Classical Field Modes and Canonical Variables

A free field becomes a collection of independent oscillators only after one fixes its equation, spatial domain, boundary conditions, canonical structure, and mode normalization. This page derives that statement for a real Klein–Gordon field. It also isolates the reality condition that prevents the complex plane-wave basis from counting every degree of freedom twice.

Required background. Quantum Mechanics supplies canonical pairs and oscillator quantization; Classical Field Theory supplies action variation and canonical field data; Special Relativity supplies inertial time and Minkowski geometry. Helpful background. Calculus and Linear Algebra supplies eigenvectors, orthonormal bases, and inner products; Waves, Green Functions, and Geometric Optics supplies hyperbolic evolution. The explicit decomposition parallels Fourier Modes, Dispersion, and Group Velocity.

Canonical data for a real scalar field

Work on Minkowski spacetime with a fixed inertial time coordinate and a static spatial domain. We take $m^2\ge0$ and impose boundary conditions that make spatial integrations by parts legitimate. For a real scalar field,

$$S[\phi] =-\frac12\int d^4x\, \left(\partial_\mu\phi\,\partial^\mu\phi+m^2\phi^2\right).$$

In the site’s $(-,+,+,+)$ convention,

$$\mathcal L =\frac12\dot\phi^2 -\frac12(\boldsymbol\nabla\phi)^2 -\frac12m^2\phi^2.$$

Varying the action gives both the field equation and the boundary term:

$$\delta S =\int d^4x\,(\Box-m^2)\phi\,\delta\phi -\int_{\partial\mathcal M}d\Sigma_\mu\, \partial^\mu\phi\,\delta\phi.$$

Fixing the field on the initial and final slices, together with Dirichlet, Neumann, periodic, or sufficiently decaying spatial data, makes the relevant boundary contribution vanish. The bulk equation is therefore

$$(\Box-m^2)\phi=0, \qquad \Box=-\partial_t^2+\boldsymbol\nabla^2.$$

The canonical momentum and Hamiltonian are

$$\pi=\frac{\partial\mathcal L}{\partial\dot\phi}=\dot\phi,$$ $$H =\int d^3x\, \left[ \frac12\pi^2 +\frac12(\boldsymbol\nabla\phi)^2 +\frac12m^2\phi^2 \right].$$

Hamilton’s field equations reproduce the Klein–Gordon equation. As a sign check, substituting $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$ gives the positive mass shell

$$\omega_{\mathbf k}^2=\mathbf k^2+m^2.$$

Tong 2006–2007, §§1.1 and 1.4, printed pp. 7–10 and 19–20, PDF develops the action and Hamiltonian using the opposite metric signature. Translating the complete kinetic term, wave operator, and mass shell gives the formulas above.

How a mode basis diagonalizes the field

First take a finite cavity or periodic box with a discrete spectrum. Choose a complete real orthonormal basis of spatial eigenfunctions $f_A(\mathbf x)$ in the domain of a self-adjoint realization of $-\boldsymbol\nabla^2$:

$$-\boldsymbol\nabla^2 f_A=k_A^2f_A, \qquad \int d^3x\,f_Af_B=\delta_{AB}.$$

The operator domain is part of this definition. Without self-adjoint boundary conditions, the eigenvalues need not be real, the modes need not be orthogonal, and flux can leak through the boundary. In all of $\mathbb R^3$, the corresponding plane waves are generalized eigenfunctions: sums become momentum integrals and Kronecker deltas become Dirac deltas. We keep the discrete notation for the following oscillator derivation. Expand the canonical data as

$$\phi(t,\mathbf x)=\sum_A q_A(t)f_A(\mathbf x), \qquad \pi(t,\mathbf x)=\sum_A p_A(t)f_A(\mathbf x).$$

Substitution into the action and one spatial integration by parts give

$$L=\frac12\sum_A \left(\dot q_A^2-\omega_A^2q_A^2\right), \qquad H=\frac12\sum_A \left(p_A^2+\omega_A^2q_A^2\right),$$

where

$$\omega_A^2=k_A^2+m^2, \qquad p_A=\dot q_A.$$

For $\omega_A\gt 0$, each independent real mode is one harmonic oscillator. A genuine $\omega_A=0$ mode instead has $H_A=p_A^2/2$ and is a free canonical degree of freedom. The diagonalization is not a metaphor: orthonormality removes the kinetic cross-terms, while the eigenvalue equation removes the gradient cross-terms. Tong 2006–2007, §§2.1–2.2, printed pp. 21–25, PDF carries the same calculation from Fourier modes to the oscillator Hamiltonian.

