will set the stage for this
Hamiltonian formalism by defining a time function, etc. Then we will examine
the evolution of qab in time. The Hamiltonian function thus obtained turns out to
be describe a totally constrained system, motivating a summary of the general
treatment of such systems in the First Interlude. Finally, we conclude the chapter
with a discussion of the constraints in the ADM Hamiltonian.
3.1
Lagrangian Gravitation (I): Einstein-Hilbert
We begin this chapter by presenting the already-familiar Einstein-Hilbert action.
Let (M, g) be an n-dimensional manifold M with a Lorentzian metric g. The
Gravity
34
Einstein-Hilbert action is:
SEH [g] =
=
Z
⋆
R
√
R −ge
ZM
(3.1)
M
√
where ⋆ is the Hodge dual operator, and −ge is the volume form on M , whose
value in a coordinate basis is given by the square root of the determinant of g. This
action reproduces the vacuum Einstein field equations (without a cosmological
constant) upon extremization in variations of g,
δSEH |g =
√
√
(δR) −ge + Rδ( −ge)
Z
(3.2)
M
!
=0
Note the following identities, [28, 3]
√
√
1
δ( −ge) = − gab δg ab −ge
2
(3.3a)
δR = Ricab δg ab + ∇a ωa
(3.3b)
where ∇a ωa is a “total divergence” of some quantity which we will eliminate by
using Stokes’ theorem and assuming that ω falls to zero “at infinity.” 1 Then,
δ
Z √
√
1
Ricab + Rgab δg ab −ge
R −ge =
2
M
M
Z
(3.4)
which is zero if and only if Ricab + 12 Rgab = 0, and we’ve recovered the vacuum
Einstein equations. For the sake of generality, we could add to the gravitational
1
This is the divergence theorem on general manifolds: given a 1-form ω, by Stokes’ theorem we
have
Z
Z
⋆
d⋆ ω =
ω
S
∂S
Which is metric-dependent: there is geometric information from g encoded into ⋆ . To recover
the abstract-index expression, note that we can use any torsion-free derivative operator ∇ in
the definition of the exterior derivative, because the total-antisymmetry property annihilates any
dependence on the connection coefficients. So we might as well use the metric-compatible ∇,
√
√
which gives the neat expression d⋆ ω = ∇[an −gω a e|a|a1 ···an−1 ] = −g∇[an ω a e|a|a1 ···an−1 ] , so that
the abstract index notation expresses straightforwardly:
Z
Z
a
∇ ωa =
ωa n a
S
∂S
3.2 Hamiltonian Gravitation: ADM formalism
35
Lagrangian LG a Lagrangian LM encapsulating the totality of gravitating matter,
and impose
Z
√
δ
(LM + LG ) −ge = 0
(3.5)
M
This gives the general Einstein field equations (with factors of 8πG restored,)
1
Gab := Ricab + Rgab = 8πGTab
2
for
8πGTab := −
∂LM
1
Lgab
+
∂g ab
2
(3.6)
(3.7)
In what follows, however, we will always assume Tab = 0.
3.2
Hamiltonian Gravitation: ADM formalism
A notion of “time evolution” does not arise naturally in the context of general
relativity. When coupled to general matter fields, the Einstein field equations
solve everything that happens in spacetime “all at once,” as though the matter
and gravitational fields were conspiring together to create the illusion of causality:
space bending and matter moving, intertangled but predetermined, dancing to a
well-scripted choreography. There are no dynamics in this picture: the river does
not flow, but rather water is rained into it from above. This is quite distinct to the
usual notion of time evolution in physics, wherein an instantaneous shapshot of a
physical system is mapped to the next instant, in accordance with the evolution
law as decreed by the Hamiltonian of the system, who is the sole sovereign.
But this is to be expected of a general spacetime, wherein a global definition
of time coordinate does not necessarily exist− without this, the very concept
of a “time evolution” becomes ill-defined. However, there exists a condition on
the causal structure of spacetime that is equivalent to the existence of such a
coordinate, through which we may hope to restore a notion of time evolution. This
condition is global hyperbolicity.
3.2.1
ADM Formalism (I): Statics
Let (M, g) be a 4-dimensional Lorentzian spacetime that satisfies global hyperbolicity. That is, there exists a diffeomorphism φ : R × Σ → M such that for every
Gravity
36
t ∈ R, φ−1 (M )|t ∼
= Σ, and the 1-form dφ is dual to a timelike vector. In other
words, we can introduce a time function that slices spacetime into “time zones”
t → Σt := φ−1 (M )|t such that t is strictly increasing for future-bound observers.
Let n be the timelike vector field normal to each Σt on M , defined by φp∗ dtd for
p ∈ Σ. It is no loss to assume g(n, n) = −1, so that we may define a Riemannian
metric on Σt :
qab := gab + na nb
(3.8)
This gives a “spatial projection” of Tp M to the subspace Tp Σt of spacelike vectors:2 ,
given by v a → qba v b . With a coordinate system {xµ } on Σ, φ gives us spatial coordinates (with a little abuse of notation,) φt∗ (∂/∂xµ ). This completes a basis on T M
with n.
We can define the extrinsic curvature of Σt as:
1
Kab = £n qab
2
(3.9)
£n being the Lie derivative with respect to n. This tensor field quantifies the nonintrinsic curvature of the embeddings of Σ into 4-dimensional M . It complements
the 3-dimensional intrinsic curvature on Σ given by qab , which we will denote
(3)
Rabc d .
Let us now introduce a suitably differentiable congruence γ(τ, x) of general
(that is, they need not be normal to Σt , nor to be geodesics.) timelike curves on M
parametrized by τ , with timelike tangents ∂τ .
∂τ |t need not be normal to Σt , but by a suitable reparametrization we can make
the congruence satisfy γ(T, Σ) = ΣT . That is, γ replicates the slicing of the spacetime into time zones generated by φ, but the flow of the curves produces some
“spatial translation” across each slice Σt .
From these definitions, it is evident that
∂τ = N n + N
2
(3.10a)
The index on qab is raised with gab . Unless explicitly stated, we will always raise and lower
indices with gab .
3.2 Hamiltonian Gravitation: ADM formalism
37
for some spacelike N , and a normalization constant N , defined by
N = −g(∂τ , n)
(3.10b)
N = ∂τ + g(∂τ , n)n
(3.10c)
so then we can write
1
(∂τ − N )
(3.10d)
N
N is called the lapse, and N the shift, because they quantify the infinitesimal
motions through time and space, respectively, generated by ∂τ .
n=
Fig. 3.1 The decomposition of ∂τ . Figure taken from [3].
Using the Riemannian metric qab , we may define on each Σt the covariant differential operator (3) ∇a , the curvature tensor (3)Rabc d , the Ricci tensor, (3)Ricab , and
the Ricci scalar (3)R, all of which are defined in the usual way on Σ by the metric
(φ∗t q)ab , then pushed forward.
Together with the extrinsic curvature, the 4-dimensional curvature tensor can
be expressed as in terms of these objects: (see [3], p. 422)
nd Rabc d = (3) ∇a Kbc − (3) ∇b Kac
(3.11a)
qed Rabc e = (3)Rabc d + Kbc Ka d − Kac K d b
(3.11b)
These are collectively known as the Gauss-Codazzi equations. These lead to the
following expressions for the Einstein tensor on M :
Gab na nb = −
1 (3)
R + (Kaa )2 + Kab K ab
2
(3.12a)
Gravity
38
Gac na qbc = (3) ∇a Kba − (3) ∇b Kaa
(3.12b)
These cannot describe any dynamics in the EFEs, since the right-hand side of
these equations only depend on qab in Σt . Setting the left-hand side to zero, as in a
vacuum, they can instead be interpreted as “constraints” imposed on the Cauchy
data which q must satisfy to be a legal initial state, much like the constraint on the
electric field divE = 0 in a vacuum Maxwell theory.
3.2.2
ADM Formalism (II): Dynamics
Let us now interpret the congruence γ(τ, x) as the world lines of a set of timelike
observers. The crux of the ADM formalism, then, is to formulate general relativity as
a description of the evolution in τ of the 3-metric qab as a field theory on Σ. To do this
(in a vacuum, and without a cosmological constant, to keep things from getting out
of hand,) we must first produce a Hamiltonian out of the Einstein-Hilbert action.
And to do that, we start by taking as our canonical coordinate qab , and defining its
time derivative
q̇ab : = £τ qab
= 2NKab + £N qab
(3.13)
The Einstein-Hilbert action gives the Lagrangian, in terms of qab :
L(q, q̇, t) =
Z
(3)
Σt
√
R + (Kaa )2 + Kab K ab N q (3)e
(3.14a)
√
√
√
where it may be verified that N q = g, and so q (3)e is the volume form on Σt .
