National University of Singapore Science Faculty / Physics Department Masters Thesis 2008/2009 In Partial Fulfilment of M.S.c Loop Quantum Gravity With Matter Fields Extension Supervisor: Dr Kuldip Singh Co-Supervisor: Prof Wayne Michael Lawton Ching Chee Leong HT050426L Abstract In this thesis, we attempt to review the full theory of Loop Quantum Gravity (LQG) with Standard Model(SM) type of matter fields extension Firstly, we briefly discuss the old canonical gravity by using Arnowitt, Deser and Misner (ADM) formulation in conventional metrical variables These will eventually lead to Wheeler-DeWitt super-Hamiltonian form of Einstein’s General Relativity (GR) as first class Dirac constraint system As we all know, this old formulation facing the difficulties to be promoted to quantum theory due to the non-polynomial structure of the constraints Also, perturbative studies shown that it is renormalized (under conventional Quantum Field Theory QFT) at most up to two loops level Next, we start our main discussion from the famous Ashtekar reformulation of gravity in term of self-dual SL(2,C) connection dynamics, and thus directly introduce the so called Ashtekar-Romano-Tate (ART) model to couple the standard model matter (classically) into the formalism In both cases, genuine Lagrangian formulation are presented as well For the matter coupling case, as expected, the effective theory comes out to be the EinsteinCartan-Sciama-Kibble-Dirac(ECSKD) 1st order theory rather than Einstein-Dirac(ED) 2nd order theory All the constraint algebras are modified due to matter degree of freedom and the nontrivial features are brought forward by fermionic fields For the consistency check, we consider the ECSKD theory and convinced that it is equivalent to the self-dual EinsteinDirac theory In the next part of the thesis, we consider Ashtekar-Barbero-Immirzi (ABI) formulation (with real SU(2) gauge connection) of GR and couple the system minimally/non-minimally to the spinorial matter fields At the level of effective theory, the theory turns out to be similar to Einstein-Cartan type (and Nieh-Yan topological term appeared) with interaction term similar to Fermi-like contact interaction With the non-minimal coupling scheme, one can study the gravity induced parity violation in gravity-fermion sector Meanwhile, torsions are induced by the spinorial current since the fundamental structure of spacetime is “modified” by the torsion source, i.e Grassmanian valued fermionic fields As the historical motivation, we perform the review on famous Rovelli-Smolin loop representation By assuming the reconstruction theorem hold, now the canonical variables are the so called gauge invariant Wilson Loop and it’s conjugate momentum (in which defined with a hand operator along the loop) The quantum loop representation can be realized by constructing a linear representation of a deformation of this loop algebra Afterward, following the ideas from lattice gauge theory, fermionic loop variables can be realized by considering an open path with fermions associated at the node This is a natural extension of the matter-free loop representation Finally, we summarized up our discussion by giving a brief outline on the modern LQG in Spin Network basis Some results in Spin network basis (i.e quantization of geometrical operator) and phenomenological aspects of the theory i.e Loop Quantum Cosmology (LQC), Black hole entropy etc will be highlighted Meanwhile, we will briefly mention on the complication of the quantization of Scalar constraint in Thiemann formalism due to the fermionic torsion contribution Acknowledgement First of all, I sincerely thank my supervisor Dr Kuldip Singh and my co-supervisor A/P Wayne Lawton who devoted so much effort in guiding me along until the completion of this thesis and allowed me to work under their supervision I am very grateful for their kindness and patience I would like to take this opportunity to show my appreciation to all the lecturers who taught me at the graduate level, especially Prof Baaquie Belal E, Prof Chong Kim Ong, Prof Ser Choon Ng and Prof Hong Kok Sy Thanks to my course mates We have so much laughter throughout these three years in NUS and it will be part of my sweet memory Special thanks for my senior Andreas Keil, who willing to share his idea with me regarding the project and help me along the way while I stuck with the conceptual difficulties and derivations Also, thanks to my friends and colleagues A Dewanto, W.K Ng, M.L Leek , H.S Poh, Z.H Lim and S.Y Ng for their valuable discussions I am also grateful to my family members especially my parents for their greatest support At last, but most importantly to my dear