Hamiltonian Initial Data and Phase Space

Choosing a time slicing turns a Lagrangian field theory into a branch test. First compute the canonical momentum density and the velocity Hessian. If the Hessian is locally invertible, the velocities can be exchanged for momenta, a Hamiltonian can be formed, and the equal-time Poisson bracket generates evolution. If the Hessian is singular, the ordinary inversion stops at relations among the canonical data; those primary relations begin a constraint analysis but do not yet identify gauge variables or physical degrees of freedom.

This page treats smooth, first-derivative classical bosonic fields on a fixed Minkowski time slicing, with boundary or falloff data sufficient for every spatial integration by parts. The real scalar supplies the regular branch and Maxwell theory supplies the singular diagnostic. Global infinite-dimensional symplectic analysis, variable-rank constraint geometry, full Dirac–Bergmann stabilization and reduction, gauge generators, fermionic graded brackets, higher-derivative theories, quantization, and analytic PDE well-posedness are outside the present scope.

Required background. The Action Principle and Field Equations supplies the scalar and Maxwell Lagrangians, their signs, and the Euler–Lagrange equations that Hamilton’s equations must reproduce.

Helpful background. Symplectic Forms, Hamiltonian Flows, and Poisson Brackets supplies the geometric meaning of Hamiltonian flow and equal-time Poisson brackets. Constraints, Dirac Brackets, and Symplectic Reduction supplies the analysis required after a singular Legendre map is diagnosed.

Canonical data from a regular Legendre map

Write a first-order action relative to a fixed spatial slice $\Sigma$ as

$$\begin{aligned} S[\Phi] &=\int_{t_i}^{t_f}\mathrm dt\,L(t), \\ L(t) &=\int_\Sigma\mathrm d^{d-1}x\, \mathcal L(\Phi^A,\dot\Phi^A,\partial_i\Phi^A). \end{aligned}$$

The canonical momentum density conjugate to $\Phi^A$ is

$$\pi_A(\mathbf x) \equiv \frac{\partial\mathcal L}{\partial\dot\Phi^A(\mathbf x)}.$$

It is also the coefficient of the field variation on a time cap. Omitting the already analyzed lateral-wall term, the first variation contains

$$\delta S = \text{bulk term} + \left[ \int_\Sigma\mathrm d^{d-1}x\, \pi_A\delta\Phi^A \right]_{t_i}^{t_f}.$$

Thus a candidate canonical datum on one slice is a pair of functions $(\Phi^A,\pi_A)$ with the declared regularity and boundary behavior. Whether all such pairs are allowed is decided by the Legendre map

$$(\Phi,\dot\Phi) \longmapsto (\Phi,\pi)$$

and its velocity Hessian

$$W_{AB}(\mathbf x) \equiv \frac{\partial^2\mathcal L} {\partial\dot\Phi^A\partial\dot\Phi^B}.$$

For this elementary local class, full rank of $W_{AB}$ is the pointwise test for a locally regular Legendre map. On a regular patch one can solve

$$\dot\Phi^A = v^A(\Phi,\pi,\partial_i\Phi)$$

and define

$$\begin{aligned} \mathcal H &=\pi_Av^A - \mathcal L(\Phi,v,\partial_i\Phi), \\ H[\Phi,\pi] &=\int_\Sigma\mathrm d^{d-1}x\,\mathcal H. \end{aligned}$$

This is a local statement. An everywhere nonsingular Hessian does not by itself prove that the velocity–momentum map is globally one-to-one, and a rank-changing Hessian does not define one smooth primary-constraint surface without further analysis. The regular scalar Legendre transform and its invertibility condition are displayed in Schwartz 2014, § 3.1, pp. 29–30.

To see why the Legendre transform reproduces the Lagrangian dynamics, vary $H$. The terms proportional to $\delta v^A$ cancel because $\pi_A=\partial\mathcal L/\partial v^A$. Introduce

$$\begin{aligned} P_A^i &\equiv \frac{\partial\mathcal L} {\partial(\partial_i\Phi^A)}, \\ K_A &\equiv -\frac{\partial\mathcal L}{\partial\Phi^A} + \partial_iP_A^i. \end{aligned}$$

After one spatial integration by parts,

$$\begin{aligned} \delta H ={}& \int_\Sigma\mathrm d^{d-1}x\, \bigl(v^A\delta\pi_A+K_A\delta\Phi^A\bigr) \\ &- \int_{\partial\Sigma}\mathrm dS\, n_iP_A^i\delta\Phi^A. \end{aligned}$$

