Witten’s Problems, Set Two — N^{o}. 1 (solution written by P. Deligne)

We will write ℝ^{d+1} for ℝ^{n}, coordinates t = x_{0}, x_{1}, …, x_{d}.

(a) In the Minkowski signature, ∗ is defined as follows:

1. Extend the given bilinear form on V to ∧^{p} V, by

(v_{1} ∧ … ∧ v_{p}, w_{1} ∧ … ∧ w_{p}) = det(v_{i}, w_{j}).

Same for V^{∗}, identified with V.

2. An orientation has to be chosen. Together with the metric, it defines a form ω of maximal degree: in standard coordinates, dt∧dx_{1}∧ … ∧dx_{d}.

3. The operation ∗ is defined by α∧ ∗ β = (α, β).ω.

The lagrangian density ℒ is

−\frac{1}{4e^{2}} (F, F).(standard density dv)

In this way of writing, no orientation is needed. However, in the computations which follow, we choose an orientation to identify densities and forms of maximal degree. The variation of the lagrangian density by A ↦ A + δA is

− \frac{1}{2e^{2}} (dδA, dA)dv = −\frac{1}{2e^{2}} dδA ∧ ∗ dA
=d (−\frac{1}{2e^{2}} δA ∧ ∗ dA) − \frac{1}{2e^{2}} δA ∧ d ∗ dA;

the variational derivative δℒ is −1/2e^{2} δA∧d ∗ dA, giving as Euler-Lagrange equation for the extremals (a.k.a. equation of motion)

d ∗ dA = 0,

i.e., for d^{∗} the adjoint of d, d^{∗}dA = 0.

(b) The general rule for defining ω_{0} is as follows:

1. Pick a space like hypersurface Γ (e.g. t = 0).

2. Let a be the d-form whose exterior derivative is used to justify the integration by part leading to Euler-Lagrange equations. Here,

a = −\frac{1}{2e^{2}} δA ∧ ∗ dA.

it is a 1-form on the space of fields, with values in d-forms, and δA ↦ a(δA) is linear over C^{∞}-functions.

3. Define a 1-form α_{Γ} on the space of fields by integrating a on Γ.

4. Restrict α_{Γ} to the space of extremals, and define

ω_{0} = dα_{Γ} .

For this to make sense, one needs some control at spatial infinity. it is not unreasonable to consider only A which vanish at spatial infinity: we will see in (c) that if at t = 0, A and Ȧ vanish for (∑ x_{i}^{2})^{1/2} > R, and if A obeys the equations of motion d ∗ dA = 0, then some gauge transform of A vanishes for (∑ x_{i}^{2})^{1/2} > R + |t|.

On the space of extremals vanishing at spatial infinity, the 1-form α_{Γ} makes sense. Its exterior derivative is

ω_{0}(δ_{1}A, δ_{2}A) = ∫_{Γ} −\frac{1}{2e^{2}} (δ_{2}A ∧ ∗ dδ_{1}A − δ_{1}A ∧ ∗ dδ_{2}A).

The space of extremals is an affine space, and ω_{0} is translation invariant. The general lagrangian formalism ensures that this 2-form is independent of Γ.

(c) Gauge transformations A ↦ A + dϕ preserve the lagrangian density, hence the space of extremals. Of course, if one imposes to A to vanish at spatial infinity, one has to impose to ϕ to be locally constant at infinity (hence constant at spatial infinity if d + 1 ≠ 2). Suppose d + 1 ≠ 2: one can impose to ϕ to be zero at spatial infinity, by subtracting a constant. The formula for ω_{0} makes it clear that if A ↦ A + δ_{1}A is an infinitesimal gauge transformation with support disjoint from Γ, then ω_{0}(δ_{1}A, δ_{2}A) = 0. As Γ can be chosen as it pleases, it follows that ω_{0} comes from the quotient by gauge transformations: decompose an infinitesimal gauge transformation δA_{1} in gauge transformations with support in t < 1, −1 < t < 1 and −1 < t, and for each evaluate ω_{0}(δ_{1}A, δ_{2}A) using a Γ for which it obviously vanishes.

Direct proof: if B is a tangent vector to the space of extremals: d ∗ dB = 0 and B vanishing at spatial infinity, then

∫_{Γ}(dϕ) ∧ ∗ dB + B ∧ ∗ d(dϕ) = ∫_{Γ} d(ϕ. ∗ dB) = 0 .

To investigate the quotient of the space of extremals by the group of gauge transformation, we now fix a decomposition of space-time as space × time. Coordinates x_{0} = t, x_{i}. Using ϕ = ∫_{0}^{t} A_{0}(x, τ)dτ (vanishing at spatial infinity if A does), one gauge transforms A into a new A with A_{0} = 0. The equation d ∗ dA becomes

∂_{0} (∑ ∂_{i}A_{i}) = 0 (1) □ A = −d ∑ ∂_{i}A_{i} (2)

if one gives A at t = 0, (1) gives f := ∑ ∂_{i}A_{i} at all times: it is independent of t. If Ȧ is given at t = 0, one can then solve □ A = −df with the given Cauchy data (A, Ȧ) at t = 0 (hyperbolicity of □). The support of the resulting A is contained in what can be reached by a time-like vector from the support of (A, Ȧ) at t = 0.