Yvonne, she know why Once again, many thanks to all of you Contents Introduction and Motivation Canonical Formulation of G.R : Geometro-dynamics 10 2.1 Lagrangian of General Relativity: Standard Einstein-Hilbert Action and Variational Principle 10 2.2 Geometro-dynamical Variables: ADM Formulation and Wheeler De-Witt equation 14 (New)-Canonical Gravity: Connection-dynamics 33 3.1 Ashtekar Hamiltonian Formulation on the Extended ADM Phase Space: SO(3, C) self-dual connection and SL(2, C) soldering form Representation 33 3.2 Covariant Self-Dual Lagrangian Approaches: Jacobson-Smolin-Samuel Action SJSS 69 (New)-Canonical Gravity with Standard Model 88 4.1 Ashtekar Variables with Matter (Standard Model) Coupling 88 4.2 Equivalence between self-dual Einstein-Dirac theory and Einstein-Cartan-SciamaKibble-Dirac theory 128 (Modern)-Canonical Gravity: Real SU (2) Connection 138 5.1 Barbero Hamiltonian Formulation: Arbitrary real Immrizi-Barbero Variables 138 5.2 Covariant Lagrangian Formulation of Barbero Hamiltonian Formulation: Holst action SHolst 165 Immirzi-Barbero Parameter and Effective Theory 184 6.1 Physical effect of Immirzi Parameter, Torsion, Parity Violation etc in Gravitational Sector 184 6.2 Effective Theory and Nieh-Yan Invariant 206 (Modern)-Canonical Gravity with Fermionic Coupling 213 7.1 Holst’s Action with Fermionic coupling 213 Loop Representation: Towards Spin Network 8.1 Classical Dynamics of Gravity 8.1.1 Loop Variables and small T’s -algebra Representation (classical theory) 8.2 Quantum Dynamics of Gravity 8.2.1 Quantum Theory: The Connection Representation 8.2.2 The Quantum Loop Representation 8.3 Matter Coupling in Loop Representation: Fermionic Loop Variables 8.3.1 Classical and Quantum Fermions in Loop Space Representation 247 247 247 261 261 267 272 272 CONTENTS Spin Networks: Modern Quantum Theory of Gravity 283 9.1 Spin Network basis 283 A Torsion-Freeness Extrinsic Curvature B SL(2, C) and SU (2) Spinors: Concepts and Some Useful Relations B.1 General Setting B.2 SL(2,C) Spinors B.3 SU (2) Spinors B.4 Relation between SL(2, C) spinors and SU (2) spinors B.5 Sen Connection B.6 Dictionary: From SU (2) spinors to Triads 294 296 296 297 303 306 309 313 C Poisson Bracket of Ashtekar Free-Field Theory 320 D Poisson Bracket for the ART Matter Coupling Model 334 E Dirac Gamma Matrices and Some Useful Relations 343 Chapter Introduction and Motivation Nowadays, we know that modern physics rests on two most fundamental building blocks, namely: Einsteinian General Relativity (GR) and Quantum Mechanical (QM) theory General relativity is a geometrical interpretation of gravity where degrees of freedom of gravitational field are encoded in the geometry of the spacetime, while Quantum Mechanics governs all the microscopic behavior of matters According to Einstein viewpoint and his famous Einstein’s field equations, geometry is curved when and where matter is localized Therefore, in General Relativity, geometry is a dynamical quantity that cannot be prescribed a priori but is in interaction with matter The equations of nature are background independent in this sense; there is no spacetime geometry on which matter propagates without backreaction of matter on geometry In other words, the gravitational field defines the geometry on top of which its own degrees of freedom and those of matter fields propagate General Relativity is not a theory of fields moving on a curved background geometry; general relativity if a theory of fields moving on top of each other This is the gist of General Relativity: Diffeomorphism invariant or background independent Since matter is described by Quantum Mechanics, which in turn couples to geometry, we need a quantum theory of gravity The absence of a viable quantum gravity theory to date is due to the fact that conventional quantum field theory (i.e Minkowskian QFT) as currently formulated assumes that a background geometry is available, thus being inconsistent with the principles of general relativity In order to construct quantum gravity, one must reformulate Quantum Mechanics in a background-independent way In other words, in quantum gravity, geometry and matter should both be “born quantum mechanically” In contrast to approaches developed by particle physicists, one does not begin with quantum matter on a background geometry and use perturbation theory to incorporate quantum effects of gravity1 There is assumed to be a manifold to begin with, but no metric or indeed any other fields in the background As a result, by taking the principle of general relativity seriously, it is necessary for us to the quantum physics of topological manifold From the foresight of Ashtekar, we can