Vector Fields Tangent to the Surfaces of Foliation. Note Quote.

Although we are interested in gauge field theories, we will use mainly the language of mechanics that is, of a finite number of degrees of freedom, which is sufficient for our purposes. A quick switch to the field theory language can be achieved by using DeWitt’s condensed notation. Consider, as our starting point a time-independent first- order Lagrangian L(q, q ̇) defined in configuration-velocity space TQ, that is, the tangent bundle of some configuration manifold Q that we assume to be of dimension n. Gauge theories rely on singular as opposed to regular Lagrangians, that is, Lagrangians whose Hessian matrix with respect to the velocities (where q stands, in a free index notation, for local coordinates in Q),

Wij ≡ ∂²L/∂q^.i∂q^.j —– (1)

is not invertible. Two main consequences are drawn from this non-invertibility. First notice that the Euler-Lagrange equations of motion [L]_i = 0, with

[L]_i : = α_i − W_ijq ̈^j

and

α_i := ∂²L/∂q^.i∂q^.j q^.j

cannot be written in a normal form, that is, isolating on one side the accelerations q ̈ = f (q, q ̇). This makes the usual theorems about the existence and uniqueness of solutions of ordinary differential equations inapplicable. Consequently, there may be points in the tangent bundle where there are no solutions passing through the point, and others where there is more than one solution.

The second consequence of the Hessian matrix being singular concerns the construction of the canonical formalism. The Legendre map from the tangent bundle TQ to the cotangent bundle —or phase space— T ∗Q (we use the notation pˆ(q, q ̇) := ∂L/∂q ̇),

FL : TQ → T ∗ Q —– (2)

(q, q ̇) → (q, p=pˆ) —– (3)

is no longer invertible because ∂pˆ/∂q ̇ = ∂L/∂q ̇∂q ̇ is the Hessian matrix. There appears then an issue about the projectability of structures from the tangent bundle to phase space: there will be functions defined on TQ that cannot be translated (projected) to functions on phase space. This feature of the formalisms propagates in a corresponding way to the tensor structures, forms, vector fields, etc.

In order to better identify the problem and to obtain the conditions of projectability, we must be more specific. We will make a single assumption, which is that the rank of the Hessian matrix is constant everywhere. If this condition is not satisfied throughout the whole tangent bundle, we will restrict our considerations to a region of it, with the same dimensionality, where this condition holds. So we are assuming that the rank of the Legendre map FL is constant throughout T Q and equal to, say, 2n − k. The image of FL will be locally defined by the vanishing of k independent functions, φ_μ(q, p), μ = 1, 2, .., k. These functions are the primary constraints, and their pullback FL ∗ φ_μ to the tangent bundle is identically zero:

(FL ∗ φ_μ)(q, q ̇) := φ_μ(q, pˆ) = 0, ∀ q, q ̇—– (4)

The primary constraints form a generating set of the ideal of functions that vanish on the image of the Legendre map. With their help it is easy to obtain a basis of null vectors for the Hessian matrix. Indeed, applying ∂/∂q^. to (4) we get

W_ij = (∂φ_μ/∂p_j)|_p=pˆ = 0, ∀ q, q ̇ —– (5)

With this result in hand, let us consider some geometrical aspects of the Legendre map. We already know that its image in T∗Q is given by the primary constraints’ surface. A foliation in TQ is also defined, with each element given as the inverse image of a point in the primary constraints’ surface in T∗Q. One can easily prove that the vector fields tangent to the surfaces of the foliation are generated by

Γμ= (∂φ_μ/∂p_j)|_p=pˆ = ∂/∂q^.j —– (6)

The proof goes as follows. Consider two neighboring points in TQ belonging to the same sheet, (q, q ̇) and (q, q ̇ + δq ̇) (the configuration coordinates q must be the same because they are preserved by the Legendre map). Then, using the definition of the Legendre map, we must have pˆ(q, q ̇) = pˆ(q, q ̇ + δq ̇), which implies, expanding to first order,

∂pˆ/ ∂q ̇ δ q ̇ = 0

which identifies δq ̇ as a null vector of the Hessian matrix (here expressed as ∂pˆ/∂q ̇). Since we already know a basis for such null vectors, (∂φ_μ /∂p_j)|p=pˆ, μ = 1, 2, …, k, it follows that the vector fields Γ_μ form a basis for the vector fields tangent to the foliation.

The knowledge of these vector fields is instrumental for addressing the issue of the projectability of structures. Consider a real-valued function f^L: TQ → R. It will — locally— define a function f^H: T∗Q −→ R iff it is constant on the sheets of the foliation, that is, when

Γ_μf^L = 0, μ = 1,2,…,k. (7)

Equation (7) is the projectability condition we were looking for. We express it in the following way:

Γ_μf^L = 0, μ = 1,2,…,k ⇔ there exists f^H such that FL ∗ f^H = f^L

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