Quantization

P. Deligne

School of Mathematics Institute for Advanced Study Princeton, NJ 08540 e-mail: deligne@math.ias.edu

The present notes do not pretend to any originality. We have tried to present different aspects of what quantization can mean, including the case of odd variables. Complex polarizations are not considered.

1. Let (M, ω) be a symplectic manifold of dimension 2d. We will use vague words “big” and “small”. For this, we need some notion of size ∼ 1 on M. For instance M could carry a Riemannian metric, with curvature of size at most ∼ 1 and injectivity radius of size at least ∼ 1. The symplectic form should be of size ∼ 1, and ℏ is a small number. The 2-form which really matters is ω/ℏ.

Our sign convention for the Poisson bracket corresponding to ω is the following: the Hamiltonian vector field X(f) defined by a function f is given by X(f)g = {f, g}, and df = −i_{X(f)}ω. If M = ℝ^{2} with coordinates p, q and if ω = dp∧dq, then {p, q} = 1.

A quantization of M consists in a complex Hilbert space ℋ and in a rule to attach to functions f on M operators f^{∧} acting on ℋ. The rule f ↦ f^{∧} should be ℂ-linear, for f real, f^{∧} should be hermitian and, in a sense I will not try to make precise, f ↦ f^{∧} should almost be an homomorphism: 1^{∧} should be the identity and for slowly varying functions f and g,

(1.1)  (fg)^{∧} ∼ f^{∧}g^{∧}  mod O(ℏ).

If (1.1) did hold exactly, the f^{∧} would commute the one with the other. The symplectic structure controls the failure of commutativity

(1.2)  [ f^{∧}, g^{∧}] ∼ −iℏ{f, g}^{∧}  mod O(ℏ^{2}).

Formula (1.2) can be rewritten

(1.2)′  (X(f)g)^{∧} ∼ i/ℏ[ f^{∧}, g^{∧}]  mod O(ℏ).

The symplectic vector field X(f), if it is well behaved at infinity, exponentiates to an automorphism exp(X(f)) of (M, ω), with Taylor’s formula reading

exp(X(f))^{∗}g = exp(derivation X(f))(g).

Formula (1.2)′ should hold in its integrated form

(1.3)  (exp(X(f))^{∗}g)^{∧} ∼ exp(i/ℏf^{∧})g^{∧} exp(−i/ℏf^{∧})  mod O(ℏ).

We don’t care about the exact rule f ↦ f^{∧}. It matters mainly mod O(ℏ).

In the deformation quantization point of view, (1.1) is the first term in an expansion (∗-product)

(1.1)′  f^{∧}g^{∧} = (fg)^{∧} + c_{1}(f, g)^{∧}ℏ + c_{2}(f, g)^{∧}ℏ^{2} + ⋯,

with c_{i}(f, g) given by a ℂ-bilinear differential operator in f and g. The formula (1.2) becomes c_{1}(f, g)−c_{1}(g, f) = −i{f, g}. If one disposes of quantizations ℋ(ℏ),with ℏ tending to 0, (1, 1)′ can be an asymptotic expansion in ℏ. if one redefines f ↦ f^{∧} to be f^{∧} + d_{1}(f)^{∧}ℏ + ⋯ with d_{i}(f) a linear differential operator in f, (1.1)′ changes to an equivalent ∗-product.

Assumption (1.1) means that ℋ almost localizes on M: an element h of ℋ can be decomposed as a sum ∑ h_{i} of elements localized in small regions U_{i} of M: if x ∈ U_{i}, f^{∧}h_{i} is about f(x).h_{i}. Because of the non commutativity (1.2), localization does not make sense for regions smaller than expressed by the uncertainty principle ΔpΔq ∼ 2πℏ.

If we were only to assume (1.1) and (1.2), a multiple ℋ^{n} of a quantization ℋ of M would again be a quantization. A “finite multiplicity” assumption is that if f is a C^{∞} function with compact support on M, the operator f^{∧} is of trace class. If this holds, I would expect that for some integer m (rather: one for each connected component of M), one has

Tr(f^{∧}) ∼ m ∫ f(x)dx

where, in Darboux local coordinates, dx is the product of the dp_{i} dq_{i}/2πℏ (Liouville measure). We want m = 1:

(1.4)  Tr(f^{∧}) ∼ ∫ f(x)dx

2. Example: cotangent bundles.

The basic example of almost localization, in the sense meant above, is when M is a cotangent bundle T^{∗}V and ℋ the space of half densities on V. If f is peaked at x, say in a gaussian fashion, and if g is a real function, with dg slowly varying, then fe^{ig/ℏ} is localized around (x, dg_{x}) in M. If f is slowly varying, then f.e^{ig/ℏ} is localized around the section x ↦ dg_{x} of M → V, above the support of f.

