Canonical Actions on Bundles – Philosophizing Identity Over Gauge Transformations.

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In physical applications, fiber bundles often come with a preferred group of transformations (usually the symmetry group of the system). The modem attitude of physicists is to regard this group as a fundamental structure which should be implemented from the very beginning enriching bundles with a further structure and defining a new category.

A similar feature appears on manifolds as well: for example, on ℜ2 one can restrict to Cartesian coordinates when we regard it just as a vector space endowed with a differentiable structure, but one can allow also translations if the “bigger” affine structure is considered. Moreover, coordinates can be chosen in much bigger sets: for instance one can fix the symplectic form w = dx ∧ dy on ℜ2 so that ℜ2 is covered by an atlas of canonical coordinates (which include all Cartesian ones). But ℜ2 also happens to be identifiable with the cotangent bundle T*ℜ so that we can restrict the previous symplectic atlas to allow only natural fibered coordinates. Finally, ℜ2 can be considered as a bare manifold so that general curvilinear coordinates should be allowed accordingly; only if the full (i.e., unrestricted) manifold structure is considered one can use a full maximal atlas. Other choices define instead maximal atlases in suitably restricted sub-classes of allowed charts. As any manifold structure is associated with a maximal atlas, geometric bundles are associated to “maximal trivializations”. However, it may happen that one can restrict (or enlarge) the allowed local trivializations, so that the same geometrical bundle can be trivialized just using the appropriate smaller class of local trivializations. In geometrical terms this corresponds, of course, to impose a further structure on the bare bundle. Of course, this newly structured bundle is defined by the same basic ingredients, i.e. the same base manifold M, the same total space B, the same projection π and the same standard fiber F, but it is characterized by a new maximal trivialization where, however, maximal refers now to a smaller set of local trivializations.

Examples are: vector bundles are characterized by linear local trivializations, affine bundles are characterized by affine local trivializations, principal bundles are characterized by left translations on the fiber group. Further examples come from Physics: gauge transformations are used as transition functions for the configuration bundles of any gauge theory. For these reasons we give the following definition of a fiber bundle with structure group.

A fiber bundle with structure group G is given by a sextuple B = (E, M, π; F ;>.., G) such that:

  • (E, M, π; F) is a fiber bundle. The structure group G is a Lie group (possibly a discrete one) and λ : G —–> Diff(F) defines a left action of G on the standard fiber F .
  • There is a family of preferred trivializations {(Uα, t(α)}α∈I of B such that the following holds: let the transition functions be gˆ(αβ) : Uαβ —–> Diff(F) and let eG be the neutral element of G. ∃ a family of maps g(αβ) : Uαβ —–> G such

    that, for each x ∈ Uαβγ = Uα ∩ Uβ ∩ Uγ

    g(αα)(x) = eG

    g(αβ)(x) = [g(βα)(x)]-1

    g(αβ)(x) . g(βγ)(x) . g(γα)(x) = eG

    and

    (αβ)(x) = λ(g(αβ)(x)) ∈ Diff(F)

The maps g(αβ) : Uαβ —–> G, which depend on the trivialization, are said to form a cocycle with values in G. They are called the transition functions with values in G (or also shortly the transition functions). The preferred trivializations will be said to be compatible with the structure. Whenever dealing with fiber bundles with structure group the choice of a compatible trivialization will be implicitly assumed.

Fiber bundles with structure group provide the suitable framework to deal with bundles with a preferred group of transformations. To see this, let us begin by introducing the notion of structure bundle of a fiber bundle with structure group B = (B, M, π; F; x, G).

Let B = (B, M, π; F; x, G) be a bundle with a structure group; let us fix a trivialization {(Uα, t(α)}α∈I and denote by g(αβ) : Uαβ —–> G its transition functions. By using the canonical left action L : G —–> Diff(G) of G onto itself, let us define gˆ(αβ) : Uαβ —–> Diff(G) given by gˆ(αβ)(x) = L (g(αβ)(x)); they obviously satisfy the cocycle properties. Now by constructing a (unique modulo isomorphisms) principal bundle PB = P(B) having G as structure group and g(αβ) as transition functions acting on G by left translation Lg : G —> G.

The principal bundle P(B) = (P, M, p; G) constructed above is called the structure bundle of B = (B, M, π; F; λ, G).

Notice that there is no similar canonical way of associating a structure bundle to a geometric bundle B = (B, M, π; F), since in that case the structure group G is at least partially undetermined.

Each automorphism of P(B) naturally acts over B.

