2013 2014 2015 2016 2017 2018

Rethinking Foundations of Physics 2016

Saturday March 12 - Saturday March 19 2016

 

Description:

Traditional conferences and subject-specific workshops offer little room for in-depth discussions about the foundations of physics in an open, creative and speculative way. In tradition of the meetings in 20132014 and 2015, this workshop offers a platform to young scientists for engaging in such discussions.

The major part of the workshop consists in discussion sessions in small groups, aiming at new approaches and ways of thinking about fundamental physics. These discussion sessions will be inspired by talks given by the participants. The specific topics of both talks and discussions will be chosen according to the expertise and interests of all participants, revolving around the following questions:


- Which mathematical, conceptual, and experimental paradigms underlie modern formulations of QM, GR, and QFT?
- Can they be relaxed or changed? And if so, how?
- Which new mathematical developments could be relevant for future foundations?
- Are there promising new or non-standard experimental possibilities?

The workshop is addressed to PhD students in physics, mathematics and related fields, as well as young researchers who have completed their PhD in the last few years. It is not required that participants have substantial knowledge in all modern physical theories, but a general interest and openness towards other fields and ideas is expected. Applications are open until January 15, 2016.

Date: Saturday, March 12 - Saturday, March 19, 2016
Place:
Mountain Cabin "Amoseralm", Dorfgastein, Austria
Participants: 18
Workshop fee: 300 Euros (Accommodation and food included. Financial support available.)

In case of questions, please contact application@rethinking-workshop.org.

 

Participants:

 

Emily Adlam
University of Cambridge, Department of Applied Mathematics and Theoretical Physics

Timo Alho
University of Iceland, Science Institute

Natacha Altamirano
University of Waterloo & Perimeter Institute for Theoretical Physics, Department of Physics and Astronomy

Lorenzo Catani
University College London, Physics and Astronomy Department

Luca Curcuraci
University of Trieste, Department of Theoretical Physics

Franz Gmeineder
University of Oxford, Mathematical Institute

Dan Goldwater
UCL & Imperial College London, Department of Physics

Matthew E. Hogan
Wells College, Department of Mathematical and Physical Sciences

Johannes Kleiner
University of Regensburg, Department of Mathematics

Lena Kononenko
DESY (German Electron-Synchrotron)

Robin Lorenz
Karlsruhe Institute of Technology (KIT)

Tim Ludwig
Karlsruher Institute of Technology, Institute for Theory of Condensed Matter

Joan Vazquez Molina
Universitat Politècnica de Valencia, Instituto Universitario de Matemática Pura y Aplicada

Jonathan Richens
UCL & Imperial College London, Department of Physics

Jens Salomon
Karlsruhe Institute of Technology, Institut für Philosophie

Alexander Smith
University of Waterloo, Department of Physics and Astronomy

Jan-Hendrik Treude
Karlsruhe Institute of Technology (KIT)

Federico Zalamea
ERC Philoquantumgravity, Laboratoire SPHERE, Université Paris Diderot

Carlos Zapata-Carratala
University of Edinburgh, School of Mathematics

Vaclav Zatloukal
Czech Technical University in Prague, Dept. of Physics & FU Berlin & MPI for History of Science

 

 

Talks:

 
The Problem of Confirmation in the Everett Interpretation
Emily Adlam


I propose that the information-theoretic nature of quantum mechanics is due to perspectival effects which arise because experiments on locally accessible variables can only uncover a certain subset of the correlations exhibited by an underlying deterministic theory. I show that the no-signalling principle, information causality, and strong subadditivity can be derived in this way; I then use these methods to propose a resolution of the black hole information paradox, and discuss the possibility of empirical tests. This approach paves the way to a unification of the modern information theoretic approach with traditional scientific realism.