The operator fˆ has a natural definition when f is affine linear on the fibers of T^{∗}V over V. If f is the pull back of a function on V, fˆ is multiplication by f. If f is linear on each cotangent space, hence identified with a vector field F on V, fˆ = −iℏℒ_{F}. For those f, (1.2) holds exactly. To define fˆ for more complicated functions requires auxiliary choices.

On T^{∗}V, we have a canonical 1-form α: in local coordinates q^{i} on V, giving local coordinates (p_{i}, q^{i}) on T^{∗}V, α is ∑ p_{i}dq^{i}. The trivial unitary line bundle ℒ, with the connection −iα/ℏ: ∇f = df − iα/ℏf, is a prequantization line bundle: its curvature is −iω/ℏ, for ω the symplectic form dα. A real function g on V defines the lagrangian section L(g): x ↦ (dg)_{x} of T^{∗}V. The pull back to L(g) of e^{ig/ℏ} is an horizontal section of ℒ.

For any symplectic manifold M, let Λ_{M} or simply Λ be the fiber space over M whose fiber Λ_{m} at m ∈ M is the space of lagrangian (= maximal isotropic) subspace of the tangent space T_{m} of M at m. To take into account the Maslov index story, it is best to take prequantization line bundles as living on Λ. They should verify (a) (b) below.

(a) The curvature 2-form R is the pull back from M of −iω/ℏ:

∇^{2} = −iω/ℏ.

In particular, on each fiber Λ_{m} of Λ → M, (ℒ, ∇) is flat. One has π_{1}(Λ_{m}) = ℤ and

(b) the monodromy of (ℒ∇) on Λ_{m} is multiplication by i. Remarks: (i) In the same way that an hamiltonian vector field X(f) acts on a prequantization line bundle, with the action depending on f, and not only on X(f), it also acts on (ℒ, ∇) as above: X(f) is symplectic, hence lifts to Λ, the question is local on Λ, and locally on Λ (ℒ, ∇) is the inverse image of a prequantization line bundle on M. If we exponentiate, we see that exp(X(f)) acts by an automorphism of (M, ω, ℒ, ∇). The infinitesimal action on sections of ℒ is given by

(2.1)  ∂_{X(f)} = ∇_{X(f)} − if/ℏ

(ii) To make sense of (b), we should make precise our sign convention identifying π_{1}(Λ_{m}) with ℤ. For the symplectic vector space ℝ^{2}, form dp∧dq, the generator is given by the path

θ ⟼ line spanned by (q, p) = (cos θ, sin θ) (0 ≤ θ ≤ π).

(iii) Let V be a symplectic vector space, and Λ_{0} be the space of lagrangian linear subspaces of V. If S_{1} and S_{2} in Λ_{0} intersect transversally, one defines as follows a preferred homotopy class of path from S_{1} to S_{2}. Identify S_{2} to S1^{∨} by s_{2} ↦ ω(s_{2}, s_{1}). Choose a basis e_{i} of S_{1} and let e′_{1} be the dual basis of S_{2}. The path is

θ ⟼ subspace spanned by the cos θe_{i} + sin θe′_{i} (0 ≤ θ ≤ π/2).

It depends only on the quadratic form for which the basis (e_{i}) is orthonormal. The choice of this quadratic form running over a contractible set, the homotopy class of the path does not depend on the choice. If V is of dimension 2d, the preferred path from S_{1} to S_{2}, followed by the preferred path from S_{2} to S_{1}, is d times the preferred generator of π_{1}(Λ_{0}).

We now come back to the case where M is a cotangent bundle T^{∗}V. Let Λ^{0} ⊂ Λ be the subbundle, with Λ_{m}^{0} ⊂ Λ_{m} consisting of those lagrangian subspaces L of T_{m} intersecting transversally the vertical subspace of T_{m}^{∗} (tangent to the fiber of T^{∗}V → V). If we pull back the prequantization line bundle on T^{∗}V to Λ^{0}, it extends uniquely to ℒ_{Λ} on Λ satisfying the conditions (a) (b).