We have

∑ ∂_{i}A_{i} = f at t = 0, f independent of t
□ A_{i} = −∂_{i}f

It follows that

□ (∑ ∂_{i}A_{i} − f) = ∑ ∂_{i}□A_{i} − □f = ∑ −∂_{i}∂_{i}f − □f = Df − Df = 0

and if \frac{d}{dt} (∑ ∂_{i}A_{i} − f) = 0 at t = 0, i.e. if (1) holds at t = 0, then ∑ ∂_{i}A_{i} − f = 0 always, i.e. (1) holds.

It follows that we have a bijection

A ⟼ (A, Ȧ) at t = 0

from the space of extremals A (vanishing at spatial infinity), with A_{0} = 0, to the space of Cauchy data (A, Ȧ) at t = 0, vanishing at infinity with A_{0} = 0, Ȧ_{0} = 0, ∑ ∂_{i}Ȧ_{i} = 0. The last condition is what keeps making sense, on the Cauchy data, of the equation of motion d ∗ dA = 0.

Transported to the affine space of Cauchy data, the 2-form ω_{0} becomes the translation invariant 2-form

(3)  (A, Ȧ), (B, Ḃ) ⟼ ∫ (A, Ḃ) − (Ȧ, B)

(up to a constant). It is degenerate, as we still have to divide by the time independent gauge transformations ϕ: we have vanishing for (A, Ȧ) = (dϕ, 0). Conversely, if (A, Ȧ) is in the kernel, we have ∫ (Ȧ, B) = 0 for any compactly supported B, hence Ȧ = 0, and ∫ (A, Ḃ) = 0 if d^{∗}Ḃ = 0. One can take Ḃ = d^{∗}C and integration by part gives ∫ (dA, C) = 0, hence dA = 0 and A = dϕ for some ϕ.

This can be presented as follows: one the space of all (A, Ȧ) at t = 0 (d^{∗}Ȧ = 0 not required), we have the symplectic structure (3). We took the symplectic quotient L^{⊥}/L by the isotropic linear subspace L of the (dϕ, 0).

Geometrically, the hyperbolicity modulo gauge of the equation of motion d∗dA = 0 has the following meaning.

One thinks of the 1-form A as being a connection ∇ on the trivial ℝ-torsor (= principal ℝ-bundle) T. For electromagnetism, it might be better to use rather ℝ/ℤ = U^{1}, but if one wants a change of trivializations to be given by a function ϕ with values in ℝ, not ℝ/ℤ, one needs ℝ. The gauge transform A + dϕ of A is the 1-form describing the same connection ∇, in the trivialization ϕ of the torsor T.

Call “space” the hypersurface t = 0 and let U be a region of space. Suppose given a 1-jet P_{1} of ℝ-torsor along U. If s_{1} is a trivialization of P_{1}, any other trivialization can be written as s_{1} + ϕ_{1}, for ϕ_{1} a 1-jet of function along U: ϕ_{1} = (ϕ, ϕ̇) on U. Suppose a connection ∇_{1} is given on P_{1}. Given a trivialization of P_{1}, it is the data of A_{0}, A_{i}, A·_{i} (i ≠ 0) in U (hence at t = 0). Suppose lastly that what makes sense of the equation of motion holds. In coordinates:

∑_{i≠0} ∂_{i}^{2}A_{0} − ∂_{i}Ȧ_{i} = 0

This is the “Cauchy data”.

Let V be the region of space time which, in terms of causality

for t > 0: is influence only by U;

for t < 0: influences only U. Otherwise said: v is in V if all time like trajectories through v hit t = 0 in U. For U the ball (∑ x_{i}^{2})^{1/2} < R, of space, i.e. at t = 0, V is the region (∑ x_{i}^{2})^{1/2} < R − |t|.

Hyperbolicity gives that the Cauchy data (P_{1}, ∇_{1}) extends uniquely to a torsor with connection obeying the equation of motion on V: an extension exists, any two extensions are isomorphic by an isomorphism whose first jet along U is the identity, and such an isomorphism is unique.

Example. If U is the region ∑ x_{i}^{2} > R, then V is the region ∑ x_{i}^{2} > R + |t|. If (P, ∇) obeys the equation of motion and has a 1-jet along U isomorphic ot the trivial flat torsor, it follows that (P, ∇) is isomorphic to the trivial flat torsor on V.

In the preceding stories, one can replace “space” t = 0 by a space like hypersurface. Proofs are similar. One begins by mapping V to U by sending v = (t, x) to the point of U with the same x-coordinates, and by using the connection along the time-lines to write the torsors with connection on V as pull back, as torsors, of a torsor on U (“gauge transformation making A_{0} = 0”).

In this geometric language, the restriction to A which vanish at spatial infinity, means considering only (P, ∇) such that P admits a trivialization which is horizontal at spatial infinity: flatness at spatial infinity, and, for d = 2, trivial monodromy at spatial infinity and vanishing of the integral of the curvature on a space like hypersurface, hence on all of them.