see that there lines of attack to formulate a quantum theory of gravity; the particle physicists approach, the mathematical physicists’ approach and the general relativists approach The particle physicists have pertubative (relativistic) quantum field theory as their main success This can be seen via the remarkable experimental success of the Standard Model in describing of Seemingly, this is the approach taken by the other promising candidate of quantum gravity: Superstring Theory [1] CHAPTER INTRODUCTION AND MOTIVATION fundamental interactions including electromagnetism, weak and strong interactions2 For the gravitational sector, by considering a perturbed background metric3 , they have quanta of mass zero and spin-2 and these are the gravitons However the theory fails to be renormalizable When Supersymmetry (SUSY) is in-cooperated (the so-called Super-Gravity or SUGRA model), it appeared renormalizable, but it turns out that detailed calculations revealed non-renormalizability at the two loop level String theory developed in another direction but turns out to be promising as a theory of everything with gravity and many other fields included in it However, the question is whether perturbative methods is the way to go or not Obviously Super-String theory at current moment is not capable of addressing the non-pertubative behavior and the diffeomorphism nature of the gravitational interactions The mathematical physicists would try define axioms to construct a theory For quantum gravity, keeping with the spirit of general relativity of background independence, there is no clue on how to construct axioms without reference to any metric (at least so far) Canonical quantization could be a possible strategy because we can have a Hamiltonian theory without introducing specific background fields Dirac’s constraint analysis will take care of the diffeomorphism invariance of the theory However we lose manifest covariance and there are ambiguities in how the quantum theory is constructed The general relativists regard Einstein’s discovery that gravity is essentially a consequence of the geometry of spacetime, as the most important principle to uphold Hence in formulating a quantum theory of gravity, there should not be any splitting of the metric into a kinematical part and a dynamical part, or generally, there should not any introduction of background fields into the theory Dirac’s constraint analysis (genuine canonical quantization method) and path integral method are two methods that allow treatment of the theory with its symmetries taken into account systematically Thus, in other words, one needs to realize the so-called diffeomorphism invariance or background independent principle at the quantum mechanical level and employ it to single out the meaningful physical quantum gravity states Loop Quantum Gravity (LQG) or Quantum General Relativity (QGR) is an attempt of a canonical quantization method on General Relativity (GR) to construct the quantum theory that respects the diffeomorphism symmetries of GR Dirac’s constraint analysis is a systematic way to construct the Hamiltonian version of the theory with the symmetries of the theory fully taken into account The methodology of quantization in Dirac’s constraint analysis is quite well laid out as well In LQG scheme, we have a few conservative assumptions as the following: Background Independent Principle or Diffeomorphism Invariance: We take the gist of Einstein’s general relativity viewpoint seriously Although there is no conceptual reason to believe that the Einstein classical description of gravitational interaction manifested in terms of spacetime curvature is generally true even at the quantum level, however canonical quantum gravity treats the diffeomorphism invariance seriously as the basic language of nature, just similar to the Gauge principles in fundamental interactions Four dimensional Spacetime: The spacetime dimension turned out to be 4-dimensions This is determined by the consistency check of the theory and obviously there is no extra dimensions concept here as contrast to the Super-String /M-theory This is related to the Local Gauge Principle in dictating the dynamics of the gauge theories Whereby normally one split the metric becomes background non-dynamical part and perturbation, i.e (4) gµν = (4) ηµν + (4) hµν (4) ηµν is set as the Minkowskian metric and hµν is the perturbation (normally assumed to be small) CHAPTER INTRODUCTION AND MOTIVATION Supersymmetry (SUSY) is not a necessary tool: In certain models, one can include the Supergravity (Supersymmetric generalization of Standard Model matter), but Supersymmtery principle does not play a crucial role