Hamiltonian systems with constraints

Audience: graduate students with prior exposure to Lagrangian/Hamiltonian mechanics and basic field theory.
Goal: learn the Dirac–Bergmann treatment of constrained Hamiltonian systems and see it in action for Maxwell theory, the Proca (massive vector) field, and general relativity (ADM).

Conventions

We use units $c=1$. Spacetime signature is $\eta_{\mu\nu}=\mathrm{diag}(-,+,+,+)$.
Spatial indices $i,j,k=1,2,3$. A dot denotes $\partial_t$.
We work on a fixed-time hypersurface $\Sigma_t\cong\mathbb{R}^3$ unless stated otherwise.

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1. Why constraints appear

In ordinary (regular) Lagrangian mechanics, the Legendre map

$$(q^a,\dot q^a)\mapsto (q^a,p_a),\qquad p_a=\frac{\partial L}{\partial\dot q^a}$$

is invertible because the Hessian

$$W_{ab}(q,\dot q)=\frac{\partial^2L}{\partial\dot q^a\,\partial\dot q^b}$$

has $\det W\neq 0$. One can solve $\dot q=\dot q(q,p)$ and define the Hamiltonian $H=p\dot q-L$.

A constrained system arises when the Hessian is singular: $\det W=0$. Then the momenta are not independent; they satisfy relations

$$\phi_\alpha(q,p)\approx 0,$$

called primary constraints. The symbol $\approx$ (“weak equality”) means the relation holds on the constraint surface in phase space, but you should not use it inside Poisson brackets until you have computed them.

In gauge theories the singularity is not an accident: it reflects redundancy in the description (gauge symmetry). In massive theories (e.g. Proca) singularity can also occur because some variables are nondynamical multipliers even without gauge symmetry.

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2. Dirac–Bergmann algorithm in a nutshell

2.1 Canonical Poisson brackets for fields

For canonical fields $(q^a(\mathbf{x}),p_a(\mathbf{x}))$,

$$\{q^a(\mathbf{x}),p_b(\mathbf{y})\}=\delta^a{}_b\,\delta^{(3)}(\mathbf{x}-\mathbf{y}), \qquad \{q^a,q^b\}=\{p_a,p_b\}=0.$$

2.2 Total Hamiltonian and consistency

Given a Lagrangian density $\mathcal{L}(q,\dot q,\nabla q)$:

1. Define canonical momenta

   $$p_a=\frac{\partial\mathcal{L}}{\partial \dot q^a}.$$

   Relations among $(q,p)$ that do not determine velocities are primary constraints $\phi_\alpha\approx 0$.

2. Canonical Hamiltonian (Legendre transform where possible)

   $$\mathcal{H}_c = p_a\dot q^a - \mathcal{L}.$$

3. Total Hamiltonian

   $$H_T = \int d^3x\left(\mathcal{H}_c + u^\alpha(\mathbf{x})\,\phi_\alpha(\mathbf{x})\right),$$

   where $u^\alpha$ are Lagrange multipliers enforcing the primary constraints.

4. Consistency conditions Require constraints be preserved under time evolution:

   $$\dot\phi_\alpha(\mathbf{x})=\{\phi_\alpha(\mathbf{x}),H_T\}\approx 0.$$

   This may:

   • produce secondary constraints, tertiary, etc.; and/or
   • fix some multipliers $u^\alpha$.

Continue until closure.

2.3 First-class vs second-class

Let $\{\Phi_A\}$ be the complete set of constraints.

• First-class constraint: $\{\Phi_A,\Phi_B\}\approx 0$ for all $B$.
  These generate gauge transformations (redundancies).

• Second-class constraint: the matrix (kernel)

  $$C_{AB}(\mathbf{x},\mathbf{y})=\{\Phi_A(\mathbf{x}),\Phi_B(\mathbf{y})\}$$

  is (functionally) invertible on the constraint surface.
  These do not generate gauge; they simply remove phase-space directions.