Let L be a lagrangian subvariety of M. Let s be the section of Λ over L: x ↦ tangent space of L at x. The section s is with values in Λ^{0} if and only if L is locally of the form L(g). As L is lagrangian, s^{∗}ℒ_{Λ} is flat and it makes sense ot speak of a slowly varying section of s^{∗}ℒ on L. Generalizing the f.e^{ig/ℏ} considered previously, to a slowly varying half density u on L, with values in s^{∗}ℒ, corresponds [u] in ℋ, with

(2.2)  ∥[u]∥^{2} ∼ ∫_{S} ⟨u, u⟩ .

It is localized near Supp(u) ⊂ S ⊂ M = T^{∗}V.

A function f defines an automorphism exp(X(f))of (M, ℒ, ∇) (Remark (ii) above), and an automorphism exp(−i/ℏfˆ ) of ℋ. Those automorphisms should (almost) preserve the construction u ↦ [u]:

(2.3)  [exp(X(f))^{∗}u] ∼ exp(i/ℏfˆ )[u].

Inner products are given as follows. Fix (S_{i}, u_{i}) (i = 1, 2) as above, and assume that S_{1} and S_{2} intersect transversally. Let x be an intersection point. The lagrangian subspaces s_{1}(x) and s_{2}(x) of T_{x}M intersect transversally, and the preferred class of path γ from s_{1}(x) to s_{2}(x) (Remark (iii) above) gives an isomorphism γ^{∗}: ℒ_{s_{2}(x)} → ℒ_{s_{1}(x)}. The symplectic structure puts s_{1}(x) and s_{2}(x) in duality, identifying the line of half densities of S_{1} at x with the dual of the similar line for S_{2}. The inner product ⟨u_{1}(x), γ^{∗}u_{2}(x)⟩ is hence just a number. The stationary phase gives

(2.4)  ⟨[u_{1}], [u_{2}]⟩ ∼ ∑_{x∈S_{1}∩S_{2}}(2πiℏ)^{d/2} ⟨u_{1}(x), γ^{∗}u_{2}(x)⟩

for M of dimension 2d.

Our convention for inner products ⟨a, b⟩ is: antilinear in a, linear in b.

3. The flat case is when M is an affine space, with a translation invariant symplectic form. In the flat case, one can define a quantization as the data of f ↦ fˆ , just for f a linear function, with (1.2) holding exactly. The purely imaginary linear functions form a Lie algebra, for −iℏ{f, g}, and we want a unitary representation f ↦ fˆ of this Lie aglebra, with iˆ = i.Id. It is more convenient to ask for a unitary representation of the corresponding Lie group. Formula (1.4) becomes a request for irreducibility. Weyl’s quantization f ↦ fˆ , for a general f, is given by

(3.1)  exp(ℓ)ˆ = exp(ℓˆ )

for ℓ a purely imaginary linear function, a generalfˆ being deduced from (3.1) by considering f as a superposition of exp(ℓ), by Fourier transform. For Weyl’s quantization, formula (1.4) holds exactly.

Understood in those terms, quantization in the flat case is unique, up to isomorphisms. If M is the cotangent bundle to an affine space V, 2. gives its Schrodinger model as L^{2}(V): if V_{0} is the vector space of translations of V, one has M = V0^{∨} × V; for f (the pull back of) a function on V, fˆ is multiplication by f. For ℓ a linear form on V0^{∨}, identified with a vector v ∈ V_{0}, ℓˆ is −iℏ∂_{v}, and exp(iℓ)ˆ is ψ(x) ↦ ψ(x + ℏv).

This unicity assertion depends on the assumption that the flat symplectic variety M is of finite dimension. Remark. In the flat case, the data on Λ_{M} of a prequantization line bundle defines a quantization ℋ, as defined above, up to unique isomorphism. This is made plausible by the fact that the group of automorphisms of

(affine symplectic space M, ℒ_{Λ}, ∇)

and of

(affine symplectic space M, ℋ, f ↦ fˆ )

are the same extension of the affine symplectic group (Sp ⋉ translations) by U^{1}.

