269. Hamiltonian formulation of tetrad gravity: three dimensional case
Save This Post as PDFThis is a guest blog post by Natalia Kiriushcheva and Sergei Kuzmin from the U. of Western Ontario. Dmitry.
Thank you, Dmitry, for the invitation to write a guest post about our recent work co-authored with Alexei Frolov and Sergei Kuzmin: “Hamiltonian formulation of tetrad gravity: three dimensional case”, arXiv: 0902.0856 [gr-qc]. We are curious about your interest in this paper because our result ought to be of interest primary to LQG practitioners rather than to people working in strings, unless, of course, there are some connections or points, unknown to us, where “loops meet strings”. Maybe, if our article can spark a discussion, someone will explain this interest. So, why did we consider the Hamiltonian formulation of tetrad gravity and what did we find so far?
The motivation for our work (to consider the Hamiltonian formulation of tetrad gravity) originated from a result in metric gravity that puzzled us: why its conventional Hamiltonian formulation leads to only a spatial diffeomorphism if the Lagrangian is invariant, as it should be for a generally covariant theory, under the full diffeomorphism. According to Hawking (in “General Relativity. An Einstein Centenary Survey”, 1979)
the split into three spatial dimensions and one time dimension seems to be contrary to the whole spirit of relativity
and according to Isham and Kuchar (Ann. Phys. 164 (1985) 316)
disappearance of Diff…
is
…the problem that has worried many people working in geometrodynamics for so long.
The resolution to this problem (disappearance of diffeomorphism) and the Hamiltonian formulation of metric GR, which leads to the full diffeomorphism, was given in arXiv: 0809.0097, 0809.1198, 0808.2623 [gr-qc].
The tetrad formulation of General Relativity nowadays is more popular than the metric formulation. One reason is that it lies in the basis of LQG. It is known that LQG did not escape from the same problem: the invariance under spatial diffeomorphisms only. There are many articles and chapters in LQG books with discussion on Hamiltonian and diffeomorphism constraints, how to treat them and what to do with them (e.g., “Master Constraint programme”).
So, we decided to reconsider the tetrad formulation of General Relativity with the expectation of finding a Lorentz plus full (not only spatial) diffeomorphism invariance for tetrads. We thought that, as in the metric formulation, if we did not make any non-canonical change of variables (as was done in ADM gravity, see arXiv: 0809.0097 [gr-qc]), we would be able to recover the full diffeomorphism. However, the result of the Hamiltonian formulation shows that the diffeomorphism constraint (neither spatial nor full) did not arise at all in the course of the Dirac procedure. The complete Hamiltonian formulation leads to the gauge invariance of tetrad gravity, which is Poincare symmetry. The Poisson brackets among first class constraints form the true Lie algebra.
This result is not completely unexpected. Witten, in his paper (Nucl. Phys. B 311 (1988) 46), without recourse to the Hamiltonian formulation, constructed a gauge theory for the Poincare group ISO(2,1) as a Chern-Simons action and showed that it coincides with the three dimensional gravity. The distinction between Witten’s and our work is that we started from the first order Lagrangian of tetrad gravity, without any assumption of what a gauge symmetry should be, and derived the gauge transformations using the Dirac procedure. Only when all steps of the procedure are performed, can the gauge symmetry be found unambiguously.
So, the Hamiltonian formulation of tetrad gravity gives a unique result: the algebra of Poisson brackets among secondary first class constraints coincides with Lie algebra of the ISO(2,1) Poincare group. The gauge parameters, responsible for rotational and translational invariance, have internal indices (which refer to the “Lorentz frame”).
Witten showed how to establish the “equivalence” of translational invariance of tetrads and diffeomorphism invariance. Such “equivalence” needs the imposition of additional conditions:
(i) the use of a field-dependent redefinition of the gauge parameter;
(ii) keeping only translational invariance and disregarding rotational invariance,
(iii) the imposition of equations of motion (on-shell invariance).
The question is: why do we need to replace the true gauge invariance with Lie algebra of first class constraints that is derived in the course of the Dirac procedure by another “equivalent” invariance which demands severe restrictions (i)-(iii) and has a non-local algebra of secondary constraints?
In addition, because we used a first order formulation of tetrad gravity, which is valid in all dimensions higher than two (and for some additional technical reasons which we gave in the last section of our paper), we can conclude that the spatial “diffeomorphism constraint” should also not arise in the Hamiltonian formulation of tetrad gravity in dimensions higher than three. The algebra of constraints in the presented Hamiltonian formulation is ordinary Lie algebra in three dimensions and the same algebra should be expected in all higher dimensions. Without any doubt it is impossible to derive a diffeomorphism constraint in any dimension without using a non-canonical change of variables; and by so doing, losing equivalence with the original formulation (that makes such formulations independent of its background).