2.4 Dirac bracket (for second-class constraints)

If $\chi_A\approx 0$ are second-class and $C_{AB}$ is invertible with inverse $C^{AB}$,

$$\{F,G\}_D = \{F,G\} - \int d^3x\,d^3y\; \{F,\chi_A(\mathbf{x})\}\,C^{AB}(\mathbf{x},\mathbf{y})\,\{\chi_B(\mathbf{y}),G\}.$$

Then $\{F,\chi_A\}_D=0$ for all $F$, so you may set $\chi_A=0$ strongly after switching to Dirac brackets.

2.5 Counting physical degrees of freedom

Let $N$ be the number of configuration fields per space point (so phase space has dimension $2N$). Let $N_1$ be the number of first-class constraints and $N_2$ the number of second-class constraints (per point, in an appropriate local sense). Then

$$N_{\text{phys}} = N - N_1 - \frac12 N_2.$$

Equivalently, in phase space:

$$\dim \Gamma_{\text{phys}} = 2N - 2N_1 - N_2.$$

Interpretation

A first-class constraint does two things: it restricts you to a constraint surface and it generates a gauge orbit that must be quotiented out.
A second-class constraint only restricts you to the surface; there is no gauge orbit.

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3. Worked example I: Maxwell theory (massless spin-1)

3.1 Lagrangian and 3+1 split

Take vacuum Maxwell in flat space:

$$\mathcal{L} = -\frac14 F_{\mu\nu}F^{\mu\nu}, \qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$

With our signature,

$$F_{\mu\nu}F^{\mu\nu}=-2F_{0i}F_{0i}+F_{ij}F_{ij}.$$

Define

$$F_{0i}=\dot A_i-\partial_iA_0, \qquad B^k=\frac12\epsilon^{kij}F_{ij}, \qquad F_{ij}F_{ij}=2\mathbf{B}^2.$$

Then

$$\mathcal{L} = \frac12(\dot A_i-\partial_iA_0)^2-\frac12\mathbf{B}^2 = \frac12\mathbf{E}^2-\frac12\mathbf{B}^2, \quad E_i\equiv F_{0i}.$$

3.2 Canonical momenta and primary constraint

Define

$$\pi^\mu=\frac{\partial\mathcal{L}}{\partial \dot A_\mu}.$$

There is no $\dot A_0$ in $\mathcal{L}$, hence

$$\pi^0 = 0 \quad\Rightarrow\quad \phi_1(\mathbf{x})\equiv \pi^0(\mathbf{x})\approx 0 \qquad\text{(primary constraint).}$$

For $i=1,2,3$,

$$\pi^i=\frac{\partial\mathcal{L}}{\partial \dot A_i} =\dot A_i-\partial_iA_0 =F_{0i} =E_i.$$

Thus $\pi^i$ is the electric field.

3.3 Canonical Hamiltonian

Compute the Hamiltonian density

$$\mathcal{H}_c=\pi^i\dot A_i-\mathcal{L}.$$

Solve $\dot A_i=\pi^i+\partial_iA_0$:

$$\pi^i\dot A_i = \pi^2+\pi^i\partial_iA_0.$$

Since $\mathcal{L}=\tfrac12\pi^2-\tfrac12\mathbf{B}^2$,

$$\mathcal{H}_c = \frac12\pi^2+\frac12\mathbf{B}^2+\pi^i\partial_iA_0.$$

Integrating by parts (dropping boundary terms),

$$\int d^3x\,\pi^i\partial_iA_0 = -\int d^3x\,A_0\,\partial_i\pi^i,$$

so the canonical Hamiltonian is

$$H_c = \int d^3x\left[ \frac12(\pi^2+\mathbf{B}^2) - A_0\,\partial_i\pi^i \right].$$

3.4 Total Hamiltonian and Gauss constraint

Add the primary constraint with multiplier $u(\mathbf{x})$:

$$H_T=H_c+\int d^3x\,u(\mathbf{x})\,\pi^0(\mathbf{x}).$$

Preserve $\pi^0\approx 0$:

$$\dot\pi^0(\mathbf{x}) = \{\pi^0(\mathbf{x}),H_T\} = -\frac{\delta H_T}{\delta A_0(\mathbf{x})} = \partial_i\pi^i(\mathbf{x}) \approx 0.$$

This yields the secondary constraint

$$\phi_2(\mathbf{x})\equiv \partial_i\pi^i(\mathbf{x})\approx 0,$$

i.e. Gauss’s law in vacuum.