in the theory As contrast, String theory definitely required the SUSY properties to obtain some consistency criteria, i.e divergence free, anomalies cancelations etc SUSY is not the key ingredient in LQG due to the diffeomorphism principles In fact, under certain regularization schemes, LQG is shown to be UV finite and it is highly related to the important culprit, diffeomorphism symmetry No aims of Unification so far: As we mentioned, LQG is start of as conventional approach to tackle quantum gravity problem There is not aims in unification of four fundamental interactions of nature As the founder of the program, Ashtekar himself argue that even if quantum general relativity did exist as a mathematically consistent theory, there is no a prior reason to assume that it would be the “final” theory [107] In fact, requirement of background independence and general covariance restrict the form of interaction between gravity and matter fields and among matter fields themselves, LQG would not have a built-in principle which determines these interactions as contrast to standard Local gauge principle principle in Yang-Mills like interactions (Standard Model) We will describe the historical development of the canonical quantization of LQG (together with matter sector as well) to recent times We believe in understanding the historical development of any theory because it serves to illustrate the conceptual development of a theory and the need for such a development4 We will only cover briefly, for more detailed coverage of the history, see Rovelli’s book [19] and Thiemann’s book [20] 1949 - Peter Bergmann forms a group that studies systems with constraints Bryce DeWitt applied Schwinger’s covariant quantization to gravity Dirac publishes Constraint Analysis for Hamiltonian systems [36] 1958 - The Bergmann group and Dirac completes the hamiltonian theory of constrained systems The double classification into primary and secondary constraints and into first- and second-class constraints reflects that Dirac and Bergmann’s group initially worked separately 1961 - Arnowitt, Deser and Misner wrote the seminal paper on ADM formulation of GR [43] The ADM formulation is simply the (incomplete) constraint analysis of GR in terms of metric variables Or more importantly, now the GR is discussed under 3+1 decomposition form The introduction of hypersurfaces (which satisfy Cauchy initial data and assumed to be spacelike) is naturally defined Einstein equations turn out to determine how these hypersurfaces evolve under “time” parameter There is an important issue of “problem of time” to address 1964 - R.Penrose invents the spin networks and it is published in 1971 Of course, it appears to be unrelated to canonical quantization of gravity at that time5 1967 - Bryce DeWitt publishes the “Einstein-Schrodinger equation” which is the imposition of the Hamiltonian (scalar) constraint on the physical state which is the last step in the constraint analysis [44] But everybody else has been calling it the “Wheeler-DeWitt equation” See [19] for the historical reason Wheeler came up with the idea of space of 3-geometries, known as We particularly agree on the Philosophical idea from Carl Sagan, “Science is a way of thinking (upon the time) much more than it is a body of knowledge” The original Penrose article is found here: http://math.ucr.edu/home/baez/penrose/PenroseAngularMomentum.pdf CHAPTER INTRODUCTION AND MOTIVATION “Superspace” Thus, Wheeler-DeWitt Superhamiltonian constraint turns out to describe how the 3-geometries evolve in the Superspace, see page 27 of [23] 1969 - Charles Misner starts the subject “quantum cosmology” 1976 - Supergravity (SUGRA) and Supersymmetric (SUSY)-string theory are born from the study of Quantum Chromodynamics in describing the strong interaction 1983 - Stephen Hawking and James Hartle introduces the Euclidean quantum gravity with their view on Wave function of Universe [45] 1986, 1987 - Ashtekar realizes that the Sen connection [46] (an extension of the covariant derivative to SL(2, C) spinors give rise to an antiself-Hodge dual connection) is suitable as a configuration variable for GR [47] The constraints simplify into polynomial form by using these variables and these are the so-called Ashtekar New variables [13], [16] 1987, 1988 - Samuel, Jacobson and Smolin independently found the Lagrangian formulation of Ashtekar New variables [48] Jacobson and Smolin found loop-like solutions to the Scalar constraint written in the connection variables [77] Rovelli and Smolin brought loop variables formulation to maturity [78], hence known as “Loop Quantum Gravity” However, reality conditions in Ashtekar formulation is intractable