COVARIANT HAMILTONIAN GENERAL RELATIVITY

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A NOTE ON THE FOUNDATION OF RELATIVISTIC MECHANICS:
COVARIANT HAMILTONIAN GENERAL RELATIVITY
CARLO ROVELLI
Centre de Physique Théorique, Luminy, F-13288 Marseille, EU
Physics Department, Pittsburgh University, PA-15260, USA
I illustrate a simple hamiltonian formulation of general relativity, derived from the work of Esposito,
Gionti and Stornaiolo, which is manifestly 4d generally covariant and is defined over a finite dimensional
space. The spacetime coordinates drop out of the formalism, reflecting the fact that they are not
related to observability. The formulation can be interpreted in terms of Toller’s reference system
transformations, and provides a physical interpretation for the spinnetwork to spinnetwork transition
amplitudes computable in principle in loop quantum gravity and in the spin foam models.
1
The problem
In the companion paper1 I have discussed the possibility of a relativistic foundation of mechanics and I have argued that the usual notions of state and observable have to be modified
in order to work well in a relativistic context. Here I apply this point of view to field theory.
In the context of field theory, the relativistic notion of observable suggests to formulate the
hamiltonian theory over a finite dimensional space, for two reasons. First the space of the
relativistic (“partial”2 ) observables of a field theory is finite dimensional. Second, the infinite
dimensional space of the initial values of the fields, which is the conventional arena for hamiltonian field theory, is based on the notion of instantaneity surface, which has little general
significance in a relativistic context. The possibility of defining hamiltonian field theory over
a finite dimensional space has been explored by several authors (See Refs. 3,4,5, and ample
references therein), developing the classic works of Weil6 and De Donder7 on the calculus of
variations in the 1930’s. In section 2, I briefly illustrate the main lines of this formulation
using the example of a scalar field, and I discuss its relation with the relativistic notions of
state and observable considered in Ref. 1 I then apply these ideas to general relativity (GR)
in Section 3.
Unraveling the hamiltonian structure of GR has taken decades. The initial intricacies
faced by Dirac8 and Bergmann9 were slowly reduced in various steps, from the work of
Arnowit, Deser, and Misner10 all the way to the Ashtekar formulation11 and its variants.
Here, I discuss a far simpler hamiltonian formulation of GR, constructed over a finite dimensional configuration space and manifestly 4d generally covariant. The formulation is largely
derived from the work of Esposito, Gionti and Stornaiolo in Ref. 12 (On covariant hamiltonian formulations of GR, see also Refs. 13,14,15,16,17,18.) I discuss two interpretations of
this formalism. The first uses the coordinates, while the second makes no direct reference
to spacetime. The four spacetime coordinates drop out from the formalism (as the time coordinate drops out from the ADM formalism) in an appropriate sense. I find this feature
particularly attractive: the general relativistic spacetime coordinates have no relation with
observability and a formulation of the theory in which they do not appear was long due.
I expect this formulation of GR to generalize to any matter coupling and to any diffeomorphism invariant theory. I think that it sheds some light on the coordinate-independent
physical interpretation of the theory and helps clarifying issues that appear confused in the
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hamiltonian formulations which are not manifestly covariant. In particular: what are “states”
and “observables” in a theory without background spacetime, without external time and without an asymptotic region? I close discussing the relevance of this analysis for the problem
of the interpretation of the formalism of quantum gravity. The formulation presented can
be interpreted in terms of Toller’s reference system transformations,19 and provides a physical interpretation for the spinnetwork to spinnetwork transition amplitudes which can be
computed in principle in loop quantum gravity20 and in the spinfoam models.21
2
Relativistic hamiltonian field theory
There are several ways in which a field theory can be cast in hamiltonian form. One possibility
is to take the space of the fields at fixed time as the (nonrelativistic) configuration space Q.
This strategy badly breaks special and, in a general covariant theory, general relativistic
invariance. Lorentz covariance is broken by the fact that one has to choose a Lorentz frame
for the t variable. I find far more disturbing the conflict with general covariance. The very
foundation of general covariant physics is the idea that the notion of a simultaneity surface
all over the universe is devoid of physical meaning. Seems to me that it is better not to found
hamiltonian mechanics on a notion devoid of physical significance.
