Phase-Space Formulation from QM to QFT

For a mechanical system, phase space is coordinatized by canonical pairs $(q^i,p_i)$ . For a classical field, the corresponding data at one time are a field configuration and its canonical momentum:

\phi(\mathbf x), \qquad \pi(\mathbf x).

This replacement is the shortest phase-space bridge from quantum mechanics to field theory:

(q^i,p_i) \quad \longrightarrow \quad \bigl( \phi(\mathbf x), \pi(\mathbf x) \bigr).

The longer bridge requires a regulator, infinitely many degrees of freedom, local operator products, constraints, and renormalization. Phase-space functions become functionals, ordinary derivatives become functional derivatives, and finite-dimensional measures become regulated functional measures.

This page is an orientation. The canonical mechanical tools remain Weyl Transform, Wigner Function, and Phase-Space Dynamics. The detailed construction of field path integrals belongs beyond this quantum-mechanical bridge.

Classical Phase Space in Mechanics and Fields

For $N$ canonical coordinates, a classical state is a point

z = \left( q^1,\ldots,q^N, p_1,\ldots,p_N \right).

The Poisson bracket is

\{F,G\}_{\mathrm{PB}} = \sum_{i=1}^{N} \left( \frac{\partial F}{\partial q^i} \frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i} \frac{\partial G}{\partial q^i} \right).

A scalar field is the continuum limit of a many-coordinate system. On a spatial lattice with $N$ sites, the field values $\phi_n$ are $N$ coordinates and their conjugate momenta $\pi_n$ supply the other $N$ phase-space coordinates. The regulated phase space is therefore $2N$ -dimensional.

In continuum notation, a functional $F[\phi,\pi]$ has field Poisson bracket

\begin{aligned} \{F,G\}_{\mathrm{PB}} &= \int d^d x \left[ \frac{\delta F}{\delta\phi(\mathbf x)} \frac{\delta G}{\delta\pi(\mathbf x)} \right. \\ &\qquad\left. - \frac{\delta F}{\delta\pi(\mathbf x)} \frac{\delta G}{\delta\phi(\mathbf x)} \right]. \end{aligned}

Hamilton’s field equations are

\dot\phi(\mathbf x) = \frac{\delta H}{\delta\pi(\mathbf x)}, \qquad \dot\pi(\mathbf x) = - \frac{\delta H}{\delta\phi(\mathbf x)}.

The continuum formulas summarize a regulated limit. Boundary conditions, spatial topology, lattice spacing or momentum cutoff, and the definition of the canonical momentum are part of the phase-space structure.

A field configuration at one time is not a spacetime history. The canonical phase space labels instantaneous data; a path integral integrates over trajectories through that phase space.

Canonical Commutation Relations

Canonical quantization replaces the field Poisson bracket with equal-time commutators. For a real bosonic scalar field,

\left[ \hat\phi(t,\mathbf x), \hat\pi(t,\mathbf y) \right] = i\hbar \delta^{(d)}(\mathbf x-\mathbf y),

with

\left[ \hat\phi(t,\mathbf x), \hat\phi(t,\mathbf y) \right] = 0, \qquad \left[ \hat\pi(t,\mathbf x), \hat\pi(t,\mathbf y) \right] = 0.

The delta distribution is the continuum counterpart of the Kronecker delta for finitely many canonical pairs. Its normalization depends on the Fourier and finite-volume conventions.

For a free scalar field, the Hamiltonian is schematically

H = \frac12 \int d^d x\, \left[ \pi^2 + (\nabla\phi)^2 + m^2\phi^2 \right].

After a mode decomposition and suitable reality conditions, each independent mode behaves like a harmonic oscillator:

H \sim \sum_{\mathbf k} \left[ \frac12p_{\mathbf k}^2 + \frac12\omega_{\mathbf k}^2q_{\mathbf k}^2 \right].

This is the canonical content of Harmonic Oscillator to Fields. The oscillator ladder operators become mode creation and annihilation operators. An interacting field is not a collection of independent oscillators: interaction terms couple modes and modify the vacuum and excitations.

Canonical commutation relations are a starting point, not a complete field theory. Operator-valued distributions cannot generally be multiplied at the same point without regularization, and different continuum settings can admit representation issues absent for finitely many canonical pairs.