When the declared boundary data make the last line vanish or a boundary contribution to $H$ cancels it, the functional derivatives are

$$\frac{\delta H}{\delta\pi_A}=v^A, \qquad \frac{\delta H}{\delta\Phi^A}=K_A.$$

The Euler–Lagrange equation then makes the second expression equal to $-\dot\pi_A$. Hence

$$\dot\Phi^A = \frac{\delta H}{\delta\pi_A}, \qquad \dot\pi_A =- \frac{\delta H}{\delta\Phi^A}.$$

These are Hamilton’s field equations. The Hamiltonian depends on the chosen time slicing and is not manifestly Lorentz scalar; equivalence with the Lagrangian equations is the relevant check.

The general field-theory definitions of conjugate momentum, the Lagrangian-to-Hamiltonian Legendre transform, and the equivalence of the two sets of equations are derived in Weinberg 1995, § 7.2, pp. 299–302.

Equal-time phase space and the classical Poisson bracket

On the regular branch, the cap term defines the canonical field-space one-form and two-form

$$\begin{aligned} \Theta_\Sigma &=\int_\Sigma\mathrm d^{d-1}x\, \pi_A\,\boldsymbol\delta\Phi^A, \\ \Omega_\Sigma &=-\boldsymbol\delta\Theta_\Sigma = \int_\Sigma\mathrm d^{d-1}x\, \boldsymbol\delta\Phi^A \wedge \boldsymbol\delta\pi_A, \end{aligned}$$

where $\boldsymbol\delta$ is the exterior derivative on field space. For differentiable functionals $F$ and $G$, the corresponding equal-time classical Poisson bracket is

$$\begin{aligned} \{F,G\} ={}& \int_\Sigma\mathrm d^{d-1}x \frac{\delta F}{\delta\Phi^A(\mathbf x)} \frac{\delta G}{\delta\pi_A(\mathbf x)} \\ &- \int_\Sigma\mathrm d^{d-1}x \frac{\delta F}{\delta\pi_A(\mathbf x)} \frac{\delta G}{\delta\Phi^A(\mathbf x)}. \end{aligned}$$

Its safest elementary statement is smeared. With test functions $f_A$ and $g^A$, let

$$\begin{aligned} \Phi[f] &=\int_\Sigma\mathrm d^{d-1}x\,f_A\Phi^A, \\ \Pi[g] &=\int_\Sigma\mathrm d^{d-1}x\,g^A\pi_A. \end{aligned}$$

Then

$$\{\Phi[f],\Pi[g]\} = \int_\Sigma\mathrm d^{d-1}x\,f_Ag^A.$$

The familiar pointwise relation is distributional shorthand for this identity:

$$\{\Phi^A(t,\mathbf x),\pi_B(t,\mathbf y)\} = \delta^A{}_B\, \delta^{(d-1)}(\mathbf x-\mathbf y),$$

with the field–field and momentum–momentum brackets equal to zero. For any explicitly time-dependent functional,

$$\frac{\mathrm dF}{\mathrm dt} = \frac{\partial F}{\partial t} + \{F,H\}.$$

Weinberg 1995, § 7.6, pp. 326–327 defines the classical Poisson bracket while treating a field label together with its spatial point as a compound index. The functional and smeared formulas above make that continuum interpretation explicit.

This bracket is not a quantum commutator. Quantization requires a separate choice of algebra, representation, operator domains, and state.

Functional differentiability is essential here: if the boundary term in $\delta H$ remains, the displayed formula does not yet define a Hamiltonian vector field on the proposed phase space. The available repairs—restricting boundary traces, imposing falloff, or adding a boundary Hamiltonian—are the canonical counterpart of Boundaries, Variations, and Well-Posed Actions.

Nor is a phase-space pair automatically a solution. A solution restricts to data on a time slice; reconstructing a solution from those data requires a separately well-posed evolution problem. The formulas above specify the formal canonical evolution once that analytic problem has been defined.