Discussion Proposals:
1.) I begin by raising a number of provocative questions about the need for explanations of apparently 'fine-tuned' features of quantum theory. This is a natural opportunity to raise more general questions about which features of a physical theory call for further explanation and the role played by fine-tuning arguments in physics.
2.) Other people will no doubt have alternative ideas about how the features I consider may be explained, so my talk would be a using starting point for a discussion of different explanatory stances for the foundations of quantum mechanics.
3.) The perspectival effects I propose can be usefully compared to perspectival effects in other physical theories, such as special relativity; this could lead to interesting conversations about which effects in physics may be viewed as perspectival and whether we should treat such effects differently from more 'objective' effects.

 

Geometric Algebra and the Mathematical Formulation of Fundamental Physics
Timo Alho


Geometric Algebra is the mathematical system defined by imbuing the vectors of an inner product space directly with the Clifford algebra generated by the inner product, without considering a separate representation of the algebra operating on the vectors. This simple change in point of view gives a system which generalizes exterior algebra, quaternions, spinors, and many other algebras used in theoretical physics, while simplifying both concepts and calculations and allowing their coordinate-free formulation. For example the Dirac equation becomes just the simplest equation one can write for a field in the even subalgebra of GA of Minkowski space. Further insights arise by studying the meaning of coordinate freedom in the description of the Hilbert space of a theory. Given that such a simplification exists, the question is if the usual tensor algebra language used for constructing QFT and GR is the most efficient or proper for describing the fundamental laws of nature, or is some other formulation such as GA more suited. If the answer is that GA or some other non-standard formulation is better or more intuitive in some sense, then the question becomes what new things can we learn by reformulating known theories such as the Standard Model or general relativity, and could such developments lead to better understanding of open questions such as quantum gravity.

Discussion Proposals:
- In geometric algebra many of the concepts, such as spinors, usually considered to belong to quantum mechanics or quantum field theory, appear already in an elementary treatment of geometry, and therefore GA seems to provide a unification of concepts across physics. However, since GA has a matrix representation, it is not strictly richer than ordinary matrix algebra. Can we nonetheless learn something new by studying quantum field theories using GA, due to it's greater intuitive power and unification of concepts?
- Expanding on the above, should we consider the standard mathematical formulations of QFT and GR as good enough, and concentrate only on directions which add (or remove!) new physical assumptions? Or is there value in attempting to understand existing theories in a deeper way by attempting to formulate their mathematical foundations in alternative, possibly more enlightening, ways?

 

Gravity as an Emergent Quantum Measurement: Cosmological Applications
Natacha Altamirano


In this work we present how we can understand the gravitational force as an effective feature of continuous quantum measurements proceeded by a feedback interaction between two quantum degrees of freedom. We will show that this type of measurements will not just simulate gravity but also introduce decoherence in the system accountable for an effective heating in the system. In the second part of the talk I will concentrate on the application of this model to a Friedmann universe and show that it allows us to predict an universe that is expanding with a power law and filled by an effective dark energy whose equation of state asymptotes to w=-1/3.

Discussion Proposals:
1.) The introduction of ancillary system that measures the scale factor breaks naturally time translation invariance in cosmology at the quantum level, a problem that shares a lot of similarities with the Hamiltonian constraint problem in quantum Gravity, and it is one of the most important puzzles QG. How can we 'cure' this feature at the macroscopical level?, what are the consequences of this problem? are the tools currently available to tackle the problem in QG helps us in our continuous quantum measurement framework?
2.) As we discussed, the decoherence present in the system due to the continuous quantum measurement introduces heat in the system. How can we use current experiments in Gravity, such as equivalence principle type of measurements to constrain the decoherence parameter?
3.) Measurements of the gravitational constant G have shown to disagree with each other. Can we explain this disagreement using the features of the continuous quantum measurement model?
4.) What happens if the decoherence parameter is time dependent? How is it different from the natural time scale of cosmology ' the Hubble parameter'?