Phase-Space Path Integrals Preview

A mechanical Hamiltonian path integral has the formal form

K = \int \mathcal Dp\,\mathcal Dq\, \exp\left[ \frac{i}{\hbar} \int dt\, \left( p_i\dot q^i-H(p,q) \right) \right].

For a bosonic field, the analogous expression is

\begin{aligned} Z &= \int \mathcal D\pi\,\mathcal D\phi\, \\ &\quad\times \exp\left\{ \frac{i}{\hbar} \int dt\,d^d x\, \left[ \pi\dot\phi - \mathcal H(\phi,\pi) \right] \right\}. \end{aligned}

At finite spatial lattice and finite time slicing, this is a many-coordinate phase-space integral. The functional notation is its continuum shorthand.

For a standard quadratic momentum dependence,

\mathcal H = \frac12\pi^2 + \mathcal V(\phi,\nabla\phi),

the $\pi$ integral is Gaussian. Completing the square,

\pi\dot\phi-\frac12\pi^2 = - \frac12 \left( \pi-\dot\phi \right)^2 + \frac12\dot\phi^2,

formally converts the Hamiltonian integral into a configuration-space integral with Lagrangian density

\mathcal L = \frac12\dot\phi^2 - \mathcal V.

This equivalence is not automatic for arbitrary systems. Position-dependent kinetic terms, constraints, nontrivial measures, first-order actions, and gauge redundancy can leave determinants or constraint factors. Gauge fields require constraint and gauge-fixing machinery. Fermionic fields use Grassmann variables rather than ordinary commuting phase-space coordinates.

From Path Integrals in QM to Field Path Integrals owns the regulated path-to-field construction. The phase-space point here is that the first-order term $\pi\dot\phi$ records the canonical symplectic structure before momenta are integrated out.

Coherent-State Path Integrals

Bosonic coherent states are labeled by complex amplitudes $\alpha_j$ and satisfy

\int \frac{d^2\alpha_j}{\pi} \lvert\alpha_j\rangle \langle\alpha_j\rvert = I_j

for each mode. Repeatedly inserting these overcomplete identities produces a path integral over complex phase-space labels.

For finitely many modes, the action has the schematic form

S = \int dt\, \left[ i\hbar \sum_j \alpha_j^*\dot\alpha_j - H_{\mathrm{symb}} \left( \alpha^*,\alpha \right) \right].

The Hamiltonian symbol and finite-slice rule depend on operator ordering. Normal, antinormal, and Weyl symbols are not interchangeable. Boundary terms also matter because coherent-state path integrals use an overcomplete, nonorthogonal basis.

For a nonrelativistic bosonic field, the mode amplitudes can be traded for a complex field $\psi(\mathbf x,t)$ :

S \sim \int dt\,d^d x\, \left[ i\hbar\psi^*\partial_t\psi - \mathcal H \left( \psi^*,\psi \right) \right].

This language is central in many-body theory, quantum optics, condensate physics, and nonequilibrium methods. It does not mean $\psi$ is a one-particle wavefunction in every use; in a field path integral it is an integration variable associated with field-mode amplitudes.

Fermionic coherent states are labeled by Grassmann variables. Their measures, signs, and thermal boundary conditions differ fundamentally from the bosonic case. Replacing them by ordinary complex numbers destroys fermionic anticommutation.

Coherent States in Phase Space gives the one-mode geometry. Coherent-State Path Integrals Preview gives the finite-slice bosonic and fermionic thermal construction. Second Quantization gives the occupation-number bridge that makes the multimode notation natural.

Wigner Functional Preview

For a lattice-regulated bosonic field, the density operator can be represented by a finite-dimensional Wigner function over all canonical site or mode variables. The continuum notation is a Wigner functional

W_\Lambda[\varphi,\pi],

where $\Lambda$ denotes the regulator data.

A formal field-basis definition is

\begin{aligned} W_\Lambda[\varphi,\pi] &= \mathcal N_\Lambda \int\mathcal D\eta\, \exp\left[ - \frac{i}{\hbar} \int d^d x\, \pi(\mathbf x)\eta(\mathbf x) \right] \\ &\quad\times \left\langle \varphi+\frac{\eta}{2} \middle| \rho \middle| \varphi-\frac{\eta}{2} \right\rangle. \end{aligned}

The normalization $\mathcal N_\Lambda$ depends on the regulated measure and Fourier convention. It should not be guessed from a one-coordinate formula after the cutoff has been removed.