Then we have a Lagrangian density:
L(q, q̇) =
(3)
√
R + (Kaa )2 + Kab K ab N q
(3.14b)
and a conjugate momentum density to qab :
δL
δ q̇ab
√ ab
= q K − Kcc q ab
pab =
(3.15)
It is crucial to note that these are not scalars or tensors, but rather the duals thereof:
scalar densities and tensor densities, whose natural habitats are under integral
signs. Then we perform a Legendre transform on L to obtain the Hamiltonian
3.2 Hamiltonian Gravitation: ADM formalism
39
density:
H(q, p) = pab q̇ab − L
√
= q (N C + N a Ca )
where
−1
(3)
C := − R + |q|
1
p pab − (pcc )2
2
ab
(3.16)
(3.17a)
Ca := −2(3) ∇b |q|−1/2 pab
(3.17b)
R
√
Now that we have a Hamiltonian H = Σ H q (3)e, we can define Poisson brackets
on functionals of q, p.
Z
{f, g} :=
Σt
!
δf
δg
δf
δg
√ (3)
− ab
q e(z)
ab
δqab (z) δp (z) δp (z) δqab (z)
(3.18)
which gives the familiar relations:
n
o
qab (x), pcd (y) = δac δbd + δad δbc δ (3) (x − y)
(3.19a)
{qab , qcd } = 0
(3.19b)
n
(3.19c)
o
pab , pcd = 0
and, finally, we obtain the time evolution of the state coordinates:
q̇ab = {qab , H}
(3.20a)
ṗab = {pab , H}
(3.20b)
By the equivalence of Hamiltonian and Lagrangian mechanics, these must be
equivalent to the remaining Einstein field equations, Gab qca qdb = 0, complementary
to equation 3.12. These expressions can be written out explicitly, but they are rather
long without being particularly enlightening, and we have no real use for them in
what follows, so we will omit them here (they are in Appendix C of [28]).
Before we proceed any further, though, the Hamiltonian, equation 3.16, has
a rather disturbing property. That is, for on-shell states, it is uniformly zero. For
consider gab , a solution to the vacuum Einstein equations: Gab [g] = 0. Writing out
equation 3.12 in terms of the phase space parametres (qab , pcd )|g , we can see that
Gravity
40
this translates into
na nb Gab [g] = 0 ⇐⇒ C|g = 0
(3.21a)
nb qac Gbc [g] = 0 ⇐⇒ Ca |g = 0
(3.21b)
H|g = 0
(3.22)
so then,
Equation 3.22 is not to be interpreted, however, as a statement that our theory is trivial. It is not any more trivial than, say, finding points on the surface
P
P
2 1/2
2
3
{(zi ) ∈ CP82 |( 33
− ( 82
i=1 (zi + zi+1 ) )
i=34 zi−1 zi ) = 0}, which is an analogous probP
lem where the equation H = 0 plays the role of the defining equation ( 33
i=1 (zi +
P82
2 1/2
2
zi+1 ) ) − ( i=34 zi−1 zi ) = 0. This requires some elaboration, provided below.
First Interlude: Constrained Hamiltonian Systems
The idea of constraint equations on phase space should not be too alien to us; the
form in which a physical problem is presented to us is generally not expressed
without extraneous degrees of freedom. Consider for example a particle that is constrained to move on the surface of a 2-sphere in 3-space: {⃗q ∈ R3 |q12 + q22 + q32 = 1}.
The apparent number of degrees of freedom is 3, but really there are only 2: ( qθ , qφ ),
the longitudinal and latitudinal ones. But even without resorting to this rather
artificial case, there is already a sufficiently general case of a constrained system
with which we are familiar: Yang-Mills theory. In this section we will converse
about general constrained systems, and then we will invoke the specific case of
Yang-Mills theory as an analogy to shed some light on GR.
A general theory of constrained Hamiltonian systems was pioneered by Dirac
in [5], in the context of (sub)atomic physics on flat or curved spaces. However, this
formalism has sufficient generality to find an application even in a description of
the gravitational field.
We begin our discussion with a system with finitely many degrees of freedom,
described by a Lagrangian L with phase coordinates (qi , q̇i ), and a canonical choice
of time parametre t. Though manifest Lorentz invariance is violated by this choice,
it can be restored in the equations of motion, as we will see later.
3
zi are homogeneous coordinates on CP. As far as I know this is just an arbitrary, convoluted
surface.
3.2 Hamiltonian Gravitation: ADM formalism
41
We introduce the canonical momenta,
pi =
∂L
∂ q̇i
(3.23)
and a set of constraints on the parametres,
(3.24)
φm (qi , pi ) = 0
which could be φ(qi , pi ) = q12 + q22 + q32 − 1, for example. Performing a Legendre
transform, we obtain the Hamiltonian from the Lagrangian:
H=
X
pi q̇i − L
(3.25)
i
and define Poisson brackets:
{f, g} :=
X
i
∂f ∂g
∂f ∂g
−
∂qi ∂pi ∂pi ∂qi
!
(3.26)
which gives
{qi , pj } = δij
(3.27a)
{qi , qj } = 0
(3.27b)
{pi , pj } = 0
(3.27c)
{f, H} =:
d
f
dt
(3.27d)
In particular, the last equation gives
d
∂H
qi =
dt
∂pi
(3.28a)
d
∂H
pi = −
dt
∂qi
(3.28b)
The Poisson brackets thus defined are anti-symmetric, linear in both arguments,
and satisfy a Jacobi equation, etc.. All of this is already familiar to us.
To impose the constraints {φm }, we crudely tack on to the system a set of new
“artificial coordinates,” the Lagrange multipliers λm , and their conjugate momenta
Gravity
42
µm , and write a new Hamiltonian:
HT (qi , pi , λm , µm ) := H(qi , pi ) +
X
λm φm
(3.29)
m
This new Hamiltonian does not depend on µm , the conjugate momenta; they manifest themselves in the dynamics through their absence: Re-defining the Poisson
brackets, etc., with respect to HT now, the equation of motion in λm gives, through
equation 3.28:
φm = 0
(3.30a)
(3.30b)
λ̇ = 0
In the picture described by the “total Hamiltonian” HT , the constraints φm = 0
are slipped into the physics as the equations of motion of the (artificial) state
coordinates λm . A crucial subtlety is that in this picture, the constraint φm = 0 is
“emergent,” and falls out of the equations of motion, instead of being imposed
“from above”. In other words, we cannot assume that φm = 0 before we “press the
play button” and evolve the system in time by imposing the equations of motion.
Applying the constraint prematurely will result in incorrect dynamics. Dirac calls
these such equations “weak equations” in [5], and the symbol ≈ denotes such a
weak equality. e.g., φm (q, p) ≈ 0. In particular, a set of constraints {φm } are said to
be of first-order if it defines a canonical algebra:
{φi , φj }T =
X
Cijk φk
(3.31)
k
where the Poisson bracket { , }T is defined with respect to the total Hamiltoi
nian. The “structure constants”Cjk
are antisymmetric in j, k, because of the antisymmetry of the Poisson brackets.
Of course, the additional terms in HT , λm φm alter the bracket structure, and therefore the time evolution, of the original physical parametres (qi , pi ) in a non-trivial
way. The first-order correction to the time evolution thusly produced is quantified
by
X
f (1) =
{f, λm φm }T
(3.32)
m
3.2 Hamiltonian Gravitation: ADM formalism
43
which is just {f, HT }T − {f, H}, to first order.4
As an example, say we have a particle on the real plane, parametrized by ⃗q, p⃗,
that is constrained to move on a circular rail of radius 1, φ(⃗q, p⃗) = qx2 +qy2 −1. We take
the classical Hamiltonian for free particles H = p⃗ 2 /2m and tack on the Lagrange
multiplier λ, so
p2x + p2y
HT (⃗q, p⃗, λ, µ) =
+ λ qx2 + qy2 − 1
(3.33)
2m
and obtain
pi
q̇i =
(3.34a)
m
ṗi = −2λqi
(3.34b)
meaning
λ
qi
m
The equations in λ become constraints on the boundary conditions,
q̈i = −2
(3.34c)
qx2 + qy2 = 1
(3.35a)
λ̇ = 0
(3.35b)
giving the correct solutions of the type
(qx (t), qy (t)) = (cos(ωt + φ0 ), sin(ωt + φ0 )) for
q
a constant angular frequency ω := 2λ/m.
We could invert this argument and identify as Lagrange multipliers the canonical coordinates B that have vanishing or absent conjugate momenta. Then the
terms in the Hamiltonian that are multiples of B can be collected together to form
a constraint equation. The simplest way to identify such cases is when the corresponding Lagrangian L does not depend on Ḃ. That is, pB := ∂L/∂ Ḃ = 0
If we make the jump right away to infinite degrees of freedom (i.e., field theory)
without looking down at the gaping crevasse below (covered in Chapter 13 of [9]),
then we see that Yang-Mills theory is a perfect example of such a system.
The SU (N ) Yang-Mills action on 4-dimensional Minkowski space may be written
4
In the cases that we review, the first order correction is sufficient to determine the entire corrected dynamics. In general cases, however, constraints of the second order may occur, producing
non-linear corrections to the dynamics, of which equation 3.32 is just the linear correction. See [5].