The relation between ℒ_{Λ} and the quantization ℋ can be fixed as follows. Let ℋ_{∞} be the C^{∞}-vectors in ℋ, for the action of the Heisenberg group. In the Schroedinger model: the Schwartz functions on V. Let ℋ_{−∞} ⊃ ℋ be the dual of ℋ_{∞}. Then, a linear lagrangian subspace L of M, given with u on L of the form (translation invariant half density). (flat section s^{∗}ℒ_{Λ}), define [u] in ℋ_{−∞}, and the inner product formula (2.4) holds exactly, to give the inner product of continuous superpositions of [u]. Formula (2.3) holds exactly, for f quadratic and u as above. In particular, if a linear form ℓ vanishes on L, one has ℓˆ [u] = 0.

4. We now go over to the superworld. A super Hilbert space is a mod 2 graded complex vector space H, with an even sesquilinear form (u, v), antilinear in u and linear in v, for which

(v, u) = (−1)^{p(u)p(v)}(u, v)^{−},

with a positivity and a completeness condition. As positivity condition, we take

(4.1)

(u, u) > 0 for u even, u ≠ 0
−i(u, u) > 0 for u odd, u ≠ 0.

This positivity condition is stable by tensor product, if ( , ) for H′ ⊗ H′′ is defined by

(u′ ⊗ u′′, v′ ⊗ v′′) = (−1)^{p(u′′)p(v′)}(u′, v′)(u′′, v′′),

in accordance with the sign rule. The adjoint T^{†} of an operator T is defined by

(4.2)  (T u, v) = (−1)^{p(T)p(u)}(u, T^{†}v).

The physicist don’t like (4.1), and prefer to work with the ordinary mod 2 graded Hilbert space with the inner product ⟨ , ⟩ defined by

(4.3)

(u, v) = ⟨u, v⟩ for u, v even
(u, v) = i ⟨u, v⟩ for u, v odd.

The adjoint T^{†} is related to the corresponding ordinary adjoint T^{∗} by

(4.4)

T^{†} = T^{∗} for T even
T^{†} = iT^{∗} for T odd.

The definition (4.2) has the unsettling effect that the eigenvalues of an odd self adjoint operator are in i^{1/2}.ℝ. However, the eigen spaces being neither even nor odd, there is not much wish to consider them.

In a quantization of a supermanifold M, to functions f on M should correspond operators fˆ , acting on a super Hilbert space ℋ, with fˆ of the same parity as f. If f is real, one should have fˆ = f ^{ ˆ †}. Because of (3.4), if M is a supermanifold, the physicists will declare to be “real” the real even f, and the i^{−1/2}η for η real. With this rule, if α_{1} and α_{2} are odd and “real”, iα_{1}α_{2} is “real” (as well as real, being even). Example: on ℝ^{1,1}, with coordinates (x, η), if we put θ = i^{−1/2}η, the vector field ∂_{θ} + iθ∂_{x} is a multiple of a real vector field:

∂_{θ} + iθ∂_{x} = i^{1/2}(∂_{η} + η∂_{x}).

5. Let now M be a super affine space, with a translation invariant symplectic form. In imitation of 3., we define a quantization as the data of a super Hilbert space ℋ, and of ℓ ↦ ℓˆ attaching to a linear function on M an operator on ℋ, with ℓ and ℓˆ of the same parity. The quantization ℓ ↦ ℓˆ should be linear, real: ℓˆ = (ℓˆ )^{†} for ℓ real, map 1 to Id, and obey

(5.1)  [ℓˆ , mˆ ] = −iℏ{f, g}ˆ .

One should also require some irreducibility.

The condition (5.1) requires that for real linear odd functions ℓ, the Poisson bracket {ℓ, ℓ} (a constant) be negative:

(5.2)  {ℓ, ℓ} ≤ 0.

Indeed, [ℓˆ , ℓˆ ] = 2ℓˆ 2 = ℓ ^{ˆ †}ℓˆ must be −iℏ{ℓ, ℓ}. The sign in (5.2) is due to the choices of sign in (4.1) and (5.1), the latter repeating (1.2).

As in 3., one extends ℓ ↦ ℓˆ to polynomial functions by requiring that for a product of even or odd linear functions, one has

(ℓ_{1} … ℓ_{n})ˆ = 1/n! ∑_{σ} ± ℓ_{σ̂(1)} … ℓ_{σ̂(n)} (σ ∈ S_{n})

with ± given by the sign rule. In particular, ± is + if the ℓ_{i} are all even, and ε(σ) if they are all odd.