 

Insights into Spekkens Toy Model
Lorenzo Catani


Spekkens toy model is a non-contextual hidden variable model built to support the epistemic view of quantum mechanics, where quantum states are seen as mere states of incomplete knowledge of a deeper underlying reality. The aim of the model is to replace quantum mechanics by a classical theory with the addition of an epistemic restriction (i.e. a restriction on what an observer can know about reality). This model has found many results that were thought to belong only to quantum mechanics (e.g. the no-cloning theorem and teleportation). These results single out also the quantum features that are not reproducible by the model. In particular it is not possible to find a violation of a Bell-like inequality. This highlights which are the features that properly characterize quantumness, namely non-locality and contextuality.
Moreover by using discrete Wigner functions it is possible to relate Spekkens toy model to stabilizer quantum mechanics (a subtheory of quantum mechanics, a sort of 'maximal one'). In particular it can be proven that stabilizer quantum mechanics and Spekkens theory are equivalent in odd dimensions.
As a consequence Spekkens toy model also sheds light onto the mysterious question about the different behavior of stabilizer quantum mechanics in odd/even dimensions. Why does contextuality, the inherent feature of quantum mechanics, denote only the even case?

Discussion Proposals:
1.) Epistemic vs ontic views of quantum mechanics. Why prefer one to the other? Would it be possible to construct a theory of physics without the assumption of the existence of an external underlying reality (i.e an alternative to ontological models)?
2.) Are nonlocality and contextuality actually the signature of quantumness? What else?
3.) Are Wigner functions a good tool to describe quantum mechanics? Which insights into quantum mechanics can we gain from using them? What about other tools?
4.) Are there physical reasons why stabilizer quantum mechanics in odd and even dimension is not the same (being respectively non-contextual and contextual)?

 

Why do we need Hilbert Spaces?
Luca Curcuraci


Quantum mechanics in its actual formulation is an extremely successful theory. In its mathematical formulation starting from quantum logic considerations reveals that we cannot avoid to use an Hilbert space. In addition, the Stone-von Neumann theorem select only a particular Hilbert space when we want describe a real physical systems. The same things happens if we start from operational considerations and choose to formulate the quantum mechanics algebraically. This lead to the following questions: why we need to use Hilbert spaces? Does they have a physical meaning? Are they the signature of some underlying physical structure hidden in the actual formulation? In the talk, after a brief presentation of the two aforementioned approach, I will try to give a possible answer to these questions (despite it will be not conclusive) using random distribution of points that changes in time and a jump-dynamics for a point particle.

Discussion Proposals:
- Is quantum mechanics really fundamental?
- How an underlying theory may look like?
- Should we find any sign of this in actual observations?

 

Collapse Theories: What should we Think of Them?
Dan Goldwater


Collapse models seek to resolve the measurement problem through a modification of the Schrodinger equation. The reward promised by these models is a theory capable of describing phenomena from the quantum scale up to the classical, without the need to introduce the subjective distinction between 'measurement apparatus' and 'quantum system', and which avoids the troubles of Many Worlds Theory. I will give an introduction to the motivation behind collapse models and their mathematical form, as well as a brief overview of the experiments seeking to test them. I will then cover some of the problems associated with these theories, and some of the open questions presented by them.

Discussion Proposals:
- Collapse theories may not be compatible with reversibility, or even energy conservation. Are they still plausible?
- Collapse theories are purely phenomenological - they modify the mathematics to fit with a particular world view, and then search for the corresponding physical effects. If a collapse effect is found, it is not clear if it could be interpreted as having a 'cause'. What should we make of this?

 

Dynamics and Symmetry in Quantum Gravity: Toward a Unified Theory of Pregeometry
Matthew E. Hogan


Physical theories of quantum gravity have undergone a resurgence of development over the last thirty years, from the 'First Superstring Revolution' and the development of Ashtekar's 'New Variables' in the mid-1980's to the modern EPRL-FK-KKL formulations of spinfoams, group field theory, SL(2,ℂ) Chern-Simons Theory, and promising techniques for N = 4 SYM scattering amplitudes, to name just a few. All background-independent theories of quantum gravity, including loop quantum gravity and spinfoams, group field theory, causal dynamical triangulation, quantum graphity, Regge calculus, causal sets, and many others, have both their merits, as well as certain disadvantages, but their degrees of success seem to be pointing to a syncretic convergence, or a kind of duality (as is seen amongst the different flavors of (super)string theory / M-theory), towards a deeper theory. Are these theories reconcilable, and what modifications to their current structures may be necessary? Can we find a unified theory of pregeometry, and describe the process of geometrogenesis, or are there insurmountable obstacles in the way?