To summarize, we have the consistent Hamiltonian formulation of tetrad gravity in three dimensions. The Poisson brackets among secondary first class constraints obey Poincare algebra, which is true Lie algebra. Thus, the Hamiltonian formulation of tetrad gravity in three dimensions passes the crucial test of the consistency of the field theory posed by Nicolai, Peeters and Zamaklar in (Class. Quantum Grav. 22 (2005) R193): it has a local algebra of constraints with off-shell closure.
Should we work with the gauge symmetry which is uniquely produced by the Hamiltonian formulation and for which first class constraints form a true Lie algebra or should we continue by inertia to struggle with non-local algebra of the Hamiltonian and spatial diffeomorphism constraints, the ordering problem, etc.?
We think it is important to answer this question before even starting any discussion about quantisation. Otherwise, we should ask: what are we trying to quantise?
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Dear Natalia and Sergey,
thanks for the post. I would like to ask you a couple of questions regarding the physical content. Here they go:
a) you say that ADM use non-canonical change of variables. Can you comment more on that?
b) what is the matter content? In pure 3d gravity, there are no propagating degrees of freedom, but dynamics reappears if one takes matter into account (matter particles create conical defects and in a sense the quantum theory of gravity in this case is the scattering theory for these conical defects on each other). If there is no matter, how your considerations change if I add one (or just simple negative/positive cosmological constant)?
c) You’ve said that rotational invariance is neglected in the procedure of establishing equivalence between 3d gravity and ISO(2,1) Chern-Simons theory. What happens with the equivalence if I take rotational invariance into account?
d) you say that Witten uses field-dependent redefinition of gauge parameter, what exactly makes you unhappy about that?
Cheers,
Dmitry.
Dear Dmitry,
Here are our answers:
a) In general, a canonical change of variables has to preserve the structure of fundamental Poisson brackets (as in ordinary classical mechanics) and for systems with constraints there are some additional restrictions. The discussion of canonicity in case of metric gravity is given in our “Myths”: one non-canonical Poisson bracket for change from ADM variables to metric is given in Eq. (152), p. 62, and discussed on pp. 59-62. The example of canonical change of variables for metric gravity is presented in arXiv:0809.1198 where two formulations due to Pirani et al and Dirac were considered. Canonical transformations are explicitly given and it was shown that both formulations lead to the same invariance.
b) To find the effect of additional terms, e.g. due to matter, the Hamiltonian formulation of a new model has to be reconsidered. Constraints will be modified, they might preserve the algebra of Poisson brackets and gauge transformations or lead to something different. We did not explore this problem (maybe, somebody else will do this). 3D is not of our interest. Our concern is the Hamiltonian formulation in N dimensions and 3D is just technically simplier case (but we consider it using variables and form of the Lagrangian which are valid in all dimensions higher than two).
d) Our unhappiness with field-dependent redefinition of parameters is simple: it creates a wrong impression that it is possible to obtain different gauge invariances for the same model by redefining parameters or by changing variables. Our point: the Hamiltonian approach (or Dirac-Hamiltonian) is the mathematical procedure that gives a unique answer about gauge invariance of any system. By the way, Witten’s transformations and our are the same (after some algebra because he specialized to 3D) and our “unhappiness” is only about field-dependent redefinition of parameters which are needed to relate a true gauge invariance (Poincare) to diffeomorphism invariance, or rather to call diffeomorphism also a gauge invariance of tetrad gravity. Gauge invariance is unique and it is uniquely derived from the constraint structure of a theory.
c) The rotational invariance is neglected by Witten not in establishing equivalence of 3D gravity and Chern-Simons theory, but, again (see (d)), in establishing “equivalence” of Poincare symmetry with diffeomorphism invariance that needs the imposition of additional restrictions.
We discussed this in detail in Section VI of our paper.
Natalia and Sergei
Dear Ms Natalia,
thanks for your provoking guest post. When “loops meet strings”, it is often an explosive encounter.
1) The Hamiltonian treatment, by its very definition, always treats time differently than other coordinates. That doesn’t mean that the underlying theory breaks the actual 4D diffeomorphisms.
The local time translations change the time so they obviously change the slicing, too. But in GR, physical states must be invariant both under spatial and temporal diffeomorphisms, therefore including H=0, so the physical subspace is constrained by all of them in the same way. The construction is not manifestly 4D diff invariant but when done properly, the physics is 4D diff invariant like in other approaches.