6. Let W be a real vector space with a positive defined symmetric bilinear form B. Let M be the affine space defined by ΠW: it is of dimension (0, dim W), and linear forms on W are odd functions on M. Define a Poisson bracket by

(6.1)  {ℓ, m} = −B^{−1}(ℓ, m)

for ℓ, m ∈ W^{∨}.

If e_{i} is an orthonormal basis of W, with dual basis e^{i}, and if e^{i} is veiwed as an odd function on M, the Poisson bracket (6.1) is given by the super symplectic form

(6.2)  ω = \frac{1}{2} ∑ de^{i}de^{i}.

One W^{∗}, let Q be the quadratic form \frac{1}{2}B^{−1}(ℓ, ℓ). If ℋ is a mod 2 graded module over the (mod 2 graded) Clifford algebra C(W^{∗}, Q), with the module structure written cℓ(ℓ) for ℓ ∈ W^{∗}, (5.1) holds for

(6.3)  ℓˆ = (iℏ)^{1/2}cℓ(ℓ).

For dim W = 1, one can take ℋ = (ℂ even) ⊕ (ℂ odd), with cℓ(e^{1}) = ( \begin{matrix}0 & 1/\sqrt{2} \\ 1/\sqrt{2} & 0 \end{matrix} ). For the obvious Hilbert space structure of ℋ, and the corresponding super Hilbert space structure, one has then cℓ(e^{1})^{∗} = cℓ(e^{1}) and for any ℓ in V^{∗},

(6.4)  ℓˆ = (ℓˆ )^{†}.

A general W can be written as an orthogonal direct sum of lines, and the corresponding tensor product of super Hilbert spaces is a Clifford module for which (6.4) holds, if ℓˆ is defined by (6.3). If ℋ_{1} is a graded submodule with the induced Hilbert space structure, (6.4) continues to hold for ℋ_{1}. This proves the existence of a super Hilbert space ℋ with a Clifford module structure, irreducible as a graded module, for which (6.4) holds when ℓˆ is defined by (6.3).

We now assume W of even dimension. They are the two isomorphy classes of ℋ as above, exchanged by the parity change ℋ ↦ ℋ ⊗ L, for L a (0, 1)-dimensional super Hilbert space. Let (e^{1}, …, e^{n}) be an orthonormal basis (for B^{−1}), and [e^{1}, …, e^{n}] be the corresponding density on M. Another orthonormal basis would give the same, or the opposite density. The choice of the density [e^{1}, …, e^{n}] picks out one of the two isomorphy classes of ℋ: the one for which the following analogue of (1.4) holds. For any f on M, the supertrace Tr(fˆ ) is given by

(6.5)  Tr(fˆ ) = ℏ^{n/2} ∫ [e^{1}, …, e_{n}]f

Justification: writing ℋ as a tensor product, one reduces to the case n = 2. For n = 2, cℓ(e^{1})cℓ(e^{2}) has square −1/4 and eigenvalues ±i/2. It follows that (e_{1}e_{2})ˆ = \frac{1}{2}(e_{ˆ1}e_{ˆ2} − e_{ˆ2}e_{ˆ1}) = e_{ˆ1}e_{ˆ2} = iℏcℓ(e_{1})cℓ(e_{2}) has eigenvalues ∓ℏ/2, hence supertrace ±ℏ.

7. A super symplectic flat space M as in 5. can be decomposed as M^{+} × M^{−}, with M^{+} even and M^{−} odd, as in 6. The tensor product of quantizations of M^{+} and M^{−} gives one for M. If M^{+} is the cotangent bundle of an affine space V, and if M^{−} is obtained as in 6. from an even dimensional quadratic vector space W, this tensor product can be realized as the space of L^{2}-density on V with values in a Clifford module for W^{∨}, and an analogue of 1.4, 6.5 holds.

8. Let V be a manifold and W be an orthogonal vector bundle on V, with a connection ∇ respecting the structural symmetric bilinear form B. We take M to be the fiber product over V of T^{∗}V and ΠW. Over any basis B, a section of M over V is the data of an even section p of T^{∗}V and of an odd section ψ of W. A section e of W^{∨} defines an odd function ψ ↦ e(ψ) on M. A local coordinate system (q^{i}) on V and a local basis (e_{α}) of W give a local coordinate system (q^{i}, p_{i}, ψ^{α}) on M.

We assume that W is even dimensional, oriented and Spin. It gives rise to a bundle of super Hilbert spaces ℋ_{V} on V, reproducing point by point what we got in 6., and the connection ∇ on W gives one on ℋ_{V}.