A second alternative is to formulate mechanics on the space of the solutions of the equations of motion. The idea goes back to Lagrange. In the generally covariant context, a
symplectic structure can be defined over this space using a spacelike surface, but one can
show that the definition is surface independent and therefore it is well defined. This strategy
as been explored by Witten, Ashtekar and several others.18 The structure is viable in principle, but very difficult to work with in practice. The reason is that we do not know the space
of the solutions of an interacting theory. Therefore we must either work over a space that we
can’t even coordinatize, or coordinatize the space with the initial data on some instantaneity
surface, and therefore, effectively, go back to the conventional fixed time formulation. Thus,
this formulation has the merit of telling us that the hamiltonian formalism is actually intrinsically covariant, but it does not really indicate how to effectively deal with it in a covariant
manner.
The third possibility, which I consider here, is to use a covariant finite dimensional space
for formulating hamiltonian mechanics. I noticed in the companion paper1 that in the relativistic context the double role of the phase space, as the arena of mechanics and the space of
the states, is lost. The space of the states, namely the phase space Γ is infinite dimensional in
field theory, virtually by definition of field theory. But this does not imply that the arena of
hamiltonian mechanics has to be infinite dimensional as well. In particular, a main result of
Ref. 1 is that the natural arena for relativistic mechanics is the extended configuration space
C of the partial observables. Is the space of the partial observables of a field theory finite or
infinite dimensional?
Consider a field theory for a field φ(x) with k components, defined over spacetime M
with coordinates x, and taking values in a k dimensional target space T
φ : M −→ T
x 7−→ φ(x).
(1)
~ B),
~
For instance, this could be Maxwell theory for the electric and magnetic fields φ = (E,
where k = 6. In order to make physical measurements on the field described by this theory
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we need k measuring devices to measure the components of the field φ, and 4 devices (say
one clock and three devices giving us the distance from three reference objects) to determine
the spacetime position x. Field values φ and positions x are therefore the partial observables
of a field theory. Therefore the operationally motivated extended configuration space for a
field theory is the finite dimensional 4+k dimensional space
C = M × T.
(2)
A correlation is a point (x, φ) in C. It represents a certain value (φ) of the fields at a
certain spacetime point (x). This is the obvious generalization of the (t, α) correlations of the
pendulum of the example in Ref. 1.
A motion is a physically realizable ensemble of correlations. A motion is determined by
a solution φ(x) of the field equations. Such a solution determines a 4-dimensional surface
in the (4+k dimensional) space C: the surface is the graph of the function (1). Namely the
ensemble of the points (x, φ(x)). The space of the solutions of the field equations, namely
the phase space Γ, is therefore an (infinite dimensional) space of 4d surfaces in the (4+k)-d
configuration space C. Each state in Γ determines a surface in C.
Hamiltonian formulations of field theory defined directly on C = M × T are possible
and have been studied. The main reason is that in a local field theory the equations of
motion are local, and therefore what happens at a point depends only on the neighborhood
of that point. There is no need, therefore, to consider full spacetime to find the hamiltonian
structure of the field equations. I refer the reader to Refs. 3 and 5, the ample references
therein, and especially the beautiful and detailed Ref. 4. What comes below is a very simple
and self-contained illustration of the formalism.
The difference with the finite dimensional case is that curves in C are replaced by 4d
surfaces. Thus, we need a hamiltonian formalism determining these four dimensional surfaces
in C. At a point p of C, a curve has a tangent, leaving in Tp C, the tangent space of C at p. A 4d
surface has four independent tangents Xµ , or a “quadritangent” X = µνρσ Xµ ⊗Xν ⊗Xρ ⊗Xσ .
Consider a self interacting scalar field φ(x) defined on Minkowski space M, with interaction potential V (φ). Its field equation
∂µ ∂ µ φ + m2 φ + V 0 (φ) = 0
(3)
can be derived from a hamiltonian formalism as follows. To test the theory (3) we need five
measuring devices: a clock reading x0 , three devices that give us the spatial position ~x, and a
device measuring the value of the field φ. Therefore, the space C of the partial observables is
the the five dimensional space C = M × T , with coordinates (x, φ). Here T = IR is the target
space of the field: φ ∈ T . Let Ω be a 10d space with coordinates (x, φ, p, pµ ) carrying the the
Poincaré-Cartan four-form
θ = p d4 x + pµ dφ ∧ d3 xµ .