Let, in fact, {σ(α)}α∈I be a trivialization of PB together with its transition functions g(αβ) : Uαβ —–> G defined by σ(β) = σ(α) . g(αβ). Then any principal morphism Φ = (Φ, φ) over PB is locally represented by local maps ψ(α) : Uα —> G such that

Φ : [x, h]α ↦ [φ(α)(x), ψ(α)(x).h](α)

Since Φ is a global automorphism of PB for the above local expression, the following property holds true in Uαβ.

φ(α)(x) = φ(β)(x) ≡ x’

ψ(α)(x) = g(αβ)(x’) . ψ(β)(x) . g(βα)(x)

By using the family of maps {(φ(α), ψ(α))} one can thence define a family of global automorphisms of B. In fact, using the trivialization {(Uα, t(α)}α∈I, one can define local automorphisms of B given by

Φ(α)B : (x, y) ↦ (φ(α)(x), [λ(ψ(α)(x))](y))

These local maps glue together to give a global automorphism ΦB of the bundle B, due to the fact that g(αβ) are also transition functions of B with respect to its trivialization {(Uα, t(α)}α∈I.

In this way B is endowed with a preferred group of transformations, namely the group Aut(PB) of automorphisms of the structure bundle PB, represented on B by means of the canonical action. These transformations are called (generalized) gauge transformations. Vertical gauge transformations, i.e. gauge transformations projecting over the identity, are also called pure gauge transformations.

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Is General Theory of Relativity a Gauge Theory? Trajectories of Diffeomorphism.

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Historically the problem of observables in classical and quantum gravity is closely related to the so-called Einstein hole problem, i.e. to some of the consequences of general covariance in general relativity (GTR).

The central question is the physical meaning of the points of the event manifold underlying GTR. In contrast to pure mathematics this is a non-trivial point in physics. While in pure differential geometry one simply decrees the existence of, for example, a (pseudo-) Riemannian manifold with a differentiable structure (i.e., an appropriate cover with coordinate patches) plus a (pseudo-) Riemannian metric, g, the relation to physics is not simply one-one. In popular textbooks about GTR, it is frequently stated that all diffeomorphic (space-time) manifolds, M are physically indistinguishable. Put differently:

S − T = Riem/Diff —– (1)

This becomes particularly virulent in the Einstein hole problem. i.e., assuming that we have a region of space-time, free of matter, we can apply a local diffeomorphism which only acts within this hole, letting the exterior invariant. We get thus in general two different metric tensors

g(x) , g′(x) := Φ ◦ g(x) —– (2)

in the hole while certain inital conditions lying outside of the hole are unchanged, thus yielding two different solutions of the Einstein field equations.

Many physicists consider this to be a violation of determinism (which it is not!) and hence argue that the class of observable quantities have to be drastically reduced in (quantum) gravity theory. They follow the line of reasoning developed by Dirac in the context of gauge theory, thus implying that GTR is essentially also a gauge theory. This then winds up to the conclusion:

Dirac observables in quantum gravity are quantities which are diffeomorphism invariant with the diffeomorphism group, Diff acting from M to M, i.e.

Φ : M → M —– (3)

One should note that with respect to physical observations there is no violation of determinism. An observer can never really observe two different metric fields on one and the same space-time manifold. This can only happen on the mathematical paper. He will use a fixed measurement protocol, using rods and clocks in e.g. a local inertial frame where special relativity locally applies and then extend the results to general coordinate frames.

We get a certain orbit under Diff if we start from a particular manifold M with a metric tensor g and take the orbit

{M, Φ ◦g} —– (4)

In general we have additional fields and matter distributions on M which are transformd accordingly.

Note that not even scalars are invariant in general in the above sense, i.e., not even the Ricci scalar is observable in the Dirac sense:

R(x) ≠ Φ ◦ R(x) —– (5)

in the generic case. Thus, this would imply that the class of admissible observables can be pretty small (even empty!). Furthermore, it follows that points of M are not a priori distinguishable. On the other hand, many consider the Ricci scalar at a point to be an observable quantity.

This winds up to the question whether GTR is a true gauge theory or perhaps only apparently so at a first glance, while on a more fundamental level it is something different. In the words of Kuchar (What is observable..),

Quantities non-invariant under the full diffeomorphism group are observable in gravity.

The reason for these apparently diverging opinions stems from the role reference systems are assumed to play in GTR with some arguing that the gauge property of general coordinate invariance is only of a formal nature.

In the hole argument it is for example argued that it is important to add some particle trajectories which cross each other, thus generating concrete events on M. As these point events transform accordingly under a diffeomorphism, the distance between the corresponding coordinates x, y equals the distance between the transformed points Φ(x), Φ(y), thus being a Dirac observable. On the other hand, the coordinates x or y are not observable.