Reality in a complex plane-wave basis

Plane waves are convenient but complex. In a periodic box of volume $V$, write

$$\phi(t,\mathbf x) =\frac{1}{\sqrt V}\sum_{\mathbf k} \varphi_{\mathbf k}(t)e^{i\mathbf k\cdot\mathbf x}.$$

The physical field is real precisely when

$$\varphi_{-\mathbf k}=\varphi_{\mathbf k}^*.$$

The reduced Lagrangian can be written

$$L=\frac12\sum_{\mathbf k} \left( \dot\varphi_{\mathbf k}\dot\varphi_{-\mathbf k} -\omega_{\mathbf k}^2 \varphi_{\mathbf k}\varphi_{-\mathbf k} \right).$$

The coefficients at $\mathbf k$ and $-\mathbf k$ are therefore not independent complex oscillators. For every nonzero pair $\{\mathbf k,-\mathbf k\}$, the real and imaginary parts of $\varphi_{\mathbf k}$ are the two real cosine and sine modes; the zero mode is one real degree of freedom. Working in the real basis $f_A$ builds this constraint in from the start. Working with all complex plane waves is equally valid only if the relation $\varphi_{-\mathbf k}=\varphi_{\mathbf k}^*$ is retained.

Symplectic and Klein–Gordon normalization

The canonical symplectic form on two variations of the initial data is

$$\Omega(\delta_1,\delta_2) =\int d^3x\, \left( \delta_1\phi\,\delta_2\pi -\delta_2\phi\,\delta_1\pi \right).$$

For complex solutions, our sign convention gives the sesquilinear Klein–Gordon form

$$(f,g)_{\mathrm{KG}} =i\,\Omega(f^*,g) =i\int d^3x\, \left(f^*\dot g-\dot f^*g\right).$$

Its conservation is a useful independent check. Differentiating and using the field equation gives

$$\frac{d}{dt}(f,g)_{\mathrm{KG}} =i\int d^3x\, \left(f^*\boldsymbol\nabla^2g -\boldsymbol\nabla^2f^*\,g\right),$$

which is a boundary flux. It vanishes for the boundary conditions assumed above. The normalization is therefore independent of the constant-time slice.

With

$$u_{\mathbf k}(x) =\frac{e^{-i\omega_{\mathbf k}t+i\mathbf k\cdot\mathbf x}} {\sqrt{(2\pi)^3\,2\omega_{\mathbf k}}},$$

direct substitution gives

$$(u_{\mathbf k},u_{\mathbf q})_{\mathrm{KG}} =\delta^{(3)}(\mathbf k-\mathbf q), \qquad (u_{\mathbf k}^*,u_{\mathbf q}^*)_{\mathrm{KG}} =-\delta^{(3)}(\mathbf k-\mathbf q), \qquad (u_{\mathbf k},u_{\mathbf q}^*)_{\mathrm{KG}}=0.$$

The full complex solution space therefore has an indefinite form. Minkowski time translations select the positive-frequency subspace, on which the form is positive and can become the one-particle inner product. Wald 1995, printed pp. 6–9, especially Eqs. (12)–(16), PDF develops the conserved phase-space form and Fock construction; Wald orders the two symplectic arguments oppositely, so his $\Omega$ has the opposite overall sign from the one displayed here.