No further constraints arise; the multiplier $u(\mathbf{x})$ remains undetermined.

3.5 Constraint algebra and first-class nature

Using canonical brackets,

$$\{\pi^0(\mathbf{x}),\phi_2(\mathbf{y})\}=0, \qquad \{\phi_2(\mathbf{x}),\phi_2(\mathbf{y})\}=0.$$

Thus $\phi_1$ and $\phi_2$ are first-class.

3.6 Gauge generator (one gauge function, two constraints)

A gauge parameter is a function $\epsilon(t,\mathbf{x})$. The appropriate generator can be written (Castellani’s algorithm) as

$$G[\epsilon] = \int d^3x\Big(-\dot\epsilon(\mathbf{x})\,\pi^0(\mathbf{x})+\epsilon(\mathbf{x})\,\phi_2(\mathbf{x})\Big).$$

Then

$$\delta A_0=\{A_0,G\}=-\dot\epsilon, \qquad \delta A_i=\{A_i,G\}=-\partial_i\epsilon,$$

so $\delta A_\mu=-\partial_\mu\epsilon$, the usual $U(1)$ gauge symmetry.

Why “two first-class constraints” but “one gauge function”?

The gauge generator involves $\epsilon$ and $\dot\epsilon$. One gauge function typically comes with a constraint chain (primary $\pi^0$ and secondary Gauss law).

3.7 Degrees of freedom

We have $N=4$ configuration fields $A_\mu$. Constraints: $N_1=2$ first-class and $N_2=0$. Hence

$$N_{\text{phys}} = 4 - 2 = 2,$$

the two transverse photon polarizations.

3.8 Optional: explicit reduction in Coulomb gauge

To see the “constraint + gauge” removal explicitly, decompose

$$A_i = A_i^T + \partial_i\alpha, \qquad \partial_iA_i^T=0,$$ $$\pi^i = \pi_T^i + \partial^i\beta, \qquad \partial_i\pi_T^i=0.$$

Then Gauss’s constraint is $\partial_i\pi^i=\nabla^2\beta\approx 0$, which removes the longitudinal momentum $\beta$ (up to boundary conditions). The longitudinal coordinate $\alpha$ is removed by gauge.

If you impose Coulomb gauge $\chi\equiv \partial_iA_i\approx 0$, then $(\chi,\phi_2)$ form a second-class pair:

$$\{\chi(\mathbf{x}),\phi_2(\mathbf{y})\} = \{\partial_iA_i(\mathbf{x}),\partial_j\pi^j(\mathbf{y})\} = \nabla^2\delta^{(3)}(\mathbf{x}-\mathbf{y}),$$

which is invertible (as an operator) after specifying boundary conditions. The reduced Hamiltonian becomes a Hamiltonian for the transverse fields only:

$$H_{\text{red}}=\int d^3x\,\frac12\big(\pi_T^2+\mathbf{B}^2\big).$$

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4. Worked example II: Proca (massive spin-1)

4.1 Lagrangian and sign conventions

With our signature $(-+++)$, a convenient Proca Lagrangian that leads to a positive-energy Hamiltonian is

$$\mathcal{L} = -\frac14 F_{\mu\nu}F^{\mu\nu} -\frac12 m^2 A_\mu A^\mu.$$

The field equation is

$$\partial_\mu F^{\mu\nu} - m^2 A^\nu = 0,$$

and taking $\partial_\nu$ gives the on-shell constraint

$$\partial_\nu A^\nu = 0 \quad (m\neq 0).$$

If you prefer the opposite sign in the equations of motion

Different metric/sign conventions shuffle a sign between the mass term and the equation $\partial_\mu F^{\mu\nu}\pm m^2A^\nu=0$. The constraint/Hamiltonian structure (first-class vs second-class, and the DOF count) is unchanged.