(4)
Here d4 x ≡ dx1 ∧dx2 ∧dx3 ∧dx4 and d3 xµ ≡ d4 x(∂µ ). (Geometrically, Ω is not the cotangent
space of C; it is a subspace of the bundle of its four-forms, or a dual first jet bundle.4 ) Consider
the constraint
H = p + ( 12 pµ pµ + 12 m2 φ2 + V (φ)) = 0
(5)
on Ω. (H–p is the DeDonder-Weyl hamiltonian function.) Let ω be the restriction of the five
form dθ to the surface Σ defined by H = 0. Then the solutions of (3) are the orbits of ω. An
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orbit of ω is an integral surface of its null quadrivectors. That is, it is a 4d surface immersed
in Σ whose quadri-tangent X satisfies
ω(X) = 0.
(6)
For simplicity, let’s say that this surface can be coordinatized by xµ , that is, it is given by
(x, φ(x), pν (x)). Then φ(x) is a solution of the field equations (3). Thus, (6) is equivalent to
the field equations (3).
A state determines a 4d surface (x, φ(x)) in the extended configurations space C. It
represents a set of combinations of measurements of partial observables that can be realized
in nature. The phase space Γ is the space of these states, and is infinite dimensional.
In the classical theory, a state determines whether or not a certain correlation (x, φ), or
a certain set of correlations (x1 , φ1 ) . . . (xn , φn ), can be observed. They can be observed iff
the points (xi , φi ) lie on the 4d surface that represents the state. Viceversa, the observation
of a certain set of correlations gives us information on the state: the surface has to pass by
the observed points. In the quantum theory, a state determines the probability amplitude of
observing a correlation, or a set of correlations.
Notice that there is an important difference between a finite dimensional system and a
field theory. For the first, the measurement of a finite number of correlations can determine
the state. In the quantum theory, a single correlation may determine the state. For instance,
if we have seen the pendulum in the position α at time t, we then know the quantum state
uniquely. It follows that quantum mechanics determines uniquely the probability amplitude
W (α0 , t0 ; α, t) for observing a correlation (α0 , t0 ) after having observed a correlation (α, t).
Clearly
W (α0 , t0 ; α, t) = hα0 , t0 |α, ti
(7)
where |α, ti is the eigenstate of the Heisenberg operator α̂(t) with eigenvalue α. In field theory,
on the other hand, an infinite number of measurement is required in principle to uniquely
determine the state. We can measure any finite number of correlations, and still do not know
the state. Predictions in field theory are therefore always given on the basis of some a priori
assumption on the state. In quantum theory, this additional assumption, typically, is that
the field is in a special state such as the vacuum. Thus, a prediction of the quantum theory
takes the following form: if the system is in the vacuum state and we observe a certain set of
correlations (x1 , φ1 ) . . . (xn , φn ), what is the probability amplitude
W (x01 , φ01 . . . x0n0 , φ0n0 ; x1 , φ1 . . . xn , φn )
(8)
of observing a certain other set (x01 , φ01 ) . . . (x0n0 , φ0n0 ) of correlations? The quantities (8) are
directly related to the usual n point distributions of field theory. The relation is the same as
the one that transforms the position basis to the energy basis for an harmonic oscillator, that
is, for instance
X
ψn (φ0 )ψm (φ) hn, x0 |m, xi
(9)
W (x0 , φ0 ; x, φ) =
n,m
where |n, xi ∼ (a† (x))n |0i is the state with n particles in x. The distributional character of
these quantities will be studied elsewhere.
For later comparison with GR, notice that the spacetime component M of the relativistic
configuration space C = M × T is essential in the description, since the predictions of the
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theory regard precisely the dependence of the partial observable φ on the partial observables
xµ .
The simplicity, covariance and elegance of this hamiltonian formulation is quite remarkable. I find it particularly attractive from the conceptual point of view, because the notions
of observable and state on which it is based are operationally well founded, relativistic and
covariant. I now apply these ideas to GR.
