# Thiemann's time

1. Jul 26, 2006

### Danny

Recently, I read a very intereting paper by Thomas Thiemann,

It is based on this idea that if the Lagrangean for a scalar field coupled to General Relativity only contains derivatives, then it is possible to completely deparametrise the theory. This means that

1. Physical observables, i.e. functions which Poisson commute with the spatial diffeomorphism and Hamiltonian constraints of General Relativity, can be easily constructed.

2. The physical time evolution of those observables is generated by a natural physical Hamiltonian which is (constrained to be) positive.

The mechanism by which this works is due to Brown and Kuchar.

In order that the physical Hamiltonian is close to the Hamiltonian of the standard model and the one used in cosmology, the required Lagrangean must be that of a Dirac-Born-Infeld type.

Such matter has been independently introduced previously by cosmologists in the context of k - essence due to Armendariz-Picon, Mukhanov and Steinhardt in order to solve the cosmological coincidence (dark energy) problem. We arrive at it by totally unrelated physical considerations originating from quantum gravity. Our manifestly gauge invariant approach leads to important modifictaions of the interpretation and the the analytical appearance of the standard FRW equations of classical cosmology in the late universe. In particular, our concrete model implies that the universe should recollapse at late times on purely classical grounds.

http://uk.arxiv.org/abs/astro-ph/0607380

Danny

2. Jul 26, 2006

Staff Emeritus
Danny, it is good to have a dedicated thread on this interesting paper. Marcus had previously noted and entered it at post #501 in his catalog of QG papers, confusingly titled "Intuitive content of Loop Gravity -- Rovelli's program", and I did a preliminary reading of it, but discussion died. Let's not let that happen again!

3. Jul 26, 2006

### marcus

Yes, that other thread has become our non-string QG bibliography---for stashing links to new research so we don't lose track as much. But to discuss stuff we should have a thread devoted to that particular paper, as per Danny's here. It is an interesting paper. It would be exciting if some people would start taking seriously the idea that there could be this unobservable "Phantom Negative Energy" (canceled out by a positive "k-essence") that Thiemann supposes. It seems to have more personality the usual "Dark Energy". I want to quote from Thiemann's conclusions section at the end:

===exerpt===
7 Conclusions and Outlook

It has been known for a long time that the problem of time can be solved in principle by the relational framework due to Rovelli and others. This has never been appreciated as much as it should have been because, while the conceptual, physical framework was clear, the analytical implementation remained largely undeveloped for a long time. With the appearance of [8], analytical methods became available for the first time. Still the framework, in its full generality, remains discouragingly difficult in particular when applied to General Relativity due to the complexity of the analytical expressions...

The first main message of the present paper is that by adding appropiate, albeit hypothetical, matter, the complexity of these formulas is drastically reduced. In contrast to the general case, there is only one series to sum, there are no matrices to invert, there is only one kind of iterated Poisson bracket to compute. Hence the formulas that we obtain are remarkably simple. In fact, the classical time evolution in a background dependent theory, say in QCD on Minkowski space, of some observable O such as a Wilson loop function, would also be given by the series

$$O(\tau) = \sum_{n=0}^{\infty} \frac{\tau^n}{n!} \{H_{QCD}, O \}_{(n)}$$

where HQCD is the QCD Hamiltonian. Comparing with (4.2) we see that the complexity of the classical time evolution in both theories is comparable!

The second main message is that, in contrast to the general case, the physical observables we obtain are strong observables and there is just one natural, physical Hamiltonian which does not depend on the physical time parameter. That Hamiltonian is (constrained to be) positive and at least in the physically interesting region in phase space, that Hamiltonian reduces to the canonical Hamiltonian that one usually uses in cosmology and the standard model when the metric is flat. In fact, we manage to completely deparametrise the system irrespective of the other matter present.

The third main message is that the scalar type of matter that we considered here, from the mathematical (to be able to solve algebraic equations) and physical (to obtain a physical Hamiltonian which is close to that of the standard model) perspective naturally leads to Dirac – Born – Infeld (DBI) negative energy phantom fields with constant potential. This negative energy phantom must be compensated for by positive energy matter, most naturally by a k – essence field. Such matter was discussed independently in the cosmology literature in order to provide a candidate for inflation and dark energy. Hence the scalar matter we consider here might actually really exist!

The fourth main message is that, at least for the scalar model we have used here and for which we gave strong motivations, the usual interpretation of the cosmological framework, although fundamentally wrong because gauge transformations of gauge dependent objects are interpreted as actual physical evolution equations of observables, remains valid when analysed in the correct way, that is, by computing the physical evolution of gauge invariant observables. The domain of validity of these equations can be tuned to be arbitrarily long, however, it is manifestly finite when using the physical time parameter corresponding to the physical Hamiltonian. The actual evolution at late times apparently leads to a recollapse.

===end quote===

Last edited: Jul 26, 2006