Where the oscillator recipe stops

The scalar kinetic Hessian is nondegenerate, so $\pi=\dot\phi$ can be inverted for the velocity. Gauge fields differ. For electromagnetism, the Lagrangian contains no independent $\dot A_0$, so

$$\pi^0=0$$

is a primary constraint rather than an evolution equation. Gauss’s law supplies a secondary restriction, and gauge-related canonical data represent the same physical state. Quantizing all four components as four unconstrained oscillators would therefore overcount. Gauge Symmetry, Constraints, and Physical States performs the reduction.

Checks and limits

• The Hamiltonian is nonnegative for $m^2\ge0$ and boundary conditions that remove the surface term. A negative $\omega_A^2$ signals an instability, not an oscillator with a ground state.
• A genuine $\omega_A=0$ mode is a free canonical degree of freedom, not an oscillator with well-defined ladder operators. It must be treated separately.
• A finite box can be a normalization aid or a physical cavity. Those are different models, and the large-volume limit must be stated when the box is only auxiliary.
• Completeness belongs to the chosen operator and domain. Writing a Fourier integral does not establish completeness on a bounded, time-dependent, or curved background.
• A time-dependent mode basis can mix positive and negative frequencies. Generic curved spacetimes need not provide the preferred global time translation used here.

Exercises

Normalize a periodic-box mode

For a cubic periodic box of volume $V$, let

$$u_{\mathbf k}(x) =\frac{e^{-i\omega_{\mathbf k}t+i\mathbf k\cdot\mathbf x}} {\sqrt{2\omega_{\mathbf k}V}}.$$

For $\omega_{\mathbf k}\gt 0$, verify $(u_{\mathbf k},u_{\mathbf q})_{\mathrm{KG}}=\delta_{\mathbf k\mathbf q}$.

Solution

The time derivatives are $\dot u_{\mathbf k}=-i\omega_{\mathbf k}u_{\mathbf k}$ and $\dot u_{\mathbf k}^*=+i\omega_{\mathbf k}u_{\mathbf k}^*$. Hence

$$(u_{\mathbf k},u_{\mathbf q})_{\mathrm{KG}} =\frac{\omega_{\mathbf k}+\omega_{\mathbf q}} {\sqrt{4\omega_{\mathbf k}\omega_{\mathbf q}V^2}} \int_Vd^3x\,e^{i(\mathbf q-\mathbf k)\cdot\mathbf x}.$$

The integral is $V\delta_{\mathbf k\mathbf q}$. On its support $\omega_{\mathbf q}=\omega_{\mathbf k}$, so the remaining prefactor is one.

Diagnose double-counting

Starting from the periodic-box Fourier series, show that a real field requires $\varphi_{-\mathbf k}=\varphi_{\mathbf k}^*$. Explain why treating both coefficients as independent complex variables would double the physical real data associated with one nonzero momentum pair.

Solution

Complex conjugation gives

$$\phi^*(t,\mathbf x) =\frac1{\sqrt V}\sum_{\mathbf k} \varphi_{\mathbf k}^*e^{-i\mathbf k\cdot\mathbf x} =\frac1{\sqrt V}\sum_{\mathbf k} \varphi_{-\mathbf k}^*e^{i\mathbf k\cdot\mathbf x}.$$

Uniqueness of the Fourier coefficients makes $\phi^*=\phi$ equivalent to $\varphi_{\mathbf k}=\varphi_{-\mathbf k}^*$. One nonzero pair then contains the two real numbers $\operatorname{Re}\varphi_{\mathbf k}$ and $\operatorname{Im}\varphi_{\mathbf k}$. Two unconstrained complex coefficients would instead contain four real numbers, twice the physical count for that pair. Relative to one real cosine or sine oscillator, it would contain four times as much real configuration data.

Continue to Free Fields, Creation Operators, and Fock Space to quantize these canonical modes.

References

• David Tong, Quantum Field Theory, University of Cambridge Part III lecture notes, Michaelmas Terms 2006 and 2007, §§1.1, 1.4, and 2.1–2.2, printed pp. 7–10 and 19–25. Official author PDF.
• Robert M. Wald, “Quantum Field Theory in Curved Spacetime,” arXiv:gr-qc/9509057 (1995), printed pp. 6–9. Author-manuscript PDF.