3
Covariant hamiltonian GR
GR can be formulated in tetrad-Palatini variables as follows. I indicate the coordinates of
the spacetime manifold M as xµ , where µ = 0, 1, 2, 3. The fields are a tetrad field eIµ and a
Lorentz connection field AIJ
µ (antisymmetric in IJ) where I = 0, 1, 2, 3 is a 4d Lorentz index,
raised and lowered with the Minkowski metric. The action is
Z
KL µνρσ
S = eIµ eJν Fρσ
IJKL d4 x
(10)
IJ
where Fµν
is the curvature of AIJ
µ . The field equations turn out to be
Dν (eJρ eJσ )µνρσ = 0,
JK µνρσ
eIµ Fνρ
IJKL = 0.
(11)
where Dµ is the covariant derivative of the Lorentz connection. The first equation implies
that AIJ
µ is the spin connection determined by the tetrad field. Using this, the second is
equivalent to the Einstein equations. That is, if (eIµ (x), AIJ
µ (x)) satisfy (11), then the metric
I
tensor gµν (x) = eµ (x)eνI (x) satisfies the Einstein equations. I shall thus refer at (11) as the
Einstein equations. Below, these equations are derived from a hamiltonian formalism, derived
from Ref. 12.
3.1
First version
˜
Consider the (4+16+24) dimensional space C˜ with coordinates (xµ , eIµ , AIJ
µ ). We have C =
M × T , where T is the target space on which the tetrad-Palatini fields of GR take value and
M is the spacetime manifold on which they are defined.
eI = eIµ dxµ ,
µ
AIJ = AIJ
µ dx
(12)
˜ For any function or form on C˜ with Lorentz indices, the Lorentz covariant
are one-forms on C.
differential is defined by
Dv I = dv I + AIJ v J .
(13)
I now introduce the main objects of the formalism: the form on C˜
θ=
1
IJKL eI ∧ eJ ∧ DAKL
2
(14)
µ
IJ
µ
and the presymplectic form ω = dθ. (Notice that dAKL = d(AIJ
µ dx ) = dAµ ∧ dx , beµ
˜
cause AIJ
µ and x are independent coordinates on C.) Now, the remarkable fact is that the
presymplectic form ω defines GR completely. In fact, its orbits, defined by
ω(X) = 0
(15)
where X is the quadritangent to the orbit, are the solutions of the Einstein equations. That
is: assuming for simplicity that the xµ coordinatize the orbit, namely that the orbit is given
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I
IJ
by (xµ , eIµ (x), AIJ
µ (x)), then it follows from (15) that (eµ (x), Aµ (x)) solve the Einstein equaI
IJ
tions. Viceversa, if (eµ (x), Aµ (x)) solve the Einstein equations, then (xµ , eIµ (x), AIJ
µ (x)) is
an orbit of ω. Equation (15) is equivalent to the Einstein equations. The demonstration is a
straightforward calculation given in the Appendix.
3.2
Second version
The simplicity of the formulation of GR described above is quite striking. I find the following
I
observations even more remarkable. The space C˜ contains the field variables AIJ
µ and eµ as well
µ
as the spacetime coordinates x . Since the theory is coordinate invariant, we are in a situation
analogous to the finite dimensional cases of the relativistic particle, or the cosmological model,
described in Ref. 1. In those examples, the unphysical lagrangian evolution parameter t
dropped out of the formalism; not surprisingly, since it had nothing to do with observability.
Here, similarly, we should expect the coordinates xµ to drop out of the formalism. In fact,
it is well known that the gauge invariant quantities of GR are coordinate independent. They
are obtained by solving away the coordinates from quantities constructed out of the fields.
Therefore the theory should actually live on the sole field space T with coordinates A IJ
µ and
I
µ
eµ , without reference to the spacetime coordinates x . Is this possible?
The remarkable aspect of the expression (14) of the form θ is that the differentials of the
µ
spacetime coordinates dxµ appear only within the one-forms eI = eIµ dxµ and AIJ = AIJ
µ dx .