4.2 3+1 split

Using $A_\mu A^\mu=-A_0^2+A_i^2$, we get

$$-\frac12 m^2 A_\mu A^\mu = +\frac12 m^2 A_0^2 - \frac12 m^2 A_i^2.$$

Therefore

$$\mathcal{L} = \frac12(\dot A_i-\partial_iA_0)^2 -\frac12\mathbf{B}^2 +\frac12 m^2 A_0^2 -\frac12 m^2 A_i^2.$$

4.3 Canonical momenta and primary constraint

Exactly as in Maxwell,

$$\pi^0=\frac{\partial\mathcal{L}}{\partial\dot A_0}=0 \quad\Rightarrow\quad \phi_1(\mathbf{x})\equiv \pi^0(\mathbf{x})\approx 0,$$

and

$$\pi^i=\frac{\partial\mathcal{L}}{\partial\dot A_i}=\dot A_i-\partial_iA_0.$$

4.4 Canonical Hamiltonian

Compute

$$\mathcal{H}_c=\pi^i\dot A_i-\mathcal{L}, \qquad \dot A_i=\pi^i+\partial_iA_0.$$

One finds

$$\mathcal{H}_c = \frac12\pi^2+\frac12\mathbf{B}^2 +\frac12 m^2 A_i^2 +\pi^i\partial_iA_0 -\frac12 m^2 A_0^2.$$

Integrating by parts,

$$H_c=\int d^3x\left[ \frac12(\pi^2+\mathbf{B}^2) +\frac12 m^2 A_i^2 - A_0\,\partial_i\pi^i -\frac12 m^2 A_0^2 \right].$$

4.5 Total Hamiltonian and secondary constraint

Total Hamiltonian:

$$H_T = H_c+\int d^3x\,u(\mathbf{x})\,\pi^0(\mathbf{x}).$$

Preserve $\pi^0\approx 0$:

$$\dot\pi^0(\mathbf{x}) = -\frac{\delta H_T}{\delta A_0(\mathbf{x})} = \partial_i\pi^i(\mathbf{x}) + m^2 A_0(\mathbf{x}) \approx 0.$$

So the secondary constraint is

$$\phi_2(\mathbf{x})\equiv \partial_i\pi^i(\mathbf{x})+m^2A_0(\mathbf{x})\approx 0.$$

4.6 Second-class structure (no gauge symmetry)

Compute the constraint bracket:

$$\{\phi_1(\mathbf{x}),\phi_2(\mathbf{y})\} = \{\pi^0(\mathbf{x}),\partial_i\pi^i(\mathbf{y})+m^2A_0(\mathbf{y})\} = -m^2\delta^{(3)}(\mathbf{x}-\mathbf{y})\neq 0.$$

Thus $\phi_1,\phi_2$ are second-class: there is no gauge symmetry. Consistency now fixes the multiplier $u$ rather than leaving it arbitrary.

4.7 Eliminating $A_0$ and the reduced Hamiltonian

The constraint $\phi_2=0$ is algebraic in $A_0$:

$$A_0 = -\frac{1}{m^2}\partial_i\pi^i.$$

Substitute into the Hamiltonian. The $A_0$-dependent terms combine as

$$- A_0\,\partial_i\pi^i - \frac12 m^2 A_0^2 = \frac{1}{2m^2}(\partial_i\pi^i)^2.$$

Hence the reduced Hamiltonian is

$$H_{\text{red}} = \int d^3x\left[ \frac12(\pi^2+\mathbf{B}^2) +\frac12 m^2 A_i^2 +\frac{1}{2m^2}(\partial_i\pi^i)^2 \right],$$

which is manifestly bounded below.

4.8 Degrees of freedom

Here $N=4$, $N_1=0$, $N_2=2$. Thus

$$N_{\text{phys}}=4-\frac12\cdot 2 = 3,$$

corresponding to helicities $-1,0,+1$ of a massive spin-1 particle.

