The Hamiltonian constraint NH = 0 defines a diffeomorphism structure on spatial manifolds by the lapse function N, but this is not identical to time. ❋ Unknown (2009)
The main problem for the latter is the general covariance of the field equations of General Relativity: any spacetime model and its image under a diffeomorphism (a infinitely differentiable, one-one and onto mapping of the model to itself) are in all observable respects equivalent to one another; all physical properties are expressed in terms of generally covariant relationships between geometrical objects. ❋ Ladyman, James (2009)
Figure 5: “Hole” diffeomorphism shifts contents of spacetime ❋ Hoefer, Carl (2008)
Other: It's not nonperturbative quantization of diffeomorphism-invariant gauge theory. ❋ Slimbolala (2006)
For mathematical reasons not relevant here, it turns out to be possible to take a given model spacetime and perform a mathematical operation called a “hole diffeomorphism” h* on it; the diffeomorphism's effect is to shift around the matter content ❋ Hoefer, Carl (2008)
As is well known already in 1D, interesting lowest-energy representations of the diffeomorphism algebra are necessarily anomalous (Virasoro algebra), and the higher-dimensional case is not different (multi-dimensional Virasoro algebra). ❋ Sean (2007)
Maybe this is the resolution for the representation facts which you refer to from CFT, insofar as those facts perhaps pertain to continuous (projective) representations of the diffeomorphism group of the circle. ❋ Sean (2007)
Non-trivial unitary lowest-energy reps of diffeomorphism and gauge algebras are necessarily anomalous, and a quantized observer jumps right out of their representation theory. ❋ Sean (2007)
In AdS/CFT, not only the bulk spacetime but also the bulk diffeomorphism group are emergent: the CFT fields are completely invariant under the bulk diffeomorphisms (this is also what happens in the much more common phenomenon of emergent gauge symmetry). ❋ Sean (2006)
Manifest diffeomorphism invariance seems too limited if we want to describe string theory. ❋ Sean (2006)
More interestingly, can I ask you why you think loop methods, or more generally methods aiming to achieve manifest diffeomorphism invariance, should be useful for providing a non-perturbative definition of string theory? ❋ Sean (2006)
It says that if you want a diffeomorphism invariant quantization of a gauge theory on a manifold with no metric, in which the algebra of Wilson loops and electric flux is realized, there is a unique hilbert space to work in for each gauge group and manifold. ❋ Sean (2006)
They can be shifted between the Weyl and diffeomorphism sectors, but they cannot be removed. ❋ Sean (2005)
Successful quantization will lead to diffeomorphism anomalies. ❋ Sean (2005)
Exactly how these anomalies manifest themselves depends on the formalism; Jackiw showed that if you start by gauge-fixing Weyl transformations, the anomaly pops up the the diffeomorphism sector instead. ❋ Cjohnson (2005)
There are certainly diff anomalies in 4D in the sense that the 4D diffeomorphism algebra has non-trivial extensions which generalize the Virasoro algebra. ❋ Cjohnson (2005)
I believe these are all models, necessary steps from which we learn how to do non-trivial calculations in background independent, diffeomorphism invariant quantum field theories. ❋ Mark (2005)
If you cannot construct non-trivial, anomalous reps of the diffeomorphism group, you cannot construct the correct Hilbert space of quantum gravity, because it must carry such a rep. ❋ Sean (2005)
What has prevented the acceptance of this discovery is the widespread belief that since the diffeomorphism symmetry is gauge in general relativity, only the trivial rep matters. ❋ Sean (2005)
They must contain qualitatively analogous ingredients that turned out to be important for the description of actual physics around: unitarity, gauge symmetries, diffeomorphism symmetries, Lorentz invariance, dependence of physics on the scale, Higgs mechanism, gauge anomaly cancellation, and so forth. ❋ Unknown (2010)