µ
This fact indicates that the sole role of the x is to arbitrarily coordinatize the orbits in the
40d space of the fields (eIµ , AIJ
µ ). We can therefore reinterpret the formalism of the previous
˜
section dropping the spacetime part of C.
Let V be a 4d vector space. Clearly V is not spacetime; it can be interpreted as a “space
of directions”. Let C be the 40d space of the one-forms (eI , AIJ ) on V . Notice that C is the
space C = V ∗ ⊗ P of the 4d one-forms with value in the algebra P of the Poincaré group.
Choosing a basis aµ in V , the coordinates on C are (eIµ , AIJ
µ ) and C can be identified with
the target space T of the tetrad-Palatini fields. Consider a 4d surface immersed in C. The
tangent space Tp to this surface at a point p = (eI , AIJ ) is a 4d vector space. This space can
be identified with V . In particular, given an arbitrary choice of coordinates xµ on the surface,
we identify the basis ∂µ of Tp with the basis aµ of V . Therefore we have immediately the ten
one-forms (eI , AIJ ) on the tangent space Tp . That is, eI (∂µ ) = eIµ and AIJ (∂µ ) = AIJ
µ .
Consider now a form α = αI deI + αIJ dAIJ on C. α is a one-form on Tp (valued in the
one-forms over C), and we can write α(∂µ ) = αI deIµ + αIJ dAIJ
µ . But notice that α determines
also a two-form on Tp by
α(∂µ ⊗∂ν ) = αI ∂µ eIν + αIJ ∂µ AIJ
ν .
(16)
It follows that ω = dθ, with θ given by (14), acts on multivectors in Tp . In particular, it acts
on the quadritangent X = µνρσ ∂µ ⊗∂ν ⊗∂ρ ⊗∂σ of Tp . The vanishing (15) of ω(X) is equivalent
to the Einstein equations (see Appendix).
Therefore the theory is entirely defined on the 40d space C. The states are the 4d surfaces
in C, whose tangents are in the kernel of dθ. This is all of GR. The spacetime part M of
C˜ ∼ M × C is eliminated from the formalism. The only residual role of the xµ is to arbitrarily
parametrize the states, precisely as for the unphysical lagrangian parameter in the examples
of Ref. 1. Below I study the physical interpretation of C.
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4
4.1
Physical interpretation of C
Classical theory: reference system transformations
As discussed in Ref. 1, in general, coordinates of the extended configuration space C are the
partial observables of the theory. A point in C represents a correlation between these observables, that is, a possible outcome of a simultaneous measurements of the partial observables,
which give information on the state of the system, or that can be predicted by the theory.
What are the partial observables and the correlations of GR captured in the space C? Can we
give the space of the Poincaré algebra valued 4d one-forms C a direct physical interpretation?
Assume that the measuring apparatus includes a reference system formed by physical objects defining three orthogonal axes and a clock. Following Toller,19 we take a transformation
T of the Poincaré group (in the reference system) as the basic operation defining the theory.
That is, assume that the basic operation that we can perform is to displace the reference system by a certain length in a spacial direction, or wait a certain time, or rotate it by a certain
amount, or boost it at a certain velocity. The operational content of GR can be taken to be
the manner in which the transformations T fit together.19 That is, we assume that a number
of these transformations, T1 , ..., Tn , can be performed and that it is operationally meaningful
to say that two transformations Ti and Tj start at the same reference system, or arrive at the
same reference system, or one starts where the other arrives. We identify one transformation
as the measurement of a partial observable (the value of the partial observable is given by the
Poincaré group element giving the magnitude of the transformation).
A set of such measurements is therefore an oriented graph γ where each link l carries
an element Ul of the Poincaré group. Arbitrarily coordinatize the nodes of the graph with
coordinates y. In the classical theory, we assume that arbitrarily many and arbitrarily fine
transformations can be done, so that we can take the elements of the Poincaré group as
infinitesimal, namely in the algebra, and the coordinates y as smooth. An infinitesimal transformation can therefore be associated to an infinitesimal coordinate change dy. As there is a
10d algebra of available transformations, there is a 10d space of infinitesimal transformations
and the coordinates y are ten dimensional. However, it is an experimental fact –coded in
the theory– that rotations and boosts close and realize the Lorentz group. That is, the relations between sets of physical rotations or boosts are entirely determined kinematically by
the Lorentz group. The same is not true for displacements. Therefore the space of the y’s is
naturally fibrated by 6-d fibers isomorphic to the Lorentz group.19 Coordinatize the remaining 4d space (the space of the fibers) with 4d coordinates xµ . It follows that the non-trivial
infinitesimal transformations assign an element of the Poincaré algebra P to an infinitesimal
displacement dxµ . A single correlation is then the determination of a Poincaré algebra element for each dxµ . The space of the correlations is therefore the space of the Poincaré algebra
valued 4d one-forms, which is precisely C.