Gravity
44
in the language of chapter 2:
1Z ⋆ J
SY M [ω] =
Ω I ∧ ΩI J
2 R4
(3.36a)
or in more familiar notation, (assuming the bundle structure is trivial,)
Z
1Z
s
SY M [A] = −
dt
d3xFsab Fab
3
4 R
R
(3.36b)
where Fab = Fsab λs is the expansion with respect to a basis 5 {λs } on su(n) satisfying
tr(λs λt ) = −δ st /2. In particular, we have broken manifest relativistic invariance
with a choice of time t, which will define time in what follows.
Taking the state coordinates to be (As0 (⃗x), Asi (⃗x)), we have a set of conjugate momenta (πs (⃗x), Esi (⃗x)):6
∂L
=0
∂ Ȧ0
∂L
Ei =
= η ij Ȧj − ∂j A0 + [A0 , Aj ]
∂ Ȧi
π=
(3.37)
and we immediately see that we have n-many continua of Lagrange multipliers
{As0 (x)}. From the Legendre transform, we obtain a Hamiltonian density:
H = Esi Ȧsi − L
1 i s
=
Es Ei + Bsi Bis − As0 (Di E i )s
2
(3.38)
where Bi = εijk Fjk is a function of spatial derivatives of the state parametres {Ai },
and Di Ei := −(D⋆ F)0 = ∂i Ei + [Ai , Ei ]. Manifest gauge invariance is lost, because
of the second term in H, but we will recover it eventually.
Defining
δF (x)
= δ (3) (x − y)
(3.39)
δF (y)
5
We can make this manifestly representation-independent by defining Killing forms on su(n), if
we want. The result is identical.
6
To avoid any possible confusion, we reiterate that {abc} are abstract indices, and {ijk} are
spatial indices on Minkowski space with an explicit choice of time. The argument of the fields is
also purely spatial, because time is now something we evolve in, and not merely a (continuous)
index. We have denoted the argument as ⃗x here to make that fact explicit, but it will be just x in the
following.
3.2 Hamiltonian Gravitation: ADM formalism
and
45
δF
δG
δG
δF
{F, G} =
dz
−
a
s
a
δAs (z) δEa (z) δAs (z) δEas (z)
R3
Z
!
3
(3.40)
with the understanding that E0 = π in the sum. It can then be seen that the
equation of motion for π is equivalent to the time component of the Yang-Mills
equation of motion (D⋆ F)0 = 0:
π̇s (x) = {πs (x), H}
δπs (x) δH
δH δπs (x)
=
dz
−
r
a
3
δAa (z) δEr (z) δAra (z) δEra (z)
R
δH
=− s
δA0 (x)
Z
!
3
(3.41)
= Di Esi
where H = R3 d3 xH. For reference, the evolution of Esi gives the spatial Yang-Mills
equations, and the evolution of Asi gives the definition of Esi we started with. So
then we have a “generalized Gauss’ law” constraint, (Di E i )s = 0 at every point in R3 .
R
We are already done, but generally working with a continuum of constraints “asis” is somewhat dangerous, because divergent expressions involving Dirac deltas
may occur. So the constraints are typically written in scalar form, “smeared out”
with an su(n)-valued distribution:
G(λ) =
Z
R3
P(µ) =
d3zλs (Di E i )s
Z
R3
d3zµs πs
(3.42a)
(3.42b)
for any su(n)-valued distributions λ, µ. The constraints are indeed “of first order”:
{G(λ1 ), G(λ2 )} = 0
(3.43a)
{P(λ1 ), G(λ2 )} = 0
(3.43b)
{G(λ1 ), G(λ2 )} = G([λ1 , λ2 ])
(3.43c)
and they “alter” the dynamics, in the sense of equation 3.32, in an interesting way.
{G(λ), Ai (x)} = −Di λ(x)
(3.44a)
Gravity
46
n
o
G(λ), Ei (x) = [Ei (x), λ(x)]
(3.44b)
Recall that when we perform infinitesimal gauge transformations U ≈ 1 + λ on Aa
generated by small λ, equation 2.7 yields precisely
δAa (x) = Di λ(x)
(3.45a)
δFab (x) = [λ(x), Fab (x)]
(3.45b)
Thus, G(λ) teaches Hamiltonian Yang-Mills theory, which is a priori “ignorant” of
gauge symmetry, how to perform infinitesimal gauge transformations in λ. When
all possible λ are thus considered, the concept of gauge symmetry re-emerges
at last. But the gauge invariance is only on the spatial components of the fields,
because the zeroth component has “privileged character” due to our choice of time.
The other constraint has only one non-trivial bracket, and addresses this:
{P(µ), A0 (x)} = −µ(x)
(3.46)
which is just the statement that A0 is a Lagrange multiplier, and can be taken to be
any constant.
(End of First Interlude.)
3.2.3
ADM Formalism (III): Constraints
Returning to our discussion of the ADM Hamiltonian, we may now interpret ( C, Ca )
as constraints, and (N , N a ) as Lagrange multipliers. Following the typical convention on constrained Hamiltonian field theories, we define the “smeared out”
constraints:
C(N ) :=
Z
−1
(3)
N − R + |q|
Σt
C(N ) := −2
Z
Σt
1
p pab − (pcc )2
2
ab
Na (3) ∇b |q|−1/2 pab
√
q (3)e
√
q (3)e
(3.47a)
(3.47b)
That N and N a are arbitrary (fields of) Lagrange multipliers should come as no
surprise to us, because they arose through our selection of ∂τ , a fiducial choice of
“time” which is not an intrinsic description of the actual physical system.
The constraints are of first order:
{C(N ), C(M )} = C(£M N )
(3.48a)
3.2 Hamiltonian Gravitation: ADM formalism
47
{C(N ), C(M)} = C(£N M)
(3.48b)
{C(N ), C(M)} = C((N ∂ i M − M∂ i N )∂i )
(3.48c)
And analogously to the Yang-Mills case, the constraints “generate” a symmetry of
the system:
{f (q, p), C(M )} = £M f (q, p)
(3.49)
This is exactly the diffeomorphism invariance that distinguishes general relativity
from other theories of nature. Specifically, it is the invariance of the system with
respect to “Lie dragging” the fields with the one-parametre family of spatial diffeomorphisms φt : Σ → Σ, along the integral curves of N ∈ Γ(Σ, T Σ).
The scalar constraint complements C as an expression of the invariance of the
physics under reparametrization of the observers γ in τ , thus giving diffeomorphisminvariance in the temporal direction as well. [3]
Indeed, the argument on the right-hand side of equation 3.48c is of the form of
a Lie derivative,£(N n) (Mn), projected onto Σ− which suggests that the constraints
( C, Ca ) might be interpreted as one set of constraints that expresses invariance
under Lie dragging in four dimensions. It is very satisfying to see this symmetry
emerge out of our formalism, without any additional effort on our part.
So then, of the ten “independent” values of gab , four ( N , N a ) are Lagrange multipliers, and there are four algebraic constraints ( C, Ca ) that must be set to zero,
leaving us with just two “true” degrees of freedom. A field theorist would identify
these with the spin-up and spin-down polarizations of the spin-2 graviton field.
The task of obtaining a quantum theory out of the formalism thus presented
seems daunting, to put things mildly: the Hamiltonian contains terms that are seri√
ously non-polynomial in the fields (e.g., q). There are also nigh-insurmountable
problems concerning the definition of an inner product on the Hilbert space, and
√
so on, but the problem with factors of q will suffice to dissuade us from going
down this particular path to quantizing gravity.
But suppose, for now, that we have obtained a quantum theory out of the ADM
formalism, and we have a Hamiltonian Ĥ. Then the constraints C(N ), C(N ) ≈ 0
ˆ )ψ = Ĉ(N )ψ = 0. In other
translates to a statement on the physical states: C(N
Gravity
48
words,
Ĥψphys = 0
(3.50)
This is an example of a Wheeler-DeWitt equation. In this context, the entirely of
the physics is specified by the kinematics- there is no room for the dynamics (time)
to have any say in the physics! The entirely of the universe, and all of the events that
ever occur within it, are determined in one fell swoop. This is the same situation
as in the interpretation we offered at the beginning of this section, of solving the
Einstein equations in their original form. Just when it seemed that we obtained a
fully dynamical formalism of general relativity, it slips away like sand between our
fingers.
Chapter 4
Quantum Gravity
In this chapter, we begin our incursion into quantum gravity. We hinted in section
2.6 that we could formulate general relativity in the context of frame bundles as a
gravitational gauge theory on the frame bundle F (M ), and we will do that now. We
start again with the Einstein-Hilbert action, but written in terms of frame fields and
the connection this time. This is done by rewriting the Ricci scalar with equation
2.54b.
This action will not suffice, but it turns out that a modified version of it will
yield a Hamiltonian that does the trick for quantization. We will summarize the
algebraic details that make this possible in section 4.1.1. The Hamiltonian function
thus obtained will be expressed in terms of canonical coordinates (A, E), named in
analogy to the gauge and electric fields in Maxwell theory. We will discuss the form
of the theory in terms of these new variables in section 4.2.