One should note that this observation is somewhat tautological in the realm of Riemannian geometry as the metric is an absolute quantity, put differently (and somewhat sloppily), ds2 is invariant under passive and by the same token active coordinate transformation (diffeomorphisms) because, while conceptually different, the transformation properties under the latter operations are defined as in the passive case. In the case of GTR this absolute quantity enters via the equivalence principle i.e., distances are measured for example in a local inertial frame (LIF) where special relativity holds and are then generalized to arbitrary coordinate systems.

Odd symplectic + Odd Poisson Geometry as a Generalization of Symplectic (Poisson) Geometry to the Supercase

A symplectic structure on a manifold M is defined by a non-degenerate closed two-form ω. In a vicinity of an arbitrary point one can consider coordinates (x1, . . . , x2n) such that ω = ∑ni=1 dxidxi+n. Such coordinates are called Darboux coordinates. To a symplectic structure corresponds a non-degenerate Poisson structure { , }. In Darboux coordinates {xi,xj} = 0 if |i−j| ≠ n and {xi,xi+n} = −{xi+n,xi} = 1. The condition of closedness of the two-form ω corresponds to the Jacobi identity {f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0

for the Poisson bracket. If a symplectic or Poisson structure is given, then every function f defines a vector field (the Hamiltonian vector field) Df such that Dfg = {f,g} = −ω(Df,Dg).

A Poisson structure can be defined independently of a symplectic structure. In general it can be degenerate, i.e., there exist non-constant functions f such that Df = 0. In the case when a Poisson structure is non-degenerate (corresponds to a symplectic structure), the map from T∗M to T M defined by the relation f → Df is an isomorphism.

One can straightforwardly generalize these constructions to the supercase and consider symplectic and Poisson structures (even or odd) on supermanifolds. An even (odd) symplectic structure on a supermanifold is defined by an even (odd) non-degenerate closed two-form. In the same way as the existence of a symplectic structure on an ordinary manifold implies that the manifold is even-dimensional (by the non-degeneracy condition for the form ω), the existence of an even or odd symplectic structure on a supermanifold implies that the dimension of the supermanifold is equal either to (2p.q) for an even structure or to (m.m) for an odd structure. Darboux coordinates exist in both cases. For an even structure, the two-form in Darboux coordinates

zA = (x1,…, x2p1,…, θq) has the form ∑i=1p dxi dxp+i + ∑a=1q εaaa,

where εa = ±1. For an odd structure, the two-form in Darboux coordinates zA = (x1,…,xm1,…,θm) has the form ∑i=1m dxii.

The non-degenerate odd Poisson bracket corresponding to an odd symplectic structure has the following appearance in Darboux coordinates: {xi, xj} = 0, {θij} = 0 for all i,j and {xij} = −{θj,xi} = δji. Thus for arbitrary two functions f, g

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where we denote by p(f) the parity of a function f (p(xi) = 0, p(θj) = 1). Similarly one can write down the formulae for the non-degenerate even Poisson structure corresponding to an even symplectic structure.

A Poisson structure (odd or even) can be defined on a supermanifold independently of a symplectic structure as a bilinear operation on functions (bracket) satisfying the following relations taken as axioms:

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where ε is the parity of the bracket (ε = 0 for an even Poisson structure and ε = 1 for an odd one). The correspondence between functions and Hamiltonian vector fields is defined in the same way as on ordinary manifolds: Dfg = {f, g}. Notice a possible parity shift: p(Df) = p(f) + ε. Every Hamiltonian vector field Df defines an infinitesimal transformation preserving the Poisson structure (and the corresponding symplectic structure in the case of a non-degenerate Poisson bracket).

Notice that even or odd Poisson structures on an arbitrary supermanifold can be obtained as “derived” brackets from the canonical symplectic structure on the cotangent bundle, in the following way.

Let M be a supermanifold and T∗M be its cotangent bundle. By changing parity of coordinates in the fibres of T∗M we arrive at the supermanifold ΠT ∗M. If zA are arbitrary coordinates on the supermanifold M, then we denote by (zA,pB) the corresponding coordinates on the supermanifold T∗M and by (zA,z∗B) the corresponding coordinates on ΠT∗M: p(zA) = p(pA) = p(z∗A) + 1. If (zA) are another coordinates on M, zA = zA(z′), then the coordinates z∗A transform in the same way as the coordinates pA (and as the partial derivatives ∂/∂zA):

pA = ∂zB(z′)/∂zA pB and z∗A = ∂zB(z′)/∂zA z∗B

One can consider the canonical non-degenerate even Poisson structure { , }0 (the canonical even symplectic structure) on T∗M defined by the relations {zA,zB}0 = {pC,pD}0 = 0, {zA,pB}0 = δBA, and, respectively, the canonical non-degenerate odd Poisson structure { , }1 (the canonical odd symplectic structure) on ΠT∗M defined by the relations {zA,zB}0 = {z∗C,z∗D}0 = 0, {zA,z∗B}0 = δAB.