Let ℋ be the super Hilbert space of half densities on V with values in ℋ_{V}. To any function f on M, of degree ≤ 1 on the fibers of M/V, when p_{i} is viewed as of degree 1 and ψ^{α} as of degree 1/2, one attaches an operator fˆ as follows.

(a) For (the pull back of) a function f on V, fˆ is multiplication by f.

(b) For the odd function defined by a section e of W^{∨},

eˆ = (iℏ)^{1/2}cℓ(e)

as in 6.3. This is extended as in 5. to define fˆ for any function f on ΠW.

(c) If f is a function on T^{∗}V, linear on each fiber of T^{∗}V/V and identified with a vector field F,

fˆ = −iℏ∇_{F} .

In (c), if x in ℋ is the product of a section h of ℋ_{V} by a half density v, ∇_{F} (x) := ∇_{F} (h).v + h.ℒ_{F} (v).

The super vector space of operators fˆ , for f of degree ≤ 1 on the fibers of M/V, is stable by bracket. By (1.2), the bracket of operators corresponds to a Poisson bracket, which we now compute.

For functions on ΠW, it is as in 6.: functions on V commute with functions on ΠW and for ℓ, m odd linear functions, identified with sections of W^{∨}, one has

{ℓ, m} = −B^{−1}(ℓ, m).

For f a function on T^{∗}V, linear on the fibers and corresponding to a vector field F on V, {f, } is ∇_{F} on functions on ΠW.

It remains to compute [fˆ , gˆ ] as being of the form −iℏ{f, g}ˆ when f and g are linear functions on T^{∗}V, corresponding to vector fields F and G. We have

[fˆ , gˆ ] = [−iℏ∇_{F} , −iℏ∇_{G}] = (−iℏ)^{2} (∇_{[F,G]} + R(F, G))
= −iℏ ({f, g}_{T̂^{∗}V} − iℏR(F, G))

for { , }_{T^{∗}V} the Poisson bracket on T^{∗}V and F the curvature 2-form of W, with values in SO(W) (which acts on ℋ_{V}).

Let S^{β}α be the matrix of S ∈ SO(W), relatively to a local basis e_{α} of W: S is (x^{α}) ↦ (S^{α}_{β}x^{β}). Let e^{α} be the dual basis, and move the indices up and down using B. The action of S on ℋ_{V} is then the Clifford multiplication by S_{αβ}e^{α}e^{β} (sum on α and β) in the Clifford algebra. This follows from

[S_{αβ}e^{α}e^{β}, e^{γ}] = S_{αβ}(e^{α}[e^{β}, e^{γ}] − [e^{α}, e^{γ}]e^{β}) (superbrackets) = \frac{1}{2}S_{αβ}(e^{α}b^{β,γ} − b^{α,γ}e^{β}) = −S_{αβ}b^{αγ}e^{β} = −S^{γ}βe^{β} = S(e^{γ})

for the Lie algbra action.

If we apply this to the curvature, and if ψ^{α} are the odd functions on M corresponding to the e^{α}, this gives [fˆ , gˆ ] = −iℏ{f, g}ˆ for

{f, g} = {f, g}_{T^{∗}V} − R(F, G)_{α,β}ψ^{α}ψ^{β}

This Poisson bracket corresponds to the following symplectic 2-form Ω. Let α_{T^{∗}V} be the canonical 1-form pdq on T^{∗}V, as well as its pull back to M. Let α_{ΠW} be the 1-form on ΠW whose inverse image by a section ψ is \frac{1}{2}(ψ, ∇ψ). If (e_{α}) is a local basis of W, and if the connection is given by the endomorphism valued 1-form γ^{α}_{β}:

∇(x^{α}e_{α}) = (dx^{α} + γ^{α}_{β}x^{β})e_{α},

one has

α_{ΠW} = \frac{1}{2}ψ_{α}(dψ^{α} + γ^{α}_{β}ψ^{β}).

With those notations,

Ω = dα_{T^{∗}V} + dα_{ΠV} .

If (e_{α}) is an orthonormal basis, one has

dα_{ΠV} = \frac{1}{2} ∑_{α}(dψ^{α} + γ^{α}_{β}ψ^{β})^{2} + ∑_{α,β} R^{α}_{β}ψ^{α}ψ^{β}.