The real scalar as the regular branch

For a real scalar field,

$$\mathcal L = \frac12\dot\phi^2 - \frac12(\boldsymbol\nabla\phi)^2 - V(\phi).$$

The momentum and Hessian are

$$\pi = \frac{\partial\mathcal L}{\partial\dot\phi} = \dot\phi, \qquad W = \frac{\partial^2\mathcal L}{\partial\dot\phi^2} =1.$$

The velocity is therefore recovered uniquely. Define

$$\mathcal H_\phi = \frac12\pi^2 + \frac12(\boldsymbol\nabla\phi)^2 + V(\phi).$$

Then

$$H[\phi,\pi] = \int_\Sigma\mathrm d^{d-1}x\, \mathcal H_\phi.$$

Under boundary conditions that make $H$ differentiable,

$$\frac{\delta H}{\delta\pi}=\pi, \qquad \frac{\delta H}{\delta\phi} =- \boldsymbol\nabla^2\phi + V'(\phi).$$

Hamilton’s equations become

$$\dot\phi=\pi, \qquad \dot\pi = \boldsymbol\nabla^2\phi - V'(\phi).$$

Eliminating $\pi$ recovers

$$\ddot\phi - \boldsymbol\nabla^2\phi + V'(\phi)=0,$$

exactly the Euler–Lagrange equation. For $V(\phi)=m^2\phi^2/2$ with $m^2\geq0$, every term in the displayed Hamiltonian density is nonnegative, which is an independent sign check. The canonical initial data at $t=t_0$ are the functions

$$\phi_0(\mathbf x)=\phi(t_0,\mathbf x), \qquad \pi_0(\mathbf x)=\dot\phi(t_0,\mathbf x),$$

subject to the declared regularity and boundary conditions. The regular Legendre map imposes no further primary relation between them. Schwartz gives this scalar Legendre transform in Schwartz 2014, § 3.1, pp. 29–30; the canonical scalar pair and Hamiltonian also appear in Weinberg 1995, § 7.1, pp. 293–297.

Maxwell theory stops the ordinary inversion

For the Maxwell potential $A_\nu$,

$$\begin{aligned} \mathcal L_M &=-\frac14F_{\mu\nu}F^{\mu\nu}, \\ F_{\mu\nu} &=\partial_\mu A_\nu-\partial_\nu A_\mu. \end{aligned}$$

The momenta conjugate to the four components are

$$\pi^\nu \equiv \frac{\partial\mathcal L_M}{\partial\dot A_\nu} =- F^{0\nu}.$$

Consequently,

$$\pi^0=0, \qquad \pi^i=F_{0i}=\dot A_i-\partial_iA_0.$$

No $\dot A_0$ appears in the density. In the component ordering $(A_0,A_i)$, the velocity Hessian has the block form

$$W = \begin{pmatrix} 0 & 0 \\ 0 & \delta^{ij} \end{pmatrix}, \qquad \operatorname{rank}W=d-1.$$

The spatial velocities can be recovered as

$$\dot A_i = \pi^i+\partial_iA_0,$$

but $\dot A_0$ cannot. The image of the Legendre map therefore lies in the relation

$$\pi^0=0,$$

which is a primary constraint when the image has the required constant-rank smoothness. This conclusion comes directly from the momentum definition, before imposing Maxwell’s equations.

The electromagnetic canonical momenta and the primary relation $\pi^0=0$ are obtained directly in Weinberg 1995, § 8.2, pp. 344–345. The component Hessian and its rank above are the corresponding calculation in the notation used here.

One can still form the canonical density from the velocities that were invertible. Spatial contractions in the following expression use the positive Euclidean metric $\delta_{ij}$ on the slice; in particular, $F^{ij}=F_{ij}$ for spatial field-strength components in the inherited spacetime convention:

$$\mathcal H_c = \frac12\delta_{ij}\pi^i\pi^j + \frac14F_{ij}F^{ij} + \pi^i\partial_iA_0,$$

but this is not an ordinary unconstrained Hamiltonian on freely specifiable pairs $(A_\nu,\pi^\nu)$. Integrating its final term by parts also produces a spatial boundary contribution that must be retained or canceled under the chosen boundary data.

The Maxwell density used in this calculation is given in Schwartz 2014, § 8.2.3, pp. 118–119. The general singular-Hessian-to-primary-constraint step is worked explicitly in Brown 2022, § IV, p. 5; the later consistency analysis is developed separately in Brown 2022, §§ VI–VII, pp. 6–8.

Model	Momentum map	Hessian rank	Correct conclusion
Real scalar	$\pi=\dot\phi$	$1$ of $1$	Invert locally and use ordinary Hamilton evolution.
Maxwell potential	$\pi^0=0$, $\pi^i=\dot A_i-\partial_iA_0$	$d-1$ of $d$	Stop the ordinary inversion and begin constraint analysis.

Singularity alone does not prove that a variable is pure gauge or that a constraint is first class. The massive Proca field with $m\ne0$ has the same primary relation from its kinetic term, yet its complete constraint pair is second class; Weinberg 1995, § 7.6, pp. 325–328 gives that counterexample. Brown’s worked system likewise shows why primary versus secondary and first class versus second class are different classifications in Brown 2022, §§ IV–X, pp. 5–11.