4.9 Optional: Stückelberg trick and “gauge vs second-class”

Introduce a scalar $\varphi$ and replace

$$A_\mu \to A_\mu + \frac{1}{m}\partial_\mu\varphi.$$

Then the Proca mass term becomes gauge invariant under

$$\delta A_\mu=\partial_\mu\lambda, \qquad \delta\varphi = -m\lambda.$$

The theory is now gauge invariant (first-class constraints reappear), but it contains an extra field $\varphi$. After gauge fixing (e.g. $\varphi=0$) you recover Proca with 3 physical DOF. This is a useful conceptual bridge: second-class constraints can be viewed as gauge-fixed first-class systems (under appropriate extensions).

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5. General relativity as a constrained Hamiltonian system (ADM)

The Hamiltonian formulation of GR is the prototype of a field theory with:

• singular Lagrangian (lapse and shift are nondynamical),
• first-class constraints (encoding diffeomorphism invariance),
• a nontrivial constraint algebra with structure functions (hypersurface deformation algebra).

We sketch the derivation carefully enough to see where each ingredient comes from.

Scope

A full derivation requires careful handling of boundary terms (Gibbons–Hawking–York term and asymptotic/boundary contributions). We highlight the core steps and give the final canonical expressions, indicating where boundary terms enter.

5.1 Einstein–Hilbert action and 3+1 split

Start from the Einstein–Hilbert action (with cosmological constant $\Lambda$)

$$S = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,(R-2\Lambda) + S_{\text{boundary}}.$$

The boundary term (e.g. Gibbons–Hawking–York) ensures a well-posed variational principle when fixing the induced metric on the boundary.

Assume spacetime is foliated by spacelike hypersurfaces $\Sigma_t$, with coordinates $x^i$ on each $\Sigma_t$. The spacetime metric can be written in ADM form:

$$ds^2 = -N^2dt^2 + h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$$

where:

• $h_{ij}(t,\mathbf{x})$ is the induced spatial metric on $\Sigma_t$,
• $N(t,\mathbf{x})$ is the lapse,
• $N^i(t,\mathbf{x})$ is the shift (with $N_i=h_{ij}N^j$).

Useful identities:

$$\sqrt{-g}=N\sqrt{h},$$

where $h=\det(h_{ij})$.

5.2 Extrinsic curvature and kinematics

Define the covariant derivative $D_i$ compatible with $h_{ij}$: $D_k h_{ij}=0$.

The extrinsic curvature of $\Sigma_t$ embedded in spacetime is

$$K_{ij} = \frac{1}{2N}\left(\dot h_{ij}-D_iN_j-D_jN_i\right), \qquad K=h^{ij}K_{ij}.$$

This shows explicitly that $\dot h_{ij}$ appears linearly in $K_{ij}$, while $\dot N$ and $\dot N^i$ do not appear at all.

5.3 ADM Lagrangian density

A standard result of the Gauss–Codazzi decomposition (up to total derivatives absorbed by $S_{\text{boundary}}$) is:

$$\sqrt{-g}\,R = N\sqrt{h}\left({}^{(3)}R + K_{ij}K^{ij} - K^2\right) +\text{(total derivative)}.$$

Therefore, dropping total derivatives already accounted for by boundary terms, the ADM Lagrangian density is

$$\mathcal{L}_{\text{ADM}} = \frac{\sqrt{h}}{16\pi G}\,N\left({}^{(3)}R + K_{ij}K^{ij} - K^2 - 2\Lambda\right).$$

5.4 Canonical momenta

The canonical momentum conjugate to $h_{ij}$ is

$$\pi^{ij}(\mathbf{x}) = \frac{\partial \mathcal{L}_{\text{ADM}}}{\partial \dot h_{ij}(\mathbf{x})}.$$

Since $\dot h_{ij}$ enters only through $K_{ij}$, and

$$\frac{\partial K_{kl}}{\partial \dot h_{ij}}=\frac{1}{2N}\delta^i{}_{(k}\delta^j{}_{l)},$$

one finds

$$\pi^{ij} = \frac{\sqrt{h}}{16\pi G}\left(K^{ij}-h^{ij}K\right).$$

Taking the trace $\pi\equiv h_{ij}\pi^{ij}$ gives

$$\pi = -\frac{\sqrt{h}}{16\pi G}\,2K \quad\Rightarrow\quad K = -\frac{8\pi G}{\sqrt{h}}\,\pi.$$

