41 research outputs found
Local Thermal Equilibrium on Curved Spacetimes and Linear Cosmological Perturbation Theory
In this work the extension of the criterion for local thermal equilibrium by Buchholz, Ojima and Roos to curved spacetime as introduced by Schlemmer is investigated. Several problems are identified and especially the instability under time evolution which was already observed by Schlemmer is inspected. An alternative approach to local thermal equilibrium in quantum field theories on curved spacetimes is presented and discussed. In the following the dynamic system of the linear field and matter perturbations in the generic model of inflation is studied in the view of ambiguity of quantisation. In the last part the compatibility of the temperature fluctuations of the cosmic microwave background radiation with local thermal equilibrium is investigated.:1. Introduction 5
2. Technical Background 10
2.1. The Free Scalar Field on a Globally Hyperbolic Spacetime . . . . . . 10
2.1.1. Construction of the Scalar Field . . . . . . . . . . . . . . . . . 10
2.1.2. Algebra of Wick Products . . . . . . . . . . . . . . . . . . . . 13
2.1.3. Local Covariance Principle . . . . . . . . . . . . . . . . . . . . 17
2.2. Local Thermal Equilibirum . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1. Global Thermodynamic Equilibrium - KMS States . . . . . . 21
2.2.2. Local Thermal Observables . . . . . . . . . . . . . . . . . . . 24
2.2.3. LTE on Flat Spacetime . . . . . . . . . . . . . . . . . . . . . . 29
2.2.4. LTE in Cosmological Spacetimes . . . . . . . . . . . . . . . . 32
2.3. Linear Scalar Cosmological Perturbations . . . . . . . . . . . . . . . . 34
2.3.1. Robertson-Walker Cosmology . . . . . . . . . . . . . . . . . . 35
2.3.2. Mathematical Background . . . . . . . . . . . . . . . . . . . . 38
2.3.3. Technical Framework and Formulae . . . . . . . . . . . . . . . 40
2.3.4. The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 46
2.3.5. The Sachs-Wolfe Effect for Adiabatic Perturbations . . . . . . 49
3. Towards a Refinement of the LTE Condition on Curved Spacetimes 54
3.1. Non-Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.1. Commutator Distribution . . . . . . . . . . . . . . . . . . . . 55
3.1.2. KMS Two-Point Function . . . . . . . . . . . . . . . . . . . . 57
3.1.3. Balanced Derivatives . . . . . . . . . . . . . . . . . . . . . . . 61
3.2. Conformally Static Spacetimes . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1. Conformal KMS States . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2. Extrinsic LTE in de Sitter Spacetime . . . . . . . . . . . . . . 71
3.3. Massive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3.1. Properties of the Model . . . . . . . . . . . . . . . . . . . . . 78
3.3.2. Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . 80
3.3.3. Thermal Observables . . . . . . . . . . . . . . . . . . . . . . . 82
3.4. Towards an Alternative Concept . . . . . . . . . . . . . . . . . . . . . 91
3.4.1. Problems and Open Questions Concerning LTE . . . . . . . . 92
3.4.2. Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . 94
3.4.3. Positivity Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96
3.4.4. Macroobservable Interpretation . . . . . . . . . . . . . . . . . 100
3.5. An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4. Cosmological Perturbation Theory 105
4.1. Dynamics of Perturbations in Inflation . . . . . . . . . . . . . . . . . 106
4.1.1. CCR Quantisation is Ambiguous . . . . . . . . . . . . . . . . 106
4.1.2. Canonical Symplectic Form . . . . . . . . . . . . . . . . . . . 111
4.1.3. The Algebraic Point of View . . . . . . . . . . . . . . . . . . . 117
4.2. LTE States in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 120
4.2.1. The Link to Fluid Dynamics . . . . . . . . . . . . . . . . . . . 120
4.2.2. Incompatibility of LTE with Sachs-Wolfe Effect . . . . . . . . 125
5. Conclusion and Outlook 131
A. Technical proofs 136
A.1. Proof of Lemma 3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.2. Proof of Lemma 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.3. Proof of Lemma 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.4. Idea of Proof for Conjecture 3.4.3 . . . . . . . . . . . . . . . . . . . . 144
B. Introduction to Probability Theory 146
Bibliography 150
Correction of Lemma 3.1.2 155In dieser Arbeit wird die von Schlemmer eingeführte Erweiterung des Kriteriums für lokales thermisches Gleichgewicht in Quantenfeldtheorien von Buchholz, Ojima und Roos auf gekrümmte Raumzeiten untersucht. Dabei werden verschiedene Probleme identifiziert und insbesondere die bereits von Schlemmer gezeigte Instabilität unter Zeitentwicklung untersucht. Es wird eine alternative Herangehensweise an lokales thermisches Gleichgewicht in Quantenfeldtheorien auf gekrümmten Raumzeiten vorgestellt und deren Probleme diskutiert. Es wird dann eine Untersuchung des dynamischen Systems der linearen Feld- und Metrikstörungen im üblichen Inflationsmodell mit Blick auf Uneindeutigkeit der Quantisierung durchgeführt. Zuletzt werden die Temperaturfluktuationen der kosmischen Hintergrundstrahlung auf Kompatibilität mit lokalem thermalem Gleichgewicht überprüft.:1. Introduction 5
2. Technical Background 10
2.1. The Free Scalar Field on a Globally Hyperbolic Spacetime . . . . . . 10
2.1.1. Construction of the Scalar Field . . . . . . . . . . . . . . . . . 10
2.1.2. Algebra of Wick Products . . . . . . . . . . . . . . . . . . . . 13
2.1.3. Local Covariance Principle . . . . . . . . . . . . . . . . . . . . 17
2.2. Local Thermal Equilibirum . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1. Global Thermodynamic Equilibrium - KMS States . . . . . . 21
2.2.2. Local Thermal Observables . . . . . . . . . . . . . . . . . . . 24
