Covariant Phase Space for Linear Fields

Covariant phase space is the space of classical solutions rather than the space of fields and momenta on one chosen time slice. A conserved bilinear current equips that solution space with a symplectic form. The value is unchanged when the Cauchy hypersurface is moved because the current has zero divergence on shell and spacelike-compact solutions carry no flux through spatial infinity.

For the local PDE bridge, a spacelike-compact solution has compact support when restricted to every Cauchy hypersurface, and a distributional Green identity is integration by parts tested against compact smooth sections. These support conditions make the surface integrals and the passage between source representatives well defined.

Required background. Partial Differential Equations and Green Functions provides Green identities, spacelike-compact support, and causal propagators; Classical Field Theory supplies variational currents, canonical momenta, and gauge degeneracy. Helpful background. Flat-Spacetime Quantum Field Theory provides canonical phase space and mode normalization; Causality and Global Structure establishes Cauchy hypersurfaces and the compactness properties used by Stokes’ theorem.

The Klein–Gordon symplectic current

For two real solutions $\phi_1,\phi_2$ of

$$P\phi=(\Box-m^2-\xi R)\phi=0,$$

the antisymmetrized variation of the action’s boundary term gives

$$j^\mu(\phi_1,\phi_2) =\phi_1\nabla^\mu\phi_2 -\phi_2\nabla^\mu\phi_1.$$

Its divergence is an off-shell identity,

$$\nabla_\mu j^\mu =\phi_1P\phi_2-\phi_2P\phi_1.$$

The mass and curvature potential cancel because $P$ is formally self-adjoint and both entries see the same real potential. Hence $\nabla_\mu j^\mu=0$ when both fields solve the equation. This calculation generalizes: for a formally self-adjoint linear system, the Green identity supplies the bilinear current, while for a complex system one uses the appropriate sesquilinear adjoint pairing.

Choose a smooth spacelike Cauchy hypersurface $\Sigma$ with future unit normal $n^\mu$ and induced positive measure $d\Sigma$. Define

$$\Omega_\Sigma(\phi_1,\phi_2) =\int_\Sigma d\Sigma\, n^\mu \left( \phi_1\nabla_\mu\phi_2 -\phi_2\nabla_\mu\phi_1 \right).$$

The future-normal convention is consequential. In Minkowski coordinates with $\Sigma:t=\text{constant}$, it gives

$$\Omega_\Sigma =\int d^3x\, \left(\phi_1\dot\phi_2-\phi_2\dot\phi_1\right),$$

the standard canonical sign.

Why the Cauchy surface does not matter

Let $\Sigma_1$ and $\Sigma_2$ be Cauchy hypersurfaces, with $\Sigma_2$ to the future, and let $U$ be the slab between them. Assume the solutions are spacelike compact, so a sufficiently large timelike cylinder contributes no boundary flux. Stokes’ theorem gives

$$0 =\int_U dV_g\,\nabla_\mu j^\mu =-\int_{\Sigma_2}d\Sigma\,n_\mu j^\mu +\int_{\Sigma_1}d\Sigma\,n_\mu j^\mu.$$

Therefore

$$\Omega_{\Sigma_2}(\phi_1,\phi_2) =\Omega_{\Sigma_1}(\phi_1,\phi_2).$$

The signs include both boundary orientation and the timelike normal’s norm. In flat coordinates, $n_\mu j^\mu=-j^0$, so the displayed expression is exactly $\int j^0(t_2)\,d^3x-\int j^0(t_1)\,d^3x$. Both symplectic forms themselves use future normals. If the spacetime has a timelike boundary, or if the fields have nondecaying flux at infinity, an additional boundary term remains. Surface independence then requires boundary conditions that make the symplectic flux vanish, or an enlarged phase space that includes boundary degrees of freedom.

For a complex scalar, the conserved Klein–Gordon form is

$$(\phi_1,\phi_2)_{\mathrm{KG}} =i\int_\Sigma d\Sigma\,n^\mu \left( \phi_1^*\nabla_\mu\phi_2 -\phi_2\nabla_\mu\phi_1^* \right).$$

It is Hermitian and surface independent, but it is not positive on the full complex solution space. Positivity appears only after a compatible positive-frequency or one-particle choice; conservation alone does not select a vacuum.