In the Second Interlude, we will outline the program of obtaining a quantum
theory from a Hamiltonian theory, through a procedure called canonical quantization. The role of the constraints in the resultant quantum theory will be covered in
4.2.1.
Quantum Gravity
50
4.1
Lagrangian Gravitation (II): Frame Fields
We may rewrite the vacuum Einstein-Hilbert action in terms of the frame fields
{eIµ } and the connection ω:
SP [g] =
→ S[e, ω] =
Z
ZM
M
⋆
R
⋆
µ ν
ΩIJ
µν [ω]eI eJ
(4.1)
This action is called the Palatini action. A priori, ω can be any gl(n, R)-valued
1-form, but a simple derivation shows that, for variations in ω, this action is extremized by the Levi-Civita connection. On the other hand, variations in eµI are
extremized by solutions to the Einstein field equations (see Appendix C of [28]).
Thus this action produces equivalent descriptions of the same physics, and so
we might want to perform a Legendre transform on this new action as a second
attempt at quantizing gravity.
But as it turns out, this new formalism obtains the same equations as does the
ADM formalism, expressed in terms of the frame fields, ([1] p. 62) and gives rise
to second-order constraints. We mainly present it here to motivate the “self-dual
action,” described in the following.
4.1.1
Self-Duality, Complexification
⃗ fields and the
In classical Maxwell theory, we were able to combine the electric E
⃗ fields together to form the Riemann-Silberstein vector [13]
magnetic B
⃗ := E
⃗ + iB
⃗
S
(4.2)
which satisfies the complexified Maxwell equations (in a vacuum),
i
∂ ⃗
⃗ + iB
⃗
S = curl E
∂t
(4.3a)
⃗=0
divS
(4.3b)
This vector transforms neatly under general Lorentz transformations, and its vector
⃗ ·S
⃗ = (E 2 − B 2 ) + 2i(E
⃗ · B)
⃗ is a sum of invariant scalars. This may seem like
norm S
a miraculous coincidence, but all that we’ve really done here is rewrite the Maxwell
4.1 Lagrangian Gravitation (II): Frame Fields
51
field strength tensor F by grouping it into three “anti-self-dual” components,
⃗ = (F0i + iFjk ) dx0 ∧ dxi
S
(4.4a)
with the understanding that ε0ijk = 1. In Lorentz-covariant notation,
S = FIJ + iεIJ KL FKL dxI ∧ dxJ
(4.4b)
As we will see in what follows, there is an analogous way of rewriting the fields
in GR that will prove to be extremely useful.
To see what we mean by “(anti-) self-duality”, consider a 2n-dimensional Riemannian manifold. Here the Hodge dual operator satisfies (⋆ )2 = 1, so we can
decompose arbitrary n-forms as
T = +T + −T
(4.5)
for self-dual + T satisfying ⋆ (+ T ) = + T , and anti-self-dual − T satisfying ⋆ (− T ) =
−− T
1
±
T := (T ± ⋆ T )
(4.6)
2
On our 4-dimensional Lorentzian manifolds we are not so lucky; (⋆ )2 = −1, so the
best we can do for a definition of self-duality is ⋆ (+ F ) = i+ F , and anti-self-duality,
⋆ −
( F ) = −i− F :
1
±
(4.7)
F := (F ∓ i⋆ F )
2
and verily, the Riemann-Silberstein “vector” satisfies ⋆ S i = −iS i . What has happened here is that we complexified the Lorentz Lie algebra, and decomposed it
into two mutually commuting complex-valued su(2) modules:
so(1, 3) → so(1, 3)C
∼
= + su(2)C ⊕ − su(2)C
(4.8)
∼
= + sl(2, C) ⊕ − sl(2, C)
because sl(2, C) ∼
= su(2)C , where gC is shorthand for g ⊗ C.
Let us make this decomposition explicit: recall the familar basis vectors of
so(1, 3), Mst , defined with components Mst I J = −(η sI δJt − η tI δJs ); recall also that we
Quantum Gravity
52
could define (see, e.g., Section 4.2 of [16], )
1
Ji := εijk Mjk
2
(4.9)
Ki := M0i
satisfying
[Ji , Jj ] = εijk Jk
[Ki , Kj ] = −εijk Jk
[Ji , Kj ] = εijk Kk
(4.10)
Then complexifying,
±
Ai :=
1
(Ji ± iKi )
2
(4.11)
we obtain two commutating copies of su(2)C :
[+Ai , +Aj ] = +Ak εijk
[−Ai , −Aj ] = −Ak εijk
[+Ai , −Aj ] = 0
(4.12)
or, writing 4.11 more suggestively, in terms of Mst ,
i
i
Mst ∓ εst uv Muv
Ast :=
2
2
±
(4.13)
But we want (anti)-self-duality on the I, J, K algebra indices, not on their labels
s, t, u. So we define instead:
±
MstIJ
i
i
MstIJ ∓ εIJ KL MstKL
:=
2
2
(4.14)
But the two definitions are in fact equivalent, because MstIJ = −(δIs δJt −δJs δIt ), which
IJ
KL
= εst uv Muv
.
implies εIJ KL Muv
We can take complex linear combinations of +Mi := +M0i as a basis for + su(2)C ,
or, we can take real linear combinations of +Mst as a basis for + sl(2, C) that is
labeled by two antisymmetric indices. Both choices give 6 independent parametres,
as we would expect. Crucially, the self-dual basis transforms “correctly” under
Lorentz transforms, so that the basis indices s, t, u can be treated as Lorentz indices.
′
′
This can be seen by noting that Λaa Λbb εa′ b′ c′ d′ = Λcc′ Λdd′ εabcd in equation 4.14, so we
have
′
′
Λss Λtt ±Ms′ t′ = ±Mst
(4.15)
In the same vein, we complexify the tangent bundle T M and denote the complexification T M C , and similarly for the frame bundle: F (M ) → F (M )C . So, local
4.1 Lagrangian Gravitation (II): Frame Fields
53
sections of the complexified frame bundle map the “internal” Minkowski C4 vectors at a point p ∈ M to tangents on the complexified tangent fibre Tp M C .
Then consider a gl(n, C)-valued connection 1-form, Aa I J , on F (M )C . We can
antisymmetrize in the internal indices it by defining
1
A′a IJ := (AaIJ − AaJI )
2
(4.16)
with the I, J indices lowered with the internal metric ηIJ . This puts A′a in the
complexified Lorentz algebra. We can then define its self-dual (in the internal
indices) part:
1
i KL ′
′
+ ′
(Aa )IJ :=
Aa IJ − εIJ AaKL
(4.17)
2
2
We can expand an antisymmetric, self-dual connection 1-form Aa I J with respect
to the basis +Mst I J on sl(2, C):
+
I
Aa I J = Ast
a ( Mst ) J
(4.18)
This will come in handy later on.
IJ
A self-dual curvature +Fab
can be built out of a general connection by taking
its self-dual anti-symmetrized part +A′a , then building it as usual. Writing out the
components explicitly,
+ IJ
Fab [A]
IJ + ′
= Fab
[ A]
IJ
= ∂a +A′ b − ∂b +A′
IJ
+ [+A′a , +A′b ]IJ
(4.19)
Linear combinations and derivatives of self-dual A are easily verified to be self-dual.
Commutators are self-dual too, because the construction 4.14 satisfies 4.12. Thus
IJ
we can see that the curvature Fab
[A] of self-dual, antisymmetric connections A is
itself self-dual. Then we define a self-dual action out of the complexifed Palatini
action:
Z
⋆ a b + IJ
S[e, A] =
eI eJ Fab [A]
(4.20)
M
Or, we could just restrict the space of connections to those that are already self-dual
and antisymmetric and write instead,
SSD [e, +A] =
Z
M
⋆
IJ +
eaI ebJ Fab
[ A]
(4.21)
Quantum Gravity
54
We shall favour the latter expression, because it eliminates equivalent solutions.
This action is, quite unusually, complex-valued.
Note that, a priori, we have no constraints on the tetrad that extremizes this
action, and so the metric gab = eIa eJb ηIJ may very well be complex. We will not
concern ourselves with this peculiarity. Instead, we will satisfy ourselves with the
knowledge that the self-dual action is extremized by solutions (eS , AS ) that solve
the self-dual half of the complexified vacuum EFEs ([3], p. 441-443):
0 = + Ricab [eS , AS ] +
1+
Rgab [eS , AS ]
2
(4.22)
where + Ric, and + R are defined with +F [AS ] contracted in the usual way with the
frame field eS . This in turn implies that + Ric = 0. Furthermore, one also obtains
that AS is the self-dual part of the complexified spin connection: ([1], p. 45-47)
(deS )I = −(ωS )I J ∧ (eS )J
i
1
(ωS )aIJ − εIJ KL (ωS )aKL
=
2
2
(AS )aIJ
(4.23a)
(4.23b)
with this in mind, it would be temping to demand all physical quantities be built out
of self-dual objects. However, we do not currently have a good way of associating
the frame field eIa with self-duality.