due to the complex structure of the Ashtekar connection 1989 - Ashtekar, Romano, Tate (ART) consider Standard Model matter fields extension under the self-dual gravity framework The model is well-defined and free from inconsistency Of course, they are some changes in terms of constraint symmetries and constraint algebras contributed by the matter fields [55] 1992 - Functional Analysis is applied to LQG by Ashtekar and Isham Abelian C∗ algebra and GNS construction are used to handle distributional connections [89] 1993, 1994 - Ashtekar and Lewandowski found a measure that is Gauss gauge invariant and 3D diffeomorphism invariant They apply projective techniques to set up calculus on the space of distributional connections [90] 1995 - Morales-Tecotl and Rovelli includes Fermionic coupling in loop theoretic language It is an immediate extension of pure gravity dynamics to open loops Fermions are placed at the end of the open path as similar to Lattice Gauge Theory [83] 1994, 1995, 1996 - Barbero formulates the real-valued connection version of LQG [58] This formulation has trivial reality conditions and has a parameter that Immirzi has considered earlier Polynomiality of the scalar constraint is lost and one needs to accept more complicated scalar constraint to recover real, Lorentzian GR Thiemann starts to realize that polynomiality of the scalar constraint is inconsistent with background independence Rovelli and Smolin discovered that spin network basis is a complete basis for LQG [92] They calculated area and volume operator eigenvalues [93] and these operators turn out to have discrete quantum spectrum (at least kinematically) 1996, 1997 - Thiemann publishes the remarkable Quantum Spin Dynamics (QSD) series of papers and a major stumbling block is cleared The (weight +1) Barbero scalar constraint finally becomes well defined as an operator expression via Thiemann’s tricks and Thiemann’s regulariza- APPENDIX D POISSON BRACKET FOR THE ART MATTER COUPLING MODEL 337 these, the diffeomorphism constraints can be further rewritten as simpler form as: → := C− N d3 x N a C a − Σ d3 x Tr N a √ N a −i 2Tr = (3) b (A,3) σ Σ − η abc Tr √ −i 2Tr N a = (3) Eb (3) b (A,3) σ −N a π∂a φ − Tr N a (3) AaAB √ −i Tr N a = (3) Db (3) Bc − (3) (Ash) Aa C Da ξ A + ωA AaAB CB A − (3) Fab − N a (3) (3) (Ash) Da η A − π ∂a φ AaAB CBA (3) AaAB ξ B + ωA ∂a η A − ωA √ AaAB −i (Ash) Da (3) a A σ B (3) AaAB η B + ξB π A + η B ω A EbBA (3) b (A,3) σ (3) (3) Fab − N a πA ∂a ξ A − πA Eb Σ = Aa C + Fab − πA (3) Σ +N a (3) Σ Fab − πA N a ∂a ξ A + ωA N a ∂a η A − N a π ∂a φ √ −Tr N a (3) Eb (3) Fab + i 2N a (3) AaAB (Ash) Da (3) σ aB A + N a √ → (ash,3) Ab + πA L− → ξ A + ωA L− →ηA − i Tr (3) σ b L− N N N (3) AaAB (3) Db (3) EbBA Σ → φ + Tr +π L− N (YM,3) → Eb L− N (YM,3) Ab , (D.18) where we have used the fact that Lie derivative treat all the internal SU (2) indices as scalar and the identity below: √ √ i 2Tr N a (3) σ b (A,3) Fab − i 2N a (3) AaAB (Ash) Db (3) σ bB A √ = i N a (3) σ bA B (A,3) FabB A − N a (3) AaAB ∂b (3) σ bB A + (3) Ab , (3) σ b √ A = i N a (3) σ bA B ∂a (3) AbB A − ∂b (3) AaBA + (3) Aa , (3) Ab B − N a −N a √ = i Na (3) AaAB (3) AbB C ∂a (3) AbB − ∂b (3) AaBA − Na (3) AaAB ∂b − Na (3) b B (3) σA AbB C (3) AaC A −N a (3) + Na (3) AbC A (3) b A σB (3) b A σB (3) b B (3) σA AaB C (3) AbC A (3) AaAB ∂b let A → C → B → A A +N a AaB C (3) (3) b A σ C + N a (3) AaAB (3) σ bB C (3) AbC A let A → B → C → A (3) b B σA A B (3) b B σA AaC A (3) b B (3) σA AbB C | Last terms cancel out and we rewrite the 2nd term as total boundary term √ = i N a (3) σ bA B ∂a (3) AbB A − ∂b N a (3) σ bA B (3) AaBA + (3) b B (3) σA AaBA ∂b N a + Na (3) AaBA ∂b (3) b B σA − Na (3) AaAB ∂b (3) b A σB | Drop the boundary term via fall-off condition of N a Also, last terms cancel each other √ = i (3) σ bA B N a ∂a (3) AbB A + (3) AaBA ∂b N a √ √ → (ash,3) AbB A = i Tr (3) σ b L− → (ash,3) Ab , = i (3) σ bA B L− (D.19) N N APPENDIX D POISSON BRACKET FOR THE ART MATTER COUPLING MODEL 338 and similarly Tr N a (3) = Tr N a − (3) Eb (3) (3) Fab − N a Eb 2∂[a Db N a (3) = (3) EbAB N a ∂a = (3) → EbAB L− N (3) Aa (3) (3) (3) (3) Ab] + (3) (3) Aa Eb + AbBA + AbBA = Tr Aa , (3) (3) (3) Db (3) Eb Ab Db N a (3) Aa (3) Eb AaBA ∂b N a (YM,3) → Eb L− N (YM,3) Ab (D.20) In the last equality of both (D.19) and (D.20), we have used the Lie derivative of connection one-form (space-time one form) which is well known from mathematical physics, i.e Differential Form, → ω ≡ L− →ω L− N N × dxa = N b ∂b ωa + ωb ∂a N b × dxa a (D.21) → as constructed is the diffeomorIn the following, in order to justify that phase space functional C− N phism constraint, we perform the computation of