Discussion Proposals:
- What elements of modern (background-independent) theories of quantum gravity must be modified for unification, and how?
- What are the meta-theoretic considerations that must be taken into account (e.g. incompleteness)?
- I would argue that a movement towards a Bohmian style of holism might be appropriate — can Bohm's approach be updated in light of current developments, and can it be of use?

 

Plasma Wakefield Acceleration and Controlled Injection: a New Vision of the Particle Accelerators
Lena Kononenko


Waves in the plasma, created by the special charge distribution of electrons and ions, open a new age of the particle accelerators. Discovered in the last century this technology uses an ultrashort lasers or a dense particle beams to create charge distribution. Plasma wakefields can sustain high field gradients (> 10 GV/m) allowing particle acceleration to ultrarelativistic energies over small distances (~mm). This new acceleration concept allows the particles to be generated directly inside plasma via different methods like colliding two laser pulses, or by adding a gas with different ionisation threshold or by tailoring the density profile of the plasma target. Control over the electron bunch phase-space during the process of injection into the accelerating wakefield is of crucial importance for the production of high-quality electron beams in plasma-based schemes. Electron beams created and accelerated in plasma could be used in the free electron lasers and have been demonstrated as a compact source of the x-rays, which could be used for the high contrast imaging of the biological samples.

Discussion Proposals:
- current status of the plasma wakefield accelerators. What have been achieved and what could be improved
- science: what we can measure with a plasma wakefield accelerator and why it would be much cheaper than via conventional accelerator
- applications: how can we use the plasma accelerators and how far we are from this point

 

Transition from Quantum to Classical Mechanics
Tim Ludwig


Classical mechanics is a well established physical theory with a broad domain of application. Within this domain its predictions are confirmed quite well. For that reason it is sometimes asserted that for quantum mechanics to be correct, it must be in agreement with classical mechanics in the appropriate limit, which is often loosely stated as 0. The aim of this talk is to set the stage for general discussions about this limiting process. There are, of course, already several standard methods that allow for a recovery of classical features from the quantum mechanical description by a well defined process. Examples are the Ehrenfest theorem, the stationary phase approximation in the path integral formulation, and the effects of decoherence for open systems. For several standard methods a short survey will be given. Then, for each presented method, it will be discussed which classical features are recovered and where the method has some difficulties in recovering classical mechanics.

Discussion Proposals:
- What is the aim of a transition from quantum mechanics to classical mechanics?
- Does any of the presented methods allow for a complete recovery of classical mechanics from quantum mechanics?
- Is there really a transition from quantum mechanics to classical mechanics or is there a more subtle relation between both?

 

Can Thermodynamics be reduced to Mechanics?
Joan Vazquez Molina


A dynamical explanation of the second law would solve the puzzle of the arrow of time. Following From Being to Becoming, we will review Boltzmann's attempt to tame irreversibility through the H-theorem, as well as his peers' criticism and paradoxes. Unfortunately, no ordinary function of the phase space variables can be a Lyapunov function. Therefore, Prigogine argued for the introduction, in classical mechanics, of a star-hermitian operator representing entropy. This beautifully connects to mathematical analysis: may the question be grounded into our choice of mathematical (Hilbert) spaces? The absence of a scalar product or dual vectors could be seen as a feature of irreversibility, since there is no interchange between incoming and outgoing states.