With matched conventions, integrating over all momentum fields gives the diagonal field-configuration probability functional:

\int\mathcal D\pi\, W_\Lambda[\varphi,\pi] = \langle\varphi\rvert\rho\lvert\varphi\rangle.

As in ordinary quantum mechanics, the full Wigner functional is a quasiprobability and can be negative.

The exact evolution can be written with a functional Moyal bracket:

\frac{\partial W_\Lambda}{\partial t} = \{H_\Lambda,W_\Lambda\}_{M,\mathrm{func}}.

Its leading derivative term is the classical field Poisson bracket, with higher functional derivatives supplying quantum corrections. Dropping those derivatives leads to classical-statistical or truncated-Wigner-type approximations in suitable bosonic regimes.

Such a truncation is not universally controlled. Its accuracy depends on occupation, coupling, initial state, cutoff, observable, and evolution time. Vacuum fluctuations contribute roughly half a quantum per bosonic mode in common Wigner conventions, so increasing the ultraviolet cutoff can inject cutoff-dependent noise unless the approximation and renormalization are handled consistently.

From Modes to Continuum Physics

The finite-mode dictionary is direct:

Quantum mechanics	Regulated field theory
canonical pairs $(q_i,p_i)$	lattice or mode pairs $(\phi_n,\pi_n)$
$[q_i,p_j]=i\hbar\delta_{ij}$	equal-time field commutator
Wigner function $W(q,p)$	multimode Wigner function or functional
coherent amplitude $\alpha_i$	mode amplitude or complex field
star product	multimode or functional star product
Moyal equation	functional Wigner evolution

The continuum limit adds problems that have no finite-dimensional analogue:

infinitely many ultraviolet modes;
singular local products of fields;
regulator and renormalization dependence;
gauge constraints and redundant variables;
Grassmann integration for fermions;
possible infrared and infinite-volume limits;
dependence of particles and modes on the chosen background or basis.

The formal replacement of sums by integrals does not solve these issues. A continuum phase-space functional is trustworthy only after the regulated theory, observables, and limiting procedure are specified.

Continue to Field Theory

The most durable ideas carried from quantum mechanics are:

canonical coordinates and momenta encode a symplectic structure;
quantization replaces Poisson brackets by noncommutative operator algebra;
coherent states label bosonic mode phase space;
Wigner transforms turn density operators into quasiprobabilities;
Moyal brackets organize quantum corrections to classical flow.

The detailed field-theory continuation branches by purpose:

For field Legendre transforms, equal-time brackets, fermions, and gauge constraints, continue with From Phase Space to Canonical Quantization.
For functional measures and lattice-to-continuum limits, continue with From Path Integrals in QM to Field Path Integrals.
For mode occupation and creation operators, continue with Second Quantization: Bridge to QFT.
For the oscillator origin of free-field modes, use Harmonic Oscillator to Fields.
For path-integral sources and correlation functions, use Path Integrals from QM to QFT.

These bridges provide orientation, not a substitute for a regulated field-theory treatment.

Common Mistakes

Identifying a field configuration at one time with an entire spacetime history.
Writing continuum functional measures without first specifying a lattice, mode cutoff, or other regulator.
Treating each free-field oscillator mode as a distinct particle rather than a mode whose excitations define quanta.
Assuming interacting fields remain independent oscillators.
Integrating out canonical momenta without checking determinants, constraints, or boundary terms.
Using an ordinary complex coherent-state field for fermions instead of Grassmann variables.
Treating a Wigner functional as a positive probability distribution.
Calling truncated-Wigner evolution exact because it follows classical field equations.
Adding half a quantum to infinitely many modes without tracking cutoff dependence.
Ignoring gauge fixing, ultraviolet renormalization, or local-operator definitions in the continuum.