4.2
Ashtekar’s Variables
Let us reinstate the formalism of section 3.2.1. That is, let us bring back the foliation of M with Σ and its normals n, and the timelike observers γ(t, x) with their
corresponding tangent vectors ∂τ , etc. But now instead of working with EinsteinHilbert action and the phase space coordinates (qab , pab ) defined by the 3-metric,
we will use the self-dual action, and “new” phase space coordinates (A, E), which
are analogous to the vector potential and electric fields.
In this section, and what follows it, we will often without impunity inter-mix our
indices, especially the index types {i, j, k}, {I, J, K, L} and {s, t, u, v}, because they
all correspond to Minkowski indices in the present context, raised and lowered by η.
4.2 Ashtekar’s Variables
55
Furthermore, consider the expansion of a 2-form Fab in the form of 4.18 with
st
i
0i
components Fab
. We will often write Fab
for spatial index i when we mean Fab
.
We can do this unambiguously because we now have a privileged choice of time
provided by n. Then, an index of 0 will always correspond to contraction with
nI := eaI na . e.g., ε0JKL := nI εIJKL .
A is simply the spatial part of the projection of the self-dual connection onto
each Σt :
Aia := iqab A0i
(4.24)
b
On the other hand, E will be the “self-dual-compatible” part of a spatial triad
tangent to Σ built from the frame field:
EIa := qba ebI
and
a
ELI
:=
√
q
n[L EI]a
i
a
− εLI JK nJ EK
2
(4.25a)
(4.25b)
nI := eaI na has by construction nI = δ0I , so this simplifies into
a
Eia : = E0i
√
= qEia
(4.25c)
The (A, E) are called the Ashtekar (new) Variables. They relate to the ADM
variables thusly:
Aia = (3) Γia − iKai
(4.26a)
E ia Eib = |q|q ab
(4.26b)
I
bI
where (3) Γia := (3) Γ0i
a and Ka = Kab e .
We can perform a Legendre transform on the self-dual Lagrangian and obtain a
Hamiltonian ([8], p. 176),
H(Aia , Eia ) = −
Z
Σ
i
e |q|−1/2 N εijk Eia Ejb Fabk + τ aAia (Db E b )i + N a Eib Fab
(3)
(4.27)
where Da = (3)∇a is the covariant derivative on Σ. τ a := ∂τ is an independent
parametre, so we might as well write τ a Aia = λi . The Hamiltonian is once again a
sum of constraints, with Lagrange multipliers N , N a , and λi . These new constraints
Quantum Gravity
56
are just the usual suspects in disguise. Smeared out, One is related to the scalar
ADM constraint C(N ), the Hamiltonian or scalar constraint,
H(N ) :=
Z
e|q|−1/2 N εijk Eia Ejb Fabk
(3)
Σ
(4.28a)
and the vector constraint,
V(N ) :=
Z
i
eN a Eib Fab
(4.28b)
eλi (Da E a )i
(4.28c)
(3)
Σ
and the last one is a Gauss law,
G(λ) :=
Z
Σ
(3)
The last constraint excludes the additional redundancy produced by our transfer
from a geometrodynamic theory to an SU (2) gauge theory: qab has 6 independent
components at each point in Σ, with C providing 1 constraint and Ca , 3, for a total
of 2 degrees of freedom. On the other hand, Aia has 9 components; Gi provides 3
constraints, Va , 3, and H, 1. This leaves 2 degrees of freedom again.
The constraints are, once again, of the first order. We can show this explicitly, but
a slight re-arrangement will make things considerably neater: linear combinations
of constraints are again constraints, so we can define
C(N ) := V(N ) − G(N a Aa )
(4.29)
This is the diffeomorphism constraint. It is related to the ADM constraint C(N )
in a way that way that shall be elucidated later.
We define the Poisson brackets,
{f, g} =
Z
Σ
δG
δG
δF
δF
−
e(z)
a
δAia (z) δEi (z) δAia (z) δEia (z)
!
(3)
(4.30)
and this yields
{Aia (x), Ejb (y)} = iδab δji δ (3) (x − y) {Aia , Ajb } = 0 {Eia , Ejb } = 0
{C(N ), C(M )} = C(£N M )
(4.31)
(4.32a)
4.2 Ashtekar’s Variables
57
{C(N ), H(M)} = H(£N M)
(4.32b)
{G(λ), G(µ)} = G([λ, µ])
(4.32c)
{C(N ), G(µ)} = G(£N µ)
(4.32d)
{G(λ), H(M)} = 0
(4.32e)
{H(N ), H(M)} = C(K)
(4.32f)
K a = |q|1/2 Eia E bi (N ∂b M − M∂b N )
(4.33)
where
Observe that the ADM constraints C, C generate the same algebra as the new
constraints H, C So, similarly to the ADM case, the bracket of C(N ) with a scalar
function f in the canonical coordinates is proportional to its Lie derivative in N ,
so the C(N ) constraint generates infinitesimal “Lie dragging” 3-diffeomorphisms
across Σ. e.g.,
{C(N ), f (A)} = £N f (A)
(4.34)
and the scalar constraint generates diffeomorphism invariance in the temporal
direction parallel to n, Lie dragging the fields into the future, or the past. This
can be interpreted as a form of time evolution, hence the association of H with
Hamiltonian functions.
As in the Yang-Mills case, G generates an SU (2) gauge symmetry, which corresponds to the invariance of the theory under local spatial rotations.
Second Interlude: Canonical Quantization
Consider a free non-relativistic particle on the real line R. The canonical formalism
offers a relatively straightforward path from the classical system to a quantum
theory: we “promote” the phase space coordinates q, p into elements q̂, p̂ of an
algebra A, and functions f (q, p) are promoted to algebra elements as well, denoted
fˆ. The commutation relations on the promoted algebra is subject to conditions
given by the Poisson bracket:
1
\
{f,
g} = [ fˆ, ĝ ]
i~
(4.35)
Quantum Gravity
58
Then we let the elements of the algebra act on some vector space of functions on
R, which will make up a Hilbert space H. In the “position basis” for the particle on
the line, we find that the condition
1
\
{q,
p} = [ q̂, p̂ ] = 1
i~
(4.36)
is satisfied by operators q̂, p̂ defined by
q̂Ψ(x) = xΨ(x)
(4.37a)
∂Ψ
(x)
(4.37b)
∂x
A natural choice of inner product on H that satisfies the necessary axioms of
bilinearlity, symmetry and positive-definiteness is
p̂Ψ(x) = −i~
⟨Ψ, Φ⟩ =
Z
dx Ψ̄(x)Φ(x)
(4.38)
R
so we might identify H with the space of square-integrable functions L2 (R), and
define its inner product with the L2 product.
In some situations, however, the Hilbert space axioms may be too strict. Say we
want to make use of the basis of momentum eigenfunctions ψk (x) = √12π eikx . But
this basis is not square integrable on the line with the usual integration measure.
However, in the context of distributions/generalized functions, we can write
Z
dx ψ̄k1 (x)ψk2 (x) = δ (3) (k1 − k2 )
(4.39)
R
which is, in a sense, a limit on the L2 norm of ⟨ψ1 , ψ2 ⟩ where ψ1 → ψk1 and ψ2 → ψk2 .
This motivates the idea of a “rigged Hilbert space,” or a Gefland triple, which
defines the Hilbert space in terms of a dense subspace Φ ⊂ H, and its dual, Φ∗ ⊃ H.
Then we “rig up” H by defining its inner products with the duality of Φ, Φ∗ . In
other words, ⟨Ψ, Φ⟩H = Ψ(Φ). This is well defined because Ψ ∈ H ⊂ Φ, and
Φ ∈ H = H∗ ⊂ Φ∗ , by the Riesz representation theorem. For a more detailed
discussion, see chapter 5, or [4].
4.2 Ashtekar’s Variables
59
Let us make this whole procedure more precise. In [1], Ashtekar outlines a
program for the quantization of a general constrained Hamiltonian system, based
on the method of Dirac in [5]. The key steps are as follows:
1. Choose a subspace of the functions on the classical phase space associated
with classical variables, denoted S. Each element in S is to be promoted to
a quantum operator (on a yet-to-be defined Hilbert space H). In particular,
S has to be general enough to parametrize the phase space, and restrictive
enough to make promotion unambiguous, without factoring problems.
2. Promote each F ∈ S to an operator F̂ , and define a commutation relation
[·, ·] such that
1
\
{F,
G} = [ F̂ , Ĝ ]
(4.40)
i~
3. Find a representation of the resulting algebra A as the set of endomorphisms
on a complex vector space V.
4. Restrict V to the set of vectors annihilated by all of the promoted constraints
φ̂m . This linear subspace will represent the space of physical quantum states
Vphys
5. Define an inner product ⟨· , ·⟩ on Vphys , associated with an involution operator
⋆
on A, which defines self-adjointedness on the resulting Hilbert space.
6. Patch up the set of operators A so that they, and their adjoints, A⋆ , commute
with the constraints under the inner product. Then Vphys is closed under the
action of elements in A.
Each of these must be addressed in order to obtain a canonically quantized theory.
(End of Second Interlude.)