We can restrict the partial observables to a smaller number. First, we are using a first
order formalism with configuration variables as well as momenta. Either the eI or the AIJ
alone suffice to characterize a solution. With the first choice, we can take physical lengths
and angles associated to the dxµ displacements as partial observables. With the second, we
can take the Lorentz rotation part of the transformations T . This gives infinitesimal Lorentz
µ
transformations RIJ = AIJ
µ dx as partial observables. We can also exploit the internal
Lorentz gauge invariance of the theory to partially gauge fix the Lorentz group. As well
known, for instance, AIJ can be gauge fixed to an element of so(3). Finally, one can further
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gauge fix by solving explicitly the dependence of some partial observables as functions of
others; see for instance Ref. 22 for a realistic way of doing this. I shall not deal with this
possibility here.
4.2
Quantum theory: spin networks
In the quantum theory, quantum discreteness does not allow us to go to the continuous
description. A finite set of partial measurements must therefore be represented by the graph
γ with elements of the Poincaré group Ul associated the links l. If we restrict to configuration
observables we can take the Ul to be in the Lorentz group, or, in gauge fixed form, in the
rotation group. In general, quantum theory gives the probability amplitude to observe a
certain ensemble of partial observables, given that a certain other ensemble of observables
has been observed. See Section IV of Refs. 1 and 23. Therefore, we should expect that the
predictions of a quantum theory of GR can be cast in the form of probability amplitudes
W (γ 0 , Ul00 ; γ, Ul ).
Now, the quantities W (γ 0 , Ul00 ; γ, Ul ) are precisely of the form spinnetwork to spinnetwork
transition amplitudes which can be computed, in principle, in loop quantum gravity (see Ref.
20, and references therein) and in the spinfoam models (see Ref. 21 and references therein).
More precisely, we can write, in analogy with (9)
X
(17)
ψj 0 (Ul00 ) ψk (Ul ) hγ 0 , j 0 |γ, ji
W (γ 0 , Ul00 ; γ, Ul ) =
j 0 ,j
where j 0 (respectively j) represents the possible labels of a spinnetwork with graph γ 0 (respectively γ), ψj (Ul ) is the spinnetwork function on the group, and hγ 0 , j 0 |γ, ji is the (physical)
spinnetwork to spinnetwork transition amplitude. See Ref. 24 for details. Therefore the
hamiltonian structure illustrated here provides a conceptual framework for the interpretation
of these transition amplitudes. Notice that no trace of position or time remains in these
expressions.
5
Conclusion
The shift in perspective defended in the companion paper1 is partially motivated by special
relativity, but it is really forced by general relativity. The notion of initial data spacelike
surface conflicts with diffeomorphism invariance. A generally covariant notion of instantaneous state, or evolution of states and observables in time, make little physical sense. In a
general gravitational field we cannot assume that there exists a suitable asymptotic region,
and therefore we do not have a notion of scattering amplitude and S matrix. In this context,
it is not clear what we can take as states and observables of the theory, and what is the
meaning of dynamics. In Ref. 1 and in this paper I have attempted a relativistic foundation
of mechanics, that could provide clean notions of states and observables, making sense in an
arbitrary general relativistic situation, as well as in quantum theory.
I have argued that mechanics can be seen as the theory of the evolution in time only
in the nonrelativistic limit. In general, mechanics is a theory of relative evolution of partial
observables with respect to each other. More precisely, it is a theory of correlations between
partial observables. Given a state, classical mechanics determines which correlations are
observable and quantum mechanics gives the probability amplitude (or probability density)
for each correlation.