Poisson bracket between this functional with the rest of fundamental variables on the phase space Γ For gravitational part, we have (3) a σ AB (y) → (x), C− N √ = −i σ , (3) (3) c D b σ C N ∂b (3) AcDC Σ √ = −i (3) a P.B + (3) c D b (3) σC N AcDC ∂b Σ Aa (3) c D (3) σC AbDC ∂c N b , (3) a σ AB (y) (3) −N b (3) AcDC ∂b σ cC D − P.B (3) c D (3) σC AcDC ∂b N b → by fall-off condition + √ = −i Σ − (3) c D (3) σC AbDC ∂c N b N b ∂b (3) c σ CD (3) c σ CD ∂c N b (3) (3) a σ AB (y) , AcCD (x), (3) (3) a σ AB (y) AbCD (x), (3) σ aAB (y) √ = −i P.B P.B + (3) c σ CD ∂b N b (3) AcCD (x), (3) a σ AB (y) P.B P.B i i C D C D N b ∂b (3) σ cCD √ δc a δ(A δB) δ (x, y) + (3) σ cCD ∂b N b √ δc a δ(A δB) δ (x, y) 2 Σ i C D − (3) σ cCD ∂c N b √ δb a δ(A δB) δ (x, y) → (3) σ aAB , = N b ∂b (3) σ aAB + (3) σ aAB ∂b N b − (3) σ cAB ∂c N a ≡ L− (D.22) N and similarly, → (x), C− N = √ =i AaAB (y) √ i 2Tr − √ =i (3) (3) b Σ Σ Σ → L− N → L− N (3) a P.B (3) (3) → σ L− N Ab Ab CD CD (3) σ , (3) Ab , (3) Aa AaAB (y) P.B (3) b σ CD (x), (3) AaAB (y) P.B i A B → − √ δa b δ(C δD) δ (x, y) ≡ L− N (3) Aa AB (D.23) APPENDIX D POISSON BRACKET FOR THE ART MATTER COUPLING MODEL 339 Thus, we see that all these are expected and similar to the source-free gravity section This is because from (D.18) we observe that the gravitational fundamental variables appeared explicitly only in the first term, which is independent of the other matter fields variables This is further motivated by the physical point of view since we believe that physically the classical matter (even the fermionic fields) will not “modify” the diffeomorphism symmetries of the gravitational degree of freedom at the classical level Of course this is a subtle issue in quantum gravity level, (i.e in the Spin Network or Spin Foam formulation [92], [101]) Subsequently, Poisson bracket between diffeomorphism constraints functional and various matter fields can be carried out trivially without any ambiguous They are, → From C− =− N ,Dirac Σ → ξ A + ωA L− →ηA πA L− N N A → (x), ξ (y) C− N =− Σ =− Σ = → ξ B + ωB L− → η B (x), ξ A (y) πB L − N N → ξ B (x) πB (x), ξ A (y) L− N → ξ A (y) L− N Σ =− Σ = Σ = Σ P.B = Σ P.B → ξ B (x) δB A δ (x, y) L− N , (D.24) → (x), πA (y) C− N =− P.B P.B → ξ B + ωB L− → η B (x), πA (y) πB L − N N P.B ∂a N a πB ξ B −πB ξ B ∂a N a − N a ξ B ∂a πB (x), πA (y) P.B → πB ∂a N a ξ B (x), πA (y) P.B + N a ∂a πB ξ B (x), πA (y) P.B πB ∂a N a −δ BA δ (x, y) + N a ∂a πB −δ BA δ (x, y) → πA = πA ∂a N a + N a ∂a πA (y) ≡ L− N (D.25) We remind readers that in the last step of second Poisson bracket, we used the fact that √ √ √ → πA = L− → ( qπA ) = πA ∂a (N a q) + qL− → πA L− N N √ a N √ a √ a = πA N ∂a q + πA q∂a N + qN ∂a πA √ = πA ∂a N a + N a ∂a ( qπA ) = πA ∂a N a + N a ∂a (πA ) (D.26) By the same token, we can obtain the Poisson bracket between the other set of Dirac fields and APPENDIX D POISSON BRACKET FOR THE ART MATTER COUPLING MODEL 340 diffeomorphism constraint functional, → (x), η A (y) C− N P.B → ξ B + ωB L− → η B (x), η A (y) πB L− N N =− Σ → η B (x) L− N Σ → η A (y), L− N ωB (x), η A (y) =− = → (x), ωA (y) C− N Σ Σ = Σ Σ → η B (x) δB A δ (x, y) L− N (D.27) P.B P.B ∂a N a ωB η B −ωB η B ∂a N a − N a η B ∂a ωB (x), ωA (y) =− Σ = → ξ B + ωB L− → η B (x), ωA (y) πB L− N N =− = P.B P.B P.B → ωB ∂a N a η B (x), ωA (y) P.B + N a ∂a ωB η B (x), ωA (y) P.B ωB ∂a N a −δ BA δ (x, y) + N a ∂a ωB −δ BA δ (x, y) → ωA = ωA ∂a N a + N a ∂a ωA (y) ≡ L− N (D.28) Also, trivially for Klein-Gordon part, we have → From C− =− N ,K.G → (x) , φ(y) C− N =− Σ =− Σ Σ =− Σ →φ π L− N P.B → φ (x), φ(y) πL− N → φ(x) −δ (x, y) L− N → (x), π(y) C− N =− Σ P.B =− Σ → φ(x) {π(x), φ(y)}P.B L− N → φ(y) = L− N P.B → φ (x), π(y) πL− N P.B ∂a N a πφ −πφ∂a N a − N a φ∂a π (x), π(y) P.B → = Σ = Σ π∂a N a φ(x), π(y) P.B + N a ∂a π φ(x), π(y) P.B → π(y) , π∂a N a δ (x, y) + N a ∂a πδ (x, y) ≡ L− N where in last step we also used, √ √ √ → π = L− → → q + qL− →π L− qπ = πL− N N N N √ a √ √ a√ = π∂a N q + qN ∂a π = π∂a N a + πN a ∂a q + qN a ∂a π √ √ = π∂a N a + N a π∂a q + q∂a π = π∂a N a + N a ∂a π (D.29) (D.30) APPENDIX D POISSON BRACKET FOR THE ART MATTER COUPLING MODEL 341 Lastly for the Yang-Mills fields, the computation will be similar to the source free part since both of them have the same “Yang-Mills like” feature on the phase space (recall that in Ashtekar formulation, gravity appears as complex SU (2) Yang-Mills like theory) From (D.18), → (Y.M.) = − C− N →, C− N (3) EaAB (y) (3) =− Σ ∂c =− Σ Tr − Eb L→ N (3) Ab = − (3) Σ EbAB N a ∂a (3) AbBA + (3) AaBA ∂b N a P.B N c ∂c (3) EbCD N c (3) EbCD (3) (3) Σ (3) AbDC + AbDC − AcDC ∂b N c , (3) EbCD (3) (3) EaAB (y) P.B AbDC ∂c N c − N c (3) AbDC ∂c (3) EbCD ⇒ by