4.2.1
Quantum Constraints
Before we set sail for quantum seas, though, let us meditate a bit on what a hypothetical quantum theory of gravity must look like. The bracket relations 4.31
afford a straightforward representation of the promoted phase coordinates. In the
Quantum Gravity
60
“connection representation,” where states are functionals of the connection A,
Âia (x)Ψ[A] = Aia (x)Ψ[A]
(4.41a)
δ
Ψ[A]
δAia (x)
(4.41b)
Êia (x)Ψ[A] =
so then
[ Âia (x), Êjb (y) ] = δab δji δ (3) (x − y)
(4.42)
Let us consider the Gauss constraint G(λ). By construction, a physical state Ψ
must be annihilated by Ĝ(λ) for every su(2)-valued distribution λ. Without even
explicitly choosing a promotion, the Poisson bracket (similar to equation 3.44a)
and equation 5.8 already show that
h
i
1 + εĜ(λ) Ψ[A] = Ψ[A − εDλ]
!
(4.43)
= Ψ[A]
for small ε. This result is very good; it means that the space of physical states does
not distinguish between expressions that differ by a gauge transformation.
Ignoring for now the constraint algebra 4.32, we can promote this to a quantum
constraint, if only formally:
Ĝ(λ)Ψ[A] =
Z
eλi Da
(3)
Σ
δΨ[A]
δAia (z)
(4.44)
Here we’ve made the choice of ordering the factor of A in D to the left of E.
The diffeomorphism constraint receives a similar treatment: from the Poisson
bracket and equation 5.8 again,
h
i
1 + εĈ(N ) Ψ[A] = Ψ[A] + ε£N Ψ[A]
≈ Ψ[A + ε£N A]
(4.45)
!
= Ψ[A]
This means that the physics is undisturbed if we slide the fields on Σ along the
flows of N .
4.2 Ashtekar’s Variables
61
To promote C(N ), we may formally promote the vector constraint V(N ) first,
V̂(N )Ψ[A] =
Z
i
eN a F̂ab
(3)
Σ
δΨ[A]
δAib (z)
(4.46)
which gives
Z
i
eN a F̂ab
+ Âia Db
(3)
Σ
δΨ[A]
δAib (z)
(4.47)
i
might be ∂a Âib − ∂b Âia − εijk Aˆja Âkb .
where a good operator ordering on F̂ab
Up to operator-ordering, the promotion of the constraints C, G yielded terms
that were merely polynomial in the fields and their derivatives. The Hamiltonian
constraint is a lot more troublesome. The integrals that smear the constraints
G and C are integrated with a coordinate-dependent 3-form (3)e(z) 1 rather than
√
the coordinate-independent volume form q (3)e This is because we absorbed the
√
factors of q into E, saving us from having to deal with the non-polynomial terms
that plagued the ADM formalism. The Hamiltonian constraint, however, has a
factor of |q|−1/2 which cannot be dealt with thusly. So it seems that we cannot even
formally write down a promotion of H, never mind ordering the operators to fit the
constraint algebra.
There exists, however, a clever trick with which we may rewrite H(N ), introduced by Thiemann, which may be found in section 10 of [26]. First we define a
classical-valued function V purely in terms of the triad 2 :
V (E) : =
Z
s
(3)
e
Σ
=
Z
1 εijk εabc Eia Ejb Ekc 3!
√ (3)
q e
(4.49)
Σ
1
2
e.g., in Cartesian coordinates, (3)e = dx ∧ dy ∧ dz; in spherical coordinates, (3)e = dr ∧ dθ ∧ dφ.
√
We can work out factors of q quick and dirty, as follows:
ea eb = qab
=⇒ det(ea eb ) = det qab
√
=⇒ det(ea ) = sgn(e) q
a
3/2
=⇒ |det(E )| = |q|
(4.48)
a
|det(e )|
a
=⇒ |det(E )| = |q|
√
because E a = qea , and the identities det(ea ) = det(ea )−1 and det(c E a ) = c3 det(E a ), the latter
holding for 3 × 3 matrices.
Quantum Gravity
62
which outputs the 3-volume of the spatial manifold Σt . When promoted into a
quantum operator V̂ , it is called the volume operator. The trick, then, is in noticing
that
{Aia , V } = |q|−1/2 εijk εabc Ebj Eck
(4.50)
so that H(N ) can be rewritten
H(N ) =
Z
eN {Akc , V }Fkab εabc
(3)
(4.51)
After quantization, we can work with eigenfunctions of V̂ so that {Akc , V } promotes to a polynomial or Taylor series in A. We will not concern ourselves too
much with the specifics of the Hamiltonian constraint in what follows, however.
Chapter 5
Loop Quantum Gravity
Our treatment of the constraints in the First Interlude is quite contrasted to that
of the Second. In the former, we slipped in to the Hamiltonian a set of artificial
coordinates λm , and the constraints φm = 0 emerged as an extra set of equations
of motion in the associated momenta µm . In the quantum theory, however, the
constraints were strong-armed in to the theory by banishing from the space of
physical states Vphys all that which is not annihilated by the quantized constraints
φ̂m , “by hand”.
In the case of canonical quantum gravity, a hybrid approach may be used.
We construct our space of states by selecting those that are annihilated by the
diffeomorphism and Gauss constraints, and the Hamiltonian constraint is imposed
through a “dynamical” equation on the resulting space:
0 = ĤΨ[A]
= εijk F̂abk
δ δ
Ψ[A]
δAia δAjb
(5.1)
The right-hand side of this is reminiscent of the non-relativistic Hamiltonian for a
free particle,
1
∂ ∂
Ĥ1p ψ(x) = − δ ab a b ψ(x)
(5.2)
2 ∂x ∂x
which perhaps illustrates the purely kinematic nature of the symmetry generated
by the Hamilton constraint1 .
1
States that solve ĤΨ = 0 are annihilated by the Hamiltonian constraint, but the converse might
not be true.
Loop Quantum Gravity
64
In this context, equation 5.1 is the Wheeler-DeWitt equation, because all the
other constraints that make up the full Hamiltonian are satisfied by definition. It
can be thought of as an equation describing time evolution, since it is related to
the Hamiltonian constraint, the generator of temporal diffeomorphisms. However,
it is not a “true” dynamical equation because there is no explicit reference to time,
and thus no parametre relative to which evolution occurs.
We note that, up to now, we have been using the self-dual part of a complexified
Lorentz algebra to define our field A:
AaIJ
i
i
=
ωaIJ − εIJ KL ωaKL
2
2
(5.3)
where ωa is the connection 1-form that solves equation 2.47. This simplified matters because A was then a connection 1-form, with a straightforward projection on
to Σ by taking its 0i components. On Σ, A has a very simple su(2) ∼
= so(3) representation, which is just the angular momentum subalgebra of the (self-dual) Lorentz
algebra. However, we also had to deal with the unhappier consequences of pushing
our real physics out onto the complex plane, leading to complex Einstein equations.
One might find it more convenient to work with the real-valued field
A′aIJ
= ωaIJ
1
+ εIJ KL ωaKL
2
(5.4)
This field would be self-dual if we were working with a Riemannian metric instead
of a Lorentzian one. It is manifestly real-valued because complexification does
not occur in the first place. However, A′ does not transform as a connection under
general Lorentz transformations, because we lose the transformation property
from 4.15. i.e., A′ does not belong to a representation of the Lorentz algebra.
Nevertheless, this A′ is still a field on R × Σ, and it transforms properly under
spatial SU (2) rotations because the field has an su(2) representation when projected on Σ: the two terms in the definition 5.4 transform the same way under pure
rotations, but not under boosts.
It must be observed however that using this field will lead to a severe change in
our physics, because the theory obtained from this field would turn out to describe
65
a Riemannian 4-dimensional spacetime metric. Because the field A no longer
represents a Lorentz algebra, we lose the hyperbolic structure of time in our theory,
and obtain instead a Euclidean “time” [20]. On the other hand, problems with
the non-compactness of SL(2, C) give rise to unsolved problematic complications
which prevent us from using the self-dual connection in the theory [7].
We seem to have found ourselves in a bit of a quandary, where neither treatment
is acceptable. However, we will be more modest with our ambitions in our coverage
of quantum gravity: we shall set the stage with the connection representation of
the kinematical space of states that are annihilated by the diffeomorphism and
Gauss constraints, and then simply bow out.
Our construction of the constrained space of physical states H will be simplified because we already know that our constraints C(N ) and G(λ) are generators of diffeomorphisms and gauge rotations, e.g., from equations 4.43 and
4.45. Thus we need only to select functionals Ψ[A] that are manifestly gauge- and
diffeomorphism-invariant to fill up the space H. In what follows, we will formulate
H as a Gefland triple S ⊂ H ⊂ S ∗ , rather than deal with the stricter laws of Hilbert
spaces.