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In this paper I have applied the ideas of Ref. 1 to field theory. I have argued that the
relativistic notions of state and observable lead naturally to the formulation of field theory
over a finite dimensional space. The application of this formulation to general relativity leads
to a remarkably simple hamiltonian formulation, in which the physical irrelevance of the
spacetime coordinates becomes manifest.
General relativity can be formulated simply as the pair (C, dθ). C is the 40d space of
the Poincaré valued 4d one-forms, and θ is given by (14). The orbits of dθ in C, solutions of
equation (15), are the solutions of the Einstein equations and form the elements of the phase
space Γ. This is all of general relativity.
The compactness and simplicity of this hamiltonian formalism is quite remarkable. Notice for instance that the target space T , the extended configuration space C, the space Ω
that carries the presymplectic form defining the theory and the constraint surface Σ are all
identified. The form θ codes the dynamics as well as the symplectic structure of the theory.
The disappearance of the spacetime manifold M and its coordinates xµ –which survive
only as arbitrary parameters on the orbits– generalizes the disappearance of the time coordinate in the ADM formalism and is analogous to the disappearance of the lagrangian evolution
parameter in the hamiltonian theory of a free particle.1 It simply means that the general relativistic spacetime coordinates are not directly related to observations. The theory does not
describe the dependence of the field components on xµ , but only the relative dependence of
the partial observables of C on each other.
We can give C a direct physical interpretation in terms of reference systems transformations. This interpretation is illustrated in Section 4. In the quantum domain, it leads directly
to the spin-network to spin-network amplitudes computed in loop quantum gravity.
Appendix
Here we prove the claim that equations (15) is equivalent to the Einstein equations (11). Let
us first write ω explicitly. We have from (14)
θ=
=
1
IJKL eIµ dxµ ∧ eJν dxν ∧
2
(dAKL
∧ dxσ + AKM
dxρ ∧ AσM L dxσ )
σ
ρ
1
µνρσ IJKL eIµ eJν dAKL
∧ dxσ
ρ
2
+µνρσ IJKL eIµ eJν AKM
AσM L d4 x.
ρ
(18)
It follows
ω = dθ = µνρσ IJKL eIµ deJν ∧ dAKL
∧ dxσ
ρ
AσM L ) ∧ d4 x.
+µνρσ IJKL d(eIµ eJν AKM
ρ
(19)
Coordinatize an orbit with the xµ . The tangents are then
Xµ =
∂
∂
∂
+ ∂µ eIν I + ∂µ AIJ
.
ν
∂xµ
∂eν
∂AIJ
ν
(20)
The components of X = µνρσ Xµ ⊗Xν ⊗Xρ ⊗Xσ that give a nonvanishing contribution when
contracting with ω are the ones with at least two ∂µ = ∂/∂xµ components. These are (I leave
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9
For Publisher’s use
the
⊗
understood in the notation)
Xµ = µνρσ [ ∂µ ∂ν ∂ρ ∂σ + ∂µ ∂ν ∂ρ (∂σ eIτ )
+∂µ ∂ν (∂ρ eIτ )
∂
∂
+ ∂µ ∂ν ∂ρ (∂σ AIJ
+
τ )
∂eIτ
∂AIJ
τ
∂
∂
(∂σ AIJ
+... ]
)
∂eIτ
∂AIJ
(21)
From (19) and (21), we obtain
µ
µ
ω(X) = KIµ deIµ + KIJ
dAIJ
µ + Kµ dx ,
(22)
where
JK L
KIµ = IJKL µνρσ Fνρ
eσ ,
µ
L µνρσ
KIJ
= IJKL Dν eK
.
ρ eσ KIµ
(23)
µ
KIJ
while Kµ vanishes if
and
do. It follows immediately that ω(X) = 0 give the Einstein
equations (11).
The Einstein equations are obtained even more directly in the second version of the
formalism. Of the four ∂µ , three contract the three eI and AIJ forms, giving their components,
and one contracts either deIµ or dAIJ
ν , leaving simply
µ
ω(X) = KIµ deIµ + KIJ
dAIJ
µ
(24)
µ
where KIµ and KIJ
are again given by (23).
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