Discussion Proposals:
- Is irreversibility an illusion? Or, on the contrary, it is a fundamental feature of the world and it is mechanics (classical or quantum) what is an idealization?
- Therefore, should the fundamental laws of nature incorporate some kind of irreversibility?
- Is reduction the aim of physics or merely an aesthetic dream? Is reduction really possible or, as argued by Anderson, we will always need extra concepts at each scale of description?
- To what extent do our mathematics condition our physics? In particular, is it a coincidence that L2 has so many unique properties?
- (if time and interest) Thurner's generalized entropy and its derivation from 3 axioms, ain't it super cool?

 

Deriving the Phenomenon of Entanglement from Dynamical Reversibility
Jonathan Richens


One of the most striking features of quantum theory is the existence of entangled states, responsible for Einstein's so called "spooky action at a distance". These states emerge as part of the mathematical formalism of quantum theory, but to date there exists no set of simple physical axioms that predict their existence. A feature shared by both quantum and classical theory is reversible transitivity, which ensures that the information content of a closed system is conserved by its dynamics and that the state space can be fully explored. Is quantum theory unique in that it is reversibly transitive and exhibits entanglement? I will consider all reversibly transitive interacting theories and show that all, with the exception of classical theory, must have dynamics that generate entangled states. By relaxing the postulate of transitivity I will show that reversibility and ''information gain implies disturbance'' (closely related to the no-cloning theorem) are sufficient to imply the existence of entangled states. These postulates derive the phenomenon of entanglement from purely physical principles, and establish entanglement as an inevitable feature of any natural non-classical probabilistic theory of nature. I will use these results as a springboard to discuss the general structure of probabilistic theories, the notion of information in physical theories.

Discussion Proposals:
- is reversibility necessary for a consistent theory of information, or can there exist reasonable information theories that allow for information to be destroyed or cloned?
- closely related: what is the minimal set of properties that define a bit?
- in quantum theory, all entangled states display some (hidden) Bell non-locality, what is it about quantum theory that makes this true, whereas it is not true in Spekkens toy model?
- is it enough to attempt to reconstruct quantum theory by just considering bipartite interactions between bits (i.e. assuming "pancomputationalism" - that all systems can be simulated with interacting bits). Would a theory that doesn't satisfy this make sense, and could it be aesthetically pleasing?
- is it reasonable to assume that the state spaces of a theory are given by the set of states that can be reached under the theories dynamics? could there be theories that are physically different although they have the same set of "reachable" states?
- classical theory is the only transitive interacting theory without entanglement - does this observation have any physical or philosophical meaning / use?
- classical states can be "cloned" and broadcasted, how does this limit entanglement and non-locality in theories with these properties?

 

Newton's Paradigm: Mechanics
Jens Salomon


I reconstruct Newton's argumentation for the gravitational interaction of celestial bodies, focusing on the role of Newton's understanding of the manual arts. In a greater context, these considerations strengthen the point of view that the creation and understanding of (both modern and classical) physical theories rests not only on "intrinsic" elements (here: Newton's Axioms) but also on "extrinsic" knowledge.

Discussion Proposals:
- We could discuss either Newton's Principia Mathematica and compare his argumentation for the law of gravitation with the account given in modern textbooks.
- We could compare empiricist and realist with idealistic and constructivistic stances towards physics. As physicists usually tend to embrace at least one of the former, I could try and provide arguments for the latter.

 

Quantum Frames of Reference
Alexander Smith


Progress in physics, from Aristotelian physics, to Galilean and Newtonian physics, and then to both special and general relativity, can be viewed as a continual refinement of the notion of a reference frame. The next natural step in this progression is the idea of a quantum reference frame. In this talk, I will introduce the basic tools that have been developed to study quantum reference frames and examine how they may be applied to relativistic scenarios. In particular I will look at how two observers in different Lorentz frames that are partially correlated can communicate via the exchange of a single massive spin-1/2 particle. I will then construct an alternative approach to quantum reference frames involving a trace over global degrees of freedom, rather than an average over all possible orientations of a system with respect to an external reference frame. This approach is anticipated to help deal with reference frames associated with non-compact groups, such as the Galilean group and Poincar ́e group.