Following Chapter 6 of [20], we will start in section 5.1 with a space of select
functionals Ψ[A], denoted K, which will be the base space upon which we build our
space of states. Then in section 5.2, we build gauge invariant linear combinations
of the basis states of K to obtain a basis that spans a manifestly gauge-invariant
space K0 . Finally, we identify states that are “Lie-diffeomorphic” to each other on
Σt to obtain KD = H, the kinematic, constrained space of states in section 5.3 . This
is the final destination in our coverage of Canonical Quantum Gravity: a rigged
Hilbert space of functionals Ψ[A] on Σ, with argument A, a su(2)-valued 1-form
that transforms as an SU (2) connection under local gauge rotations.
As an aside: if had we started from the Palatini action and defined our canonical
variable A as
A = (3) Γia + βKai
(5.5)
Loop Quantum Gravity
66
the Hamiltonian constraint has the more convoluted general form2
H(N ) =
Z
"
(3)
−1/2
e|q|
Σ
N Eia Ejb
!
β2 + 1
i
Kb]j
ε Fabk + 2
K[a
2
β
ijk
#
(5.6)
β is called the Barbero-Immirzi parametre. For the choice β = −i, we get the
Ashtekar variable A, and the second term in the sum vanishes; for β = 1 we get
the real valued field A′ . Had we chosen the real field, the quantum commutator in
equation 4.42 would have instead been
[ Âia (x), Êjb (y) ] = iδab δji δ (3) (x − y)
(5.7)
which would have yielded the operators
Âia (x)Ψ[A] = Aia (x)Ψ[A]
Êia (x)Ψ[A] = −i
δ
Ψ[A]
δAia (x)
(5.8a)
(5.8b)
analogous to the one-particle non-relativistic theory. Since we will be working
with the real valued field A′ (henceforth, simply A,), we will need to deal with a
much more complicated expression of the Wheeler-DeWitt equation.
We will not need to concern ourselves with this additional complexity, or with
commutators in general, in what follows; we only state these for the sake of completeness.
5.1
The Base Space
On Σ, the field Aa has an su(2)-valued representation:
Aa = Aia
2
σi
2i
(5.9)
The form of the diffeomorphism and Gauss constraints would also be changed, of course.
However, it will not change the fact that they would still be generators of diffeomorphism and gauge
symmetries, so their exact form will not matter.
5.1 The Base Space
67
Observe that a family of manifestly gauge-invariant functionals in A may be constructed out of the set of holonomies, by simply taking their trace:
Hγ [A] = Tr P exp −
I
γ
Ai
(5.10)
where γ is any closed loop with an orientation. That this is gauge invariant can be
seen from equation 2.27 for a loop.
It turns out that these functionals span the space of of gauge invariant functionals on Σ; indeed, this property was the historical impetus for the study of loops in
canonical quantum gravity, hence the umbrella term of Loop Quantum Gravity.
With this basis, the (smooth 1-parametre) diffeomorphism invariance on A can
be transferred over as an equivalence relation on the set of loops γ:
P exp −
I
∗
φ A = P exp −
γ
!
I
A
(5.11)
φ(γ)
So we need only γ which are homotopically distinct to parametrize our basis on the
space of states. i.e., classes of knots and links that cannot be smoothly deformed
into one another. This leads to the surprising cameo appearance of knot theory in
quantum gravity. [8, 3]
However, even with this equivalence relation, this basis is over-complete; this
fact would lead to undesirable redundancies in describing physical states. Instead,
we will end up considering the better-tempered basis of spin network states, which
will be a continuation of the spirit underlying this line of inquiry.
We start by defining a set of “cylindrical” functions, S, which will be a dense
subspace of the base space K in the Gefland triple. Consider:
1. A function f : SU (2)L → C for L > 0 ∈ N,
2. An ordered3 collection Γ of L oriented non-trivial paths in Σ, (γ1 , · · · , γL ),
3. A map (A, γ) → U (A, γ) ∈ SU (2) given by P exp −
3
That is, (γ1 , γ2 ) ̸= (γ2 , γ1 ) unless γ1 = γ2 , and so on.
H
γ
A
Loop Quantum Gravity
68
then we can define a set of functionals Ψ[A]:
ΨΓ,f [A] := f (U (A, γ1 ), · · · , U (A, γL ))
(5.12)
Choosing f that are not trivial (i.e., constant) in any of their variables, S is defined
as the space of ΨΓ,f for all sufficiently differentiable f and Γ.
We may be well-concerned about the enormous size of this space, since there
are very many distinct choices of oriented paths γ in Σ, and the fact that we are
taking ordered collections of these loops can only make things worse. We will see
later when we build KD that diffeomorphism invariance drastically reduces the
size of this space, in a way similar to the reduction afforded by 5.11.
We can define an inner product right away:
⟨ΦΓ′ ,g , ΨΓ,f ⟩ =



Z
dU1 · · · dUL g(U1 , · · · UL )f (U1 , · · · UL ) if Γ = Γ′


0
if Γ ̸= Γ′
(5.13)
where dU is the Haar measure on SU (2), which is just a normalized integration
measure on the 3-sphere.
We define K as the limit of sequences {Ψn }n∈N ⊂ S for which ||Ψn || = ⟨Ψn , Ψn ⟩
converges. S ∗ , being the dual space to S, has the natural definition as the limits
of {Ψn }n∈N for which ⟨Ψn , Ψ⟩ converges, for every Ψ ∈ S.4 This makes it manifestly
clear that S is a dense subset of K with the metric topology defined by the L2 norm.
The definition of K as the completion of S with respect to the L2 norm means
that we can take not just finite linear combinations of the “basis” states ΨΓ,f , but
in fact arbitrary sums, so long as the sum has a finite norm. That we should be
allowed to do this in our space of states is completely reasonable. For example, in
the 1D infinite square well in NRQM, we often had to express wavefunctions in
P
n2 π 2
terms of series of the energy eigenfunctions: ψ(x) = ∞
n=1 an sin( 2L2 x), because
merely finite sums would generally not suffice.
4
In L2 , we identify functions that differ by a function of norm zero. i.e., ||f − g|| = 0 ⇐⇒ f ∼ g.
In other words, the limits are well-defined and unique up to an equivalency.
5.2 The Gauge Invariant Space
69
Similarly, S ∗ accommodates “states” that do not have a well-defined finite norm,
but do have a well-defined inner product with physical states. Examples of these
are the momentum and position eigenfunctions of a free particle on the line, eipx
and δ(x − y), respectively.
By our definition of the inner product in 5.13, the base space K is already neatly
divided into a set of mutually orthogonal subspaces parametrized by the ordered
collections Γ, KΓ ∼
= L2 [SU (2)L ]. In what follows, it will be convenient to find an
orthonormal basis for each KΓ .
There is a ready-made such basis given by the Peter-Weyl theorem, which implies that L2 [SU (2)] decomposes into an orthogonal direct sum of finite-dimensional
irreducible representations (irreps) of SU (2). In other words, the space is spanned
by the (2j +1)×(2j +1) matrix representations of SU (2), R(j)a b [U ], for non-zero halfintegers j. The j labels the particular representation to which the matrix belongs.
For computational simplicity, we will label these basis states directly with their
matrix components. So then a basis |j, a, b⟩ of L2 [SU (2)] may be defined, satisfying
⟨U |j, a, b⟩ = R(j)a b [U ]
(5.14)
for a fixed, faithful spin-j representation R(j) , for each positive half-integer j. The
extension to L2 [SU (2)L ] is just the tensor product of L-many such basis vectors,
|Γ, J, A, B⟩ := |j1 , a1 , b1 ⟩ · · · |jL , aL , bL ⟩, for multi-indices J, A, B. And thus, in the
connection representation,
⟨A|Γ, J, A, B⟩ = R(j1 )a1 b1 [U (A, γ1 )] · · · R(jL )aL bL [U (A, γL )]
When all distinct Γ are considered, this becomes a basis on K =
5.2
L
Γ
(5.15)
KΓ .
The Gauge Invariant Space
To construct K0 , we need to find linear combinations of the basis vectors |Γ, J, A, B⟩
that are invariant under local SU (2) transformations. The fact that these are associated vector spaces to SU (2) irreps makes this rather straightforward: consider
the action of a local SU (2) rotation λ : Σ → SU (2) on |Γ, J, A, B⟩. From equations
Loop Quantum Gravity
70
2.27 and 5.15,
⟨A|Uλ |Γ, J, A, B⟩ =⟨λ(A + d)λ−1 |Γ, J, A, B⟩
′
=R(j1 )a1 a′1 [λ(γ1,1 )] R(j1 )b1 b1 [λ−1 (γ1,0 )] · · ·
′
R(jL )aL a′L [λ(γL,1 )] R(jL )bL bL [λ−1 (γL,0 )]
(5.16a)
⟨A|Γ, J, A′ , B ′ ⟩
where γn,0 is the initial point of γn , and γn,1 is its endpoint, and by unitarity, ⟨A|Uλ =
⟨Uλ−1 A| = ⟨λ(A + d)λ−1 |. Since this holds for arbitrary fields A, and since neither
the basis nor the rotation matrices depend on A, we can write
′
Uλ |Γ, J, A, B⟩ = R(j1 )a1 a′1 [λ(γ1,1 )] R(j1 )b1 b1 [λ−1 (γ1,0 )] · · ·
′
R(jL )aL a′L [λ(γL,1 )] R(jL )bL bL [λ−1 (γL,0 )]
′
(5.16b)
′
|Γ, J, A , B ⟩
Now, if we could find a collection of SU (2) tensors I that trivialize rotations, e.g.,
I[jm , · · · jn ]am ···an R(jm )a1 a′m [U ] · · · R(jn )an a′n [U ] = Ia′m ···a′n [jm , · · · jn ]
(5.17)
then by a suitable identification of initial and end points of the curves γn , we can
write down gauge-invariant states as, for example,
′
′
|S⟩ :=I[j1 , · · · jL ]a′1 ···a′L I[j1 , · · · jL ]b1 ···bL |Γ, J, A, B⟩
(5.18)
This state corresponds to the particular case where all the curves in Γ start at the
same point, p, and all end at another point q. If we wanted to join end points then I
would have raised b indices mixed in. In that case, the b indices would correspond
to the conjugate representations, R−1 .