2) The tetrad/vielbein formulation of GR is more useful than the metric-only language not because for the reason you mention but primarily because it makes it more straightforward to couple the theory to spinors. A local “Minkowski” basis of the tangent space is kind of needed to define a natural spinor representation.
The advantage of the tetrads you mention is completely unphysical. In LQG, one just decides that an SU(2) gauge field should encode gravity. No rational justification of this goal is known.
In 3D, the spatial components of the SU(2) gauge field have 3×3 = 9 components which makes it easier to map them, by an equally unjustified and arbitrary map, to 3×3 = 9 doubly spatial components of a vielbein rather than a symmetric matrix that has 6 or 10 degrees of freedom. But besides this numerological agreement, 3×3 = 9, there exists no evidence whatsoever that 4D gravity can be rewritten in terms of a well-defined bulk gauge theory. And as argued in the following point, it doesn’t even work in 3 spacetime dimensions nonperturbatively.
3) Even in three dimensions, gravity is not equivalent to a (Chern-Simons) gauge theory in the bulk at the nonperturbative level, when the vielbein is allowed to be non-invertible. See e.g. around page 3 of Witten’s more recent paper
http://arxiv.org/PS_cache/arxi.....3359v1.pdf
The signs of the Lagrangian density disagree once you can change them in the CS description: the gravity Lagrangian has an absolute value, so to say. Even more obviously, the bulk of the modern analysis of the 3D gravity systems in the last decade includes the research of black hole microstates.
The CS theory in 3D knows pretty much nothing about these black hole microstates, so it can’t fully describe gravity at the quantum level. A holographic description on the boundary is more well-defined. Various 2D conformal field theories capture the physics perfectly – and some of their aspects are hidden inside all the “integrable” calculations of black hole entropy in any dimension (BTZ black holes are “factors” in all simple enough black holes that have been fully explained).
4) Because the CS theory doesn’t give these correct microstates, this whole exercise seems to be somewhat vacuous because 3D gravity has no propagating degrees of freedom (Ricci flatness implies Riemann flatness in 3D) – no gravitational waves or gravitons. Adding the BH microstates is the only – sufficient and necessary – way to make the theory locally nontrivial while keeping the “pure gravity” status of the theory.
What the perturbative agreement between CS and 3D gravity really amounts to is the agreement between some perturbative approximations of topological invariants for certain 3-manifolds. You don’t seem to study these things. So the best thing you can do for the Nicolai et al. “challenge” is to argue that you have two languages to describe an empty Hilbert space {0}.
I don’t think that it really solves anything about their serious puzzles – which are, frankly speaking, pretty much complete proofs that nothing along the lines of LQG can ever work (and there are many more independent proofs of the same assertion). In trying to avoid this otherwise inevitable conclusion, it would be a better idea to realize that they were talking about 4D gravity where gravity has local propagating degrees of freedom that actually bring UV divergences to physical observables in gravity if it is not properly quantized (i.e. quantized according to the rules of string theory).
Best wishes
Lubos
The promissed follow up article showed up today on arxiv.org:
http://arxiv.org/abs/0907.1553
The Hamiltonian formulation of N-bein, Einstein-Cartan, gravity in any dimension: the Progress Report (Extended version of a talk given on CAIMS-2009, June 11-14, London, Canada)
Authors: N. Kiriushcheva, S.V. Kuzmin
(Submitted on 9 Jul 2009)
Abstract: The Hamiltonian formulation of N-bein, Einstein-Cartan, gravity, using its first order form in any dimension higher than two, is analyzed. This Hamiltonian formulation allows to explicitly show where peculiarities of three dimensional case (\textit{A.M.Frolov et al, 0902.0856 [gr-qc]}) occur and make a conjecture, based on presented in this report results, that there is one general for \textit{all} dimensions characteristic of N-bein formulation of gravity: after elimination of second class constraints the algebra of Poisson brackets among remaining first class secondary constraints is the Poincare algebra and in all dimensions N-bein, Cartan-Einstein, gravity \textit{is the Poincare gauge theory}. The gauge symmetry corresponding to the algebra of first class constraints has two parameters – rotational (Lorentz) and translational. Translational invariance is common to all dimensions but some terms in general expressions for gauge transformations of N-beins and connections are zero in a particular, three dimensional, case.
The proof of our conjecture is outlined in detail. Some straightforward but tedious calculations remain to be completed to call our conjecture – a theorem and will be reported later.
That post above is mine… I hope Dmitry ressurects as soon as possible.