Now consider Hamiltonians on T∗M or on ΠT∗M that are quadratic in coordinates of the fibres. An arbitrary odd quadratic Hamiltonian on T∗M (an arbitrary even quadratic Hamiltonian on ΠT∗M):

S(z,p) = SABpApB (p(S) = 1) or S(z,z∗) = SABz∗Az∗B (p(S) = 0) —– (1)

satisfying the condition that the canonical Poisson bracket of this Hamiltonian with itself vanishes:

{S,S}0 = 0 or {S,S}1 = 0 —– (2)

defines an odd Poisson structure (an even Poisson structure) on M by the formula

{f,g}Sε+1 = {f,{S,g}ε}ε —–(3)

The Hamiltonian S which generates an odd (even) Poisson structure on M via the canonical even (odd) Poisson structure on T∗M (ΠT∗M) can be called the master Hamiltonian. The bracket is a “derived bracket”. The Jacobi identity for it is equivalent to the vanishing of the canonical Poisson bracket for the master Hamiltonian. One can see that an arbitrary Poisson structure on a supermanifold can be obtained as a derived bracket.

What happens if we change the parity of the master Hamiltonian in (3)? The answer is the following. If S is an even quadratic Hamiltonian on T∗M (an odd quadratic Hamiltonian on ΠT∗M), then the condition of vanishing of the canonical even Poisson bracket { , }0 (the canonical odd Poisson bracket { , }1) becomes empty (it is obeyed automatically) and the relation (3) defines an even Riemannian metric (an odd Riemannian metric) on M.

Formally, odd symplectic (and odd Poisson) geometry is a generalization of symplectic (Poisson) geometry to the supercase. However, there are unexpected analogies between the constructions in odd symplectic geometry and in Riemannian geometry. The construction of derived brackets could explain close relations between odd Poisson structures in supermathematics and the Riemannian geometry.

Quantum Geometrodynamics and Emergence of Time in Quantum Gravity

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It is clear that, like quantum geometrodynamics, the functional integral approach makes fundamental use of a manifold. This means not just that it uses mathematical continua, such as the real numbers (to represent the values of coordinates, or physical quantities); it also postulates a 4-dimensional manifold M as an ‘arena for physical events’. However, its treatment of this manifold is very different from the treatment of spacetime in general relativity in so far as it has a Euclidean, not Lorentzian metric (which, apart from anything else, makes the use of the word ‘event’ distinctly problematic). Also, we may wish to make a summation over different such manifolds, it is in general necessary to consider complex metrics in the functional integral (so that the ‘distance squared’ between two spacetime points can be a complex number), whereas classical general relativity uses only real metrics.

Thus one might think that the manifold (or manifolds!) does not (do not) deserve the name ‘spacetime’. But what is in a name?! Let us in any case now ask how spacetime as understood in present-day physics could emerge from the above use of Riemannian manifolds M, perhaps taken together with other theoretical structures.

In particular: if we choose to specify the boundary conditions using the no-boundary proposal, this means that we take only those saddle-points of the action as contributors (to the semi-classical approximation of the wave function) that correspond to solutions of the Einstein field equations on a compact manifold M with a single boundary Σ and that induce the given values h and φ0 on Σ.

In this way, the question of whether the wave function defined by the functional integral is well approximated by this semi-classical approximation (and thus whether it predicts classical spacetime) turns out to be a question of choosing a contour of integration C in the space of complex spacetime metrics. For the approximation to be valid, we must be able to distort the contour C into a steepest-descents contour that passes through one or more of these stationary points and elsewhere follows a contour along which |e−I| decreases as rapidly as possible away from these stationary points. The wave function is then given by:

Ψ[h, φ0, Σ] ≈ ∑p e−Ip/ ̄h

where Ip are the stationary points of the action through which the contour passes, corresponding to classical solutions of the field equations satisfying the given boundary conditions. Although in general the integral defining the wave function will have many saddle-points, typically there is only a small number of saddle-points making the dominant contribution to the path integral.

For generic boundary conditions, no real Euclidean solutions to the classical Einstein field equations exist. Instead we have complex classical solutions, with a complex action. This accords with the account of the emergence of time via the semiclassical limit in quantum geometrodynamics.

On the Emergence of Time in Quantum Gravity