The tensors I[J] “shrug off” the effects of a local SU (2) rotation, leaving the
overall state |S⟩ unchanged. Thus the collection of tensors I[J], the intertwiners
of the representations of SU (2), find for us states |S⟩ that are gauge invariant.
These intertwiners are in fact no strangers to us: when only joining initial points,
they are simply the Clebsch-Gordan coefficients cm1 ···mn that produce trivial spin-0
representations out of tensor products of spin-(jm , · · · jn ) irreps. For example, for
5.3 The Diffeomorphism-Invariant Space
71
j1 = 1/2, j2 = 1/2, this is the tensor cm1 m2 that gives the singlet state combination5
|0, 0⟩ =
X
cm1 m2 |m1 , 1/2⟩|m2 , 1/2⟩
1
= √ (| ↑↓⟩ − | ↓↑⟩)
2
(5.19)
The connection is apparent: it is only the spin-0 representation that is invariant
under general spatial rotations [10].
This motivates the definition of spin network states |Γ, J, In ⟩, each labeled by
a triplet (Γ, J, IC ) and associated pictorially with a web with L links and N nodes.
This state assigns a spin-jn irrep (called a colouring,) and an orientation to each
“link” γn , and an intertwiner IC to each node, labeled by an index C, which trivializes the (tensor product of the) irreps carried by each of the γi that meet on the
node. These states comprise an orthonormal set of states, as states with different
choices of colouring J or intertwiners I are orthogonal [20]. We can use the method
of Young tableaux to take the product j1 ⊗ · · · ⊗ jn ; a set of links can only form a
node if their irreps form a trivial representation.
Then the span of spin network basis states |Γ, J, In ⟩ gives us gauge-invariant S0 ,
and K0 and S0∗ can be defined with the same completions with respect to the L2
inner product as last time.
5.3
The Diffeomorphism-Invariant Space
The last piece of the puzzle is brought to us by means of an equivalence relation.
Consider an arbitrary cylindrical function ΨΓ,f [A] ∈ S, and a family of diffeomorphisms φt , smooth in t ∈ [0, 1], for which φ0 is the identity map. Then from the
definition, we have
ΨΓ,f [φt∗ A] = Ψφ−1
[A]
(5.20)
t (Γ),f
5
When end points are involved, we will need to invert the signs on the corresponding index m,
because they are in the conjugate representation. If we are exclusively joining endpoints, the result
will be unchanged, but mixing endpoints and initial points requires extra care to the signs on the
matrix indices m.
Loop Quantum Gravity
72
where φ(Γ) is the ordered collection of oriented curves (φ(γ1 ), · · · φ(γn )) for Γ =
(γ1 , · · · γn ). This can be seen by pulling back the integral, as in equation 5.11.
So then, the condition of diffeomorphism-invariance suggests that we identify
basis states in S0
|Γ, J, In ⟩ ∼ |Γ′ , J, In ⟩
⇐
∃ smooth φt : φ0 (Γ) = Γ; φ1 (Γ) = Γ′
(5.21)
This relation identifies a large number of mutually orthogonal subspaces S0Γ′ ∼ S0Γ ,
drastically reducing the size of the space of distinct physical states; it saves our
physical space of states from being spanned by a continuum of basis vectors to
just countably many of them.
To make this more clear, we define an equivalence on the oriented graphs Γ
instead:
∃φt : φt (Γ) = Γ′ ⇐⇒ Γ ∼ Γ′
(5.22)
Since φt is smooth in t, it must preserve the ordering and orientation of the links.
These equivalence classes are called (oriented) s-knot classes, in analogy to knot
classes. We will denote s-knots classes as [K] = Γ/ ∼, and a representative of [K]
as K. Indeed, the class of s-knots classes reduces to the class of knot classes when
we restrict it to graphs consisting of only one (necessarily closed) curve, and Σ ∼
= R3 .
So now we can label states on KD by identifying spin network states with equivalent oriented graphs K, as |[K], J, IC ⟩. With this identification, the inner product
on cylindrical functions must then undergo the modification
⟨ΦΓ′ ,g , ΨΓ,f ⟩ =



Z


0
dU1 · · · dUL g(U1 , · · · UL )f (U1 , · · · UL )
if Γ ∼ Γ′
if Γ Γ′
(5.23)
so that |[K], J, In ⟩ forms an orthonormal basis on SD . Taking the same L2 comple∗
tions as before for KD and SD
, then, we are done.
So for states in the physical space of states, only the inter-weaving of the oriented, ordered links γ, their colourings J and the choice of intertwiners I determine
the state− not the points on Σ to which they are mapped. In fact, the only role that
Σ plays in defining KD is as the topological background through which the graphs
5.3 The Diffeomorphism-Invariant Space
73
Γ are deformed and identified.
Then we no longer need to think of spin networks as being embedded in Σ and
occupying space, they are space− we are at last free of the crutch of the “chessboard”
that is the manifold Σ, and we can now play the game directly, without extraneous
descriptions of the state pertaining to absolute positions and gauge.
Fig. 5.1 Two inequivalent knots, and a pair of equivalent s-knots.
Chapter 6
Conclusions
∗
Now that we have constructed the kinematic space of states SD ⊂ KD ⊂ SD
, we
are finished, so far as this dissertation is concerned. But in a much broader sense,
we have only just started. The board is set, and the pieces assembled− now it’s
finally time to play. But alas, we can only cover so much material in the amount of
space/time allocated to us, and we will have to sign off here.
Proceeding further down the path set out by our narrative, the next step would
logically be a coverage of the geometric operators  and V̂ , respectively the quantum area and volume operators. The (unquantized) area operator has the expression
Z
q
A(σ) = (2)e na Eia nb E bi
(6.1)
σ
q
where σ is a 2-dimensional manifold embedded in Σ, and (2)e na Eia nb E bi is its
volume form. The spectrum of the quantum operator can be determined, and is of
the form (restoring physical constants,)
−3
q
Â(σ)|S⟩ = 8πG~c γ j(j + 1)|S⟩
(6.2)
where γ is again the Barbero-Immirzi parametre. The spectrum of the area operator
thus gives a quantitative and testable prediction of loop quantum gravity. Its
smallest permissible eigenvalue, for γ = 1, is equal to
√
A0 = 4 3πG~c−3
(6.3)
Conclusions
76
and corresponds to the quantum of area. Thus spacetime is discretized on the
Planck scale, cutting the Gordian Knot on the problem of renormalization.
Perhaps also surprising is the fact that the loop quantum gravity formalism
already has a known exact solution when the cosmological constant Λ is non-zero.
It is given by
6
(6.4)
ΨCS [A] = exp − SCS (A)
Λ
where
2
SCS (A) = Tr A ∧ dA + A ∧ A ∧ A
3
Σ
Z
(6.5)
This is the Chern-Simons state [3]. It is gauge- and 4-diffeomorphism-invariant,
and therefore it solves
ĤΨCS [A] = 0
(6.6)
The physical interpretation of this state, however, is still not entirely clear.
Though still an immature science, loop quantum gravity has already found
applications, in cosmology and black hole thermodynamics. Very surprisingly, in
LQG cosmology, the inflationary phase can be derived analytically through the
quantum properties of the gravitational field itself, without the use of, say, a scalar
inflation field. Furthermore, the initial cosmological (big bang) “singularity” is no
longer singular, due to a lower bound on the permissible values of volume. The
latter property is exactly the what we would like to see in a consistent quantum
theory of gravity that describes the early universe. [20]
In black hole thermodynamics, an expression for the entropy of a black hole in
terms of its area is usually given as
SBH =
kA
4G~
(6.7)
This result is commonly known as the Bekenstein-Hawking entropy. A calculation
of the same quantity through LQG seems to set the Immirzi-Barbero parametre
√
at γ = ln(2)/π 3, but the interpretation of this result is not presently very well
understood, either. [20]
77
There is also a formulation of the theory in terms of path integrals and transition
amplitudes, which gives rise to spin foams, a generalization of spin networks...
All of this is to say that we have only here covered one edge of the tip of the
iceberg. Research in LQG remains extremely active today, with multiple teams
worldwide working in collaboration to develop it further.
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