fall-off condition + (3) (3) = Σ − Σ − = (3) EbCD ∂c N c (3) (3) = EbCD ∂b N c (3) (3) EbCD ∂b N c AcDC , Ab (3) CD (3) (x), EaAB (y) (3) AcCD (x), P.B EaAB (y) (3) P.B EaAB (y) EbCD ∂c N c δb a δ (x, y)δAB CD + N c ∂c (3) + N c ∂c (3) EbCD (3) AbCD (x), (3) EaAB (y) P.B P.B (3) EbCD δb a δ (x, y)δAB CD EbCD ∂b N c δc a δ (x, y)δAB CD EaAB ∂c N c + N c ∂c (3) EaAB − (3) → EbCD ∂b N a ≡ L− N (3) EaAB (D.31) and also, −, C→ N (3) =− AaAB (y) Tr (3) Σ =− Σ =− Σ → L− N (3) → L− N (3) P.B → Eb L− N AbDC (3) (3) Ab , (3) AaAB (y) EbCD (x), (3) P.B AaAB (y) P.B → AbDC −δa b δ (x, y)δCD AB = L− N (3) AaAB (D.32) → and fundamental phase space variables we found out, we With all the Poisson brackets between C− N are confident to conclude that indeed the canonical transformations generated by the new constraint → corresponds precisely to the 3D-diffeomorphisms These symmetries are generated functional C− N by the smearing field N a on the 3D hyper-surfaces Σ With the geometrical interpretation of canonical transformation in mind, we can make use of this nice geometrical feature to deduce that: → , CM,M C− N P.B = −C[L→ − M, N → , C− → C− N M P.B → = −C[− N → , CM C− N P.B = −CL→ − M N − → , M] − M] L→ N − → − → , such that N , M a → M a ≡ L− N (D.33) Finally, we consider the last and more complicated total scalar constraint Before proceed, we recall that we are interested to consider the situation in which Minkowski space can be thought APPENDIX D POISSON BRACKET FOR THE ART MATTER COUPLING MODEL 342 of as the “classical vacuum” Thus, in this section we set the Einstein cosmological term, Λ to be zero The scalar constraint functional (smeared by “negative” weighted tensor density lapse field N ) is given by CN d3 xN C = Σ = (3) a (3) b (A,3) N −Tr σ σ Σ +im (3) σ + (3) √ Fab + i 2 σ ξ A η A − π A ωA − 4πTr −2 Tr (3) a (3) c σ σ Tr (3) a B σ A (3) a (3) b Da ξ A + ωB σ ∂a φ∂b φ + σ (3) b (3) d σ (Ash) πB (3) σ Tr Eab (3) (Ash) π + 4π 16π Ecd + (3) Bab (3) (3) Da η A σ µ2 φ2 Bcd (D.34) Recall that from source-free self-dual gravity case, we have the following Poisson bracket CN (3) a CN (3) a σ , σ , (3) (3) Aa , (3) a B σ A Aa , (3) AaAB P.B P.B √ = i = (Ash) i √ N Db N (3) b σ , (3) [a (3) b] σ (A,3) Fba σ B A , B A (D.35) so we aspect the inclusion of classical matter does modify it Since overall only Dirac fields coupled to the Ashtekar self-dual connection, we shall see that the first Poisson bracket will be modified mainly due to the Dirac fields It is given by, CN , (3) σ aAB P.B √ (Ash) Db N (3) σ [a (3) σ b] AB =i √ (3) b D +i σ C πD (Ash) Db ξ C + ωD Σ (Ash) A A (ash) (Ash) Db η C , (3) a σ AB P.B AaAB ξ B | Since D ξ = ∂a ξ − √ (Ash) a =i Db N (3) σ [a (3) σ b] AB √ (3) b D +i σ C πD (3) AbCE ξE , (3) σ aAB + (3) σ bC D ωD (3) AbCE η E , (3) σ aAB P.B P.B Σ √ (Ash) (3) [a (3) b] σ AB =i Db N σ √ i i C E C E (3) b D +i σ C πD ξE √ δb a δ(A δB) δ (x, y) + (3) σ bC D ωD η E √ δb a δ(A δB) δ (x, y) 2 Σ √ (Ash) (3) a D (3) [a (3) b] (3) a D σ (A η B) ωD σ AB − σ (A ξB) πD + =i Db N σ (D.36) Besides that, we also have another fundamental Poisson bracket We are not manage to show it explicitly, so we just quote the result from [16], CN , CM P.B → + CK·A,K·A = C− K where the K is similar to the free field case (D.37) Appendix E Dirac Gamma Matrices and Some Useful Relations In this section, we want to show the useful relations of the Dirac matrices It is useful when we couple the canonical quantum general relativity together with the minimal/non-minimal Dirac spinorial fields In the main text, we use the particle physics conventions for the representation of Dirac matrices Recall that Dirac matrices satisfy the so-called Clifford algebra {γ I , γ J }+ = 2η IJ , where η IJ is the flat Minkowskian metric η (IJ) = diag(+1, −1, −1, −1) Also, γ5 = γ = iγ γ γ γ The reality condition are γI† = γ0 γI γ0 , (iγ5 )† = γ0 (iγ5 )γ0 The Dirac-γ matrices also satisfy γ I γ [J γ K] = −i IJKL γ5 γL + 2η I[J γ K] Firstly, one can show that, γ [J γ K] γ I J K I γ γ γ − γK γJ γI J = γ 2η KI − γ I γ K − γ K 2η JK − γ I γ J 1 = η KI γ J − 2η JI − γ I γ J γ K − η JI γ K + 2η KI − γ I γ K γ J 2 = 2η IK γ J − 2η IJ γ K + γ I γ [J γ K] = = 2η IK γ J − 2η IJ γ K + −i IJKL γ5 γL + 2η I[J γ K] = 2η IK γ J − 2η IJ γ K + −i IJKL γ5 γL + η IJ γ K − η IK γ J = −i IJKL γ5 γL + 2η I[K γ J] 343 (E.1) APPENDIX E DIRAC GAMMA MATRICES AND SOME USEFUL RELATIONS 344 With these, we can obtain two useful identities as following, Identity {γK , γ[I γJ] }+ := γK γ[I γJ] + γ[I γJ] γK = −i KIJL γ5 γL = −2i KIJL γ5 γL + 2ηK[I γJ] + − i KIJL γ5 γL + 2ηK[J γI] (E.2) Identity [γK , γ[I γJ] ]− := γK γ[I γJ] − γ[I γJ] γK = −i KIJL γ5 γL = 4ηK[I γJ] + 2ηK[I γJ] − − i KIJL γ5 γL + 