Discussion Proposals:
- Reference frames naturally have a group structure associated with them, as changes of reference frames form a group; almost all of the previous literature has focused on reference frames associated with compact groups. How do we successfully apply the developed quantum reference frame formalism to physically relevant non-compact groups like the Poincar ́e group? What new mathematical tools are required to deal with non-compact groups? And what can we hope to learn from such an endeavour?
- Quantum reference frames have a natural size associated with them (arXiv:0812.5040), and the limit in which this size becomes infinite, a classical reference frame is obtained. Can we understand the quantum to classical transition through quantum reference frames?
- Can we introduce a relativity principle for quantum mechanics? In special relativity, the transformations that change between inertial reference frames preserve the Minkowski line element. Changes of quantum reference frames have been previously examined (arXiv:1307.6597), and it would be interesting to ask if there is an object, analogous to the line element that is preserved or approximately preserved (and perhaps in the classical limit exactly preserved) when changing quantum reference frames?
- Is the framework developed to study quantum reference frames compatible with Carlo Rovelli's relational interpretation of quantum mechanics (arXiv:quant-ph/9609002)? Alternatively, what does Rovelli's interpretation say about the framework of quantum reference frames?

 

The Difference between Classical and Quantum Kinematics and the Double Role of Physical Properties
Federico Zalamea


One of the most important problems in theoretical physics is, undisputedly, the quantization of gravity. In this talk, instead of moving ahead and exploring one of the several tentative routes to answer the problem (Loop Quantum Gravity, String Theory, Causal Fermion Systems), I would like to pause and reflect on the meaning of the problem of quantization. When we pass from the classical description of a given physical system to its quantum description, what are we exactly doing? Even for the simplest non-­‐relativistic systems (e.g. a free particle moving on Euclidean space, or the harmonic oscillator), it appears to be quite difficult to capture the technical and/or conceptual core of what quantization amounts to. The transition from the Classical to the Quantum is often related to some of the following transitions: from the continuum to the discrete, from real numbers to complex numbers, from functions to operators, from commutative to noncommutative algebras. All of these analogies exert a strong influence on the chosen strategies to tackle the problem of quantizing gravity. Yet, they all appear to be wrong. In the face of this, how should we then think of the difference between Classical and Quantum Kinematics?
I propose to approach this question by paying special attention to the algebraic structures describing the set of properties of a system (Jordan and Lie structures) and the geometric structures describing the space of states (symplectic forms, transition probabilities). From this perspective, it appears that both Kinematics are extremely similar and one can precisely pinpoint the technical difference between them. The difficult part is to give a conceptual interpretation of this difference. I will try to show that this is related to an often-unnoticed fact in both Classical and Quantum Mechanics: the double role of physical properties. Indeed, properties allow to assign numbers to states (the values of the properties) but they also generate transformations of the state space. Understanding the interplay between the numerical and transformational role of properties may, I hope, shed some light into the real difference between Classical and Quantum Kinematics.

Discussion Proposals:
Besides the main question ("How should we think of the difference between Classical and Quantum Kinematics?"), which surely will lead to an interesting discussion, I think of at least two other possibilities:
- What defines a physical property? For example, what defines linear momentum: the fact that it generates translations, the fact that it measures the amount of inertia of a system, or something else (e.g., being the canonical conjugate of position in the Lagrangian formalism)?
- Can non-­‐relativistic Mechanics shed some light on the problem of quantizing gravity, or should we solely treat the problem of quantizing fields? Put differently, the quantization of fields is technically much more involved, but is it conceptually different?