Source representatives and the causal propagator

The causal propagator uses the convention

$$E=G_{\mathrm{ret}}-G_{\mathrm{adv}}.$$

Every spacelike-compact solution can be written $\phi=Ef$ for some $f\in C_c^\infty(M)$, and $f$ is unique only modulo $Ph$ with compact $h$. With the surface convention above, Green’s identity gives

$$\Omega(Ef,Eh) =\int_M dV_g\,f(Eh) =-\int_M dV_g\,(Ef)h.$$

To see the sign without mixing sourced and homogeneous equations, apply the current identity to $u=G_{\mathrm{ret}}f$ and $v=Eh$. Then $Pu=f$, $Pv=0$, and $\nabla_\mu j^\mu(u,v)=-fv$. On a surface before the source, $u$ and its normal derivative vanish; on a surface after it, $u=Ef$. The preceding divergence theorem therefore gives $-\Omega(Ef,Eh)=-\int fEh\,dV_g$. Formal self-adjointness gives the second expression. This check ties the surface orientation to the same retarded-minus-advanced convention used in the quantum commutator.

Because $EPh=0$, the pairing is independent of each source representative. Because the exact Green-operator sequence identifies $\ker E=P C_c^\infty(M)$, the induced form on

$$C_c^\infty(M)/P C_c^\infty(M)$$

is nondegenerate and agrees with the surface form on the solution space. This quotient construction, including formal self-adjointness and the Green pairing, is developed in Bär, Ginoux, and Pfäffle 2007, §4.3, pp. 123–130. The equivalent Cauchy-surface smearing formula appears in Hollands and Wald 2015, §2.1, Eqs. (5)–(10).

Symplectic form, presymplectic form, and gauge

For the real Klein–Gordon field, uniqueness of the Cauchy problem makes $\Omega$ nondegenerate: if $\Omega(\phi,\psi)=0$ for every solution $\psi$, one may vary the two independent Cauchy data of $\psi$ to show that both Cauchy data of $\phi$ vanish, hence $\phi=0$.

Gauge theories differ. A gauge variation may pair to zero with every admissible perturbation, so the form on raw solutions is presymplectic. One must restrict to the constraint surface and quotient gauge directions, while also checking boundary and topological sectors. Calling the unquotiented form symplectic would silently assume away physical degeneracies. This distinction prepares the Maxwell comparison on the next page.

Checks in canonical variables

In Minkowski spacetime write Cauchy data as $(\varphi,\pi)=(\phi,\dot\phi)$. Then

$$\Omega((\varphi_1,\pi_1),(\varphi_2,\pi_2)) =\int d^3x\, (\varphi_1\pi_2-\varphi_2\pi_1).$$

This checks the normalization against canonical Hamiltonian mechanics. Conservation follows directly by differentiating in time, substituting $\ddot\phi=\nabla^2\phi-m^2\phi$, and integrating the spatial Laplacian by parts. The mass terms cancel algebraically and the spatial term becomes a boundary flux, exactly mirroring the covariant proof.

Check your understanding

Suppose a solution pair on a spacetime region with timelike boundary satisfies $\nabla_\mu j^\mu=0$. Is that alone enough to conclude that $\Omega_\Sigma$ is independent of $\Sigma$?

Solution

No. Apply Stokes’ theorem to the slab between two Cauchy slices. If $s^\mu$ is the outward spacelike unit normal on the timelike wall, our convention gives $0=-\Omega_{\Sigma_2}+\Omega_{\Sigma_1}+\int_{\mathrm{wall}}s_\mu j^\mu\,dS$. Thus the difference of surface forms equals the outward wall-current integral. Surface independence follows only if the boundary condition forces this term to vanish, if suitable falloff removes it at infinity, or if boundary degrees of freedom contribute a compensating term. The local conservation equation is therefore necessary but not sufficient when the integration region has an additional boundary.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society, 2007. DOI 10.4171/037.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. DOI 10.1016/j.physrep.2015.02.001.