Cross-Links

References

J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley, 1988.
J. R. Klauder and B.-S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics, World Scientific, 1985.
A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
A. Kamenev, Field Theory of Non-Equilibrium Systems, Cambridge University Press, 2011.
A. Polkovnikov, “Phase space representation of quantum dynamics,” Annals of Physics 325, 1790–1852 (2010).
J. Berges, “Introduction to nonequilibrium quantum field theory,” AIP Conference Proceedings 739, 3–62 (2004).

Exercises

A scalar field is regulated on a lattice with $N$ sites. What is the dimension of its canonical phase space, and why must the regulator be retained before taking a continuum limit?

Solution

Each site has one field coordinate $\phi_n$ and one conjugate momentum $\pi_n$ . The regulated phase space therefore has dimension

2N.

At finite $N$ , Poisson brackets, commutators, measures, and Wigner transforms are finite-dimensional. Sending the lattice spacing to zero while holding physical volume fixed sends $N$ to infinity. Ultraviolet divergences and regulator-dependent normalization can then appear, so the limiting observables and renormalization conditions must be specified before the continuum notation is treated as defined.

Show that the field Poisson bracket generates Hamilton’s field equations.

Solution

Take the functional $F[\phi,\pi]=\phi(\mathbf y)$ . Its functional derivatives are

\frac{\delta F}{\delta\phi(\mathbf x)} = \delta^{(d)}(\mathbf x-\mathbf y), \qquad \frac{\delta F}{\delta\pi(\mathbf x)} = 0.

Therefore

\begin{aligned} \{\phi(\mathbf y),H\}_{\mathrm{PB}} &= \int d^d x\, \delta^{(d)}(\mathbf x-\mathbf y) \frac{\delta H}{\delta\pi(\mathbf x)} \\ &= \frac{\delta H}{\delta\pi(\mathbf y)}. \end{aligned}

Thus $\dot\phi=\{\phi,H\}_{\mathrm{PB}}=\delta H/\delta\pi$ . Repeating the calculation with $F=\pi(\mathbf y)$ gives

\dot\pi = \{\pi,H\}_{\mathrm{PB}} = - \frac{\delta H}{\delta\phi}.

Integrate the momentum dependence of a scalar phase-space path integral formally.

Solution

For

\mathcal H = \frac12\pi^2 + \mathcal V(\phi,\nabla\phi),

the first-order integrand is

\pi\dot\phi-\mathcal H = - \frac12 \left( \pi-\dot\phi \right)^2 + \frac12\dot\phi^2 - \mathcal V.

The regulated integral over the shifted Gaussian variable $\pi-\dot\phi$ contributes a normalization or determinant factor. What remains in the exponent is

\mathcal L = \frac12\dot\phi^2 - \mathcal V.

The simple result depends on the quadratic, field-independent coefficient of $\pi^2$ . More general kinetic terms or constraints require additional factors.

Derive the field-configuration marginal of the regulated Wigner functional.

Solution

Start from

\begin{aligned} W_\Lambda[\varphi,\pi] &= \mathcal N_\Lambda \int\mathcal D\eta\, \exp\left[ - \frac{i}{\hbar} \int d^d x\,\pi\eta \right] \\ &\quad\times \left\langle \varphi+\frac{\eta}{2} \middle| \rho \middle| \varphi-\frac{\eta}{2} \right\rangle. \end{aligned}

With the normalization matched to the regulated Fourier transform, integration over every momentum variable produces a functional delta:

\mathcal N_\Lambda \int\mathcal D\pi\, \exp\left[ - \frac{i}{\hbar} \int d^d x\,\pi\eta \right] = \delta_\Lambda[\eta].

The delta functional sets $\eta=0$ , leaving

\int\mathcal D\pi\, W_\Lambda[\varphi,\pi] = \langle\varphi\rvert\rho\lvert\varphi\rangle.

Why is truncated-Wigner evolution not automatically exact when the classical field equations are solved without numerical error?

Solution

The exact Wigner-functional equation contains a functional Moyal bracket. Classical field evolution keeps only its leading functional Poisson bracket and drops higher functional derivatives. Solving the retained equation exactly removes numerical integration error but does not restore the omitted quantum terms.

The truncation error depends on occupation, coupling, state gradients, cutoff, observable, and time. Classical flow can also generate fine structure that makes higher derivatives important. Numerical convergence of the truncated equation is therefore necessary but not sufficient for convergence to the exact quantum theory.