The stop rule is therefore strict: when the Hessian is singular or changes rank, record the relations defining the local image of the Legendre map and do not claim an unconstrained or reduced phase space. For Maxwell theory, preservation of the primary relation, classification of the resulting constraints, and the physical initial-data count belong to the later worked analysis.

Candidate, constrained, and physical phase spaces

The word “phase space” can refer to several different stages:

Stage	What defines it	What is not yet guaranteed
Candidate canonical space	Regular pairs $(\Phi,\pi)$ with declared boundary behavior	Equations of motion, constraints, or global existence
Primary constraint surface	Local image of a singular Legendre map	Preservation under evolution or final constraint set
Final constraint surface	All consistency conditions imposed	Which null directions are genuine redundancies
Reduced physical phase space	Final surface quotiented only by demonstrated gauge redundancies	Smoothness, Hausdorffness, or global coordinates

This page constructs the first stage for the scalar and diagnoses the second for Maxwell. The later stages require the Poisson geometry of the complete constraint set and the model’s boundary-sensitive distinction between gauge redundancy and physical symmetry. They cannot be inferred from the velocity Hessian alone.

Common pitfalls

Inverting before checking the Hessian. Writing every velocity as a function of momenta assumes the step that must first be tested. Compute the velocity Hessian and inspect its rank before forming an ordinary Hamiltonian.

Treating $\pi^0=0$ as a gauge choice or polarization count. It is a primary relation in canonical data. Gauge generators, secondary constraints, reduction, and physical degrees of freedom require later steps.

Replacing a Poisson bracket by a commutator on a classical page. The bracket here is a classical operation on differentiable functionals. Canonical quantization is an additional postulate with operator-domain and representation questions.

Dropping the boundary term in $\delta H$. A formal bulk functional derivative is not enough. The Hamiltonian must be differentiable under the actual phase-space boundary conditions before it generates a Hamiltonian vector field.

Calling formal initial data a solved evolution problem. Canonical equations organize time evolution, but do not prove existence, uniqueness, continuous dependence, or constraint propagation for the chosen function spaces.

Check your understanding

1. Starting from the scalar Hamiltonian, compute both functional derivatives and recover the Euler–Lagrange equation.

   Solution

   Varying $\pi$ gives $\delta H/\delta\pi=\pi$. Varying the gradient term and integrating by parts gives $\delta H/\delta\phi=-\boldsymbol\nabla^2\phi+V'(\phi)$ when the boundary term is controlled. Hence $\dot\phi=\pi$ and $\dot\pi=\boldsymbol\nabla^2\phi-V'(\phi)$, so $\ddot\phi-\boldsymbol\nabla^2\phi+V'(\phi)=0$.

2. Derive the smeared scalar bracket $\{\Phi[f],\Pi[g]\}$ and explain what the delta-function shorthand means.

   Solution

   The functional derivatives are $\delta\Phi[f]/\delta\phi=f$, $\delta\Pi[g]/\delta\pi=g$, and the crossed derivatives vanish. Substitution gives $\int_\Sigma\mathrm d^{d-1}x\,fg$. The relation $\{\phi(\mathbf x),\pi(\mathbf y)\}=\delta^{(d-1)}(\mathbf x-\mathbf y)$ means precisely that this result is obtained after smearing in both variables.

3. Compute the Maxwell velocity Hessian and state exactly where the ordinary Legendre procedure stops.

   Solution

   Since $\pi^0=0$, the row and column associated with $\dot A_0$ vanish. Since $\pi^i=\dot A_i-\partial_iA_0$, the spatial block is $\delta^{ij}$. The rank is therefore $d-1$, so the spatial velocities can be recovered but $\dot A_0$ cannot. One records the primary relation $\pi^0=0$ and hands the system to constraint analysis; no gauge quotient or degree count follows yet.

The complete constraint algorithm is developed in Constraints, Dirac Brackets, and Symplectic Reduction and applied physically on Maxwell Constraints as a Worked Application. The interpretation of Gauss constraints and gauge orbits belongs to Gauge Orbits, Gauss Constraints, and Stabilizers. Quantization begins only at Canonical Quantization: Algebra, Representation, and State.

References

• Brown, J. David. “Singular Lagrangians, Constrained Hamiltonian Systems and Gauge Invariance: An Example of the Dirac–Bergmann Algorithm.” Universe 8, no. 3 (2022): 171. DOI.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.