You can invert to express $K_{ij}$ in terms of $\pi^{ij}$:

$$K_{ij} = \frac{16\pi G}{\sqrt{h}}\left(\pi_{ij}-\frac12 h_{ij}\pi\right).$$

Primary constraints: because $\dot N$ and $\dot N^i$ do not appear in $\mathcal{L}_{\text{ADM}}$,

$$\pi_N(\mathbf{x}) \equiv \frac{\partial\mathcal{L}}{\partial \dot N}=0,\qquad \pi_i(\mathbf{x}) \equiv \frac{\partial\mathcal{L}}{\partial \dot N^i}=0.$$

Thus,

$$\pi_N\approx 0,\qquad \pi_i\approx 0$$

are primary constraints.

5.5 Canonical Hamiltonian and the ADM constraints

The canonical Hamiltonian is

$$H_c=\int d^3x\,\left(\pi^{ij}\dot h_{ij}-\mathcal{L}_{\text{ADM}}\right),$$

where $\dot h_{ij}$ should be expressed using

$$\dot h_{ij}=2NK_{ij}+D_iN_j+D_jN_i.$$

A key computation uses integration by parts:

$$\int d^3x\,\pi^{ij}(D_iN_j+D_jN_i) = -2\int d^3x\,N_j D_i\pi^{ij}$$

(up to boundary terms). After rewriting $K_{ij}$ in terms of $\pi^{ij}$, one arrives at the standard ADM form

$$H_c = \int d^3x\left( N\,\mathcal{H}_\perp + N^i\,\mathcal{H}_i\right) + H_{\partial\Sigma}.$$

Here $H_{\partial\Sigma}$ is a boundary term (e.g. ADM energy for asymptotically flat spacetimes). The bulk constraint densities are:

Momentum (diffeomorphism) constraint

$$\boxed{\;\mathcal{H}_i = -2 D_j \pi_i{}^{j}\;}$$

Hamiltonian (scalar) constraint

$$\boxed{\; \mathcal{H}_\perp = \frac{16\pi G}{\sqrt{h}}\left(\pi_{ij}\pi^{ij}-\frac12\pi^2\right) -\frac{\sqrt{h}}{16\pi G}\left({}^{(3)}R-2\Lambda\right) \;}$$

(again, up to convention-dependent signs/factors).

Where did the constraints come from?

$N$ and $N^i$ appear in the Hamiltonian without conjugate momenta; they multiply $\mathcal{H}_\perp$ and $\mathcal{H}_i$.
They act as Lagrange multipliers enforcing $\mathcal{H}_\perp\approx 0$ and $\mathcal{H}_i\approx 0$.

5.6 Total Hamiltonian and secondary constraints

The total Hamiltonian adds the primary constraints:

$$H_T = H_c + \int d^3x\left(u\,\pi_N + u^i \pi_i\right).$$

Preserving $\pi_N\approx 0$ gives

$$\dot\pi_N(\mathbf{x}) = \{\pi_N(\mathbf{x}),H_T\} = -\frac{\delta H_T}{\delta N(\mathbf{x})} = -\mathcal{H}_\perp(\mathbf{x}) \approx 0,$$

hence the Hamiltonian constraint

$$\mathcal{H}_\perp(\mathbf{x})\approx 0.$$

Preserving $\pi_i\approx 0$ gives

$$\dot\pi_i(\mathbf{x}) = -\frac{\delta H_T}{\delta N^i(\mathbf{x})} = -\mathcal{H}_i(\mathbf{x}) \approx 0,$$

hence the momentum constraints

$$\mathcal{H}_i(\mathbf{x})\approx 0.$$

No new independent constraints appear beyond these (for pure GR); instead, consistency fixes nothing because the theory is gauge invariant (diffeomorphisms).