 

The Soundness of Hamiltonian General Relativity - a reassessment of the mathematical models underlying modern physics
Carlos Zapata-Carratala


Both classic and quantum mechanics strongly depend on the notion of Hamiltonian mechanics or, in the mathematician's parlance, symplectic manifolds, for their axiomatic formulation. A similarly axiomatic approach to relativity, as I will try to illustrate, leads to a basic mathematical structure that differs considerably, at least at first sight, from a symplectic manifold. In particular, the formulations of dynamics in each setting, albeit being somehow connected, are radically different; however, it is simple graduate text bookwork to formulate "Hamiltonian General Relativity". The question I pose is simple: does it really make sense to do so? Is it both mathematically and philosophically consistent to formulate General Relativity as a Hamiltonian Field Theory in a similar way to what is done with Classical Electromagnetism? I will try to present the background where I frame this question, as it requires for a somewhat unconventional axiomatic presentation of relativity - that, of course, recovers the usual theory but one that amends some of the conceptual gaps that a hasty, standard textbook presentation usually contains - and a careful review of Lagrangian-Hamiltonian mechanics.

This talk will then tackle of a series of issues that arise as a result of trying to give solid mathematical and philosophical foundations to what could be considered one of the "hard problems" in modern physics, namely, Quantum Gravity. I strongly believe that there are conceptual problems that are underestimated when focusing too much on the computational or technical aspects of the current working theories and I will attempt to outline and bring them to light.

As it is usually the case when addressing the conceptual foundations of physical or mathematical theories, there is a great deal of freedom of choice and the constraints are not always obvious. This is where dialogue and discussion really become main tools of progress. I will try to initiate the conversation on the following list of topics:

Discussion Proposals:
- We usually take the mathematical and philosophical foundations of classical mechanics for granted, what is our view on this topic? Can everybody give a more or less consistent axiomatic formulation of classical mechanics off the top of their heads?
- Can we do the same with Special and General relativity?
- How strong is the correlation Symplectic Manifolds ~ Classical Phase Spaces?
- What geometric structures can be derived from relativistic dynamics drawing parallels with what is done with classical dynamics and symplectic manifolds?
- Is it illegitimate to try to give a global, consistent Hamiltonian formulation of General Relativity?
- If that was to be the case, a canonical quantum formulation is no longer a possibility, does it still make sense to talk about Quantum Gravity? How should we approach this problem, which clearly still remains in the form of gravitational effects becoming relevant at small scales due to extreme phenomena, if such a formulation is not possible?

 

Classical and Quantum Field Theories from Hamiltonian Constraint
Vaclav Zatloukal


Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined by a (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, I derive local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton-Jacobi equation applicable in the field theory then follows readily. A field-theoretic Hamiltonian version of the Noether theorem is also presented to provide an interpretation of the generalized momentum. The general method is illustrated with three examples: non-relativistic Hamiltonian mechanics, De Donder-Weyl scalar field theory, and string theory.
Throughout, I use the mathematical formalism of geometric algebra and geometric calculus, which allows to perform completely coordinate-free manipulations, and which I briefly introduce.

Discussion Proposals:
With the Hamiltonian formulation of classical field theory outlined above, the question arises, how to proceed to quantum field theory. Apart from a rather straightforward path-integral quantisation, I would like to discuss the possibility to quantise in analogy with the Schroedinger's formulation of quantum mechanics, that is, to try to find a differential operator corresponding to the generalized momentum, and a field-theoretic Schroedinger-like equation for a generalized wave function.
Another question goes back to treating the spacetime points and field values on equal footing. This is analogous to what happens in relativistic mechanics, which combines time and space. From the formal point of view, the relativistic setting is elegant and compact, and it can be used to study both relativistic and non-relativistic theories. The physically significant point is, however, that it allows to formulate the special relativity. Therefore, is there any physical significance in combining the spacetime with the space of fields, i.e., in regarding the spacetime coordinates as some kind of fields?

 

Poster download:

JPG or PDF

 

Supported by:
        
University of Regensburg - Faculty of Mathematics, German Research Foundation: DFG Graduate School GRK 1692 "Curvature, Cycles, and Cohomology"

Organisation:
          
Johannes Kleiner, Jan-Hendrik Treude, Robin Lorenz, Franz Gmeineder

Scientific adivsor:
Prof. Dr. Felix Finster, Department of Mathematics, University of Regensburg

 

What is more, we wish to note the International Spring School on Causal Fermion Systems
which takes place in Regensburg in the week before the workop: