Klein-Gordon Equation

Prerequisites: Schrödinger Equation, Special Relativity. Full path (16 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• Definitions

Intuitive

The Klein-Gordon equation describes how the state of a relativistic (= fast moving) quantum system without spin changes in time.

The analogous equation for systems with half-integer spin is the dirac_equation.

If the system only moves slowly, the Klein-Gordon equation becomes the schroedinger_equation.

Concrete

The Klein-Gordon equation can be derived from the Lagrangian

$$\begin{equation} \mathscr{L}= \frac{1}{2}( \partial _{\mu} \Phi \partial ^{\mu} \Phi -m^2 \Phi^2) \end{equation}$$

using the Euler-Lagrange equations.

Solutions

The most general solution of the Klein-Gordon equation is$$\begin{equation} \Phi(x)= \int \mathrm{d }k^3 \frac{1}{(2\pi)^3 2\omega_k} \left( a(k){\mathrm{e }}^{ -i(k x)} + a^\dagger(k) {\mathrm{e }}^{ i(kx)}\right) .\end{equation}$$

• A nice discussion can be found in chapter 3 of Klauber's Student Friendly QFT book
• For an elementary derivation of the Klein-Gordon equation see Physics from Symmetry by Schwichtenberg

Graphical Summary

The diagram below shows the Klein-Gordon equation and its Lagrangian in various forms. For a more detailed explanation see Fun with Symmetry.

• No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, section "Free Scalar fields and the Klein Gordon Equation"

Abstract

We use $\hbar = c = 1$ and the signature $(+,-,-,-)$, as in the Definitions below. The operator $\partial_\mu\partial^\mu$ is (up to sign) the image of the quadratic Casimir $P_\mu P^\mu$ of the Poincaré group, with $P_\mu = i\partial_\mu$ acting on functions on Minkowski space. The Klein-Gordon equation is therefore the statement that the field carries the value $m^2$ of this Casimir. Wigner's classification of the unitary irreducible representations of the Poincaré group makes this precise. For $m > 0$ and spin $0$, the representation space is

$$\mathcal H_m = L^2\!\left(X_m^+, \frac{d^3k}{(2\pi)^3\, 2\omega_k}\right), \qquad X_m^+ = \{k \in \mathbb R^{1,3} : k_\mu k^\mu = m^2,\ k^0 > 0\},$$

with $\omega_k = \sqrt{\vec k^2 + m^2}$. The measure $d^3k/(2\omega_k)$ is the Lorentz-invariant measure on the mass hyperboloid $X_m^+$, which is why it appears in the general solution on this page. The positive-frequency solutions of the Klein-Gordon equation are exactly the Fourier transforms of elements of $\mathcal H_m$.

On solutions, the Klein-Gordon product

$$(\phi_1,\phi_2) = i\int d^3x\,\left(\phi_1^*\,\partial_t\phi_2 - \phi_2\,\partial_t\phi_1^*\right)$$

is independent of time, because the current $j^\mu = i(\phi^*\partial^\mu\phi - \phi\,\partial^\mu\phi^*)$ is conserved. It is positive on positive-frequency solutions and negative on negative-frequency ones, so it does not define a Hilbert space on all solutions and $j^0$ cannot be read as a probability density. The consistent interpretation is a quantum field: the field operator is built from $\mathcal H_m$ by second quantization (the bosonic Fock space), with the negative-frequency part carrying the creation operators.

The space of real solutions carries the symplectic form $\Omega(\phi_1,\phi_2) = \int d^3x\,(\phi_1\partial_t\phi_2 - \phi_2\partial_t\phi_1)$, which is also independent of time, and the classical theory is a Hamiltonian system on it. The Pauli-Jordan function $\Delta(x) = \Delta_{\text{ret}}(x) - \Delta_{\text{adv}}(x)$, the difference of the retarded and advanced Green's functions, vanishes at spacelike separation and determines the commutator $[\phi(x),\phi(y)] = i\Delta(x-y)$ (with the sign convention for $\Delta$ fixed by this equation). This is the statement of microcausality.

On a globally hyperbolic Lorentzian manifold $(M,g)$ the equation generalises to $(\Box_g + m^2 + \xi R)\phi = 0$, where $\Box_g$ is the Laplace-Beltrami operator, $R$ the scalar curvature and $\xi$ a coupling constant ($\xi = 0$ is minimal coupling). The Cauchy problem is well posed, with unique advanced and retarded fundamental solutions, so the symplectic structure of the solution space survives. There is in general no preferred split into positive and negative frequencies, and the choice of vacuum is no longer unique, which is the origin of particle creation by gravitational fields.

Why is it interesting?

The Klein-Gordon equation is the correct equation of motion that describes free spin $0$ particles. For a spin-1 generalization see the Duffin-Kemmer-Petiau equation.

Definitions

• $\partial _{\mu}$ denotes the partial derivative and $\partial _{\mu} \partial ^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\partial _{\mu} \partial ^{\mu} = \partial _0 \partial^0 + \partial _1 \partial^1 +\partial _2 \partial^2 +\partial _3 \partial^3$,
• $m$ denotes the mass of the particle,
• $\Phi$ describes the spin $0$ field if we work in a field theory.
• Note: $\Phi$ cannot be interpreted as a wavefunction because it is a real valued field; it is its own anti-particle like the Majorana fermion. Only in the case that it is the U(1)-charged (requires 2 independent real Klein Gordon fields that are symmetry transform into each other) is a naive wavefunction interpretation possible. Basically, you get a relativistic scalar superfluid field. Nevertheless, there are single particle wavefunctions lurking in the single real Klein-Gordon theory. But you need to use the coherent state representation to see the 1st quantized operators from the complex annihilation and creation operators. Essentially undoing the second quantization.

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Legendre Transformation
6. Newton's Second Law
7. Vector Calculus
8. Newtonian Mechanics
9. Variational Calculus
10. Configuration Space
11. Euler-Lagrange Equations
12. Special Relativity
13. Lagrangian Formalism
14. Phase Space
15. Hamiltonian Formalism
16. Schrödinger Equation

Leads to: Dirac Equation, Proca Equation.