5.7 Smeared constraints and the constraint algebra

It is cleaner to use smeared functionals:

$$\mathcal{H}[N] \equiv \int d^3x\,N(\mathbf{x})\,\mathcal{H}_\perp(\mathbf{x}), \qquad \mathcal{D}[\vec{N}] \equiv \int d^3x\,N^i(\mathbf{x})\,\mathcal{H}_i(\mathbf{x}).$$

Then (schematically) the Poisson brackets close as:

$$\{\mathcal{D}[\vec{N}],\mathcal{D}[\vec{M}]\} = \mathcal{D}[\mathcal{L}_{\vec{N}}\vec{M}],$$ $$\{\mathcal{D}[\vec{N}],\mathcal{H}[M]\} = \mathcal{H}[\mathcal{L}_{\vec{N}}M],$$ $$\{\mathcal{H}[N],\mathcal{H}[M]\} = \mathcal{D}\!\left[h^{ij}(N\partial_j M - M\partial_j N)\right].$$

This is the hypersurface deformation algebra (often called “Dirac algebra”). It is not a Lie algebra with constant structure constants; it has structure functions involving $h^{ij}$.

The closure implies that $\mathcal{H}_\perp$ and $\mathcal{H}_i$ are first-class (together with the primary constraints $\pi_N,\pi_i$).

5.8 Gauge interpretation: diffeomorphisms

• $\mathcal{D}[\vec{N}]$ generates spatial diffeomorphisms on $\Sigma_t$: $$\{h_{ij},\mathcal{D}[\vec{N}]\}=\mathcal{L}_{\vec{N}}h_{ij}, \qquad \{\pi^{ij},\mathcal{D}[\vec{N}]\}=\mathcal{L}_{\vec{N}}\pi^{ij}.$$
• $\mathcal{H}[N]$ generates normal deformations of the hypersurface (time reparametrizations / refoliations).

A precise mapping between $(\mathcal{H}_\perp,\mathcal{H}_i)$ and spacetime diffeomorphisms requires care, because the algebra closes with structure functions; nonetheless, the standard viewpoint is that these first-class constraints encode the redundancy under diffeomorphisms.

5.9 Degrees of freedom of GR

In 3+1D, the configuration variable $h_{ij}$ is a symmetric $3\times 3$ tensor: $N=6$ per point.

Constraints:

• primary: $\pi_N\approx 0$ (1), $\pi_i\approx 0$ (3),
• secondary: $\mathcal{H}_\perp\approx 0$ (1), $\mathcal{H}_i\approx 0$ (3).

Altogether there are 8 constraints; however, the standard DOF counting for GR focuses on the true canonical pair $(h_{ij},\pi^{ij})$ and treats $N,N^i$ as multipliers.

On the $(h_{ij},\pi^{ij})$ phase space:

• $2N = 12$ phase-space dimensions per point,
• there are $4$ independent first-class constraints $(\mathcal{H}_\perp,\mathcal{H}_i)$.

Thus

$$\dim \Gamma_{\text{phys}} = 12 - 2\times 4 = 4, \qquad N_{\text{phys}} = \frac{4}{2}=2.$$

These are the two polarizations of the graviton (gravitational waves) in 4D.

5.10 Linearized check (TT gauge intuition)

Linearize around Minkowski: $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. In harmonic gauge one can reduce to transverse-traceless (TT) components $h_{ij}^{\text{TT}}$ satisfying a wave equation. The TT condition removes gauge redundancy and constraints, leaving two propagating modes—consistent with the Hamiltonian count above.

Details: harmonic gauge ⇒ TT wave equation and the “2 polarizations” count

Start from

$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\qquad |h_{\mu\nu}|\ll 1.$$

Infinitesimal diffeomorphisms $x^\mu\to x^\mu-\xi^\mu(x)$ act as a gauge symmetry:

$$\delta h_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.$$

It is convenient to use the trace-reversed field

$$h\equiv \eta^{\mu\nu}h_{\mu\nu},\qquad \bar h_{\mu\nu}\equiv h_{\mu\nu}-\frac12\,\eta_{\mu\nu}h.$$

The harmonic (Lorenz) gauge condition is

$$\partial^\mu \bar h_{\mu\nu}=0.$$

In this gauge, the vacuum linearized Einstein equations simplify to

$$\Box\,\bar h_{\mu\nu}=0,\qquad \Box\equiv \eta^{\rho\sigma}\partial_\rho\partial_\sigma.$$

So the radiative degrees propagate as massless waves.