2ηK[J γI] 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(3) q ab and (3) K ab are fields with zero component9 in the direction orthogonal to 8 This justifies the arabic number (3) stuck to them Also, Latin symbols are used rather than Greek indices when we are dealing with 3D tensor fields on Σt When there is no confusion appears, we may sometime use the notation sloppily 9 In other words, we can interpret (3) qab and (3) Kab as fields on M which happen... of the theory Also, there are many phenomenological aspect of the theory in terms of parity violation, Neutrino oscillations, effective action of gravity + fermion system etc have been addressed This ends the historical development of LQG with matter field extension We would like to note that viewing Ashtekar variables as a special case of the Immrizi-Barbero parameter is clean mathematically but rather... formulation (free field case and matter fields inclusion) Consistency check is imposed on the different action proposed to make sure we are dealing with the same physical theories Immirzi-Barbero formulation is discussed next to lay the foundations of the modern theory of LQG or QGR Effective theory is then take place whereby minimal/non-minimal coupling of fermions to canonical gravity is considered One... fermions to canonical gravity is considered One realize that it is useful to decompose all the variables and constraints into torsion-freeness and torsional parts Then a brief of overview loop representation (with loop quantum fermions) and Spin Network basis are given to close the thesis In the thesis, logical development of concepts is emphasized And wherever we can, we tried to justify completely the... By using Spin Network basis, Ashtekar et al study isolated horizon and Black-hole entropy is shown to be finite with condition that real Immirzi-Barbero parameter must be fixed compatible with Hawking-Bekenstein semi-classical black-hole entropy 2000 onwards - Martin Bojowald started Loop Quantum Cosmology” (LQC) based on the modern LQG type of Hilbert space Big-Bang singularity is removed and replaced... Cauchy surface in the case of pure gravity Chapter 3 (New)-Canonical Gravity: Connection-dynamics 3.1 Ashtekar Hamiltonian Formulation on the Extended ADM Phase Space: SO(3, C) self-dual connection and SL(2, C) soldering form Representation The main reference in this section are Ashtekar book [16] chapter 6, 7 and 8, together with his seminal paper on self-dual gravity [47] In this section we cover... and Misner at around 60s The motive was to obtain a Hamiltonian formulation of General Relativity together with the hope of applying Dirac canonical quantization scheme (in which it works well in ordinary “background dependent” quantum field theory) to constraint system likes GR and thus obtain a quantum theory of GR [43], [8], [53] To arrive at the Hamiltonian formulation of GR, we need to consider... (3) Kab as fields on M which happen to be orthogonal to (4) nµ , implying that they lie on Σt By keeping this in mind, we can take the indices to run from 0, 1, 2, 3 and are raised and lowered with (4) gµν without any ambiguity CHAPTER 2 CANONICAL FORMULATION OF G.R : GEOMETRO-DYNAMICS 17 Σt or simply they only defined on Σt In the following, we shall write the (3+1) decomposition of the metric... (4) N (4) nβ (4) qaβ use: (4) nβ (4) qaβ = 0 (3) Na (4) gµν (4) qaµ (4) qbν (3) qab + (4) N (4) α n (4) β qa (2.30) So, with these decomposition we can obtain a way to understand the Lapse function, shift vectors and the splitting by writing down an infinitesimal element dxµ on M with its proper length ds given by ds2 := (4) gµν dxµ ⊗ dxν = (3) qab dxa + (3) N a dt dxb + (3) N b dt − s (4) 2 N dt ... (3) ab [ q + s NNN2 ] (2.32) Next, with the same token we can split important geometrical objects on M into Σt by using the (3+1) decomposition scheme Define the 3-covariant derivative, (3) ∇ on Σ by the projection operator as (3) ∇a (3) Tb c m n := (4) µ (4) ν qa qb (4) qcα (4) qβm (4) qγn (4) ∇µ (4) Tν α β γ (2.33) where (4) Tν α β γ is any arbitrary tensor fields on M and (3) Tb c m n is its ... attempt to review the full theory of Loop Quantum Gravity (LQG) with Standard Model(SM) type of matter fields extension Firstly, we briefly discuss the old canonical gravity by using Arnowitt, Deser... 8.2.2 The Quantum Loop Representation 8.3 Matter Coupling in Loop Representation: Fermionic Loop Variables 8.3.1 Classical and Quantum Fermions in Loop Space Representation... to construct quantum gravity, one must reformulate Quantum Mechanics in a background-independent way In other words, in quantum gravity, geometry and matter should both be “born quantum mechanically”