2.2.3. LTE on Flat Spacetime . . . . . . . . . . . . . . . . . . . . . . 29
2.2.4. LTE in Cosmological Spacetimes . . . . . . . . . . . . . . . . 32
2.3. Linear Scalar Cosmological Perturbations . . . . . . . . . . . . . . . . 34
2.3.1. Robertson-Walker Cosmology . . . . . . . . . . . . . . . . . . 35
2.3.2. Mathematical Background . . . . . . . . . . . . . . . . . . . . 38
2.3.3. Technical Framework and Formulae . . . . . . . . . . . . . . . 40
2.3.4. The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 46
2.3.5. The Sachs-Wolfe Effect for Adiabatic Perturbations . . . . . . 49
3. Towards a Refinement of the LTE Condition on Curved Spacetimes 54
3.1. Non-Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.1. Commutator Distribution . . . . . . . . . . . . . . . . . . . . 55
3.1.2. KMS Two-Point Function . . . . . . . . . . . . . . . . . . . . 57
3.1.3. Balanced Derivatives . . . . . . . . . . . . . . . . . . . . . . . 61
3.2. Conformally Static Spacetimes . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1. Conformal KMS States . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2. Extrinsic LTE in de Sitter Spacetime . . . . . . . . . . . . . . 71
3.3. Massive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3.1. Properties of the Model . . . . . . . . . . . . . . . . . . . . . 78
3.3.2. Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . 80
3.3.3. Thermal Observables . . . . . . . . . . . . . . . . . . . . . . . 82
3.4. Towards an Alternative Concept . . . . . . . . . . . . . . . . . . . . . 91
3.4.1. Problems and Open Questions Concerning LTE . . . . . . . . 92
3.4.2. Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . 94
3.4.3. Positivity Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96
3.4.4. Macroobservable Interpretation . . . . . . . . . . . . . . . . . 100
3.5. An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4. Cosmological Perturbation Theory 105
4.1. Dynamics of Perturbations in Inflation . . . . . . . . . . . . . . . . . 106
4.1.1. CCR Quantisation is Ambiguous . . . . . . . . . . . . . . . . 106
4.1.2. Canonical Symplectic Form . . . . . . . . . . . . . . . . . . . 111
4.1.3. The Algebraic Point of View . . . . . . . . . . . . . . . . . . . 117
4.2. LTE States in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 120
4.2.1. The Link to Fluid Dynamics . . . . . . . . . . . . . . . . . . . 120
4.2.2. Incompatibility of LTE with Sachs-Wolfe Effect . . . . . . . . 125
5. Conclusion and Outlook 131
A. Technical proofs 136
A.1. Proof of Lemma 3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.2. Proof of Lemma 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.3. Proof of Lemma 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.4. Idea of Proof for Conjecture 3.4.3 . . . . . . . . . . . . . . . . . . . . 144
B. Introduction to Probability Theory 146
Bibliography 150
Correction of Lemma 3.1.2 15
The Filament Sensor for Near Real-Time Detection of Cytoskeletal Fiber Structures
A reliable extraction of filament data from microscopic images is of high
interest in the analysis of acto-myosin structures as early morphological
markers in mechanically guided differentiation of human mesenchymal stem cells
and the understanding of the underlying fiber arrangement processes. In this
paper, we propose the filament sensor (FS), a fast and robust processing
sequence which detects and records location, orientation, length and width for
each single filament of an image, and thus allows for the above described
analysis. The extraction of these features has previously not been possible
with existing methods. We evaluate the performance of the proposed FS in terms
of accuracy and speed in comparison to three existing methods with respect to
their limited output. Further, we provide a benchmark dataset of real cell
images along with filaments manually marked by a human expert as well as
simulated benchmark images. The FS clearly outperforms existing methods in
terms of computational runtime and filament extraction accuracy. The
implementation of the FS and the benchmark database are available as open
source.Comment: 32 pages, 21 figure
Diffusion Means and Heat Kernel on Manifolds
We introduce diffusion means as location statistics on manifold data spaces.
A diffusion mean is defined as the starting point of an isotropic diffusion
with a given diffusivity. They can therefore be defined on all spaces on which
a Brownian motion can be defined and numerical calculation of sample diffusion
means is possible on a variety of spaces using the heat kernel expansion. We
present several classes of spaces, for which the heat kernel is known and
sample diffusion means can therefore be calculated. As an example, we
investigate a classic data set from directional statistics, for which the
sample Fr\'echet mean exhibits finite sample smeariness.Comment: 8 pages, 1 figure, conference paper submitted to GSI 202
Dynamical Backreaction in Robertson-Walker Spacetime
The treatment of a quantized field in a curved spacetime requires the study
of backreaction of the field on the spacetime via the semiclassical Einstein
equation. We consider a free scalar field in spatially flat Robertson-Walker
space time. We require the state of the field to allow for a renormalized
semiclassical stress tensor. We calculate the sigularities of the stress tensor
restricted to equal times in agreement with the usual renormalization
prescription for Hadamard states to perform an explicit renormalization. The
dynamical system for the Robertson Walker scale parameter coupled to the
scalar field is finally derived for the case of conformal and also general
coupling.Comment: Obtained equation of motion for non-conformal coupling, not just
counter terms as in previous version. Typos fixed, renormalization term
proportional to R adde