Harmonic gauge does not fully fix the gauge: it is preserved by residual transformations with

$$\Box\,\xi^\mu=0.$$

For a plane wave $\bar h_{\mu\nu}=\varepsilon_{\mu\nu}e^{ik\cdot x}$, the field equation gives $k^2=0$ and the gauge condition gives transversality $k^\mu\varepsilon_{\mu\nu}=0$. Using the residual gauge freedom one can impose the stronger TT conditions (for a wave moving along $z$):

$$h_{0\mu}=0,\qquad \partial^i h_{ij}=0,\qquad h^i{}_i=0.$$

The only nonzero components then live in the $2\times2$ block transverse to the propagation direction and satisfy

$$\Box\,h^{TT}_{ij}=0.$$

A standard basis is

$$h^{TT}_{xx}=-h^{TT}_{yy}\equiv h_+,\qquad h^{TT}_{xy}=h^{TT}_{yx}\equiv h_\times,$$

i.e. the two gravitational-wave polarizations.

This is the linearized counterpart of the Hamiltonian counting: the constraints remove non-propagating components and the gauge symmetry quotients out redundancies, leaving two physical modes (in 4D), exactly as in §5.9.

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6. Conceptual comparison: Maxwell vs Proca vs GR

6.1 Why first-class “removes more” than second-class

A single first-class constraint does two things:

1. it restricts to the constraint surface, and
2. it generates a gauge flow—points along that flow are physically equivalent.

Therefore each first-class constraint removes two phase-space dimensions (one for the surface, one for the orbit), while each second-class constraint removes only one.

• Maxwell: constraints are first-class $\Rightarrow$ gauge symmetry $\Rightarrow$ 2 physical DOF.
• Proca: constraints are second-class $\Rightarrow$ no gauge symmetry $\Rightarrow$ 3 physical DOF.
• GR: constraints are first-class $\Rightarrow$ diffeomorphism redundancy $\Rightarrow$ 2 physical DOF in 4D.

6.2 Nondynamical fields and constraints

Both Maxwell and Proca have $\pi^0=0$ because $A_0$ has no time derivative.
The difference is what happens next:

• Maxwell: preservation produces Gauss law $\partial_i\pi^i\approx 0$ (first-class).
• Proca: preservation produces $\partial_i\pi^i+m^2A_0\approx 0$ (second-class with $\pi^0$).

In GR, lapse and shift are nondynamical: $\pi_N=\pi_i=0$. Their preservation produces Hamiltonian/momentum constraints, all first-class.

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7. Exercises

1. Maxwell with sources: Add $-A_\mu J^\mu$ to the Maxwell Lagrangian and show that Gauss’s law becomes $\partial_i\pi^i=\rho$. Discuss which parts of the constraint structure change.

2. Coulomb gauge Dirac bracket: In Maxwell theory, impose Coulomb gauge $\partial_iA_i=0$ and compute the Dirac bracket for the transverse fields.

3. Proca Dirac bracket: Treat $\pi^0$ and $\partial_i\pi^i+m^2A_0$ as second-class and compute the Dirac bracket for $A_0$ and $A_i$.

4. ADM constraint algebra: Verify at least one of the ADM bracket relations using smeared constraints and integration by parts.

5. GR DOF in $D$ dimensions: Generalize the ADM DOF count to $D$ spacetime dimensions and show that the number of propagating graviton DOF is $D(D-3)/2$.

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8. References for further study

• P. A. M. Dirac, Lectures on Quantum Mechanics (constrained Hamiltonian systems).
• M. Henneaux & C. Teitelboim, Quantization of Gauge Systems (modern, systematic).
• K. Sundermeyer, Constrained Dynamics (classic).
• R. M. Wald, General Relativity (ADM, constraints, canonical structure).
• E. Poisson, A Relativist’s Toolkit (3+1 tools, extrinsic curvature).
• Arnowitt–Deser–Misner (ADM) original papers/lectures (historical source).
• 梁灿彬, 周彬. 微分几何入门与广义相对论（下册）第二版. 科学出版社.